Flux-controlled wall model for large eddy simulation integrating the compressible law of the wall

Abstract

Recent advances in velocity and temperature transformations have enabled recovery of the law of the wall in compressible wall-bounded turbulent flows. Building on this foundation, a flux-controlled wall model (FCWM) for Large Eddy Simulation (LES) is proposed. Unlike conventional wall-stress models that solve the turbulent boundary layer equations, FCWM formulates the near-wall modeling as a control problem applied directly to the outer LES solution. It consists of three components: (1) the compressible law of the wall, (2) a feedback flux-control strategy, and (3) a shifted boundary condition. The model adjusts the wall shear stress and heat flux based on discrepancies between the computed and target transformed velocity and temperature, respectively, at the matching location. The proposed wall model is evaluated using LES of turbulent channel flows across a broad range of conditions, including quasi-incompressible cases with bulk Mach number \(M_b = 0.1\) and friction Reynolds number \(Re_τ= 180 \sim 10{,}000\), and compressible cases with \(M_b = 0.74 \sim 4.0\) and bulk Reynolds number \(Re_b = 7667 \sim 34{,}000\). The wall-modelled LES reproduce mean velocity and temperature profiles in agreement with direct numerical simulation data. For all tested cases with \(M_b \leq 3\), the wall model achieves relative errors of \(|ε_{C_f}| < 4.1\%\), \(|ε_{B_q}| < 2.7\%\), and \(|ε_{T_c}| < 2.7\%\) in friction coefficient, non-dimensional heat flux, and centerline temperature, respectively. In the quasi-incompressible regime, the wall model achieves \(|ε_{C_f}| < 1\%\). Compared to the conventional equilibrium wall model, the proposed FCWM achieves higher accuracy in compressible turbulent channel flows without solving the boundary layer equations, thereby reducing computational cost.

Figures (20)

• Figure 1: Schematic of the flux-controlled wall model. (a) WMLES setup of compressible wall-bounded turbulent flow. (b) SL-type transformed velocity profile. (c) SL-type transformed temperature profile. When $\bar{\tau}_w < \tau_{ref}$, it follows that $\Delta U^+_{SL} > 0$, and vice versa. Analogously, $\bar{q}_w < q_{ref}$ implies $\Delta T^+_{SL} > 0$, or equivalently $\bar{T}(y) > T_{ref}(y)$, and vice versa. Note that the blue and red curves in panels (b, c) represent the transformed velocity and temperature with extended logarithmic profile.
• Figure 2: Dependence of log-law intercepts on $Re^*_\tau$ for (a) transformed velocity and (b) transformed temperature. The shaded areas represent an error margin of $\pm \, rms$ for $B$ and $B_T$. Note that $Re^*_\tau = Re_\tau$ for incompressible flows.
• Figure 3: Evolution of (a) log-law intercept $B$ for transformed velocity, (b) log-law intercept $B_T$ for transformed temperature, (c) friction coefficient $C_f$, (d) non-dimensional heat flux $B_q$, and (e) mean centerline temperature $\tilde{T}_c/\tilde{T}_w$ for WMLES of compressible turbulent channel flow at $M_b = 1.57$ and $Re_b = 25{,}216$, using a uniform grid of $104 \times 40 \times 64$.
• Figure 4: Converged profiles of (a) $U^+_{SL}$ and (b) $T^+_{SL}$ from WMLES of compressible turbulent channel flow at $M_b = 1.57$ and $Re_b = 25{,}216$, using FCWM-base and $f^{outer}_{VD}$. The black dashed lines: $U^+_{SL} = \frac{1}{\kappa} \log (y^*) + 5.27$ and $T^+_{SL} = \frac{Pr_t}{\kappa} \log (y^*) + 3.64$. The red crosses $\boldsymbol{\times}$ mark the matching location ($y_m = 0.3h$). DNS data from Gerolymos2023Gerolymos2024aGerolymos2024b are included for comparison.
• Figure 5: Profiles of (a) velocity and (b) temperature in WMLES of compressible turbulent channel flow at $M_b = 0.74, Re_b = 21{,}092$ and $M_b = 1.57, Re_b = 25{,}216$, using $f^{inner}_{VD}$ and $f^{outer}_{VD}$ defined in Eqs. (\ref{['eq:ALDM_inner_damping']}) and (\ref{['eq:ALDM_outer_damping']}). DNS data from Gerolymos2023Gerolymos2024aGerolymos2024b are included for comparison.