Strategy Analysis of Seamlessly Resolving Turbulent Flow Simulations
Abstract
1. Introduction
2. Generic Two-Equation Turbulence Model
3. Hybridization Frameworks
3.1. Hybridization of the Generic Two-Equation Model
3.2. Viscosity Hybridization of the Generic Two-Equation Model
3.3. The One-Equation Hybrid Generic Model
4. Analysis of Hybridization Strategies
4.1. Simple Empirical Strategies
4.2. Unbounded Extended Range
4.3. Bounded Full Range
5. Performance of Hybridization Strategies
5.1. Simple Empirical Strategies
5.2. Unbounded Extended Range
5.3. Bounded Full Range
6. Conclusions
- Turbulence modeling. An essential ingredient of turbulence models is a scale equation, e.g., an or equation. There is no theoretical basis for such an equation; these equations are designed by taking reference to empirical arguments. There exists a huge variety of such equations: the model [2,121], models [2,122], models [123,124,125], models [5,9,10,126], models [5,12], and stand-alone models without a k-equation [13,127,128]. On top of that, a variety of equation structures (including or excluding cross-diffusion terms) is applied. The analysis presented here cannot provide strict guideline for further developments, but it leads to very valuable conclusions. First, the requirement for turbulence models to be applicable to various scaling regimes excludes the inclusion of cross-diffusion terms, which prohibits the hybridization of equations. Second, the diffusivities of the k and scale equations (e.g., the and applied) should be the same. Third, the analysis presented clarifies the structure of dissipation terms; it provides corresponding evidence that is missing in SAS and DES approaches.
- Usually applied hybrid RANS-LES. Based on the fundamental conceptual shortcomings of RANS and LES (to often provide unreliable predictions of separated turbulent flows or to be inapplicable to very high- flows seen in reality because resolution requirements cannot be met), the use of hybrid RANS-LES is the no-alternative approach to deal with these problems. Most hybrid RANS-LES applications are performed with DES-type or WMLES-type models. WMLES is known to be relatively inaccurate [6,110,111]; explicit evidence for this view is provided by the discussions in Section 5. In addition, WMLES faces significant uncertainty of predictions caused by different ways to combine RANS and LES elements in WMLES equations. The situation is different in regard to DES. There is research over decades hoping to improve DES predictions by empirical model improvements combined with variations of how DES is used. For the first time, this paper provides explicit mathematical evidence that DES cannot systematically span a range of modeled-to-resolved flow regimes because of the scaling applied (the setting of ). As specified in Figure 2, DES only triggers uncontrolled instabilities. This is often helpful, but is no guarantee for systematic improvements of simulation results compared to RANS. As shown here, the same issue applies to SAS. The flow simulation results presented in Section 5 fully confirm this view.
- Optimal solution: core CES: A way to overcome these problems is the use of the full-range strategy, which is equivalent to CES methods that apply to explicitly drive the hybrid model in between fully modeled (RANS) and fully resolved (LES) regimes. Applications of this approach [110,111,119] demonstrate an impressive ability of this approach to provide very good flow predictions at relatively low computational cost. A specific feature of CES is its well balanced predictive ability, in contrast to the DES and WMLES methods that can provide some flow characteristics well at the cost of other flow characteristics. Comparisons with WRLES and WMLES results (see, e.g., the comparisons presented in Section 5) show that CES performs clearly better than WMLES and at least as good or better than WRLES at a small fraction of the WRLES cost. It is essential to note that CES methods include via their own flow resolution indicator in contrast to LES methods. This matters in regard to the known difficulty of assessing the resolution ability of LES [24,25]. It is also worth mentioning that the methods presented here (based on the generic model consideration) enable the use of LES in conjunction with a variety of turbulence models that may be considered according to specific requirements.
- Interesting option: WMLES-type CES. In contrast to the usually applied hybrid methods, the CES core methods require the explicit calculation of by processing statistics of resolved motion. Currently available experience shows that this adds only a little fraction to computational costs, but it requires corresponding computational code modifications. An interesting alternative pointed out here is the use of the unbounded extended range strategy, which takes explicitly reference to LES scaling in terms of the filter width . In this way, the involvement of additional simulation ingredients (like ) can be avoided. This concept (which represents a version of consistently formulated WMLES) may be seen to safeguard an LES simulation performed on relatively coarse grids (grids that cannot ensure an appropriate LES resolution). Nevertheless, because of the reference scaling applied, there is no guarantee that this concept works well in regimes on coarse grids, well away from the LES regime. A variant of using this approach has been discussed here by taking reference to GAS simulation results; see Section 5. The results are much better than the corresponding RANS, DES, and SAS results, but not as good as the CES results. It is worth noting that the GAS concept differs from the corresponding result obtained here by variational analysis by applying, for example, an inaccurate scaling with (see the discussion related to Equation (29)).
