Turbulent flow computations on 3D unstructured grids

*Corresponding author.

Finite Elements in Analysis and Design 30 (1998) 353—363

Turbulent flow computations on 3D unstructured grids

M.T. Manzari*, O. Hassan, K. Morgan, N.P. WeatherillDepartment of Civil Engineering, University of Wales, Swansea SA2 8PP, UK

Received 21 November 1997

Abstract

An edge-based finite element method is presented for the simulation of compressible turbulent flows on unstructuredtetrahedral grids. A two equation k—u turbulence model is employed and the standard Galerkin approach is used forspatial discretisation. Stabilisation of the resulting procedure is achieved by the addition of an appropriate diffusion. Anexplicit multistage time-stepping scheme is used to advance the solution in time to steady state. The performance of thealgorithm is demonstrated for the simulation of a high Reynolds number transonic separated flow over a wing. ( 1998Elsevier Science B.V. All rights reserved.

Keywords: Finite element method; Compressible turbulent flow; Unstructured grids

1. Introduction

Simulation methods based upon an unstructured grid approach are now in widespread usewithin the aerospace industry, where important application areas are computational aerodynamicsand computational electromagnetics. A major attraction of the approach is the relative easewith which computational domains may be discretised, even when the geometry involved isextremely complex in shape [1]. At present, the approach is employed routinely in industryfor the simulation of inviscid compressible flows over general configurations, and research atten-tion is now concentrated on extending its range of application. In this context, the development ofmethods which enable the simulation of transonic turbulent flows is an active area of currentresearch [2].

One solution approach which can be naturally implemented on a general discretisation is thefinite element method. For the equations of compressible flow, a successful algorithm may be

0168-874X/98/$19.00 ( 1998 Elsevier Science B.V. All rights reservedPII: S 0 1 6 8 - 8 7 4 X ( 9 6 ) 0 0 0 3 9 - 0

constructed by employing a general unstructured tetrahedral grid, defined via an edge-based datastructure, and achieving the spatial discretisation by the Galerkin method [3]. Spatial stabilisationand shock capturing can be achieved by the addition of an appropriate level of diffusion,constructed by using one-dimensional concepts along each edge to replace the physical fluxfunction by a consistent numerical flux function. With an appropriate discretisation in time, thisapproach has been employed for the simulation of both general inviscid [3] and two-dimensionallaminar viscous [4] flow problems.

In this paper, this basic approach is extended to produce an initial capability for the computa-tion of three-dimensional turbulent aerodynamic flows. The approach which is adopted is basedupon the experience gained in an earlier study covering a wide range of flow regimes, of thebehaviour of two equation turbulence models implemented on general triangular grids [5,6]. Thisstudy confirmed that, subject to the well-known drawbacks of these models, turbulent flowfeatures, such as boundary layers and shock-boundary layer interactions, could be well-representedon an unstructured triangular grid. However, the accuracy of the computed solution was found todegrade significantly if adequate attention was not given to the quality of the grid or to theconstruction of the added diffusion.

For the three-dimensional extension, stabilisation and discontinuity capturing is accomplishedby an unstructured grid implementation of the JST scheme [7], which involves a blend offourth- and second-order operators, and the selected turbulence model is the low Reynoldsnumber k—u model [8]. An explicit multistage time-stepping scheme is used to advance thesolution of the resulting equation system towards steady state. The CPU requirements ofthe resulting procedure can be expected to be significant when realistic flows are simulated and,for this reason, the parallelisation of the solution algorithm is described. The generation of anunstructured grid, which is appropriate for a turbulent viscous flow simulation over a generalthree dimensional body, is a non-trivial task and the manner in which this can be accomplishedis only briefly considered in this paper. The numerical performance of the flow algorithm isdemonstrated by considering the simulation of a transonic separated flow over an ONERA M6wing.

