A Unif Fem Form Comp and Incomp Fl Aug Con Var

8/14/2019 A Unif Fem Form Comp and Incomp Fl Aug Con Var

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c~ Computer methodsin applied

mechanics andengineering

ElSEVIER Comput. Methods App1. Mech. Engrg. 161 (1998) 229-243

A unified finite element ormulation for compressible andincompressible lows using augmented onservation variables

s. Mittala, T. Tezduyarb,*"Department of Aerospace Engineering, Indian Institute of Technology, Kanpur, UP 208 016, India

bDepartment of Aerospace Engineering and Mechanics, Anny High Performance Computing Research Center, University of Minnesota,1100 Washington Avenue South, Minneapolis, MN 55415, USA

Received 6 November 1997

Abstract

A unified approach to computing compressible and incompressible flows is proposed. The governing equation for pressure is selectedbased on the local Mach number. In the incompressible limit the divergence-free constraint on velocity field determines the pressure, while itis the equation of state that governs the pressure solution for the compressible flows. Stabilized finite element formulations, based on thespace-time and semi-discrete methods, with the 'augmented' conservation variables are employed. The 'augmented' conservation variablesconsist of the usual conservation variables and pressure as an additional variable. The formulation is applied to various test problemsinvolving steady and unsteady flows over a large range of Mach and Reynolds numbers. @ 1998 Elsevier Science S.A. All rights reserved.

1. Introduction

Computational methods o solve flow problems all mainly into two categories: a) methods or compressibleflows and (b) methods or incompressible lows. These wo cases of methods are quite different from each otherwith respect o the choice of variables, ssues related to numerical stability and choice of solvers. Variousresearchers n the past have proposed deas for a unified approach o compressible and incompressible lows.Turkel [1] suggested preconditioning method to accelerate he convergence o a steady state for both thecompressible and incompressible low equations. Hauke and Hughes [2] and Hauke [3] presented a finiteelement ormulation for solving the compressible Navier-Stokes equations with different sets of variables. Theyalso showed hat in the context of primitive or entropy variables, he incompressible imit is well behaved andtherefore, one formulation can be used or solving both compressible nd ncompressible lows. Weiss and Smith[4] proposed a preconditioning echnique n conjunction with a dual time-step procedure o compute unsteadycompressible and incompressible lows with density-based ariables. Karimian and Schneider 5] presented acollocated pressure-based ethod that works in both compressible and incompressible egimes.

In this article we present an alternate, unified approach o computing compressible nd ncompressible lowsusing 'augmented' conservation variables formulation. Compressible lows have been computed by severalresearchers n the past with the conservation ariables ormulation [6-13]. It was shown by Hauke [3], that inthe incompressible imit, the Euler Jacobians or the formulation employing conservation ariables s not wellbehaved. n the incompressible imit, since density becomes constant, some of the coefficients must go toinfinity to accommodate inite variations in pressure. t has also been shown by Panton [14] that in the

* Corresponding author.

0045-7825/98/$19.00 @ 1998 Elsevier Science S.A. All rights reserved.PII: SO045-7825(97)00318-6


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230 S. Mittal, T. Tezduyar I Camput. Methods Appl. Mech. Engrg. 161 (1998) 229-243

incompressible imit, the state equation degenerates o a result according o which the density of a fluid particleis constant. This relation along with the mass balance equation eads to the divergence-free onstraint on thevelocity field that can be used to determine he pressure n incompressible lows. The formulation that wepropose n this article is based on the philosophy hat pressure s determined by the equation of state when theflow is compressible, whereas t is determined by the divergence-free onstraint when the flow is incompress-ible. To this end we employ the 'augmented' conservation variables which consist of the usual conservationvariables (density, momenta and energy) and pressure as an additional variable.

