Part 31

Technical Notes

Numerical Solution of Boundary ControlProblems for Boussinesq Model of HeatConvection

Gennady Alekseev and Dmitry Tereshko

Much attention has recently been given to statement and investigation of new prob-lems for the models of hydrodynamics and heat convection. The control problemsfor Navier-Stokes and Boussinesq equations are examples ofsuch kind problems(see [1, 2]). This interest to control problems is connectedwith a variety of techni-cal applications in science and engineering such as the crystal growth process, theaerodynamic drag reduction, the suppression of a turbulence and flow separation.

In this work we study a two-parameter boundary control problem for stationarymodel of heat convection. LetΩ be a bounded domain in the spaceRm,m = 2,3 withLipschitz boundaryΓ . We consider the boundary value problem for the stationaryBoussinesq equations

−ν∆u+(u ·∇)u+ ∇p = −β TG, divu = 0 in Ω , u = g onΓ , (1)

−λ ∆T + u ·∇T = f in Ω , T = ψ onΓD, λ ∂T/∂n = χ onΓN (2)

which describes the steady flow of the viscous incompressible heat conducting fluidin the domainΩ . Hereu, p andT denote the velocity, pressure and temperaturefields respectively,ν is the kinematic viscosity coefficient,β is the volumetric ther-mal expansion coefficient,λ is the thermal diffusivity coefficient,ΓN = Γ \ΓD.

The boundary control problems consist in minimization of certain cost functionaldepending on the state(u, p,T ) and controls(g,χ). We assume that the controlsg and χ vary in some closed convex setsK1 ⊂ H1/2(Γ ) and K2 ⊂ L2(ΓN). Themathematical statement of the optimal control problem is asfollows: find a pair(x,u), wherex = (u, p,T ) ∈ X , andu = (g,χ) ∈ K1×K2 = K such that

Gennady AlekseevInstitute of Applied Mathematics FEB RAS, 7, Radio St., Vladivostok, 690041, Russia, e-mail:alekseev@iam.dvo.ru

Dmitry TereshkoInstitute of Applied Mathematics FEB RAS, 7, Radio St., Vladivostok, 690041, Russia e-mail:ter@iam.dvo.ru

1

2 Gennady Alekseev and Dmitry Tereshko

Fig. 1 Uncontrolled flow

Fig. 2 Controlled flow

J(x,g,χ) = I(u,T )+µ1

2‖g‖2

H1/2(Γc)+

µ2

2‖χ‖2

L2(ΓN ) → inf, F(x,u) = 0. (3)

HereF(x,u) = 0 is the operator constraint in the form of the weak formulation of(1), (2);µ1, µ2 are nonnegative constants,I(u,T ) is a cost functional.

For solution of boundary control problems we propose an iterative algorithmbased on Newton’s method (see details in [2]). Following example is connectedwith the vortex reduction in the 2D viscous fluid flow around a cylinder in a channelby means of the heat flux control on some parts of the boundary.The initial uncon-trolled flow is the solution of the nonlinear Navier-Stokes problem (1) withβ = 0and Reynolds number Re=100. The streamlines for this case are shown in Fig. 1.

In order to laminarize a fluid flow, the vorticity functionalI1(u) = ‖curlu‖2L2(Ω)

is minimized using an optimal control approach for the mathematical model (1), (2).In this case we control the heat fluxχ on the cylinder surface and nearest parts ofthe channel. The stream lines for controlled flow are shown inFig. 2. The results ofanother numerical experiments can be found in [2].

References

1. Alekseev, G.V., Tereshko, D.A.: Analysis and Optimization in Hydrodynamics of ViscousFluid. Dalnauka, Vladivostok (2008)

2. Alekseev, G.V., Tereshko, D.A.: Extremum problems of boundary control for steady equa-tions of thermal convection. J. Appl. Mech. Tech. Phys.51, 453–463 (2010)

Blade design effects on the performance of a centrifugal pump impeller John S. Anagnostopoulos School of Mechanical Engineering, National Technical University of Athens, Greece 9 Heroon Polytechniou, 15780 Zografou, Athens, e-mail: anagno@fluid.mech.ntua.gr Abstract. This work performs numerical simulations of the flow in a centrifugal pump impeller in order to investigate the influence of blades parametric design on the impeller performance and efficiency for a wide operation range. The combined use of orthogonal unstructured grids and of a specific cut cell boundary method enhances the results accuracy and provides a fully automated computational environment suitable for numerical design optimization. The results showed the potential of remarkable increase in efficiency by optimizing the impeller design. Numerical Methodology The flow simulation in the impeller (Fig. 1a) is performed by solving the RANS equations with the finite volume method, while turbulence closure is obtained by the standard k-ε model. Key feature of the present approach is the use of unstructured orthogonal grids that can be constructed in a fast and fully automated way. The accuracy of the results at the flow boundaries is enhanced by the use of a conservative cut-cell sharp-interface approach [3], along with an adaptive grid refinement technique [1], applied here at the blades’ region (Fig. 1b).

The impeller is parameterized using a reduced number of geometric variables, while Bezier polynomials are used for the blades’ design (Fig. 1c). The hydraulic efficiency of the impeller equals to the net energy (head) added to the fluid, divided by the energy given at the shaft. The net fluid head is obtained by energy balancing at the inlet and outlet, whereas the calculation of torque developed on the blades is computed from the resulting pressure field. The region and the magnitude of the minimum pressure are also monitored to assess the suction efficiency of the impeller in respect to cavitation.

ω

P1

P2 P3

P4

P5

(a) (b) (c)

β1

β2

Figure 1: Sketch of the centrifugal pump impeller (a); Computational domain and the adaptively refined grid (b); Blade shape parameterization (c).

Application and Results The numerical method is applied to study the flow development in the impeller for various designs and operation conditions (nominal and off-design). The results can provide detailed pictures of the flow, facilitating the analysis of the impeller performance (Fig. 2). The algorithm is then used to conduct various parametric

studies of the effect of blades geometry on the impeller performance characteristics. Blade length and chord angle, number of blades, and solidity factor are among the tested design variables, whereas inlet and outlet blade angles and height are fixed to optimum values from previous studies [2].

(a) (b) (c) (d)

Figure 2. Computed pressure field (a, c) and velocity field (b, d) for: off-design (low

load) conditions (a, b), and near BEF operation (c, d).

Figure 3: Effects of blades length: a) Hydraulic efficiency curves; b) Minimum pressure values in the impeller.

Fig. 3 shows the resulting impeller performance curves for different blade

length. This parameter affects both the maximum efficiency and the BEF point location (Fig. 3a). On the other hand, the minimum pressure data in Fig. 3b represent the cavitation characteristics of the corresponding impellers, which may be considerably different in some operation regions.

From the above and the rest parametric studies it is concluded that a remarkable increase in hydraulic efficiency of the impeller can be achieved by adjusting the value of certain design parameters, hence there is a potentiality to improve further the hydrodynamic performance of centrifugal pumps. The fully automated parameterization and grid construction techniques used in this work are very suitable for such numerical design optimization.

References 1. Anagnostopoulos J (2003) Discretization of transport equations on 2D Cartesian

unstructured grids using higher-order schemes for the convection terms. Int J Num Meth Fluids, 42(3):297-321

2. Anagnostopoulos J (2006) CFD analysis and design effects in a radial pump impeller. WSEAS Trans. on Fluid Mechanics 1(7):763-770

3. Anagnostopoulos J (2007) A Cartesian grid method for the simulation of flows in complex geometries. 3rd Intl. Conference ADMOS 2007, Göteborg, Sweden.

The Local Flux Correction method

Bolat Baimirov

Abstract A new method for localization of oscillations and their suppression isproposed. It is shown that the cause of the oscillations in Lax-Wendroff scheme isan incorrect calculation of the fluxes at cell boundaries. Correcting these errors byusing the exact solution of the Riemann problem can eliminate the oscillations. Asan example 1-D hybrid non-oscillatory second order accuracy scheme is proposedon the basis of Lax-Wendroff and Godunov schemes.

Introduction

Creating difference methods for solving hyperbolic equations is usually faced withtwo major difficulties. The first one deals with the calculation of discontinuous so-lutions. An important role is played here by Godunov-type methods , in which theexact solution of the Riemann problem is used. The second problem is connectedwith the minimization of computational work. In this paper, we propose to use hy-brid schemes which can combine various schemes for simple and complex parts ofthe flow. This simplifies the algorithm and significantly reduces the computationtime.

Great importance for the hybrid scheme is a good selection of switch. In the caseof poor choice, the switch will be switched on either very rarely keeping the oscil-lations, or too often smearing non-oscillatory plots. For example, in schemes suchas TVD, derivatives are totally (on the whole grid) diminished. Perhaps because ofthis more sophisticated versions TVD - ENO, UNO, etc. have been developed. Butthe main drawback of TVD has not disappeared. The total variation diminishing isnot quite correct generalization of the oscillation’s suppression problem. We have toact locally, i.e. suppress errors in specific cell, but not all the derivatives on the grid.

Bolat M. BaimirovKazakh National University, Almaty, e-mail: bolatbay2000@gmail.com

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2 Bolat Baimirov

A total (on the whole grid) approach to solve the Riemann problem is also oneof the main problems of Godunov-type schemes. For example, to reduce the com-putation time P.Roe has proposed a simplified Riemann solver. As a consequence,we have to contend with the results of an approximate calculation.

The Local Flux Correction method

Let’s consider the Euler equation

∂U∂ t

+∂F∂x

= 0,

where U is a vector of parameters , F is a vector of fluxes.On a predictor step by Lax-Wendroff scheme, the flux is averaged not only in

time but also in space. For a smooth flow it is permissible, but averaging flux overspace close to the discontinuity brings up a substantial error.

How can we find out in which cell the flux has been calculated incorrectly? It isa key point in constructing a mechanism of suppressing oscillations. It is supposedempirically, the function of the flux usually behaves monotonically. Grudnitskiyshowed that almost the entire flow can be described using the generalized charac-teristics. The emergence of extremum of the flux function in one cell is unusual andassociated either with incorrect calculations, or with a complex flow in this place(for example, the interaction of two shock waves). In both cases a flux correction isrequired in a given cell.

Thus, the emergence of extreme flow is a good indicator of suspicious cells,in which fluxes should be calculated more accurately. The application of Riemannsolver avoids mathematical reflection on the minima and maxima of functions.

Results of numerical experiment

The basis of numerical experiment is taken from article of G. Sod, in which a com-parative analysis of difference schemes is provided on the example of the Riemannproblem for one-dimensional Euler equations.

At Figure 1 (a density graph) Lax-Wendroff scheme and a new hybrid schemeLWG (Lax-Wendroff-Godunov) have been compared. The rectangles show the po-sition of the flux correction with the Riemann solver for the scheme LWG, in theremaining cells Lax-Wendroff scheme is used. New hybrid scheme uses a simpleswitch, which is based on finding new extrema of the fluxes, i.e. seeks the emer-gence of oscillations. It is shown that the Godunov scheme is rarely used (in twocells, at the shock wave and contact discontinuity) and not only it does not spoil theorder of approximation, but also corrects the errors of the Lax-Wendroff scheme.

The Local Flux Correction method 3

Fig. 1 Comparison of thescheme of Lax-Wendroff withLWG scheme (density). Therectangles show the positionof the flux correction (in twocells, at the shock wave andcontact discontinuity)

The speed of calculation for the scheme LWG is not too different from the Lax-Wendroff scheme and depends on the frequency of implementation of costly Rie-mann solver. Since the switch is triggered quite rarely, the scheme LWG is slowerthan Lax-Wendroff scheme only for a few percent. This allows to call a hybridscheme LWG the fastest non-oscillatory second order schemes. Separately, the Lax-Wendroff and Godunov cannot bring to such a result as a hybrid scheme LWG.