2. Governing equations

The time averaged form of the unsteady compressible Navier—Stokes equation is considered. Inthe time-averaging process, the mass average is used for the velocity components and the totalenergy, while the conventional time average is employed for the density and the pressure. Asa result, the governing equations are expressed in terms of mean flow variables and closure isachieved by the addition of transport equations for the turbulence kinetic energy k and the specificturbulence dissipation u. The resulting system of equations, non-dimensionalised in terms of thedensity, velocity and molecular viscosity of the free stream and a characteristic length of theproblem, is expressed in the conservation form

LULt

#

LFj

Lxj

"

LGj

Lxj

#S, (1)

354 M.¹. Manzari et al. / Finite Elements in Analysis and Design 30 (1998) 353—363

where summation over the repeated index j"1,2,3 is implied and

U"

o

ou1

ou2

ou3

oE

ok

ou

, Fj"

ouj

ou1uj#pd

1jou

2uj#pd

2jou

3uj#pd

3j(oE#p)u

joku

jouu

j

, Gj"

0

q1j

q2j

q3j

uiqij!q

j#D

kjD

kjDuj

, S"

0

0

0

0

0

Pk!D

kPu!Du

. (2)

In these equations, t denotes time, Ox1x2x3

is a Cartesian coordinate system, uiis the fluid velocity

in direction xi, o is the fluid density, p is the fluid pressure and d

ijis the Kronecker delta. The total

energy per unit mass of the fluid, E, is defined according to

E"e#uiui/2#k, (3)

where e is the mass averaged specific internal energy. The fluid is assumed to be an ideal gas, withconstant specific heat ratio c, which obeys the equation of state

p"(c!1)

co¹, (4)

where ¹ is the dimensionless temperature. The stress tensor, qij, includes the effect of both the

molecular and the eddy viscosity and is defined as

qij"

2 kRe Aeij!

13Lu

kLx

k

dijB#q5

ij, e

ij"

12A

Lui

Lxj

#

Luj

LxiB. (5)

In this expression, the turbulent stress tensor, q5ij, is determined, from the Boussinesq hypothesis, in

the form

q5ij"

2k5

Re Aeij!13

Luk

Lxk

dijB!

23okd

ij. (6)

The molecular dynamic viscosity, k, varies according to Sutherland’s law, while k5"a*okRe/u is

the turbulent viscosity and Re is the free stream Reynolds number. The component, qj, of the heat

flux vector in the direction xjis defined as

qj"!

1ReA

kPr

#

k5

Pr5B

L¹Lx

j

, (7)

where Pr and Pr5denote the molecular and turbulent Prandtl numbers, respectively. The quantities

Dkj

and Dujare defined by the expressions

Dkj"

1Re Ak#

k5

pkB

LkLx

j

, Duj"

1ReAk#

k5

puBLuLx

j

, (8)

M.¹. Manzari et al. / Finite Elements in Analysis and Design 30 (1998) 353—363 355

while the production and dissipation of k and u are represented by the terms

Pk"q5

ij

Lui

Lxj

, Pu"auk

Pk, D

k"b*ouk, Du"bou2 (9)

The standard values

b"3/40, b*"0.09, a"5/9, a*"1, pk"pu"2 (10)

are assigned to the closure constants.

2.1. Initial and boundary conditions

Suppose that the solution is required in a spatial domain, X, which is bounded by a closedsurface, C, with unit outward normal vector n"(n

1, n

2, n

3). The correct specification of a problem

governed by Eq. (1) will then require the definition of an initial condition and appropriateboundary conditions. For the initial condition, it will be assumed that free stream values areimposed everywhere in X at some time t"t

0. At a wall boundary, the no slip condition u

i"0 is

imposed. If the wall is assumed to be isothermal, the temperature of the wall will also be prescribed,while a zero heat flux condition is imposed if the wall is assumed to be adiabatic. The turbulencekinetic energy, k, is set to zero at a wall while u is required to demonstrate the correct asymptoticbehaviour as the wall is approached. At a far-field boundary, the conditions which have to beimposed will depend upon the local nature of the flow.

