High Order Finite Element Discretization of the Compressible
Euler and Navier-Stokes Equations
J.S. Wong, D.L. Darmofal, and J. Peraire
Department of Aeronautics and Astronautics,
Massachusetts Institute of Technology, Cambridge, MA 02139
Abstract
We present a high order accurate streamline-upwind/Petrov-Galerkin (SUPG) algorithm for the solution of
the compressible Euler and Navier-Stokes equations. The ow equations are written in terms of entropy
variables which result in symmetric ux Jacobian matrices and a dimensionally consistent Finite Element
discretization. We show that solutions derived from quadratic element approximation are of superior quality
next to their linear element counterparts. We demonstrate this through numerical solutions of both classical
test cases as well as examples more practical in nature.
Key Words: Euler and Navier-Stokes equations, Petrov-Galerkin, entropy variables, symmetric ux jacobian matrices, high order accuracy
1 Introduction
This paper centers upon a high-order accurate, stabilized, nite element method for the numerical solution
of the compressible Euler and Navier-Stokes equations. The SUPG nite element method for compressible
ow simulations was initially developed and analyzed by Hughes et al. [3, 4, 5, 2] and has since gained
signicant popularity. Its relation to multidimensional upwinding was elucidated in [9, 10] and higher order
implementations for inviscid ows were presented in [6]. In [1], the SUPG algorithm was extentded to cover
the simulation of near-incompressible ows by employing a stabilization matrix which exhibits proper scaling
over the entire range of Mach numbers. Here, we focus on the higher order implementation of the algorithm
developed in [1] for invicid and viscous ows.
1
2 Compressible ow governing equations
We start from the time dependent two dimensional compressible Euler equations in conservation form
U;t + (F
where
2
3
66
7
66 u1 777
U=6
;
64 u2 775
E
and
2
66
6
Fv1 = 6
66
4
Fv )1;1 + (F
2
66 u1
6 u21 + p
F1 = 6
66 u u
1 2
4
u1 (E + p)
0
11
12
u1 11 + u212 + q1
3
77
77
77 ;
5
Fv )2;2 = 0;
3
77
77
77 ;
5
2
66
6
Fv2 = 6
66
4
(1)
2
66 u2
6 u1u2
F2 = 6
66 u2 + p
2
4
u2 (E + p)
0
21
22
u121 + u2 22 + q2
3
77
77
77 :
5
3
77
77
77 :
5
In the above expressions, is the density; u = [u1 ; u2]T is the velocity vector; E is the specic total energy;
p is the pressure; and the comma denotes partial dierentiation (e.g. U;t = @ U=@t, the partial derivative
with respect to time, Fi;j = @ Fi =@xj , the partial derivative with respect to the j -th spatial coordinate). The
system of equations is closed once the pressure is related to the problem variables through the equation of
state, p = ( 1)e, where e = E juj2 =2, is the internal energy. Here, is the ratio of specic heats and
is the absolute viscosity, both of which are assumed to be constant. Following the usual assumptions:
11 = 2
@u1
Re @x1
12 = 21 =
22 = 2
and
@u2
Re @x2
2 @u1 @u2
+
3 Re @x1 @x2
@u1 @u2
+
Re @x2 @x1
2 @u1 @u2
+
3 Re @x1 @x2
@T
@T
; q2 =
:
ReP r @x1
ReP r @x2
We will assume that all the above quantities have been non-dimensionalized using reference, or free stream,
values for density , velocity u , and length L. Thus, the dimensional variables, denoted with an overbar,
are related to the non-dimensional variables introduced above as
E
u
p
x
u t
= ; ui = i ; i = 1; 2; p =
; E = 2 ; = ; xi = i ; i = 1; 2; and t = :
2
u
u
u
L
L
We note that the equation system (1) can be written as
q1 =
U;t + A1 U;1 + A2 U;2 = (K11 U;1 );1 + (K12 U;2 );1 + (K21 U;1 );2 + (K22 U;2 );2 ;
2
(2)
where the Jacobian matrices Ai = Fi;U , i = 1; 2, are unsymmetric but have real eigenvalues and a complete
set of eigenvectors. Kij = Fvi;U are the viscous ux jacobians. The above equation may be symmetrized
through a change of variables, for details, we refer the reader to [5, 7].
