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Sergey Smirnov

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Sergey Smirnov

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Page 1: Sergey Smirnov

Acknowledgment

I would like to thank several people who have helped me to complete thisthesis� First of all� I need to mention my advisor� Professor Chris Lacor�who has given me the opportunity to do this research and whose continuoussupport and patience made this work possible�

I would also like to thank my collegues for their scienti�c support andfriendship� namely� Jan Ramboer �with who I shared an o�ce for severalyears�� Bamdad Lessani� Ghader Gorbaniasl� TimBroeckhoven� Mark Brouns�Kris Van den Abeele� Patryk Widera and Santhosh Jayaraju� I am alsothankful to Alain Wery� our system administrator� who has always been ofgreat help� and Jenny D�haes for her administrative support�

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Executive Summary

The objective of this thesis is to develop a high�order Pad�e�type �nitevolume method that can be applied on irregular structured meshes� ThePad�e type schemes were originally formulated in the �nite dierence con�text� Their application on uniform Cartesian meshes is relatively simple� buttheir extension to general meshes is not straightforward� In the recent yearsthere appeared several papers dealing with formulation of Pad�e schemes oncurvilinear meshes� both in the �nite dierence and �nite volume contexts�However� they all treated the problem of curvilinearity of the grid in the com�putational space� i�e� by the method that requires calculation of a Jacobiantransformation of coordinates� which can be an extra source of numericalinaccuracies on meshes with non�smoothly varying mesh spacing� This iswhy� in this thesis� a �nite volume Pad�e scheme has been formulated in thephysical space� taking into account the irregularity of the mesh in the moststraightforward way�

In the �rst chapter the general idea of the Pad�e discretization is intro�duced along with the present state of the art� The second chapter� whichis the core of this work� contains the detailed description of an original ��nite volume formulation of a Pad�e scheme on a general structured mesh�The irregularity of the mesh is taken into account directly in the physicalspace through coordinates of the grid nodes� As the main objective was toconstruct a scheme for unsteady calculations� the discrete equations are de�rived for cell�averaged values of the solution as this approach does not requirehigh�order quadrature formulas to calculate volume integrals containing timederivatives of the unknown quantities�

The maximum order of the scheme was limited to in order to obtaina scheme on the smallest stencil possible for a central Pad�e discretization�Although it is possible to construct schemes of higher order of accuracy byusing bigger stencils� this task requires much more work� both in terms of thetreatment of boundary conditions and in terms of formulating it on generalmeshes� This was beyond the scope of the thesis�

In the �nite volume method the discretized equations are derived fromthe integral form of partial dierential equations written for a cell� As thecell averaged values of the solution are used in the proposed approach� the

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evaluation of the volume integrals containing time derivatives of the solutionis straightforward� Therefore� the main task is to construct a high�orderPad�e formula for evaluation of surface �in �D� or line �in D� integrals of�uxes over the interfaces of the cells� The variable averaged approach forcalculating these integrals has been chosen because it is simpler to apply incase of �uxes being a non�linear function of the unknowns� First surfaceor line integrals of the unknowns are calculated by means of the Pad�e typeinterpolation formula� then the integrals of the �uxes are calculated based theintegrals of the unknowns through a reconstruction procedure that maintainsthe high�order accuracy of the scheme�

The coe�cients of the Pad�e interpolation formula depend on the localqualities of the mesh� They are calculated through a multidimensional Taylorseries analysis of a truncation error of the formula� On a stencil chosen in thisthesis the maximum order of the formula on a general mesh is �� althoughthe leading term of the truncation error is minimized by means of the leastsquare approach� which ensures that on grids that are not very distorted thescheme is almost th order accurate� On the Cartesian mesh� this leadingterm of the truncation error vanishes� which means that on the Cartesianmesh the scheme is th order accurate�

As the resulting scheme turns out to be unstable on extremely distortedgrids� another scheme with formal accuracy instead of � �on a general mesh�has been constructed by freezing the coe�cients in the left hand side of theformula� Again the leading truncation error was minimized by means of theleast square approach� This scheme also retrieves the th order accuracy onthe Cartesian uniform meshes� To demonstrate the accuracy of the obtainedscheme� it is tested on a number of model linear convection problems

In chapter three� an original Discontinuous�Galerkin�like �nite volumemethod is proposed� Unlike the classical DGmethod that represent a solutionas a linear combination of basis function� the proposed method utilizes the therepresentation of the solution by the cell averaged values� This allows the useof the highest order possible central discretization of the �uxes� which makesthe order of the method twice as big as in the classical DG method� Themethod is� however� inspired by the DG method as far as the derivation ofextra equations is concerned� The advection equations for spatial derivativesof the solution are obtained by multiplying the original PDEs by a �rstorder polynomial and integrating a product over the cell� The method wasformulated for a linear convective equation for the use on a Cartesian uniformmesh and numerically compared to the second order �nite volume method�second order DG method and th order Pad�e scheme� Although� its extensionto non�linear equations and general grids is possible� this was beyond thescope of this thesis�

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The fourth chapter deals with the issues of time integration of the incom�pressible Navier�Stokes equations� A fractional step method� which a subsetof pressure correction methods is described for a fully incompressible �ow�The satisfaction of the continuity equation is achieved by solving a Poissonequation for pressure� the right hand side of which is a divergence of theintermediate velocity �eld that is updated from the momentum equationswithout pressure gradients�

For the solution of the Poisson equation� two original methods are in�troduced� One is a single grid implicit iterative approximate factorizationmethod� the other one is a multigridmethod based on a Guass�Seidel smoother�Both methods are designed to be used in conjunction with a Pad�e type dis�cretization� which does not allow for an explicit one discrete equation thatnumerically represents the Poisson equation�

The �fth chapter is devoted to arti�cial dissipation on the non�uniformgrids� As central schemes suer from stability and non�monotonicity prob�lems� an arti�cial dissipation term is often introduced to tackle them� Whetherit is classical th order dissipation or a more recent invention�Arti�cial Se�lective Damping �which ensures that only the high frequency componentsof the solution are eectively damped out��the issue of applying it on thenon�uniform grids is very important� In the chapter �ve it is shown thatthe application of arti�cial dissipation on the non�uniform grids without tak�ing the non�uniformity into account� may lead to undesirable eects� suchas negative dissipation� The method that accounts for the irregularity ofthe mesh is introduced for both th order dissipation and Arti�cial SelectiveDamping� As opposed to the existing methods that take into account thenon�uniformity of the grid in the computation space� the proposed methoddoes it in the physical space as this approach is more suitable for the use onhighly irregular grids�

The last chapter of the thesis contains a number of simulations of in�compressible �ows� both steady and unsteady� The calculations have beenperformed with the developed in this thesis Pad�e �nite volume method aswell as the classical nd order scheme� The objective was to assess the gainin accuracy in e�ciency of the proposed approach as compared to the con�ventional low order schemes� The third test case�propagation of a vortexin otherwise uniform �ow�was chosen to test the ability of the method toaccurately represent the time development of vortical structures of the �ow�This ability is very important in such applications as DNS and LES of tur�bulent �ows� where the unsteady development of the �ow containing a widerange of physical scales is followed over a long period of time� Accurate res�olution of all these scales is a very important issue in such simulations� Thatis why Pad�e schemes� which have a better spectral resolution as compared

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to the conventional numerical methods� are a good candidate for such prob�lems� The assess the quality of the proposed method in the LES context� anLES of a turbulent periodic channel �ow has been considered as the last testcase in this thesis� The importance of applying the Pad�e scheme in dierentdirections of the �ow has been evaluated by performing calculations in whichthe compact scheme has only been used in one or two directions�

The original contribution of this thesis to the existing state of the art canbe summarized as follows�

� A formulation of a Pad�e type �nite volume method applicable on gen�eral structured meshes and taking the irregularity of the mesh intoaccount in the physical space �chapter ��

� A formulation of a DG�like �nite volume method applicable to thelinear convection equation on the Cartesian meshes �chapter ���

� Development of e�cient iterative methods for the solution of the Pois�son equation for pressure used in the context of pressure correctionmethods �chapter ��

� A formulation of Arti�cial SelectiveDamping applicable on non�uniformgrids by taking the non�uniformity into account in the physical spaceas opposed to doing it in the computational space �chapter ���

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Contents

� Introduction �

��� Numerical solution of PDE � � � � � � � � � � � � � � � � � � � � �

�� Classical central schemes � � � � � � � � � � � � � � � � � � � � � �

��� Pad�e type schemes � � � � � � � � � � � � � � � � � � � � � � � � �

�� Discontinuous Galerkin Method � � � � � � � � � � � � � � � � �

� Finite Volume Formulation ��

�� Finite Volume Method � � � � � � � � � � � � � � � � � � � � � � �

� �D case � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Pad�e Type Finite Volume Schemes in Multidimensions � � � � �

� Discretization of Viscous Fluxes � � � � � � � � � � � � � � � � � ��

�� Boundary Conditions � � � � � � � � � � � � � � � � � � � � � � � �

�� Non�linear Equations � � � � � � � � � � � � � � � � � � � � � � � �

� DG�like Finite Volume Method ��

��� �D case � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Multidimensional formulation � � � � � � � � � � � � � � � � � � ��

� Solution of Unsteady Incompressible Flows ��

�� Simulation of turbulent �ows � � � � � � � � � � � � � � � � � � � ��

� Time Integration � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Incompressible Formulation � � � � � � � � � � � � � � � � � � � �

���� Iterative Poisson solver � � � � � � � � � � � � � � � � � � �

��� The Multigrid Method � � � � � � � � � � � � � � � � � � ��

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� CONTENTS

� Arti�cial Selective Damping ��

��� Arti�cial Dissipation on Stretched Meshes � � � � � � � � � � � ��

����� Arti�cial Selective Damping � � � � � � � � � � � � � � � ��

Numerical Results �

��� Flat Plate Flow � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Lid�Driven Cavity � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Vortical Flow � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� LES of channel �ow � � � � � � � � � � � � � � � � � � � � � � � � ��

���� The description of the test case � � � � � � � � � � � � � ��

��� Results � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Conclusions ���

A Point�wise Approach ���

B Relations Determining Coe�cients ���

C Author s List of Publications ���

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Chapter �

Introduction

��� Numerical solution of PDE

The importance of solution of partial dierential equations in engineeringas well as fundamental research can hardly be overestimated� The rangeof practical and theoretical problems that are described by PDEs is vast�encompassing aeroacoustics� electro�magnetism� �uid and solid mechanics�thermodynamics� to name just a few� It comes as no surprise that the theoryof PDEs is one of the most developed areas in mathematics� But despite itsrelative success� very few problems of practical interest that require solutionof PDEs can be solved analytically� Therefore� the development and useof methods allowing to obtain an approximate solution of PDEs have beena topic of intensive research ���� ���� Typically� these methods consist inrepresenting a continuous �or piece�wise continuous� solution of a PDE by adiscrete set of parameters� for which a set of algebraic equations is obtained�All the existing methods can be categorized in a few groups according tothe representation of solution and the way in which the algebraic equationsthat determine them are derived� Among them the most popular are �nitedierence� �nite volume� �nite element and spectral methods�

Historically� it was the �nite�dierence approach that was �rst appliedto solution of dierential equations �the solution of initial value problems bymeans of �nite dierence method dates back to the ��th century�� It consistsin introducing a grid covering the computational domain and a discrete func�tion representing solution in grid points� The set of algebraic equations todetermine this discrete function is then obtained by replacing partial deriva�tives by its discrete analogues��nite dierences� This is one of the mostpopular methods up to date because of its simplicity and straightforwardcharacter� One of the drawbacks of the method is that it requires a high

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� CHAPTER �� INTRODUCTION

degree of regularity of the mesh� The mesh must be set up in a structuredway� i�e� each grid point can uniquely be identi�ed by n indices for an ndimensional problem� one for each mesh direction�

The �nite volume method is dierent from the �nite�dierence approachin its way of discretizing PDEs� Instead of replacing derivatives by �nitedierences� the integral form of a PDE is used for a set of typically non�overlapping control volumes determined by the grid covering the computa�tional domain� The Gauss theorem is used to replace volume integrals ofterms that are written in a divergence form by surface integrals� which arethen discretized using the discrete function representing the solution� Themost important advantage of this method is that the discrete analogues ofPDEs represent the integral form of the PDE� the aspect that is very impor�tant for applications with discontinuities and where the conservation of suchquantities as mass� momentum or energy is necessary�

The spectral methods dier from the methods described above both intheir way of representing the solution and in obtaining the descrete equations�Global spectral methods use a single representation of the continuous solu�tion thoughout the computational domain by a truncated series expansion�i�e� a sum of basis functions multiplied by unknown coe�cients� This seriesis substituted into a dierential equation and through the minimization ofthe residual function the unknown coe�cients are computed� The basis func�tions can be a set of orthogonal polynomials �e�g� Chebyshev or Legendrepolynomials� or Fourier basis functions� Spectral methods can be devidedinto two subsets� collocation �or peusdospectral� and Galerkin methods� The�rst approach uses a grid� i�e� a set of nodes �collocation points�� where theresidual function is reguired to be exactly zero to obtain the unknown co�e�cients of the exansion series representing the solution� In the Galerkinapproach the residual function is weighted with a set of test functions andafter integration is set to zero�

One of the most important issues of using such approximate methods isthe accuracy of the solution they provide� With the exception of some par�ticular cases� a numerical solution of a PDE diers from its exact solution�The quality of the numerical method is� therefore� assessed by how smallthe error of the numerical solution it provides is� If the exact solution of aPDE is available� this error can easily be calculated and the quality of thenumerical solution estimated� This� however� more often than not� is notthe case� as has been mentioned above� Therefore� one needs another way ofestimating the quality of a numerical method� There are two conditions thathave to be imposed on a numerical scheme in order to obtain an acceptableapproximation of the solution of a PDE�consistency and stability� Consis�tency means that the discretized equations should tend to the PDE to which

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���� NUMERICAL SOLUTION OF PDE �

they are related when the mesh is re�ned� i�e� when the size of mesh cellstends to zero� The rate of this convergence as a function of mesh spacing isa crude estimation of the accuracy of the scheme� The approximation is saidto be nth order accurate if the dierence between the discretized operator ofthe PDE and the exact operator is proportional to the size of mesh cells tothe power n at the limit�

This condition� however� is not su�cient for obtaining an accurate numer�ical solution� Another requirement is stability of the method� which meansthat small disturbances in initial and boundary conditions as well as in theright hand side of discretized equations lead to only small disturbances inthe solution of the discretized equations� This is an internal quality of thescheme not related to a PDE that it approximates�

If the numerical scheme is both consistent and stable� the dierence be�tween solutions of the discretized equations and PDEs tends to zero whenthe mesh is re�ned �Equivalence Theorem of Lax� for a proof see e�g� ������The order of convergence of the numerical solution to the exact solution isthen the same as the order of approximation�

The satisfaction of the afore�mentioned conditions guarantees that onthe in�nitely �ne mesh the solution of the numerical method is the sameas the exact solution of the PDE� In practice� however� if the mesh is not�ne enough� the solution of discretized equations can still be quite dierentfrom the exact solution� even if the scheme is both consistent and stable�To obtain a more accurate solution one can use a �ner mesh� which leadsto increased computational costs� Another strategy that can be adopted isto use schemes of higher order of accuracy� The schemes that are typicallyused in engineering applications are second order accurate� For some ofthese applications they are su�ciently accurate to obtain acceptable resultson relatively coarse grids� especially for steady equations� However� it isoften desirable to use schemes that have a higher order of accuracy� as theyallow to use much coarser grids and� as a result� reduce computational costs�The construction of such higher�order schemes can be achieved by extendinga stencil of the scheme� i�e� a number of grid points or control volumesused to approximate derivatives� �uxes� etc� Extended stencils� however�are more di�cult to treat on general �non�Cartesian� grids and close to theboundaries� That is why so called compact higher order schemes have becomevery popular ��� �� ��� ��� ��� ��� ���� They allow to construct a scheme ofhigh order of accuracy without extending the stencil� The most popular ofthem are probably those that belong to the Pad�e type of schemes� to whichthis thesis is devoted� In the remainder of this thesis� the term �compactscheme� will be meant to refer to Pad�e schemes� unless explicitly mentionedotherwise�

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�� CHAPTER �� INTRODUCTION

The development of high�order numerical methods has been a topic ofintensive reearch� One of the most accurate approaches to date is a spectraldiscretization technique ���� mentioned above� Unlike �nite dierence and�nite volume methods they have an exponential �or spectral convergence��This means that as a number of collocation points or the number of basisfunctions is doubled� the error of the numerical solution decreases by atleast two orders of magnitude and not a �xed factor as in �nite dierenceand �nite volume methods� The main disadvantage of this method is thatits application is limited to problems in simple domains and with simpleboundary conditions �the type of the boundary conditions that can be treatedby the global spectral method is determined by the basis functions used in themethod� However� it has been successfully applied to a number of problemsincluding Direct Numerical Simulation of turbulence ���

A spectral elementmethod� whose delelopment occured in the early eight�ies �� �� combines the exponential convergence of global spectral methods and�exibility of �nite volume and �nite element methods� This is an extensionof the classical �nite element method� in which the domain is divided into anumber of elements� to each of which the so called Ritz�Rayleigh procedureis applied to derive a set of discrete equations� In the classical �nite elementmethod the points that are used to de�ne the geometry of elements �intowhich the computational domain is divided� are also used for the interpola�tion of the solution� In the spectral element method� the solution is expressedin terms of higher�degree Lagrange polynomials on Gauss�Lobatto�Legendrepoints� One of the most important properties of the spectral element methodis that the mass matrix is exactly diagonal by construction� which drasticallysimpli�es the implementation and reduces computational costs because onecan use an explicit time integration scheme without having to solve a linearsystem of equations with non�diagonal matrix�

Another extension of the classical �nite element method is a Discontinu�ous Galerkin Method �DGM�� which is explained in more detail later in thethesis� with proper references to the literature�

In applications with discontinuities a family of essentially non�oscillatory�ENO� �nite volume schemes have become a popular tool � �� The idea ofENO schemes is to use the �smoothest� stencil among several candidatesto approximate the �uxes at cell boundaries to avoid spurious oscillationsnear shocks� The cell avaraged version of ENO schemes involves a procedureof reconstructing point�wise values from cell�averaged values ����� The socalled weighted ENO �WENO� schemes ���� unlike ENO schemes� employa combination of all possible stencils �instead of only one�� Each of thecanditate stencils is assigned a weight which determines the contribution ofthis stencil to the �nal approximation of the numerical �ux� The weights

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���� NUMERICAL SOLUTION OF PDE ��

can be de�ned in such a way that in smooth regions it approaches certainoptimal weights to achieve a higher order of accuracy� while in regions neardiscontinuities� the stencil which contains the discontinuities are assigneda nearly zero weight� The ENO�WENO ideology has also been applied tocompact �Pad�e� schemes ��� ��� ����

Originally� compact schemes were de�ned in the �nite dierence context�in one dimension and on a uniform grid� One of the �rst applications ofcompact �nite�dierencing to solving dierential problems can be found inthe book by Collatz � �� and � �� In ��� Pad�e schemes have been formulatedon dierent stencils for �rst and second order derivatives as well as interpo�lation operators� The important conclusion of that paper was that not onlyare Pad�e schemes high order accurate on compact stencils� they also resolveshorter scales of the solution better than classical �nite dierence schemes�which brings them closer to highly accurate spectral methods� This aspect ofcompact schemes has made them very popular in applications such as directnumerical simulation �DNS� and large eddy simulation �LES� of turbulenceand computational aeroacoustics �CAA� because of importance of properresolution of a wide range of time and space scales of the solution in thesesimulations�

An important issue of application of compact schemes is their use inmultidimensions and their accuracy on general� i�e� non�Cartesian and alsonon�uniform� grids� In a �nite dierence context� this is usually dealt withby formulating the compact schemes in the computational space� e�g� �� �� � ��� ���� However� as mentioned in ����� the use of the Jacobian trans�formation can lead to an important reduction of the accuracy of the schemein case of non�smoothly varying mesh spacing because of numerical errorsin the determination of the derivatives of the transformation appearing inthe Jacobians� Gamet et al� ���� take the non�uniformity of the mesh intoaccount by adapting the coe�cients of the compact stencils on the Cartesiangrids� If the grid is non�Cartesian� a Jacobian transformation still has to beused� Note that a similar approach was formulated earlier by Goedheer andPotters ���� for a second order partial dierential equation�

Although� in the framework of the �nite�dierence approach the compactschemes are relatively easy to construct on irregular grids� special attentionmust be paid to the conservation properties of the scheme� as conservationis not automatically guaranteed�

The �nite volume method� on the other hand� is inherently conservative�One of the �rst papers dealing with formulation of compact schemes in ��nite volume context is by Gaitonde and Shang ����� The scheme is basedon an implicit reconstruction step� relating cell face values� A similar ap�

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� CHAPTER �� INTRODUCTION

proach is proposed more recently by Kobayashi ���� Both papers deal withlinear advection equations on uniform Cartesian grids� In ��� ��� Kobayashiand co�workers developed an extension to �nite volume compact scheme toNavier�Stokes equations and curvilinear non�uniform grids� Their approachmade use of Jacobian transformation of coe�cients� i�e� the scheme was for�mulated in the computational space� The disadvantage of this has alreadybeen mentioned�

In an attempt to formulate a higher�order accurate �nite volume schemeon arbitrary structured meshes� another route was taken in ���� ��� � �� In�stead of formulating the scheme in the computational space� it was con�structed in the physical space� coe�cients of the scheme taking into accountthe irregularity of the mesh directly�

In this thesis� this approach is described and applied to a variety of prob�lems� such as linear advection equations� laminar and turbulent �ows�

��� Classical central schemes

Consider an approximation of the �rst order derivative on a uniform mesh�

u�i ��

�x

MXj��M

ajui�j �����

where M de�nes the stencil width� ui�j is a solution in a grid point i � j�and �x is a distance between two neighboring mesh nodes� To determine thecoe�cients aj and the order of the scheme� one can decompose the ui�j in aTaylor expansion around i and equate left� and right�hand�side of equation����� to the maximum possible order� One �nds that a� � � and aj � �a�j�i�e� the scheme is central� and that the order is M � The simplest schemeis the well known second order central scheme �M � �� the stencil width isthree grid nodes��

u�i ��

�x�ui�� � ui��� ��� �

Although this scheme is routinely used in various applications� its accuracyoften turns out insu�cient� In applications such as DNS and LES of tur�bulence and CAA� where accurate simulation of unsteady development of�elds containing a wide range of scales is important� higher order schemesare usually used� As has been mentioned above� the extension of a stencilis one of the possible strategies of constructing higher order schemes� A thorder scheme can be obtained� if M � �

u�i ��

��x

���ui�� � ui�� � ui�� � �

ui��

������

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���� CLASSICAL CENTRAL SCHEMES ��

Theoretically� by extending a stencil even further �M � �� one can con�struct schemes of any order� which is only restricted by the size of the mesh�In practice� however� stencils with M � � are hardly used�

In central schemes� the discretization stencil is symmetric with respectto the point where the derivative is evaluated� There exist upwind typeschemes� in which the discretization stencil is upwind �or downwind� biased�i�e� upwind �downwind� points have a higher contribution weight� E�g� a �storder scheme for the �rst order derivative would be given by ����� if M � ��a� � �� a� � � and a�� � ���

u�i ��

�x�ui � ui��� ����

These schemes do not have the highest possible order of accuracy on a givenstencil and were not considered in this thesis�

The formal accuracy of the �nite�dierence scheme �the order of the trun�cation error� does not provide all the information about the numerical error�To get a better insight into the error characteristics� one can use the Fourieranalysis of dierencing errors� By comparing the modi�ed wavenumber ofthe numerical approximation to the wavenumber of the exact dierentiation�resolution characteristics of the scheme can be observed in a Fourier space�

Consider a harmonic u � eIkx �I �p��� and its discrete representation

ui � eIkxi on a uniform mesh� The exact �rst derivative of the harmonicu� � IkeIkx can be compared with its discrete analogue �u�i�num � Ik�eIkxi�The more accurate the numerical approximation of the derivative� the closethe modi�ed wavenumber k� is to k� By inserting eIkxi into ����� one canobtain the function ������ where � � k�x and �� � k��x�

�� ��

I

MXj��M

ajeIj� �����

For a central scheme the imaginary part of �� equals zero� therefore one canwrite

�� � MX

j��M

aj sin�j�� �����

The maximum order of the scheme on a given stencil is M � However�for each stencil with M � � there exists a family of schemes of order less orequal to ��M���� Among all these schemes that do not have the maximumorder on a chosen stencil� one can choose the one that have better spectralcharacteristics� Such are� for instance� the Dispersion Relation Preserving�DRP� schemes� which have been developed by Tam and Webb� ����� In

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� CHAPTER �� INTRODUCTION

contrast to the central schemes that have a maximumorder on chosen stencil�only part of the available coe�cients aj of the stencil are used to �x the orderof the scheme� The remaining coe�cients are tuned so as to minimize theresolution error� This is the dierence between the physical wave number kand the numerical wave number k� which is given by ������

The error� to be optimized� is expressed as

E �Z �

��j� � ��j�d��� �����

The value of �� determining the range of the integral� is originally chosen as�� by Tam and Webb� ����� This means that optimization is performed onlyfor waves with a wavelength larger than �x� The reason for not optimizingover the complete wave number range �������� is that� irrespective of thescheme used� the numerical wave number goes to zero for k�x � �� It there�fore makes no sense to try to optimize up to the highest wavenumbers� seealso ��� for optimization in the framework of compact schemes� Accordingto Lockard et al�� � ��� the choice � � �� places already a very stringent re�quirement on the minimization which can only be met by schemes with somesigni�cant overshoots �the numerical wave number is larger than the actualone� in the lower wavenumber range of the wavenumber diagram� The betterresolution for high wave numbers is then at the cost of a reduced resolutionfor lower wave numbers� This was also recognized by Tam in later work� �����In � �� the integral therefore extends only up to waves with wavelengths of��x or larger� i�e� � � ���� The DRP scheme is often used in combinationwith the Linearized Euler �LEE� approach� e�g� �� for acoustic scattering bya fuselage and ���� for the aeroacoustics of a wind turbine blade�

��� Pad�e type schemes

The idea of a Pad�e type approximation of any unknown quantity is to usean implicit �nite�dierence or interpolation formula as opposed to classicalmethods that only make use of explicit relation between approximated quan�tity and other variables� For instance� for the �rst derivative of any scalarquantity u the Pad�e type approximation can be written as follows�

�u�

i�� � �u�

i�� � u�

i � �u�

i�� � �u�

i�� � cui�� � ui��

�h� �����

bui�� � ui��

h� a

ui�� � ui�� h

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���� PAD�E TYPE SCHEMES ��

where h is the grid�spacing�in the assumption of uniform mesh� and �� ��a� b� c are the coe�cients that determine the accuracy of the approximationand which can be derived by developing all the variables in a Taylor seriesabout point i� and requiring all the coe�cients of the resulting expansion tovanish� up to some de�nite term� The �rst non�zero coe�cient will determinethe formal truncation error and the order of the scheme�

One of the advantages of Pad�e approximation is that it can be made highorder accurate on only a compact stencil� The price to pay� though� is thatone has to solve an additional set of linear algebraic equations� which makesthe numerical algorithm computationally more expensive�

The second order approximation of the �rst order derivative can be de�rived by satisfying the following relation�

a� b� c � � � � � � �����

The fourth order�

a� �b� ��c � ��

���� ��� ������

The sixth order�

a� �b� ��c � ��

���� ��� ������

The eighth order�

a� �b� ��c � ��

����� ��� ���� �

The tenth order�

a� �b� ��c � ��

����� ��� ������

The compact approximation of the second derivative can be written inthe form�

�u��

i�� � �u��

i�� � u��

i � �u��

i�� � �u��

i�� � cui�� � ui � ui��

�h�� �����

bui�� � ui � ui��

h�� a

ui�� � ui � ui��h�

One gets the second order approximation� if the following relation is sat�is�ed�

a� b� c � � � � � � ������

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�� CHAPTER �� INTRODUCTION

The fourth order�

a� �b� ��c ��

���� ��� ������

The sixth order�

a� �b� ��c ���

���� ��� ������

The eighth order�

a� �b� ��c ���

����� ��� ������

The tenth order�

a� �b� ��c ����

���� � ��� ������

As has already been mentioned in the previous section� the formal accu�racy of the �nite�dierence scheme does not� however� provide all the infor�mation about the numerical error� The Fourier analysis gives a better insightinto dierencing errors�

The modi�ed wavenumber of the discretization ����� is given by �for thede�nition see the previous section� the part on DRP schemes�

����� �a sin��� � �b� � sin� �� � �c��� sin����

� � � cos��� � � cos� ����� ��

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

ω

ω′

exact2nd order4th order explicit4th order compact

Figure ���� Plot of modi�ed wavenumber vs� wavenumber of the �rst deriva�tive approximations� Schemes� nd order explicit� th order explicit and thorder compact�

Plots of the modi�ed wavenumber�� against wavenumber � are presentedin Figures ��� and �� for a variety of schemes� As one can see from picture

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���� PAD�E TYPE SCHEMES ��

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

ω

ω′

exact6th order explicit6th order compact8th order compact

Figure �� � Plot of modi�ed wavenumber vs� wavenumber of the �rst deriva�tive approximations� Schemes� �th order explicit� �th order compact and �thorder compact�

��� the spectral resolution of the th order Pad�e schemes is better than forclassical schemes of the same order� The same is true for the �th orderschemes� This can also be shown by comparing numerical solutions of themodel problems for both classical and compact schemes of the same order�In Fig� ��� the numerical solutions of the convection equation

�u

�t��u

�x� � ��� ��

with a Gauss signal e����x�����

as an initial condition are presented for bothth order compact and classical �nite�dierence schemes� The computationaldomain is � � x � � �x � �� �� the solution is set to zero at the boundaries�For the time integration a stage low storage RK scheme �with coe�cient��� ���� �� � ��� which for linear problems with constant coe�cients is thorder accurate� was used� and the time step was taken su�ciently small��t � ������ to ensure that the errors coming from time integration aremuch smaller than those of spatial discretization� The signal was propagatedfor � second� As is seen from the plots� although the formal accuracy of theschemes is the same� the compact scheme gives the result that is closer tothe exact solution of the problem�

Thus the advantage of compact schemes as compared to classical schemesis twofold� First� they are higher order schemes on a more compact stencilthan classical schemes of the same order of accuracy� which makes theminteresting from the implementation point of view� Second� they actuallyhave a better resolution than the classical schemes with the same formalaccuracy� This feature brings them closer to the highly accurate spectral

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�� CHAPTER �� INTRODUCTION

1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X

U

Compact 4th orderClassical 4th orderexact

Figure ���� Plots of the solution of the convection equation for th ordercompact and classical �nite�dierence schemes

methods� while preserving the �exibility of the �nite�dierence and �nite�volume methods in terms of grid topology and boundary conditions�

The stability analysis of these schemes when applied to a linear convectionequation can be found in ��� and leads to the following condition�

�s

�max

��� �

with the CFL number� s the segment on the imaginary axis� where the timeadvancement scheme is stable� and �max the maximummodi�ed wavenumberof the scheme� For the th order Pad�e scheme �max �

p�� Assuming a

standard fourth order four stage RK scheme for time integration �s � ����this leads to the following stability criterion� � ���

The same idea as in DRP schemes can be used to obtain compact schemeswith better resolution characteristics� Lele ��� proposed giving up on theformal accuracy of the scheme in favor of getting a scheme spectral resolution�By requiring the modi�ed wavenumber of the compact approximation shouldbe equal to the wavenumber of the exact dierentiation in three points� hegot the additional constraints to replace ������� ���� �� �������

������ � ��� ������ � ��� ������ � �� ��� ��

As a result the fourth�order accurate scheme for approximation of the �rstderivative with a better representation of the short scales as compared tothe tenth�order accurate scheme was obtained �with �� � � �� � ��

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���� PAD�E TYPE SCHEMES ��

�� � ��

� � �������� � � �������� a � ��� ����� b � ������� c � ������ ���� �

From Figure �� it can be seen that the proposed scheme has better resolutioncharacteristics than the tenth�order scheme with the same stencil�

.00 .79 1.57 2.36 3.14 .00

1.05

2.09

3.14

AB

C

Figure ��� Plot of modi�ed wavenumber vs� wavenumber of the �rst deriva�tive approximations� �A� tenth�order compact approximation �� � ��� � ����� a � ���� � b � �������� c � ���� �b� spectral�like compact approxi�mation ��� � � �������� � � �������� a � ��� ����� b � ������� c ������� ��� �C� exact dierentiation�

Analoguewise� the fourth�order accurate scheme with improved resolutioncharacteristics can be obtained for approximation of the second derivative�

Although the above approach allows to construct schemes with improvedresolution characteristics� it seems to be somewhat empirical� Kim and Lee��� suggested a more general optimization of the resolution characteristicsof compact schemes based on the minimalization of the integrated dispersive�phase� errors in the wave number domain� They de�ned an integrated erroras

E �Z r�

��� � ����W ��d��� ��� ��

where � is the modi�ed wave number of the approximation ����� or ������W ���x� is a weighting function� and r is a factor to determine the optimiza�tion range �� � r � ��� The weighted function proposed in their paper waschosen to make ��� �� integrable� and weight the integrated error more thanenough in the high wavenumber range close to ��

W ��� �h�� � � cos� � � cos ��e�

i���� ��

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� CHAPTER �� INTRODUCTION

The integrated error introduced is a function of the coe�cients �� �� a� b�c� To minimize the error the following constraints� which provide a locationof a local minimum of E� can be used

�E

��� � ��� ��

�E

��� � ��� ��

�E

�a� � ��� ��

�E

�b� � ������

�E

�c� � ������

Equations ���������� and ��� ������� provide a system of linear algebraicequations by which the optimum coe�cients can be determined� The set ofequation to be solved is optional� Kim and Lee chose to search for optimizedschemes with dierent orders of formal accuracy�

�� Tridiagonal �� � ��

� Second order� Eqs� ������ ��� ��� ��� ��� �������� Fourth order� Eqs� ������ ������� ��� ��� �������� Sixth order� Eqs� ������ ������� ������� ������� � Pentadiagonal �� �� ��� Second order� Eqs� ��� ��� ��� ��������� Fourth order� Eqs� ������ ������� ��� ��������� Sixth order� Eqs� ������ ������� ������� ������� �������� Eighth order� Eqs� ������ ������� ������� ���� �� �������It was found to be necessary to reduce the optimization range factor r

due to the fact that schemes optimized over the whole range �r � ��� hadundesirable overshoots in the high wave number range� Since the errors inthe range close to � are uncontrollable� it is preferable that this wavenumberrange should be omitted in the optimization�

Kim and Lee found the optimization range factors of each scheme needto be readjusted to obtain maximum characteristics as follows�

�� Tridiagonal �� � ��

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���� PAD�E TYPE SCHEMES �

� Second order� r � �� � Fourth order� r � ���� Sixth order� r � ���� � Pentadiagonal �� �� ��� Second order� r � ��� Fourth order� r � ���� Sixth order� r � ����� Eighth order� r � ����The above procedure allowed to obtain the following optimized compact

schemes�

Tridiagonal a b c � �

Second order �������� �� �� � ����� ���� ��������� �Fourth order ��������� ����� ���� ��� ������ �������� �Sixth order ������� � � �������� ��� ������ ������ �� �

Pentadiagonal a b c � �Second order � ������ � ������ �� ��������� �������� � ���������Fourth order � ���� ��������� ������ �������� � ������ ���Sixth order �� �� ��� ���� � �� ������� ������ �� ���� �� � Eighth order �������� � ������� ���������� ���� ���� ���������

Note that a similar procedure was applied by Tam and Webb ���� to theclassical compact schemes� which have been discussed in the previous section�

An alternative way for optimization is proposed by Tang and Baeder������ Their optimization is based on the description of the compact schemesvia Hermitian interpolation� Their approach is illustrated below for a threepoint scheme� involving ui���ui�ui���

First consider a Lagrangian interpolation� which will lead to standardcentral schemes� The Lagrangian interpolation function q�x� for a � pointscheme is a polynomial of power �

q�x� � a� � a��x� xi� � a��x� xi�� ���� �

Equating q�x� in the nodal points i � �� i� i � � to resp� ui���ui�ui�� allowsto determine the coe�cients ai� The discretization formula for u�i is thenobtained as u�i � q��xi� � a�� One obtains the standard central scheme�

u�i �ui�� � ui�� �x

������

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CHAPTER �� INTRODUCTION

The compact scheme is obtained via Hermitian interpolation� q�x� is now apolynomial of order �

q�x� � a� � a��x� xi� � a��x� xi�� � a��x� xi�

� � a��x� xi�� �����

Two additional conditions are needed to determine ai� These conditions are�

q��xi��� � u�i�� ������

q��xi��� � u�i�� ������

The discretization formula for u�i is again obtained as u�

i � q��xi� � a�� It caneasily be checked that the th order compact scheme given by ������ ������������ and � � b � c � � is retrieved�

Tang and Baeder propose the use of trigonometric functions instead ofpower polynomials for the interpolation� leading to so�called Fourier dier�ence schemes as opposed to Taylor dierence schemes� For the same exampleas above the interpolation function becomes�

q�x� � a� � a� cos

���x� xi�

�x

�� a� sin

���x� xi�

�x

�������

where the half wavelength of the cosinus and sinus functions corresponds tothe stencil width of �x� Equating q�x� in the nodal points i� �� i� i� � toresp� ui���ui�ui�� allows to determine the coe�cients ai� The discretizationformula for u�i is then obtained as u

i � q��xi� ��

��xa�� One obtains

u�i ��

ui�� � ui�� �x

������

Comparing the numerical wavenumber of this scheme with the one of ������it is seen that the numerical wavenumber is multiplied with �

�� As a result

the dispersion errors are more uniformly distributed in the frequency domain�but there is also a large accuracy contamination in the low frequency range�

As a result this low order trigonometric interpolation does not lead to areasonable scheme� Although the �th�order Fourier dierence scheme seemssuperior to the �th�order Taylor scheme� it has to be noted that the Fourierdierence scheme has always a zero�order truncation error� which is of coursenot acceptable from a mesh re�nement point of view�

Tang and Baeder therefore propose to combine the Fourier dierenceapproach with the Taylor dierence approach� by replacing only the higherorder power polynomials of the Hermitian interpolation with higher order

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���� PAD�E TYPE SCHEMES �

trigonometric functions� For the compact scheme above� the interpolationfunction ����� is replaced with

q�x� � a� � a��x� xi� � a��x� xi��� a� cos

�l�x� xiL

�� a� sin

�l�x� xiL

�������

with L the stencil width� i�e� �x� and l a free parameter� The expression foru�i is obtained in exactly the same way as for the th order compact scheme�By varying l� the dispersive behavior of the scheme can be improved� Notethat l� corresponds to the maximum wavenumber visible on the presentstencil �wavelength of �x�� Experience shows that l must be well below thismaximum wavenumber value in order to produce useful optimized schemes�Tang and Baeder also further re�ne this strategy by including trigonometricfunctions of dierent orders� l� and l�� in the interpolation� Results shownin ���� are limited however to �D convection of a Gaussian wave�

These approaches allow to obtain schemes with a yet better spectral reso�lution than Pad�e schemes with highest order of accuracy possible on a chosenstencil� This yields the schemes that are even closer to spectral methods interms of accuracy� However� a straightforward application of this method�ology is only possible on uniform meshes� The use of such schemes on non�uniform grids requires their formulation in the computational space� which�as has already been mentioned� is a less accurate approach than formula�tion of schemes in the physical space ����� For this reason� only the schemesobtained through requiring the maximum formal accuracy are considered inthis thesis�

To even further increase the accuracy of the compact scheme on a givenstencil one needs a discretization formula with more coe�cients that can betuned to achieve the desirable accuracy� In ���� Mahesh proposed to evaluatethe �rst and second derivatives simultaneously� This procedure allows foreven higher order of accuracy on a given stencil that standard Pad�e schemescan provide� For instance� on a � point stencil Mahesh�s scheme is �th orderaccurate� while the standard Pad�e scheme is only th order accurate�

�u�i�� � ��u�

i � �u�

i�� ��x�u��

i�� � u��i��� � ��ui�� � ui�� �x

�����

��u�i�� � u�i�����x�u��i�� � �u��i � u��i��� � ui�� � ui � ui��

�x�����

Equations ����� and ����� represent a system of linear equation that canbe solved by the Thomas matrix algorithm� Note that in the standard Pad�eapproach� the systems of equations for the �rst and second derivatives are de�coupled from each other and therefore can be solved separately� In Mahesh�s

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CHAPTER �� INTRODUCTION

scheme ����� and ����� are coupled� therefore even if the second derivativeis not present in the PDE� it still has to be calculated� It is obvious that thisapproach is more interesting for the second order PDEs�

��� Discontinuous Galerkin Method

The discontinuous Galerkin �DG� method is a compact �nite�element pro�jection method that provides a practical framework for the development ofhigh�order methods using unstructured grids� The higher�order accuracy isobtained by representing the solution in each cell as a high�degree polyno�mial whose time evolution is governed by a local Galerkin projection� Incontrast with standard �nite�element methods� the DG methods enforce theconservation law only locally� This allows them to have a mass matrix thatcan be easily made to be the identity� and therefore does not necessitate�lumping� nor matrix inversion� while being highly accurate and nonlinearlystable� Because of the local character� the solution may be discontinuousacross dierent elements� As a result the basis for the polynomials is es�sentially unrestricted� However� for computational e�ciency� an orthogonalbasis is chosen relative to the inner product� Note however that� even if thepolynomials are not orthogonal� one still only needs to invert a small massmatrix and there is never a global mass matrix as in a typical �nite elementmethod� The DG method can also be interpreted as an extension of a �nitevolume method� incorporating notions such as approximate Riemann solvers�numerical �uxes and slope limiters into a �nite element framework� However�instead of only one degree of freedom per cell as in a �nite volume method�namely the cell average of the solution� there are� in D� �n � ���n � �� degrees of freedom for a polynomial of order n� These degrees of freedom arechosen as the coe�cients of the polynomial when expanded in a local basis�

Consider for instance the Euler system�

�U

�t�rF � � ��� �

In each cell U is expanded in terms of the polynomials bk� k � �� � � � � N �

U �NXk��

Ukbk �����

The discretized equation is obtained by multiplying ��� � with bj and inte�grating over the cell� Z

V

�U

�tbjdV �

ZVrFbjdV � � ����

Page 27: Sergey Smirnov

���� DISCONTINUOUS GALERKIN METHOD �

Using the expansion ����� and applying Green�s theorem one obtains�

NXk��

�Uk

�t

ZVbkbjdV �

ZVFrbjdV �

IFbj �dS � � �����

This equation shows that the derivative of Uk is retrieved by inverting amatrix M with components Mkj �

RV bkbjdV � The surface integral requires

an expression for the �ux on the cell faces� Since the solution is discontinuousacross cell faces �as in �nite volume methods� numerical �uxes based onRiemann solvers and incorporating limiters can be used� One can showthat for polynomials of degree p� the order of accuracy is at least p � �

������� Equation ����� is usually discretized by evaluating the integrals usingquadrature formulas of the required order which is p for the volume integraland p�� for the edge integral� � ��� This limits the usefulness of the method�as the number of terms in the quadrature summations signi�cantly exceedsthe number of unknowns� making the method computationally expensive�However� Atkins and Shu� ��� ��� described a quadrature free formulation�

The application of the DG method to hyperbolic systems� in combina�tion with Runge�Kutta methods� the so�called Runge�Kutta DiscontinuousGalerkin �RKDG� method� was thoroughly investigated by Cockburn andShu� see � �� � for a detailed survey�

Hu et al�� ���� study the wave propagation properties of the DG method�Based on an analysis of scalar convection� they �nd that in a formulation withupwind �uxes the dissipative errors are dominant� whereas the dispersiveerror is negligible for wave numbers up to a value equal to the order of themethod� In a centered �ux formulation� there is no dissipative error� butthe range of wave numbers for which the dispersion error is negligible issmaller than with the upwind �ux� Since DG methods are especially usefulon unstructured grids� the in�uence of the mesh was also investigated� Basedon an analysis of the D wave euqation� it is found that an unstructured�like triangular mesh has better dispersion and dissipation properties thana structured quadrilateral grid or than triangular grids derived from suchgrid� Also the properties are less prone to anisotropy� i�e� vary less with theorientation of the Fourier waves�

Atkins and Lockard� ���� use the RKDG method to solve the LinearizedEuler Equations �LEE� to study acoustic scattering from a two�dimensionalslat and a three�dimensional blended�wing�body combination� In addition�they show� for a three�dimensional wave propagation in a cube� that theRKDG method is insensitive to the mesh smoothness� on a smooth tetrahe�dral mesh and the same mesh with � random perturbation of each gridpoint� the two solutions are almost indistinguishable�

Page 28: Sergey Smirnov

� CHAPTER �� INTRODUCTION

Stanescu at al�� ����� use the spectral element implementation of DG ofKopriva et al�� ���� for the computation of sound radiation from aircraftengine sources to the far��eld� The non�linear Euler equations are solvedand only the radiation of inlet noise into a quiescent �uid is modeled� Bothnacelle alone and fuselage�nacelle� fuselage�wing�nacelle con�gurations areconsidered� In a later paper� ����� the same methodlogy is applied to anactual two�engine jet aircraft and compared to a spectral element solution inthe frequency domain� The results show that trends of the noise �eld are wellpredicted by both methods� In ��� the DG method is applied to a benchmarkproblem of the CAA workshop of �����

Page 29: Sergey Smirnov

Chapter �

Finite Volume Formulation

��� Finite Volume Method

In the �nite volume method� the computational domain is divided in a num�ber of time invariant� non�overlapping volumes �later called cells�� This vol�ume can in principle be of any shape� but the most popular are tetrahedra�triangles in D� or hexahedra �quadrilaterals in D� or prisma� The formerare almost exclusively used in unstructured grid solvers� which is beyondthe scope of this thesis� The hexahedra cells� however� can be used on bothstructured and unstructured meshes� In a structured grid each cell can beuniquely identi�ed by n indices� where n is a number of spatial dimensions ofthe problem solved� The neighboring cells can be found directly via a unityincrease or decrease of one of the indices� Due to this� the data managementin the codes employing structured meshes is straightforward and requires lessmemory as compared to unstructured meshes� For the Pad�e schemes it isespecially important as the existence of grid directions along which all butone indices are constant makes the construction of the equations for approx�imated values� e�g� ������ straightforward� A similar situation is found in theapplication of implicit methods� which are also mostly used on structuredmeshes�

Once the mesh dividing the domain of interest in the cells is introduced�the idea behind the �nite volume method is the following� the dierentialequations are integrated over each cell and the Gauss theorem is appliedto terms that are written in divergence form� The resulting volume andsurface integrals �area and line integrals in case of a D problem� are thenapproximated using the discrete solution stored around the cell� Thus a setof algebraic equations is obtained� the solution of these equations giving thedesired numerical solution to the problem�

Page 30: Sergey Smirnov

� CHAPTER �� FINITE VOLUME FORMULATION

Two alternative strategies can be used for storing the discrete data inthe �nite volume method� cell�centered or cell�vertex approach� In the cell�vertex approach the solution is stored in the vertices of the cells� while inthe cell�centered approach the discrete solution is either considered in thecenters of each control volume or associated with cell�averaged values of thesolution�

��� �D case

Let us now consider a �nite�volume approximation of a �D advection equation

�u

�t��f�u�

�x� � � ���

Assuming a uniform mesh and integrating � ��� over the mesh cells� oneobtains�

Z xi����

xi����

�u

�tdx� f�xi������ f�xi����� � � � � �

This is still an exact relation� To discretize it� both the integral containingthe time derivative and the �uxes on the interfaces x � xi���� and x � xi����must be approximated� Before doing so� one has to choose whether point�wise �de�ned� for example� in the centre of the cell� or cell averaged valuesof u

ui ��

�x

Z xi����

xi����

u�x�dx � ���

are used� In the !point�wise� approach� a scheme of higher order accuracyin space does not only require the �uxes to be approximated with high�orderaccuracy but also the integral containing the time derivative� One thereforehas to use a complex high order quadrature formula for the calculation ofthis integral in order to maintain the high�order accuracy for unsteady calcu�lations� Note that strictly speaking this is only necessary in case of unsteadycalculations� as at the steady state the time derivative and� as a result� theintegral vanish and the spatial accuracy only depends on the accuracy withwhich the cells face �uxes are approximated�

Therefore for unsteady problems the !cell�averaged� approach is prefer�able� as it allows to calculate the integral exactly by a simple formula�

Z xi����

xi����

�u

�tdx �

�ui�t�x � ��

Page 31: Sergey Smirnov

���� �D CASE �

As the major concern in this work is unsteady calculations� the cell�averaged approach will be assumed in the remainder of this thesis� For abrief discussion of pointwise high�order �nite volume method see AppendixA�

To evaluate the �uxes f�xi����� and f�xi����� two approaches exist� The�rst one �so�called �ux averaging approach� consists in �rst calculating the�uxes associated with the cells �fi� and then using an interpolation formulato calculate the �uxes on the interfaces�

�fi���� � �fi���� � fi���� � �fi���� � �fi��� � � ���

� afi�� � fi

� bfi�� � fi��

� c

fi�� � fi��

The disadvantage of this approach in combination with cell�averaged val�ues of u is that the calculation of the �ux fi � �

�x

R xi����xi����

fdx is not straight�

forward� since the relation

fi � f�ui� � f

��

�x

Z xi����

xi����

udx

�� ���

is exact only if f is a linear function of u �otherwise fi � f�ui� �O��x����

In the second approach �so�called variable averaging approach�� one �rstevaluates the values of u on the interfaces �ui����� ui����� by means of asimilar interpolation formula

�ui����� �ui���� � ui���� � �ui���� � �ui��� � � ���

� aui�� � ui

� bui�� � ui��

� c

ui�� � ui��

where the right�hand side contains cell�averaged values of u� Then the �uxes

fi���� and fi���� can be calculated in a straightforward way as�

fi���� � f�ui����� fi���� � f�ui����� � ���

Therefore� if one solves a non�linear problem and uses cell�averaged valuesof u� the variable averaging approach is preferable�

The coe�cients a� b� c and �� � in � ��� can be derived by developingall the variables in a Taylor series about point i � �� � and requiring all

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�� CHAPTER �� FINITE VOLUME FORMULATION

the coe�cients of the resulting expansion to vanish up to some de�nite term�The �rst non�zero coe�cient will determine the formal truncation error� Theapproximation is second order accurate provided the following relation issatis�ed�

a� b� c � � � � � � � ���

For higher order accuracy additional relations have to be ful�lled�

For fourth order�

a� � � � ��b� ��� � ��c � �� ��� � ��� � ����

For sixth order�

a� � � ��b� �� � �c � ����� � ��� � ����

For eighth order�

a� � � ��b� �� � �c � ������ � ��� � �� �

For tenth order�

a� � � � ��b� ��� � ��c � ������ � ��� � ����

E�g� for th order one gets�

ui���� � ui���� �

ui���� �

�ui�� � ui� � ���

The classical second order scheme is given by � � � � b � c � � and a � ��

ui���� ��

�ui�� � ui� � ����

Note that when using a point�wise approach the coe�cients are deriveddierently ��� due to the fact that in the right hand side of the implicitformula there are pointwise values as opposed to cell�averaged values rep�resenting an integral of the function over the cell divided by the volume�E�g� the fourth order interpolation formula in case of point�wise approachbecomes�

Page 33: Sergey Smirnov

���� �D CASE ��

�ui���� � ui���� �

�ui���� �

��ui�� � ui� � ����

The second order scheme in case of the point�wise approach� however� isidentical to its cell�averaged counterpart�

Although one can obtain Pad�e schemes of order higher than four� thisrequires stencils bigger than that used for classical second order centralschemes� The th order scheme � ��� however� uses the smallest stencilpossible for central schemes �there are only cells involved�� It� of course�makes it attractive from the practical point of view� Although the formalaccuracy of this scheme is th order� it is in fact more accurate than theclassical th order central scheme constructed on a four cell stencil� Thiscan be con�rmed by comparing the spectral resolution of both interpolationformulas�

Consider a harmonic u � eikx� On a uniform �D grid a linear interpolationformula applied to this harmonic gives a discrete function de�ned on theinterfaces of the cells� which corresponds to the the same harmonic with adierent amplitude� T ���eikx� where � � k�x� One can �gure out T ���for any linear scheme on a uniform mesh� For the spectral interpolationT ��� � �� For the classical and Pad�e schemes T ���� however� deviates fromunity� especially at the high frequency range� i�e� for high values of �� Thespectral resolution of the th order classical scheme

ui���� ��

ui�� � ui

� ��

ui�� � ui��

� ����

is given by

T ��� �

sin���

�� ��

sin� ��

�� ����

This relation is obtained by applying the right hand side operator of � ����

to the harmonic eikx� Note that the right hand side of � ���� is writtenfor cell average values� i�e� for the harmonic eikx� one must e�g� considerui�� �

��x

R xi����xi����

eikxdx

The resolution of the th order Pad�e scheme � ��� is

T �w� ��

sin�w�x�

w�x�� � ��cos�w�x��

� ����

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� CHAPTER �� FINITE VOLUME FORMULATION

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

w dx

T

ExactClassical 4th orderCompact 4th order

Figure ��� Spectral resolution of classical and Pad�e th schemes

In Fig� �� the resolutions of both schemes are compared� As is seen� thePad�e scheme has better spectral properties in a high frequency range�

��� Pad�e Type Finite Volume Schemes in Mul�

tidimensions

Now let us consider the linear D scalar advection equation

ut � fx � gy � � � � ��

where f � au and g � bu� a and b being constants�

Consider the structured mesh of �gure � � Integrating equation � � ��over the cells and making use of the Gauss theorem� one can writeZ Z

Si�j

�u

�tdxdy �

IABB�A�A

��gdx� fdy� � � � � ��

To discretize this equation both surface and line integrals must be ap�proximated� Before doing this� as for the �D case described above� one hasto choose whether point�wise or cell�averaged �ui�j � �

Si�j

R RSi�j

u�x� y�dxdy�

values of u will be used� For the reasons mentioned above� again the cell�averaged approach is chosen� Let us now consider the discretization of theline integrals in � � ��� As all the integrals are approximated in the same

Page 35: Sergey Smirnov

���� PAD�E TYPE FINITEVOLUME SCHEMES INMULTIDIMENSIONS��

A’’

i,j

i,j+1

i+1,j+1

i,j-1

i+1,j-1

O

B

A

x

y

normal

i+1,j

ß

B’

A’

B’’

Figure � � Mesh

manner� we will only considerRBA fdy� Following the strategy of constructing

the Pad�e type schemes� the idea is to use an implicit formula for evaluatingthe line integrals over the cell faces� First assume that the grid is Cartesianand uniform� Once the cell�averaged values of u are given in the cells� one cancalculate the line integral

RBA udy by means of the following implicit formula

ui�����j � ui�����j �

ui�����j � � � �

��

�ui���j � ui�j�

where

ui�����j ��

yB� � yA�

Z B�

A�

udy � � ��

ui�����j ��

yB � yA

Z B

Audy

ui�����j ��

yB�� � yA��

Z B��

A��

udy

Note that u represents the cell face averaged value of u� It can be easilyshown that � � � is th order accurate on a Cartesian� uniform mesh bydeveloping u in a Taylor series around point O� However� if one uses thisscheme on general �i�e� non�Cartesian� non�uniform� meshes� the truncationerror will be of order zero� Although� on slightly distorted meshes the errors�associated with the fact that � � � does not account for irregularity of themesh� can be small� in general one may need to use a formula that takes

Page 36: Sergey Smirnov

� CHAPTER �� FINITE VOLUME FORMULATION

the metrics of the grid into account� In this thesis the following scheme isproposed

�ui�����j � ui�����j � �ui�����j � � � �

��X

n��

�Xm���

an�mui�n�j�m

Coe�cients an�m� �� � can be chosen so that � � � has the highest accuracypossible� To �t all the coe�cients of the Taylor series of both right�handand left�hand sides of � � � up to order one has to satisfy �� relations�the details of which are given in the Appendix B� However� there are only� coe�cients in � � �� which means that the linear system of equations tobe solved is overde�ned� Therefore� it is not possible to approximate

RBA udy

with th order accuracy on the stencil given by � � �� Third order accuracycan be obtained if the �rst � relations �B���B��� cf� Appendix B� are satis�ed�This leaves degrees of freedom in de�ning the � coe�cients� It was chosento use this freedom to minimize the leading truncation error of order � bymeans of the least square approach� This method allows to obtain a �rdorder accurate formula for the calculation of the line integrals� Note thaton Cartesian uniform meshes the leading truncation error of this schemevanishes� so that the scheme becomes th order accurate and corresponds tothe formula � � �� i�e� � � � � ��� a��� � a��� � ��� a��� � a��� � a���� �a���� � �� Numerical tests have shown however that on very distorted gridsthis third order accurate scheme shows a strong oscillatory behavior� makingthe calculations unstable�

To overcome this problem� a scheme with formal accuracy of one orderless was constructed� � and � were set to �

�� their values in the formulation

for a uniform Cartesian mesh� and the remaining � coe�cients an�m werechosen by imposing nd order accuracy of � � �� In this case only the �rst� constraints �B���B��� cf Appendix B� have to be satis�ed leaving � degreesof freedom� Again the least square approach was used to reduce the leadingtruncation error�

Although� the formal order of accuracy of the proposed scheme is only � this order is guaranteed on arbitrary grids� whereas the standard centralscheme may even have an order less than one on such grids� In addition� it isnot the formal order of accuracy which is important but rather the resolutionof the scheme� An insight into this can be� for instance given by the spectralanalysis of the scheme �cf� the spectral analysis in the previous sections��This can be done in �D case� where the present scheme reduces to standardth order compact and therefore behaves much better than the standard ndorder central scheme� especially for the higher wave number components of

Page 37: Sergey Smirnov

���� PAD�E TYPE FINITEVOLUME SCHEMES INMULTIDIMENSIONS��

the solution� This can also be analyzed by testing the scheme on modelproblems�

Once the line integrals of u are calculated over all interfaces� one cancalculate the line integrals of �uxes f and g needed in � � ��� For example

Z B

Afdy � a

Z B

Audy � � ��

On Cartesian non�uniform meshes the coe�cients a���� a���� a����� a����turn out to be zero� whereas �� �� a���� a��� are non�zero� The resultingscheme can also be used for �D calculations�

To make it easier to refer to dierent schemes� let us denote the proposedformulation of the compact scheme as CS� and the standard nd order central� nd order only on Cartesian uniform meshes� scheme as CE�

To test the present formulation of compact schemes� a series of numer�ical experiments has been carried out for a number of D linear advectionproblems�

ut � �au�x � �bu�y � � � � ��

In all simulations a six stage low storage RK scheme was used to advancein time� The RK coe�cients were taken as ���� ���� ��� ���� �� � �� Thisscheme is �th order accurate in time for linear problems� Since only thespatial accuracy was to be examined� the time step was taken very small tomake sure that the errors� occurring due to time integration� are negligible�as compared to those arising from spatial discretization�

In the �rst experiment a D signal was propagated in a domain ��� �x � ��� ��� � y � ��� A D Gaussian wave of the following shape

u��x� y� ��

exp

h������x� � y��

i� � ��

was used as an initial solution� and periodic boundary conditions were im�posed in both x and y directions� For each test case the wave was propagatedduring � seconds with a speed given by a � ��� b � ���

In Figures ��� � the results obtained on the Cartesian uniform mesh������ cells� are presented both for the classical second order central schemeand the present th order compact �nite�volume scheme� Results with thecompact scheme are in excellent agreement with the exact solution �circularisolines�� while the second order scheme shows important errors both in termsof location of the peak of the solution and symmetry�

To show the importance of accounting for the non�uniformity of the mesh�a �D test case �propagation of a Gaussian wave for � second� was calculated

Page 38: Sergey Smirnov

�� CHAPTER �� FINITE VOLUME FORMULATION

Scalar - contours :

x

y

-.60 -.20 .20 .60 -.60

-.20

.20

.60 1 -.0800 2 -.0400 3 .0000 4 .0400 5 .0800 6 .1200 7 .1600 8 .2000 9 .240010 .2800

Figure ��� Isolines of the solution � nd order scheme� Cartesian mesh��

Scalar - contours :

x

y

-.60 -.20 .20 .60 -.60

-.20

.20

.60 1 -.0400 2 .0000 3 .0400 4 .0800 5 .1200 6 .1600 7 .2000 8 .2400 9 .280010 .3200

Figure �� Isolines of the solution �th order compact scheme� Cartesianmesh��

on a non�uniform grid consisting of alternating bigger and smaller cells withthe size ratio ����� The periodic boundary conditions were imposed� In Fig

Page 39: Sergey Smirnov

���� PAD�E TYPE FINITEVOLUME SCHEMES INMULTIDIMENSIONS��

�� the results� obtained with a scheme with constant coe�cients that doesnot take into account non�uniformity of the grid� are compared with those ofthe compact scheme proposed in this thesis� It is seen that� in spite of therelatively small non�uniformity� the former scheme performs poorly�

X

U

-.50 -.17 .17 .50 -.050

.100

.250

.400 123

Figure ��� Solution of a �D convection equation� �� �Constant coe�cient�scheme� � �Non�uniform� scheme� �� exact solution�

In Figures ��� ��� �� the isolines of the solution on a curvilinear mesh��� � �� cells� �Fig ��� for the compact scheme with constant coe�cients�the scheme proposed in the this section and the explicit central scheme

ui�����j ��

ui���j � ui�j

� ��

ui���j � ui���j

� � ��

�which is th order accurate on a Cartesian uniform mesh� are shown� Theimprovement in accuracy with the present method� although less pronouncedthan on the clustered grid in the previous test case� is still visible from the iso�line plot� The explicit scheme provides the worst results� while the compactscheme that takes the irregularity of the mesh into account performs betterthan the compact scheme with constant coe�cients� An analysis of the or�der of accuracy and the absolute error on a sequence of meshes with dierentlevels of resolution is provided in Fig ���� The measured order of accuracyfor both schemes is � This can be explained by the smoothness of the grid�which� being re�ned� locally becomes straight and uniform� although non�orthogonal� Nevertheless� the absolute error is smaller �by approximately afactor of � for the present scheme�

Page 40: Sergey Smirnov

�� CHAPTER �� FINITE VOLUME FORMULATION

Scalar - contours :

x

y

-.200 -.100 .000 .100 .200 -.200

-.067

.067

.200

Figure ��� Zoom of the mesh used in calculations

Scalar - contours :

x

y

-.60 -.20 .20 .60 -.60

-.20

.20

.60 1 -.0400 2 .0000 3 .0400 4 .0800 5 .1200 6 .1600 7 .2000 8 .2400 9 .280010 .3200

Figure ��� Isolines of the solution �compact scheme with constant coe��cients��

��� Discretization of Viscous Fluxes

Now let us consider the linear D scalar advection diusion equation

ut � aux � buy � � ux�x � � uy�y � � ��

where is diusion coe�cient�

Page 41: Sergey Smirnov

���� DISCRETIZATION OF VISCOUS FLUXES ��

Scalar - contours :

x

y

-.60 -.20 .20 .60 -.60

-.20

.20

.60 1 -.0400 2 .0000 3 .0400 4 .0800 5 .1200 6 .1600 7 .2000 8 .2400 9 .280010 .3200

Figure ��� Isolines of the solution �compact scheme with non�constant coef��cients��

Scalar - contours :

x

y

-.60 -.20 .20 .60 -.60

-.20

.20

.60 1 -.0400 2 .0000 3 .0400 4 .0800 5 .1200 6 .1600 7 .2000 8 .2400 9 .280010 .3200

Figure ��� Isolines of the solution �explicit �th order� scheme with constantcoe�cients��

Consider the structured mesh of Figure � � Integrating equation � � ��over the cells and making use of the Gauss theorem� one can writeZ Z

Si�j

�u

�tdxdy �

IABB�A�A

��budx� audy� �IABB�A�A

�� uydx� uxdy�

� ����Approximation of the left hand side of � ���� has already been considered in

Page 42: Sergey Smirnov

� CHAPTER �� FINITE VOLUME FORMULATION

log(h)

log(error)

-2.40 -2.10 -1.80 -1.50 -6.00

-4.33

-2.67

-1.00 12

Figure ���� Order of accuracy on the curvilinear mesh ��� �constant coe��cient� scheme� � �non�uniform� scheme��

the previous section� The right hand side is an integral form of the viscouspart of the equation � � ��� Unlike its convective counterparts� it containsline integrals of gradients of solution multiplied by diusion coe�cient �This means that an extra procedure must be introduced to calculate thesegradients� The cell�averaged values of these gradients are �rst calculated bymeans of the �nite�volume approach� For example�

��u

�x

�i�j

� �

Si�j

ZSi�j

�u

�xdxdy �

Si�j

IABB�A�A

udy � ����

The line integrals in � ���� are approximated by using the same interpolationroutines as were developed for discretization of convective �uxes� Once thecell�averaged values of gradients are known� the interface�averaged values arecalculated by means of the aforementioned interpolation formulas� The accu�racy of the discretization of the viscous �uxes is determined by the accuracyof the interpolation formula� On a Cartesian uniform mesh the scheme willbe th order accurate� while on curvilinear grids the formal accuracy willreduce to third or second order accuracy� depending on the accuracy of theinterpolation scheme�

Page 43: Sergey Smirnov

���� BOUNDARY CONDITIONS �

��� Boundary Conditions

The application of boundary conditions for compact schemes in the �nitevolume context is relatively simple� The Dirichlet and periodic boundaryconditions are imposed in a straightforward way� For the Dirichlet bound�ary conditions the interface averaged values are imposed on the interfaceslying on the boundary of the computational domain� which closes a systemof linear equations � � � that relates the interface averaged values to cellaveraged values of the solution� In case of a periodic condition� with peri�odicity assumed along the grid lines� a simple use of the periodic Thomasalgorithm ���� solves the problem�

In case of the outlet boundary condition� some values of the solution mustbe extrapolated from the inside of the computational domain� This can beachieved by using forward or backward Pad�e approximations of the necessaryvariables� E�g�

u��� � �u��� � au� � bu� � �� �

where u��� is an interface value of the solution on the boundary� u� and u�are the cell averaged values of the solution in the �rst two cells� and u��� is avalue of the solution on the interface between them� The coe�cients �� a andb can be tuned to achieve the desirable order of accuracy in the same wayas was described in the previous sections� The formula � �� � can be made�rd order accurate on the Cartesian mesh� the coe�cients being dependenton the local properties of the mesh�

��� Non�linear Equations

Now consider the D Navier�Stokes system written in the form

�U

�t��Fx

�x��Fy

�y�

�Gx

�x��Gy

�y� ����

where U is the set of conservative variables and Fx� Fy and Gx� Gy arecomponents of the advective and diusive �ux vectors respectively�

U �

�BBB���u�v�E

�CCCA � ���

Fx �

�BBB��u�u� � p�uv��E � p�u

�CCCA Fy �

�BBB��v�uv�v� � p��E � p�v

�CCCA � ����

Page 44: Sergey Smirnov

CHAPTER �� FINITE VOLUME FORMULATION

with �� u� v� p� E respectively density� x and y components of velocity�pressure� total energy per unit mass�

As in the previous section� consider the structured mesh of Figure � �Integrating equation � ���� over the cells and making use of Gauss theorem�one can writeZ Z

Si�j

�U

�tdxdy �

IABB�A�A

��Fydx� Fxdy� �IABB�A�A

��Gydx�Gxdy�

� ����As was described above� before discretizing this equation� one has to choosewhether point�wise or cell�averaged ��i�j � �

Si�j

R RSi�j

��x� y�dxdy� values of

primitive variables

V �

�BBB��uvp

�CCCA � ����

will be used� In this thesis the cell�averaged approach is chosen� as it ispreferable for unsteady calculations� as mentioned above� Let us now considerthe discretization of the line integrals in � ����� As all the integrals areapproximated in the same manner� we will only consider

RBA Fxdy� Following

the strategy of constructing the Pad�e type schemes� the idea is to use animplicit formula for evaluating the line integrals over the cell faces� Firstassume that the grid is Cartesian and uniform� Once the cell�averaged valuesof primitive variables V are given in the cells� one can calculate the lineintegral

RBA V dy by means of the following implicit formula

V i�����j � V i�����j �

V i�����j �

�Vi���j � Vi�j� � ����

or for a scalar variable

�i�����j � �i�����j �

�i�����j �

��i���j � �i�j� � ����

where

V i�����j ��

yB� � yA�

Z B�

A�

V dy � ���

V i�����j ��

yB � yA

Z B

AV dy

V i�����j ��

yB�� � yA��

Z B��

A��

V dy

Page 45: Sergey Smirnov

���� NON�LINEAR EQUATIONS �

It can be easily shown that � ���� is th order accurate on Cartesian uniformmeshes� To account for irregularity of the mesh� a similar approach as forthe scalar equation is adopted� cf� eq� � � ��

�V i�����j � V i�����j � �V i�����j ��X

n��

�Xm���

an�mVi�n�j�m � ���

or for a scalar variable

��i�����j � �i�����j � ��i�����j ��X

n��

�Xm���

an�m�i�n�j�m � � �

Coe�cients of this interpolation formula are chosen as it was explained forthe scalar equation�

Once the interface�averaged values of the primitive variables have beencalculated� one can use the following procedure to evaluate the integrals ofthe �uxes which� in contrast to the scalar convective case� contains non�linearterms� Let us consider an approximation of the line integral

RBA u�x� y�v�x� y�dy�

The following formula may be used�

yB � yA

Z B

Auvdy � � ���

��

yB � yA

Z B

Audy

yB � yA

Z B

Avdy �O�h��

oruvi�����j � ui�����jvi�����j �O�h�� � ��

The leading truncation error of this approximation �which would be exact�if either u or v was a constant� is of order � In ����� where only calculationson Cartesian uniform meshes were considered� it was proposed to recoverth order accuracy by approximating the leading truncation error of order by the �nite�dierence formula of second order accuracy� This method canbe generalized for the use on arbitrary meshes� By developing u and v ina Taylor series around point O� �gure � � one gets the following formula�indices i� j are omitted here��

uv � uOvO � uO

�v��xx �x� �

v��xy� �x�y�

v��yy �y�

�� � ���

vO

�u��xx �x� �

u��xy� �x�y�

u��yy �y�

��

u�yv�

y

�y�

� � �u�xv

y � u�yv�

x��x�y

� �O�h��

Page 46: Sergey Smirnov

CHAPTER �� FINITE VOLUME FORMULATION

u � v � uOvO � uO

�v��xx �x� �

v��xy� �x�y�

v��yy �y�

�� � ���

vO

�u��xx �x� �

u��xy� �x�y �

u��yy �y�

�Comparing � ��� and � ���� one arrives at the following reconstruction for�mula�

uv � u � v � u�xv�

x

�x�

� � u�yv

y

�y�

� � � ���

��u�xv�

y � u�yv�

x��x�y

� �O�h��

Similarly� one can write�uv � � � u � v� � ���

����xu�

xvO � ��xuOv�

x � �Ou�

xv�

x��x�

� �

����yu�

yvO � ��yuOv�

y � �Ou�

yv�

y��y�

� �

�h���xuy � ��yux�vO � ��

xvy � ��yvx�uO�

��u�xvy � u�yvx��Oi�x�y�

�O�h��

Therefore� to calculate uv or �uv with th order accuracy one has to ap�proximate the derivatives ��x� u

x� v�

x� ��

y� u�

y� v�

y �in the point O� see Fig� � �and also �O� uO� vO in � ��� and � ��� with nd order formulas� This canbe accomplished by a classical �nite�volume �or �nite�dierence� method� Inthis thesis the cell averaged derivatives were calculated by means of a Gausstheorem� where the line integrals of the variables �� u� v are calculated usingformula � ���� The derivatives in point O as well as �O� uO� vO are thenobtained by means of the standard second order interpolation� as it is donein the classical �nite�volume method�

To demonstrate the importance of the correction of the �uxes on theinterfaces described above� the following test case is considered� A Gaussianwave rotating around the origin x � �� y � � is described by equation

ut � �au�x � �bu�y � � � ���

with a � � �y� b � �x �varying advection coe�cients�� This problem hasbeen solved with the classical Finite Volume method and the method pre�sented in this thesis� The grid used in all calculations was an !O� type grid�In circumferential direction periodicity was assumed� while at boundaries thesolution was put to zero� The isolines of the solution after one rotation of

Page 47: Sergey Smirnov

���� NON�LINEAR EQUATIONS �

Scalar - contours :

x

y

-3.0 -1.0 1.0 3.0 -3.0

-1.0

1.0

3.0 1 -.1200 2 -.0800 3 -.0400 4 .0000 5 .0400 6 .0800 7 .1200 8 .1600 9 .200010 .2400

Figure ���� Isolines of the solution �classical scheme��

Scalar - contours :

x

y

-3.0 -1.0 1.0 3.0 -3.0

-1.0

1.0

3.0 1 -.0400 2 .0000 3 .0400 4 .0800 5 .1200 6 .1600 7 .2000 8 .2400 9 .280010 .3200

Figure �� � Isolines of the solution �compact scheme��

the signal are presented in Figures ���� �� for both the classical centralscheme and the compact scheme derived in the previous section�

Note that in the CE scheme no arti�cial dissipation was used� The CEscheme is nd order accurate on uniform Cartesian grids� but on distortedmeshes the accuracy is less� The CS scheme is nd order accurate on arbitrarymeshes� but becomes th order accurate on a uniform Cartesian mesh� Hence�if the mesh is not too irregular the accuracy of the CS scheme will be closeto th order�

Page 48: Sergey Smirnov

� CHAPTER �� FINITE VOLUME FORMULATION

J

U

20. 40. 60. 80. -.100

.050

.200

.350 exact2nd ordercompact

Figure ���� The solution along the circular grid line passing through thecenter of the Gaussian wave�

The signi�cant improvement in accuracy for the compact scheme is clearlyseen� The isolines of the exact solution correspond to circular contours� whichis much better approximated by the compact approach� Note that both theCE scheme and the CS scheme are of the central type and as such non�monotone� This results in some wiggles in the solution� corresponding to theradial isolines in Figures ���� �� � Note� however� that the amplitude ofthese wiggles is much less for the compact scheme� as shown in Figure ����which plots the solution along a circumferential line through the center ofthe Gaussian wave� Note the eect of the dispersive error of the CE schemecausing the numerical solution to lag behind the physical solution�

Again radial isolines represent oscillations due to dispersive errors of thescheme� In case of the compact calculation� their amplitude is extremelysmall� After a su�ciently long time �about �� rotations for this particulartest case� these oscillations can build up and cause the simulation to diverge�

The achieved order of accuracy is studied numerically by calculations ona range of grids with decreasing mesh size� The results are shown in Figure ��� where the logarithm of the average error is plotted as a function of thelogarithm of the mesh size� From these results the actual order of accuracycan be calculated and corresponds to the inclination of the curves in Figure ��� The measured order of accuracy depends on the way the mesh is re�ned�The present mesh locally becomes uniform and Cartesian when re�ned� andtherefore the measured order of accuracy is the same as the order obtainedon uniform Cartesian meshes� Since the advection coe�cients in � ��� are

Page 49: Sergey Smirnov

���� NON�LINEAR EQUATIONS �

not constant� the leading truncation error of formulas such as

yB � yA

Z B

Aaudy � � ����

��

yB � yA

Z B

Aady

yB � yA

Z B

Audy �O�h��

must be approximated with second order of accuracy to maintain the highorder of the resulting scheme� as it was described above� In Figure �� theresults for the classical Finite Volume method as well as the compact schemewith and without correction of the �uxes on the interfaces are presented� Itcan be seen that the compact scheme without the aforementioned correctionfails to maintain the th order of accuracy�

log10(h)

log10(Error)

-2.00 -1.60 -1.20 -.80 -6.0

-4.0

-2.0

.0 classicalcompactcompact w/o correction

Figure ��� The absolute error of the numerical solution obtained on mesheswith dierent levels of resolution�

Page 50: Sergey Smirnov

� CHAPTER �� FINITE VOLUME FORMULATION

Page 51: Sergey Smirnov

Chapter �

DG�like Finite Volume Method

As one can see the higher order of accuracy of Pad�e schemes is achieved bysolving extra equations for derivatives or �uxes� This equations are indepen�dent of the dierential equation being solved� However� one could considerintroducing extra equations for derivatives of the solution that can be usedto increase the order of the method from the dierential equation itself�A Discontinuous�Galerkin�like �nite volume method is an example of thisapproach� In this section this methodology is described for a linear advec�tion equation and compared with the �nite volume formulation of compactschemes� The extension of this approach to irregular meshes and non�linearequations is possible� but it is beyond the scope of this thesis�

Discontinuous Galerkin �DGM� method is a variational approach to solv�ing partial dierential equations� It diers from the classical Finite Element�FEM� method in its representation of solution� While in FEM the solutionis considered continuous in the whole computational domain� in DGM solu�tion is allowed to have discontinuities on the element interfaces� The solutionin any element is a linear combination of basis functions� which are normallya set of orthogonal polynomials� By multiplying the PDE by basis functions�integrating the products over each element and making use of the Gausstheorem� one obtains a system of algebraic equations for the coe�cients ofa linear combination of basis functions that represents the solution in eachelement�

In this section a �nite volume formulation that employs a similar idea forobtaining extra transport equations for the spatial derivatives of the solutionis considered� Again the maximum order of the formulation is limited to �although it is possible to achieve a higher order of accuracy by introducingadditional equations�

Page 52: Sergey Smirnov

�� CHAPTER �� DG�LIKE FINITE VOLUME METHOD

��� �D case

Consider a �D linear convection equation

�u

�t� c

�u

�x� � �����

Assuming a uniformmesh and integrating ����� over the cells� one obtains�

�ui�t�x� c�ui���� � ui����� � � ��� �

where ui ���x

R xi����xi����

udx and ui���� and ui���� are the values of the solution

on the interfaces� ��� � is an exact relation� Once ui���� and ui���� areapproximated� one gets a spatial discretization of ������ The classical centralapproximation is

ui���� �ui�� � ui

�����

This is� however� only a second order approximation� If one needs a higherorder scheme� a bigger stencil can be used

ui���� � aui�� � ui

� bui�� � ui��

����

An alternative is a compact scheme considered in the previous sections�

�ui���� � ui���� � �ui���� �a

�ui�� � ui� �����

If the values of space derivatives were known� one could use the followingapproximation of interface values of the solution�

ui���� �ui�� � ui

� b�x�u�i�� � u�i� �����

where u�i are the pointwise values of�u�xin the centers of the cells and ui are

cell averaged values of the solution� By developing all terms of ����� in aTaylor series around point x � xi���� and requiring all the coe�cients of theresulting expansion to vanish up to term O��x��� one obtains the value ofb � ��� that makes ����� th order accurate�

These derivatives can be made available� if an extra equation for u�i is

solved� This equation can be obtained by multiplying ����� by �x�xi��x

� where

Page 53: Sergey Smirnov

���� �D CASE ��

xi is the coordinate of the center of the cell i� and integrating the productover the cell�

�t

Z xi����

xi����

�x� xi�

�xudx� c

Z xi����

xi����

�x� xi�

�x

�u

�xdx � � �����

The solution in the cell i can be written in the following form�

u�x� t� � u�� � u�i�x� xi� �u��i �x� xi�

� �HOT �����

where u�� is the value of u in the center of the cell� Using ����� one can showthat

�t

Z xi����

xi����

�x� xi�

�xudx� c

Z xi����

xi����

�x� xi�

�x

�u

�xdx � �����

�u��i

�t

�x�

�� c�ui���� � ui���� � ui� �O��x��

where u�i is a value of space derivative in the center of the cell� ui���� andui���� are the values of the solution on the interfaces� and ui is a cell averagedvalue of u� Thus one obtains the following equation for u�i�

�u�i�t

�x�

�� c�ui���� � ui���� � ui� � � ������

Equations ��� � and ������ along with

ui���� �ui�� � ui

� ���x�u�i�� � u�i� ������

give an extension of FVM analogous to DGM�

To test the performance of the proposed formulation �D advection equa�tion ����� is solved on the uniform mesh� An initial solution is a Gaussiansignal� which is propagated with a constant velocity c � �� In �gure ���the results obtained with the proposed method are compared to those ob�tained with the classical nd order and th order Pad�e schemes� The resultsobtained with DG method are in the best agreement with the exact solution�

Note that� unlike in the classical DG method in which the �uxes arecalculated by an upwind method� here the �uxes are approximated by acentral interpolation formula ������� As a result� the proposed method hasa higher order of accuracy� i�e� instead of � would be the order of theclassical DGM for the same number of polynomials��

Page 54: Sergey Smirnov

� CHAPTER �� DG�LIKE FINITE VOLUME METHOD

1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6

−0.2

0

0.2

0.4

0.6

0.8

1 Exact solution2nd order schemeDG methodcompact scheme

Figure ���� Solution of �D linear advection equation

1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7

0

0.2

0.4

0.6

0.8

1

Exact solution1st order upwind schemeDGM 4th orderDGM 2nd order

Figure �� � Solution of �D linear advection equation

In Figure �� the solution of the proposed th order method is comparedwith the classical nd order DGM and the �rst order upwind scheme� Forthis simple equation ����� in the classical DGM the following �ux is used

ui���� � ui � u�i�x

���� �

As is clearly seen from the picture the nd order DG is more dissipative thanthe th order one� although it is much more accurate than the �rst orderupwind scheme�

Page 55: Sergey Smirnov

���� MULTIDIMENSIONAL FORMULATION ��

��� Multidimensional formulation

Consider a D linear convection equation

�u

�t� a

�u

�x� b

�u

�y� �s ������

Assuming a uniform Cartesian mesh and integrating ������ over the cells�one obtains�

�ui�j�t�x�y� a�ui�����j � ui�����j��y � b�ui�j���� � ui�j������x � � �����

where ui�j ��

�x�y

R yj����yj����

R xi����xi����

udxdy and ui�����j and ui�j���� are the inter�

face averaged values of the solution� ����� is an exact relation� Once ui�����jand ui�j���� are approximated� one gets a spatial discretization of �������

To calculate these �uxes a formula similar to ����� can be used� E�g�

ui�����j �ui���j � ui�j

� � � vi�����j�� �

��

� vi�����j �

vi�����j����� ������

where vi�����j � �x�uxi���j � uxi�j�� This formula is dierent from ����� be�cause uxi�j represent a pointwise value� while the formula is used to calculateinterface averaged value� which in the D case in a line integral divided bythe length of the interface� The last � terms represent the th order approx�imation of the line integral by pointwise values �see Appendix A��

To obtained equations for uxi�j and uyi�j the same procedure as in �D casecan be used� By using a D Taylor expansion of the solution� one obtainsthe following relations

�uxi�j�t

�a

��x��ui�����j�ui�����j� ui�j�� b

�y�uxi�j�����uxi�j����� � � ������

�uyi�j�t

�b

��y��ui�j�����ui�j����� ui�j�� a

�x�uyi�����j�uyi�����j � � � ������

In Figures ���� ��� ��� the results for a D convection equation ������with a � � b � � in a domain � � x � �� � � y � � with periodic boundaryconditions are shown� A D Gaussian signal is propagated for � second �thetime needed for the signal to return to its original position�� The isolines ofthe solution and �D plot along the horizontal line crossing the center of thedomain are shown� In Figure ��� the numerical solutions are compared tothe exact one� It is clearly seen that the present formulation of DGM gives asigni�cant improvement in the accuracy compared to the nd order scheme�

Page 56: Sergey Smirnov

� CHAPTER �� DG�LIKE FINITE VOLUME METHOD

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure ���� Solution of D linear advection equation� nd order scheme

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure ��� Solution of D linear advection equation� DGM

Page 57: Sergey Smirnov

���� MULTIDIMENSIONAL FORMULATION ��

0.2 0.3 0.4 0.5 0.6 0.7

−0.2

0

0.2

0.4

0.6

0.8

1

x

u

exact2nd orderDG

Figure ���� Solution of D linear advection equation� DGM and nd orderscheme� �D plot

Page 58: Sergey Smirnov

�� CHAPTER �� DG�LIKE FINITE VOLUME METHOD

Page 59: Sergey Smirnov

Chapter �

Solution of Unsteady

Incompressible Flows

��� Simulation of turbulent ows

There exist two contrasting methods for the simulation of turbulent �ows�One of themmakes use of so called Reynolds Averaged Navier�Stokes �RANS�equations� Historically it was the �rst method that was used to study turbu�lent �ows� To obtain this set of equations an ensemble averaging satisfyingso called Reynolds conditions is applied to the Navier�Stokes equations� Onethus gets equations describing the behavior of the mean velocity �eld insteadof the actual �ow �eld containing a very wide range of time and space scales�The advantage of this method consists in its relatively low computationalcosts� The averaged quantities of the �ow contain a much smaller range ofscales� which allows the use of relatively coarse grids� Besides� if the �ow isstatistically steady� the RANS equations also become steady even though aturbulent �ow is an inherently unsteady phenomenon� The disadvantage ofthe method is that RANS equations contain more unknowns than there areequations� In addition to the �rst order statistical properties� i�e� mean val�ues� they contain second order statistics � so called Reynolds stresses� whichhave to be modeled� Although a big progress has been made in RANS tur�bulence modeling since the RANS equations had been introduced� RANSapproach still fails to give satisfactory results for the more complex �ows�especially those containing detachment and recirculation zones�

An alternative is to solve unsteady �D Navier�Stokes equation on a meshthat can properly capture all the physical scales of the turbulent �ow� Thisis the most accurate and direct method to study turbulence� hence the nameDirect Numerical Simulation �DNS�� The obvious disadvantage of this ap�

��

Page 60: Sergey Smirnov

��CHAPTER �� SOLUTION OFUNSTEADY INCOMPRESSIBLE FLOWS

proach is that it is extremely expensive in terms of both data storage andCPU� So far only the low Reynolds number �ows have been studied by meansof DNS� and it is safe to say that the scope of its applicability is going to belimited to academic problems for many years to come�

The compromise is the so called Large Eddy Simulation �LES�� In thisapproach the larger scale turbulence is resolved directly by solving unsteady�D �ltered Navier�Stokes equations� The in�uence of so called subgrid scales�SGS� of the turbulent �ow �i�e� those scales of the �ow that cannot beresolved directly due the grid resolution� on the resolved �ow �eld is modeled�Because of the fact that small scale turbulence is less dependent on geometryof the �ow� this modeling is simpler and more accurate than that in RANSapproach� In LES much coarser meshes can be used compared to DNS� whichmakes it possible to apply to �ows that cannot yet be solved by means ofDNS� On the other hand� LES provides more satisfactory and detailed resultsthan RANS approach�

The governing equations for the resolved �elds of LES solution are ob�tained after �ltering� The �ltering operation can be written in terms of aconvolution integral�

f �x� �ZDG�x � x��f�x��dx� ����

Where f is a turbulent �eld� G is some spatial �lter and D is the �ow domain�Note that the minimum width of the spatial �lter could be equal to the gridspacing� The important quality that a �lter must have in order to obtainLES equations is that a �lter and a spatial derivative can commute� i�e�

�f

�x�

�f

�x�� �

This condition is generally speaking not satis�ed� However� the top hat �lter�i�e� if G is unity divided by the volume of the cell over which the variable is�ltered and zero elsewhere� does satisfy this condition� and so does the sharpcut�o �lter performed in the Fourier space�

The �ltered Navier�Stokes equations for incompressible �ows are�

�ui�xi

� � ����

��ui�t� �

�uiuj�xj

� � �p

�xi��i�j�xj

���

The momentum equation can be further worked out as�

��ui�t� �

�uiuj�xj

� � �p

�xi��"ij�xj

���ij � "ij�

�xj� �z A

� ��ij�xj� �z B

Page 61: Sergey Smirnov

���� SIMULATION OF TURBULENT FLOWS ��

Two unclosed terms appear� term A is due to the nonlinearity of the viscousstresses� and term B is the divergence of the SGS stresses

�ij � ��uiuj � uiuj� ����

ij and "ij are de�ned as�

ij � ���ui�xj

��uj�xi�

"ij � ���ui�xj

��uj�xi�

If the viscosity is constant� the term A equals zero� otherwise it is neglected�The term B is modeled as follows

�ij � ���kk�ij � � �tSij ����

Where Sij ����ui�j � uj�i� is the magnitude of large�scale stress�rate tensor�

the eddy viscosity is��t � #�C�

�jSj ����

with � � ��x�y�z���� the �lter width and jSj � � SijSij����� The traceof the SGS stresses� �ii� can be either modeled� Moin et al� ����� or simplyneglected� Erlebacher et al� � ��� In present thesis it is neglected�

Equation ���� is an eddy viscosity model by Smagorinsky ���� TheSmagorinsky model can be derived in a number of ways including a sortof mixing�length assumption in which the eddy viscosity is assumed to beproportional to the SGS characteristic length scale �also called �lter width��� and to a characteristic turbulent velocity v� � �j "Sj� Where j "Sj �� "Sij "Sij���� is a typical velocity gradient at �� determined with the aid ofthe large scale ��ltered��eld� deformation tensor "Sij� In the context of �nitevolume no explicit �ltering is used� and the �lter is implicit in the numericalgrid which does not support scales smaller than the grid size� The �lterwidth used in the Smagorinsky model is taken as the third root of the controlvolume� The eddy viscosity model can �nally be written as�

t � C�d���jSj ����

where d �� � exp

�� y�

�� ���� The coe�cient C is a constant which

should be combined with a near wall damping d to vanish near the solidboundaries� The near wall damping is a function of the wall coordinatey� � u� yw