3. Solution algorithm

3.1. Spatial discretisation

The problem which has been described is first cast in the alternative weak variational form: findU such that

PX

LULt

¼dX"PXFj

L¼Lx

j

dX!PCFM jn

j¼dC

!PXGj

L¼Lx

j

dX#PCGM jn

j¼dC#PX

S¼dX

(11)

for all suitable weighting functions, ¼, and for all t't0. In this expression, an overbar represents

a prescribed normal boundary flux.The domain X is discretised using tetrahedral elements, with nodes numbered 1,2, p located at

the element vertices. A piecewise linear approximate solution is assumed in the form

U+U (p)"UJ(t)N

J(x), (12)


where J takes the values 1,2, p in the implied summation, NJrepresents the linear finite element

shape function associated with node J and UJ

is the value of the approximation at node J.A Galerkin approximate solution is produced by using the variational formulation of the problemin the form: find U (p) such that

PX

LU (p)

LtN

IdX "PX

Fj(p)LN

ILx

j

dX!PCFM j(p)n

jN

IdC

!PXGj(p)

LNI

Lxj

dX#PCGM j(p)n

jN

IdC#PX

S(p)NIdX

(13)

for I"1,2,2,p and for all t't0. The integrals appearing in this Galerkin statement are evaluated,

using an edge-based data structure [3], in the form

CMdUdt D

I

"!

mI

+s/1

C jIIs2

M(F jI#F j

Is)!(G j

I#G j

Is)N#[M

L]ISI

#TlI

+f/1

DfM(6FM n

I#FM n

If1#FM n

If2)!(6GM n

I#GM n

If1#GM n

If2)NU

I

,

(14)

where the summations here extend over the mIedges and the l

Iboundary faces which are connected

to node I, and edge s connects nodes I and Is, while boundary face f has nodes I, I

f1and I

f2. Note

that the terms S ) TIare then only non-zero when node I is a boundary node. The weights Cj

IIsand

Df

are computed as

C jIIs"! +

E|IIs

XE

2 CLN

ILx

jDE

#T +f|IIs

Cf

12njU, D

f"!

Cf

24. (15)

Here njis the component in the x

jdirection of the unit normal to the boundary face f, X

Eis the

volume of element E and Cf

is the area of face f. An advantage of this edge data structure is that itleads to savings in both CPU and memory requirements for 3D simulations. For the steady flowanalysis which is of interest here, the consistent finite element mass matrix M which appears inEq. (4) is replaced by the diagonal lumped mass matrix M

L. The nodal values of the gradients of U,

which are required before the viscous fluxes G jI

can be evaluated for use in Eq. (14), are alsoobtained in a variational form [4] as

CML

LULx

jDI

"

mI

+s/1

C jIIs2

(UI#U

Is)!T

lI

+f/1

Df(6UM n

I#UM n

If1#UM n

If2)U

I

. (16)

Eq. (14) represents a central difference type of approximation to the spatial derivatives and, assuch, will require stabilisation before it can be applied to the simulation of smooth compressibleflows. In addition, a discontinuity capturing capability is required if general flows are to besuccessfully modelled. In the current context, stabilisation and discontinuity capturing is achievedby the explicit addition of diffusion to this semi-discrete equation.


3.2. Stabilisation and discontinuity capturing

In computational aerodynamics, many successful algorithms have used an added diffusion of theform that was originally proposed in the JST scheme [7]. Following these ideas, in the presentcontext, the diffusion D

Iadded at a general node I is constructed as a blend of approximations to

second- and fourth-order operators as

DI"

mI

+s/1Ae(2)IIs

UIs!U

Im

I

!e (4)IIs

(+ 2UI s!+ 2U

I)B

36 min(jI, j

Is)

mI#m

Is

, (17)

where the second-order operator is approximated according to

+ 2UI+

1m

I

mI

+s/1

(UIs!U

I). (18)

Here j is the maximum eigenvalue of the Jacobian matrix ljLFj/LU in absolute value, where

l"(l1, l2, l3) is the unit vector in the direction of the edge II

s. The parameters e (2)

IIsand e (4)

IIsare

defined by

e(2)IIs

"i(2)max(PI,P

Is) e(4)

IIs"max(0,i(4)!i(2)max(P

I,P

Is)), (19)

where

PI"

mI

+s/1

(pIs!p

I)/

mI

+s/1

(pIs#p

I) (20)

is the nodal value of a pressure switch.