Conclusions

• The Local Flux Correction method and a new switch for hybrid schemes havebeen presented. Analysis of fluxes on the formation of extrema allows to trackcritical grid cells in which oscillations are generated.

• This method allows to create simple hybrid non-oscillatory scheme of high order• As an example, a hybrid non-oscillatory scheme LWG of second-order approxi-

mation is developed.

References

Grudnitskiy V.G., ”Sufficient conditions of stability for discontinuous solutions of the Euler equa-tions”, Computational Fluid Dynamics Journal, 2001, v.10, No.2, pp.334-337.Godunov S.K., Zabrodin A.V., Ivanov M.Ya., Kraiko A.N., and Prokopov G.P., Numerical Solutionof Multidimensional Problems of Gas Dynamics Nauka, Moscow, 1976 [in Russian].Sod G., ”A Survey of Several Finite Difference Methods for Systems of Nonlinear HyperbolicConservation Laws”, Journal of Computational Physics, Vol.27, 1978, pp 1-31

Adaptive Meshes to Improve the Linear StabilityAnalysis of the Flow Past a Circular Cylinder

Iago C. Barbeiro, Julio R. Meneghini & J. A. P. Aranha

Abstract Adaptive mesh schemes have been employed to simplify lots of numericalproblems, from hard to capture multi-scale to complex transient phenomena. Thisstudy integrates a mesh adaptation scheme to the linear stability analysis of the flowpast a circular cylinder. Preliminary results for Re = 100 are presented to illustratethe benefits.

This work comes to emphasize the benefits of employing an adaptive mesh schemeto solve the problems related to the linear stability analysis of a stationary flow.

The viscous flow past a circular cylinder has been chosen for a case study andsome preliminary results are presented. It is one of the most celebrated prototypesfor bluff body flows and presents a rich sequence of transitions growing in complex-ity with the Reynolds number (Re). The first transition occurs around Re = 47, whenthe solution becomes transient and periodic. This transition is a Hopf bifurcation,where a pair of complex conjugated eigenvalues of the linearized perturbation equa-tion crosses the imaginary axis to the positive unstable half plan of the spectrum.

The discrete incompressible Navier-Stokes equation is obtained by the Finite El-ement Method and the computational code is built based on Getfem++, an efficientC++ library for Finite Element Methods, see Pommier & Renard. The spatial do-main is split in a linear triangular mesh and the shape functions are of Taylor-Hoodtype, quadratic for velocity and linear for pressure.

All the meshes are generated using the computational code BAMG, from Hecht(1998). It is a free research code capable of building unstructured meshes after agiven map of elements sizes and orientation. It has also the ability of calculatingthe desirable map of sizes and orientation of the elements using a previous solutionassociated to a previous mesh, allowing its use as part of an adaptive loop.

NDF, Escola Politecnica, University of Sao Paulo, Brazil e-mail: iago.barbeiro@gmail.com

1

2 Iago C. Barbeiro, Julio R. Meneghini & J. A. P. Aranha

Another important feature of BAMG is the possibility to calculate the intersec-tion of many metrics, see Frey & Alauzet (2005). The intersection metric minimizesthe maximum interpolation error of a set of metrics.

The mesh adaptation loop alternates between the stationary solver and BAMG,until the the level of interpolation error or the maximum number of elements isachieved (this adaptation loop is called A).

An Arnoldi method is employed and however the pair of eigenvalues is obtainedwith reasonable precision, the respective eigenvectors present poor spatial resolu-tion, as it is illustrated in figure 1(c).

To overcome this issue the alternative is to include the eigensolver in the meshadaptation loop (this modified adaptation loop is called B). Figure brings the refine-ment detail of the meshes delivered by loop A and loop B. The streamwise stretchedelements of figure 1(a) are good to represent the stationary wake, which is almostparallel after a certain distance from the cylinder, but not to accommodate the un-stable eigenvector, that is not parallel at all. The mesh on figure 1(d) has a finerstreamwise refinement and is better to represent the eigenmode wavelength.

(a) Mesh (916 nodes) - A . (b) Stationary flow - A . (c) Eigenmode - A

(d) Mesh (2432 nodes) - B. (e) Stationary flow - B . (f) Eigenmode - B

Fig. 1 (a), (b) and (c): results obtained with loop A. (d), (e) and (f): results obtained with loop B

Acknowledgements The authors are grateful to FINEP-CTPetro, FAPESP and Petrobras (TheBrazilian Oil Company) for providing the research grant for this project.

References

1. Frey, P. J. & Alauzet, F. 2005 Anisotropic adaptation for CFD computations. Comput. Meth-ods Appl. Mech. Engrg., 194:5068-5082

2. Hecht, F. 1998 BAMG: Bidimensional Anisotropic Mesh Generator.http://www.inria.fr/valorisation/logiciels.

3. Huang, W. 2005 Metric tensors for anisotropic mesh generation. J. Comput. Phys., 204:633-665

4. Pommier, J. & Renard, Y. Getfem++ http://home.gna.org/getfem.

Migration of species into a particle underdifferent flow conditions

Carella, Alfredo R. and Dorao, Carlos A.

Abstract Fractional derivatives constitute a wider generalization of Fickian modelsthat allows describing more complex behaviors of diffusing species. In this work,the resolution of an anomalous transport model based on fractional derivatives isdiscussed. Spectral Least Squares Finite Element Method is used along with Gauss-Jacobi quadrature in order to improve the numerical convergence.

1 Fractional diffusion model

Different formulations of Fick’s law are the most common approach for model-ing diffusion processes. Nevertheless, a growing collection of experimental resultsshows that diffusion often proceeds faster or slower than Fickian models predict.The models based on fractional derivatives constitute a wider generalization of Fick-ian models that allows describing more complex behaviors of diffusing species. As aparticular case, these models allow a continuous variation of a constituent law fromadvection to dispersion [3].

The evolution of the concentration is described by the fractional derivative model

∂C∂ t

= k(

ν b1Dαx (C) +(1−ν) b2 Dα

x (C))

(1)

Alfredo R. CarellaNorwegian University of Science and Technology, N-7491 Trondheim, Norway, e-mail: al-fredo.r.carella@ntnu.no

Carlos A. DoraoNorwegian University of Science and Technology, N-7491 Trondheim, Norway, e-mail: car-los.dorao@ntnu.no

1

2 Carella, Alfredo R. and Dorao, Carlos A.

where C is the species concentration, k is the diffusion coefficient, b1 and b2 are theleft and right boundaries of the problem domain [0,1], θ is the asymmetry constantruled by 0 ≤ θ ≤ 1 and Dα is the fractional derivative operator defined as

bDαx (C)≡ 1

Γ (n−α)

dn

dxn

∫ x

b

C (x′)

(x− x′)α−n+1 dx′ n−1 < α ≤ n (2)

The definition of the fractional derivative produces a singularity in the limit x′ → x,which causes the integral operand to become infinity. In order to improve the conver-gence, [1] proposed to construct Jacobi polynomials incorporating the singularityinto the weight function for axisymmetric problems. The solution is calculated us-ing a least–squares approach. The discretization statement consists in searching thesolution in a reduced subspace similar to the Galerkin method.

2 Numerical results and discussion

Figures 1(a), 1(b) and 1(c) show the time evolution of the concentration through-out the 1D domain for two different values of the derivative exponent α . As thederivative exponent increases the behavior goes from convective to diffusive.

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

αLR

= 1.1

t = 0.0t = 0.4t = 0.8t = 1.2t = 1.6t = 2.0t = 2.4

(a) fig1a0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

αLR

= 1.5

t = 0.0t = 0.4t = 0.8t = 1.2t = 1.6t = 2.0t = 2.4

(b) fig1b0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

αLR

= 1.9

t = 0.0t = 0.4t = 0.8t = 1.2t = 1.6t = 2.0t = 2.4

(c) fig1c

Fig. 1 Convection (U = 0.17) and fractional diffusion: (a):α = 1.1; (b): α = 1.5; (c): α = 1.9.

References

1. Fernandino, M., Dorao, C.A. & Jakobsen, H.A.: Jacobi Galerkin spectral method for cylin-drical and spherical geometries. Chemical Engineering Science. 62 (23), 6777–6783 (2007)

2. Jiang, B.: The Least-Square Finite Element Method: Theory and Applications in Computa-tional Fluid Dynamics and Electromagnetics. Springer. (1998)

3. El-Sayed A.M.A., Behiry S.H., Raslan W.E.: A numerical solution of an intermediate frac-tional advection dispersion equation. Commun Nonlinear Sci Numer Simulat , Vol. 15, pp.1253-1258 (2010)

On the role of numerical dissipation in unsteadylow Mach number flow computations

Yann Moguen, Tarik Kousksou, Erik Dick and Pascal Bruel

Abstract Numerical dissipation is necessary in low Mach number flow compu-tations on collocated grids to avoid checkerboard decoupling, which correspondsto non-physical pressure oscillations. A commonly used technique to cancel outcheckerboard decoupling consists in introducing a relation between the mass fluxand the pressure difference on cell faces, which involves a dissipation coefficientscaling as 1/M2, where M denotes a representative Mach number in the flow. Wedemonstrate that a typical algorithm with such an explicit pressure-velocity cou-pling results in unphysical build-up of kinetic energy as a consequence of excitedunsteadiness at boundaries. This happens despite the correct Mach number scalingof the dissipation coefficient and despite the correct steady flow behaviour. We fur-ther demonstrate that the stabilisation of the checkerboard decoupling by the Rhie-Chow interpolation does not have this deficiency. We give an explanation for theerroneous behaviour and for the correct behaviour of the coupling methods.

Yann MoguenUniversite de Pau et des Pays de l’Adour - Laboratoire de Sciences Appliquees au Genie Civilet Cotier, ISA-BTP avenue du Parc Montaury, 64 600 Anglet, France, e-mail:yann.moguen@free.fr

Tarik KousksouUniversite de Pau et des Pays de l’Adour - Laboratoire de Thermique,Energetique et Procedes,ENSGTI, rue Jules Ferry - 64 075 Pau, France, e-mail:tarik.kousksou@univ-pau.fr

Erik DickGhent University, Department of Flow, Heat and Combustion Mechanics, Sint-Pietersnieuwstraat,41 - 9 000 Gent, Belgium, e-mail:erik.dick@ugent.be

Pascal BruelCNRS and Universite de Pau et des Pays de l’Adour - Laboratoire de Mathematiques et de leursApplications, UMR 5142 CNRS-UPPA, avenue de l’Universite, BP 1155 - 64 013 Pau, France,e-mail:pascal.bruel@univ-pau.fr

1

2 Yann Moguen, Tarik Kousksou, Erik Dick and Pascal Bruel

1 Introduction

This study aims at investigating the physical foundation ofthe pressure-velocitycoupling that is commonly introduced through numerical dissipation in low Machnumber flow computations.

It is recognized that some numerical dissipation is necessary in low Mach num-ber flow computations on collocated grids. This allows to avoid checkerboard de-coupling, which corresponds to non-physical pressure oscillations.

The checkerboard decoupling is an undesirable phenomenon that can be ex-plained by considering, in the inviscid case, a convective time and space scalesemi-discrete asymptotic analysis. According to this approach, the variables aresubstituted into appropriately non-dimensionalized Euler equations, as expansionsin powers of a repesentative Mach number M. From the momentumequation, itappears that the checkerboard decoupling concerns zeroth and first order pressurefields, which can be interpreted as thermodynamic and acoustic pressures.