We begin by reviewing the governing equations or compressible nd ncompressible luid flow in Section 2.The equations are cast in a non-dimensional orm and a parameter z, based on the local Mach number, isintroduced that governs the choice of equations for compressible and incompressible lows locally in thecomputational domain. The stabilized space-time variational formulation of these equations n terms of theaugmented onservation variables s presented n Section 3. The SUPG (streamline-upwind/Petrov-Galerkin)stabilization technique 6,7,9,13,15,16] s employed to stabilize our computations against spurious numericaloscillations. n Section 4 we present some numerical esults o test the performance f the proposed ormulation.We begin with the computation of the shock-reflection roblem that involves three low regions separated y anoblique shock and ts reflection rom a wall. The exact solution or this problem s known and s compared withthe computed solution. Next, supersonic low past a cylinder at Mach 2 and Re 2000 is computed with theunified formulation and with the compressible low formulation based on the equation of state. Finally, resultsare presented or unsteady low past a cylinder at Re 100. These computations are carried out for differentsubsonic Mach numbers ncluding the incompressible imit.

2. The governing equations

Let {},tC ~n'd and (0, T) be the spatial and temporal domains, espectively, where nod s the number of spacedimensions, and et 1; denote he boundary of ,{},t'The spatial and temporal coordinates are denoted by x and ,The Navier-Stokes equations governing he fluid flow, in conservation orm, are

iJpiJt-+V'(pu)=O on,{},tfor(O,T), (1)

iJ(pu) + v. (puu) + Vp - V. T =0 on ll, for (0, T) , (2)ata(pe)at + V. (peu) + V. (pu) -V. (Tu) + V. q = 0 on n, for (0, T) (3)

Here, p, U, p, T, e, and q are he density, velocity, pressure, iscous stress ensor, otal energy per unit mass, andheat flux vector, respectively. The viscous stress ensor s define~ as

T = ,uVu) + (VU)T) + }"(V. u)1 . (4)

where ,u and },. are the viscosity coefficients. t is assumed hat ,u and },. are related by

2},.= -3 ,u . (5)

Pressure s related o the other variables via the equation of state. For ideal gases, he equation of state assumesthe special orm

p=(y-l)pi, (6)

where y is the ratio of specific heats, and i is the internal energy per unit mass which is related to the totalenergy per unit mass and kinetic energy as

(7)

The heat flux vector is defined as


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S. Mittal, T. Tezduyar I Comput. Methods Appl. Mech. Engrg. 161 (1998) 229-243 231

q = - KVO (8)where K is the heat conductivity and fJ s the temperature. he temperature s related o the internal energy by thefollowing relation:

(J=~ (Q)Cv

where Cv is the specific heat of the fluid at constant volume. For an ideal gas

RCv =-:y-=-T' (IV)

where R is the ideal gas constant. Prandtl number (P )' assumed o be specified, elates he heat conductivity ofthe fluid to its viscosity according o the following relation:

JLCp(11)= p ,r

where Cp is the specific heat of the fluid at constant pressure. For an ideal gas

yRC =p-y-l

In the limit of incompressible lows, i.e. when the Mach number approaches ero, the above-mentioned et ofequations assume new form. It can be shown 14], that the state equation along with the mass balance equationlead to the following relation:

pV'u =0. (13)

Using the relation, Eqs. (1), (2) and (3) can be modified for incompressible lows as

iJpiJt-+ V. (pu) - pV' u = 0 on fl, for (0, T) , (14)

+ v. (puu) - puV. u + Vp -v. T=O on [}" for (O,T),

a(pe)-ae-+ V. (peu) - peV.u + V. (pu) -pV' u -v. (Tu) + V. q = 0 on {It for (0, T). (16)

tU~x*=L'

u* =-:::- t* = p*=Lp~u

U",' L '

(8 - 8~)CpU2

p-p~2 .

p~U~p*= 8* =

where all the quantities with the subscript 00' refer to the free-stream alues of the flow variables. hegoverning equations n the non-dimensional ariables hat are valid over the entire range of compressible ndincompressible lows are

iJp*at*"+ V*. (p*u*) - (1 - z)p*V*. u* = 0 on [J~ for (0, T*), (19)

iJ(p*u*) + v* . (p*u*u*) - (1 - z)p*u*V* . u* + V* p* - V* . T* = 0 on [J~ for (0, T*) ,at

a(pu)at

In this situation, the viscous stress ensor, given by Eq. (4) can be rewritten as

T = jLVu) + (VU)T) . (17)

It is possible o combine he two sets of governing equations or the compressible nd ncompressible lows andexpress hem in terms of non-dimensional ariables. The non-dimensional ariables hat we choose are