3.3. Time discretisation

A three-stage scheme is employed to advance the solution from time level t"tn

to time levelt"t

n`1"t

n#Dt. Within each time step, this scheme is implemented in the form

U (0)I

"U nI,

U (k)I

"U nI!a

kDt [M

L]~1(R(k~1)

I!D(0)

I) for k"1, 2, 3,

U n`1I

"U (3)I

. (21)

Here R(k~1)I

represents the right-hand side of Eq. (14) computed at the stage k!1, while the addeddiffusion D

Iis held constant at the value computed at tn. The values a

1"0.6, a

2"0.6 and a

3"1

are adopted for the coefficients in Eq. (21).

4. Parallel solution algorithm

Before parallel version of this solver can be employed, the mesh which has been designed for thesimulation must be partitioned into an appropriate number of smaller meshes. This is achieved bycolouring the nodes in the mesh, using recursive spectral bisection [9]. Interior and interface edgesare identified for each sub-domain. An edge which connects two nodes of the same colour, I, is an


interior edge for sub-domain I. An edge which connects a node of colour I and a node of colourJ, where I(J, will be an interface edge in sub-domain I. In this case, the node of colour I andthe node of colour J will be interior nodes for the sub-domains I and J, respectively. The node ofcolour J will be duplicated as an interface node in sub-domain I. Local numbering of nodes,elements, edges and boundary faces is employed within each sub-domain. The communicationarrays, which are necessary for the transformation of information between the sub-domains, areevaluated during the domain partitioning stage.

A parallel implementation of the flow solver is achieved by using the single program multipledata concept, in conjunction with the use of standard PVM routines for message passing. Noattempt has been made to optimise the coding for performance on any particular computerplatform. In the serial version of the algorithm, the main computation consists of loops over edgesin the mesh. Information from points is gathered to edges and edge information is scattered andadded to points. In the parallel implementation, at the start of each time step, the interface nodesobtain contributions from the interface edges. These partially updated interface nodal contribu-tions are then broadcast to the corresponding interior nodes in the neighbouring sub-domains.A loop over the interior edges is followed by the receiving of the interface node contributions andthe subsequent updating of all interior nodal values. The sending of the updated values back to theinterface nodes completes a time step of the procedure. The process is implemented in such a waythat it attempts to allow computation and communication to take place concurrently.

5. Mesh generation

The efficient computation of problems involving compressible turbulent flow at realisticReynolds numbers will require the use of meshes having regions of highly stretched elements closeto any solid boundary. The approach which is being employed to generate unstructured meshes ofthis form involves three separate phases [10]. In the first phase, the individual boundary surfacecomponents are triangulated, with the desired variation in the element size distribution beingcontrolled by a user-specified mesh distribution function. In the second phase, unstructured layersof stretched tetrahedral elements are generated adjacent to those boundary surface componentswhich represent solid walls. The height and number of these layers is specified by the user in sucha way that the expected boundary layer profile should be capable of being adequately represented.The layers are constructed by generating points along prescribed lines and connecting thegenerated points, by using advancing front mesh generation concepts, to form tetrahedral elements.Point generation will cease before the prescribed number of layers is reached if the local mesh size isclose to that specified in the user-specified mesh distribution function. In the third phase, theremainder of the domain is discretised, according to the requirements of the user-specified meshdistribution function, using a Delaunay procedure.

6. Example

To demonstrate the performance of the solution algorithm, transonic flow over an ONERA M6wing is considered. The case which is chosen has been extensively studied by other authors [2] and


Fig. 1. Computational surface grid.