Based on this observation, a natural attempt to cancel out checkerboard decou-pling consists in introducing a relation between the mass flux and the pressuredifference on cell faces. According to the asymptotic approach previously men-tionned, this relation involves a dissipation coefficient scaling as 1/M2. Thus, thenon-physical pressure oscillations are removed.

This highly effective approach is commonly used (see for example Refs. [1, 2,3]). However, one of its obvious drawbacks is the involvement of flow dependentdissipation coefficients that require calibration. As the pressure-mass flux couplingleads to the addition to the zeroth-order conservation equations of diffusion termsinvolving these dissipation coefficients, any error on the dissipation coefficients im-plies directly consistency errors of the conservation equations. Moreover, the phys-ical foundation of the pressure-mass flux coupling remains questionable.

In the present study, we compare the approach previously described to the so-called Rhie-Chow interpolation method [4].

2 Governing equations and SIMPLE-type algorithm

We consider the simple case of a one-dimensional flow of a perfect and ideal gas ina nozzle with a variable section. From now on,x denotes the coordinate in the flowdirection. The flow model is given by the Euler equations,

∂t(ρS)+ ∂x(ρvS) = 0 (1a)

∂t(ρvS)+ ∂x((ρv2 + p)S) = pdxS (1b)

∂t(ρES)+ ∂x(ρvHS) = 0 (1c)

E = e +12

v2 , ρH = ρE + p , ρe =p

γ −1(1d)

On the role of numerical dissipation in unsteady low Mach number flow computations 3

wheret, ρ , p, v, e, E and H represent time, density, pressure, velocity, internalenergy, total energy and total enthalpy per unit mass, respectively. Furthermore,γdenotes the specific heats ratio andS the cross-section area of the nozzle.

To solve the set of equations (1) in the collocated finite volume formulation, theenergy based SIMPLE-type algorithms that we shall considertakes the following“guess-and-correct” form (seee.g. Ref. [3]), where the superscripts⋆ and′ denotepredicted and correction quantities, respectively:

1. Prediction step (first iteration:k = n): p⋆i = pk

i ; ρ⋆i and (ρv)⋆i are calculated

usingρ⋆i = ρn

i − δSi

(ρ⋆i vk

i+1/2Si+1/2− ρ⋆i−1vk

i−1/2Si−1/2) whereδ = ∆ t/∆x. Formomentum, the calculation differs for the two algorithms considered. It will bedetailed in the following. Once this calculation is achieved, the predicted total

energy is(ρE)⋆i =pk

iγ−1 + 1

2(ρv)⋆i(ρv)⋆i

ρ⋆i

and the face quantities, such as the mass

flux (ρv)⋆i+1/2, are calculated. The face pressure is centrally interpolated.

2. Correction step: The pressure correctionp′ is computed by solving the en-ergy equation using an Euler-implicit discretization. Forcorrection, the totalenergy is expanded as(ρE)k+1

i = (ρE)⋆i + (∂p(ρe))⋆i p′i. In the case of a per-fect gas,(∂p(ρe))⋆i = 1

γ−1. The correction for momentum is (simplified expres-sion obtained from the Euler-implicit discretization of the momentum equa-tion): (ρv)′i+1/2 = −δ (p′i+1 − p′i). Flux terms are expanded as (approximate

expansion):(ρvH)k+1i+1/2 = (ρH)⋆i v⋆

i+1/2 + H⋆i (ρv)′i+1/2 + (ρH)′i+1/2v⋆

i+1/2, with

(ρH)′i+1/2 = (ρE)′i+1/2+ p′i+1/2 =p′i+1/2γ−1 + p′i+1/2 = γ

γ−1p′i+p′i+1

2 . AsρH is a con-vected quantity,(ρH)⋆i+1/2 is upwinded as(ρH)⋆i . Finally, the energy equationreads:

(ρE)⋆i +1

γ −1p′i − (ρE)n

i

+δSi

[(ρH)⋆i v⋆i+1/2Si+1/2− (ρH)⋆i−1v⋆

i−1/2Si−1/2]

+δSi[

γγ −1

p′i + p′i+1

2v⋆

i+1/2− δH⋆i (p′i+1− p′i)]Si+1/2

− [γ

γ −1

p′i−1 + p′i2

v⋆i−1/2− δH⋆

i−1(p′i − p′i−1)]Si−1/2 = 0 (2)

3. Updates:pk+1i = p⋆

i + p′i, ρk+1i = ρ⋆

i , (ρv)k+1i = (ρv)⋆i +(ρv)′i, (ρE)k+1

i =pk+1

iγ−1 +

12(ρv)k+1

i(ρv)k+1

i

ρk+1i

. The face quantities are finally updated, again with a central

interpolation for the pressure.

4 Yann Moguen, Tarik Kousksou, Erik Dick and Pascal Bruel

3 A generic explicit pressure-mass flux coupling

A particular form of the preceding algorithm is now considered, such as to involvean explicitly given level of numerical dissipation. We utilize it as a generic – ormodel – algorithm for the pressure-mass flux coupling involving a dissipation coef-ficient which scales as 1/M2.

In this algorithm, the predicted momentum is given by

(ρv)⋆i = (ρv)ni −

δSi

((ρv)⋆i vk

i+1/2Si+1/2− (ρv)⋆i−1vki−1/2Si−1/2

)− δ (pk

i+1/2− pki−1/2) (3)

and the pressure-mass flux coupling is written as

(ρv)⋆i+1/2 =(ρv)⋆i +(ρv)⋆i+1

2−D(pk

i+1− pki ) (4)

where the dissipation coefficientD is defined such as to scale as the reciprocal ofthe Mach number square. This can be achieved by an appropriate nondimensionali-sation of the variables [3]. Practically,D is of the formα/vref, whereα is a positiveconstant andvref a reference velocity. In the test hereafter,α = 0.5 andvref is theinlet velocity.

4 Rhie-Chow based pressure-mass flux coupling

A second pressure correction algorithm is also considered,involving a pressure-mass flux coupling through the Rhie-Chow interpolation method,

(ρv)⋆⋆i = (ρv)n

i −δSi

((ρv)⋆⋆

i vki+1/2Si+1/2− (ρv)⋆⋆

i−1vki−1/2Si−1/2

)(5)

and

(ρv)⋆i+1/2 =(ρv)⋆⋆

i +(ρv)⋆⋆i+1

2− δ (pk

i+1− pki ) (6)

Let us note that, in contrast to the first algorithm, this one is free of flow dependentdissipation coefficients, and so no calibration is required.

5 Numerical experiments: pressure outflow perturbation

The comparison between the two approaches is carried out through numerical ex-periments on flows in a converging-diverging nozzle (cf. [3]) where unsteadiness isachieved through a time-varying outflow pressure, written as

On the role of numerical dissipation in unsteady low Mach number flow computations 5

pout(t) = p0out(1+ Asin(2π f t))

For the test-case considered in this paper, the initial density and pressure are1.2046 kg/m3 and 101 300 Pa, respectively, and the initial velocity is zero. At theinlet, the density and velocity are imposed, and the pressure is extrapolated. At theoutlet, the pressure is imposed, and the density and velocity are extrapolated. SeeTable 1 for more details. Note that these boundary conditions are absorbing, for theacoustic and the entropic waves.

(a) f = 0 Hz. (b) Kinetic energy distributionfrom x = 0 tox = 0.9 m.

(c) f = 104 Hz. (d) Kinetic energy distributionfrom x = 0.9 m tox = 1 m.

Fig. 1: Left: evolution of kinetic energy in the nozzle. Right: distribution of kinetic energy alongthe nozzle after the last time step,f = 104 Hz.

Figure 1, left, shows the evolution in time of the kinetic energy in the nozzle.For f = 0, a steady flow is obtained and the final kinetic energy by bothmethodsis the same and is of order 10−5 J, as it should be. For largef (104 Hz), the kineticenergy by the Rhie-Chow method remains at low level, of order10−5 J. With the

6 Yann Moguen, Tarik Kousksou, Erik Dick and Pascal Bruel

generic algorithm, an explosion is observed after a large number of time steps. Theunphysical growth of the kinetic energy happens in the outlet zone of the nozzle, asis obvious from the distribution along the nozzle, shown in figure 1, right. Appar-ently, the imposed pressure oscillation at the outlet generates an erroneous mass fluxdue to the fixed dissipation coefficientD in (4). This does not happen with the Rhie-Chow method, where the coefficient of the pressure difference in (6) goes to zeroas the time step goes to zero. Remark that with the generic method, the upstreampropagation of the kinetic energy is completely unphysical.

Table 1: Physical and numerical settings for the test-case

Nozzle length (m) ρin (kg/m3) vin (m/s) Mthroat p0out (Pa) A CFLv

1 1.2046 3.1×10−3 10−5 101 300 10−6 10−3

6 Conclusion

We demonstrated that a typical algorithm with explicit pressure-velocity couplingresults in unphysical build-up of kinetic energy as a consequence of excited un-steadiness at boundaries. This happens despite the correctMach number scaling ofthe dissipation coefficient and despite the correct steady flow behaviour. We fur-ther demonstrated that the stabilisation of the checkerboard decoupling by the Rhie-Chow interpolation does not have this deficiency.

References

1. Edwards, J. R. and Liou, M.-S.: Low-Diffusion Flux-Splitting Methods for Flows at AllSpeeds. AIAA J.36(9), 1610–1617 (1998)

2. Liou, M.-S.: A Sequel to AUSM, part II: AUSM+-up for all speeds. J. Comp. Phys.214,137–170 (2006)

3. Nerinckx, K., Vierendeels, J. and Dick, E.: Mach-uniformity through the coupled pressureand temperature correction algorithm. J. Comp. Phys.206, 597-623 (2005)

4. Rhie, C. M. and Chow, W. L.: Numerical Study of the Turbulent Flow Past an Airfoil withTrailing Edge Separation. AIAA J.21(11), 1525-1532 (1983)

Unsteady Flow around Two Tandem Cylinders Using Advanced Turbulence Modeling Method Jian LIU, Zhixiang XIAO, Song FU

Introduction: Advanced turbulence modeling method, called as DDES based on k- ω-SST turbulence model, is chosen to predict the flow past tandem cylinders (TCs) in streamwise direction with a critical spacing (L/D=3.70)[ -3]1 . Effect of scheme dissipation has been investigated through predicting the DIT (Decay of Isotropic Turbulence) and the time-averaged and instantaneous flow fields around the TCs. To obtain more accurately unsteady characteristics, the numerical dissipation should be effectively decreased in the regions where the flow should be almost fully dominated by the turbulence.

DIT is always used to validate the unsteady turbulence modeling methods. The numerical dissipation of our in-house code is tested and CDES of DDES[4] is re-determined through predicting DIT. In DDES, CDES of k-ε and k-ω branches are calibrated, respectively. The final definition of CDES is written as equation（1）, where F1 is a blending function of SST model.

( )1 k1 1

kDES DES DESC F C FCε ω− −= − + （1）

(a) 4th order central scheme (b) different schemes with new CDES Figure 1: Calibration of CDES

Figure 1 present the new CDES in our code. We find that smaller CDES of both k-ω and k-ε branches are achieved, which are 0.4 and 0.3.