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S. Mittal, T. Tezduyar I Comput. Methods Appl. Mech. Engrg. 161 (1998) 229-24332

The Mach number (M) is defined as the ratio of the flow speed o the speed of sound; M~ refers to thefree-stream Mach number. n the above equations (M) E [0, 1] is a function of Mach number such hat z(O) = 0and z(M ~ Mc) = 1, where Mc is a 'cut-off Mach number, decided a priori. All the results eported n this articleare with Mc = 0.3 and with the following definition of z:

MMc

M~M, (23)='M>Mc

(24)

The non-dimensional iscous stress ensor and heat flux vector are defined as

1 2T* = - (V*u*) + (V*U*)T- - z(V* .u*)1Re I 3

1(25)* = --=- V*(J*

R~Pr

Here, Re s the Reynolds Number defined as

p~U~LR.= (26)C JL

In the rest of the article we will work with the non-dimensional ariables only and therefore, he superscript' * '

will be dropped. The governing equations 19)-(22) can be written in the augmented onservation variables

.. a~ aF; aE; + B.!!!!. (27)I ax.

I

+ SU = 0 on n, for (0, T) ,+~ I ax;

(28),=

where U = (p, pu., puv p, pe), is the vector of augmented onservation ariables and M is a diagonal matrixdefined as M = diag(l, I, 1,0, I). The various erms nvolving z and (I - z) in Eqs. (19)-(22) contribute o theterms nvolving Bj and S in Eq. (27). Fj and Ej are, respectively, he Euler and viscous lux vectors defined as

u;p \u;pu. + 5jlPujpu2 + 5;2P

0uj(pe +p) I

-q

(29)i=

Here, Ui and qj are the components f the velocity and heat flux vectors, respectively, and Tik are the componentsof the viscous stress tensor. In the quasi-linear form, Eq. (27) is written as

(30)au au a ( au )at+(Aj+Bj)~-~ Kjj~ +su=O on {},t for (O,T),I I Jwhprp

0TilTi20

'.+7J ikU


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S. Mittal, T. Tezduyar I Comput. Methods Appl. Mech. Engrg. 161 (1998) 229-243 233

of.--!.Ai= au'

is the Euler Jacobian Matrix, and Kij is the diffusivity matrix satisfying

auK.. -;-- =E.

IJ ux. I

J

Corresponding o Eq. (30), appropriate boundary and initial conditions are chosen

3. Finite element formulation

In order to construct he finite element function spaces or the space-time method, we partition the timeinterval (0, T) into subintervals n = (tn, n+,), where tn and tn+1 belong to an ordered series of time levels0 = to < t, < ... < tN = T. Let {In = {J, and 1;: = r;. We define he space-time slab Q as the domain enclosed

n n nby the surfaces In' {In+'' and P n' where P n is the surface described by the boundary r; as t traverses n' The

surface Pn is decomposed nto (Pn)g and (P n)h with respect o the type of boundary condition (Dirichlet orNeumann) being imposed. For each space-time slab we define he corresponding inite element unction spaces.:;'hand rho Over the element domain, his space s formed by using first-order polynomials n space and time.Globally, the interpolation functions are continuous n space but discontinuous n time.

The stabilized space-time formulation is written as follows: given (Uh)-, find Uh Er such that 'v' Wh Eh nr,

f auh f (aWh)e.h.Mh_dQ+ -Qn at Qn aXj

( aWh ) [(A:)T -a;; . Mh-

B' au', ax;

(-F~+E~)dQ+f Q,

j" auk-, + (A h + Bh) - -\&, 'ax

Wh.(

a ( ,at - -- Kh~\.-, ,- ',aXi iia;;) +ShUh] dQ- , ;. , ","

Qe "

( Wh . hh dPJ(Pn)h

(33)

Here, hh represents he Neumann boundary condition imposed and (P n)h is the part of the slab boundary withsuch onditions. he solution o (33) s obtained equentially or all space-time labs QQ' QI' Q2"'" QN-I andthe computations start with

h- UU )0 = 0' (34)where U 0 is the specified initial condition.

