Abstract

A new partially averaged Navier–Stokes (PANS) closure is derived based on the kkL (KSKL) model. The aim of this new model is to incorporate the desirable features of the KSKL model, compared to the kω shear stress transport model, into the PANS framework. These features include reduced eddy-viscosity levels, a lower dependency on the cell height at the wall, well-defined boundary conditions, and improved iterative convergence. As well as the new model derivation, the paper demonstrates that these desirable features are indeed maintained, for a range of modeled-to-total turbulence kinetic energy ratios (fk), and even for multiphase flow.

References

1.
Nallasamy
,
M.
,
1987
, “
Turbulence Models and Their Applications to the Prediction of Internal Flows: A Review
,”
Comput. Fluids
,
15
(
2
), pp.
151
194
.10.1016/S0045-7930(87)80003-8
2.
Pope
,
S.
,
2000
,
Turbulent Flows
,
Cambridge University Press
, London.
3.
Argyropoulos
,
C.
, and
Markatos
,
N.
,
2015
, “
Recent Advances on the Numerical Modelling of Turbulent Flows
,”
Appl. Math. Modell.
,
39
(
2
), pp.
693
732
.10.1016/j.apm.2014.07.001
4.
Pereira
,
F.
,
Eça
,
L.
,
Vaz
,
G.
, and
Girimaji
,
S.
,
2021
, “
Toward Predictive RANS and SRS Computations of Turbulent External Flows of Practical Interest
,”
Arch. Comput. Methods Eng.
,
28
(
5
), pp.
3953
4029
.10.1007/s11831-021-09563-0
5.
Smagorinsky
,
J.
,
1963
, “
General Circulation Experiments With the Primitive Equations: I. The Basic Experiment
,”
Mon. Weather Rev.
,
91
(
3
), pp.
99
164
.10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2
6.
Spalart
,
P.
,
1997
, “
Comments on the Feasibility of LES for Wings, and on a Hybrid RANS/LES Approach
,”
Proceedings of 1st AFOSR International Conference on DNS/LES
, Ruston, LO, Aug. 4–8,
Greyden Press
, pp.
1
11
.https://www.researchgate.net/publication/236888805_Comments_on_the_Feasibility_of_LES_for_Wings_and_on_a_Hybrid_RANSLES_Approach
7.
Girimaji
,
S.
, and
Abdol-Hamid
,
K.
,
2005
, “
Partially Averaged Navier-Stokes Model for Turbulence: Implementation and Validation
,”
AIAA
Paper No. 2005-502.10.2514/6.2005-502
8.
Girimaji
,
S.
,
2006
, “
Partially-Averaged Navier-Stokes Model for Turbulence: A Reynolds-Averaged Navier-Stokes to Direct Numerical Simulation Bridging Method
,”
ASME J. Appl. Mech.
,
73
(
3
), pp.
413
421
.10.1115/1.2151207
9.
Pereira
,
F.
,
Vaz
,
G.
,
Eça
,
L.
, and
Girimaji
,
S.
,
2018
, “
Simulation of the Flow Around a Circular Cylinder at Re = 3900 With Partially-Averaged Navier-Stokes Equations
,”
Int. J. Heat Fluid Flow
,
69
, pp.
234
246
.10.1016/j.ijheatfluidflow.2017.11.001
10.
Klapwijk
,
M.
,
Lloyd
,
T.
, and
Vaz
,
G.
,
2019
, “
On the Accuracy of Partially Averaged Navier-Stokes Resolution Estimates
,”
Int. J. Heat Fluid Flow
,
80
, p.
108484
.10.1016/j.ijheatfluidflow.2019.108484
11.
Klapwijk
,
M.
,
Lloyd
,
T.
,
Vaz
,
G.
, and
Van Terwisga
,
T.
,
2020
, “
Evaluation of Scale-Resolving Simulations for a Turbulent Channel Flow
,”
Comput. Fluids
,
209
, p.
104636
.10.1016/j.compfluid.2020.104636
12.
Abdol-Hamid
,
K.
