Constrained large-eddy simulation of turbulent flow over rough walls

Recent research indicates that the structure of the wall-bounded turbulence depends on the mean shear created by the wall. To examine the response of the wall turbulence to the rough-wall-like mean shear, we performed numerical tests of the channel flows in the framework of the constrained large-eddy simulation without resolving the roughness elements. Once the rough-wall-like mean shear is imposed, the normalized contribution of the sweep motions to the Reynolds stress increases in the near-wall region and the two-point correlation of the streamwise velocity in the outer layer is enhanced. The constraint of the rough-wall-like mean shear provides extra resistance just like the surface roughness, which is observed via the decomposition of the skin friction coefficient. All the statistics indicate that the effect of the rough-wall-like mean shear on the turbulent flow is similar to the effect of roughness elements. The large-eddy simulations of the rough-wall flows using the conventional equilibrium-wall model are also performed for comparison. The statistical results in the outer layer are qualitatively consistent with those obtained by constraining the mean shear, which demonstrates the latter approach can be used as a new wall model for the simulation of the rough-wall flows.

Figure 1

(a),(b) Mean streamwise velocity profile of the channel flows at ${\text{Re}}_{\tau }=180$, ${\text{Re}}_{\tau }=590$; the mean shear constraint is imposed for the cases of $k_s^{+}>0$. (c) The mean velocity profile obtained from the simulation with the equilibrium-wall model at ${\text{Re}}_{\tau }=590$. The solid lines are the DNS results of the smooth-wall flows at ${\text{Re}}_{\tau }=180$ and 550 [57]. The dashed lines are the expected logarithmic distribution $U^{+}={\kappa }^{-1}ln\left(y^{+}\right)+B-\mathrm{\Delta }{U}^{+}$ for each case. The dash-dotted lines represent the linear distribution $U^{+}=y^{+}$.

Figure 2

(a),(b) The distribution of the velocity fluctuation intensities of the channel flows at (a) ${\text{Re}}_{\tau }=180$ and (b) ${\text{Re}}_{\tau }=590$; the mean shear constraint is imposed for the cases of $k_s^{+}>0$. (c) The fluctuation intensities obtained from the simulation using the equilibrium wall model at ${\text{Re}}_{\tau }=590$. The solid lines are the DNS results of the smooth-wall channel flow at ${\text{Re}}_{\tau }=180$ and ${\text{Re}}_{\tau }=550$, respectively [57].

Figure 3

Mean stress distribution for the channel flows at ${\text{Re}}_{\tau }=180$, (a) $k_s^{+}=0$, (b) $k_s^{+}=15$, and (c) $k_s^{+}=30$. The mean shear constraint is imposed for the cases of $k_s^{+}>0$.

Figure 4

The normalized spectrum of the streamwise velocity fluctuation obtained from the smooth-wall cases at (a) ${\text{Re}}_{\tau }=180$ and (b) ${\text{Re}}_{\tau }=590$, and the mean-shear-constrained cases (c) ${\text{Re}}_{\tau }=180$, $k_s^{+}=30$ and (d) ${\text{Re}}_{\tau }=590$, $k_s^{+}=60$.

Figure 5

(a)–(c) The joint probability distribution function $P(u^{\prime },v^{\prime })$, and (d)–(f) the covariance integrand $u^{\prime }{v}^{\prime }P(u^{\prime },v^{\prime })$ at ${\text{Re}}_{\tau }=590$, $k_s^{+}=0$ (black dash-dotted contour lines) and $k_s^{+}=60$ (red solid contour lines) at wall distances of (a),(d) $y^{+}=148$, (b),(e) $y^{+}=68$, and (c),(f) $y^{+}=17$. Contour levels are from 0.0005 to 0.003 for (a)–(c) and from $-0.003$ to 0.001 for (d)–(f) with the same contour spacing 0.0005.

Figure 6

(a) Quadrant contributions of the resolved Reynolds stress throughout the channel at ${\text{Re}}_{\tau }=590$ in the smooth-wall case (lines: solid, Q1; dashed, Q2; dash-dot-dotted, Q3; dash-dotted, Q4) and the constrained case ($k_s^{+}=60$; square, Q1; circle, Q2; up triangle, Q3; down triangle, Q4). Fractional quadrant contributions of the resolved Reynolds stress filtered by the hyperbolic hole at wall-normal distances of (b) $y^{+}=148$, (c) $y^{+}=68$, and (d) $y^{+}=17$. Lines and symbols are consistent with (a).