REMARKS(1) In the variational onnulation given by Eq. (33), the first three ntegrals and he right-hand side constitute

the Galerkin fonnulation of the problem. Both the Euler and viscous lux tenns are integrated by parts.This fonn of the variational fonnulation ensures hat, in the presence f shocks, he method gives rightjump conditions and shock ocation. The Neumann boundary condition at the outflow boundary nvolvesthe nonnal components of the stress vector and momentum lux. In the limit of incompressible lows(where he pressure s specified only upto a constant and one needs o define a datum pressure) his fixesthe pressure at the boundary. To compute lows that involve free surfaces he weak fonn given by Eq.(33) has to be modified by carrying out the integration-by-part of the time-dependent enn.

(2) The first series of element-level ntegrals n Eq. (33) are added o the variational onnulation to stabilizethe computations against numerical nstabilities. In the advection-dominated ange, hese enns preventthe node-to-node scillations of the flow variables. n the limit of incompressible lows, the inclusion of


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234 S. Minai, T. Tezduyar I Comput. Methods Appi. Mech. Engrg. 161 (1998) 229-243

these terms allows one to employ equal-order-interpolation or velocity and pressure. The choice ofstabilization coefficient T s quite different for compressible nd ncompressible lows. In the context of aunified formulation, one would like to design the stabilization coefficient 'T such that it reduces o theappropriate definitions n the two limits. The second series of element evel integrals n Eq. (33) are theshock capturing erms that stabilize the computations n the presence f sharp gradients. The coefficientof shock-capturing operator, J is same as defined n [13]. The stabilization coefficient T is defined as

'T = max[O, 'Ta 'TpJ

where 'Ta s a diagonal matrix defined as 'Ta = diag(T) T), T), T2' T),

*) 2) -1/2+',= (36)

(37)

R ~3uReu > 3

A= (38)

In the above equations Reu s the cell Reynolds number, c is the wave speed, luhll s the flow speed, nd his the element ength. (M) E [0, 1] is a function of Mach number such hat (0) = 0 and (M ~ Mc) = 1.All the results reported n this article are with the following definition of :

'-N:) ZM~M,

z=M>M,

where Mc = 0.1. Matrix T/3 s subtracted rom Ta o account or the shock-capturing erm as shown n Eq.(35). It is defined as

i/3(40)

Consider he computation of compressible low past a solid body using the unified formulation as n Eq.(33). In the regions where he mach number s low, for example n the boundary ayer, the flow is almostincompressible and density assumes, pproximately, he free-stream value. Under these conditions thecontinuity equation, Eq. (19), behaves ike an advection equation for density. Farther away from theboundary ayer, the flow is in the compressible egime and the density variations are quite significant. Theterm in Eq. (33) that involves Nh provides numerical stability to the density ield in the region where heabove-mentioned ransition takes place. This term is not needed n the formulation if one is seekingsolutions governed by either the compressible r the incompressible low equations only. The matrix Nhis a diagonal matrix defined as Nh = diag(l, 0, 0, 0, 0).

AhIn the stabilizing terms in Eq. (33), components of the A k matrix are defined as

Ah h

[Ak ]. .=[A k]. .+z8 k + 1 8. 5 C',J '.J I, J. (41)

where C is a constant. This term provides stability to the computations for compressible flows. Noticethat such a term is not explicitly added to the formulation in terms of the conservation variables; a similar

term is already present n the definition of A ~. In the case of augmented onservation variables such aterm is not present n the original definition of A~. It has been our experience, ith augmentedconservation ariables ormulation, hat in the absence f this term the velocity field develops oscillationsthat grow with time. In our computations we choose C = 2.

'1",8~~w M


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S. Mi/tal, T. Tezduyar I Comput. Methods Appl. Mech. Engrg. 161 (1998) 229-243 235

(3) The sixth integral enforces weak continuity of the velocity field across he space-time slabs.(4) If one is interested n strictly incompressible lows, Eq. (33) can still be used. However, to reduce he

computational ost, one can prescribe he density ield in the entire domain and herefore, not solve or it.Additionally, if one s not interested n the temperature ield, one can drop the energy equation oo. Thisis possible because n the incompressible imit, the energy equation s decoupled rom the rest of the flowequations.

