Abstract

This work performs systematic studies for code verification for turbulence modeling in our research CFD code SENSEI. Turbulence modeling verification cases including cross term sinusoidal manufactured solutions and a few exact solutions are used to justify the proper Spalart–Allmaras and Menter’s SST turbulence modeling implementation of the SENSEI CFD code. The observed order of accuracy matches fairly well with the formal order for both the 2D/3D steady-state and 2D unsteady flows when using the cross term sinusoidal manufactured solutions. This work concludes that it is important to keep the spatial discretization error in a similar order of magnitude as the temporal error in order to avoid erroneous analysis when performing combined spatial and temporal order analysis. Since explicit time marching scheme typically requires smaller time-step size compared to implicit time marching schemes due to stability constraints, multiple implicit schemes such as the singly diagonally implicit Runge–Kutta multistage scheme and three point backward scheme are used in our work to mitigate the stability constraints.

References

1.
Allmaras
,
S. R.
, and
Johnson
,
F. T.
,
2012
, “
Modifications and Clarifications for the Implementation of the Spalart-Allmaras Turbulence Model
,” Proceedings of the Seventh International Conference on Computational Fluid Dynamics (
ICCFD7
),
Big Island, HI
, July 9–13, pp.
1
11
.https://www.iccfd.org/iccfd7/assets/pdf/papers/ICCFD7-1902_paper.pdf
2.
Chien
,
K.-Y.
,
1982
, “
Predictions of Channel and Boundary-Layer Flows With a Low-Reynolds-Number Turbulence Model
,”
AIAA J.
,
20
(
1
), pp.
33
38
.10.2514/3.51043
3.
Wilcox
,
D. C.
,
2008
, “
Formulation of the k-w Turbulence Model Revisited
,”
AIAA J.
,
46
(
11
), pp.
2823
2838
.10.2514/1.36541
4.
Menter
,
F. R.
,
1994
, “
Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications
,”
AIAA J.
,
32
(
8
), pp.
1598
1605
.10.2514/3.12149
5.
Oberkampf
,
W. L.
, and
Roy
,
C. J.
,
2010
,
Verification and Validation in Scientific Computing
,
Cambridge University Press
, Cambridge, UK.
6.
Oberkampf
,
W. L.
, and
Trucano
,
T. G.
,
2002
, “
Verification and Validation in Computational Fluid Dynamics
,”
Prog. Aerosp. Sci.
,
38
(
3
), pp.
209
272
.10.1016/S0376-0421(02)00005-2
7.
Roy
,
C. J.
,
2005
, “
Review of Code and Solution Verification Procedures for Computational Simulation
,”
J. Comput. Phys.
,
205
(
1
), pp.
131
156
.10.1016/j.jcp.2004.10.036
8.
Al-Saif
,
A.-S. J.
, and
Harfash
,
A. J.
,
2019
, “
A New Approximate Analytical Solutions for Two-and Three-Dimensional Unsteady Viscous Incompressible Flows by Using the Kinetically Reduced Local Navier–Stokes Equations
,”
J. Appl. Math.
,
2019
, pp.
1
19
.10.1155/2019/3084394
9.
Spiegel
,
S. C.
,
Huynh
,
H.
, and
DeBonis
,
J. R.
,
2015
, “
A Survey of the Isentropic Euler Vortex Problem Using High-Order Methods
,”
AIAA
Paper No. 2015-2444.10.2514/6.2015-2444
10.
Hui
,
W.
,
1987
, “
Exact Solutions of the Unsteady Two-Dimensional Navier-Stokes Equations
,”
Z. Angew. Math. Phys. ZAMP
,
38
(
5
), pp.
689
702
.10.1007/BF00948290
11.
Shah
,
A.
,
Yuan
,
L.
, and
Khan
,
A.
,
2010
, “
Upwind Compact Finite Difference Scheme for Time-Accurate Solution of the Incompressible Navier–Stokes Equations
,”
Appl. Math. Comput.
,
215
(
9
), pp.
3201
3213
.10.1016/j.amc.2009.10.001
12.
Tavelli
,
M.
, and
Dumbser
,
M.
,
2016
, “
A Staggered Space–Time Discontinuous Galerkin Method for the Three-Dimensional Incompressible Navier–Stokes Equations on Unstructured Tetrahedral Meshes
,”
J. Comput. Phys.
,
319
, pp.
