Fluid-kinetic multiscale solver for wall-bounded turbulence

TL;DR

This work addresses the challenge of simulating wall-bounded turbulence across laminar, transitional, and turbulent regimes by merging kinetic near-wall physics with continuum bulk dynamics via a two-level coupling of DSMC and high-order HOLB. The method leverages Grad's 13-moment framework to enable macro-micro information exchange in a buffer zone between a DSMC near-wall layer and a bulk RD3Q41 HOLB region, capturing strong non-equilibrium effects while maintaining computational feasibility. Validation on plane Poiseuille and Couette flows demonstrates accurate macroscopic fields and smooth interfaces, and simulations of Minimal Turbulent Couette Flow at ${\rm Re} \approx 1318$ reveal regeneration cycles and coherent structures that are hard to reproduce with either method alone. This fluid-kinetic multiscale framework offers a practical path toward high-fidelity wall-bounded turbulence simulations and opens avenues to study wall micro-roughness and high thermal-gradient scenarios with improved physical realism.

Abstract

We present a two-level (fluid-kinetic) coupling procedure for the simulation of wall-bounded flows at Reynolds numbers up to thousands. The method combines a kinetic Direct Simulation Monte Carlo (DSMC) treatment of the near-wall layer, with a high-order Lattice-Boltzmann (HOLB) scheme as a fluid solver in the bulk flow. Given the kinetic nature of HOLB, this coupling is expected to provide a physically accurate treatment of the near-wall instabilities which trigger the transition to turbulence above a critical threshold around $Re_c \sim 750$. The coupled DSMC-HOLB solver is validated by simulating plane Poiseuille and Couette flows far from equilibrium, i.e at finite Knudsen number regimes. Based on this validation, we provide the first preliminary evidence that the combination of HOLB and DSMC permits to observe the regeneration cycles of coherent structures which arise above a critical value of the Reynolds number. This task would be hardly attainable by either of the two solvers separately; while DSMC can capture strong near-wall non-equilibrium effects, it lacks the compute power to deal with both near-wall and bulk flow at the same time. We look to HOLB to make it computationally feasible to perform such a simulation. The present two-level coupling procedure may pave the way to a new generation of fluid-kinetic simulations of wall-bounded turbulent flows, thus helping to gain deeper insights into the role of wall micro-corrugations in triggering the dynamic instabilities that drive the transition to turbulent regimes.

Figures (7)

• Figure 1: DSMC-HOLB coupling geometry.
• Figure 2: (\ref{['fig:coupledCouette_Re100']}) Normalized transient velocity profile ($u_x/U_c$, where $U_c$ is the wall velocity) in the continuum regime at various diffusion times compared against analytical results for a planar Couette flow at Re $\approx$ 132, Ma = 0.2. (\ref{['fig:coupledCouette_Knp1']}) Steady-state velocity ($u_x$) profile for a Couette flow at ${\rm Kn} = 0.1$. A zoomed-in version of the velocity profile at the upper coupling region in the inset. (\ref{['fig:coupledPoiseuille100_ux']}) Developing stream-wise velocity profile compared with the analytical for a plane Poiseuille flow at Re $\approx$ 132, Ma = 0.2. (\ref{['fig:coupledPoiseuille100_stress']}) Normalized transient shear stress ($\sigma_{xy}/\sigma_{xy}^0$), where $\sigma_{xy}^0$ is the wall shear stress) profile compared with the analytical solution obtained from a model kinetic equation for the same simulation.
• Figure 3: Steady-state velocity ($u_x$) profiles for a Poiseuille flow using an LBM RD3Q41 modelkolluru2020lattice at (\ref{['fig:lbCompMap05Kn']}) ${\rm Ma} = 0.05$ and ${\rm Kn} = 0.01,0.1$, (\ref{['fig:lbCompMap2Kn']}) ${\rm Ma} = 0.2$ and ${\rm Kn} = 0.05,0.1$.
• Figure 4: Steady-state (\ref{['fig:knp1poiseuille_ux']}) velocity ($u_x$) and (\ref{['fig:knp1poiseuille_stress']}) Shear stress ($\sigma_{xy}$) profiles for a Poiseuille flow at ${\rm Kn} = 0.1$. Steady-state (\ref{['fig:knp1stressTensor_theta']}) temperature ($\theta$) and (\ref{['fig:knp1stressTensor_sigmaDiag']}) stress tensor components ($\sigma_{xx}, \sigma_{yy}, \sigma_{zz}$) profiles for a poiseuille flow at ${\rm Kn} = 0.1$.
• Figure 5: (\ref{['fig:velocityCouplingComp_meanVel']}) Turbulent steady-state mean velocity profile from the coupled solver compared against the numerical profile from komminaho1996very and experimental profile from ref.aydin1979novel. Here, $U_c$ represents the imparted wall velocity. (\ref{['fig:velocityCouplingComp_interfaceLower_instVel']}) & (\ref{['fig:velocityCouplingComp_interfaceUpper_instVel']}) A zoomed-in version of the lower (left) and upper (right) coupling interface in the instantaneous stream-wise velocity profile.