SUPG finite element computation of viscous ...

This is the updated version of the article that appeared as… S.K. Aliabadi, S.E. Ray, and T.E. Tezduyar, “SUPG finite element computation of viscous compressible flows based on the conservation and entropy variables formulations”, Computational Mechanics, 11 (1993) 300–312, http://dx.doi.org/10.1007/BF00350089

Updated References 1. S.K. Aliabadi and T.E. Tezduyar, "Space-Time Finite Element Computation of Compressible Flows Involving Moving Boundaries and Interfaces", Computer Methods in Applied Mechanics and Engineering, 107 (1993) 209-223. 6. T.J.R. Hughes and T.E. Tezduyar, "Finite Element Methods for First-order Hyperbolic Systems with Particular Emphasis on the Compressible Euler Equations", Computer Methods in Applied Mechanics and Engineering, 45 (1984) 217-284. 9. G.J. Le Beau, S.E. Ray, S.K. Aliabadi and T.E. Tezduyar, "SUPG Finite Element Computation of Compressible Flows with the Entropy and Conservation Variables Formulations", Computer Methods in Applied Mechanics and Engineering, 104 (1993) 397-422. 10. G.J. Le Beau and T.E. Tezduyar, "Finite Element Computation of Compressible Flows with the SUPG Formulation", Advances in Finite Element Analysis in Fluid Dynamics, FED-Vol. 123, ASME, New York (1991) 21-27. 15. T.E. Tezduyar and T.J.R. Hughes, "Finite Element Formulations for Convection Dominated Flows with Particular Emphasis on the Compressible Euler Equations", AIAA Paper 83-0125, Proceedings of AIAA 21st Aerospace Sciences Meeting, Reno, Nevada (1983).

Computational Mechanics (1993) 11, 300-312

Computational Mechanics O Springer-Verlag 1993

SUPG finite element computation of viscous compressible flows based on the conservation and entropy variables formulations S. K. Aliabadi, S. E. Ray and T. E. Tezduyar Department of Aerospace Engineering and Mechanics, Army High Performance Computing Research Center, and Minnesota Supercomputer Institute, University of Minnesota, 1200 Washington Avenue South, Minneapolis, MN 55415, USA

Abstract. In this article, we present our investigation and comparison of the SUPG-stabilized finite element formulations for computation of viscous compressible flows based on the conservation and entropy variables. This article is a sequel to the one on inviscid compressible flows by Le Beau et al. (1992). For the conservation variables formulation, we use the S U P G stabilization technique introduced in Aliabadi and Tezduyar (1992), which is a modified version of the one described in Le Beau et al. (1992). The formulation based on the entropy variables is same as the one introduced in Hughes et al. (1986). The two formulations are tested on three different problems: adiabatic flat plate at Mach 2.5, Reynolds number 20,000; Mach 3 compression corner at Reynolds number 16,800; and Mach 6 NACA 0012 airfoil at Reynolds number 10,000. In all cases, we show that the results obtained with the two formulations are very close. This observation is the same as the one we had in Le Beau et al. (1992) for inviscid flows.

1 Introduction One of the challenges involved in computation of compressible flows is stabilization of the numerical formulation to prevent spurious oscillations generated by the dominance of the advection terms in the differential equation. The challenge becomes even more formidable in the presence of shocks and sharp layers, and for high-speed flows. Improper stabilizations may result in excessive numerical diffusion (i.e., "over-stabilization") and therefore loss of accuracy. The SUPG (streamline-upwind/Petrov-Galerkin)stabilization technique was first introduced, for the advection-diffusion equation and for incompressible flows, by Hughes and Brooks (1979). The technique was investigated in detail and applied to various problems in Brooks and Hughes (1982). The SUPG techniques for compressible flows were first introduced, in the context ofinviscid flows, by Tezduyar and Hughes (1982, 1983). These techniques were applicable to general hyperbolic systems. The article by Hughes and Tezduyar (1984), which might be easier for the interested reader to locate, is the journal version of Tezduyar and Hughes (1982), and includes some additional numerical examples. The SUPG techniques are consistent stabilization methods, in the sense that the exact solution still satisfies the stabilized formulation, just like it satisfies the Galerkin formulation of the problem. The SUPG formulations involve minimal numerical diffusion, and therefore result in good spatial accuracy. Many stabilization techniques for compressible flows similar to those introduced in Tezduyar and Hughes (1982) have appeared in the literature during the past decade. For example, the Taylor-Galerkin stabilization method promoted by Donea (1984) is very similar (under certain conditions identical) to one of the stabilization methods introduced in Tezduyar and Hughes (1982). The SUPG stabilization introduced in Tezduyar and Hughes (1982) is based on the conservation variables formulation of the compressible flow equations, and does not involve any shock-capturing terms. A similarly stabilized formulation, based on the entropy variables, was introduced in Hughes et al. (1986), this one including a shock-capturing term. Recently, it was shown by Le Beau and Tezduyar (1991) and Le Beau et al. (1992) that the SUPG stabilization technique introduced in Tezduyar and Hughes (1982), supplemented with a shock-capturing term, gives solutions which are as good as those obtained with the entropy variables. For all test problems

