Abstract

Although most flows in maritime applications can be modeled as incompressible, for certain phenomena like sloshing, slamming, and cavitation, this approximation falls short. For these events, it is necessary to consider compressibility effects. This paper presents the first step toward a solver for multiphase compressible flows: a single-phase compressible flow solver for perfect gases. The main purpose of this work is code verification of the solver using the method of manufactured solutions. For the sake of completeness, the governing equations are described in detail including the changes to the SIMPLE algorithm used in the incompressible flow solver to ensure mass conservation and pressure–velocity–density coupling. A manufactured solution for laminar subsonic flow was therefore designed. With properly defined boundary conditions, the observed order of grid convergence matches the formal order, so it can be concluded that the flow solver is free of coding mistakes, to the extent tested by the method of manufactured solutions. The performance of the pressure-based SIMPLE solver is quantified by reporting iteration counts for all grids. Furthermore, the use of pressure–weighted interpolation (PWI), also known as Rhie–Chow interpolation, to avoid spurious pressure oscillations in incompressible flow, though not strictly necessary for compressible flow, does show some benefits in the low Mach number range.

References

References
1.
Roache
,
P. J.
,
1998
,
Verification and Validation in Computational Science and Engineering
,
Hermosa Publishers
,
Albuquerque, NW
.
2.
Roy
,
C. J.
,
Smith
,
T. M.
, and
Ober
,
C. C.
,
2002
, “
Verification of a Compressible CFD Code Using the Method of Manufactured Solutions
,”
AIAA
Paper No. 2002-3110.10.2514/6.2002-3110
3.
Roy
,
C. J.
,
Nelson
,
C. C.
,
Smith
,
T. M.
, and
Ober
,
C. C.
,
2004
, “
Verification of Euler/Navier—Stokes Codes Using the Method of Manufactured Solutions
,”
Int. J. Numer. Methods Fluids
,
44
(
6
), pp.
599
620
.10.1002/fld.660
4.
Eça
,
L.
,
Hoekstra
,
M.
,
Hay
,
A.
, and
Pelletier
,
D.
,
2007
, “
Verification of RANS Solvers With Manufactured Solutions
,”
Eng. Comput.
,
23
(
4
), pp.
253
270
.10.1007/s00366-007-0067-9
5.
ASME
,
2007
, “
Guide for Verification and Validation in Computational Solid Mechanics
,”
American Society of Mechanical Engineers
,
New York
, Standard No. VV 10–2006.
6.
ASME
,
2009
, “
Standard for Verification and Validation in Computational Fluid Dynamics and Heat Transfer
,”
American Society of Mechanical Engineers
,
New York
, Standard No. VV 20–2009.
7.
Oberkampf
,
W. L.
, and
Roy
,
C. J.
,
2010
,
Verification and Validation in Scientific Computing
,
Cambridge University Press
,
Cambridge, UK
.
8.
Roache
,
P. J.
,
2002
, “
Code Verification by the Method of Manufactured Solutions
,”
ASME J. Fluid Eng.
,
124
(
1
), pp.
4
10
.10.1115/1.1436090
9.
Knupp
,
P.
, and
Salari
,
K.
,
2002
,
Verification of Computer Codes in Computational Science and Engineering
,
Chapman and Hall/CRC
,
Boca Raton, FL
.
10.
Maritime Research Institute Netherlands
,
2020
, “ReFRESCO Web Page,” Maritime Research Institute Netherlands, Wageningen, The Netherlands, accessed Aug. 24, 2020, http://www.refresco.org
11.
Demirdžić
,
I.
,
Lilek
,
Perić
,
M.
,
Peric
,
M.
,
Lilek
,
Z.
, and
Demirdzic
,
L.
,
1993
, “
A Collocated Finite Volume Method for Predicting Flows at All Speed
,”
J. Numer. Methods Fluids
,
16
(
12
), pp.
1029
1050
.10.1002/fld.1650161202
12.
Lien
,
F.-S.
, and
Leschziner
,
M. A.
,
1993
, “
A Pressure-Velocity Solution Strategy for Compressible Flow and Its Application to Shock/Boundary-Layer Interaction Using Second-Moment Turbulence Closure
,”
ASME J. Fluid Eng.
,
115
(
4
), pp.
717
725
.10.1115/1.2910204
13.
Rincon
,
J.
, and
Elder
,
R.
,
1997
, “
A High-Resolution Pressure-Based Method for Compressible Flows
,”
Comput. Fluids
,
26
(
3
), pp.
217
231
.10.1016/S0045-7930(96)00037-0
14.
Moukalled
,
F.