, and
Girimaji
,
S.
,
2004
, “
A Two-Stage Procedure Toward the Efficient Implementation of PANS and Other Hybrid Turbulence Models
,” NASA Langley Research Center, Hampton, VA, Technical Memorandum.
13.
Basu
,
D.
,
Hamed
,
A.
, and
Das
,
K.
,
2007
, “
Assessment of Partially Averaged Navier Stokes (PANS) Multiscale Model in Transonic Turbulent Separated Flows
,”
ASME
Paper No. FEDSM2007-37630.10.1115/FEDSM2007-37630
14.
Basara
,
B.
,
Krajnovic
,
S.
, and
Girimaji
,
S.
,
2008
, “
PANS vs. LES for Computations of the Flow Around a 3D Bluff Body
,”
Proceedings of 7th International Symposium on Engineering, Turbulence, Modelling and Measurements
, Vol.
2
, Limassol, Cyprus, June 4–6, p.
3
.
15.
Jeong
,
E.
, and
Girimaji
,
S.
,
2010
, “
Partially Averaged Navier-Stokes (PANS) Method for Turbulence Simulations—Flow Past a Square Cylinder
,”
ASME J. Fluids Eng.
,
132
(
12
), p.
121203
.10.1115/1.4003153
16.
Davidson
,
L.
, and
Friess
,
C.
,
2019
, “
A New Formulation of fk for the PANS Model
,”
J. Turbul.
,
20
(
5
), pp.
322
336
.10.1080/14685248.2019.1641605
17.
Friess
,
C.
, and
Davidson
,
L.
,
2020
, “
A Formulation of PANS Capable of Mimicking IDDES
,”
Int. J. Heat Fluid Flow
,
86
, p.
108666
.10.1016/j.ijheatfluidflow.2020.108666
18.
Luo
,
D.
,
Yan
,
C.
,
Liu
,
H.
, and
Zhao
,
R.
,
2014
, “
Comparative Assessment of PANS and DES for Simulation of Flow Past a Circular Cylinder
,”
J. Wind Eng. Ind. Aerodyn.
,
134
, pp.
65
77
.10.1016/j.jweia.2014.08.014
19.
Bensow
,
R.
, and
van den Boogaard
,
M.
,
2019
, “
Using a PANS Simulation Approach for the Transient Flow Around the Japan Bulk Carrier
,”
J. Ship Res.
,
63
(
2
), pp.
123
129
.10.5957/JOSR.09180064
20.
Kamble
,
C.
, and
Girimaji
,
S.
,
2020
, “
Characterization of Coherent Structures in Turbulent Wake of a Sphere Using Partially Averaged Navier-Stokes (PANS) Simulations
,”
Phys. Fluids
,
32
(
10
), p.
105110
.10.1063/5.0024854
21.
Saroha
,
S.
,
Sinha
,
S.
, and
Lakshmipathy
,
S.
,
2019
, “
Evaluation of PANS Method in Conjunction With Non-Linear Eddy Viscosity Closure Using OpenFOAM
,”
Int. J. Numer. Methods Heat Fluid Flow
,
29
(
3
), pp.
949
980
.10.1108/HFF-09-2018-0529
22.
Saroha
,
S.
,
Chakraborty
,
K.
,
Sinha
,
S.
, and
Lakshmipathy
,
S.
,
2020
, “
An OpenFOAM-Based Evaluation of PANS Methodology in Conjunction With Non-Linear Eddy Viscosity: Flow Past a Heated Cylinder
,”
J. Appl. Fluid Mech.
,
13
(
5
), pp.
1453
1469
.10.1115/1.4045209
23.
Basara
,
B.
,
Pavlovic
,
Z.
, and
Girimaji
,
S.
,
2018
, “
A New Approach for the Calculation of the Cut-Off Resolution Parameter in Bridging Methods for Turbulent Flow Simulation
,”
Int. J. Heat Fluid Flow
,
74
, pp.
76
88
.10.1016/j.ijheatfluidflow.2018.09.011
24.