involves flow features such as separation and shock-boundary layer interaction. The free streamconditions are defined by a Mach number of 0.8447 and an angle of attack 5.06°, while theReynolds number is 11.7]106 based on the mean aerodynamic chord. The surface grid used forthis computation is shown in Fig. 1 and contains 48 056 triangular elements on the wing. In theviscous mesh generation process, 14 layers of stretched elements were generated about the wing,with the height of the elements in the first layer being 10~5 the mean aerodynamic chord, so thatthis layer lies totally inside viscous sub-layer. The complete volume grid consists of 2 788 768tetrahedral elements and 464 736 nodal points. The simulation was performed on a CRAY T3D,using 256 processors, and required 12]10~6 s per time step per nodal point. The computeddistribution of the pressure contours, after 25 000 time steps, on the wing surface and on thesymmetry plane is shown in Fig. 2. The computed variation in the pressure across the wing, atdifferent spanwise sections, is compared with the experimental data [11] in Fig. 3. It is seen that the


Fig. 2. Pressure coefficient contour plot.

computed results are in good agreement with the experimental data, considering the coarseness ofthe grid which has been employed.

7. Conclusions

A finite-element-based algorithm has been presented for the simulation of three-dimensionalcompressible turbulent flows on unstructured tetrahedral grids. Turbulence is modelled with the


Fig. 3. Comparison of computed Cp

(line) with experimental data (circle) at 20%, 44%, 65%, 80%, 90% and 95% ofwing span.


two-equation k—u model and the resulting numerical scheme is implemented in terms of an edge-based data representation of the mesh. The numerical results which are obtained for a flow over anONERA M6 wing, using a relatively coarse grid, give confidence that the method can besuccessfully applied to simulations involving more complex geometries. Although a parallel versionof the full procedure has been successfully developed, it is apparent that some form of implicitnesswill also need to be added before this approach can be routinely employed in the industrialenvironment.

Acknowledgements

The authors gratefully acknowledge the support of British Aerospace AIRBUS Ltd, BritishAerospace Military Aircraft Division, and DERA, Farnborough. Access to the CRAY T3D at theEdinburgh Parallel Computing Centre was provided by the UK Engineering and Physical SciencesResearch Council under Research Grant GR/K42264.

References

[1] J. Peraire, K. Morgan, J. Peiro, Unstructured finite element mesh generation and adaptive procedures for CFD, in:Application of Mesh Generation to Complex 3D Configurations, Conf. Proc. No. 464, AGARD, Paris, 1990, pp.18.1—18.12.

[2] N.T. Frink, Assessment of an unstructured-grid method for predicting 3D turbulent viscous flows, AIAA Paper96—0292, 1996.

[3] J. Peraire, J. Peiro, K. Morgan, Multigrid solution of the 3-D compressible Euler equations on unstructuredtetrahedral grids, Int. J. Numer. Methods Eng. 36 (6) (1993) 1029—1044.

[4] P.R.M. Lyra, M.T. Manzari, K. Morgan, O. Hassan, J. Peraire, Upwind side-based unstructured grid algorithmsfor compressible viscous flow computations, Int. J. Eng. Anal. Des. 2 (1995) 197—211.

[5] M.T. Manzari, K. Morgan, O. Hassan, Compressible turbulent flow computations on unstructured grids, in:M. Hafez (Ed.), Computational Fluid Dynamics Reviews, Wiley, New York, 1997, in press.

[6] M.T. Manzari, K. Morgan, O. Hassan, Transonic flow computations using two-equation turbulence models, Int.J. Numer. Methods Eng. (1998), submitted.

[7] A. Jameson, W. Schmidt, E. Turkel, Numerical simulation of the Euler equations by the finite volume method usingRunge—Kutta time stepping schemes, AIAA Paper 81-1259, 1981.

[8] D.C. Wilcox, Reassessment of the scale determining equation for advanced turbulence models, AIAA J. 26 (11)(1988) 1299—1310.

[9] H.D. Simon, Partitioning of unstructured problems for parallel processing, Comput. Systems Eng. 2 (1991)135—148.

[10] O. Hassan, E.J. Probert, K. Morgan, J. Peraire, Mesh generation and adaptivity for the solution of compressibleviscous high speed flows, Int. J. Numer. Methods Eng. 38 (7) (1995) 1123—1148.

[11] V. Schmitt, F. Charpin, Pressure distributions on the ONERA M6 wing at transonic mach numbers, AGARDReport AR-138, Paris, 1979.


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Turbulent flow computations on 3D unstructured grids

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