At the same time, we compare the numerical dissipation of different numerical scheme, such as Roe with MUSCL, 6th order symmetric plus 5th order WENO with 3% numerical dissipation of the original scheme and 4th central scheme plus 0.0001 of 4th artificial viscosity. From the results, Roe with MUSCL can not

Jian LIU: liujian08@mails.tsinghua.edu.cnZhixiang XIAO: xiaotigerzhx@tsinghua.edu.cnSong FU: fs-dem@tsinghua.edu.cnSchool of Aerospace Engineering, Tsinghua University, 100084, Beijing, P.R. China

predict most of the energy of turbulence, while other two low-dissipation scheme can capture the DIT very well. It indicates that lower dissipation can predict more accurate turbulence results.

TCs: In figure 2, some spanwise vorticities by numerical simulation are compared, where the turbulence model is DDES with Roe plus MUSCL and 6th symmetric plus only 12% of dissipation of 5th WENO. From these figures, we can effectively distinguish between the instantaneous Ωz predicted by 12% of dissipation of S6WENO5 and by Roe with MUSCL. The former can give more reasonable turbulence structures which display as small scale of vortex. The later can only present very large scale of structures and small structures are almost smoothed completely.

From figure 3, we also observe very strong Karman vortex shedding which are presented using Q criterion obtained by Roe with MUSCL. The similar result is that more detailed and more spanwise and streamwise small scale of structures are captured through high order scheme. In fact, the time-averaged results are not available at this time because the test case is on-going.

(a)Roe with MUSCL (b) modified STVD (c) Experiment data

Figure 2: Influence of different schemes on spanwise vorticity

Figure 3: Influence of different schemes on Q around TCs

Acknowledgements This project is supported by EU-FP7-ATAAC project. The author also thanks

Prof. M. Strelets for his useful suggestion about the numerical scheme. Reference [1]. Luther N. Jenkins et al., 2005, Characterization of Unsteady Flow Structures

around Tandem Cylinders for Component Interaction Studies in Airframe Noise. AIAA paper 2005-2812, 2005.

[2]. Luther N. Jenkins et al., 2006, Measurements of Unsteady Wake Interference between Tandem Cylinders. AIAA paer 2006-3202, 2006.

[3]. Dan H. Neuhart et al. Measurements of the Flowfield Interaction between Tandem Cylinders. AIAA paper 2009-3275, 2009.

[4]. Xiao Z.X et al. Study of delayed-detached eddy simulation with weakly nonlinear turbulence model. J. of Aircraft. Vol. 43(5), 1377-1385, 2006.

High Order Versions of the Collocations andLeast Squares Method for Navier-StokesEquations

Vadim Isaev and Vasily Shapeev

Abstract We propose new high order versions of the collocations and least squares(CLS) method for 2D stationary Navier-Stokes equations. In these versions, com-ponents of the solution are sought in the form of piecewise-polynomial functions.Computational domain is covered by a grid with rectangular cells. Velocity com-ponents are approximated by polynomials of degree mv at most in both spatial di-rections in each cell of the grid. Polynomials for velocity v are chosen so that theapproximate solution exactly satisfies continuity equation div v = 0 inside each cell.Pressure is approximated by polynomials of degree mp at most respectively. Newversions of the CLS method were implemented on regular and nonregular grids withrectangular cells here. The method was verified on a problem with analytical exactsolution and benchmark lid-driven cavity flow problem. Computational results forthe latter one coincide with highly accurate solutions obtained by other researchers(Barragy E. et al., Botella O. et al., Erturk E. et al., Shapeev A. et al., etc.) with theaccuracy 10−7 ∼ 10−8 for Re = 1000.

We consider a boundary value problem for 2D stationary Navier—Stokes (N-S)equations in rectangular domain Ω = [0,L1]× [0,L2]

v1∂v j∂x1

+ v2∂v j∂x2

+ ∂ p∂x j

− 1Re ∆v j = f j, div v = 0, v|∂Ω = V,

∫Ω

pdΩ = 0, (1)

where v j are velocity components, j = 1, 2, p is the pressure, Re is Reynolds number.Here, domain Ω is covered by a grid with rectangular cells Ω1, . . . , ΩN , where Nis the grid size. Approximate solution of problem (1) is constructed with the use ofiterations by subdomains (domain decomposition method). At an iteration by cell Ωi

Vadim IsaevNovosibirsk State University, Pirogova str. 2, Novosibirsk, 630090, Russia,e-mail: issaev.vadim@gmail.com

Vasily ShapeevKhristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of RussianAcademy of Sciences, Institutskaya str. 4/1, Novosibirsk, 630090, Russia,e-mail: vshapeev@ngs.ru

1

2 Vadim Isaev and Vasily Shapeev

Table 1 Intensity ψmin and Position of the Center (x1,x2) of the Primary Eddy, Re = 1000

ψmin x1 x2Barragy E., Carey G. F. −0.118930 — —Erturk E. et al. −0.118938 0.5300 0.5650Botella O., Peyret R. −0.1189366 0.5308 0.5652Shapeev A., Lin P. −0.1189366 0.5307901 0.5652406Present Work −0.11893658 0.53079011 0.56524057

we consider an auxiliary problem for N-S equations with the following boundaryconditions and condition for pressure(

vn+hi∂vn∂n − p

)∣∣∂Ωi\∂Ω = vn+hi

∂ vn∂n − p,

(vτ +hi

∂vτ∂n

)∣∣∂Ωi\∂Ω = vτ +hi

∂ vτ∂n , (2)

v|∂Ωi∩∂Ω = V,∫

Ωi

pdΩ =−∫

Ω\Ωi

pdΩ . (3)

Here, vn, vτ , p are values of the solution for neighbour cells found at previous iter-ations by subdomains, n, τ are unit normal and tangent vectors for boundary ∂Ωi,vn = v ·n, vτ = v · τ , hi = (h1,i h2,i)

1/2, 2 h1,i and 2 h2,i are width and height ofcell Ωi. Approximate solution of the auxiliary problem is sought as a linear combi-nation of polynomial basis functions. For velocity we use solenoidal basis elementsof degree mv at most. Pressure is approximated by polynomials of degree mp atmost. For each cell we have (mv +1)(mv +4)/2 and (mp +1)(mp +2)/2 unknowncoefficients for velocity and pressure respectively. They are found at each iterationby subdomain from an overdetermined system. It consists of collocations equa-tions — requirements that the approximate solution must satisfy N-S equations atcertain points of the cell, matching (interface) conditions — requirements thatequations (2) are satisfied at certain points of sides between adjacent cells, bound-ary conditions — boundary conditions from (1) written at certain points for sideslying on the boundary ∂Ω and condition for pressure. Here, we distributed collo-cation, matching and boundary points uniformly in each cell. The overdeterminedsystem in each cell is solved by means of the least squares method. The latter wasimplemented with the use of orthogonal transformations. They do not worsen theconditionality of the system during the process of solving.

We carried out a series of numerical experiments on a sequence of grids. It isshown that, if mp = mv − 1 and exact solution of the problem is smooth enough,then the error of the approximate solution is not worse than O(hmv) for even mv andO(hmv−1) for odd mv, where h is the maximal linear size of grid cells.

Lid-driven cavity flow is a well-known benchmark problem for numerical meth-ods for N-S equations. Its solution has singularities at two upper corners. Here, wesubtracted the singular part of the solution as it was done in the paper of O. Botellaand R. Peyret. Computations were carried out on nonregular grids with a local re-finement near cavity corners. The results obtained with the use of the CLS methodare given in Table 1. It is shown that the approximate solution of the cavity flowproblem converges to the exact one with a high order, when h → 0.

The work was supported by the RFBR (08-08-00249, 10-01-00575); SB RASint. project 26.

Large-Eddy Simulation of the flow over a ThinAirfoil at Low Reynolds Number

Ryoji Kojima, Taku Nonomura, Akira Oyama and Kozo Fujii

Abstract The performance of airfoil NACA0002 at Reynolds number of 2.3×104

is investigated with large-eddy simulation (LES). The angle of attack is 3, 6, or9 degree. The behavior of a laminar separation bubble which appears over a thinairfoil and its effects on aerodynamic characteristics are mainly discussed.

1 IntroductionThe experimental results by E. V. Laitone at Re = 2.07× 104 show the possibilitythat the larger camber, smaller thickness, or sharper leading edge of an airfoil, thebetter performance it has at low Re than an ordinary airfoil [1]. However, the causehas not been clarified yet and it was assumed that a laminar separation bubble exists.In this study, we focus on the flow characteristics of a thin airfoil, and discuss thebehavior of a laminar-separation bubble resulting in its superior aerodynamic char-acteristics.

2 Numerical settings and methodsMach number is 0.2 and Reynolds number is 2.3× 104. NACA0002 is adopted asthe representative of a thin and symmetric airfoil, for simplifying the problem. An-gle of attack α is 3, 6 or 9 degrees.Three-dimensional unsteady compressible Navier-Stokes equations are solved. Theconvection and viscous terms are evaluated by sixth-order compact scheme andtenth-order tridiagonal filter at αcoef. = 0.495, respectively. ADI-SGS is used for timeintegration, while triple inner iterations maintain second-order in time.The grid is C-type, and the number of grid points is approximately 12 million.

Ryoji KojimaDepartment of Aeronautics and Astronautics, University of Tokyo, Sagamihara, Kanagawa, 252-5210, Japan., e-mail: kojima@flab.isas.jaxa.jp

Taku Nonomura, Akira Oyama and Kozo FujiiInstitute of Space and Astronautical Science, JAXA, Sagamihara, 252-5210, Japan.

1

2 Ryoji Kojima, Taku Nonomura, Akira Oyama and Kozo Fujii

3 ResultsIt is found in Fig.1 that the flow separates from the leading edge at each α includingas low as 3 degrees due to the sharp leading edge of NACA0002 airfoil. The sharpleading edge keeps separation point at the leading edge at all the α .In the separated shear layer, two-dimensional vortices are shed and the flow tran-sits to generate three-dimensional structured or complex vortices. Transition causesthe separated flow to reattach at α = 3 and 6 degrees and, as the result a laminar-separation bubble is formed as shown in Fig.2. A recirculation zone is observed onthe upper surface at each α in Fig.2, and the separated shear layer at α = 3, 6 degreesreattaches to upper surface to form a laminar-separation bubble. At α = 9 degrees,the leading edge stall occurs. Moreover, as α increases, the size of a laminar sepa-ration bubble increases. This bubble is classified into ”long bubble.”From the corresponding airfoil part in Fig.2 and Fig.3 (a), it is noticeable that alaminar-separation bubble enhances the negative pressure. Despite the existence ofa laminar separation bubble, the dependence of lift on α , shown in Fig.3 (b), islinear.

Fig. 1 Instantaneous vortical structure colored by chord-direction vorticity at each angle of attack

Fig. 2 Span and time averaged chord-direction velocity and stream lines at each angle of attack

reattachment

separation 3D LES2D RANS3D LES2D RANS

(a) (b)Fig. 3 (a) Cp distribution and (b) CL,CD −α dependence of span and time averaged flow

References

1. Laitone, E. V. : Aerodynamic Lift at Reynolds Numbers Below 7×104. Berkeley, California(1996).

Adverse Free-Stream Conditions for TransonicAirfoils with Concave Arcs

Alexander Kuzmin

Abstract Turbulent transonic flow past airfoils with concave arcs is analyzed. Thestudy is focused on adverse free-stream conditions, in which the lift coefficient ex-hibits jumps under slight variations of the free-stream velocity or angle of attack. Fora Drela Apex 16 airfoil, the instability occurs at negative angles of attack α < 0.For a Whitcomb airfoil with a bump on the upper surface, the instability is obtainedat 0 < α < 1 deg.