294
323
.10.1016/j.jcp.2016.05.009
13.
Tsoutsanis
,
P.
,
Kokkinakis
,
I. W.
,
Könözsy
,
L.
,
Drikakis
,
D.
,
Williams
,
R. J.
, and
Youngs
,
D. L.
,
2015
, “
Comparison of Structured-and Unstructured-Grid, Compressible and Incompressible Methods Using the Vortex Pairing Problem
,”
Comput. Methods Appl. Mech. Eng.
,
293
, pp.
207
231
.10.1016/j.cma.2015.04.010
14.
Panton
,
R. L.
,
2013
,
Incompressible Flow
,
Wiley
, Hoboken, NJ.
15.
White
,
F. M.
, and
Corfield
,
I.
,
2006
,
Viscous Fluid Flow
, Vol.
3
,
McGraw-Hill
,
New York
.
16.
Tryggeson
,
H.
,
2007
, “
Analytical Vortex Solutions to Navier-Stokes Equation
,” Ph.D. thesis,
Växjö University Press
, Växjö, Sweden.
17.
Wang
,
C.
,
1989
, “
Exact Solutions of the Unsteady Navier–Stokes Equations
,”
ASME Appl. Mech. Rev.
,
42
(
11S
), pp.
S269
S282
.10.1115/1.3152400
18.
Deissler
,
R. G.
,
1965
, “
Unsteady Viscous Vortex With Flow Toward the Center
,” NASA Lewis Research Center, Cleveland, OH,
Report No. NASA-TN-D-3026.
19.
Sengupta
,
T. K.
,
Sharma
,
N.
, and
Sengupta
,
A.
,
2018
, “
Non-Linear Instability Analysis of the Two-Dimensional Navier-Stokes Equation: The Taylor–Green Vortex Problem
,”
Phys. Fluids
,
30
(
5
), p.
054105
.10.1063/1.5024765
20.
Wang
,
H.
,
Tyson
,
W. C.
, and
Roy
,
C. J.
,
2019
, “
Discretization Error Estimation for Discontinuous Galerkin Methods Using Error Transport Equations
,”
AIAA
Paper No. 2019-2173.10.2514/6.2019-2173
21.
Tyson
,
W. C.
,
Yan
,
G. K.
,
Roy
,
C. J.
, and
Ollivier-Gooch
,
C. F.
,
2019
, “
Relinearization of the Error Transport Equations for Arbitrarily High-Order Error Estimates
,”
J. Comput. Phys.
,
397
, p.
108867
.10.1016/j.jcp.2019.108867
22.
Taylor
,
G. I.
, and
Green
,
A. E.
,
1937
, “
Mechanism of the Production of Small Eddies From Large Ones
,”
Proc. R. Soc. London, Ser. A
,
158
(
895
), pp.
499
521
.10.1098/rspa.1937.0036
23.
DeBonis
,
J.
,
2013
, “
Solutions of the Taylor–Green Vortex Problem Using High-Resolution Explicit Finite Difference Methods
,”
AIAA
Paper No. 2013-0382.10.2514/6.2013-0382
24.
Wang
,
Z. J.
,
Fidkowski
,
K.
,
Abgrall
,
R.
,
Bassi
,
F.
,
Caraeni
,
D.
,
Cary
,
A.
,
Deconinck
,
H.
, et al.,
2013
, “
High-Order CFD Methods: Current Status and Perspective
,”
Int. J. Numer. Methods Fluids
,
72
(
8
), pp.
811
845
.10.1002/fld.3767
25.
Bo
,
Y.
,
Wang
,
P.
,
Guo
,
Z.
, and
Wang
,
L.-P.
,
2017
, “
DUGKS Simulations of Three-Dimensional Taylor–Green Vortex Flow and Turbulent Channel Flow
,”
Comput. Fluids
,
155
, pp.
9
21
.10.1016/j.compfluid.2017.03.007
26.
Kokkinakis
,
I.
, and
Drikakis
,
D.
,
2015
, “
Implicit Large Eddy Simulation of Weakly-Compressible Turbulent Channel Flow
,”
Comput. Methods Appl. Mech. Eng.
,
287
, pp.
229
261
.10.1016/j.cma.2015.01.016
27.