Zhang
,
J.
,
Minelli
,
G.
,
Rao
,
A.
,
Basara
,
B.
,
Bensow
,
R.
, and
Krajnović
,
S.
,
2018
, “
Comparison of PANS and LES of the Flow Past a Generic Ship
,”
Ocean Eng.
,
165
, pp.
221
236
.10.1016/j.oceaneng.2018.07.023
25.
Pereira
,
F.
,
Grinstein
,
F.
,
Israel
,
D.
,
Rauenzahn
,
R.
, and
Girimaji
,
S.
,
2021
, “
Modeling and Simulation of Transitional Taylor-Green Vortex Flow With Partially Averaged Navier-Stokes Equations
,”
Phys. Rev. Fluids
,
6
(
5
), p.
054611
.10.1103/PhysRevFluids.6.054611
26.
Menter
,
F.
,
Egorov
,
Y.
, and
Rusch
,
D.
,
2006
, “
Steady and Unsteady Flow Modelling Using the k k L Model
,”
ICHMT Digital Library Online
,
Begell House Inc
., Danbury, CT, pp.
1
20
.
27.
Larsson
,
L.
,
Raven
,
H.
, and
Paulling
,
J.
,
2010
,
Ship Resistance and Flow. Principles of Naval Architecture
,
Society of Naval Architects and Marine Engineers
, Jersey City, NJ.
28.
Menter
,
F.
, and
Egorov
,
Y.
,
2004
, “
Revisiting the Turbulent Scale Equation
,”
IUTAM Symposium on One Hundred Years of Boundary Layer Research
, Göttingen, Germany, Aug. 12–14,
Springer
, pp.
279
290
.
29.
Liebrand
,
R.
,
Klapwijk
,
M.
,
Lloyd
,
T.
, and
Vaz
,
G.
,
2021
, “
Transition and Turbulence Modeling for the Prediction of Cavitating Tip Vortices
,”
ASME J. Fluids Eng.
,
143
(
1
), p.
011202
.10.1115/1.4048133
30.
Eça
,
L.
,
Pereira
,
F.
, and
Vaz
,
G.
,
2018
, “
Viscous Flow Simulations at High Reynolds Numbers Without Wall Functions: Is y + 1 Enough for the Near-Wall Cells?
,”
Comput. Fluids
,
170
, pp.
157
175
.10.1016/j.compfluid.2018.04.035
31.
Menter
,
F. R.
,
1994
, “
Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications
,”
AIAA J.
,
32
(
8
), pp.
1598
1605
.10.2514/3.12149
32.
Hoekstra
,
M.
, and
Vaz
,
G.
,
2009
, “
The Partial Cavity on a 2D Foil Revisited
,”
Proceedings of 7th International Symposium on Cavitation
, MI, Aug. 16–20, pp.
1
12
.
33.
Rijpkema
,
D.
,
Baltazar
,
J.
, and
de Campos
,
J.
,
2015
, “
Viscous Flow Simulations of Propellers in Different Reynolds Number Regimes
,”
Proceedings of 4th International Symposium on Marine Propulsors
, Austin, TX, May 31-June 4, pp.
1
12
.https://www.marinepropulsors.com/proceedings/2015/T B4-1.pdf
34.
Vaz
,
G.
,
Hally
,
D.
,
Huuva
,
T.
,
Bulten
,
N.
,
Muller
,
P.
,
Becchi
,
P.
,
Herrer
,
J.
,
Whitworth
,
S.
,
Macé
,
R.
, and
Korsström
,
A.
,
2015
, “
Cavitating Flow Calculations for the E779A Propeller in Open Water and Behind Conditions: Code Comparison and Solution Validation
,”
Proceedings of 4th International Symposium on Marine Propulsors
, Austin, TX, May 31–June 4, pp.
1
16
.https://www.researchgate.net/publication/275622126_Cavitating_Flow_Calculations_for_the_E779A_Propeller_in_Open_Water_and_Behind_Conditions_Code_C
35.
Vaz
,
G.