Numerical simulations based on the Euler and RANS equations revealed tran-sonic flow bifurcations and instability of shock waves on airfoils whose middle partsare either flat or convex and nearly flat [1, 2, 3]. It was shown that the bifurcationsand instability are caused by interaction of local supersonic regions. As to airfoilswith concave arcs, they received little attention in the literature [2, 4, 5].

In this work, we examine 2D turbulent transonic flow past the Drela Apex 16 air-foil, whose lower surface is concave near the leading edge, and the Whitcomb airfoilwith a thin bump placed on the upper surface. In both cases, a numerical analysisdemonstrates the existence of adverse free-stream conditions, which provoke theflow instability and abrupt changes of the lift coefficient. Solutions of the RANSequations are obtained with a finite volume solver of the second-order accuracy [6].Initial data are either a uniform state or a flow field obtained previously for othervalues of the free-stream Mach number M∞ and the angle of attack α . We employa standard k−ω SST turbulence model, which predicts the boundary layer sep-aration quite reasonably. Computations were performed on unstructured meshes ofabout 2×105 grid points, which were clustered in vicinities of the shock waves, inthe wake, and in the boundary layers to meet the condition y+ < 1.

For transonic flow over the Drela Apex 16 airfoil [7], numerical solutions showedthe existence of double supersonic regions on the lower surface at negative angles ofattack (see Fig. 1). The calculated lift coefficient CL versus the angle α is presented

Alexander KuzminSt Petersburg State University, 28 University Ave., St. Petersburg, 198504 Russia, e-mail: alexan-der.kuzmin@pobox.spbu.ru

1

2 Alexander Kuzmin

in Fig. 2. Abrupt changes of CL in the interval −4.6<α ,deg<−3.8 are explainedby the instability of shock wave locations, which is caused by a coalescence/ruptureof the local supersonic regions.

A second example addresses transonic flow past the Whitcomb airfoil modifiedby a smooth bump, which is placed on the upper surface. The modified portion ofthe surface is given by the expression

y(x) = (1+(2x−1)2 − (2x−1)4) · ywhitcomb, 0.5 < x < 0.9962,

where ywhitcomb refers to the original Whitcomb airfoil. The bump yields a slightconcavity of the profile in the midchord region.

Figure 3 shows a flow field obtained for initial data determined by the uniformfree stream at M∞ = 0.833, α = 0.4 deg , Re=5× 106. As seen, there is a doublesupersonic region on the upper surface of the airfoil. For the same free-stream con-ditions, there exists another steady flow, which exhibits a single supersonic regionon the upper surface. The latter flow can be obtained by solving the problem forinitial conditions determined by the uniform free stream at M∞ = 0.833,α = 1 degand then gradually reducing the angle α to 0.4 deg.

Figure 4 illustrates the obtained dependence of CL on the angle α and the free-stream velocity U∞. As seen, the surface CL(α,U∞) is discontinuous and consistsof two parts. Projections of the upper and lower parts onto the plane of free-streamparameters overlap. The overlapping area (shown in dark grey in Fig. 5) is a bifurca-tion region, in which transonic flow is non-unique and realization of a certain flowfield depends on the time history of α and U∞.

There are slight oscillations of the flow in the near wake region. In contrast totransonic flow past symmetric airfoils [3], computations have not revealed periodicoscillations of shock waves on the airfoil at the considered small angles of attack.

This example demonstrates that, though a modification of an airfoil by a bumpcan weaken shock waves and enhance the aerodynamic performance at some valuesof M∞ and α , as a number of studies showed [8, 9, 10], however, this provokes flowinstability and bifurcations at slightly different values of M∞ and α .

Fig. 1 IsoMachlines in transonic flow past the DrelaApex 16 airfoil at M∞ = 0.7493, α =−3.9 deg , Re=2×107.

Adverse Free-Stream Conditions for Transonic Airfoils with Concave Arcs 3

Fig. 2 Lift coefficient as a function of the angle α for transonic flow past the Drela Apex 16 airfoil.

Fig. 3 IsoMachlines in tran-sonic flow past the Whit-comb airfoil with the bump atM∞ = 0.833, α = 0.4 deg.

Fig. 4 Lift coefficient as a function of U∞ and α for transonic flow past the Whitcomb airfoil withthe bump.

4 Alexander Kuzmin

Fig. 5 The domain of adversefree stream conditions (shownin dark grey), in which thereexist flow bifurcations.

References

1. Jameson, A.: Airfoils admitting non-unique solutions of the Euler equations. AIAA Paper,no. 91–1625, 1–13 (1991).

2. Hafez, M., Guo, W.: Some anomalies of numerical simulation of shock waves. Part II: effectof artificial and real viscosity. Computers and Fluids. 28, 721–739 (1999).

3. Kuzmin, A.: Aerodynamic surfaces admitting jumps of the lift coefficient in transonic flight.In: Hafez, M.M., Oshima, K., Kwak, D. (eds.) Computational Fluid Dynamics Review 2010.World Scientific Publishing Co. (2010).

4. Kuzmin, A., Ivanova, A.: 2004 The structural instability of transonic flow associated withamalgamation/splitting of supersonic regions. J. of Theoretical and Computational Fluid Dy-namics. 18, 335–344 (2004).

5. Babarykin K.V.: Bifurcation of a transonic ideal-gas flow around a symmetric airfoil with aconcavity. J. of Engineering Physics and Thermophysics. 80, 702–707 (2007).

6. Kuzmin, A. & Shilkin, A.: Transonic buffet over symmetric airfoils. In: H. Deconinck, E.Dick (eds.) Computational Fluid Dynamics 2006. 849–854 (2009).

7. Greer, D., Hamory, P., Krake, K., Drela, M.: Design and predictions for a high-altitude (low-Reynolds-number) aerodynamic flight experiment. NASA/TM-1999-206579, 17 p. (1999).

8. Reneaux, J. et. al.: Overview on drag reduction technologies for civil transport aircraft. In: P.Neitaanmaki, T. Rossi et al. (eds) Proc. of ECCOMAS, Univ. of Jyvaskyla, Finland, 18 p.,(2004)

9. Konig, B., Patzold, M., Lutz T., Kramer E.: Shock Control Bumps on Flexible and TrimmedTransport Aircraft in Transonic Flow. Notes on Numerical Fluid Mechanics and Multidisci-plinary Design. 96, 80–87 (2008).

10. Lee D.S., et al.: Shock Control Bump Design Optimization on Natural Laminar Aerofoil. In:A. Kuzmin (ed.) Computational Fluid Dynamics 2010 (present book), (2011).

Development of a Conservative Overset Mesh Method on Unstructured Meshes

Mun Seung Jung1 and Oh Joon Kwon2

Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology(KAIST), Daejeon, Korea 1chimera@kaist.ac.kr 2ojkwon@kaist.ac.kr

Recently, several moving-grid CFD techniques have been developed to handle time-accurate unsteady flows involving multiple bodies in relative motion. Among them, overset mesh technique has been regarded as the most prominent approach. In spite of the successful application of these overset mesh techniques to realistic flow problems, several practical problems still exist. For conventional overset mesh schemes, transfer of flow variables between mesh blocks is achieved through interpolation, and therefore the conservation of fluxes across the overset mesh block boundary is usually not guaranteed. Because of this problem, numerical difficulties arise when severe flow gradients exist [1, 2].

In the present study, a conservative overset mesh scheme based on unstructured meshes has been developed for the efficient and accurate numerical simulation of two and three-dimensional unsteady time-accurate flows around multiple objects in relative motion. In this new approach, after the hole-cutting, the mesh blocks are separated by blank region with a gap size of local cells, and this blank region is refilled with new triangular and tetrahedral elements for 2-D and 3-D flow problems, respectively, by connecting the intergrid boundary vertices of the adjacent mesh blocks. By executing this procedure, all overset mesh blocks are reconnected instantaneously at each time step and can be treated as a single block. Thus the conservation property of flow over the global computational domain is automatically satisfied, even when the objects are in an arbitrary relative motion, without any spurious mass, momentum and energy production from inside of the flow domain. For this purpose, an intergrid boundary reconnection technique was developed to enhance the efficiency and the robustness of generating new elements inside the blank region. In the case of the flow problems without mesh movement, the overset mesh restores to single block unstructured mesh.

As the 2-D validation case, a subsonic turbulent flow around an oscillating NACA0012 airfoil [3] was calculated. In Fig. 1, comparison of the computational meshes between the present conservative scheme and the conventional non-conservative scheme is presented. Also, vorticity contours of the single-block mesh calculation and the conservative and conventional non-conservative overset mesh schemes are compared at the instantaneous peak angle of attack of 22 degrees. While the results of the single-block mesh and the conservative mesh scheme are similar to each other, the results by the non-conservative mesh scheme

show significant difference in the flow pattern. This difference is attributed to the numerical error activated across the mesh block boundary when a non-conservative interpolation of the conventional overset mesh scheme is applied.

Fig. 1 Comparison of computational meshes and vorticity contours between conservative overset mesh scheme (a) and conventional non-conservative overset mesh scheme (b).

In Fig. 2, the vortical structure and the vorticity contours at a selected streamwise location are presented for a 3-D rotating sphere. The axis of rotation was set parallel to the free stream direction. The figure shows that the vorticity was diffused across the block boundary in a much quicker rate by the non-conservative method. The results indicate that the conventional non-conservative overset mesh method is more diffusive than the present conservative overset mesh method.

Fig. 2 Comparison of vorticity contours at a streamwise location x/D=2 of a rotating sphere. Acknowledgements This work was supported by the National Research Foundation of Korea (NRF) grant No.2009-0083510 funded by the Korean government (MEST) through Multi-phenomena CFD Engineering Research Center. References [1] Wang, Z. J.: A fully conservative interface algorithm for overlapped grids.

Journal of Computational Physics, Vol. 122, No. 1, pp. 96-106 (1995) [2] Wright, J., Shyy, W.: A pressure-based composite grid method for Navier-

Stokes equations. Journal of Computational Physics, Vol. 107, No. 2, pp 225-238 (1993)

[3] Hilaire, A. O., Carta, F. O.: Analysis of unswept and swept wing chordwise pressure data from an oscillating NACA0012 airfoil experiment. NACA CR-3567 (1983)

Symmetrical Vortex Fragmenton as a VortexElement for Incompressible 3D Flow Simulation

I.K. Marchevsky, G.A. Scheglov

Abstract Problems of numerical flow simulation around bluff bodies using mesh-free Lagrangian vortex element method are discussed. New type of vortex elements— symmetrical vortex fragmenton — is developed and its properties are investi-gated. Some results of computations with vortex fragmentons are shown.

Vortex element method is well-known. It should be noted, that this meshfreemethod is very attractive because of sufficiently low computational cost in compar-ison with other CFD methods. The motion of vortex elements is described by ODEsystem.

Vortex element is an element of vorticity field, which generates velocity field inthe whole space. Total vorticity field is the superposition of elementary fields gener-ated by certain vortex elements. There are some different types of vortex elements,for example, vortex filament, vortex ring, Novikov’s vorton [4], vortex blob etc.,each type has both some advantages and limitations.

In this research new type of vortex element is developed — Symmetrical VortexFragmenton. Each vortex fragmenton is described by its marker position r0, frag-menton vector h and intensity Γ (fig. 1).