, and
Darwish
,
M.
,
2001
, “
A High-Resolution Pressure-Based Algorithm for Fluid Flow at All Speeds
,”
J. Comput. Phys.
,
168
(
1
), pp.
101
130
.10.1006/jcph.2000.6683
15.
Menter
,
F. R.
,
Galpin
,
P. F.
,
Esch
,
T.
,
Kuntz
,
M.
, and
Berner
,
C.
,
2004
, “
CFD Simulations of Aerodynamic Flows With a Pressure-Based Method
,”
24th International Congress of the Aeronautical Sciences
, Yokohama, Japan, Aug. 29–Sept. 4, Paper No. ICAS 2004-2.4.1.
16.
Darwish
,
M.
, and
Moukalled
,
F.
,
2011
, “
A Coupled Finite Volume Solver for Compressible Flows
,”
AIP Conf. Proc.
,
1389
(
1
), pp.
187
190
.10.1063/1.3637752
17.
Przulj
,
V. P.
,
2016
, “
Generalized SIMPLE-Based Pressure Correction Method for Unstructured Colocated Grids
,”
AIAA J.
,
54
(
5
), pp.
1542
1553
.10.2514/1.J054505
18.
Choi
,
D.
, and
Merkle
,
C. L.
,
1985
, “
Application of Time-Iterative Schemes to Incompressible Flow
,”
AIAA J.
,
23
(
10
), pp.
1518
1524
.10.2514/3.9119
19.
Turkel
,
E.
,
1987
, “
Preconditioned Methods for Solving the Incompressible and Low Speed Compressible Equations
,”
J. Comput. Phys.
,
72
(
2
), pp.
277
298
.10.1016/0021-9991(87)90084-2
20.
Edwards
,
J. R.
, and
Liou
,
M.-S.
,
1998
, “
Low-Diffusion Flux-Splitting Methods for Flows at All Speeds
,”
AIAA J.
,
36
(
9
), pp.
1610
1617
.10.2514/2.587
21.
Turkel
,
E.
,
1999
, “
Preconditioning Techniques in Computational Fluid Dynamics
,”
Annu. Rev. Fluid Mech.
,
31
(
1
), pp.
385
416
.10.1146/annurev.fluid.31.1.385
22.
Ferziger
,
J. H.
, and
Perić
,
M.
,
2001
,
Computational Methods for Fluid Dynamics
, 3rd ed.,
Springer-Verlag
,
New York
.
23.
Moukalled
,
F.
,
Mangani
,
L.
, and
Darwish
,
M.
,
2016
,
The Finite Volume Method in Computational Fluid Dynamics an Advanced Introduction With OpenFOAM and Matlab
, 1st ed.,
Springer
,
Berlin, Germany
.
24.
Hirsch, C.
,
1990
,
Numerical Computation of Internal and External Flows. Vol. 2 - Computational Methods for Inviscid and Viscous Flows
,
John Wiley & Sons Ltd
,
Chichester, UK.
25.
Waterson
,
N. P.
, and
Deconinck
,
H.
,
2007
, “
Design Principles for Bounded Higher-Order Convection Schemes–a Unified Approach
,”
J. Comput. Phys.
,
224
(
1
), pp.
182
207
.10.1016/j.jcp.2007.01.021
26.
Wesseling
,
P.
,
2001
,
Principles of Computational Fluid Dynamics
,
Springer
,
Berlin, Germany
.
27.
Patankar
,
S. V.
, and
Spalding
,
D. B.
,
1972
, “
A Calculation Procedure for Heat, Mass and Momentum Transfer in Three-Dimensional Parabolic Flows
,”
Int. J. Heat Mass Transfer
,
15
(
10
), pp.
1787
1806
.10.1016/0017-9310(72)90054-3
28.
Klaij
,
C. M.
, and
Vuik
,
C.
,
2013
, “
SIMPLE-Type Preconditioners for Cell-Centered, Colocated Finite Volume Discretization of Incompressible Reynolds-Averaged Navier-Stokes Equations
,”
Int. J. Numer. Methods Fluids
,
71
(
7
), pp.
830
849
.10.1002/fld.3686
29.
Rhie
,
C. M.
, and
Chow
,
W. L.
,
1983
, “
Numerical Study of the Turbulent Flow Past an Airfoil With Trailing Edge Separation
,”
AIAA J.
,
21
(
11
), pp.
1525
1532
.10.2514/3.8284
30.
Miller
,
T. F.
, and
Schmidt
,
F. W.
,
1988
, “
Use of a Pressure-Weighted Interpolation Method for the Solution of the Incompresible Navier-Stokes Equations on a Nonstaggered Grid System
,”
Numer. Heat Transfer: Int. J. Computation Methodology
,
14
(
2
), pp.