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Constraints on Generic Model Parameters
Appendix B. Hybridization of the Generic Two-Equation Model
Appendix C. Viscosity Hybridization of the Generic Two-Equation Model
Appendix D. The One-Equation Hybrid Generic Model
References
- Kolmogorov, A.N. Equations of turbulent motion of an incompressible fluid. Izv. Akad. Nauk SSR Seria Fiz. 1942, 6, 56–58. [Google Scholar]
- Wilcox, D.C. Turbulence Modeling for CFD, 2nd ed.; DCW Industries: La Canada Flintridge, CA, USA, 1998. [Google Scholar]
- Hanjalić, K. Advanced turbulence closure models: A view of current status and future prospects. Int. J. Heat Fluid Flow 1994, 15, 178–203. [Google Scholar]
- Hanjalić, K. Will RANS survive LES?: A view of perspectives. ASME J. Fluids Eng. 2005, 127, 831–839. [Google Scholar] [CrossRef]
- Menter, F.R.; Egorov, Y. The scale-adaptive simulation method for unsteady turbulent flow prediction: Part 1: Theory and model description. Flow Turbul. Combust. 2010, 78, 113–138. [Google Scholar] [CrossRef]
- Heinz, S. A review of hybrid RANS-LES methods for turbulent flows: Concepts and applications. Prog. Aerosp. Sci. 2020, 114, 100597. [Google Scholar]
- Asinari, P.; Fasano, M.; Chiavazzo, E. A kinetic perspective on k-ϵ turbulence model and corresponding entropy production. Entropy 2016, 18, 121. [Google Scholar] [CrossRef]
- Bhatnagar, P.L.; Gross, E.P.; Krook, M. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 1954, 94, 511–525. [Google Scholar] [CrossRef]
- Rotta, J.C. Über eine Methode zur Berechnung Turbulenter Scherströmungen. Aerodyn. Vers. Gott. 1968, Rep. 69, A14. [Google Scholar]
- Rotta, J.C. Turbulente Strömumgen; BG Teubner: Stuttgart, Germany, 1972. [Google Scholar]
- Rodi, W. Turbulence modelling for boundary-layer calculations. In Proceedings of the IUTAM Symposium on One Hundred Years of Boundary Layer Research, Göttingen, Germany, 12–14 August 2004; Meier, G., Sreenivasan, K., Eds.; Springer: Berlin/Heidelberg, Germany, 2006; pp. 247–256. [Google Scholar]
- Menter, F.R.; Egorov, Y. Revisiting the turbulent scale equation. In Proceedings of the IUTAM Symposium on One Hundred Years of Boundary Layer Research, Göttingen, Germany, 12–14 August 2004; Meier, G.E.A., Sreenivasan, K.R., Eds.; Springer: Dordrecht, The Netherlands, 2006; pp. 279–290. [Google Scholar]
- Spalart, P.; Allmaras, S. A one-equation turbulence model for aerodynamic flows. La Rech. Aérospatiale 1994, 1, 5–21. [Google Scholar]
- Pope, S.B. Turbulent Flows; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Sagaut, P. Large Eddy Simulation for Incompressible Flows: An Introduction; Springer: Berlin, Germany, 2002. [Google Scholar]
- Lesieur, M.; Metais, O.; Comte, P. Large-Eddy Simulations of Turbulence; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Piomelli, U. Large-eddy simulation: Achievements and challenges. Prog. Aerosp. Sci. 1999, 35, 335–362. [Google Scholar] [CrossRef]
- Piomelli, U. Large eddy simulations in 2030 and beyond. Phil. Trans. R. Soc. A 2014, 372, 20130320/1–20130320/23. [Google Scholar] [CrossRef]
- Yang, X.I.A.; Sadique, J.; Mittal, R.; Meneveau, C. Integral wall model for large eddy simulations of wall-bounded turbulent flows. Phys. Fluids 2015, 27, 025112. [Google Scholar] [CrossRef]
- Larsson, J.; Kawai, S.; Bodart, J.; Bermejo-Moreno, I. Large eddy simulation with modeled wall-stress: Recent progress and future directions. Mech. Eng. Rev. 2016, 3, 15-00418. [Google Scholar] [CrossRef]
- Bose, S.T.; Park, G.I. Wall-modeled large-eddy simulation for complex turbulent flows. Annu. Rev. Fluid Mech. 2018, 50, 535–561. [Google Scholar] [CrossRef]
- Yang, X.I.A.; Griffin, K.P. Grid-point and time-step requirements for direct numerical simulation and large-eddy simulation. Phys. Fluids 2021, 33, 015108. [Google Scholar] [CrossRef]
- Toosi, S.; Larsson, J. Towards systematic grid selection in LES: Identifying the optimal spatial resolution by minimizing the solution sensitivity. Comput. Fluids 2020, 201, 104488. [Google Scholar] [CrossRef]
- Davidson, L. Large Eddy Simulations: How to evaluate resolution. Int. J. Heat Fluid Flow 2009, 30, 1016–1025. [Google Scholar] [CrossRef]