(5) The variational ormulation in terms of the augmented onservation ariables has an advantage hat anychange n the equation of state can be incorporated n the implementation with very little effort. Forexample, o include the real gas effects n the formulation, one needs o modify the state equation only.However, the implementation s not so straightforward f one uses he conservation variables.

(6) We also implemented this unified approach n the context of a semi-discrete ormulation. In theincompressible imit, Eq. (21) reduces o the divergence ree condition on the velocity field. It behaveslike a constraint equation and one looks for a pressure ield such that the velocity field satisfies hedivergence ree condition at each time level. In an implementation or time-accurate computation ofincompressible lows, the divergence ree equation and the pressure erms are evaluated at the n + 1 timelevel while the rest of the terms are evaluated at n + 1/2 time level. This ensures hat the velocity field ateach ime level is divergence ree. On the other hand, f the divergence ree equation s evaluated at then + 1 2 time level, the velocity field computed at later times may not be divergence ree and one cannotensure the stability of computations. Therefore, in the semi-discrete mplementation of the unifiedformulation for time-accurate omputations, he pressure and the terms nvolving divergence of velocityare evaluated at n + 1 time level while the other terms are evaluated at n + 1 2 time level. We havecomputed unsteady solutions using, both, the space-time and semi-discrete mplementations of ourformulation. The results obtained from the two implementations are almost indistinguishable. n thisarticle, therefore, we report the results only from the space-time mplementation. t must be pointed outthat the computations with the space-time method are substantially more expensive han the ones withthe semi-discrete method. However, the space-time method allows one to compute flows involvingmoving boundaries and interlaces.

4. Numerical examples

Most of the computations eported n this article were carried out on the Digital 3000/300 AXP work-stationat liT Kanpur. Some were computed on the CRAY C90 at Networking Computing Services n Minneapolis,Minnesota. For the space-time implementations, he finite-element basis functions are bilinear-in-space andlinear-in-time, and 2 X 2 X 2 Gaussian quadrature s employed or numerical ntegration. The calculations withthe semi-discrete mplementation, based on bilinear finite-element basis unctions, give almost ndistinguishableresults as the ones rom the space-time method. n this article, only the results computed with the space-timemethod are shown. The nonlinear equation systems esulting from the finite-element discretization of the flowequations are solved using the Generalized Minimal RESidual (GMRES) technique 17] in conjunction withdiagonal and block-diagonal preconditioners.

4.1. Shock-reflection roblem

This two-dimensional, nviscid, steady problem nvolves three low regions separated y an oblique shock andits reflection from a wall as shown n Fig. 1.

It is a standard benchmark problem and for more details the interested eader s referred o the work by Le

Beau and Tezduyar 9] and Shakib [18]. The motivation for this computation s to establish confidence n ourformulation and its implementation or computing flows involving shocks. The computational domain is arectangular egion of dimensions 4.1 in the x direction and 1.0 n the y direction. The mesh consists of 60 X 20rectangular elements. At the left boundary, low data corresponding o Mach 2.9 is prescribed:


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236 S. Mittal, T. Tezduyar / Comput. Methods Appl. Mech. Engrg. 161 (1998) 229-243

'M=2.9p=1u =1IU2 =0,0 = 0

Region 1 (42)

M=2.378

M=2.9 M=1.942

Region 1 Region 3

Fig. I. Shock-reflection problem: problem description.

At the top boundary, he flow conditions that are specified correspond o Mach 2.3781 and an incident shockangle of 29:

'M = 2,3781p = 1.7Ul = 0,9033Uz = -0.1746

.f} = 0.07685

Region 2 (43)

At the lower boundary, he component of velocity normal to the wall is assigned zero value. The computationsbegin with a uniform Mach 2.9 flow in the domain and continue ill the steady-state orm drops below a certain

Fig. 2. Shock-reflection problem: density and pressure fields for the steady-state solution.


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S. Mittal. T. Tezduyar / Comput. Methods Appl. Mech. Engrg. 161 (1998) 229-243 237

value. It should be pointed out that in this problem, the local Mach number in the entire domain is alwaysgreater han the 'cut-off Mach number defined n Eq. (23) and, therefore, z = 1 everywhere. This implies thatthe pressure s determined by the equation of state or perfect gas everywhere n the domain.