213
233
.10.1080/10407788808913641
31.
Mathur
,
S.
, and
Murthy
,
J.
,
1999
, “
All Speed Flows on Unstructured Meshes Using a Pressure Correction Approach
,”
AIAA
Paper No. 99-3365.10.2514/6.1999-3365
32.
Gustafsson
,
B.
, and
Sundström
,
A.
,
1978
, “
Incompletely Parabolic Problems in Fluid Dynamics
,”
SIAM J. Appl. Math.
,
35
(
2
), pp.
343
357
.10.1137/0135030
33.
Oliger
,
J.
, and
Sundström
,
A.
,
1978
, “
Theoretical and Practical Aspects of Some Initial Boundary Value Problems in Fluid Dynamics
,”
SIAM J. Appl. Math.
,
35
(
3
), pp.
419
446
.10.1137/0135035
34.
Dutt
,
P.
,
1988
, “
Stable Boundary Conditions and Difference Schemes for Navier–Stokes Equations
,”
SIAM J. Numer. Anal.
,
25
(
2
), pp.
245
267
.10.1137/0725018
35.
Nordström
,
J.
, and
Svärd
,
M.
,
2005
, “
Well-Posed Boundary Conditions for the Navier–Stokes Equations
,”
SIAM J. Numer. Anal.
,
43
(
3
), pp.
1231
1255
.10.1137/040604972
36.
Rudy
,
D. H.
, and
Strikwerda
,
J. C.
,
1981
, “
Boundary Conditions for Subsonic Compressible Navier–Stokes Calculations
,”
Comput. Fluids
,
9
(
3
), pp.
327
338
.10.1016/0045-7930(81)90005-0
37.
Moukalled
,
F.
,
Mangani
,
L.
, and
Darwish
,
M.
,
2016
, “
Implementation of Boundary Conditions in the Finite Volume Pressure-Based Method-Part I: Segregated Solvers
,”
Numer. Heat Transfer, Part B: Fundam.
,
69
(
6
), pp.
534
562
.10.1080/10407790.2016.1138748
38.
Nelson
,
C.
, and
Roy
,
C.
,
2004
, “
Verification of the Wind-US CFD Code Using the Method of Manufactured Solutions
,”
AIAA
Paper No. 2004-1104.10.2514/6.2004-1104
39.
Bond
,
R. B.
,
Ober
,
C. C.
,
Knupp
,
P. M.
, and
Bova
,
S. W.
,
2007
, “
Manufactured Solution for Computational Fluid Dynamics Boundary Condition Verification
,”
AIAA J.
,
45
(
9
), pp.
2224
2236
.10.2514/1.28099
40.
Choudhary
,
A.
,
Roy
,
C. J.
,
Luke
,
E. A.
, and
Veluri
,
S. P.
,
2016
, “
Code Verification of Boundary Conditions for Compressible and Incompressible Computational Fluid Dynamics Codes
,”
Comput. Fluids
,
126
, pp.
153
169
.10.1016/j.compfluid.2015.12.003
41.
Eça
,
L.
, and
Hoekstra
,
M.
,
2014
, “
A Procedure for the Estimation of the Numerical Uncertainty of CFD Calculations Based on Grid Refinement Studies
,”
J. Comput. Phys.
,
262
, pp.
104
130
.10.1016/j.jcp.2014.01.006
42.
Balay
,
S.
,
Abhyankar
,
S.
,
Adams
,
M. F.
,
Brown
,
J.
,
Brune
,
P.
,
Buschelman
,
K.
,
Dalcin
,
L.
,
Dener
,
A.
,
Eijkhout
,
V.
,
Gropp
,
W. D.
,
Karpeyev
,
D.
,
Kaushik
,
D.
,
Knepley
,
M. G.
,
May
,
D. A.
,
McInnes
,
L. C.
,
Mills
,
R. T.
,
Munson
,
T.
,
Rupp
,
K.
,
Sanan
,
P.
,
Smith
,
B. F.
,
Zampini
,
S.
,
Zhang
,
H.
, and
Zhang
,
H.
,
2019
, “PETSc Web Page,” Argonne National Laboratory, Lemont, IL, accessed Aug. 24, 2020, https://www.mcs.anl.gov/petsc
43.
Klaij
,
C. M.
,
2015
, “
On the Stabilization of Finite Volume Methods With co-Located Variables for Incompressible Flow
,”
J. Comput. Phys.
,
297
, pp.
84
89
.10.1016/j.jcp.2015.05.012
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