Closure modeling in near-wall region of steep resolution variation for partially averaged Navier-Stokes simulations

We seek to develop a closure model to enable scale-resolving simulation (SRS) of a turbulent flow to optimally switchover from a Reynolds-averaged Navier-Stokes (RANS) calculation at the wall to a specified degree of resolution in the wake or free-stream region. The closure model is derived by (i) using the physical principle that the total energy of resolved and unresolved scales should be conserved in the switchover region and (ii) establishing consistency with equilibrium boundary layer scaling of the partially resolved field. The model development is performed in the context of a partially averaged Navier-Stokes (PANS) scale-resolving method by quantifying and modeling the commutation terms resulting from varying resolutions in the wall-normal direction. The resulting wall-modeled PANS (WM-PANS) is used to compute the turbulent channel flow in the ${\text{Re}}_{\tau }$ range $180-8000$. The influence of the RANS-SRS switchover location on the computed flow field is examined. It is then demonstrated that the mean flow is reproduced with reasonable accuracy at modest computational effort without discernible log-layer mismatch even at the highest Reynolds number considered. While the Reynolds stresses are also recovered accurately over most of the flow domain, a noticeable computational transition from RANS to unsteady SRS flow behavior is observed and the underlying physics is examined. Irrespective of the location of computational transition, the unsteady features of the flow away from the wall are well captured. It is demonstrated that the proposed closure is able to inject the appropriate amount of resolved turbulence without the need for artificially generated synthetic turbulence. Overall, WM-PANS presents an accurate and computationally viable option for scale-resolving computations of near-wall high-Reynolds number flows.

Figure 1

RANS-SRS switchover region near the wall.

Figure 2

A priori analysis of the commutation viscosity for turbulent channel flow at ${\text{Re}}_{\tau }=8000$ using DNS data [32]. The $f_k$ variations for cases A1–A3 are presented in Table 1.

Figure 3

WR-PANS simulation of turbulent channel flow at different ${\mathrm{Re}}_{\tau }$: mean streamwise velocity profile ($f_k=0.2$).

Figure 4

WR-PANS simulation of turbulent channel flow for ${\text{Re}}_{\tau }=4200$: (a) Streamwise stress, (b) wall-normal stress, (c) spanwise stress, and (d) shear stress.

Figure 5

WOC-PANS simulation of turbulent channel flow without the commutation terms for ${\text{Re}}_{\tau }=4200$: (a) $f_k$ variation and (b) mean streamwise velocity.

Figure 6

WOC-PANS simulation of turbulent channel flow for ${\text{Re}}_{\tau }=4200$: (a) streamwise stress and (b) shear stress.

Figure 7

(a) Prescribed $f_k$ and (b) mean streamwise velocity profile for different cases for ${\text{Re}}_{\tau }=950.$

Figure 8

WM-PANS simulation of turbulent channel flow for different cases for ${\text{Re}}_{\tau }=950$: (a) streamwise stress, (b) wall-normal stress, (c) spanwise stress, and (d) shear stress.

Figure 9

Q isosurfaces (colored with streamwise velocity) (a) case B1 and (b) case B2.

Figure 10

Temporal spectra of streamwise velocity fluctuations for cases B1 and B2 at (a) $y^{+}=25$ and (b) $y^{+}=500$.

Figure 11

WM-PANS simulation of turbulent channel: (a) Recovery of $f_{\nu }$ (near-wall variation of $f_v$ is not included) and (b) mean streamwise velocity profiles for different ${\text{Re}}_{\tau }$.

Figure 12

WM-PANS simulation of turbulent channel for ${\text{Re}}_{\tau }=4200$: (a) streamwise stress, (b) wall-normal stress, (c) spanwise stress, and (d) shear stress.

Figure 13

WM-PANS simulation of turbulent channel for ${\text{Re}}_{\tau }=4200$: (a) Resolved ($\sqrt{{\overline{uu}}^R}$) and modeled ($\sqrt{{\overline{uu}}^M}$) streamwise stresses and (b) resolved (${\overline{uv}}^R$) and modeled (${\overline{uv}}^M$) shear stresses.

Figure 14

WM-PANS simulation for ${\text{Re}}_{\tau }=8000$: (a) prescribed $f_k$ variation and (b) profiles of streamwise Reynolds stresses ($\sqrt{\overline{uu}}$). DNS data of Ref. [32] are used for comparison. The results clearly demonstrate that a twitch occurs in the computational buffer layer.

Figure 15

Spanwise fluctuating vorticity (${\omega }_z^{\prime }\delta /U_c$) contours (red) in x-y plane for (a) case A1 and (b) case A3 for ${\text{Re}}_{\tau }=8000$. The yellow shaded region is the RANS region, white region is the unsteady PANS region, and the switchover region is denoted by the grey shaded area with width ($\gamma$); $U_c$ is the velocity in center of the channel.