- Wurps, H.; Steinfeld, G.; Heinz, S. Grid-resolution requirements for large-eddy simulations of the atmospheric boundary layer. Boundary Layer Meteorol. 2020, 175, 119–201. [Google Scholar] [CrossRef]
- Deardorff, J.W. A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J. Fluid Mech. 1970, 41, 453–480. [Google Scholar] [CrossRef]
- Schumann, U. Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J. Comput. Phys 1975, 18, 376–404. [Google Scholar] [CrossRef]
- Grötzbach, G. Direct numerical and large eddy simulation of turbulent channel flow. Encycl. Fluid Mech. 1987, 6, 1337–1391. [Google Scholar]
- Piomelli, U.; Ferziger, J.; Moin, P. New approximate boundary conditions for large eddy simulations of wall-bounded flows. Phys. Fluids A 1989, 6, 1061–1068. [Google Scholar] [CrossRef]
- Cabot, W.; Moin, P. Approximate wall boundary conditions in the large-eddy simulation of high Reynolds number flow. Flow Turbul. Combust. 1999, 63, 269–291. [Google Scholar] [CrossRef]
- Piomelli, U.; Balaras, E. Wall-layer models for large-eddy simulations. Annu. Rev. Fluid Mech. 2002, 34, 349–374. [Google Scholar] [CrossRef]
- Piomelli, U. Wall-layer models for large-eddy simulations. Prog. Aerosp. Sci. 2008, 44, 437–446. [Google Scholar] [CrossRef]
- Kawai, S.; Larsson, J. Wall-modeling in large eddy simulation: Length scales, grid resolution, and accuracy. Phys. Fluids 2012, 24, 015105. [Google Scholar] [CrossRef]
- Bose, S.T.; Moin, P. A dynamic slip boundary condition for wall-modeled large-eddy simulation. Phys. Fluids 2014, 26, 015104. [Google Scholar] [CrossRef]
- Park, G.I.; Moin, P. An improved dynamic non-equilibrium wall-model for large eddy simulation. Phys. Fluids 2014, 26, 015108. [Google Scholar] [CrossRef]
- Moin, P.; Bodart, J.; Bose, S.; Park, G.I. Wall-modeling in complex turbulent flows. In Advances in Fluid-Structure Interaction, Notes on Numerical Fluid Mechanics and Multidisciplinary Design 133; Braza, M., Bottaro, A., Thompson, M., Eds.; Springer: Cham, Switzerland, 2016; pp. 207–219. [Google Scholar]
- Chen, S.; Xia, Z.; Pei, S.; Wang, J.; Yang, Y.; Xiao, Z.; Shi, Y. Reynolds-stress-constrained large-eddy simulation of wall-bounded turbulent flows. J. Fluid Mech. 2012, 703, 1–28. [Google Scholar] [CrossRef]
- Chen, S.; Chen, Y.; Xia, Z.; Qu, K.; Shi, Y.; Xiao, Z.; Liu, Q.; Cai, Q.; Liu, F.; Lee, C.; et al. Constrained large-eddy simulation and detached eddy simulation of flow past a commercial aircraft at 14 degrees angle of attack. Sci. China Phys. Mech. Astron. 2013, 56, 270–276. [Google Scholar] [CrossRef]
- Xia, Z.; Shi, Y.; Hong, R.; Xiao, Z.; Chen, S. Constrained large-eddy simulation of separated flow in a channel with streamwise-periodic constrictions. J. Turbul. 2013, 14, 1–21. [Google Scholar] [CrossRef]
- Jiang, Z.; Xiao, Z.; Shi, Y.; Chen, S. Constrained large-eddy simulation of wall-bounded compressible turbulent flows. Phys. Fluids 2013, 25, 106102. [Google Scholar] [CrossRef]
- Hong, R.; Xia, Z.; Shi, Y.; Xiao, Z.; Chen, S. Constrained large-eddy simulation of compressible flow past a circular cylinder. Commun. Comput. Phys. 2014, 15, 388–421. [Google Scholar] [CrossRef]
- Zhao, Y.; Xia, Z.; Shi, Y.; Xiao, Z.; Chen, S. Constrained large-eddy simulation of laminar-turbulent transition in channel flow. Phys. Fluids 2014, 26, 095103. [Google Scholar] [CrossRef]
- Xua, Q.; Yang, Y. Reynolds stress constrained large eddy simulation of separation flows in a U-duct. J. Propul. Power Res. 2014, 3, 49–58. [Google Scholar] [CrossRef]
- Xia, Z.; Xiao, Z.; Shi, Y.; Chen, S. Constrained large-eddy simulation for aerodynamics. In Progress in Hybrid RANS-LES Modelling, Notes on Numerical Fluid Mechanics and Multidisciplinary Design 130; Girimaji, S., Haase, W., Peng, S.H., Schwamborn, D., Eds.; Springer: Cham, Switzerland, 2015; pp. 239–254. [Google Scholar]
- Jiang, Z.; Xiao, Z.; Shi, Y.; Chen, S. Constrained large-eddy simulation of turbulent flow and heat transfer in a stationary ribbed duct. Int. J. Numer. Methods Heat Fluid Flow 2016, 26, 1069–1091. [Google Scholar] [CrossRef]
- Verma, A.; Park, N.; Mahesh, K. A hybrid subgrid-scale model constrained by Reynolds stress. Phys. Fluids 2013, 25, 110805. [Google Scholar] [CrossRef]
