Abstract
Gas-kinetic schemes are derived from the BGK-Boltzmann equation. These family of schemes for CFD are computationally more demanding than conventional upwind schemes but provide a number of advantages stemming precisely from the fact that they were not derived from the Navier–Stokes equations. We highlight three peculiarities: (i) the gas evolution is derived in space and time exactly, (ii) the viscous stress tensor is built from the collision operator and not through the diffusion operator, variations of the relaxation time are thus not simply applied linearly to the strain rate (and its moments) but input through the collision operator, (iii) the order of the Chapman–Enskog expansion is a “natural” way to further improve the physical consistence of the collision operator. In practice, gas-kinetic schemes are not only more suitable to resolve vortical structures but they also handle turbulent viscosity (or, better, a turbulent relaxation time) in a physically more relevant fashion. Whereas the first advantage is exploited mostly in approaches like Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES), where the ability to resolve turbulent structures is key, the latter provides some leverage to approaches like Reynolds-Averaged Navier–Stokes (RANS) or hybrid RANS-LES where unresolved turbulence is funnelled through the collision operator. The paper aims at reviewing these advantages in the light of the results obtained by the author and those published in the recent years.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Babinsky, H., Harvey, J.K.: Shock Wave-Boundary-Layer Interactions. Cambridge University Press, New York (2011)
Bendiksen, O.O.: Review of unsteady transonic aerodynamics: theory and applications. Progr. Aerosp. Sci. 47(2), 135–167 (2011)
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. 94(3), 511–525 (1954)
Bookey, P., Wyckham, C., Smits, A.: Experimental investigations of Mach 3 shock-wave turbulent boundary layer interactions. AIAA Paper No. 2005-4899 (2005)
Cao, G., Pan, L., Xu, K.: Three dimensional high-order gas-kinetic scheme for supersonic isotropic turbulence i: criterion for direct numerical simulation. Comput. Fluids 192, 104273 (2019)
Cao, G., Su, H., Xu, J., Xu, K.: Implicit high-order gas kinetic scheme for turbulence simulation. Aerosp. Sci. Technol. 92, 958–971 (2019)
Cao, G., Pan, L., Xu, K.: Three dimensional high-order gas-kinetic scheme for supersonic isotropic turbulence II: Coarse-graining analysis of compressible Ksgs budget. J. Comput. Phys. 439, 110402 (2021)
Cao, G., Pan, L., Xu, K., Wan, M., Chen, S.: Non-equilibrium time-relaxation kinetic model for compressible turbulence modeling (2021). Preprint arXiv:2112.08873
Cao, G., Pan, L., Xu, K.: High-order gas-kinetic scheme with parallel computation for direct numerical simulation of turbulent flows. J. Comput. Phys. 448, 110739 (2022)
Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, New York (1988)
Chen, H., Kandasamy, S., Orszag, S., Shock, R., Succi, S., Yakhot, V.: Extended Boltzmann kinetic equation for turbulent flows. Science 301(5633), 633–636 (2003)
Chen, H., Orszag, S.A., Staroselsky, I., Succi, S.: Expanded analogy between Boltzmann kinetic theory of fluids and turbulence. J. Fluid Mech. 519(1), 301–314 (2004)
Chou, S.Y., Baganoff, D.: Kinetic flux–vector splitting for the Navier–Stokes equations. J. Comput. Phys. 130(2), 217–230 (1997)
Cook, P.H., McDonald, M.A., Firman, M.C.P.: Aerofoil RAE 2822–pressure distributions, and boundary layer and wake measurements. Experimental data base for computer program assessment. AGARD Advisory (1979)