Fig. 2 shows he density and pressure ields in the domain for the steady-state olution. Compared n Fig. 3are the density ields for the computed and exact solutions at y = 0.25. Our results compare quite well with thosereported by other researchers sing alternate ormulations [9,18].

4.2. Supersonic low past a cylinder

Mach 2 flow past a circular cylinder is computed or two cases. n the first case he original compressible lowequations are employed by setting z = 1 in Eqs. (19)-(22), i.e. the pressure s determined by the equation ofstate or a perfect gas. The second ase s computed with the unified formulation where z is defined by Eq. (23).The mesh employed, consists of 5120 quadrilateral elements and 5264 nodes. The Reynolds number based onthe diameter of the cylinder and the free-stream alues of the velocity and kinematic viscosity is 2000, and thePrandtl Number s 0.72. The cylinder wall is assumed o be adiabatic and the no-slip condition is specified orthe velocity on the surface of the cylinder. All the variables are specified at the upstream boundary. At the upperand lower boundaries, normal components of the velocity and heat flux are set to zero together with thetangential component of the stress vector. At the downstream boundary, we specify a Neumann-type oundarycondition for the velocity and energy hat is consistent with the variational formulation given by Eq. (33). Thecomputations are initiated with free-stream conditions n the entire domain and continue till the steady-statenorm of the solution falls below a certain desired value. Shown in Fig. 4 are the density, temperature andpressure ields for the steady-state olution computed and the augmented onservation alues ormulation with


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238 S. Minai, T. Tezduyar I Comput. Methods Appl. Mech. Engrg. 161 (1998) 229-243

//'~,

I 0"'~ "

ion ~rl1 ~#LJ

r"""" W~'.

~

f

":';/

z = 1. One can observe a strong bow shock upstream of the cylinder and a weak tail shock n the wake. Theshock stand-off distance compares quite well with experimental observations 19]. It can also be observed hatthe shock has been captured quite well within two to three elements. Solution to the same problem has beencomputed by the conservation ariables ormulation on a much finer mesh with 16000 elements and reported n[13]. On comparing the two solutions we observe that the conservation variables and the agumentedconservation variables ormulations ead to quite comparable esults.

Fig. 5 shows the density, temperature and pressure ields for the steady-state olution computed with the

unified formulation with z defined by Eq. (23). On comparing Figs. 5 and 4 we observe hat they are quitesimilar except hat the density and the temperature ields exhibit some differences close to the cylinder wall. Inthe case of unified formulation, with z defined by Eq. (23), the divergence-free onstraint kicks-in very close othe wall of cylinder where the Mach number is nearly zero. In this region, the mass balance equation (19)behaves ike an advection equation or density. As a result one observes rom the contour plots, very close o thewall of the cylinder, that the flow attempts o advect he density from the upstream o downstream ocations.This is certainly not the case n Fig. 5 where he pressure s determined by the state equation or a perfect gasand not by the divergence ree constraint. t is quite interesting o note that the pressure ields computed by thetwo formulations are almost identical. The drag coefficient computed by both the formulations s 1.48.

These est problems demonstrate hat the unified compressible-incompressible ormulation results n correctshock ocation and strength or both viscous and inviscid flows.

4.3. Subsonic low past a cylinder

The main motivation to develop the unified compressible-incompressible ormulation is to improve theperformance of compressible low algorithm at low Mach numbers. n this section we present our solutions or

~\\-'\, a. "\'-\-J \ \1

'--'--~ ,"" ,...". 1 ~ '-~ \'

Fig. 5. Mach = 2, Re = 2000 flow past a cylinder computed with z defined by Eq. (23): density, temperature and pressure fields (and their

close-ups in the lower row) for the steady-state solution.