- Xiao, Z.; Shi, Y.; Xia, Z.; Chen, S. Comment on ‘A hybrid subgrid-scale model constrained by Reynolds stress’ [Phys. Fluids 25, 110805 (2013)]. Phys. Fluids 2014, 26, 059101. [Google Scholar] [CrossRef]
- Verma, A.; Park, N.; Mahesh, K. Response to Comment on “A hybrid subgrid-scale model constrained by Reynolds stress” [Phys. Fluids 26, 059101 (2014)]. Phys. Fluids 2014, 26, 059102. [Google Scholar] [CrossRef]
- Spalart, P.R.; Jou, W.H.; Strelets, M.; Allmaras, S.R. Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. In Advances in DNS/LES; Liu, C., Liu, Z., Eds.; Greyden Press: Columbus, OH, USA, 1997; pp. 137–147. [Google Scholar]
- Travin, A.; Shur, M.L.; Strelets, M.; Spalart, P. Detached-eddy simulations past a circular cylinder. Flow Turbul. Combust. 1999, 63, 113–138. [Google Scholar] [CrossRef]
- Spalart, P.R. Strategies for turbulence modelling and simulations. Int. J. Heat Fluid Flow 2000, 21, 252–263. [Google Scholar] [CrossRef]
- Strelets, M. Detached eddy simulation of massively separated flows. In Proceedings of the 39th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 8–11 January 2001; AIAA Paper 01-0879. pp. 1–18. [Google Scholar]
- Travin, A.; Shur, M.L. Physical and numerical upgrades in the detached-eddy simulation of complex turbulent flows. In Advances in LES of Complex Flows; Friedrich, R., Rodi, W., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2002; pp. 239–254. [Google Scholar]
- Menter, F.R.; Kuntz, M.; Langtry, R. Ten years of industrial experience with SST turbulence model. Turbul. Heat Mass Transf. 2003, 4, 625–632. [Google Scholar]
- Spalart, P.R.; Deck, S.; Shur, M.L.; Squires, K.D.; Strelets, M.K.; Travin, A. A new version of detached-eddy simulation, resistant to ambiguous grid densities. Theor. Comput. Fluid Dyn. 2006, 20, 181–195. [Google Scholar] [CrossRef]
- Shur, M.L.; Spalart, P.R.; Strelets, M.K.; Travin, A. A hybrid RANS-LES approach with delayed-DES and wall-modelled LES capabilities. Int. J. Heat Fluid Flow 2008, 29, 1638–1649. [Google Scholar] [CrossRef]
- Spalart, P.R. Detached-eddy simulation. Annu. Rev. Fluid Mech. 2009, 41, 181–202. [Google Scholar] [CrossRef]
- Mockett, C.; Fuchs, M.; Thiele, F. Progress in DES for wall-modelled LES of complex internal flows. Comput. Fluids 2012, 65, 44–55. [Google Scholar] [CrossRef]
- Friess, C.; Manceau, R.; Gatski, T.B. Toward an equivalence criterion for hybrid RANS/LES methods. Comput. Fluids 2015, 122, 233–246. [Google Scholar] [CrossRef]
- Chaouat, B. The state of the art of hybrid RANS/LES modeling for the simulation of turbulent flows. Flow Turbul. Combust. 2017, 99, 279–327. [Google Scholar] [CrossRef]
- Menter, F.; Hüppe, A.; Matyushenko, A.; Kolmogorov, D. An overview of hybrid RANS–LES models developed for industrial CFD. Appl. Sci. 2021, 11, 2459. [Google Scholar] [CrossRef]
- Heinz, S. Unified turbulence models for LES and RANS, FDF and PDF simulations. Theoret. Comput. Fluid Dynam. 2007, 21, 99–118. [Google Scholar] [CrossRef]
- Heinz, S. Realizability of dynamic subgrid-scale stress models via stochastic analysis. Monte Carlo Methods Applic. 2008, 14, 311–329. [Google Scholar] [CrossRef]
- Heinz, S.; Gopalan, H. Realizable versus non-realizable dynamic subgrid-scale stress models. Phys. Fluids 2012, 24, 115105. [Google Scholar] [CrossRef]
- Gopalan, H.; Heinz, S.; Stöllinger, M. A unified RANS-LES model: Computational development, accuracy and cost. J. Comput. Phys. 2013, 249, 249–279. [Google Scholar] [CrossRef]
- Mokhtarpoor, R.; Heinz, S.; Stoellinger, M. Dynamic unified RANS-LES simulations of high Reynolds number separated flows. Phys. Fluids 2016, 28, 095101. [Google Scholar] [CrossRef]
- Mokhtarpoor, R.; Heinz, S. Dynamic large eddy simulation: Stability via realizability. Phys. Fluids 2017, 29, 105104. [Google Scholar] [CrossRef]
- Girimaji, S.; Srinivasan, R.; Jeong, E. PANS turbulence for seamless transition between RANS and LES: Fixed-point analysis and preliminary results. In Proceedings of the ASME FEDSM03, Honolulu, HI, USA, 6–11 July 2003; ASME Paper FEDSM2003-45336. pp. 1–9. [Google Scholar]