Veluri
,
S. P. K.
,
2010
, “
Code Verification and Numerical Accuracy Assessment for Finite Volume CFD Codes
,” Ph.D. thesis,
Virginia Tech
, Blacksburg, VA.
28.
Salari
,
K.
, and
Knupp
,
P.
,
2000
, “
Code Verification by the Method of Manufactured Solutions
,”
Sandia National Laboratories
,
Albuquerque, NM
, Technical Report No. SAND2000-1444.
29.
Roy
,
C. J.
,
Ober
,
C. C.
, and
Smith
,
T. M.
,
2002
, “
Verification of a Compressible CFD Code Using the Method of Manufactured Solutions
,”
AIAA
Paper No. 2002-3110.10.2514/6.2002-3110
30.
Etienne
,
S.
,
Garon
,
A.
, and
Pelletier
,
D.
,
2009
, “
Code Verification for Unsteady Flow Simulations With High Order Time-Stepping Schemes
,”
AIAA
Paper No. 2009-169.10.2514/6.2009-169
31.
Minion
,
M. L.
, and
Saye
,
R.
,
2018
, “
Higher-Order Temporal Integration for the Incompressible Navier–Stokes Equations in Bounded Domains
,”
J. Comput. Phys.
,
375
, pp.
797
822
.10.1016/j.jcp.2018.08.054
32.
Yu
,
K. R.
,
Étienne
,
S.
,
Hay
,
A.
, and
Pelletier
,
D.
,
2015
, “
Code Verification for Unsteady 3-D Fluid–Solid Interaction Problems
,”
Theor. Comput. Fluid Dyn.
,
29
(
5–6
), pp.
455
471
.10.1007/s00162-015-0367-4
33.
Xue
,
W.
,
Wang
,
H.
, and
Roy
,
C. J.
,
2020
, “
Code Verification for 3D Turbulence Modeling in Parallel SENSEI Accelerated With MPI
,”
AIAA
Paper No. 2020-0347.10.2514/6.2020-0347
34.
Krist
,
S. L.
,
1998
, “
CFL3D User’s Manual (Version 5.0)
,”
National Aeronautics and Space Administration, Langley Research Center
, Hampton, VA.
35.
Bartels
,
R. E.
,
Rumsey
,
C. L.
, and
Biedron
,
R. T.
,
2006
, “
CFL3D Version 6.4-General Usage and Aeroelastic Analysis
,” NASA Langley Research Center, Hampton, VA,
Report No. NASA/TM-2006-214301.
36.
Biedron
,
R. T.
,
Carlson
,
J.-R.
,
Derlaga
,
J. M.
,
Gnoffo
,
P. A.
,
Hammond
,
D. P.
,
Jones
,
W. T.
,
Kleb
,
B.
, et al.,
2019
, “
FUN3D Manual: 13.5
,”
Langley Research Center
,
Hampton, VA
,
Report No. NASA/TM–2019–220271.
37.
Derlaga
,
J. M.
,
Phillips
,
T.
, and
Roy
,
C. J.
,
2013
, “
SENSEI Computational Fluid Dynamics Code: A Case Study in Modern Fortran Software Development
,”
AIAA
Paper No. 2013-2450.10.2514/6.2013-2450
38.
Jackson
,
C. W.
,
Tyson
,
W. C.
, and
Roy
,
C. J.
,
2019
, “
Turbulence Model Implementation and Verification in the SENSEI CFD Code
,”
AIAA
Paper No. 2019-2331.10.2514/6.2019-2331
39.
Wang
,
H.
,
Xue
,
W.
, and
Roy
,
C. J.
,
2020
, “
Error Transport Equation Implementation in the SENSEI CFD Code
,”
AIAA
Paper No. 2020-1047.10.2514/6.2020-1047
40.
Rumsey
,
C.
,
Smith
,
B.
, and
Huang
,
G.
,
2010
, “
Description of a Website Resource for Turbulence Modeling Verification and Validation
,”
AIAA
Paper No. 2010-4742.10.2514/6.2010-4742
41.
Menter
,
F. R.
,
Kuntz
,
M.
, and
Langtry
,
R.
,
2003
, “
Ten Years of Industrial Experience With the SST Turbulence Model
,”
Turbul., Heat Mass Transfer
,
4
(
1
), pp.