Délery, J.: Experimental investigation of turbulence properties in transonic shock/boundary-layer interactions. AIAA J. 21, 180–185 (1983)
Dolling, D.S., Erengil, M.E.: Unsteady wave structure near separation in a Mach 5 compression rampinteraction. AIAA J. 29(5), 728–735 (1991)
Dupont, P., Haddad, C., Debiève, J.F.: Space and time organization in a shock-induced separated boundary layer. J. Fluid Mech. 559, 255–278 (2006)
Edwards, J.R.: Numerical simulations of shock/boundary layer interactions using time-dependent modeling techniques: a survey of recent results. Progr. Aerosp. Sci. 44(6), 447–465 (2008)
Guo, Z., Xu, K.: Progress of discrete unified gas-kinetic scheme for multiscale flows. Adv. Aerodyn. 3(1), 1–42 (2021)
Harris, C.D.: Two-dimensional aerodynamic characteristics of the NACA 0012 airfoil in the Langley 8 foot transonic pressure tunnel. NASA Technical Memorandum 81-927 (1981)
Heeg, J., Chwalowski, P.: Investigation of the transonic flutter boundary of the benchmark supercritical wing. In: 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA SciTech Forum. American Institute of Aeronautics and Astronautics, AIAA–2017–0191, 9–13 January 2017 (2017)
Li, Q., Fu, S., Xu, K.: Application of gas-kinetic scheme with kinetic boundary conditions in hypersonic flow. AIAA J. 43(10), 2170–2176 (2005)
Li, Q., Xu, K., Fu, S.: A high-order gas-kinetic Navier–Stokes flow solver. J. Comput. Phys. 229(19), 6715–6731 (2010)
Liu, H., Cao, G., Chen, W., Agarwal, R.K., Zhao, W.: Gas-kinetic scheme coupled with turbulent kinetic energy equation for computing hypersonic turbulent and transitional flows. Int. J. Comput. Fluid Dyn. 35(5), 319–330 (2021)
Liu, H., Agarwal, R.K., Chen, W.: Computation of hypersonic turbulent and transitional flows using an extended gas kinetic scheme. In: AIAA SCITECH 2022 Forum, p. 1050 (2022)
Mandal, J.C., Deshpande, S.M.: Kinetic flux vector splitting for Euler equations. Comput. Fluids 23(2), 447–478 (1994)
May, G., Srinivasan, B., Jameson, A.: An improved gas-kinetic BGK finite-volume method for three-dimensional transonic flow. J. Comput. Phys. 220(2), 856–878 (2007)
McConkey, R., Yee, E., Lien, F.-S.: A curated dataset for data-driven turbulence modelling. Sci. Data 8(1), 1–14 (2021)
Menter, F.R.: Improved two-equation k-omega turbulence models for aerodynamic flows. NASA STI/Recon Technical Report N 93, 22809 (1992)
Menter, F.R., Langtry, R.B., Likki, S.R., Suzen, Y.B., Huang, P.G., Völker, S.: A correlation-based transition model using local variables–part i: model formulation. J. Turbomach. 128, 413–422 (2006)
Mishra, A.A., Mukhopadhaya, J., Iaccarino, G., Alonso, J.: Uncertainty estimation module for turbulence model predictions in su2. AIAA J. 57(3), 1066–1077 (2019)
Ohwada, T., Xu, K.: The kinetic scheme for the full-Burnett equations. J. Comput. Phys. 201(1), 315–332 (2004)
Pan, L., Cao, G., Xu, K.: Fourth-order gas-kinetic scheme for turbulence simulation with multi-dimensional WENO reconstruction. Comput. Fluids 221, 104927 (2021)
Righi, M.: A Gas-Kinetic Scheme for Turbulent Flow. AIAA Paper No. 2014-3330 (2014)
Righi, M.: A modified gas-kinetic scheme for turbulent flow. Commun. Comput. Phys. 16(1), 239–263 (2014)
Righi, M.: A Numerical Scheme for Hypersonic Turbulent Flow . AIAA Paper No. AIAA 2015-3341 (2015)
Righi, M.: A gas-kinetic scheme for turbulent flow. Flow Turbulence Combust. 97(1), 121–139 (2016)
Righi, M.: Turbulence Modelling in Aeroelastic Problems. ERCOFTAC ETMM11 (2016)