;j
2.1 Entropy variables
We seek a new set of variables V, called entropy variables, such that the change U = U(V) applied to (1)
yields the transformed system
~ 1 V;1 + A
~ 2 V;2 = (K
~ 11 V;1 );1 + (K
~ 12 V;2 );1 + (K
~ 21 V;1 );2 + (K
~ 22 V;2 );2 ;
A0 V;t + A
(3)
~ i = Ai A0 = Fi;V , i = 1; 2, are symmetric.
where A0 = U;V is symmetric positive denite, and A
Following [7], we introduce a scalar entropy function H (U) = g(s), where s is the non-dimensional
entropy s = ln(p= ). The required change of variables is obtained by taking
2
0
2
66 e( g=g ) juj =2
u1
g0 6
V = H;TU = 6
6
e6
u2
4
1
3
77
77
77 :
5
(4)
The conditions g0 > 0 and g00 =g0 < 1, ensure that H (U) is a convex function and therefore A0 1 = V;U =
H;UU , and A0 , are symmetric positive denite. Furthermore, if we chose g(s) = s then we insure the
matrix K
2
=4
K
~ 11 K
~ 12
K
~ 21 K
~ 22
K
3
5
is symmetric as well as positive semi-denite.
3 Variational formulation for the steady state problem
We now consider the compressible steady problem in conservation form expressed in terms of symmetrizing
variables. The conservative form of the equations is taken to be the starting point because we are ultimately
interested in an algorithm that can be used over the whole range of speed regimes, including situations were
the solution may contain discontinuitites. The problem is dened in a domain with boundary by
(F + Fv )1 (V);1 + (F + Fv )2 (V);2 = 0 in ;
~nV = A
~ n g on
A
Fv n = f on
(5)
n a;
(6)
a
(7)
For simplicity, the domain boundary is assumed to be made up of a solid wall a , and a computational far
~ n = An A0 ,
eld boundary n a . In (6, 7), n = [n1 ; n2 ]T is the outward unit normal vector to , and A
~ n = An A0 , and An denotes the negative denite part of An .
An = A1 n1 + A2 n2 . Finally, A
3
S
Let the spatial domain , be discretized into non-overlapping elements Te , such that = Te , and
T
Te Te = ;, e 6= e0 . We consider the space of functions Vh, dened over the discretization and consisting of
the continuous functions which are piecewise linear over each element
0
Vh = fW j W 2 (C 0 (
))4 ;
W jT
e
2 (Pk (Te ))4 ; 8 Te 2 g:
The SUPG algorithm can then be written as: Find Vh 2 V h such that for all W 2 V h ,
B (Vh ; W)gal + B (Vh ; W)supgjgls + B (Vh ; W)bc = 0;
(8)
where the forms B (; )gal , B (; )supg and B (; )bc account for the Galerkin, SUPG stabilization, and boundary condition terms respectively, and are dened as
B (V; W)gal =
( W;1 (F Fv )1 (V) W;2 (F Fv )2 (V)) d
;
~ 1 W;1 + A
~ 2 W;2 ) (A
~ 1 V;1 + A
~ 2 V;2
(A
(K22 V;2 );2 ) d
;
B (V; W)supg =
(K21 V;1 );2
Z
Z
(K11 V;1 );1
(K12 V;2 );1
Z
(K21 W;1 );2
~ 1 W;1 + A
~ 2 W;2 (K11 W;1 );1 (K12 W;2 );1
((A
~ 1 V ;1 + A
~ 2 V;2 (K11 V;1 );1 (K12 V;2 );1
(K22 W;2 );2 ) (A
(K21 V;1 );2
(K22 V;2 );2 ) d
;
B (V; W)gls =
and
Z
Z
(9)
(10)
(11)
W (Fff
W Fv (V; f ; n) ds:
(12)
n
where is the stabilization matrix. The numerical ux function on the far eld boundary Fff , is dened by
1
1
Fff (V ; V+ ; n) = (Fn (V ) + Fn (V+ ))
jA (V (V ; V+ ))j(V+ V ):
2
2 n
Here, jAn (V)j = A+n (V) A (V) is the absolute value of An evaluated at V , and V (V+ ; V ), is the an
average between the states V+ and V . The average state, V , is chosen to ensure the global stability of the
algorithm [6]. For inviscid compuations, the viscous terms in the expressions above would of course, vanish.