- Girimaji, S.; Abdol-Hamid, K. Partially averaged Navier Stokes model for turbulence: Implemantation and validation. In Proceedings of the 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 10–13 January 2005; AIAA Paper 05-0502. pp. 1–14. [Google Scholar]
- Girimaji, S. Partially-averaged Navier-Stokes method for turbulence: A Reynolds-averaged Navier-Stokes to direct numerical simulation bridging method. ASME J. Appl. Mech. 2006, 73, 413–421. [Google Scholar] [CrossRef]
- Girimaji, S.; Jeong, E.; Srinivasan, R. Partially averaged Navier-Stokes method for turbulence: Fixed point analysis and comparisons with unsteady partially averaged Navier-Stokes. ASME J. Appl. Mech. 2006, 73, 422–429. [Google Scholar] [CrossRef]
- Lakshmipathy, S.; Girimaji, S.S. Extension of Boussinesq turbulence constitutive relation for bridging methods. J. Turbul. 2007, 8, 1–20. [Google Scholar] [CrossRef]
- Frendi, A.; Tosh, A.; Girimaji, S. Flow past a backward-facing step: Comparison of PANS, DES and URANS results with experiments. Int. J. Comput. Methods Eng. Sci. Mech. 2007, 8, 23–38. [Google Scholar] [CrossRef]
- Lakshmipathy, S.; Girimaji, S.S. Partially averaged Navier-Stokes (PANS) method for turbulence simulations: Flow past a circular cylinder. ASME J. Fluids Eng. 2010, 132, 121202/1–121202/9. [Google Scholar] [CrossRef]
- Jeong, E.; Girimaji, S.S. Partially averaged Navier–Stokes (PANS) method for turbulence simulations: Flow past a square cylinder. ASME J. Fluids Eng. 2010, 132, 121203/1–121203/11. [Google Scholar] [CrossRef]
- Basara, B.; Krajnovic, S.; Girimaji, S.S.; Pavlovic, Z. Near-wall formulation of the partially averaged Navier-Stokes turbulence model. AIAA J. 2011, 42, 2627–2636. [Google Scholar] [CrossRef]
- Krajnovic, S.; Lárusson, R.; Basara, B. Superiority of PANS compared to LES in predicting a rudimentary landing gear flow with affordable meshes. Int. J. Heat Fluid Flow 2012, 37, 109–122. [Google Scholar] [CrossRef]
- Foroutan, H.; Yavuzkurt, S. A partially averaged Navier Stokes model for the simulation of turbulent swirling flow with vortex breakdown. Int. J. Heat Fluid Flow 2014, 50, 402–416. [Google Scholar] [CrossRef]
- Drikakis, D.; Sofos, F. Can artificial intelligence accelerate fluid mechanics research? Fluids 2023, 8, 212. [Google Scholar] [CrossRef]
- Schiestel, R.; Dejoan, A. Towards a new partially integrated transport model for coarse grid and unsteady turbulent flow simulations. Theor. Comput. Fluid Dyn. 2005, 18, 443–468. [Google Scholar] [CrossRef]
- Chaouat, B.; Schiestel, R. A new partially integrated transport model for subgrid-scale stresses and dissipation rate for turbulent developing flows. Phys. Fluids 2005, 17, 065106. [Google Scholar] [CrossRef]
- Chaouat, B.; Schiestel, R. From single-scale turbulence models to multiple-scale and subgrid-scale models by Fourier transform. Theor. Comput. Fluid Dyn. 2007, 21, 201–229. [Google Scholar] [CrossRef]
- Befeno, I.; Schiestel, R. Non-equilibrium mixing of turbulence scales using a continuous hybrid RANS/LES approach: Application to the shearless mixing layer. Flow Turbul. Combust. 2007, 78, 129–151. [Google Scholar] [CrossRef]
- Chaouat, B.; Schiestel, R. Progress in subgrid-scale transport modelling for continuous hybrid nonzonal RANS/LES simulations. Int. J. Heat Fluid Flow 2009, 30, 602–616. [Google Scholar] [CrossRef]
- Chaouat, B. Subfilter-scale transport model for hybrid RANS/LES simulations applied to a complex bounded flow. J. Turbul. 2010, 11, N51. [Google Scholar] [CrossRef]
- Chaouat, B. Simulation of turbulent rotating flows using a subfilter scale stress model derived from the partially integrated transport modeling method. Phys. Fluids 2012, 24, 045108. [Google Scholar] [CrossRef]
- Chaouat, B.; Schiestel, R. Analytical insights into the partially integrated transport modeling method for hybrid Reynolds averaged Navier-Stokes equations-large eddy simulations of turbulent flows. Phys. Fluids 2012, 24, 085106. [Google Scholar] [CrossRef]
- Chaouat, B.; Schiestel, R. Partially integrated transport modeling method for turbulence simulation with variable filters. Phys. Fluids 2013, 25, 125102. [Google Scholar] [CrossRef]