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flow past a circular cylinder at Re 100 and low Mach numbers. The Prandtl Number is 0.72. The cylinderresides n a rectangular omputational domain whose upstream and downstream oundaries re ocated at 15 and35 cylinder radii, respectively, rom the cylinder's center. The upper and ower boundaries re placed at 16 radiifrom the center of the cylinder. The finite element mesh consists of 4688 quadrilateral space-time elements and4826 nodes. At each ime step, 4741.2 nonlinear equations are solved teratively to compute he flow field. Thecylinder surface s assumed o be adiabatic and the no slip condition s specified or the velocity on the cylinder

wall. At the upstream boundary, density, velocity and temperature re assigned o their free-stream alues. Atthe downstream boundary, we specify a Neumann-type oundary condition for the velocity and energy hat isconsistent with the variational ormulation given by Eq. (33). At the upper and ower computational oundaries,normal components of the velocity and heat flux are set to zero together with the tangential component of thestress vector. We first present our results for Mach 0.2 flow computed with z = 1. Fig. 6 shows he vorticity,pressure and temperature ields corresponding o the peak value of lift coefficient. Fig. 7 shows the timehistories of the lift and drag coefficients or that part of the simulation when the periodic solution is achieved.The Strouhal number corresponding o the variation of lift coefficient for this case s 0.164. Figs. 8 and 9 showthe solution for Mach 0.2 flow computed with the unified formulation with z defined by Eq. (23). The Strouhalnumber corresponding o the variation of lift coefficient or this case s 0.169. As expected, he two solutions arequite similar. On comparing Figs. 6 and 8 we observe hat there are some differences n the temperature ields ofthe two cases. As has been explained n the previous section, n the case of unified formulation, the flow veryclose o the wall is modeled by incompressible low equations and herefore, he temperature hanges ake placeonly because f viscous effects. On the other hand, n the case of computations with z = 1, density, emperatureand pressure changes ake place in accordance with the equation of state for perfect gas. Therefore, thecontribution to temperature hanges ome from, both, the viscous and compressible effects.

The formulation based purely on the compressible low equations, .e. with z = 1., ails to yield an acceptableunsteady solution at Mach 0.05 with the present mesh. However, f the mesh s refined t is possible o compute

r++

\ r

~~ ... 1\\I I.Fig. 6. Mach = 2, Re = 100 flow past a cylinder computed with z = 1: vorticity, pressure and temperature fields (and their close-ups on theright) at the peak value of the lift coefficient.

1.50

1.25

1.00

0.75"CU.0.50


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240 S. Minai, T. Tezduyar I Comput. Methods Appl. Mech. Engrg. 161 (1998) 229-243

the unsteady low at Mach 0.05. It has been our experience hat the low Mach number lows are very sensitive othe spatial efinement. For a given mesh here exists a certain Mach number below which the compressible lowformulation breaks down. On the other hand, with the unified compressible-incompressible low formulation,one s able to compute lows at any Mach number. Figs. 10 and 11 show the solution for unsteady low past acylinder at Mach 0.05 computed with z defined by Eq. (23). The Strouhal number corresponding o the variation

1.501.25

1.00

0.75C3-: 0.50U

0.25

0.00

-0.25

-0.50

~~~Jf\\ i\ ('\'\ i\ I '\

\J \ I '\ Iv vi '\ I//0 20 40 60 80 100

tFig. 9. Mach = 0.2, Re = 100 flow past a cylinder computed with z defined by Eq. (23): time histories of the lift and drag coefficients.

Fig. 8. Mach = 2, Re = 100 flow past a cylinder computed with z defined by Eq. (23): vorticity, pressure and temperature fields (and their

close-ups on the right) at the peak value of the lift coefficient.


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S. Mitral, T. Tezduyar I Comput. Methods Appl. Mech. Engrg. 161 (1998) 229-243 241

1.50

1.25

1.00

0.75"CU..: 0.50

U0.25

0.0010.25

-0.50

[JJ~~J

(\ (\(\\ (\~(\

I( '\

'\J '\ /V '\ /V 'V \ i0 20 40 60 80 100

tFig. 11. Mach = 0.05, Re = 100 flow past a cylinder computed with z defined by Eq. (23): time histories of the lift and drag coefficients.

of lift coefficient for this case s 0.170. We observe hat the solutions at Mach 0.05 and Mach 0.2 are quitesimilar except or certain differences n the temperature ields that are due to the compressibility effects. Thedemonstrate he robustness f the unified formulation we compute he unsteady low past a cylinder at Mach0.001 and n the incompressible imit. The solution at Mach 0.001 s shown n Figs. 12 and 13 while the one nthe incompressible imit is shown n Figs. 14 and 15. The incompressible low case s computed by setting z = O.The Strouhal number for vortex shedding n both the cases s 0.170. We observe hat the solutions or Machnumbers 0.05, 0.001 and in the incompressible imit are almost indistinguishable and agree quite well withresults from alternate ormulations or incompressible lows [20].

:~,I -- ~

Fig. 12. Mach = 0.001, Re = 100 flow past a cylinder computed with z defined by Eq. (23): vorticity, pressure and temperature fields (andtheir close-ups on the right) at the peak value of the lift coefficient.

.- -- -- .--t

Fig. 13. Mach = 0.001, Re = 100 low past a cylinder computed with z defined by Eq. (23): time histories of the lift and drag coefficients.


14/15

242 S. Mittal, T. Tezduyar I Comput. Methods Appl. Mech. Engrg. 161 (1998) 229-243

Fig. 14. Incompressible, Re = 100 flow past a cylinder computed withz= 0: vorticity, pressure and temperature fields (and their close-ups

on the right) at the peak value of the lift coefficient.

1.50

1.25

1.00

0.75"0O- 0.50U

0.25

0.00

-0.25

-0.50