Righi, M., Wang, R.: A gas-kinetic scheme for the simulation of turbulent flows. In: Fan, J. (ed.) Proceeding of the 29th International Symposium on Rarefied Gas Dynamics, Xi’an, pp. 1363–1370. American Institute of Physics, College Park (2014)
Rubinstein, R., Barton, J.M.: Nonlinear Reynolds stress models and the renormalization group. Phys. Fluids A Fluid Dyn. (1989–1993) 2(8), 1472–1476 (1990)
Settles, G.S., Fitzpatrick, T.J., Bogdonoff, S.M.: Detailed study of attached and separated compression corner flowfields in high Reynolds number supersonic flow. AIAA J. 17(6), 579–585 (1979)
Spalart, P., Allmaras, S.: A one-equation turbulence model for aerodynamic flows. 30th Aerospace Sciences Meeting and Exhibit (1992). https://arc.aiaa.org/doi/abs/10.2514/6.1992-439
Taghizadeh, S., Witherden, F.D., Girimaji, S.S.: Turbulence closure modeling with data-driven techniques: physical compatibility and consistency considerations. New J. Phys. 22(9), 093023 (2020)
Tan, S., Li, Q., Xiao, Z., Fu, S.: Gas kinetic scheme for turbulence simulation. Aerosp. Sci. Technol. 78, 214–227 (2018)
Wilcox, D.C.: Turbulence Modeling for CFD, 3rd edn. DCW Industries, La Canada (2006)
Wilcox, D.C.: Formulation of the kw turbulence model revisited. AIAA J. 46(11), 2823–2838 (2008)
Xiao, H., Cinnella, P.: Quantification of model uncertainty in rans simulations: a review. Progr. Aerosp. Sci. 108, 1–31 (2019)
Xu, K.: Gas-kinetic schemes for unsteady compressible flow simulations. In: VKI, Computational Fluid Dynamics, Annual Lecture Series, 29th, Rhode-Saint-Genese, Belgium (1998)
Xu, K.: A gas-kinetic BGK scheme for the Navier–Stokes equations and its connection with artificial dissipation and Godunov method. J. Comput. Phys. 171(1), 289–335 (2001)
Xu, K.: A Unified Computational Fluid Dynamics Framework from Rarefied to Continuum Regimes. Cambridge University Press, Cambridge (2021)
Xu, K., Prendergast, K.H.: Numerical Navier-Stokes solutions from gas kinetic theory. J. Comput. Phys. 114(1), 9–17 (1994)
Xu, K., Mao, M., Tang, L.: A multidimensional gas-kinetic BGK scheme for hypersonic viscous flow. J. Comput. Phys. 203(2), 405–421 (2005)
Xu, K., He, X., Cai, C.: Multiple temperature kinetic model and gas-kinetic method for hypersonic non-equilibrium flow computations. J. Comput. Phys. 227(14), 6779–6794 (2008)
Xuan, L., Xu, K.: A new gas-kinetic scheme based on analytical solutions of the BGK equation. J. Comput. Phys. 234, 524–539 (2013). https://doi.org/10.1016/j.jcp.2012.10.007
Yang, X., Shyy, W., Xu, K.: Unified gas-kinetic wave-particle method for gas-particle two phase flow from dilute to dense solid-particle limit (2021). Preprint arXiv:2112.01829
Yang, X., Ji, X., Shyy, W., Xu, K.: Comparison of the performance of high-order schemes based on the gas-kinetic and HLLC fluxes. J. Comput. Phys. 448, 110706 (2022)
Zhao, W., Wang, J., Cao, G., Xu, K.: High-order gas-kinetic scheme for large eddy simulation of turbulent channel flows. Phys. Fluids 33(12), 125102 (2021)
Zhao, F., Ji, X., Shyy, W., Xu, K.: A compact high-order gas-kinetic scheme on unstructured mesh for acoustic and shock wave computations. J. Comput. Phys. 449, 110812 (2022)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Righi, M. (2023). Gas-Kinetic Methods for Turbulent Flow. In: Barbante, P., Belgiorno, F.D., Lorenzani, S., Valdettaro, L. (eds) From Kinetic Theory to Turbulence Modeling. INdAM 2021. Springer INdAM Series, vol 51. Springer, Singapore. https://doi.org/10.1007/978-981-19-6462-6_18
Download citation
DOI: https://doi.org/10.1007/978-981-19-6462-6_18
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-19-6461-9
Online ISBN: 978-981-19-6462-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)