- Chaouat, B.; Schiestel, R. Hybrid RANS-LES simulations of the turbulent flow over periodic hills at high Reynolds number using the PITM method. Comput. Fluids 2013, 84, 279–300. [Google Scholar] [CrossRef]
- Chaouat, B. Application of the PITM method using inlet synthetic turbulence generation for the simulation of the turbulent flow in a small axisymmetric contraction. Flow Turbul. Combust. 2017, 98, 987–1024. [Google Scholar] [CrossRef]
- Menter, F.R.; Kuntz, M.; Bender, R. A scale-adaptive simulation model for turbulent flow predictions. In Proceedings of the 41st AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 6–9 January 2003; AIAA Paper 03-0767. pp. 1–12. [Google Scholar]
- Menter, F.R.; Egorov, Y. A scale-adaptive simulation model using two-equation models. In Proceedings of the 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 10–13 January 2005; AIAA Paper 05-1095. pp. 1–13. [Google Scholar]
- Menter, F.R.; Egorov, Y. The scale-adaptive simulation method for unsteady turbulent flow prediction: Part 2: Application to complex flows. Flow Turbul. Combust. 2010, 78, 139–165. [Google Scholar]
- Jakirlić, S.; Maduta, R. Extending the bounds of "steady" RANS closures: Toward an instability-sensitive Reynolds stress model. Int. J. Heat Fluid Flow 2015, 51, 175–194. [Google Scholar] [CrossRef]
- Wyngaard, J.C. Toward numerical modeling in the “Terra Incognita”. J. Atmos. Sci. 2004, 61, 1816–1826. [Google Scholar] [CrossRef]
- Juliano, T.W.; Kosović, B.; Jiménez, P.A.; Eghdami, M.; Haupt, S.E.; Martilli, A. “Gray Zone” simulations using a three-dimensional planetary boundary layer parameterization in the Weather Research and Forecasting Model. Mon. Weather Rev. 2022, 150, 1585–1619. [Google Scholar] [CrossRef]
- Heinz, S.; Heinz, J.; Brant, J.A. Mass transport in membrane systems: Flow regime identification by Fourier analysis. Fluids 2022, 7, 369. [Google Scholar] [CrossRef]
- Heinz, S. The potential of machine learning methods for separated turbulent flow simulations: Classical versus dynamic methods. Fluids 2024, 9, 278. [Google Scholar] [CrossRef]
- Yakhot, V.; Orszag, S.A. Renormalization group analysis of turbulence. I. Basic theory. J. Sci. Comput. 1986, 1, 3–51. [Google Scholar] [CrossRef]
- Yakhot, V.; Orszag, S.A.; Thangam, S.; Gatski, T.B.; Speziale, C.G. Development of turbulence models for shear flows by a double expansion technique. Phys. Fluids A Fluid Dyn. 1992, 4, 1510–1520. [Google Scholar] [CrossRef]
- Nagano, Y.; Itazu, Y. Renormalization group theory for turbulence: Assessment of the Yakhot-Orszag-Smith theory. Fluid Dyn. Res. 1997, 20, 157–172. [Google Scholar] [CrossRef]
- Zhou, Y. Renormalization group theory for fluid and plasma turbulence. Phys. Rep. 2010, 488, 1–49. [Google Scholar]
- Sadiki, A.; Bauer, W.; Hutter, K. Thermodynamically consistent coefficient calibration in nonlinear and anisotropic closure models for turbulence. Contin. Mech. Therm. 2000, 12, 131–149. [Google Scholar] [CrossRef]
- Sadiki, A.; Hutter, K. On Thermodynamics of Turbulence: Development of First Order Closure Models and Critical Evaluation of Existing Models. J. Non-Equil. Thermodyn. 2000, 25, 131. [Google Scholar] [CrossRef]
- Shih, T.H.; Liou, W.W.; Shabbir, A.; Yang, Z.; Zhu, J. A new k-ϵ eddy viscosity model for high reynolds number turbulent flows. Comput. Fluids 1995, 24, 227–238. [Google Scholar] [CrossRef]
- Shaheed, R.; Mohammadian, A.; Kheirkhah Gildeh, H. A comparison of standard k–ε and realizable k–ε turbulence models in curved and confluent channels. Environ. Fluid Mech. 2019, 19, 543–568. [Google Scholar] [CrossRef]
- Heinz, S.; Fagbade, A. Evaluation metrics for partially and fully resolving simulations methods for turbulent flows. Int. J. Heat Fluid Flow 2025, 115, 109867. [Google Scholar] [CrossRef]
- Heinz, S. Physically consistent resolving simulations of turbulent flows. Entropy 2024, 26, 1044. [Google Scholar] [CrossRef] [PubMed]
- Fagbade, A.I. Continuous Eddy Simulation for Turbulent Flows. Ph.D. Thesis, University of Wyoming, Laramie, WY, USA, 2024. Available online: https://www.proquest.com/docview/3058393461 (accessed on 1 May 2025).
