Double time-relaxation kinetic model for compressible turbulence modeling

Abstract In this paper, a double time-relaxation kinetic model (DtrKM) is proposed for compressible turbulence modeling on unresolved grids. Within the double time-relaxation framework, DtrKM is extended in the form of generalized Bhatnagar-Gross-Krook model. Based on the first-order Chapman-Enskog expansion, DtrKM connects with the six-variable macroscopic governing equations. The first five governing equations correspond to the conservative laws in mass, momentum and total energy, while the sixth equation governs the evolution of unresolved turbulence kinetic energy $$K_{utke}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow> <mml:mi>utke</mml:mi> </mml:mrow> </mml:msub> </mml:math> . The unknowns in DtrKM, including turbulent relaxation time and source term, are modeled via gradient-type assumption and standard dynamic modeling approach. The current kinetic model on unresolved grids correspondingly offers a mesoscopic understanding for one-equation subgrid-scale turbulence kinetic energy $$K_{sgs}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow> <mml:mi>sgs</mml:mi> </mml:mrow> </mml:msub> </mml:math> model for compressible large eddy simulation. To solve DtrKM accurately and robustly, a high-accuracy gas-kinetic scheme is developed, which inherits the advantages of well-established gas-kinetic scheme for simulating macroscopic governing equations. Three-dimensional decaying compressible isotropic turbulence and temporal compressible plane mixing layer on unresolved grids are simulated to evaluate the generalized kinetic model. The performance of key turbulent quantities up to second-order statistics confirms that DtrKM is comparable with the widely-used dynamic Smagorinsky model. The DtrKM provides a workable approach for compressible turbulence modeling and simulation on unresolved grids.