Main advantages of vortex fragmentons are the following: 1) closed vortexframework could be constructed from vortex fragmentons of the equal intensity;2) vortex structures in the flow could be simulated with sufficiently small number ofvortex fragmentons in comparison with Novikov’s vortons and vortex blobs; 3) vor-ticity field line reconnection process could be simulated with vortex fragmentons.

For symmetrical vortex fragmenton analytical formulas for vector potential, ve-locity and vorticity fields are obtained:

I.K. MarchevskyBauman Moscow State Technical University, Russia, 105005, Moscow, 2-nd Baumanskaya st., 5,e-mail: iliamarchevsky@mail.ru

G.A. ScheglovBauman Moscow State Technical University, Russia, 105005, Moscow, 2-nd Baumanskaya st., 5,e-mail: georg@energomen.ru

1

2 I.K. Marchevsky, G.A. Scheglov

2s

0r

1s

h

x

y

z

O

r

rV

Fig. 1 Symmetrical vortex fragmenton model and vortex wake near cylinder

V(r) = curlΨ =Γ4π

( |s1|− e · s1

|s1|(|s1|− e · s1)− |s2|− e · s2

|s2|(|s2|− e · s2)

)× e, (1)

Ψ(r) =Γ e4π

ln|s1|− e · s1

|s2|− e · s2, Ω(r) = curlV =

Γ4π

(s1

|s1|3 −s2

|s2|3)

. (2)

In (1)–(2) we denoted e = h/h, h = |h|, s1 = r0 +h, s2 = r0−h.Symmetrical vortex fragmenton can be considered as generalization of Novikov’s

vorton. It allows to use some results obtained by Alkemade [1]. It can be easilyshown that vorticity field generated by symmetrical vortex fragmenton is divergence-free and improved numerical scheme for vorticity field evolution could be used.

If we simulate the flow around a body then the surface should be approximatedwith quadrangles — closed vortex frameworks which consist of vortex fragmentons.Their intensities could be found from solution of linear equations which approxi-mates boundary conditions. Due to vorticity flux model [3] all vortex frameworksare split into separate vortex fragmentons and they become part of vortex wake.

Thus, we need only one vortex element type — symmetrical vortex fragmenton— both for flow and vorticity flux simulation. Other researchers mentioned in [2]used different types of vortex elements for vortex wake simulation and for satisfac-tion of boundary conditions. Vortex element transformation from one type to othersis non-trivial operation and can cause significant errors.

Several model problems were solved: vortex rings leapfrogging simulation, semi-elliptical vortex ring evolution, flow around axisymmetric bluff bodies investigationetc. Obtained results are close to experimental data.

References

1. Alkemade, A.J.Q.: On vortex atoms and vortons. PhD thesis. TU-Delft, Delft (1994)2. Cottet, G.-H., Koumoutsakos, P.: Vortex Methods: Theory and Practice. Cambridge Univer-

sity Press, Cambridge (2000)3. Lighthill, M.J.: Introduction. Boundary layer theory. In: Rosenhead, L.(ed.) Laminar Bound-

ary Layers, pp. 46–113. Oxford University Press, Oxford (1963)4. Novikov, E.A.: Generalized Dynamics of Three-Dimensional Vortical Singularities (vortons).

Sov. Phys. JETP, 57, 566–569 (1983)

Morphogenic computational fluid dynamics

1

Morphogenic computational fluid dynamics: Brain shape similar to the engine flow

Haruka Kawanobe1, Shou Kimura1, and Ken Naitoh1

1 Faculty of Science and Engineering, Waseda University, 3-4-1 Ookubo, Shinjuku, Tokyo 169-8555 Japan k-naito@waseda.jp

A computational fluid dynamic model that simulates the developmental process of the human brain is proposed. The bones of the skull become increasingly larger over the neck and also soup-like fluid for generating brain cells enters the skull from the body. This process is essentially similar to the intake process of an internal combustion engine, because the volume of engine cylinder, which increases according to the descent of the piston, geometrically corresponds to the development of the skull and also because the human neck resembles the intake port of the engine. A higher-order numerical computation of the Navier-Stokes equation at very low engine speed reveals the similarity between the convexoconcave forms inside the brain and the flow structure in the internal combustion engine. We will show that the present computation also simulates the nose and the eyeballs.

1. Introduction

The soup for generating flexible brain cells can be essentially modeled by using fluid dynamics for flow in an engine. The flow of air during the intake process of a piston engine and fluid flow in the brain, including water as the main component, can be approximated as being incompressible. Thus, we employ the incompressible Navier-Stokes equation as the basic governing equation that describes the flexible dynamic motion of a lot of soup-like fluid for generating brain cells. The equation can be written as

(1) where νρ andpui ,,, denote the fluid velocity, fluid density, pressure, and kinetic viscosity coefficient, respectively, while )3,2,1( =ixi denotes Cartesian coordinates in three-dimensional space. Equation (1) is computed with the finite difference method using a higher order of accuracy. The details of the computational method are given in

.0

,12

2

=∂∂

∂∂

+∂∂

−=∂∂

+∂∂

∑

∑∑

i

i

i

j

i

jij

ij

j

i

xu

andxu

xp

xuu

tu ν

ρ

2 Haruka Kawanobe, Shou Kimura, Ken Naitoh

the references. [1,2,3] In our previous report, we showed the similarity between the intake flow filed of piston engine and the brain shape, by comparing with the MRI image photographs and the computational results at the engine speeds between 2000rpm and 10rpm. [4] 2. Present results Here, we will show the computational results at the very low engine speeds less than

1.0rpm. Figure 1(a) shows an image of the human brain obtained with magnetic resonance imaging (MRI) at a horizontal cross section including the two eyes, while Fig. 1(b) is the result of the corresponding computation performed with the Navier-Stokes equation. It should be noted that the two circular regions in Fig. 1(a), which are the two human eyeballs, can also be seen in Fig. 1(b). The MRI tomogram agrees well with the computational results. Another important point is the nose simulated. (Fig. 1) The reason why the flow field in the engine cylinder at very low engine speeds is similar to the brain shape is related to the Richardson effect. References 1. Naitoh K, Kaneko Y and Iwata K (2004) Cycle-resolved Large Eddy Simulation of

actual 4-valve Engine based on Quasi-gridless approach, SAE paper 2004-01-3006. 2. Naitoh K and Kuwahara K (1992) Large eddy simulation and direct simulation of

compressible turbulence and combusting flows in engines based on the BI-SCALES method, Fluid Dynamics Research 10: 299-325.

3. Naitoh K, Nakagawa Y, and Shimiya H, (2008)Stochastic determinism approach for simulating the transition points in internal flows with various inlet disturbances. Computational Fluid Dynamics 2008, Springer-Verlag.

4. Naitoh K, (2008) Engine for cerebral development, Artificial Life Robotics, Springer, 13: 18-21.

(a) (b) Fig. 1 Brain shape and flow field in piston engine.

Computational Modeling of the Flow and Noisefor 3-D Exhaust Turbulent Jets.

Cheprasov S.A., Lyubimov D.A., Secundov A.N.,Yakubovsky K.Ya. and Birch S.F.

Abstract Several problems, connected with aircraft and engine aerodynamics, areconsidered. All the models and jet flows considered were studied jointly by Boeingand the scientific group ECOLEN. The paper contains both new unpublished resultsand a short review of some older published work.

1 Introduction

The exhaust jets from airplanes still generate a considerable part of overall airplanenoise, in spite of increases in engine by-pass ratio. The correct design of jet noisesuppressors requires the accurate prediction of both turbulence and acoustic jet char-acteristics. Also important to be able to accurately predict danger zones due to thejet-wakes behind airplanes on the ground.

There are two basic approaches to jet flow prediction. The first involves time-averaging the equations of motion - the RANS approach. This approach requiresthe use of a turbulence model. But the most popular models arenot able to accu-rately predict all complex 3-D jet flows. Other direct (DNS) or semi-direct (LES,DES) methods of jet modeling appear to offer more promise. The work of Shur(2006), Uzun (2009) and Lyubimov (2009) shows rather good progress. Such calcu-lations, however, are very expensive and are not practical for the calculation of real,complex, airplane nozzle flows.

A similar situation exists for jet noise prediction. One possible method is basedon RANS flow calculations and uses approximate semi-empirical relations. Suchan approach is described in Birch (2007) and Secundov (2007). An alternative, andpotentially more accurate approach, is based on the use of LES jet flow calculations,

Cheprasov S.A., Lyubimov D.A., Secundov A.N.,Yakubovsky K.Ya.Scientific and Research Center ECOLEN, 111116 Moscow, Russia, e-mail: secundov@rambler.ru

Birch S.F.The Boeing Company, Seattle WA 98124, USA, e-mail: Stanley F.Birch@Boeing.com

1

2 Cheprasov S.A., Lyubimov D.A., Secundov A.N., YakubovskyK.Ya. and Birch S.F.

combined with the concept of a Kirchhoff surface and some integral form of theacoustic wave equation, for example, the FWH equation.

Calculations of the noise spectra (SPL) and directivity diagrams (OASPL) showthat both approaches can give errors of about 2-3 dB and sometimes much larger.Therefore the current state of jet noise prediction methodsneeds improvement.

In this paper the authors will try to show that a promising wayout of this difficultsituation is the combined application of RANS and LES methods, together withmodel experiments.

2 Computational models

Our group uses the JET3D code that is described in Birch (2005). It uses 5th (andsometimes 9th) order finite-difference approximations (FDA). Besides, we also usethe well known Fluent code (2nd order FDA). It should be noted that both codes useinflow boundary conditions set at a station well upstream of the nozzle exit sectiondeep inside nozzle, at an X/D≈ -3 to -5. This makes it possible to account for thenozzle geometry and real wall boundary layer effects.

In order to improve the prediction of turbulent jets, a new zonal model was de-veloped, based on the well known “k− ε” model Birch (2007). The key element ofthe new turbulence model is the use of different coefficientsfor the two jet regionsand smoothly changing them near the end of the jet potential core.

An improved jet noise prediction model (JNPM), based on RANScalculations,which was briefly presented in Birch (2007) and Secundov (2007), is based on theclassical results of aeroacoustics. This noise model was tested for jets ranging fromsimple round nozzles, to 3-D, coaxial nozzles, with and without chevrons and pylonsand so on.

3 Three dimensional wall jets

The first problem considered involves the theoretical and experimental study of 3-Dwall jets that are formed behind airplanes on the ground. Themain unusual feature ofthis flow is the strong anisotropic jet mixing (Khritov, 2002). The transverse wall jetwidth is 8-10 times larger that the width normal to the wall. Special experiments andLES/DES calculations were performed with the help of the JET3D code in order tobetter understand the anisotropic features of the Reynoldsshear stress components.

This was our first experience in the LES calculations, so the grid was very courseand the initial part of jet was quasi laminar. However, far downstream the cross-section was similar to the experimental data.

Computational Modeling of the Flow and Noise for 3-D ExhaustTurbulent Jets. 3

4 Pylon effects

Real airplane nozzles have pylons that attach engines to airplane wings. Because ofthis, real exhaust jet flows are always 3-dimentional. Calculations showed that theturbulence energy on the side of the jet closest to the pylon is 1.3-1.6 time larger thanthe turbulence energy on the other side of the jet (Birch, 2005). As a consequence,far field noise has an azimuthal asymmetry of about 1-2 dB.

Fig. 1 Calculated (solid lines) 1/3 octave spectra for BPR5 nozzlewith pylon, side microphoneposition.

Predictions (fig.1) obtained using our jet noise model (JNPM) show rather goodagreement with the experimental data of Thomas (2004) for spectra and for theazimuthal noise distribution. Similarly good predictionsusing JNPM, for a range ofdifferent flows, are described in Birch (2007).