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301

computed in Le Beau et al. (1992), the solutions obtained with the two formulations are barely distinguishable. A slightly modified version of the SUPG stabilization technique given in Le Beau and Tezduyar (1991) and Le Beau et al. (1992) was introduced and applied to viscous compressible flows by Aliabadi and Tezduyar (1992). It was pointed out in Aliabadi and Tezduyar (1992) that the SUPG matrix used in Le Beau and Tezduyar (1991) and Le Beau et al. (1992) sometimes results in excessive numerical damping at high Mach numbers, and that this is due to nonoptimal selection of the SUPG matrix with respect to the spatial coordinates and the shock-capturing term. The modified SUPG matrix introduced in Aliabadi and Tezduyar (1992) takes into account the presence of the shock-capturing term and also the physical diffusion. In this article, which is a sequel to the one on inviscid flows by Le Beau et al. (1992), we investigate and compare the SUPG-stabilized finite element formulations for viscous flows, based on the conservation and entropy variables. We are particularly interested in evaluating the performances of these formulations in high Mach number and high Reynolds number flows. For this purpose, we consider three test problems: adiabatic flat plate at Mach 2.5, Mach 3 compression corner, and Mach 6 NACA 0012 airfoil.

2 The governing equations Let .(2 c R "~ and (0, T) be the spatial and temporal domains, respectively, where nsd is the number of space dimensions, and let F denote the boundary of ~. The spatial and temporal coordinates are denoted by x and t. We consider the conservation law form of the Navier-Stokes equations governing unsteady compressible flows with no source terms: 0U aFi +-~?t ~?x~

~E i

=0

on ~ x (0, T),

(1)

~x~

where U = (p, p u l , . . . , pu,s,,, pe) is the vector of conservation variables, F~ and E~ are, respectively, the Euler and viscous flux vectors defined (for nsd = 3) as

Fi

uip

0

UipU 1 -']- 6il p

Til

UipU 2 -'[- t~i2p

and

Ei =

Ti2

uipu3 -}- (~i3P

Ti3

ui(Pe + P)

--qi-~-ZikUk

(2, 3)

and repeated indices imply summation over the range of the spatial dimension. Here p, n,p, and e are, respectively, density, velocity, pressure, and the total energy per unit mass. The identity tensor is denoted by 6ij. Pressure is related to the other variables with an equation of state of the form (4)

p = p(p, i),

where i is the internal energy per unit mass, defined as i=e- 89

2.

(5)

For ideal gases, the equation of state given by (4) takes the special form p = (? - 1)pi,

(6)

where ? is the ratio of the specific heats. The viscous stress tensor -r~jand the heat flux vector q~ are defined as

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( ( c~u i

c~u j "]

2

C~Uk \

aO

(7,8)

where 0 is the temperature, and the coefficients of viscosity and thermal conductivity are assumed to be given by Sutherland's formulae:

[ 0 "~3/2O0 .~_ 110 #

=

#~

0 + 110

7R# ,

(7-- l)Pr'

(9,10)

where #o is the viscosity at the reference temperature 00, R is the gas constant, and Pr is the Prandtl number. Alternatively, Eq. (1) can be written in the quasi-linear form -c~U -+At c~t

c~U ~Xi

Ki~

=0

on D x (0, r),

(11)

i

where A~-

c~F~

~U

and

K ~ j~3U - - = Ei.