625
632
.https://cfd.spbstu.ru/agarbaruk/doc/2003_Menter,%20Kuntz,%20Langtry_Ten%20years%20of%20industrial%20experience%20with%20the%20SST%20turbulence%20model.pdf
42.
Gropp
,
W.
,
Lusk
,
E.
, and
Skjellum
,
A.
,
1999
,
Using MPI: Portable Parallel Programming With the Message-Passing Interface
, Vol.
1
,
MIT Press
, Cambridge, MA.
43.
Roe
,
P. L.
,
1981
, “
Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes
,”
J. Comput. Phys.
,
43
(
2
), pp.
357
372
.10.1016/0021-9991(81)90128-5
44.
Steger
,
J. L.
, and
Warming
,
R.
,
1981
, “
Flux Vector Splitting of the Inviscid Gasdynamic Equations With Application to Finite Difference Methods
,”
J. Comput. Phys.
,
40
(
2
), pp.
263
293
.10.1016/0021-9991(81)90210-2
45.
Van Leer
,
B.
,
1997
, “
Flux-Vector Splitting for the Euler Equation
,”
Upwind and High-Resolution Schemes
,
Springer
, Berlin, pp.
80
89
.
46.
Ascher
,
U. M.
,
Ruuth
,
S. J.
, and
Spiteri
,
R. J.
,
1997
, “
Implicit-Explicit Runge–Kutta Methods for Time-Dependent Partial Differential Equations
,”
Appl. Numer. Math.
,
25
(
2–3
), pp.
151
167
.10.1016/S0168-9274(97)00056-1
47.
Kennedy
,
C. A.
, and
Carpenter
,
M. H.
,
2016
, “
Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review
,” NASA Langley Research Center, Hampton, VA,
Report No. NASA/TM-2016-219173.
48.
Jameson
,
A.
,
Schmidt
,
W.
, and
Turkel
,
E.
,
1981
, “
Numerical Solution of the Euler Equations by Finite Volume Methods Using Runge Kutta Time Stepping Schemes
,”
Proceedings of the 14th Fluid and Plasma Dynamics Conference
, Palo Alto, CA, June 23–25, p.
1259
.
49.
Menter
,
F.
,
1993
, “
Zonal Two Equation k-w Turbulence Models for Aerodynamic Flows
,”
AIAA
Paper No. 93-2906.10.2514/6.93-2906
50.
Wu
,
J.
,
Fan
,
L.
, and
Erickson
,
L.
,
1990
, “
Three-Point Backward Finite Difference Method for Solving a System of Mixed Hyperbolic-Parabolic Partial Differential Equations
,”
Comput. Chem. Eng.
,
14
(
6
), pp.
679
685
.10.1016/0098-1354(90)87036-O
51.
Ferracina
,
L.
, and
Spijker
,
M.
,
2008
, “
Strong Stability of Singly-Diagonally-Implicit Runge–Kutta Methods
,”
Appl. Numer. Math.
,
58
(
11
), pp.
1675
1686
.10.1016/j.apnum.2007.10.004
52.
Richards
,
S. A.
,
1997
, “
Completed Richardson Extrapolation in Space and Time
,”
Commun. Numer. Methods Eng.
,
13
(
7
), pp.
573
582
.10.1002/(SICI)1099-0887(199707)13:7<573::AID-CNM84>3.0.CO;2-6
53.
Kamm
,
J.
,
Rider
,
W.
, and
Brock
,
J.
,
2003
, “
Combined Space and Time Convergence Analysis of a Compressible Flow Algorithm
,”
AIAA
Paper No. 2003-4241.10.2514/6.2003-4241
54.
Yee
,
H. C.
,
Sandham
,
N. D.
, and
Djomehri
,
M. J.
,
1999
, “
Low-Dissipative High-Order Shock-Capturing Methods Using Characteristic-Based Filters
,”
J. Comput. Phys.
,
150
(
1
), pp.
199
238
.10.1006/jcph.1998.6177
55.
Drikakis
,
D.
,
Fureby
,
C.
,
Grinstein
,
F. F.
, and
Youngs
,
D.
,
2007
, “
Simulation of Transition and Turbulence Decay in the Taylor–Green Vortex
,”
J. Turbul.
,
8
(
8
), p.
N20
.10.1080/14685240701250289
You do not currently have access to this content.