�� u� � ��w������ is the friction velocity� �w is the wall shear stress�

� is the density of the �uid and yw is the distance from closest the wall�

Page 62: Sergey Smirnov

��CHAPTER �� SOLUTION OFUNSTEADY INCOMPRESSIBLE FLOWS

In the present thesis the �lter width � � ��x�y�z����� with �x� �yand �z the dimensions of computational cell� However� generally speaking�the width of the �lter � need not have anything to do with the grid size hother than the obvious condition � � h�

A variety of other SGS models have been used by dierent researchers�such as two�point closures� scale�similar models� and one equation models� toname a few among others� Kraichnan ����� by using a two�point closure modelfor isotropic turbulence� computed the energy transfer from the resolved tothe unresolved scales� He then de�ned an eddy�viscosity in wave space� whichhampers its extension to �nite�dierence schemes and to complex geometries�To overcome this shortcoming� based on the theory of isotropic turbulence�M�etais and Lesieur ���� derived the structure function model which is verysimilar to the Smagorinsky model� see also Ducros et al� � ��� Based on theassumption that the most active subgrid scales are those closer to the cut�o� and that the scales with which they interact most are those right abovethe cuto� Bardina et al� ��� developed the scale�similar models� Anotherapproach is the one�equation models� in which in a similar way to RANS� atransport equation is solved for the subgrid�scale energy to obtain the veloc�ity scale� One�equation models have been used by Schumann ����� Horiuti��� and Carati et al� ���� among others� However� as stated by Piomelli����� based on the results obtained using one�equation models� the expenseinvolved in solving an additional equation does not seem to be justi�ed byimprovements in the accuracy� There are more methods for the subgrid�scalemodeling in the literature� The interested reader can refer to the book ofSagaut �� � for a complete review�

��� Time Integration

The study of accuracy of time integration is beyond the scope of this thesis�Here a brief description of time integration methods is given to complete thepicture� In all the simulations presented here the time step was su�cientlysmall to ensure that the errors due to time integration are much smaller thanthose arising from spatial discretization� as the latter are the subject of thiswork�

A multistage low storage Runge�Kutta �RK� scheme was used to advancein time�

U� � U� ��tC�Res�U�� ����

U� � U� ��tC�Res�U�� �����

�����

Page 63: Sergey Smirnov

���� TIME INTEGRATION ��

UN � U� ��tCNRes�UN��� ��� �

where Res�U� is the residual� For linear problems with constant propagationcoe�cients an N�th stage RK scheme is N�th order accurate provided the RKcoe�cients are chosen as follows

Ck ��

N � k � ������

where k is the RK stage number� For non�linear problems� however� theformal accuracy of the scheme is only of order �

The linear stability function for an s�stage explicit Runge�Kutta methodis given by�

P �z� � � � ��z � ��z� � ��z

� � � � �� �szs ����

�i coe�cients depend on the values of Ck� For a value of z such that jP �z�j �� the method is stable�

To provide an example� suppose the three�stage Runge�Kutta scheme isused to discretize the following model problem�

dy

dt� �y y��� � y� �����

here � is a complex constant� The ampli�cation yn��

ynis equal to�

yn��

yn� p�z� � � � ��z � ��z

� � ��z�

with �� � C�� �� � C�C�� �� � C�C�C�� and z � ��t�

For more examples about model problems� see �����

The time integration of incompressible �ows diers from that in com�pressible simulations due to the fact that there is no time derivative in thecontinuity equation� Since only velocity can be updated from the momen�tum equations� there must be an extra procedure to update pressure� Oneof the solutions to this problem is so called arti�cial compressibility method�introduced independently and under slightly dierent forms by Vladimirovaet al� and Chorin ����� The method consists in adding a time derivative ofpressure to the continuity equation

��

�p

�t��ui�xi

� � �����

where � is an arbitrary constant� This equation� strictly speaking� has nophysical meaning before the steady state solution is reached� As at the steady

Page 64: Sergey Smirnov

� CHAPTER �� SOLUTION OFUNSTEADY INCOMPRESSIBLE FLOWS

state all time derivatives vanish� the continuity equation is satis�ed� but atthe convergence only� Therefore this method is primarily used for steady�ows�

If ��� �� �� ����� is only a little dierent from the real continuityequation� therefore it is possible to consider unsteady solution of arti�cialcompressibility method as an approximation of an unsteady incompressible�ow� if ��� �� � ����� The use of large values of � leads however to a verystrict stability criterion� The time step that has to be chosen must be muchsmaller than what is needed for an accurate calculation� the problem similarto what is experienced in the solution of compressible low Mach number �ows�Therefore� for unsteady simulations the arti�cial compressibility method isnot an e�cient tool� In this thesis� however� it has been used to solve allsteady �ows� such as boundary layer on the �at plate and a lid�driven cavity�ow�

��� Incompressible Formulation

The method used to solve the unsteady incompressible Navier�Stokes equa�tions in this thesis is a subset of the pressure correction method originallyapplied by Harlow and Welch ���� for the computation of free surface incom�pressible �ows� It is called fractional step or projection method� developedindependently by Chorin ���� and Temam ������ ������

The fractional step method is slightly dierent from the original pres�sure correction method in a way that the pressure term is removed in theintermediate step� In this section the fractional step method is explained�

The governing equations for an incompressible �ow are�

�ui�xi

� � �����

�ui�t��uiuj�xj

� ���

�p

�xi�

��ui�xj�xj

�����

where xi�s are the Cartesian coordinates� and ui�s are the correspondingvelocity components� The integration method is an explicit four stage Runge�Kutta scheme�

aij �

������ � � ��� � � �� ��� � �� � �� �

����� bj �h� � � �

i

Page 65: Sergey Smirnov

���� INCOMPRESSIBLE FORMULATION ��

The fractional step method� or time�split method� is used to solve the systemof equations ������ To explain the fractional step method we assume� forthe sake of clarity� that the time derivative is discretized with an explicitforward Euler method� The extension to a multi�stage Runge�Kutta methodis straightforward as the same procedure is followed at every stage�

The discretized form of equation ����� may be written as�

un��i � uni�t

��uni u

nj

�xj� ��

�p

�xi�

��uni�xj�xj

�����

In an incompressible �ow the pressure cannot be updated from either themomentum� nor the continuity equation� However� with the following methoda pressure �eld is calculated which satis�es the continuity equation�

First� by dropping the gradient of pressure from equation ����� an inter�mediate velocity u�i is obtained�

u�i � uni�t

��uni u

nj

�xj�

��uni�xj�xj

�� ��

The velocity at the next time level un��i is equal to u�i plus a correctionvelocity ucori �

un��i � u�i � ucori �� ��

The velocity �eld at time level n� � must be divergence free� i�e��

�un��i

�xi� �� �u�i

�xi� ��u

cori

�xi�� �

By inserting the value of un��i from �� �� into equation ������

�u�i � ucori �� uni�t

��uni u

nj

�xj� ��

�p

�xi�

��uni�xj�xj

�� ��

after subtracting equation �� �� from equation �� ���

ucori

�t� ��

�p

�xi�� �

By applying the divergence operator to both sides of equation �� ��

�t

�ucori

�xi� ��

��p

�xi�xi�� ��

from �� � the value of�ucori

�xiis known and is inserted into �� ���

�t

�u�i�xi

��

��p

�xi�xi�� ��

Page 66: Sergey Smirnov

�CHAPTER �� SOLUTION OFUNSTEADY INCOMPRESSIBLE FLOWS

Once the pressure �eld� which satis�es the continuity equation� is calculatedfrom the Poisson equation �� ��� the velocity at time level n � � can beobtained from equation �� �� and �� ��

un��i � u�i ��t

�p

�xi�� ��

����� Iterative Poisson solver

Eective solution of the Poisson equation for pressure is one of the most im�portant aspects of incompressible �ow simulation by means of the projectionmethod� The most straightforward approach is the use of a direct solver� Ifthe �ow is periodic in one or more directions with mesh being uniform inthose directions� one can bene�t from this by solving the Poisson equationpartly or completely in the Fourier space� The use of Fast Fourier Transformof the right hand side of the Poisson equation makes this procedure very ef�fective� The obvious advantage of this method is that the discretized Poissonequation is solved exactly in the most direct way�

In general� however� if the �ow does not have periodic features� the directsolver becomes extremely expensive� both in terms of CPU and data storage�In this case an iterative method is a more e�cient solution�

In this section one of the simplest iterative methods that can be appliedto solve the Poisson equation for pressure when the compact schemes areused is considered�

The Poisson equation for pressure is obtained by combining the continuityand momentum equations� In the fractional step method described in theprevious section one has

un��i � u�i�t

� ���

�p

�xi�� ��

�un��i

�xi� � �� ��

By considering the divergence of �� �� one obtains the Poisson equa�tion with the right hand side proportional to the divergence of intermediatevelocity u�� This requires an application of the boundary condition to theintermediate velocity �eld� One� however� does not have to write out thePoisson equation explicitly� but can instead solve the system of equations�� ��� �� �� by introducing a pseudo time derivative in �� ���

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���� INCOMPRESSIBLE FORMULATION ��

�p

����un��i

�xi� � �����

The system �� ��� ����� allows the iterative solution of �� ��� �� ���Once an approximation of un�� and p are known� a new approximation ofpressure is obtained from ������ By using this new pressure in the right handside of �� ��� one obtains a new approximation of un��� The procedure isrepeated until �� �� is satis�ed with a prescribed criterion of accuracy�

This method does not require imposing the boundary conditions for in�termediate velocity �eld as the divergence of u� need not be calculated� Ifthe pseudo time derivative of pressure in ����� is approximated as follows

�p

���pm�� � pm

�������

where m is an iteration index� the method is explicit� which limits the maxi�mum pseudo time that can be taken by a strict stability criterion� With themaximum time step allowed by this criterion the method is identical to theJacobi method� which is know for its low e�ciency� To allow for bigger timesteps and� thus� increase the e�ciency of this approach� one can make themethod implicit by introducing an extra term in ������

�pm��

��� ���pm��

�xi�xi��un��i

�xi� � ��� �

�pm�� is pm�� � pm and the second term is the operator representing theleft hand side of the approximated �by classical low order scheme� Poissonequation applied to �pm��� By combining equations �� �� and ������ oneobtains

�p

��� ��

��p

�xi�xi� ��u

i

�xi�����

Consider an implicit time integration �in pseude time��

pm�� � pm

��� ��

��pm��

�xi�xi� ��u

i

�xi����

or�pm��

��� ���pm��

�xi�xi� ��

��pm

�xi�xi� ��u

i

�xi� � �����

Using �� �� and ������ one obtains ��� �� wich contains an extra Laplaceoperator applied to �pm��� In a general case this operator is � dimensional�which makes an exact solution of ��� � computationally very expensive�Instead of solving ��� � for �pm�� exactly� one can split the operator in� one dimensional operators� which can be successively solved by Thomasalgorithm� This approach make it possible to take a much bigger pseudotime step and as result to achieve a faster convergence�

Page 68: Sergey Smirnov

��CHAPTER �� SOLUTION OFUNSTEADY INCOMPRESSIBLE FLOWS

����� The Multigrid Method

The most e�cient iterative method to solve discretized steady equationsknown today is a multigrid approach� It was �rst introduced and applied byFedorenko � �� to solution of elliptic equations� such as the Poisson equation�and further developed by Brandt ����� There is by now a fairly well�developedtheory of multigrid methods for elliptic equations ����� ����� The methodcan also be extended to hyperbolic systems such Euler equations ���� � ������ ����� Some multigrid methods ������ ����� ����� among others� have alsobeen devised for the numerical solution of the compressible Navier�Stokesequations�

The idea of the multigrid approach emerges from the fact that most it�erative methods eliminate the high frequency errors very fast� but fail toeliminate the low frequency errors at the same rate� In addition� the notionof high and low frequency error is related to the coarseness of the grid� in thatany low frequency error on a certain grid will become a high frequency erroron another su�ciently coarser grid� Therefore� the idea is to solve the equa�tions on a sequence of grids ranging from �ne to coarse� so that the completespectrum of errors can be eliminated with a rate proper to the eliminationof the high frequency errors�

Consider the following linear problem �which can be a discretized Poissonequation��

Au � f �����

Suppose v is an approximation to the solution of ������ If we de�ne theresidual as r � f �Av and the error as �u � u�v� then the error will satisfythe residual equation A�u � r�

After applying a few relaxation sweeps of an iterative method� eective inremoving high frequency components of the error� to ������ an approxima�tion to the solution of ����� v and the error �u will be su�ciently smooth�On the coarser grid� however� the error �u will appear more oscillatory and�therefore� relaxation on this coarser grid will be more eective� To acceleratethe convergence we can go to a coarser grid and relax the residual equationA�u � r� with an initial guess �u � �� Once the residual equation is con�verged� �u can be interpolated on the �ne grid �this is called prolongation�and added to v to obtain a new approximation to the solution of ������ Afterthis the whole procedure can be repeated�

This two grid algorithm is the basic building block of any multigridmethod� Before the general multigrid algorithm is described� let us �rstgive a more detailed account of how the two grid method works� The multi�grid procedure in the cell centered �nite volume context is slightly dierent

Page 69: Sergey Smirnov

���� INCOMPRESSIBLE FORMULATION ��

Coarse Grid

b a b

1 1 2 2

C

aFine Grid

Figure ��� One dimensional restriction operator�

from that in the �nite dierence approach� The coarser grid in the �nite dif�ference method is formed by removing every second point of the �ne mesh inevery grid direction� Since the discrete solution is considered in grid points�the points in which solution is de�ned on the coarser mesh will coincide withthe points in which solution is de�ned on the �ne mesh� In the �nite volumecontext� though� the solution is considered in the centers of the cells� Thecells of the coarser grid are former by merging �in �D�� �in D� or ��in �D� neighboring cells of the �ne mesh �all of them having one commonvertex� together� Therefore� the discrete solution on the coarse grid is notde�ned in the same points as the solution on the �ne grid� This aects therestriction and prolongation operations described below�

Consider a discretization of a linear PDE Au � f on mesh $h�

Ahuh � fh �����

After applying one or more relaxation sweeps of the iterative method to������ a relatively smooth approximation vh to the solution is obtained� Toexecute a coarse grid correction to this approximation� �rst the residual rhis calculated�

rh � fh �Ahvh �����

Thus one obtains the residual equation for the error�

Ah�uh � rh �����

Instead of solving this equation on the �ne grid $h� one solves the followingequation on the coarser grid $�h�

A�h�u�h � I�hh rh ����

A�h is an operator A discretized on the coarse grid $�h and I�hh is a socalled restriction operator that brings �ne grid information about rh onto thecoarse grid� In the �nite volume approach� the restriction operator consistsin summing the residuals rh of the �ne mesh cells forming a coarse mesh cellmultiplied by the volumes �areas or lengths in D and �D respectively� of

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��CHAPTER �� SOLUTION OFUNSTEADY INCOMPRESSIBLE FLOWS

i−1

U Ua b

Ufd

d d

i

Figure � � One dimensional prolongation operator�

these cells and dividing the sum by the volume of the coarse grid cell� E�g�in �D cases on a uniform grid ��g� ����

�I�hh rh�i � ��rh��i � �rh��i���h

h����

where h is the length of �ne mesh cells� Here the information in �ne meshcells a is used to calculate the right hand side of the equation on the coarsegrid in the cell C� It is also possible to use a bigger stencil on the �ne grid�involving also cells b� but this is beyond the scope of this work�

When the solution of ���� is obtained� �u�h is interpolated on the �nemesh and added to vh�

vh � Ih�h�u�h � vh �� �

Ih�h is called a prolongation operator� The simplest prolongation is the piece�wise constant prolongation which is a �rst order operator� The correctionin the �ne mesh cell is simply the correction in the course mesh cell part ofwhich the �ne mesh cell is�

To improve the accuracy of the prolongation one can apply a more ac�curate second order prolongation that involves a bigger stencil � E�g� in �Dcase on a uniform mesh ��g� � ��

�Ih�h�u�h��i � �����u�h�i � � ���u�h�i�� ����

�Ih�h�u�h��i�� � � ���u�h�i � �����u�h�i�� ���

If the mesh used is not uniform� in order to account for the mesh non�uniformity a volume�weighted average can be used� Based upon the expe�rience from a large number of numerical simulations� Zhu ���� and Eliasson� ��� a constant linear prolongation works well for both uniform and non�uniform meshes� The weighted linear prolongation does not give signi�cantimprovement in convergence� and it is rather complicated and costly�

In the two grid approach� the coarse grid can still have a large amount ofpoints� which can make a solution of ���� very expensive� In the multigridapproach the two grid procedure is extrapolated on still coarse grids� E�g�

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���� INCOMPRESSIBLE FORMULATION ��

in a three grid method� instead of solving equation ���� exactly� a two gridmethod is used� involving a still coarser grid� Basically the exact solution ofthe residual equation is only required on the coarsest grid� which normallyinvolves only a small amount of computational eort� as this mesh containsa small number of points� A typical multigrid scheme is described below�

� Perform one or more relaxation sweeps on the �ne mesh� calculateresiduals and restrict them to a coarser grid�

� Perform one or more relaxation sweeps on the next mesh for an equationthat has a restricted residual from the previous mesh as a right handside� Calculate residuals of this equation and restrict them on yet acoarser grid�

� Repeat this procedure till the coarsest mesh is reached� Solve theequation on this mesh exactly�

� Interpolate the correction to the previous mesh �prolongation�� add itto the correction on that mesh and perform one or more relaxationsweeps�

� Repeat this procedure till the �nest mesh is reached�

Described above is a so called V�cycle multigrid strategy� More sophisti�cated strategies� which have not been investigated in the present thesis� canalso be employed�

For a successful multigrid calculation� one needs an iterative method �asmoother� that is e�cient in removing high frequency components of theerror� The simple point Jacobi method is not very good at it� The alter�nating line Gauss�Seidel method is a much better smoother� which has beenemployed in the present thesis�

Consider a discretization of D Poisson equation on a uniform mesh �thetreatment of non�uniform and �D meshes is similar��

ui���j � ui�j � ui���j�x�

�ui�j�� � ui�j � ui�j��

�y�� qi�j ����

The point Jacobi method is the following�

uni���j � un��i�j � uni���j�x�

�uni�j�� � un��i�j � uni�j��

�y�� qi�j ����

If an approximation un is known� the next approximation un�� can be directlycalculated from ����� This is typically done in a D loop over indices i and

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��CHAPTER �� SOLUTION OFUNSTEADY INCOMPRESSIBLE FLOWS

j� In this loop the solution in points �i� j � �� and �i � �� j� is alreadyupdated to level n � �� when un��i�j is calculated� One therefore can use thisthese updated values instead of uni���j and uni�j��� which gives the classicalGauss�Seidel �GS� method�

uni���j � un��i�j � un��i���j

�x��uni�j�� � un��i�j � un��i�j��

�y�� qi�j ����

This method has better convergence properties than the point Jacobi method�It can� however� still be improved by adding one extra point at which analready updated solution is used�

un��i���j � un��i�j � un��i���j

�x��uni�j�� � un��i�j � un��i�j��

�y�� qi�j ����

Here the values of the solution lying on the grid lines j � const must beupdated by solving a system of linear equations with a tridiagonal matrix�This is done by means of the Thomas algorithm or LU decomposition� Thisapproach is called the line GS method� If an updated value of solution un��i�j��

is used instead of un��i���j� one gets the line GS scheme for another direction inwhich the Thomas algorithm must be applied�

uni���j � un��i�j � un��i���j

�x��un��i�j�� � un��i�j � un��i�j��

�y�� qi�j ����

By alternating ���� and ���� one gets a method with good smoothingproperties of high frequency components of the error�

5 10 15 20 25

−14

−12

−10

−8

−6

−4

−2

0

iteration

lg(R

es)

one grid GS methodMG GS method

Figure ��� Convergence rate� Gauss�Seidel method on � and grids �uniformmesh�

Page 73: Sergey Smirnov

���� INCOMPRESSIBLE FORMULATION ��

5 10 15 20 25 30

−14

−12

−10

−8

−6

−4

−2

0

iteration

lg(R

es)

one grid line GS methodMG line GS method

Figure �� Convergence rate� Line Gauss�Seidel method on � and grids�uniform mesh�

To illustrate the performance of these smoothers� one can consider solu�tion of a D Poisson equation

��p

�x����p

�y�� �� �����

in a domain � � x � �� � � y � � with p � � imposed on the boundaries������ can be discretized with a second order scheme on a Cartesian uniformmesh� In Figures ��� � the convergence histories of iterative solution ofthese discretized equations are shown for both classical and alternating lineGS methods� The calculations are performed on a grid with � cells in bothdirections �total number of cells is �� �� The results obtained employing aMG strategy are compared to those on one grid �in MG grids are used�� Asis seen from the pictures� the convergence of a MG method is tremendouslybetter than that in single grid calculations� Only about ���� iterations areneeded to achieve a computer accuracy convergence when MG is applied�Note that the performance of both smoothers in MG is practically identical�This is� however� not the case on grids with ratios of cell lengths in dierentdirection signi�cantly dierent from unity� In Figures ��� �� the convergencehistories for the same problem are shown on a non�uniform Cartesian mesh�hyperbolic tangent distribution is used in both directions�� Due to the non�uniformity of the mesh there are cells with high aspect ratios�a well knownproblem in MG� The performance of an alternating line GS method is clearlybetter than its classical counterpart�

The explicit nature of discretized Poisson equation in case of second orderapproximation makes constructing an iterative method relatively easy� When

Page 74: Sergey Smirnov

� CHAPTER �� SOLUTION OFUNSTEADY INCOMPRESSIBLE FLOWS

5 10 15 20 25

−15

−10

−5

0

iteration

lg(R

es)

one grid GS methodMG GS method

Figure ��� Convergence rate� Gauss�Seidel method on � and grids �non�uniform mesh�

5 10 15 20 25 30

−14

−12

−10

−8

−6

−4

−2

0

2

iteration

lg(R

es)

one grid line GS methodMG line GS method

Figure ��� Convergence rate� Line Gauss�Seidel method on � and grids�non�uniform mesh�

a high order Pad�e scheme is used for discretization� a Gauss�Seidel type ofsmoothers cannot be applied in a straightforward way� Suppose one has aPad�e type approximation of the D Poisson equation on Cartesian uniformgrid�

�pxi�����j � pxi�����j � �pxi�����j � api���j � pi�j

�x�����

�pyi�j���� � pyi�j���� � �pyi�j���� � api�j�� � pi�j

�y��� �

pxi�����j � pxi�����j�x

�pyi�j���� � pyi�j����

�y� qi�j �����

Page 75: Sergey Smirnov

���� INCOMPRESSIBLE FORMULATION ��

In case of cell averaged approach this discretization is th order accurate� if� � �� and a � � �

One can write equation ����� for two neighboring interfaces� subtractthem and divide by �x� This gives the following relation�

�pxi�����j � pxi�����j

�x�pxi�����j � pxi�����j

�x� �

pxi�����j � pxi�����j�x

� ����

api���j � pi�j � pi���j

�x�

Note that the right hand side of this relation corresponds to the nd orderapproximation of the second order derivative� and the second term of the lefthand side is nothing else but the �rst term of ������

By applying the same procedure to ��� �� one gets

�pyi�j���� � pyi�j����

�y�pyi�j���� � pyi�j����

�y� �

pyi�j���� � pyi�j�����y

� �����

api�j�� � pi�j � pi�j��

�y�

Summation of ���� and ����� yields the following equation�

pi���j � pi�j � pi���j�x�

�pi�j�� � pi�j � pi�j��

�y��Qi�j

a�����

where

Qi�j � qi�j � �pxi�����j � pxi�����j

�x� �

pxi�����j � pxi�����j�x

� �����

�pyi�j���� � pyi�j����

�y� �

pyi�j���� � pyi�j�����y

Now the GS method can applied to the equation ������ The right handside of ����� contains the �uxes pxi�����j � pyi�j���� � which can be calculatedthrough relations ������ ��� � based on the approximation of the previousiteration�

In Figure �� the convergence histories of the solution of Pad�e discretiza�tion of D Poisson equation ����� are presented� The calculations wereperformed on a single grid as well as with MG � grids were used�� Theuniform mesh had � � � cells� The GS smoother described above performssatisfactory� although the convergence is slightly worse than in case of ndorder discretization of the Poisson equation� This is apparently due to thefact that the right hand side of ����� contains �uxes that in the GS are cal�culated based on the approximation of the previous iteration� which reducesthe implicit character of the method�

Page 76: Sergey Smirnov

�CHAPTER �� SOLUTION OFUNSTEADY INCOMPRESSIBLE FLOWS

0 5 10 15 20 25 30−14

−12

−10

−8

−6

−4

−2

0

2

iteration

lg(R

es)

one grid GS method, compact schemeMG GS method, compact scheme

Figure ��� Convergence rate� Gauss�Seidel method on � and grids� com�pact scheme

Page 77: Sergey Smirnov

Chapter �

Arti�cial Selective Damping

��� Articial Dissipation on Stretched Meshes

The compact central schemes may suer from stability problems or producespurious numerical oscillations arising from the lack of built�in numerical dis�sipation� To overcome this problem� a dissipative term has to be added� Theclassical fourth order Jameson type dissipation� however� may turn out toodissipative for LES or CAA calculations� In recent years� a so�called arti�cialselective damping �ASD� ����� ��� has been applied to a number of CAA ap�plications� The eect of the background smoothing term in ASD makes surethat damping occurs only in the narrow band of high wavenumbers leavingthe rest of the spectrum intact� In regions of sharp gradients �shock waves�contact discontinuities� the second order dissipation is switched on to ensurethe monotonicity of the solution�

When ASD is applied on non�uniform meshes� the non�uniformity of thegrid must be taken into account� Even central schemes� when used onstretched meshes may produce numerical dissipation� positive or negative�depending on the local quality of the grid� This aspect� which is absent onuniform meshes� must be taken into account to prevent instability and non�monotonic behavior of the solution due to stretching� As an ASD term� asproposed by Tam ����� does not explicitly contain even order derivatives� asis the case with Jameson type dissipation� the formulation of ASD on non�uniform meshes is not straightforward� But let us �rst consider the classicalth order dissipation term on stretched meshes�

Consider a �D convection equation�

�u

�t��u

�x� � �����

��

Page 78: Sergey Smirnov

�� CHAPTER �� ARTIFICIAL SELECTIVE DAMPING

Once the grid dividing the computational domain in a set of non�overlappingcells is introduced� one can integrate ����� over these cells�

Z xi����

xi����

�u

�tdx�

Z xi����

xi����

�u

�xdx � � ��� �

where xi���� and xi���� are coordinates of the grid points bounding the celli� Making use of the Gauss theorem� one obtains

�ui�t�ui���� � ui����

hi� � �����

where

ui ��

hi

Z xi����

xi����

udx ����

ui���� and ui���� are the values of u on the interfaces and hi is a size of thecell i�

To remove spurious numerical oscillations and ensure the stability of thescheme� one can add a term introducing numerical dissipation� which willdamp out the short waves of the solution� which are responsible for instabil�ities and non�monotone behavior�

�ui�t�ui���� � ui����

hi� Di �����

The arti�cial dissipation term� Di� is represented by the dierence of thenumerical dissipation �ux�

Di �di���� � di����

hi�����

If the solution does not have any discontinuities� one can consider addingonly high order dissipation� e�g� th order�

di���� � �����ui�� � �ui�� � �ui � ui��� �����

If the mesh is uniform� this term only introduces positive dissipation� If�however� the grid is stretched� ����� may add negative numerical dissipationto the scheme� as well as dispersion errors� Developing u�x� in a Taylor seriesaround x � xi����� one obtains from �����

di���� � ����Xk��

ak�i����

��ku

�xk

�i����

�����

Page 79: Sergey Smirnov

���� ARTIFICIAL DISSIPATION ON STRETCHED MESHES ��

where

ak�i���� � �

����k���hi�� � hi�k��

�k � ���hi��������k��hki�k � ���

�����k��hk��i

�k � ���hi��� �����

�hki���k � ���

�hk��i��

�k � ���hi��� �hi�� � hi���

k��

�k � ���hi��

If the mesh is uniform� a��i���� � a

��i���� � �� For a non�uniform grid it is not

the case� however� One can write

Di � ����

hi

�Xk��

��ak�i����

��ku

�xk

�i����

� ak�i����

��ku

�xk

�i����

�� ������

or

Di � ����Xk��

ak�i

hi

����ku�xk

�i����

���ku

�xk

�i����

��� ������

���

hi

�Xk��

ak�i���� � a

k�i����

����ku�xk

�i����

��ku

�xk

�i����

��where

ak�i �

ak�i���� � a

k�i����

���� �

Taking into account that �k��ui�xk��

� �hi

��ku�xk

�i����

���ku�xk

�i����

� where �kui

�xk�

�hi

R xi����xi����

�ku�xk

dx� one can write�

Di � ����Xk��

ak�i

�k��ui�xk��

����

hi

�Xk��

ak�i���� � a

k�i����

����ku�xk

�i����

��ku

�xk

�i����

��������

The terms of the �rst sum with odd indices k represent dissipation� whilethe terms with even indices are dispersion terms� For the second sum� it�sthe other way around� i�e� the terms with even indices k are dissipative andthe terms with odd indices are dispersive� Whether dissipation added byintroducing Di is positive or negative is determined by ak�� which dependon the stretching of the grid�

To demonstrate the eects of dispersion and dissipation errors introducedby the term Di on non�uniform grids� consider the following test case� Anequation

�ui�t� Di �����

Page 80: Sergey Smirnov

�� CHAPTER �� ARTIFICIAL SELECTIVE DAMPING

with a Gaussian signal �u��x� � exp�����x��� as an initial solution� is solvedon a grid clustered towards the center of the Gaussian signal� In Figure ���the solution on the stretched mesh is compared to the initial solution and thesolution obtained on the uniform grid� While on the uniform mesh �whereDi adds only positive dissipation and no dispersion� the Gaussian signal issuccessfully dissipated� it is not the cases for the solution obtained on thenon�uniform grid�

X

U

-.50 -.17 .17 .50 -.10

.37

.83

1.30 123

Figure ���� The eect of th order dissipation ��� initial solution� � solutionon stretched mesh� �� solution on uniform mesh��

To avoid this problem� let us consider the numerical dissipation �ux inthe following form

di���� � �����X

j���

bj�i����ui�j ������

where coe�cients bj�i���� are chosen in such a way that dispersion and negative

dissipation errors are reduced� Developing u�x� in Taylor series around x �xi����� one obtains from ������

di���� � �����Xk��

ak�i����

��ku

�xk

�i����

������

where

ak�i���� � ����k

�hi � hi���k�� � hk��i

�k � ���hi��b���i���� � ����k

hki�k � ���

b��i����� ������

Page 81: Sergey Smirnov

���� ARTIFICIAL DISSIPATION ON STRETCHED MESHES ��

hki���k � ���

b��i���� �

�hi�� � hi���k�� � hk��i��

�k � ���hi��b��i����

To determine the coe�cients bj�i����� one can solve the following system of

linear equations�

a��i���� � �� a

��i���� � �� a

��i���� � �� a

��i���� �

�hi�� � hi��

�h ������

where h is a characteristic mesh size� This characteristic mesh size is usedin ������ to ensure that the arti�cial dissipation is more evenly distributedover the computational domain� If this is not necessary� instead of the lastrelation in ������ one can use�

a��i���� �

�hi�� � hi��

�������

The same test problem as described above is solved with the proposedform of arti�cial dissipation taking into account the non�uniformity of themesh� In Figure �� two solutions obtained on the clustered mesh are com�pared� A much better performance of the arti�cial dissipation given by �������as compared to ������ demonstrates the importance of employing the pro�posed formulation on clustered grids�

X

U

-.50 -.17 .17 .50 -.10

.37

.83

1.30 123

Figure �� � The eect of th order dissipation� �� initial solution� � solutionon clustered mesh �arti�cial dissipation with constant coe�cients ������� ��solution on clustered mesh �clustering of the grid is taken into account��������

Page 82: Sergey Smirnov

�� CHAPTER �� ARTIFICIAL SELECTIVE DAMPING

����� Arti�cial Selective Damping

A dissipative term is introduced�

Di � � �

hi

�Xj���

cjui�j ��� ��

which can be written in a conservative form ��� as a dierence of numericaldissipation �uxes�

Di �di���� � di����

hi��� ��

where

di���� � ���X

j���

bjui�j ��� �

The coe�cients cj and bj are related through the following relation�

bj ��X

n�j

cn ��� ��

Taking into account that ��� �� has to be purely dissipative� i�e� c�k � ck�the Fourier transform of the damping in case of cell averaged values on auniform mesh is�

fD���x� � �c� �

�Xk��

ck cos �k��x�

�sinc

���x

���� �

The sinc term �sinc�x� � sinxx� appears because the Fourier waves have to be

integrated over the cells because cell�averaged values are used� To calculatethe coe�cients ck� so that the damping is eective only for high wavenumbers�the following integral is minimized�Z �

�fD ��x

� ��x

sin�

���x

���

d���x� ��� ��

where is chosen as � � ����� Both terms are multiplied by �x� because

it makes the integral easier to minimize analytically� The function sin� ischosen because the shape of this function corresponds very well to the desiredcharacteristics i�e� damping the high wavenumbers and leaving the lower onesunaected� Developing the ui�j terms in ��� � Taylor series about pointi��� and integrating over the cells one can write the dissipation �ux di����on a uniform mesh as�

di���� � ���X

j���

bjui�j � ��� ��

Page 83: Sergey Smirnov

���� ARTIFICIAL DISSIPATION ON STRETCHED MESHES ��

����a� �u

�x

�����i����

�x� a���u

�x�

�����i����

�x� � a�u

�x

�����i����

�x �

�Awhere the aj coe�cients are linear combinations of the bj�s and only odd orderderivatives appear� On a non�uniform mesh one can do the same which leadsto the following expression�

edi���� � �� �Xj���

ebi����j ui�j � ����A�ui���� �A�

�u

�x

�����i����

�A���u

�x�

�����i����

��� ��

A���u

�x�

�����i����

�A���u

�x�

�����i����

�A�u

�x

�����i����

�AThe ebj�s are now local coe�cients and the coe�cients Aj are linear com�

binations of the ebj�s and depend on the local mesh size hi� Expressions ��� ��and ��� �� are identical if the following equations are satis�ed�

Aj � aj�xj ��� ��

To have similar damping properties on the non�uniform and uniform meshthe six coe�cients ebi����j �s are determined by solving the �rst six equationsof ��� �� where a�� a� and a� are zero� These local coe�cients are stored forfurther use� Figure ��� compares the damping of the current ASD formulationwith formulations from literature�

Page 84: Sergey Smirnov

� CHAPTER �� ARTIFICIAL SELECTIVE DAMPING

Figure ���� Comparison of ASD formulation ���� ��

Page 85: Sergey Smirnov

Chapter �

Numerical Results

In this chapter the application of the �nite volume scheme to solution ofNavier�Stokes equations is considered� The test cases include a boundarylayer on the �at plate� a lid�driven cavity �ow� a propagation of a vorticalstructure in otherwise a uniform �ow� and LES of a periodic turbulent chan�nel �ow� In all these simulations the proposed compact scheme has showna signi�cant improvement in accuracy compared to the conventional secondorder schemes� Although these results are not included in the present thesis�the proposed formulation has also been applied to LES of a �ow around acircular cylinder ����� This test case was very important for the testing ofthe current formulation of the compact scheme� as the calculation requiredthe use of a curvilinear mesh� The results obtained in the cited work showedan important improvement in the quality of the results as compared to thosefound in literature� For details see �����

��� Flat Plate Flow

The �rst Navier�Stokes problem to be considered in this thesis is a D bound�ary layer on the �at plate in otherwise uniform �ow� Due to the viscous eectsthe boundary layer is formed on the plate� the further from the leading edge�the thicker� This problem has a self�similarity solution known as the Bla�sius solution� It is derived from the boundary layer equations introduced byPrandtl�

�u�

�x��uv

�y�

��u

�y������

�u

�x��v

�y� � ��� �

��

Page 86: Sergey Smirnov

� CHAPTER �� NUMERICAL RESULTS

These equations are obtained from the Navier�Stokes equations by re�moving terms that are negligible at high Reynolds numbers� Pressure isconsidered constant in the direction normal to the �at plate� and as the �owoutside of the boundary layer is uniform the pressure gradient in the steam�wise direction is also zero� Another important feature of boundary layer�ows is that the viscous eects in the streamwise direction are negligiblysmall compared to the viscous eects in the normal direction� The secondorder derivative in the streamwise direction is therefore removed from themomentum equation� which makes the system of equation parabolic�

It can be shown that the dimensionless velocity pro�le u�u� �where u�is the free�stream velocity� is a function only of the single composite dimen�sionless variable ��

u

u�� f ���� �����

� � y�u� x

����

����

where the prime denotes dierentiation with respect to �� From the Prandtlequations the following third order non�linear ordinary dierential equationis derived �Blasius equation��

f ��� ��

ff �� � � �����

The boundary conditions are f��� � f ���� � �� f ��� � ��The numerical solution of ����� is used as a reference solution and as well

as an inlet boundary condition�

The full system of Navier�Stokes equations was calculated for both ndorder classical central and th order compact schemes on a relatively coarsemesh ���� � ��� The Blasius solution was used as an inlet boundary con�dition� The inlet and outlet were located in cross sections corresponding toReynolds numbers Rex � u�x� � ���� and Rex � ����� based on distancefrom the leading edge� was chosen ��� and u� � �� Therefore� the inletand outlet were located at x � �� and x � ��� respectively �i�e� the lengthof the domain is ����� The height of the computational domain was chosenbig enough� so that the boundary layer thickness at the outlet is smaller thanthe height of the domain� The boundary layer thickness in case of a laminar�ow is given by the following formula�

� ��x

�u�x� ���������

Page 87: Sergey Smirnov

���� FLAT PLATE FLOW ��

For the chosen u� � �� � ���� the boundary layer thickness at the outlet�x � ���� is � � ����� The height of the domain was taken to be �� whichis su�ciently large�

The boundary conditions were the following�

u�x� �� � �� v�x� �� � � �����

u�x� �� � u�� vy�x� �� � � �����

At the inlet the pro�les obtained from the Blasius solution corresponding tox � �� were imposed for u and v� At the outlet the components of velocitywere extrapolated from inside of the computational domain �cf� section onboundary conditions�� Pressure at outlet and upper boundary was put zeroand extrapolated to the inlet and the �at plate�

Figures ���� �� show the pro�les of non�dimensional streamwise and nor�mal components of velocity

u� � u

u�� v� � v

� u��x���������

�where u� is a freestream velocity� as a function of the similarity variable ��

Eta

U*

.0 2.7 5.3 8.0 .00

.40

.80

1.20 Blasius profile2nd ordercompact

Figure ���� Streamwise component of velocity as a function of similarityvariable �Rex � ������

The computed velocity pro�les are presented in the cross sections corre�sponding to the Reynolds number Rex � ���� �x � ���� It is clearly seenthat for this simulation the second order scheme produces poor results� while

Page 88: Sergey Smirnov

�� CHAPTER �� NUMERICAL RESULTS

Eta

V*

.0 2.7 5.3 8.0 .00

.50

1.00

1.50 Blasius solution2nd ordercompact

Figure �� � Normal component of velocity as a function of similarity variable�Rex � ������

the results obtained with the compact scheme are in much better agreementwith the Blasius solution� It is especially pronounced for the vertical com�ponent of the velocity �v�� It is well known that second order schemes fail tosatisfactory predict it for the this test case� The compact scheme� howeveris in an excellent agreement with the reference solution�

��� Lid�Driven Cavity

A lid driven cavity �ow is considered as another example to test the abilityof the present formulation of compact schemes� The motion of �uid withina lid�driven rectangular D cavity is maintained by the continuous diusionof kinetic energy from the moving wall� This energy is initially con�nedto a very thin viscous layer of �uid next to the moving boundary� After aperiod of time� which depends on the Reynolds number� the redistributionof energy reaches an equilibrium� leading to a steady laminar �ow �in case ofnigh Reynolds number �ows this steady state solution is not possible� though�due to the instabilities leading to transition to turbulence��

The domain considered in this test case is � � x � �� � � y � �� theReynolds number based on the velocity �u� � �� with which the upper wallis moving is ���� This means the kinematic viscosity � ����

Page 89: Sergey Smirnov

���� LID�DRIVEN CAVITY ��

The boundary conditions are

u�x� �� � �� v�x� �� � � ������

u�x� �� � �� v�x� �� � � ������

u��� y� � �� v��� y� � � ���� �

u��� y� � �� v��� y� � � ������

Pressure is extrapolated on all the boundaries from inside of the domain� InFigure ��� the streamlines of the solution are shown to give an idea of the�ow con�guration�

5 10 15 20 25 30 35 40

5

10

15

20

25

30

35

40

x

y

Figure ���� Streamlines of the lid�driven cavity �ow

Figures ��� ��� show results on a very coarse mesh � � � �� with boththe nd order and the compact scheme�

The calculations were performed on the coarse mesh � � � so thatdeviation of the nd order results would be clearly seen� As a reference� thesolution obtained on a very �ne mesh was used and considered to be a veryaccurate approximation of the exact solution�

In Figures ��� ��� the u component of the velocity �in the directionparallel to the moving lid� is plotted along the line perpendicular to the lidand passing through the center of the cavity� The deviation

from the exact solution is clearly seen for the second order calculationswhile the curve obtained with the compact scheme corresponds to that ofthe reference solution reasonably well�

The same behavior is observed in Figure ��� where pressure along thesame line is plotted for both nd order and compact schemes� In addition

Page 90: Sergey Smirnov

�� CHAPTER �� NUMERICAL RESULTS

Y

U

.00 .33 .67 1.00 -.50

.00

.50

1.00 "exact"2nd ordercompact

Figure ��� Horizontal component �parallel to the moving lid� of the velocityalong the vertical line passing through the center of the cavity�

Y

U

.300 .433 .567 .700 -.300

-.200

-.100

.000 "exact"2nd ordercompact

Figure ���� Horizontal component �parallel to the moving lid� of the velocityalong the vertical line passing through the center of the cavity �zoom��

Page 91: Sergey Smirnov

���� VORTICAL FLOW ��

Y

P

.00 .33 .67 1.00 -.100

-.050

.000

.050 exactcompact scheme2nd order scheme

Figure ���� Pressure along the vertical line passing through the center of thecavity�

to a bigger deviation from the exact solution� the second order solution alsoshows some oscillatory behavior� which is due to an odd�even decouplingphenomenon�

��� Vortical Flow

The capability of the numerical schemes to accurately advect vortical struc�tures is an important issue in DNS and LES of turbulence� Therefore� one ofthe problems chosen to demonstrate the accuracy of the proposed high�ordercompact method was a subsonic invicid vortical �ow� A vortex is convectedin otherwise uniform �ow with freestream velocity U� � ��m�s� The ini�tial solution is imposed by prescribing a vortical structure centered aroundx � �� y � � as follows

u � U� � Cy

R�exp��r�� � �����

v �Cx

R�exp��r�� �

p � P� � �C�

R�exp��r��� r �

sx� � y�

R�

Page 92: Sergey Smirnov

�� CHAPTER �� NUMERICAL RESULTS

A vortex is propagated in x�direction in a domain���� � x � ���� ���� �y � ��� with the advection speed U� � ��� the shape of the vortex beingpreserved� The periodicity is imposed in both directions� As a result� ananalytical solution� with which the results of numerical simulations can becompared� is available� This test case was also used in �� � and �����

Figure ���� Isolines of vorticity� nd order scheme �Cartesian mesh�

Figure ���� Isolines of vorticity� compact scheme �Cartesian mesh�

In all the calculations presented here� the values for C and R were chosen

Page 93: Sergey Smirnov

���� VORTICAL FLOW ��

Y

Vorticity

-.50 -.17 .17 .50 -200.

167.

533.

900. ExactCompact2nd order

Figure ���� Vorticity along a vertical centerline �Cartesian mesh��

X

Vorticity

-.50 -.17 .17 .50 -200.

167.

533.

