The $YZβ$ shock-capturing technique was introduced originally for use in combination with the streamline-upwind/Petrov–Galerkin (SUPG) formulation of compressible flows in conservation variables. It is a simple residual-based shock-capturing technique. Later it was also combined with the variable subgrid scale (V-SGS) formulation of compressible flows in conservation variables and tested on standard 2D test problems. The V-SGS method is based on an approximation of the class of SGS models derived from the Hughes variational multiscale method. In this paper, we carry out numerical experiments with inviscid supersonic flows around cylinders and spheres to evaluate the performance of the $YZβ$ shock-capturing combined with the V-SGS method. The cylinder computations are carried out at Mach numbers 3 and 8, and the sphere computations are carried out at Mach number 3. The results compare well to those obtained with the $YZβ$ shock-capturing combined with the SUPG formulation, which were shown earlier to compare very favorably to those obtained with the well established OVERFLOW code.

1.
Hughes
,
T. J. R.
, and
Brooks
,
A. N.
, 1979, “
A Multi-Dimensional Upwind Scheme With No Crosswind Diffusion
,”
Finite Element Methods for Convection Dominated Flows
, AMD-Vol.
34
,
T. J. R.
Hughes
, ed.,
ASME
,
New York
, pp.
19
35
.
2.
Brooks
,
A. N.
, and
Hughes
,
T. J. R.
, 1982, “
Streamline Upwind/Petrov-Galerkin Formulations for Convection Dominated Flows With Particular Emphasis on the Incompressible Navier-Stokes Equations
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
32
, pp.
199
259
.
3.
Tezduyar
,
T. E.
, and
Hughes
,
T. J. R.
, 1983, “
Finite Element Formulations for Convection Dominated Flows With Particular Emphasis on the Compressible Euler Equations
,” AIAA Paper No. 83-0125.
4.
Hughes
,
T. J. R.
, and
Tezduyar
,
T. E.
, 1984, “
Finite Element Methods for First-Order Hyperbolic Systems With Particular Emphasis on the Compressible Euler Equations
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
45
, pp.
217
284
.
5.
Hughes
,
T. J. R.
,
Franca
,
L. P.
, and
Mallet
,
M.
, 1987, “
A New Finite Element Formulation for Computational Fluid Dynamics: VI. Convergence Analysis of the Generalized SUPG Formulation for Linear Time-Dependent Multi-Dimensional Advective-Diffusive Systems
,”
Comput. Methods Appl. Mech. Eng.
,
63
, pp.
97
112
. 0045-7825
6.
Le Beau
,
G. J.
, and
Tezduyar
,
T. E.
, 1991, “
Finite Element Computation of Compressible Flows With the SUPG Formulation
,”
Advances in Finite Element Analysis in Fluid Dynamics
, FED-Vol.
123
,
ASME
,
New York
, pp.
21
27
.
7.
Le Beau
,
G. J.
,
Ray
,
S. E.
,
,
S. K.
, and
Tezduyar
,
T. E.
, 1993, “
SUPG Finite Element Computation of Compressible Flows With the Entropy and Conservation Variables Formulations
,”
Comput. Methods Appl. Mech. Eng.
,
104
, pp.
397
422
. 0045-7825
8.
Tezduyar
,
T. E.
, 2004, “
Finite Element Methods for Fluid Dynamics With Moving Boundaries and Interfaces
,”
Encyclopedia of Computational Mechanics, Volume 3: Fluids
,
E.
Stein
,
R.
De Borst
, and
T. J. R.
Hughes
, eds.,
Wiley
,
New York
.
9.
Tezduyar
,
T. E.
, 2004, “
Determination of the Stabilization and Shock-Capturing Parameters in SUPG Formulation of Compressible Flows
,”
Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2004
10.
Tezduyar
,
T. E.
, and
Senga
,
M.
, 2006, “
Stabilization and Shock-Capturing Parameters in SUPG Formulation of Compressible Flows
,”
Comput. Methods Appl. Mech. Eng.
,
195
, pp.
1621
1632
. 0045-7825
11.
Tezduyar
,
T. E.
, and
Senga
,
M.
, 2007, “
SUPG Finite Element Computation of Inviscid Supersonic Flows With YZβ Shock-Capturing
,”
Comput. Fluids
,
36
, pp.
147
159
. 0045-7930
12.
Tezduyar
,
T. E.
,
Senga
,
M.
, and
Vicker
,
D.
, 2006, “
Computation of Inviscid Supersonic Flows Around Cylinders and Spheres With the SUPG Formulation and YZβ Shock-Capturing
,”
Comput. Mech.
0178-7675,
38
, pp.
469
481
.
13.
Buning
,
P. G.
,
Jespersen
,
D. C.
,
Pulliam
,
T. H.
,
Klopfer
,
G. H.
,
Chan
,
W. M.
,
Slotnick
,
J. P.
,
Krist
,
S. E.
, and
Renze
,
K. J.
, 2000, OVERFLOW User’s Manual, Version 1.8s, NASA Langley Research Center, Hampton, VA.
14.
Corsini
,
A.
,
Rispoli
,
F.
, and
Santoriello
,
A.
, 2005, “
A Variational Multiscale High-Order Finite Element Formulation for Turbomachinery Flow Computations
,”
Comput. Methods Appl. Mech. Eng.
,
194
, pp.
4797
4823
. 0045-7825
15.
Hughes
,
T. J. R.
, 1995, “
Multiscale Phenomena: Green’s Functions, the Dirichlet-to-Neumann Formulation, Subgrid Scale Models, Bubbles, and the Origins of Stabilized Methods
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
127
, pp.
387
401
.
16.
Rispoli
,
F.
, and
Saavedra
,
R.
, 2006, “
A Stabilized Finite Element Method Based on SGS Models for Compressible Flows
,”
Comput. Methods Appl. Mech. Eng.
,
196
, pp.
652
664
. 0045-7825
17.
Rispoli
,
F.
,
Saavedra
,
R.
,
Corsini
,
A.
, and
Tezduyar
,
T. E.
, 2007, “
Computation of Inviscid Compressible Flows With the V-SGS Stabilization and YZβ Shock-Capturing
,”
Int. J. Numer. Methods Fluids
,
54
, pp.
695
706
. 0271-2091
18.
Tezduyar
,
T. E.
, and
Osawa
,
Y.
, 2000, “
Finite Element Stabilization Parameters Computed From Element Matrices and Vectors
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
190
, pp.
411
430
.