Grid resolution requirement for resolving rare and high intensity wall-shear stress events in direct numerical simulations

Turbulent signals are intermittent with large instantaneous fluctuations. Such large fluctuations lead to small Kolmogorov scales that are hard to resolve in numerical simulations [P. K. Yeung, K. R. Sreenivasan, and S. B. Pope, Effects of finite spatial and temporal resolution in direct numerical simulations of incompressible isotropic turbulence, Phys. Rev. Fluids 3, 064603 (2018)]. The present paper follows the above basic logic, but instead of dissipation events in isotropic turbulence, we study wall-shear stress events in plane channel flow. Wall-shear stress fluctuations are increasingly more intermittent as the Reynolds number increases. Hence, one has to employ higher grid resolutions as the Reynolds number increases in order to resolve a given percentage of wall-shear stress events. The objective of this paper is to quantify effects of the grid resolutions on the rare and high intensity wall-shear stress events. We find that the standard grid resolution resolves about 99% of the wall-shear stress events at ${\text{Re}}_{\tau }=180$. A slightly higher grid resolution has to be employed in order to resolve 99% of the wall-shear stress events at higher Reynolds numbers, and if the standard grid resolution is used for, e.g., a ${\text{Re}}_{\tau }=10000$ channel flow, one resolves about 90%–95% wall-shear stress events.

Figure 1

(a), (c) Mean velocity. (b), (d) Streamwise velocity's rms. (a), (b) Results at ${\text{Re}}_{\tau }=180$ (c), (d) Results at ${\text{Re}}_{\tau }=400$.

Figure 2

PDFs of (a) the streamwise wall-shear stress and (b) the spanwise wall-shear stress in R180, F180, FF180, and LM180. The vertical lines are at ${\tau }_x^{+}=1$ and ${\tau }_z^{+}=0$. The horizontal lines are at constant PDF values above which FF180's probability density function integrates to 0.9, 0.99, and 0.999, respectively. (c), (d) ${\text{Re}}_{\tau }=400$ results.

Figure 3

Normalized premultiplied PDF. R180 results. Red lines: ${\tau }_z^{+}\times \mathrm{PDF}$. Blue lines: ${\tau }_z^{+4}\times \mathrm{PDF}$. Solid lines: Wall-shear stress events that are resolved in R180. Dashed lines: Wall-shear stress events that are not (well) resolved in R180. The premultiplied PDF is symmetric/antisymmetric with respect to ${\tau }_z^{+}=0$ and we show only the results for ${\tau }_z^{+}>0$.

Figure 4

Scaled PDFs of (a) the streamwise wall-shear stress and (b) the spanwise wall-shear stress in FF180, FF400, and the channel flow DNSs at ${\text{Re}}_{\tau }=1000$, 2000, and 5200 [41]. Details of the LM cases can be found in Ref. [41]. As we will show later in this section, the rare events, i.e., the tails of the wall-shear stress PDFs, are not accurately predicted at higher Reynolds number if one uses the standard DNS grid resolution, and therefore the tails of the wall-shear stress PDFs for LM1000, LM2000, and LM5200 are not shown here. The plotted part of the PDFs integrates to 0.9. The two insets are unscaled PDFs of the streamwise and the spanwise wall-shear stresses for FF180 and FF400. The two horizontal lines are at a constant ${\sigma }_{\tau }^{+}\times \mathrm{PDF}$ location above which the PDF integrates to 0.99.

Figure 5

The grid resolution required to resolve 99% of the wall-shear stress events. Blue lines: $\mathrm{\Delta }{x}^{+}$. Red lines: $\mathrm{\Delta }{z}^{+}$. Solid lines: Assuming ${\sigma }^{+2}\sim log\left({\text{Re}}_{\tau }\right)$. Dashed lines: Assuming ${\sigma }^{+}\sim log\left({\text{Re}}_{\tau }\right)$. Cross symbols: $\mathrm{\Delta }{x}^{+}$ in some of the channel flow DNSs. Round symbols: $\mathrm{\Delta }{z}^{+}$ in some of the channel flow DNSs. The references are as follows: For ${\text{Re}}_{\tau }=180$, Ref. [3]; for ${\text{Re}}_{\tau }=2000$, Refs. [41, 48]; for ${\text{Re}}_{\tau }=4200$, Ref. [42]; for ${\text{Re}}_{\tau }=5200$, Ref. [41]; and for ${\text{Re}}_{\tau }=8000$, Ref. [49].

Figure 6

The percentage of the resolved wall-shear stress events if one uses the standard DNS grid resolution. Blue lines are for streamwise wall-shear stress events. Red lines are for spanwise wall-shear stress events. Solid lines: Assuming ${\sigma }_{\tau }^{+2}\sim log\left({\text{Re}}_{\tau }\right)$. Dashed lines: Assuming ${\sigma }_{\tau }^{+}\sim log\left({\text{Re}}_{\tau }\right)$.

Figure 7

(a) Conditional averaged the wall-shear stress based on ${\tau }_z^{+}>1.5$ at $x^{+}=z^{+}=0$. The color contour shows the results for FF180. The line contour shows the results for R180. The four black solid lines are streamlines for the vector field $({\tau }_x,{\tau }_z)$ in FF180. The four dashed solid lines are the results for R180. The solid lines and the dashed lines start at the same $x$ and $z$ locations. (b) Same as (a) but for the results at ${\text{Re}}_{\tau }=400$.

Figure 8

PDFs of (a) the streamwise wall-shear stress and (b) the spanwise wall-shear stress.