- Fagbade, A.; Heinz, S. Continuous eddy simulation (CES) of transonic shock-induced flow separation. Appl. Sci. 2024, 14, 2705. [Google Scholar] [CrossRef]
- Fagbade, A.; Heinz, S. Continuous eddy simulation vs. resolution-imposing simulation methods for turbulent flows. Fluids 2024, 9, 22. [Google Scholar] [CrossRef]
- Heinz, S. A mathematical solution to the Computational Fluid Dynamics (CFD) dilemma. Mathematics 2023, 11, 3199. [Google Scholar] [CrossRef]
- Heinz, S. Minimal error partially resolving simulation methods for turbulent flows: A dynamic machine learning approach. Phys. Fluids 2022, 34, 051705. [Google Scholar] [CrossRef]
- Heinz, S. Remarks on energy partitioning control in the PITM hybrid RANS/LES method for the simulation of turbulent flows. Flow Turbul. Combust. 2022, 108, 927–933. [Google Scholar] [CrossRef]
- Heinz, S. From two-equation turbulence models to minimal error resolving simulation methods for complex turbulent flows. Fluids 2022, 7, 368. [Google Scholar] [CrossRef]
- Heinz, S. Theory-based mesoscale to microscale coupling for wind energy applications. Appl. Math. Model. 2021, 98, 563–575. [Google Scholar] [CrossRef]
- Heinz, S. The continuous eddy simulation capability of velocity and scalar probability density function equations for turbulent flows. Phys. Fluids 2021, 33, 025107. [Google Scholar] [CrossRef]
- Heinz, S.; Peinke, J.; Stoevesandt, B. Cutting-edge turbulence simulation methods for wind energy and aerospace problems. Fluids 2021, 6, 288. [Google Scholar] [CrossRef]
- Heinz, S.; Mokhtarpoor, R.; Stoellinger, M.K. Theory-based Reynolds-averaged Navier-Stokes equations with large eddy simulation capability for separated turbulent flow simulations. Phys. Fluids 2020, 32, 065102. [Google Scholar] [CrossRef]
- Heinz, S. The large eddy simulation capability of Reynolds-averaged Navier-Stokes equations: Analytical results. Phys. Fluids 2019, 31, 021702. [Google Scholar] [CrossRef]
- Rodi, W. Examples of calculation methods for flow and mixing in stratified fluids. J. Geophys. Res. Oceans 1987, 92, 5305–5328. [Google Scholar]
- Wilcox, D.C. Reassessment of the scale-determining equation for advanced turbulence models. AIAA J. 1988, 26, 1299–1310. [Google Scholar] [CrossRef]
- Speziale, C.G. On nonlinear kL and k-ϵ models of turbulence. J. Fluid Mech. 1987, 178, 459–475. [Google Scholar] [CrossRef]
- Smith, B. A near wall model for the k-L two equation turbulence model. In Proceedings of the Fluid Dynamics Conference, Colorado Springs, CO, USA, 20–23 June 1994; AIAA Paper 94-2386. pp. 1–9. [Google Scholar]
- Goldberg, U.; Chakravarthy, S. A kL turbulence closure for wall-bounded flows. In Proceedings of the 35th AIAA Fluid Dynamics Conference and Exhibit, Toronto, ON, Canada, 6–9 June 2005; AIAA Paper 05-4638. pp. 1–11. [Google Scholar]
- Mellor, G.L.; Yamada, T. Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys. 1982, 20, 851–875. [Google Scholar] [CrossRef]
- Wray, T.J.; Agarwal, R.K. Application of new one-equation turbulence model to computations of separated flows. AIAA J. 2014, 52, 1325–1330. [Google Scholar] [CrossRef]
- Qian, X.; Agarwal, R.K. CFD analysis of separated flow over the Gaussian bump using various turbulence models. In Proceedings of the AIAA SCITECH 2025 Forum, Orlando, FL, USA, 6–10 January 2025; AIAA Paper 25-2209. pp. 1–6. [Google Scholar]
- Umlauf, L.; Burchard, H. A generic length-scale equation for geophysical turbulence models. J. Mar. Res. 2003, 61, 235–265. [Google Scholar] [CrossRef]
- Liu, J.; Chen, J.; Xiao, Z. Improvements for the DDES model with consideration of grid aspect ratio and coherent vortex structures. Aerosp. Sci. Technol. 2024, 151, 109314. [Google Scholar] [CrossRef]
- Heinz, S. Turbulent supersonic channel flow: Direct numerical simulation and modeling. AIAA J. 2006, 44, 3040–3050. [Google Scholar] [CrossRef]
- Plaut, E.; Heinz, S. Exact eddy-viscosity equation for turbulent wall flows—Implications for computational fluid dynamics models. AIAA J. 2022, 60, 1347–1364. [Google Scholar] [CrossRef]