Figure 7

Two-point correlations of the streamwise fluctuation velocity at ${\text{Re}}_{\tau }=590$, (a)–(c) smooth and (d)–(f) constrained ($k_s^{+}=60$), with the reference wall distance of (a),(d) $y_{\text{ref}}^{+}=148$, (b),(e) $y_{\text{ref}}^{+}=68$, and (c),(f) $y_{\text{ref}}^{+}=17$. The contour levels are (0.2, 0.3, 0.4, 0.5, 0.6) from the outside to the inside. The straight solid line (red) is inclined to show the inclination angle of the correlational structure. $y_{\text{ref}}^{+}$ is represented by the horizontal dashed line.

Figure 8

Two-point correlations of the streamwise fluctuation velocity in the ($y,z$)-plane at ${\text{Re}}_{\tau }=590$, (a)–(c) smooth wall, (d)–(f) constrained ($k_s^{+}=60$) with the reference wall distances of (a),(d) $y_{\text{ref}}^{+}=148$, (b),(e) $y_{\text{ref}}^{+}=68$, (c),(f) $y_{\text{ref}}^{+}=17$. The contour levels for the positive correlation are plotted from 0.05 to 0.3 with the contour spacing of 0.05. The negative contour levels ($-0.175,-0.15,-0.1,-0.05$) are plotted if they exist. $y_{\text{ref}}^{+}$ is represented by the horizontal dashed line.

Figure 9

Two-point correlations of the streamwise fluctuation velocity in the ($y,z$) plane obtained from the simulations using the equilibrium-wall model at ${\text{Re}}_{\tau }=590$, with (a),(b) $k_s^{+}=0$ and (c),(d) $k_s^{+}=60$ with the reference wall distances of (a),(c) $y_{\text{ref}}^{+}=157$, (b),(d) $y_{\text{ref}}^{+}=65$. The contour levels for the positive correlation are plotted from 0.05 to 0.3 with the contour spacing of 0.05. The negative contour levels ($-0.2,-0.15,-0.1,-0.05$) are plotted if they exist. $y_{\text{ref}}^{+}$ is represented by the horizontal dashed line.

Figure 10

Total skin friction coefficient obtained from the present LES using the decomposition (open symbols). The correlation function for the smooth-wall flows is $C_f=0.065{\mathrm{Re}}_b^{-0.25}$ (thick solid line), and for laminar flow is $C_f=12/{\text{Re}}_b$ (dashed line). The correlation for other $k_s^{+}$ is derived from (15).

Figure 11

(a),(b) Skin friction coefficient obtained from the decomposition and (c),(d) the percentage contribution of the laminar part, the resolved turbulent part, and the SGS part, at (a),(c) ${\text{Re}}_{\tau }=180$ and (b),(d) ${\text{Re}}_{\tau }=590$.

Figure 12

Mean stream-wise velocity profile of the smooth wall channel flow at $R{e}_{\tau }=590$ simulated without constraint using (a) SDLADS model with five different discrete filters, (b) four different eddy-viscosity models with 4th order box filter. The solid line represents the log-law distribution $U^{+}={\kappa }^{-1}ln\left(y^{+}\right)+B$ for smooth wall flows.

Figure 13

Mean streamwise velocity profile of the channel flows with the constraint at ${\text{Re}}_{\tau }=590$, (a) $k_s^{+}=30$, and (b) $k_s^{+}=60$ with four basic SGS models. The dashed and solid lines are the logarithmic law for smooth and rough wall, respectively.

Figure 14

(a) Mean streamwise velocity profile and (b) fluctuation intensities for $k_s^{+}=0$, ${\text{Re}}_{\tau }=590$ obtained from various grid resolutions. The solid lines in (b) are DNS results [57].

Figure 15

(a) Mean streamwise velocity profile and (b) fluctuation intensities for $k_s^{+}=60$, ${\text{Re}}_{\tau }=590$ obtained from various grid resolutions.

Figure 16

Comparison of the normalized quadrant contribution to the resolved Reynolds stress in smooth ($k_s^{+}=0$, lines) and rough-wall ($k_s^{+}=60$, symbols) flows with the grid ($N_x\times N_y\times N_z$) (a) $80\times 65\times 80$, (b) $96\times 65\times 96$.