,
Lloyd
,
T.
, and
Gnanasundaram
,
A.
,
2017
, “
Improved Modelling of Sheet Cavitation Dynamics on Delft Twist 11 Hydrofoil
,”
Proceedings of VII International Conference on Computational Methods in Marine Engineering
, Nantes, Frances, June 15–17, pp.
1
14
.
36.
Klaij
,
C.
,
Hoekstra
,
M.
, and
Vaz
,
G.
,
2018
, “
Design, Analysis and Verification of a Volume-of-Fluid Model With Interface-Capturing Scheme
,”
Comput. Fluids
,
170
, pp.
324
340
.10.1016/j.compfluid.2018.05.016
37.
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
38.
Lopes
,
R.
,
Fernandes
,
E.
,
Eça
,
L.
,
Vaz
,
G.
, and
Kerkvliet
,
M.
,
2020
, “
Coupling Two Correlation-Based Transition Models to the k k L Eddy Viscosity Turbulence Model
,”
AIAA J.
,
59
(
5
), pp.
1735
1748
.10.2514/1.J059523
39.
Pereira
,
F.
,
Eça
,
L.
, and
Vaz
,
G.
,
2017
, “
Verification and Validation Exercises for the Flow Around the KVLCC2 Tanker at Model and Full-Scale Reynolds Numbers
,”
Ocean Eng.
,
129
, pp.
133
148
.10.1016/j.oceaneng.2016.11.005
40.
Vaz
,
G.
,
Jaouen
,
F.
, and
Hoekstra
,
M.
,
2009
, “
Free-Surface Viscous Flow Computations: Validation of URANS Code FRESCO
,”
ASME
Paper No. OMAE2009-79398.10.1115/OMAE2009-79398
41.
Germano
,
M.
,
1992
, “
Turbulence: The Filtering Approach
,”
J. Fluid Mech.
,
238
, pp.
325
336
.10.1017/S0022112092001733
42.
Hirt
,
C.
, and
Nichols
,
B.
,
1981
, “
Volume of Fluid (VoF) Method for the Dynamics of Free Boundaries
,”
J. Comput. Phys.
,
39
(
1
), pp.
201
225
.10.1016/0021-9991(81)90145-5
43.
Schnerr
,
G.
, and
Sauer
,
J.
,
2001
, “
Physical and Numerical Modeling of Unsteady Cavitation Dynamics
,”
Proceedings of ICMF, 4th International Conference on Multiphase Flow
, New Orleans, May 27–June 1, pp.
10
12
.https://www.researchgate.net/publication/296196752_Physical_and_Numerical_Modeling_of_Unsteady_Cavitation_Dynamics
44.
Reyes
,
D.
,
Cooper
,
J.
, and
Girimaji
,
S.
,
2014
, “
Characterizing Velocity Fluctuations in Partially Resolved Turbulence Simulations
,”
Phys. Fluids
,
26
(
8
), p.
085106
.10.1063/1.4892080
45.
Spalart
,
P.
, and
Rumsey
,
C.
,
2007
, “
Effective Inflow Conditions for Turbulence Models in Aerodynamic Calculations
,”
AIAA J.
,
45
(
10
), pp.
2544
2553
.10.2514/1.29373
46.
Lopes
,
R.
,
Eça
,
L.
,
Vaz
,
G.
, and
Kerkvliet
,
M.
,
2021
, “
Assessing Numerical Aspects of Transitional Flow Simulations Using the RANS Equations
,”
Int. J. Comput. Fluid Dyn.
,
35
(
3
), pp.
1
22
.10.1080/10618562.2020.1870962
47.
Klapwijk
,
M.
,
Lloyd
,
T.
,
Vaz
,
G.
, and
Van Terwisga
,
T.
,
2021
, “
On the Use of Synthetic Inflow Turbulence for Scale-Resolving Simulations of Wetted and Cavitating Flows
,”
Ocean Eng.
,
228
, p.
108860
.10.1016/j.oceaneng.2021.108860
48.
Lopes
,
R.