For viscous simulations, Dirichlet boundary conditions may replace portions of the boundary integral.
B (V; W)bc =
+ Fv )(V; g; n) ds: +
a
a
4 Denition of The denition of follows that of [1]. The following modication is made for viscous simulations:
1
1+
= i
v
1
where i is the stabilization dened in [1] and v is dened as:
v 1
~ 11 + K
~ 22 )=he 2
= 2(K
4
5 Numerical results
In this section we present some numerical results that illustrate the performance of the proposed algorithm.
Test problems were solved employing both linear and quadratic element approximations. For comparative
purposes, the meshes used for all linear element approximations were obtained by subdividing each element
of the corresponding quadratic element mesh into four linear elements. In this way, comparisons between P1
and P2 solutions involving the same number of nodes can be made. hc thus represents the distance between
two nodes in the meshes used in the numerical simulations presented herein.
5.1 Example 1: Rinleb ow
In this example, we consider a ringleb test case (an exact solution of the Euler equations, [13]). The error
is computed in the L2 entropy norm [1]. Both P1 and P2 element approximation achieved their respective
optimal convergence rate of O(h2 ) and O(h3 ) respectively, as can be seen in gure 1.
5.2 Example 2: Flow over an airfoil
In this example, the proposed scheme was used to simulate the ow over NACA 0012 airfoil at a Mach
number of 0:6, and at an angle of attack of 2Æ. In gure 2, the L2 entropy deviation for both P1 and P2
simulations are presented. The quadratic element approximation results in a much lower level of entropy
error than its linear element counterpart. The geometric singularity at the trailing edge of the airfoil requires
a much ner discretization around that point relative to the rest of the mesh for both the linear and quadratic
element approximations to achieve their optimal convegence rate. This is paticularly important for the P2
approximation since the error away from the geometric singularity vanishes far quicker, rendering the trailing
edge error as the dominant source of error for the numerical approximation.
5.3 Example 3: Flow over at plate
In this example, we consider ow over a at plate of unit length. The computational domain is [ 1:5; 1][0; 1],
with the leading edge of the plate at (0; 0). The free stream Mach number is 0:5. The Reynolds number is
raised from 8000 to 64000 in successive simulations while keeping the mesh unchanged. The results in the form
of boundary layer thickness, Æ99 (x = L), are plotted in gure 4. The quadratic element approximation yields
results very close to that of the Blasius solution while the linear element approximation shows increasing
error with rising Reynolds number. Further numerical tests have shown that it is possible to resolve the
Blasius boundary layer with only two elements when quadratic element approximation is used.
5
6 Conclusion
A high-order accurate, stabilized nite element method for the solution of compressible Euler and NavierStokes equations has been presented. The advantages of quadratic element approximation over linear representation of solution were demonstrated through a number of test problems. In particular, these numerical
tests have shown that higher-order approximation is signicantly more eective in resolving viscous boundary
layers.
References
[1] J.S. Wong, D.L. Darmofal and J. Peraire, The Solution of the Compressible Euler Equations at Low
Mach Numbers using a Stabilized Finie Element Algorithm, in preparation 2000.