(12, 13)

~xj

To rewrite Eq. (11) in terms of entropy variables, as it is done in Hughes et al. (1986), we employ the variable transformation

- Us+Pi(7+l_s+So

)-

U2 U3

U(U) = _1 pt

(14)

U4

--U 1 where the internal energy pi and the entropy s are expressed as

pi= U5

U2+U2+U2 2U, '

9 s = i n F(7--1)pi].L U~ A

(15, 16)

Then the equation system takes the form A o+-_7 &

i--

Oxl

- ! ~xi\

Kij-

~xj.]! = 0

onDx(O,T),

(17)

where A-o = QU fj,

*i = A,Ao,

Kij = KijAo

(18-20)

In this form, Ao is a symmetric and positive-definitematrix, .~'s are symmetric matrices, and the matrix [Kij] is symmetric and positive-semidefinite. It is assumed that appropriate sets of boundary and initial conditions are specified in conjunction with both Eqs. (11) and (17). 3 The finite element formulation

Consider a finite element discretization of the domain D into subdomains (elements) D e, e = 1, 2 .... , net, where net is the number of elements. Based on this discretization, respectively corresponding to the trial solutions and weighting functions, we define the finite element function spaces S h and W h for conservation variables, arm ~h and qdh for entropy variables. These function

S, K. Aliabadi et al.: SUPG finite element computation of viscous compressible flows

303

spaces are selected, by taking the Dirichlet boundary conditions into account, as subsets of [H~h(D)]"'d+2, where Hlh(D) is the finite dimensional function space over D. The stabilized finite element formulation of (11) is written as follows: find u h ~ s h such that vwh~v h

~w".\ at

axi )

~- ~\ ax i

,/ ~

'J-~xj )

-- I (wh'H)d/-'

+ ~ j" ,tA[( ~gWh']('Ouh+ OUh ~ (KuOUh)']d \-~xkj't,---~- A, ~Xi QX, \ ~Xi J J e=l a~

D

=0.

-It- e=lE ..Oe \ aX i /] k,~Xi/]

(21)

The stabilized finite element formulation of (17), on the other hand, is written as follows: find O h e g h such that VWhe~r h +

..,

R

~172-. I'I~o--+ \ Xk / \ at i,--8X i

+ e=Y',l t2j~ -I- E

+

-

~

l aqc"k / afs"h

I aAoti)'t--)

= 1 a~

c~x,

~gxl

)

~xj ) /

dD (22)

do:O"

Remark 1 In Eqs. (21) and (22), the first three integrals represent the Galerkin formulation of the problem; the third integral accounts for the Neumann boundary conditions. The first series of element level integrals are the SUPG stabilization terms added to the variational formulation, whereas the second series of element level integrals are the shock-capturing terms.

Definition of~ and ~. The SUPG matrix x, used with the conservation variables formulation, is the same as the one introduced by Aliabadi and Tezduyar (1992) for viscous compressible flows: = max (0, % - % - %),

(23)

where x. is the stabilization matrix due to the advection terms, and % and % are the matrices subtracted from % to account for the presence of the shock-capturing term and the physical diffusion, respectively. These matrices are defined as max(hilflil)I (no sum),

ra - 2(c + lu-[~l) xa =

6

%-

2I,

(24,25)

(c + lu-Pl)

/~ diag K11 +/722 diag Kz2 + fl~ diag K33

,

(26)

(c + lu.Pl) ~ where c is the acoustic speed,

1~-[iVllUll2,112,V IIU LL2,

and

hi

is the element length defined by Tezduyar and Hughes (1982),

[l"II, -- [1"112

or

IIll Xo'

(27, 28)

The SUPG matrix ~, used with the entropy variables formulation, is the same as the one described in Hughes et al. (1986), which is a generalization from the SUPG matrix defined for inviscid flows. In the inviscid case the matrix ~ is defined in Hughes et al. (1986) as

. ~ l r~, = Z, r 'l&l

(29)

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where 2g and r i are respectively the eigenvalues and eigenvectors given by (B* -- 22.~o)r I = O,

(30)

in which ~ , = (~2 + ~ + ~2)~0,

~i = ~r (31, 32) 8xj and r 1= 1, 2,..., nsd are the element coordinates. For viscous flows, to account for the physical diffusion present, the matrix ~ is modified in Hughes et al. (1986) as ~=~r'f(~ ~[2i[ ,,

where f ( ~ ) = { ~ / 3

0