900. ExactCompact2nd order

Figure ����� Vorticity along a horizontal centerline �Cartesian mesh��

as followsC � m��s� R � ��m ������

The Euler equations were solved on both Cartesian uniform and distortedmeshes with dierent levels of resolution ��x�R � ����� � �� ����� �� ���Periodic boundary conditions were imposed in both directions�

In Figures ���� ��� the isolines of vorticity are shown for the calculation onthe Cartesian mesh with grid resolution �x�R � � � for both nd order and

Page 94: Sergey Smirnov

� CHAPTER �� NUMERICAL RESULTS

J

U

0. 20. 40. 60. -5.0

5.0

15.0

25.0 exact solutioncompact2nd order

Figure ����� The streamwise component of velocity along the grid line j �jmid passing through the center of the vortex �Cartesian mesh��

log10(Dx)

log10(Error)

-1.00 -.80 -.60 -.40 -6.0

-4.0

-2.0

.0 2nd ordercompact

Figure ��� � The results obtained on meshes with dierent levels of resolution�Cartesian mesh��

compact schemes after a vortex has been propagated over a distance of ���meters� Figures ���� ���� show the vorticity along the vertical and horizontallines passing through the center of the vortex� The results obtained withthe compact scheme are in very good agreement with the exact solution�while the CE scheme produces signi�cant errors� This is clearly seen from�D graphs as well as D plots showing contours of vorticity� The isolines of

Page 95: Sergey Smirnov

���� VORTICAL FLOW ��

Figure ����� The curvilinear grid�

Figure ���� Isolines of vorticity� nd order scheme �distorted mesh�

vorticity of the exact solution correspond to circular lines with the center atx � �� y � �� The second order scheme fails to produce circular isolines��g� ����� whereas the compact scheme gives an excellent result ��g� ����� InFigure ���� the streamwise component of the velocity along the central gridline is presented and compared to the exact solution� which also con�rms thesuperior behavior of the compact scheme�

In Figure ��� a more systematic analysis of the absolute �maximum�

Page 96: Sergey Smirnov

� CHAPTER �� NUMERICAL RESULTS

Figure ����� Isolines of vorticity� compact scheme �distorted mesh�

J

U

0. 20. 40. 60. -5.0

5.0

15.0

25.0 ExactCompactClassical

Figure ����� The streamwise component of velocity along the grid line j �jmid passing through the center of the vortex �distorted mesh��

error and order of accuracy is provided� The logarithm of the average error isplotted as a function of the logarithm of the mesh size� From these results theactual order of accuracy can be calculated and corresponds to the inclinationof the curves in Figure ��� � The compact scheme performs very well withthe measured order of accuracy ��� even higher than expected theoretically�This was also reported in �� �� but the reason for this is not clear�

Page 97: Sergey Smirnov

���� VORTICAL FLOW ��

The bene�t of using the proposed compact scheme can be estimated forthis test case� From the plot in Figure ��� one can evaluate how much �nerthe mesh needs to be for the CE scheme in order to produce a solution with anerror level comparable to that of the CS scheme� One �nds that for a ���mesh in case of the CS scheme� one would need a ��� � ��� mesh for theCE scheme which has about � times more grid points� Taking into accountthat the present compact method� in its current implementation� is about � times more expensive in terms of CPU� one arrives at the conclusion that�at least for this test case� the proposed compact scheme is � to � times moree�cient than the CE scheme�

A similar series of calculations was carried out on a curvilinear grid� whichis shown in Figure ����� Figures ���� ���� show the isolines of vorticity forthe calculations employing CE and CS schemes� In Figure ���� the stream�wise component of the velocity along the central grid line is presented andcompared to the exact solution� Again� the superior behavior of the com�pact scheme in terms of accuracy is clearly con�rmed� Also note there is adeterioration of the CE scheme results when switching from the Cartesian tothe curvilinear grid� The solution shows some oscillatory behavior in front ofthe velocity peak and the overshoot is also slightly higher on the curvilineargrid� Such deterioration is not found for the CS scheme�

The performed tests con�rm the capacity of the present compact schemeformulation to accurately simulate propagation of vortical structures� Theconventional nd order central scheme gives poor quality results� Thereforeone is advised to use compact scheme instead of nd order schemes in sim�ulations such as LES and DNS of turbulence� where proper resolution ofunsteady development of vortical structures is necessary for obtaining satis�factory results� This will be further con�rmed in the next section devoted toLES of a channel �ow�

Another simulation that has been carried out is a propagation of the vor�tex on the non�uniform Cartesian grid where the size of the cells alternateswith a ratio ���� This was to test the performance of the proposed �nitevolume formulation of ASD� As is well known� central schemes tend to pro�duce unphysical oscillations� which are undesirable by themselves� but canalso lead to instability of the calculation due their uncontrollable growth�On highly irregular grids� the situation is even worse� Application of stan�dard smoothing techniques� such as when ASD or th order dissipation isadded to the numerical scheme� must take the irregularity of the mesh intoaccount� An ASD �nite volume formulation that can be applied on Cartesianbut non�uniform grids was proposed in Chapter �� Here it has been testedon the propagation of the vortex�

Page 98: Sergey Smirnov

�� CHAPTER �� NUMERICAL RESULTS

Figure ����� Results of D Euler vortex calculation for central compactschemes on non�uniform mesh

Looking at the vorticity ������ we see that some oscillations appear down�stream the vortex due to the irregularity of the mesh if ASD is not applied�The non�uniformity of the mesh introduces dispersion and dissipation errorsthat lead to this non�monotonic behavior� If ASD is added to the scheme�for details cf� Chapter ��� the spurious oscillations caused by these errorsare removed� The drawback of adding dissipation is that the vortex itself isfurther dissipated�

��� LES of channel ow

One of the dierences between numerical simulation of laminar �ows as wellas turbulence by means of RANS on the one hand and LES on the otherconsists in LES equations being mesh�dependent� The exact solution of �l�tered Navier�Stokes equations used in LES is dierent on dierent grids forthe same physical problem as the �lter width is normally determined by themesh size� Therefore the grid�re�nement strategies and the choice of thenumerical methods in LES are somewhat dierent from those in RANS andlaminar �ow simulations�

As the exact solution of Navier�Stokes and RANS equations is not mesh�dependent� there is no substantial dierence between a numerical solutionobtained with a high order scheme on a coarse mesh and a solution obtained

Page 99: Sergey Smirnov

���� LES OF CHANNEL FLOW ��

with a less accurate method but on a �ner mesh� The main reason forpreferring a high order numerical methods is computational costs� Compactschemes allow the use of coarser grids� and therefore less data storage� Also�as has been shown in the previous sections� CPU time required to obtaineda solution of the same order of accuracy is smaller for compact schemes thanfor classical low order methods� If� however� the computational costs are notan issue� one can do equally well by using low order schemes in combinationwith a �ne enough mesh�

In LES� however� the re�nement of the grid causes the change of thesolution� as �ner grids allow to represent a wider range of space scales� Inorder to resolve all the scales of such �ow �elds� highly accurate methods areneeded� The ideal approach is the use of spectral methods� which� however�are not very �exible in terms of grid topology and boundary conditions� Pad�etype schemes� therefore� become a valuable alternative� The main advantageof these schemes is that� while providing a better representation of the shorterlength scales of solution� as compared to classical �nite�dierence and �nite�volume methods� they allow to use more complex mesh geometries� whereasspectral methods are limited to applications in simple domains and withsimple boundary conditions�

The performance of the compact scheme formulated in this thesis in LEScontext is studied in this section� A fully developed periodic channel �owis solved by means of both compact and classical second order schemes ondierent grids� As Pad�e schemes are computationally rather costly� it isinteresting to study for which directions the use of high order schemes ismore crucial� This has been done by applying the compact scheme in onlyone or two directions instead of using it in all directions� As the results haveshown� the most crucial direction in terms of accuracy of the scheme is aspanwise direction� The normal direction �from wall to wall� has turned outthe least sensitive to the accuracy of the scheme applied in this direction�

����� The description of the test case

The geometry of the channel is shown in �gure ������� A mesh of ��������points is used in the x� y and z directions� the streamwise� normal andspanwise dimensions are � � � �

��� Uniform meshes with spacing �x� �

�xu��

�� and �z� � �zu��

are used in the streamwise and spanwisedirections� A non uniform mesh with hyperbolic tangent distribution is usedin the wall�normal direction� The �rst mesh point away from the wall isat yj�� � �yu�

� �� and the maximum spacing �at the centerline of the

channel� is ���� wall units� The following transformation gives the location

Page 100: Sergey Smirnov

�� CHAPTER �� NUMERICAL RESULTS

Lx

xy

z

Lz

Ly

Figure ����� Channel �ow test case�

of grid points in the y direction�

yi ��

atanh��iarctanh�a�� ������

with��i � �� � �j � ����Ny � �� �j � �� � � � � � Ny�

Here a is the adjustable parameter of the transformation �� � a � ��� a largevalue of a distributes more points near the walls� In the present computa�tions� a � ������ Ny � ���

The �ow is driven by a constant streamwise pressure gradient introducedas a source term in a momentum and energy equations�

A laminar steady�state �ow with superposed random perturbations wasused as an initial solution� Computations were done until a statisticallysteady state was reached and continued to obtain reliable statistical quanti�ties�

The pressure gradient in streamwise direction driving the �ow was chosen

so that the mean friction velocity u� �q�w�� was unity� The integration of

the momentum equation across the whole �ow domain in the assumption ofperiodic boundary conditions in streamwise and spanwise directions gives

�p Lx � �wLxLz ������

where �w is a skin friction averaged over both walls� It can be rewritten as�assuming dp

dx� �p

Lx�

dp

dx� �u�� ������

Page 101: Sergey Smirnov

���� LES OF CHANNEL FLOW ��

The pressure gradient is thus chosen to be equal to density�

The Reynolds number based on the friction velocity is Re� �u� �� ����

where � is the channel half�width� Since � and friction velocity are unity� � �

���� ������� The time step was �t � ����� For time integration� the

low storage �stage RK method was used with coe�cients �� �� � ���� ���

The subgrid�scale closure model was Smagorinsky model with a constantcoe�cient �Cs � �����

t � Cs�d���q Si�jSi�j ������

To force the subgrid�scale stresses to vanish at the walls the following damp�ing function was used�

d ��� exp

�� y� �

�� ������ ��

After a statistical steady state has been reached� the �ow statistics isaveraged for ��u� � To study the performance of the scheme� the followingsimulations have been carried out�

� a� nd order in all direction�

� b� nd order in normal direction and th order compact scheme inhomogeneous directions�

� c� nd order in normal direction and streamwise directions� th ordercompact scheme in spanwise direction�

� d� nd order in normal direction and spanwise directions� th ordercompact scheme in streamwise direction�

����� Results

In �gure ���� the pro�les of average streamwise component of velocity areshown for the cases a and b and compared to DNS data of Kim et al� ���The improved quality of the simulation b as compared to the second ordercalculation a is obvious� The LES �rst order statistics obtained with thecompact scheme in homogeneous directions is almost on top of the DNSresults� while the pro�le of mean velocity obtained with the second orderscheme in all directions is clearly overestimated�

The same is true for the second order statistics �Figures �� �� �� �� �� ��Turbulence intensities obtained in simulation b are much closer to the DNS

Page 102: Sergey Smirnov

��� CHAPTER �� NUMERICAL RESULTS

10−1

100

101

102

0

5

10

15

20

25

30

y+

u+

Mean velocity profile

DNS by KimLES 2nd orderLES compact scheme

Figure ����� The mean velocity pro�le� Simulations a and b�

results than in simulation a� However� the second order statistics is in aworse agreement with DNS results as compared to the mean velocity� Apossible explanation for the discrepancies between the LES and DNS resultsmay be suggested by considering the coherent structures near the wall� Fromthe experiments of Kline et al� ���� it has been well known that streaks ofrespectively low and high longitudinal velocity exist close to the wall� betweeny� � � and y� � � � ��� Several numerical and experimental studies haveshown that the �uid elements located above the low�speed streaks are ejectedaway from the wall� Conversely� the high speed �uid elements correspond tothe sweep events� i�e�� pumping of the high speed �ow of the outer region tothe near wall region�

There are still a lot of controversies about the interpretation of the low�and high�speed streaks� and how they are related to the sweep and ejectionevents� But what is obvious� is the important role of these structures on thetransport of parcels of �uid� In LES the inadequacy of the computationalgrid resolution in horizontal directions to resolve the streaks at their properscale� is a reason for the dierence between the LES and DNS results� In�� � it has been shown that the use of zonal embedded grids that allow fora better resolution in directions parallel to the wall in the near wall regionyields better results for the turbulent intensities�

This improvement might be explained by the better resolution in thespanwise direction� It is known that it is especially important in LES of thechannel �ow� Simulations c and d clearly demonstrate it� In Figures �� � and

Page 103: Sergey Smirnov

���� LES OF CHANNEL FLOW ���

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 00

0.5

1

1.5

2

2.5

3

3.5

4

y

rms u

DNS by KimLES 2nd orderLES CS in homogeneous dir.

Figure �� ��pu�u� turbulence intensities� Simulations a and b�

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 00

0.5

1

1.5

y

rms v

DNS by KimLES 2nd orderLES CS in homogeneous dir.

Figure �� ��pv�v� turbulence intensities� Simulations a and b�

�� the pro�les of mean velocity for simulations b� c and d are shown� Eventhough in case c he compact scheme is only used in the spanwise direction� theresults are very close to DNS� In simulation d� where the compact schemewasonly used in the streamwise direction in the results are signi�cantly worse�The same is observed for the turbulence intensities �Figures �� �� �� ���� ���

The next to simulations have been carried out on the mesh that is coarserin the normal direction ���� ��� ��� to test the in�uence of using the com�

Page 104: Sergey Smirnov

�� CHAPTER �� NUMERICAL RESULTS

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 00

0.5

1

1.5

y

rms w

DNS by KimLES 2nd orderLES CS in homogeneous dir.

Figure �� �pw�w� turbulence intensities� Simulations a and b�

10−1

100

101

102

0

5

10

15

20

25

30

y+

u+

Mean velocity profile

DNS by KimLES CS in spanwise dir.LES CS in streamwise dir.LES CS in homogeneous dir.

Figure �� �� The mean velocity pro�le� Simulations b� c and d�

pact scheme in this direction� In Figure �� � the mean velocity pro�les areshown for two simulations� in one of them the compact scheme is appliedonly in homogeneous directions� in the other in all directions including thenormal one� As is seen from the picture� the latter gives results that areslightly closer to DNS data� In Figures �� �� ��������� the turbulence in�tensities are presented� Although� for the streamwise component the secondcalculation gives slightly better results� it is not the case for normal and

Page 105: Sergey Smirnov

���� LES OF CHANNEL FLOW ���

102

12

14

16

18

20

22

24

y+

u+

Mean velocity profile

DNS by KimLES CS in spanwise dir.LES CS in streamwise dir.LES CS in homogeneous dir.

Figure �� � The mean velocity pro�le �zoom�� Simulations b� c and d�

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 00

0.5

1

1.5

2

2.5

3

3.5

4

y

rms u

DNS by KimLES CS in spanwise dir.LES CS in streamwise dir.LES CS in homogeneous dir.

Figure �� ��pu�u� turbulence intensities� Simulations a and b�

spanwise components�in the middle of the channel the turbulence intensi�ties obtained with compact scheme applied only in homogeneous directionsare closer to DNS data�

As has already been mentioned� the use of the higher�order scheme in thenormal direction for this �ow does not necessarily improve the results� Themost crucial is a spanwise direction� which has clearly been demonstrated�

Page 106: Sergey Smirnov

�� CHAPTER �� NUMERICAL RESULTS

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 00

0.5

1

1.5

y

rms v

DNS by KimLES CS in spanwise dir.LES CS in streamwise dir.LES CS in homogeneous dir.

Figure �� ��pv�v� turbulence intensities� Simulations a and b�

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 00

0.5

1

1.5

y

rms w

DNS by KimLES CS in spanwise dir.LES CS in streamwise dir.LES CS in homogeneous dir.

Figure �� ��pw�w� turbulence intensities� Simulations a and b�

Another important conclusion is that the compact schemes provide a signi��cant improvement in LES calculations� con�rming that the spatial resolutionof the numerical method is very important in such long time integrationproblems�

Page 107: Sergey Smirnov

���� LES OF CHANNEL FLOW ���

10−1

100

101

102

0

5

10

15

20

25

30

y+

u+

DNS (Kim et al.)4th order compact hom. dir.4th order compact all dir.

Figure �� �� The mean velocity pro�le�

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 00

0.5

1

1.5

2

2.5

3

3.5

4

y

rms u

DNS4th order compact hom. dir.4th order compact all dir.

Figure �� ��pu�u� turbulence intensities�

Page 108: Sergey Smirnov

��� CHAPTER �� NUMERICAL RESULTS

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 00

0.5

1

1.5

y

rms v

DNS4th order comp.4th order comp. all dir.

Figure �����pv�v� turbulence intensities�

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 00

0.5

1

1.5

y

rms w

DNS4th order comp.4th order comp. all dir.

Figure �����pw�w� turbulence intensities�

Page 109: Sergey Smirnov

Chapter �

Conclusions

The �nite volume formulation of compact schemes on irregular structuredmeshes has been presented� The governing equations are discretized in thefollowing manner� �rst the equations are integrated over the control volumesdetermined by the grid and the Gauss theorem is applied to terms writtenin the divergence form� The volume integrals containing the time derivativesof the solution do not need any approximation because the cell averagedvalues of the solution are used as a discrete representation of the continuoussolution of the PDE� The surface integrals are then calculated by means ofimplicit� Pad�e type interpolation formulas that relate cell averaged values andinterface averaged values of the solution� If the �ux is not a linear functionof the solution �the linear operator having constant coe�cients� the �ux isreconstructed to maintain the overall higher�order of accuracy of the method�

The irregularity of the mesh is taken into account directly in the physicalspace through coordinates of the grid nodes� The coe�cients of the Pad�einterpolation formula depend on the local qualities of the mesh and are cal�culated through a multidimensional Taylor series analysis of a truncationerror of the formula� On the stencil chosen in this thesis only the �rd or�der of accuracy can be achieved on a general mesh� However� the leadingtruncation error of order is minimized by the least square approach� On aCartesian mesh� the scheme becomes th order accurate�

Preliminary numerical tests have shown that the scheme with the highestorder possible �i�e� �� is unstable on very distorted meshes� The reason forthat is unclear and requires further investigation� However� the scheme ofaccuracy of one order less� which is obtained by setting the coe�cients in theleft hand side of the Pad�e formula to thier values on the Cartesian uniformgrid� does not suer from this problem� This scheme is identical to the formerone on a Cartesian uniform mesh and� although having only nd order formal

���

Page 110: Sergey Smirnov

��� CHAPTER � CONCLUSIONS

accuracy� provides very accurate results on curvilinear grids�

If a �ux is non�linear function of the unknowns or a linear function withnon�constant coe�cients� one needs a reconstruction procedure to calculatesurface or line integrals of �uxes without losing the high�order accuracy ofthe method� This reconstruction step was formulated and its importanceassessed by testing it one a model problem� where the Gaussian signal isrotated around the origin located in the center of an O�type mesh �this meansthat the coe�cients of the linear convection equation are not constant�� Bythe grid re�nement technique it has been shown that the high�order accuracyof the method can be mantained only when the reconstruction step is used�

Pad�e schemes of formal accuracy higher than can be obtained by extend�ing a stencil of the interpolation formula or by introducing an extra equationfor the surface or line integrals of gradients of the interpolated variables overthe interfaces of cells� This was beyond the scope of the thesis� and can beconsidered as a sugestion for further research�

The formulated scheme has been applied to a number of problems� suchas linear advection equation in �D and D with constant and variable coef��cients� laminar and turbulent �ows� In all the calculations the scheme hasprovided a signi�cant improvement of results as compared to conventionallow order schemes� except for the low Reynolds number lid�driven cavity �owwhere the improvement was less pronounced�

The e�ciency of using compact schemes to obtain the solution of the sameaccuracy as the one obtained by second order methods� but on coarser grids�has been demonstrated in case of propagation of a D vortical structure�The compact scheme turned out to be � to � times more e�cient than theclassical nd order method� In �D cases this improvement in e�ciency isexpected to be even higher�

One of the most important topics of this thesis was application of thecompact scheme to unsteady �ows consisting of a wide range of space scales�In LES of turbulence the �ow �elds contain vortical structures of dierentsizes the development of which has to be simulated over a long period oftime� The propagation of the vortex in an otherwise uniform �ow test casehas clearly demonstrated that the conventional nd order scheme is unable topredict this phenomenon accurately enough� The use of high order schemes�the Pad�e schemes being among them� is therefore necessary for successful LEScalculations� This has been demonstrated for the LES of a periodic turbulentchannel �ow� The results obtained by means of the proposed compact schemehave shown a tremendous improvement in quality of results as compared tothe nd order scheme�

Page 111: Sergey Smirnov

���

As the method developed in the present thesis is applicable on non�Cartesian meshes� it can be employed to LES simulations in more complexgeometries than the one found in the channel �ow� As has already been men�tioned� the proposed scheme was applied to LES of turbulent �ow aroundthe circular cylinder�the simulation that requires a curvilinear mesh �seethe reference in the beginning of chapter ��� The results obtained in thatwork have shown an important improvement compared to the results foundin literature� which were obtained with a nd order schmeme� However� fur�ther tests on general meshes must be performed in the context of LES tofully assess the the advantages and drawbacks of the method�

As the use of compact schemes is computationally more expensive� itis interesting to study whether the application of a compact schemes in alldirections is necessary for an accurate calculation� If the scheme in one ortwo of the grid directions can be a conventional nd order scheme� this maysigni�cantly reduce the computational costs of the simulation� This issue hasbeen studied for the LES of the channel �ow by applying compact schemesonly in one or two directions instead of all directions� The calculations haveshown that the use of the second order scheme in the normal direction reducesthe accuracy of results only a little� This is� of course� only true for thisparticular simulation� and every case must studied individually�

An alternative compact method has been brie�y described for �D and D convection equations� This method is a central �nite volume formulationinspired by the popular Discontinuous Galerkin method� which originallywas formulated in the �nite element context� The major dierence of theproposed formulation was that the �uxes were approximated with a centralformula� This allows to construct a scheme without a built�in numericaldissipation that has a formal accuracy twice higher than a conventional Dis�continuous Galerkin method with the same number of polynomials� Themethod was compared to nd order �nite volume method and the current�nite volume formulation of compact schemes� and showed excellent results�Although the proposed approach has only been applied to a linear convec�tion equation on uniform Cartesian grid� it can be extended to non�linearequations such as Navier�Stokes equations and be used on arbitrary meshes�This can be a topic of further research� namely formulation of the central �uxapproximation on general meshes� reconstruction of non�linear �uxes to man�tain the high�order accuracy of the scheme �here the ideas similar to thosedescribed in chapter can be used�� and discretization of viscous �uxes�

The time integration of incompressible �ows by a fractional step �or pro�jection� method requires solution of the Poisson equation for pressure� Asthis method has been applied to some simulations in this thesis� a descriptionof a Poisson solver used in these calculations has been provided� An e�cient

Page 112: Sergey Smirnov

��� CHAPTER � CONCLUSIONS

solution of the Poisson solver is needed for the successful calculation of un�steady incompressible �ows� Two iterative methods are described in chapter� a single grid implicit approximate factorization method based on spliting amultidimensional implicit operator in one�dimensional ones� which are solvedsuccessively� and a multigrid method based on the Gauss�Seidel smoother�Both methods are designed to be used in the context of Pad�e discretization�which does not allow for an explicit one discrete numerical representationof the PDE� The methods have only been applied on the Cartesian meshes�Their use on cuvileaner grids requires further research�

In chapter � an issue of superiors oscillations that arise due to a varietyof reasons including non�regularity of the mesh has been considered� Tosuppress these oscillations an arti�cial dissipation term is often added todiscretized equations that does not ruin the accuracy of the scheme� butdamps out undesirable oscillations� On irregular meshes special attentionmust be paid to formulation of these arti�cial dissipation term� This has beendemonstrated for th order dissipation� and an arti�cial selective damping�which has originally been introduced for uniformmeshes� has been formulatedfor the use on non�uniform Cartesian grids� The proposed approach tackledthe non�uniformaty of the mesh in the physical space as opposed to themethods found in literature� which treat this in the computational space� Thephysical space approach allows for the use of highly irregular grids with non�smoothly varying or even random mesh spacing� This has been con�rmed inchapter � on a vortical �ow test case�

The proposed formulation is however only valid on the Cartesian meshes�Its formulation on general structured grids can be a topic of further investi�gation�

Page 113: Sergey Smirnov

Appendix A

Point�wise Approach

In this Appendix a �nite volume discretization of the �rst derivative will beconsidered� Dierent higher order approximations will be introduced andcompared with that of the �nite dierence method�

Let us consider a continuous dierentiable function u�x� de�ned in theinterval a � x � b and introduce a mesh x��� � a� x���� � xN���� � b� thusdividing the interval into N � � subintervals

xi���� � x � xi����� i � �� � � N � � �A���

Note that the index i denotes a subinterval �later called a cell� and not a gridpoint� In the cell�centered approach �which is assumed in this appendix� thediscrete variable is de�ned in the centers of the cells�

ui � u�xi�� i � �� � � N � � �A� �

where

xi �xi���� � xi����

� i � �� � � N � � �A���

By integrating the exact value of the derivative u��x� over the cell i andapplying the �D Gauss theorem� one gets the following formulaZ xi����

xi����

u��x�dx � u�xi������ u�xi����� �A��

To discretize this equation both the integral and the �uxes u�xi����� andu�xi����� must be approximated�

Let us assume that the mesh introduced above is uniform� i�e�

x��� � x��� � x�� � x��� � � xN���� � xN���� � �x �A���

���

Page 114: Sergey Smirnov

�� APPENDIX A� POINT�WISE APPROACH

By developing u�x� and u��x� in a Taylor series about point x � xi� thefollowing expressions for all the terms of �A�� can be obtained�

Z xi����

xi����

u��x�dx � u��xi��x� u����xi��x�

�O��x� �A���

u�xi����� �ui�� � ui

� u���xi�

�x�

�� u����xi�

��x�

��O��x�� �A���

u�xi����� �ui � ui��

� u���xi�

�x�

�� u����xi�

��x�

��O��x�� �A���

Putting them in �A��� one gets

u��xi��x� u����xi��x�

�ui�� � ui��

� u����xi�

��x�

�O��x�� �A���

which can be re�written as

u��xi� � u����xi��x�

ui�� � ui�� �x

� u����xi���x�

�O��x�� �A����

If one employs the following approximations when discretizing �A��Z xi����

xi����

u��x�dx � u��xi��x �A����

u�xi����� � %ui���� � ui�� � ui

�A�� �

u�xi����� � %ui���� � ui � ui��

�A����

the resulting approximation of derivative u��xi� will be

u��xi� � ui�� � ui�� �x

�A���

By comparing it to an exact relation �A����� it is seen that this approximationis second order accurate�

Instead of �A�� � and �A���� one can use more accurate approximation%ui����� %ui���� �of order n� for the �uxes u�xi����� and u�xi������ �A��� and�A��� will be then re�written as

u�xi����� � %ui���� �O��xn� �A����

u�xi����� � %ui���� �O��xn� �A����

Page 115: Sergey Smirnov

���

and �A���� will be replaced by

u��xi� � u����xi��x�

�O��x�� �

%ui���� � %ui�����x

�O��xn��� �A����

It can be seen that if n � �� the �nite volume discretization

u��xi� � %ui���� � %ui�����x

�A����

making use of just mentioned higher order accurate interpolations �A����A����is only second order accurate� since the term u����xi�

�x�

��in the left hand side

of �A����� which represents the truncation error of approximation of the in�tegral� will determine the formal accuracy of the algorithm�

Hence� to obtain a scheme more accurate than second order� one has toeither apply a more accurate approximation of the integral �of at least thesame order as that of interpolation of the �uxes�� or use a special secondorder accurate interpolations� the truncation errors of which would cancel asecond order truncation error of approximation of the integral�

To illustrate the latter approach let us consider the following interpolation

%ui���� � Aui�� � ui

�Bui�� � ui��

�A����

%ui���� � Aui � ui��

�Bui�� � ui��

�A� ��

This interpolation will be second order accurate� provided

A�B � � �A� ��

and fourth order accurate� if

A ��

�� B � ��

��A� �

By developing u�x� and u��x� in a Taylor series about point x � xi and taking�A� �� into account� the following expressions can be obtained�

Z xi����

xi����

u��x�dx � u��xi��x� u����xi��x�

� � � � � � u��xi��x

� � � � �� �A� ��

u��xi��x

� � � � � �O��x �

u�xi����� � %ui���� ��

�� � � � ��B

� A

�x�

�u���xi�� �A� �

Page 116: Sergey Smirnov

�� APPENDIX A� POINT�WISE APPROACH

�� � � � ��B

� A

�x�

��u����xi� � �

�� � � � ��B

� A

�x�

�u���xi��

� � � ��B

� A

�x

��u��xi� �O��x��

u�xi����� � %ui���� ��

�� � � � ��B

� A

�x�

�u���xi�� �A� ��

� � �� � � � ��B

�A

�x�

��u����xi���

�� � � � ��B

� A

�x�

�u���xi��

� � � � � ��B

�A

�x

��u��xi� �O��x��

Putting the above expressions in �A��� one gets

u��xi��x� u����xi��x�

� � � � � � u��xi��x

� � � � � �O��x � � �A� ��

� %ui���� � %ui���� ��

�� � � � ��B �A

�x�

��u����xi��

�� � � ��B �A

�x

��u��xi� �O��x �

or

u��xi� � u����xi��x�

� � � � � � u���xi��x

� � � � � �O��x�� � �A� ��

�%ui���� � %ui����

�x��

�� � � � ��B �A

�x�

��u����xi��

�� � � ��B �A

�x�

��u��xi� �O��x��

Now it is easily seen that in order to make the �nite volume approximation

u��xi� � %ui���� � %ui�����x

�A� ��

which makes use of interpolations given by �A����A� �� and approximationof the integral �A����� A and B should satisfy the following relation�

�� � � � ��B �A

�x�

��u����xi� �

�x�

� � � � �u����xi� �A� ��

which can be reduced toA� �B � � �A����

Along with �A� �� it gives the following values for A and B�

A ��

�� B � ��

��A����

Page 117: Sergey Smirnov

���

Using these values for interpolations �A����A� ��� one gets the followingfourth order accurate �nite volume approximation of u��xi�

u��xi� � %ui���� � %ui�����x

� �A�� �

��

�x

Aui�� � ui

�Bui�� � ui��

�A

ui � ui��

�Bui�� � ui��

� �A�B�ui�� � ui�� �x

� Bui�� � ui���x

ui�� � ui�� �x

� ��

ui�� � ui���x

which is an exact equivalent of a th order accurate �nite dierence approx�imation� Note that �A���� makes the interpolations �A����A� �� only secondorder accurate�

The other approach mentioned above consists in using a more accurateapproximation for the integral in �A��� The drawback is that this methodis more expensive from a computational point of view� but it is more generaland can easily be extended to arbitrary meshes�

Let us again construct a fourth order �nite volume scheme �based on theexact formula �A���� For approximating the �uxes u�xi����� and u�xi������the fourth order accurate interpolation given by �A����A� �� and �A� � canbe used� A fourth order approximation of the integral can be carried out ina variety of ways� First� an explicit three point quadrature formula is con�sidered� By using a Taylor series about point x � xi the following expressioncan be written�Z xi����

xi����

u��x�dx � ��u��xi��� � �u��xi� � �u��xi�����x� �A����

���� � � ��u��xi��x���

� � � � ��u����xi�

�x�

��

��

� � � � ��u��xi�

�x

��O��x �

If � and � are taken as

� ��

� � �

��

� �A���

�A���� reduces toZ xi����

xi����

u��x�dx ��

u��xi��� �

��

� u��xi� �

u��xi���

�x� �A����

Page 118: Sergey Smirnov

��� APPENDIX A� POINT�WISE APPROACH

u��xi����x

�����O��x �

Hence� one has the following quadrature formula�

Ri ��

u��xi��� �

��

� u��xi� �

u��xi���

�x �A����

where Ri �R xi����xi����

u��x�dx� Making use of �A�� and �A����� one obtains

u�xi������ u�xi����� ��

u��xi��� �

��

� u��xi� �

u��xi���

�x� �A����

u��xi����x

�����O��x �

The following expressions for u�xi����� and u�xi����� can be derived from�A� �A� �� with A � �

�� B � ��

u�xi����� � %ui���� ��

��

�x�

�u���xi� �

�x

��u��xi� �O��x�� �A����

u�xi����� � %ui���� ��

��

�x�

�u���xi�� �

�x

��u��xi� �O��x�� �A����

where %ui���� and %ui���� are given by �A����A� �� and �A� �� Putting theabove expressions in �A����� one obtains

u��xi��� �

��

� u��xi� �

u��xi���

�x � %ui���� � %ui����� �A���

u��xi�����x

�����O��x �

or�

u��xi��� �

��

� u��xi� �

u��xi��� �

%ui���� � %ui�����x

� �A���

u��xi�����x�

�����O��x��

Hence� by using the mentioned above approximations of the integral and�uxes� the following th order implicit �nite volume scheme is obtained�

u��xi��� �

��

� u��xi� �

u��xi��� � %ui���� � %ui����

�x�A� �

If one is given the exact values for the �uxes u�xi����� and u�xi������ �A� �becomes

u��xi��� �

��

� u��xi� �

u��xi��� � u�xi������ u�xi�����

�x�A���

Page 119: Sergey Smirnov

���

which is an exact equivalent of a th order �nite dierence approximation ofthe �rst derivative in a center of a cell�

An implicit quadrature formula for calculating the integral can also beused� One can write

�Z xi����

xi����

u��x�dx�Z xi����

xi����

u��x�dx��Z xi����

xi����

u��x�dx � Au��xi��x� �A��

��� � � �A�u��xi��x��

� ���

u����xi�

�x�

� �

� � ��

u��xi�

�x

����O��x �

If � and A are taken as

� � � � �� A �

���A���

it reduces to

� � �

Z xi����

xi����

u��x�dx�Z xi����

xi����

u��x�dx� �

Z xi����

xi����

u��x�dx � �A���

��u��xi��x� u��xi�

��x

����O��x �

which gives the following implicit quadrature formula�

� � �Ri�� �Ri � �

�Ri�� � �

��u��xi��x �A���

where Ri �R xi����xi����

u��x�dx�

This approximation of the integral in �A�� and the same interpolations ofthe �uxes as in the previous case provide the following �nite volume schemeapproximating the �rst derivative�

u��xi� � � �

%ui���� � %ui�����x

���

%ui���� � %ui�����x

� �

%ui���� � %ui�����x

�A���

It can be veri�ed that this approximation is also th order accurate�

Since both the �uxes and integral in �A�� can be approximated with ath order of accuracy in a number of ways �by explicit or implicit formulae��there can be obtained a variety of th order �nite volume schemes� which arelisted in the following table�

Page 120: Sergey Smirnov

��� APPENDIX A� POINT�WISE APPROACH

N� Quadrature formula Interpolation

� Ri �hu�xi���

�� � ����u

��xi� �u�xi���

��

i�x u�xi����� � �

�ui���ui

� � ��ui���ui��

Ri �hu�xi���

��� ��

��u��xi� �

u�xi�����

i�x �

�u�xi����� � u�xi����� �

��u�xi����� � �

�ui���ui

� � ���Ri�� �Ri � �

��Ri�� � ��

��u��xi��x u�xi����� � �

�ui���ui

�� �

�ui���ui��

� ���Ri�� �Ri � �

��Ri�� � ��

��u��xi��x

��u�xi����� � u�xi����� �

��u�xi����� � �

�ui���ui

W

W’/W

.00 1.05 2.09 3.14 .00

.40

.80

1.20 exact1234

W

W’/W

.00 1.05 2.09 3.14 .00

.40

.80

1.20 exact2FD2FD4FD4C

Figure A��� Plot of modi�ed wavenumber divided by wavenumber vs�wavenumber for �rst derivative approximations �schemes �� � �� � FD � sec�ond order �nite dierence scheme� FD � th order �nite dierence scheme�FDC � th order compact �nite dierence scheme�

In Fig� A�� the spectral characteristics of these schemes are plotted alongwith those of the following �nite dierence formulae�

u��xi� � ui�� � ui�� �x

�FD � �A���

u��xi� � �

ui�� � ui�� �x

� ��

ui�� � ui���x

�FD� �A����

u��xi��� � u��xi� �

u��xi��� � �

ui�� � ui�� �x

�FDC� �A����

In the left picture all four �nite volume formulae are compared with eachother� while in the right �gure the scheme �with the best spectral repre�sentation� is compared with �nite dierence schemes� It can be observedthat interpolations of the �uxes is a very important issue in the th or�der accurate �nite volume discretization of the �rst derivative� whereas the

Page 121: Sergey Smirnov

���

choice of quadrature formula does not play any signi�cant role� Schemes and �obtained with an implicit interpolation of �uxes� have better spectralcharacteristics than schemes � and � �obtained with a less accurate explicitinterpolation of �uxes�� On the other hand the dierence between schemes and �as well as between � and �� is quite small�

In the right picture the scheme is compared with �nite dierence ap�proximations of the �rst derivative� A th order �nite dierence scheme�FDC� shows a better behavior than any �nite volume formulae� while anexplicit �nite dierence approximation �FD� is less accurate than schemes and �

To obtain more accurate than th order �nite volume schemes� one canuse yet more accurate approximations of the integral and �uxes in �A���The general quadrature formula for evaluating the integral can be written asfollows

�Ri�� � �Ri�� �Ri � �Ri�� � �Ri�� � �A�� �

�Bu��xi��� �Au��xi� �Bu��xi�����x

and an interpolation formula can be given by

�u�xi����� � u�xi����� � �u�xi����� � �A����

Aui�� � ui

�B

ui�� � ui��

� Cui�� � ui��

A large variety of schemes of dierent orders of accuracy can be obtainedby using the above quadrature and interpolation formulae� the highest orderbeing ��

Page 122: Sergey Smirnov

� � APPENDIX A� POINT�WISE APPROACH

Page 123: Sergey Smirnov

Appendix B

Relations Determining

Coecients

The �� relations that have to be satis�ed in order to achieve a th orderaccurate Pad�e type formula for approximation of the �uxes on the interfacesare

� � � � � ��X

n��

�Xm���

an�m �B���

��xi�j � ��xi���j ��X

n��

�Xm���

an�mTxi�n�j�m �B� �

��yi�j � ��yi���j ��X

n��

�Xm���

an�mTyi�n�j�m �B���

�� �x�i�j �

�X�

i����

���

�X�

i����� �B��

��� �x�i���j �

�X�

i����

��

��X

n��

�Xm���

an�mTx�

i�n�j�m�

���xi�j�yi�j �

� �Xi�����Yi����

�� �B���

��

� �Xi�����Yi�����

� �

Page 124: Sergey Smirnov

� APPENDIX B� RELATIONS DETERMINING COEFFICIENTS

����xi���j�yi���j �

� �Xi�����Yi����

��

��X

n��

�Xm���

an�mTxyi�n�j�m

�� �y�i�j �

�Y �

i����

���

�Y �

i����� �B���

��� �y�i���j �

�Y �

i����

��

��X

n��

�Xm���

an�mTy�

i�n�j�m�

��

��x�i�j �

� �xi�j�X

�i����

�� �B���

���

��x�i���j �

� �xi���j�X

�i����

��

��X

n��

�Xm���

an�mTx�

i�n�j�m��

�h�x�i�j�yi�j �

��yi�j�X

�i����� �B���

� �xi�j�Xi�����Yi�����i�

��h�x�i���j�yi���j �

��yi���j�X

�i�����

� �xi���j�Xi�����Yi�����i�

��X

n��

�Xm���

an�mTx�yi�n�j�m�

�h�xi�j�y

�i�j �

��xi�j�Y

�i����� �B���

� �yi�j�Xi�����Yi�����i�

��h�xi���j�y

�i���j �

��xi���j�Y

�i�����

� �yi���j�Xi�����Yi�����i�

��X

n��

�Xm���

an�mTxy�

i�n�j�m�

Page 125: Sergey Smirnov

� �

��

��y�i�j �

� �yi�j�Y

�i����

�� �B����

���

��y�i���j �

� �yi���j�Y

�i����

��

��X

n��

�Xm���

an�mTy�

i�n�j�m��

where�xi�j � xi�j � xO� �yi�j � yi�j � yO �B����

�xi�j and yi�j are the coordinates of the center of the cell i� j� Fig � �

�Xi���� � xB� � xA� � �Xi���� � xB � xA� �B�� �

�Xi���� � xB�� � xA��

�Yi���� � yB� � yA� � �Yi���� � yB � yA� �B����

�Yi���� � yB�� � yA��

T xi�n�j�m �

Qxi�n�j�m ��xi�n�j�mSi�n�j�m

Si�n�j�m�B���

T yi�n�j�m �

Qyi�n�j�m ��yi�n�j�mSi�n�j�m

Si�n�j�m�B����

T x�

i�n�j�m ��

Si�n�j�m

�Qx�

i�n�j�m� �B����

� �xi�n�j�mQxi�n�j�m ��x

�i�n�j�mSi�n�j�m

�T xyi�n�j�m �

Si�n�j�m

�Qxyi�n�j�m� �B����

��yi�n�j�mQxi�n�j�m ��xi�n�j�mQ

yi�n�j�m�

��xi�n�j�m�yi�n�j�mSi�n�j�m�

T y�

i�n�j�m ��

Si�n�j�m

�Qy�

i�n�j�m� �B����

� �yi�n�j�mQyi�n�j�m ��y

�i�n�j�mSi�n�j�m

T x�

i�n�j�m ��

Si�n�j�m

�Qx�

i�n�j�m� �B����

���xi�n�j�mQx�

i�n�j�m � ��x�i�n�j�mQ

xi�n�j�m�

Page 126: Sergey Smirnov

� APPENDIX B� RELATIONS DETERMINING COEFFICIENTS

��x�i�n�j�mSi�n�j�m�

T x�yi�n�j�m �

Si�n�j�m

�Qx�yi�n�j�m� �B� ��

��yi�n�j�mQx�

i�n�j�m � �xi�n�j�mQxyi�n�j�m�

� �xi�n�j�m�yi�n�j�mQxi�n�j�m�

��x�i�n�j�mQyi�n�j�m�

��x�i�n�j�m�yi�n�j�mSi�n�j�m�

T xy�

i�n�j�m ��

Si�n�j�m

�Qxy�

i�n�j�m� �B� ��

��xi�n�j�mQy�

i�n�j�m � �yi�n�j�mQxyi�n�j�m�

� �xi�n�j�m�yi�n�j�mQyi�n�j�m�

��y�i�n�j�mQxi�n�j�m�

��xi�n�j�m�y�i�n�j�mSi�n�j�m

T y�

i�n�j�m ��

Si�n�j�m

�Qy�

i�n�j�m� �B� �

���yi�n�j�mQy�

i�n�j�m � ��y�i�n�j�mQ

yi�n�j�m�

��y�i�n�j�mSi�n�j�m�

Qxsyp

i�j �Z Z

Si�j�x� xi�j�

s�y � yi�j�pdxdy �B� ��

and Si�j is an area of the cell i� j

Page 127: Sergey Smirnov

Appendix C

Authors List of Publications

Journal Publications�

� Jan Ramboer� Tim Broeckhoven� Sergey Smirnov� Chris Lacor� Opti�mization of Time Integration Schemes Coupled to Spatial Discretiza�tion for Use in CAA Applications� Journal of Computational Physics�Vol� ��� pp� ������ � ����

� Tim Broeckhoven� Sergey Smirnov� Jan Ramboer and Chris Lacor�Finite Volume Formulation of Upwind and Central Schemes with Arti��cial Selective Damping� SIAM Journal of Scienti�c Computing� Vol� �� No� �� pp� ������� ���

� Chris Lacor� Sergey Smirnov and Martine Baelmans� A Finite Vol�ume Formulation of Compact Schemes on Arbitrary Structured Grids�Journal of Computational Physics� Vol� ���� pp� �������� ���

International conferences with full proceedings�

� T� Broeckhoven� S� Smirnov� C� Lacor and E� Brizuela� Large�EddySimulation of Piloted and Blu�body Diusion Flame� th ECCOMASConference� Jyvaskyla� Finland� ���

� S� Smirnov� C� Lacor and M� Baelmans� A Finite Volume Formulationfor Compact Schemes with Applications to LES� AIAA paper ���� ��� AIAA ��th CFD Conference� ����

� B� Lessani� S� Smirnov� C� Lacor� M� Baelmans and J� Meyers� E�cientLarge�Eddy Simulation of Compressible Flows Using Multigrid� SixthEuropean Multigrid Conference� Belgium� October ����

� �

Page 128: Sergey Smirnov

� � APPENDIX C� AUTHORS LIST OF PUBLICATIONS

� S� Smirnov� C� Lacor� B� Lessani� J� Meyers and M� Baelmans� A Fi�nite Volume Formulation for Compact Schemes on Arbitrary Mesheswith Applications to RANS and LES� European Congress on Com�putational Methods in Applied Sciences and Engineering �ECCOMAS ����� Spain� ���� September ����

� Chris Lacor� Sergey Smirnov and Martine Baelmans� A Finite VolumeFormulation for Compact Schemes on Arbitrary Meshes with Applica�tions to LES� First International Conference on Computational FluidDynamics ��st ICCFD�� Kyoto Research Park� Kyoto� Japan� ��� July ���

� Jan Ramboer� Bamdad Lessani� Sergey Smirnov� Chris Lacor� Jo�han Meyers and Martine Baelmans� Extension of a Reynolds Aver�aged Navier�Stokes Code for Large Eddy Simulation Applications� �thNational Congress on Theoretical and Applied Mechanics� Louvain�la�Neuve� �� May ���

Page 129: Sergey Smirnov

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�� A� Agarwal and P�J� Morris� Direct Simulation of Acoustic Scatteringby a Rotorcraft Fuselage� AIAA ���� ���� ����

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