,
2021
, “
Simulation of Transition From Laminar to Turbulent Regime in Practical Applications of Incompressible Flow
,” Ph.D. thesis,
Universidade de Lisboa, Instituto Superior Técnico
,
Portugal
.
49.
Xu
,
C.-Y.
,
Zhang
,
T.
,
Yu
,
Y.-Y.
, and
Sun
,
J.-H.
,
2019
, “
Effect of Von Karman Length Scale in Scale Adaptive Simulation Approach on the Prediction of Supersonic Turbulent Flow
,”
Aerosp. Sci. Technol.
,
86
, pp.
630
639
.10.1016/j.ast.2019.01.030
50.
Klapwijk
,
M.
,
Lloyd
,
T.
,
Vaz
,
G.
, and
Van Terwisga
,
T.
,
2019
, “
PANS Simulations: Low Versus High Reynolds Number Approach
,”
Proceedings of VIII International Conference on Computational Methods in Marine Engineering
, Göthenburg, Sweden, May 13–15, pp.
48
59
.https://www.researchgate.net/publication/333395278_PANS_SIMULAT IONS_LOW_VERSUS_HIGH_REYNOLDS_NUMBER_APPROACH
51.
Moser
,
R.
,
Kim
,
J.
, and
Mansour
,
N.
,
1999
, “
Direct Numerical Simulation of Turbulent Channel Flow Up to R e τ = 590
,”
Phys. Fluids
,
11
(
4
), pp.
943
945
.10.1063/1.869966
52.
Pereira
,
F.
,
Eça
,
L.
,
Vaz
,
G.
, and
Girimaji
,
S.
,
2018
, “
Challenges in Scale-Resolving Simulations of Turbulent Wake Flows With Coherent Structures
,”
J. Comput. Phys.
,
363
, pp.
98
115
.10.1016/j.jcp.2018.02.038
53.
Zhang
,
Y.
,
2017
, “
Critical Transition Reynolds Number for Plane Channel Flow
,”
Appl. Math. Mech.
,
38
(
10
), pp.
1415
1424
.10.1007/s10483-017-2245-6
54.
Pennings
,
P.
,
Westerweel
,
J.
, and
Van Terwisga
,
T.
,
2015
, “
Flow Field Measurement Around Vortex Cavitation
,”
Exp. Fluids
,
56
(
11
), p.
206
.10.1007/s00348-015-2073-9
55.
Xie
,
Z.-T.
, and
Castro
,
I.
,
2008
, “
Efficient Generation of Inflow Conditions for Large Eddy Simulation of Street-Scale Flows
,”
Flow Turbul. Combust.
,
81
(
3
), pp.
449
470
.10.1007/s10494-008-9151-5
56.
Lloyd
,
T.
,
Vaz
,
G.
,
Rijpkema
,
D.
, and
Reverberi
,
A.
,
2017
, “
Computational Fluid Dynamics Prediction of Marine Propeller Cavitation Including Solution Verification
,”
Proceedings of 5th Symposium on Marine Propulsion
, Espoo, Finland, June 12–15, pp.
58
70
.https://www.researchgate.net/publication/316240685_Computational_fluid_dynamics_prediComputationalFluidDynamicsPredictionofMarinePropellerCavitationIncludingSolutionVerificationction_of_marine_propeller_cavitation_including_solution_verification
57.
Asnaghi
,
A.
,
Svennberg
,
U.
, and
Bensow
,
R.
,
2020
, “
Large Eddy Simulations of Cavitating Tip Vortex Flows
,”
Ocean Eng.
,
195
, p.
106703
.10.1016/j.oceaneng.2019.106703
58.
Reverberi
,
A.
,
2016
, “
Cavitation Modelling of E779A Propeller Accounting for Transition Effects
,” MARIN, Wageningen, the Netherlands, Technical Report.
59.
Bosschers
,
J.
,
2018
, “
Propeller tip-vortex cavitation and its broadband noise
,” Ph.D. thesis,
University of Twente
,
the Netherlands
.
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