- Wang, G.; Tang, Y.; Liu, Y. Gray area mitigation in grid-adaptive simulation for wall-bounded turbulent flows. Int. J. Mech. Sci. 2025, 296, 110305. [Google Scholar]
- Rapp, C.; Manhart, M. Flow over periodic hills—An experimental study. Exp. Fluids 2011, 51, 247–269. [Google Scholar] [CrossRef]
- Kähler, C.J.; Scharnowski, S.; Cierpka, C. Highly resolved experimental results of the separated flow in a channel with streamwise periodic constrictions. J. Fluid Mech. 2016, 796, 257–284. [Google Scholar] [CrossRef]
- Seifert, A.; Pack, L. Active flow separation control on wall-mounted hump at high Reynolds numbers. AIAA J. 2002, 40, 1362–1372. [Google Scholar] [CrossRef]
- Greenblatt, D.; Paschal, K.B.; Yao, C.-S.; Harris, J.; Schaeffler, N.W.; Washburn, A.E. Experimental investigation of separation control Part 1: Baseline and steady suction. AIAA J. 2006, 44, 2820–2830. [Google Scholar] [CrossRef]
- Wang, G.; Liu, Y. A grid-adaptive simulation model for turbulent flow predictions. Phys. Fluids 2022, 34, 075125. [Google Scholar] [CrossRef]
- Wang, G.; Tang, Y.; Wei, X.; Liu, Y. Predicting turbulent flow over a backward-facing step using grid-adaptive simulation method. Aerosp. Sci. Technol. 2025, 158, 109913. [Google Scholar] [CrossRef]
- Menter, F.R. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J. 1994, 32, 1598–1605. [Google Scholar] [CrossRef]
- Gloerfelt, X.; Cinnella, P. Large eddy simulation requirements for the flow over periodic hills. Flow Turbul. Combust. 2019, 103, 55–91. [Google Scholar] [CrossRef]
- Iyer, P.S.; Malik, M.R. Wall-modeled large eddy simulation of flow over a wallmounted hump. In Proceedings of the 2016 AIAA Aerospace Sciences Meeting, San Diego, CA, USA, 4–8 January 2016; pp. 1–22. [Google Scholar]
- Uzun, A.; Malik, M. Large-Eddy Simulation of flow over a wall-mounted hump with separation and reattachment. AIAA J. 2018, 56, 715–730. [Google Scholar] [CrossRef]
- Uzun, A.; Malik, M.R. Wall-resolved large-eddy simulation of flow separation over NASA wall-mounted hump. In Proceedings of the 55th AIAA Aerospace Sciences Meeting, Grapevine, TX, USA, 9–13 January 2017. AIAA Paper 17-0538. [Google Scholar]
, |
, |
, |
Methods | Remarks |
---|---|
GAS-SST [133] | The GAS version of CES-K*V, SST is hybridized according to GAS. |
SAS-SST [133] | SST hybridized according to scale adaptive simulation (SAS) [5,12]. |
DDES-SST [133] | SST hybridized according to delayed detached eddy simulation (DDES) [55]. |
RANS-SST [133] | SST RANS baseline model. |
CES-KOS [111,119] | CES-K*S applied to turbulence model. |
CES-KOKU [119] | CES-K*K version (equivalent to CES-KOS), model, hybridization via time scale. |
Methods | Cells | Error | Methods | Cells | Error | ||
---|---|---|---|---|---|---|---|
GAS-SST [133] | 0.216M | 3.70 | GAS-SST [133] | 0.77M | 1.18 | ||
0.392M | 3.98 | 1.53M | 1.18 | ||||
0.768M | 4.00 | ||||||
SAS-SST [133] | 0.216M | 7.49 | SAS-SST [133] | 0.77M | 1.29 | ||
0.768M | 4.45 | ||||||
DDES-SST [133] | 0.216M | 7.40 | DDES-SST [133] | 0.77M | 1.54 | ||
0.768M | 4.17 | ||||||
RANS-SST [133] | 0.216M | 7.43 | RANS-SST [133] | 0.77M | 1.27 | ||
CES-KOKU [119] | 0.5M | 3.78 | CES-KOS [119] | 1.7M | 1.11 | ||
0.12M | 3.70 | 3.9M | 1.10 | ||||
WRLES [141] | 33.6M | 4.00 | WRLES [142] | 210M | 1.095 | ||
WMLES [143] | 4.4M | 1.045 | |||||
11.6M | 1.105 |
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Heinz, S. Strategy Analysis of Seamlessly Resolving Turbulent Flow Simulations. Aerospace 2025, 12, 597. https://doi.org/10.3390/aerospace12070597
Heinz S. Strategy Analysis of Seamlessly Resolving Turbulent Flow Simulations. Aerospace. 2025; 12(7):597. https://doi.org/10.3390/aerospace12070597
Chicago/Turabian StyleHeinz, Stefan. 2025. "Strategy Analysis of Seamlessly Resolving Turbulent Flow Simulations" Aerospace 12, no. 7: 597. https://doi.org/10.3390/aerospace12070597
APA StyleHeinz, S. (2025). Strategy Analysis of Seamlessly Resolving Turbulent Flow Simulations. Aerospace, 12(7), 597. https://doi.org/10.3390/aerospace12070597