[2] F. Shakib, T.J.R. Hughes and Z. Johan A new nite element formulation for computational uid dynamics : X. The compressible Euler and Navier-Stokes equations, Comp. Meth. in Appl. Mech. and
Engnr., 89, 1991, pp. 141-219.
[3] T.J.R. Hughes and M. Mallet, A new nite element formulation for computational uid dynamics : III.
The generalized streamline operator for multidimensional advective diusive systems, Comp. Meth. in
Appl. Mech. and Engnr., 58, 1986, pp. 305-328.
[4] T.J.R. Hughes, L.P. Franca and G.M. Hulbert, A new nite element formulation for computational uid
dynamics : VIII. The Galerkin Least-Squares method for advective diusive equations, Comp. Meth.
in Appl. Mech. and Engnr., 73, 1989, pp. 173-189.
[5] T.J.R. Hughes, L.P.Franca and M. Mallet, A new nite element formulation for computational uid
dynamics : I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second
law of thermodynamics, Comp. Meth. in Appl. Mech. and Engnr., 54, 1986, pp. 223-234.
[6] T.J. Barth, Numerical methods for gasdynamic systems on unstructured meshes, Lecture Notes in
Computational Science and Engineering, 1998, pp. 195-284
[7] A. Harten, On the symmetric for of systems of conservation laws with entropy, Journal of Computational
Physics, 49, 1983, pp. 151-164.
[8] P.L. Roe, Approximate Riemann solvers, parameter vectors, and dierence schemes, J. Comp. Phys.,
43, 1981, pp. 357-72.
[9] J.C. Carette, Adaptive Unstructured Mesh Algorithms and SUPG Finite Element Method for Compressible High Reynolds Number Flows, PhD Thesis, von Karman Institute, 1997
6
[10] J.C. Carette, H. Deconinck, H. Paillere, and P.L. Roe, Multidimensional upwinding: Its relation to
nite elements, Int. J. of Num Meth. Fluids, vol.20 pp935-955, 1995.
[11] D.R. Fokkema, Subspace methods for linear, nonlinear and eigenproblems, PhD Thesis, Utrech University, 1996.
[12] G.L.G. Sleijpen, H.A. van der Vorst and D.R. Fokkema, BiCGstab(1) and other hybrid Bi-CG methods,
Numerical Algorithms, 7, 1994, pp.75-109.
[13] Test Cases for Inviscid Flow Field Methods, AGARD-AR-211, 7 Ancelle 92200, Neuilly sur Seine,
France, 1985.
7
Ringleb Flow
-4
P1
P2
-4.5
1
2
Log10(ε)
-5
-5.5
-6
3
-6.5
1
-7
-2.5
-2.25
-2
-1.75
-1.5
Log10(hc)
Intermediate Mesh-6400 elements
3
2.5
2
1.5
1
0.5
0
-2
-1.5
-1
-0.5
0
0.5
1
Figure 1: Top: L2 entropy norm error of ringleb ow solution, Bottom: Computational mesh.
8
Flow Over NACA 0012 M∝=0.6, α=2°
-5
10
P1
P2
L2 Sgen
10-6
10-7
10-8
0.0025
0.005 0.0075 0.01
hc
Coarse Mesh-1672 elements
1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
Figure 2: Top: L2 entropy error of ow over NACA 0012 airfoil, Bottom: Computational mesh.
9
P1 Solution-Coarse Mesh
1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
P2 Solution-Coarse Mesh
1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
Figure 3: Mach contour of ow over NACA 0012 airfoil, M1 = 0:6, = 2Æ
10
0.06
Blasius
P1
P2
0.05
δ99(x=1)
0.04
0.03
0.02
0.01
0.0025
0.005
0.0075
0.01
0.0125
-0.5
L
Re
Mesh - Flow Over Flat Plate
1
0.75
0.5
0.25
0
-0.5
-0.25
0
0.25
0.5
0.75
1
Figure 4: Top: Computed value of boundary layer thickness at x = 1, Bottom: Computational mesh.
11