5 “Aerodynamic chevrons”

3-D RANS calculations (Fluent code) were also performed to investigate a newmethod for jet noise suppression. The key idea in this methodis the use of specialslots in the fan nozzle wall that conduct flow of the high pressure air from insidethe nozzle to form a series of jets on the external cowl. Theseadditional small jets“work” like the more usual metal chevrons (Birch, 2009).

Smoke visualization and 3-d RANS calculation show that 16 jets from slots col-lide interact to form 8 new compact jets that spread along main exhaust jet. Thesenew small jets play the role of specific screen, which reducesmain jet noise

4 Cheprasov S.A., Lyubimov D.A., Secundov A.N., YakubovskyK.Ya. and Birch S.F.

6 Acoustic waves near jet surface

One of the main goal of our LES calculations (JET3D code with 9-th order FDA,grid is 1.2×106 meshes) is to obtain a better understanding of the reasons for theexcess high frequency noise generated by many chevron nozzles. These calculationsshow some new features of the acoustic wave generation between chevrons.

Fig. 2 Pressure contours (acoustic waves) in the plane X-Y for nozzle SMC006 with 6 chevrons.

Fig. 3 Pressure contours (acoustic waves) in the cross-section X=- 0.3D for nozzle SMC006 with6 chevrons, X=0 - corresponds to the tip of chevrons.

Computational Modeling of the Flow and Noise for 3-D ExhaustTurbulent Jets. 5

The Fig. 2 shows “unusual” waves (arrows) that radiate from the jet surface andmove in an upstream direction forming interference patterns with the more usualradial waves. Similar waves are shown in Uzun (2006), but were obtained using amuch larger 50×106 grid. Such interference seems especially strong just inside theexit of a chevron nozzle Fig. 3. Since interference between waves with frequenciesω1 andω2 can produce 4 different frequency:ω1,ω2 ,(ω1−ω2) and (ω1 + ω2) , andcould, perhaps, move some of the noise energy to high frequencies.

For a chevron nozzle, the jet perimeter near the nozzle exit is two to three timesthat of a round jet, so the total noise flux from this region is also increased. It is notyet clear if this can explain the excess high frequency noisegenerated by chevronnozzles, but LES calculations are proving to be a valuable tool in our attempts tobetter understand these complex phenomena.

7 The flow prediction problems near nozzle edge

When a jet emerges from a nozzle, the initial turbulent wall boundary layer (BL)flow, with turbulence levels of 9-12 % ,transitions to a mixing layer with turbulencelevels of 16-17 %. The length of this transition region can beup to 1000θ0, whereθ0 is momentum thickness of the initial BL (Birch 2006). This transition regioncan cover a large fraction of the jet potential core and make an important contribu-tion to the high frequency part of the total noise spectrum. This can be particularlyimportant for effective perceived noise levels (EPNL, dB).

Unfortunately, most RANS models underestimate the extent of the transition re-gion by a factor of 2-3. The problem is even more difficult withLES. Estimates byLyubimov (2009) and direct LES calculations by Uzun (2009) show that the cor-rect prediction of this transition region requires about 109 computational meshes.For more complex real nozzle/pylon configurations, this estimate increases to 1010

meshes. It is evident that such calculations are not practical using current computers,especially for the more complex problems

Fig. 4 Contours of vorticity values near plane nozzle edge inside wall BL and inside mixing layer.

6 Cheprasov S.A., Lyubimov D.A., Secundov A.N., YakubovskyK.Ya. and Birch S.F.

One possible solution to this problem is to attempt to approximately model theturbulent flow near the nozzle exit. Several such methods arecurrently being studied.However the prediction extent of transition is still too short.

More direct and hopefully more universal prediction methods are also being stud-ied. To get a better understanding of the problem involve, initial calculations arebeing run for the transition region between the boundary layer on a flat plate and amixing layer, using Fluent code with a 400×120×60 meshes grid. It is importantto note that such LES calculation require a very fine grid nearthe wall,Y + < 1. Allaveraged BL characteristics were predicted quite well. Thewall ends and the mixinglayer starts at X=0. A calculated longitudinal section of the instantaneous vorticityin the transition region is shown in the Fig. 4 .

Analysis of calculation in the Fig. 4 and obtained dependence u′(X)/U0 showsthat transitional region is still too much small.

Acknowledgements This work was supported, in part, by the Russian Foundation for Basic Re-search (grant No. 10-01-00255-a). The authors are gratefulto Dr. I.L. Petrouchenkov for his helpin the computer technology.

References

1. Birch, S.F., Lyubimov, D.A., Buchshtab, P.A., Secundov,A.N. & Yakubovsky, K.Ya. : Jet-pylon interaction effects. AIAA paper 2005-3082

2. Birch, S.F., Lyubimov, D.A., Maslov, V.P. and Secundov, A.N.: Noise Prediction for ChevronNozzle Flows. AIAA 2006-2600

3. Birch, S.F., Lyubimov, D.A., Maslov, V.P., Secundov, A.N., Yakubovsky, K.Ya. A RANSbased Jet Noise Prediction Procedure. AIAA paper 2007-3727.

4. Birch, S.F., Bukshtab, P.A., Khritov, K.M., Lyubimov, D.A., Maslov, V.P., Secundov, A.N.and Yakubovsky, K.Ya.: The Use of Small Air Jets to Simulate Metal Chevrons. AIAA paper2009-3372

5. Khritov, K.M., Lyubimov, D.A., Maslov, V.P., Mineev, B.I., Secundov, A.N., Birch, S.F. Three-dimensional wall jets: experiment, theory and application: AIAA paper 2002-0732

6. Lyubimov, D.A.: The Application of Hybrid RANS/ILES Approach for the Investigation ofthe Effect of Nozzle Geometry and Mode of Efflux on the Characteristics of Turbulence ofExhaust Jets. High Temperature, 2009, Vol. 47, No. 3, pp. 374-383.

7. Morris, S.C., Foss, J.F.: Turbulent boundary layer to single-stream shear layer: the transitionregion. J. Fluid Mech., 2003, vol. 403, pp. 187-211.

8. Secundov, A.N., Birch, S.F., and Tucker, P.G. : 2007. Propulsive jets and their acoustics. Phil.Trans. R. Soc. A, vol. 365, pp. 2443-2467.

9. Shur, M.L., Spalart, P.R., Strelets, M.Kh. and Garbaruk,A.V.: Further Steps in LES BasedNoise Prediction for Complex Jets. AIAA Paper 2006-485.

10. Thomas, R.H. and Kinzie, K.W. : Jet-Pylon Interaction ofHigh Bypass Ratio Separate FlowNozzle Configurations. AIAA Paper 2004-2827.

11. Uzun. A., Hussiani. M.Y.: High Frequency Noise Generation in the Near Nozzle Region of aJet. AIAA paper 2006-2499.

12. Uzun, A., Hussaini, M.Y. : High- Fidelity Numerical Simulations of a Chevron Nozzle JetFlow. AIAA paper 2009-3194.

Numerical Simulation of Laser Welding of ThinMetallic Plates Taking into Account Convectionin the Welding Pool

Vasily Shapeev, Vadim Isaev, Anatoly Cherepanov

Abstract A new algorithm for numerical simulation of laser welding of thin metal-lic plates was developed here. It is based on the collocations and least squaresmethod and a new quasi-three-dimensional model of laser welding process. Cal-culations for titanium plates were carried out. An influence of the convection on thewelding process was investigated.

We consider a stationary process of laser welding of thin metallic plates. Theplates are rectangular parallelepipeds tightly adjoined by thin side faces. The laseraxis is perpendicular to the plates and lies in the joint plane. The laser beam movesparallel to the plates along the joint. The welded plates are blown by inert gas toprotect the alloy from oxidation. Let us introduce Cartesian coordinate system forwhich the laser beam is immobile, while the plates move with the welding speed Vw.Axis z is directed downward along the beam axis, x axis is oriented along the jointin the direction of the plates movement, and y axis is perpendicular to the joint. Thecoordinate origin is located on the beam axis at the plates upper surface.

A new three-dimensional model of thin metallic plates laser welding process isdeveloped here. Due to the complexity of thermo- and hydrodynamic processes oc-curred in the welding pool we make some simplifying assumptions. Taking intoaccount that the two phase zone is thin enough we consider the melt solidification inStefan’s approximation. We assume that the welding speed is constant and the ther-mophysical parameters are equal to their average values. Heat transfer in the plates isdescribed by heat conduction equation in the model. The flow of liquid metal insidethe welding pool is simulated with the use of Navier-Stokes equations. Interaction

Vasily Shapeev, Anatoly CherepanovKhristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of RussianAcademy of Sciences, Institutskaya str. 4/1, Novosibirsk, 630090, Russia,e-mail: vshapeev@ngs.ru

Vadim IsaevNovosibirsk State University, Pirogova str. 2, Novosibirsk, 630090, Russia,e-mail: issaev.vadim@gmail.com

1

2 Vasily Shapeev, Vadim Isaev, Anatoly Cherepanov

0

1

2

3

4

5z, m

m

0 2 4 6 8

1

6

x, mm-2

2 3 4 5

0

1

2

3

4

5

z,

mm

0 2 4 6 8x, mm

-2

0

1

2

3

4

5

z, m

m

0 2 4 6 8

1 2 3 4

5

6

x, mm-2

a b c

Fig. 1 Pattern of the liquid metal flow and temperature field in the welding domain

of the welding zone with the environment is described by corresponding boundaryconditions. We allow for a presence of a steam channel in the vicinity of the laserbeam. We take into account convective and radiative heat losses through upper andlower surfaces of the plates, influence of surface tension forces and friction forcesof the metal vapour escaping from the channel on the liquid metal flow.

Due to essential difficulty of the 3D model proposed here we averaged its equa-tions with respect to variable y and developed a new quasi-three-dimensional model.Heat flow in the direction of y axis and viscous friction between orthogonal to axis ylayers of liquid metal are approximately estimated and taken into account here.

Numerical solution of Navier-Stokes equations was carried out by a new ver-sion of the collocations and least squares (CLS) method. Heat conduction equationwas approximately solved by a new conservative version of the CLS method. Com-putations were performed on nonuniform meshes with a local refinement near thewelding pool. Numerical simulations on a sequence of grids with decreasing stepswere carried out. Calculation parameters of the problem were observed (the tem-perature distribution, the location of isotherms and boundaries between liquid andsolid phases of the material). It is shown that the error of the approximate solutionobtained here is equal to O(h), where h is the maximal linear size of grid cells.

Fig. 1 shows some results of numerical simulation of titanium plates laser weld-ing. Here, the thickness of the plates is 5 mm, the laser beam power is 3 kW,Vw = 1 m/min. For the sake of clearness, in Fig. 1 we present results for a fragmentof the computational domain that is located near the welding pool. Black domain onthe figures crossed by line x = 0 corresponds to the steam channel. Isotherms shownin the figures are the following: T = 2800 K (1), 2400 K (2), 2150 K (3), 1944 K (4)(crystallization temperature), 1200 K (5), 1000 K (6).

In order to investigate the influence of the pattern of liquid metal flow on thetemperature distribution in the plates we made the following experiment: in thefirst computation we assumed that the flow in the welding pool is plane-parallel( Fig. 1 (a) ); in the second one the flow was simulated with the use of Navier-Stokes equations ( Fig. 1 (b), (c) ). One can see that the presence of vortex motionin the welding pool ( Fig. 1 (b) ) has an essential influence on the temperature distri-bution ( Fig. 1 (c) ) and the form of the welding pool boundary (isotherm 4). At thesame time, the volume of the welding pool is approximately the same in both cases.