~~~]

"(\ (\ f\ f\ f\"\f\ f\\; \/ \Jv v \//

20 40 60 80 100

Fig. 15. Incompressible, Re = 100 flow past a cylinder computed with z = 0: time histories of the lift and drag coefficients.

5. Conclusions

A unified formulation for compressible and incompressible lows in terms of the augmented onservationvariables has been proposed. n case of compressible lows the equation of state determines he pressure whereasit is the divergence-free constraint on velocity field that sets the pressure or incompressible lows. Theappropriate governing equations are chosen ocally based on the local Mach number. The formulation wassuccessfully applied to various numerical ests nvolving steady and unsteady lows over a range of Mach andReynolds numbers.

Acknowledgement

This work was sponsored y ARPA and by the Army High Performance Computing Research Center underthe auspices of the Department of the Army, Army Research Laboratory cooperative agreement numberDAAH04-95-2-0003 contract number DAHH04-95-0008. The content does not necessarily eflect the positionor the policy of the government, and no official endorsement hould be inferred. The CRAY time was provided,in part, by the University of Minnesota Supercomputer nstitute.

References[1] E. Turkel, Review of preconditioning methods for fluid dynamics, Technical Report 92-47, Institute for Computer Applications in

Science and Engineering, NASA Langley Research Center, September 1992.


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[2] G. Hauke and T.J.R. Hughes, A unified approach to compressible and incompressible flows, Comput. Methods Appl. Mech. Engrg. 113(1994) 389-395.

[3] G. Hauke, A unified approach to compressible and incompressible flows and a new entropy-consistent formulation of the K-epsilonmodel, Ph.D. Thesis, Department of Mechanical Engineering, Stanford University, 1995.

[4] J.M. Weiss and W.A. Smith, Preconditioning applied to variable and constant density flows, AIAA J. 33(11) (1995) 2050-2057.[5] S.M.H. Karimian and G.E. Schneider, Pressure-based control-volume finite element method for flow at all speeds, AIAA J. 33(9)

(1995) 1611-1618.

[6] T.E. Tezduyar and T.J.R. Hughes, Development of time-accurate finite element techniques for first-order hyperbolic systems withparticular emphasis on the compressible Euler equations, Report prepared under NASA-Ames University Consortium Interchange, No.NCA2-0R745-104, 1982.

[7] T .E. Tezduyar and T.J .R. Hughes, Finite element formulations for convection dominated flows with particular emphasis on thecompressible Euler equations, in: Proc. AIAA 21st Aerospace Sciences Meeting, AIAA Paper 83-0125, Reno, Nevada, 1983.

[8] G.J. Le Beau, The finite element computation of compressible flows, Master's Thesis, Aerospace Engineering, University ofMinnesota, 1990.

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field on the location of the lateral boundaries, Comput. Methods Appl. Mech. Engrg. 123 (1995) 309-316.

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