The work was supported by the RFBR, projects 08-08-00249, 10-01-00575,SB RAS integration projects 11.5, 26, 140.

Numerical Simulation of Cavitation BubbleCollapse near Wall

Byeong Rog Shin

Abstract A finite-different method for gas-liquid two-phase flow such as cavitat-ing flow with variable density is applied to investigate cavitation bubble collaps-ing behavior. The investigation of cavitation-bubble collapsing phenomena is veryimportant to clarify the mechanism of damage on materials in hydromachine sys-tems, which the bubble collapse induces deformation and erosion on the surface ofmaterials and causes reduction of durability of hydro-devices. The present methodemploys a finite-difference Runge-Kutta method and Roe’s flux difference splittingapproximation with the MUSCL-TVD scheme. A homogeneous equilibrium gas-liquid multi-phase model, which takes into account the compressibility of mixedmedia, is used. Therefore the present density-based numerical method permits sim-ple treatment of the whole gas-liquid mixed flow field, including wave propagation,large density changes and incompressible flow characteristics at low Mach number.By applying this method, cylindrical bubbles located near solid wall and incidentliquid shock wave are computed.

Cavitation is a typical gas-liquid 2-phase flow with a phase change phenomenonin the flow of hydromachines. When cavitation occurs and collapses near solid sur-faces, it causes the noise, vibration and damage to hydraulic machine systems (Bren-nen (1995)).

In the sense of reducing these unfavorable effects, therefore, technology of ac-curate prediction/estimation of cavitation is very important. In order to understandthe behavior of collapsing of cavitation bubbles, some efforts to propose cavity flowmodel for numerical simulations (Singhal et al.(1997), Kunz et al.(2000)), and, an-alytical and experimental method for shock-bubble interaction problems have beenmade. Recently, the author has proposed a mathematical cavity flow model (Shin etal. (2003), Yamamoto and Shin (2004)) based on a homogeneous equilibrium modeltaking account of the compressibility of the gas-liquid two-phase media. With thismodel, the mechanism of developing cavitation has been investigated through theapplication to a couple of cavitating flows around a hydrofoil (Iga et al. (2003),Shin et al. (2004)).

Department of Mechanical Engineering, Changwon National University, Changwon 641-773, Ko-rea, e-mail: brshin@changwon.ac.kr

1

2 Byeong Rog Shin

(a) t = t0 (b) t = t0 +3.0µS (c) t = t0 +5.0µS

(d) t = t0 +6.4µS (e) t = t0 +8.0µS (f) t = t0 +9.6µS

(g) t = t0 +11.0µS (h) t = t0 +13.0µS (i) t = t0 +15.0µS

Fig.1: Time evolution of void fraction distributions in the process of 3-arry bubble collapsing atthe fixed outlet pressure

In this paper the numerical method for cavitating flow by Shin et al. (2003) isextended to a cavitation bubble collapsing problem with a high-order Runge-Kuttamethod and MUSCL TVD solution method for stable and accurate treatment of gas-liquid interfaces considered by contact discontinuity. As numerical examples, cav-itation bubble collapsing problems between cylindrical single- or multi-cavitationbubbles located near the solid wall and incident liquid shock wave are solved andinvestigated bubble collapsing behavior and shock-bubble interaction.

In the computation of cavitating flow, the gas-liquid two-phase flow is possible tomodel into an pseudo single-phase flow by using concept of the homogeneous equi-librium model (Shin et al. (2003)) in which thermodynamic equilibrium is assumedand velocity slip between both phases is neglected.

Under this model concept, the density for gas-liquid two-phase media is deter-mined by using a combination of two equations of state for gas phase and liquidphase, that is written as follows:

ρ =p(p+ pc)

K(1−Y )p(T +Tc)+RY (p+ pc)T(1)

where, ρ, p,Y , and T are the mixture density, pressure, quality (dryness) and thetemperature, respectively. R is the gas constant and K, pc and Tc represent the liquidconstant, pressure constant and the temperature constant for water, respectively.

By using above two-phase flow model, the 2-D governing equations for the mix-ture mass, momentum, energy and the gas-phase mass conservation can be writtenin the curvilinear coordinates (ξ ,η) as follows:

Numerical Simulation of Cavitation Bubble Collapse near Wall 3

∂Q∂ t

+∂E∂ξ

+∂F∂η

=∂Ev

∂ξ+

∂Fv

∂η+S (2)

where Q is an unknown variable vector, E, F are flux vectors and Ev, Fv are viscousterms. S is the source term.

Fundamental equations (2) are solved by using a finite-difference method withTVD Runge-Kutta method. Roe’s flux difference splitting method with the MUSCL-TVD scheme is applied to enhance the numerical stability, especially for steep gra-dients in density and pressure near the gas-liquid interface. Therefore, the derivativeof the flux vector, for instance, E with respect to ξ at point i can be written withthe numerical flux as (∂E/∂ξ ) = (E i+1/2 −E i−1/2)/∆ξ and then, the approximateRiemann solver based on the Roe’s FDS is applied.

As a numerical result, Fig.1 shows time evolution of void fraction distributions inthe process of bubble collapsing by incident shock wave near the wall. As an initialcondition, 3-array bubbles are located in the stationary flow field with incident liquidshock wave with a high pressure of 100 MPa at the upstream. A low pressure of0.1 MPa is imposed at outlet boundary. When incident shock wave impacts on thebubble the bubble is asymmetrically contracted with concave shape by the pressuredifference between back and forth of shock plane. A sort of microjet is formed andeventually it impinges on the rear surface of the bubble. Eventually the microjetspenetrate bubbles one by one and flow toward down stream forming a kind of pairvortex. Bubble collapsing behavior, bubble deforming shapes and flow pattern arewell simulated in this application. %

Acknowledgements: This work was supported by the National Research Foun-dation of Korea (NRF) grant No.2009-0083510 funded by the Korean government(MEST) through Multi-phenomena CFD Engineering Research Center.

ReferencesBrennen, C.E. 1995 Cavitation and Bubble Dynamics, Oxford Univ. Press, Ox-

ford, 1995Iga, Y., et al. 2003 Numerical Study of Sheet Cavitation Break-off Phenomenon

on a Cascade Hydrofoil. ASME J. Fluid Engng, Vol. 125, pp.643-651.Kunz, R.F., et al. 2000 A Preconditioned Navier-Stokes Method for Two-Phase

Flows with Application to Cavitation Prediction. Computers & Fluids, Vol. 29,pp.849-875.

Shin, B.R., et al. 2003 A Numerical Study of Unsteady Cavitating Flows Using aHomogenous Equilibrium Model. Computational Mechanics, Vol. 30, pp.388-395.

Shin, B.R., et al. 2004 Application of Preconditioning Method to Gas-LiquidTwo-Phase Flow Computations. ASME J. Fluid Engng., Vol. 126, pp.605-612.

Singhal, A. K. et al. 1997 Multi-Dimensional Simulation of Cavitating FlowsUsing a PDF Model for Phase Change, ASME Paper FEDSM97-3272.

Yamamoto, S. & Shin, B.R. 2004 A Numerical Method for Natural Convectionand Heat Conduction around and in a Horizontal Circular Pipe. Int’l J. of Heat andMass Transfer, Vol. 47, pp.5781-5792.

Numerical Analysis of Airflow in Human VocalFolds Using Finite Element and Finite VolumeMethod

Petr Sidlof and Bernhard Muller and Jaromır Horacek

Human phonation is a complex process involving fluid-structure and fluid-acousticinteractions. Although considerable effort has been devoted to model the phonationprocess in all its complexity, see e.g. [1], there are still many open questions re-garding the fundamental aspects of glottal airflow. The current study presents twoapproaches for solution of the unsteady airflow in human larynx.

The geometry of the computational domain was specified according to exper-imental data measured on excised human larynges [2]. In the first case, the flowwas modeled by 2D laminar incompressible Navier-Stokes equations in arbitraryLagrangian-Eulerian (ALE) approach. The ALE formulation of the governing equa-tions for the flow velocity u and kinematic pressure p reads

DA

Dtu+

[(u−w) ·∇

]u+∇p−ν ∆u = 0

∇ ·u = 0 , (1)

where w is the domain velocity and DA/Dt is ALE-derivative. The strong formu-lation (1) was discretized in time (2nd order backward differences), multiplied bytest functions and integrated over the domain. Applying Green’s theorem yieldedthe weak formulation, which was discretized using finite element (FE) method ona locally refined triangular mesh (16000 P3/P2 elements). For each timestep of theimplicit scheme, the linear system was solved with a direct linear solver UMFPack.

The second approach was based on finite volume (FV) discretization of the in-compressible Navier-Stokes equations in 2D or 3D and included optionally a tur-

Petr Sidlof, Jaromır HoracekInstitute of Thermomechanics, Academy of Sciences of the Czech Republic, Dolejskova 5, 182 00Prague 8, Czech Republic, e-mail: sidlof@it.cas.cz, jaromirh@it.cas.cz

Bernhard MullerNorwegian University of Science and Technology, Department of Energy and Process Engineering,Kolbjørn Hejes vei 2, Trondheim, NO-7491 Norway e-mail: bernhard.muller@ntnu.no

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2 Petr Sidlof and Bernhard Muller and Jaromır Horacek

Fig. 1 Velocity isolines inglottis (FEM).

Fig. 2 Isolines in x-normalsections (FVM).

bulence model. The discretization schemes were as follows: first-order Euler im-plicit for the time derivative, Gauss with linear interpolation for the gradient, diver-gence and Laplacian operators. The velocity field was solved by preconditioned bi-conjugate gradient method (PBiCG), for the pressure predictor and corrector steps,faster convergence was obtained with a geometric-algebraic multigrid (GAMG).The 3D mesh consisted of 120000 prismatic elements.

Figures 1 and 2 demonstrate the results of a 2D FE computation and a 3D FVcomputation without turbulence model. In both cases, the vocal folds were staticand the airflow was driven by constant flow velocity at inlet, with Re ≈ 1500.

The results support the hypothesis that a 2D model gives reasonable results nearglottis, which has tendency to bi-dimensionalize the flow. Further downstream thenarrow cross-section, however, the 2D and 3D simulations show important differ-ences: in the 2D case the coherent turbulent structures seem to persist for a longtime, but in a 3D reality they interact mutually and with the boundaries and tend todisintegrate much faster.

It should be also noted that the laminar simulations (computations without turbu-lence model) are mesh-dependent. For a mesh-consistent mean flow solution, RANSturbulence model was used. To model the small-scale turbulent structures importantfor sound productions, large-eddy simulation approach might be more appropriate.

Acknowledgements The research has been supported by the Grant Agency of the Academy ofSciences of the Czech Rep., project KJB200760801 Mathematical modeling and experimentalinvestigation of fluid-structure interaction in human vocal folds, research plan no. AV0Z20760514.

References

1. Luo, H., Mittal, R., Zheng, X., Bielamowicz, S.A., Walsh, R.J., Hahn, J.K.: An immersed-boundary method for flow-structure interaction in biological systems with application tophonation. J. Comput. Phys. 227, 9303–9332 (2008)

2. Sidlof, P., Svec, J. G., Horacek, J., , Vesely, J., Klepacek, I. and Havlık, R.: Geometry ofhuman vocal folds and glottal channel for mathematical and biomechanical modeling of voiceproduction, J. Biomech. 41, 985–995 (2008)