Contributions to pressure drag in rough-wall turbulent flows: Insights from force partitioning

The force partitioning method [Menon and Mittal, J. Fluid Mech. 918, R3 (2021)] is employed to decompose and analyze the pressure-induced drag for turbulent flow over rough walls. The pressure drag force imposed by the rotation-dominated vortical regions ($Q>0$, where $Q$ is the second invariant of the velocity gradient tensor) and strain-dominated regions ($Q<0$) are quantified using a geometry dependent auxiliary potential field (denoted by $\varphi$). The analysis is performed on data from direct numerical simulations (DNSs) of turbulent channel flows, at frictional Reynolds number of ${\text{Re}}_{\tau }=500$, with cube and sand-grain roughened bottom walls. Results from both simulations indicate that the $Q$-induced pressure drag is the largest contributor (more than 50%) to the total drag on the rough walls. Data are further analyzed to quantify the effects of time-mean (coherent) and incoherent turbulent flow on the $Q$-induced drag force, and to discuss possible effects of roller and U-shaped structures expected to occur at the crest and midcrest locations of the roughness elements, respectively. Based on the observation that the $\varphi$ field encodes information about the surface geometry that directly impacts the drag, we provide initial evidence that it can also be used to parametrize the surface drag. Specifically, we propose and test three norms based on the $\varphi$ field (two of them related to the surface-induced potential flow), and explore the characterization of the Nikuradse equivalent sand-grain height $k_s$ based on these parameters for a number of channel flows with different roughness topologies. Data are provided from a suite of DNS cases by Aghaei-Jouybari et al. [J. Fluid Mech. 912, A8 (2021)]. An empirical correlation depending on these $\varphi$-based parameters, with five empirically tuned coefficients, is shown to predict $k_s$ with average and maximum errors of 10.5 and 26 percent, respectively. The results confirm that a purely geometric quantity, the $\varphi$ field, provides useful additional information that can be used in drag law formulations.

Figure 1

Schematics of drag generation and annihilation by vortex-dominated ($Q>0$, a) and strain-dominated ($Q<0$, b) regions in the bulk of the flow.

Figure 2

Cubical roughness elements (a), and isosurfaces of $\varphi$ (b), $\varphi =+0.01$ (red bulblike surface) and $\varphi =-0.01$ (blue bulblike surface).

Figure 3

Contours of instantaneous $u, Q, \varphi$ (constant in time), $f_Q$ and $f_{\nu }$. A vortex structure cross-section is encircled in the zoomed in figures of $Q$ and $f_Q$.

Figure 4

Decomposition of the total hydrodynamic drag as function of time. $A_b$ and $A_t$ are, respectively, the area of the regions not covered by cubes in the bottom wall and the area of top wall. “Momentum rate” refers to the volume integral of the unsteady term in NS equations. Considering the simulation is done in a constant pressure gradient mode, the mass flux inside the channel is not exactly constant (although being statistically stationary) and its time derivative fluctuates around zero. The average contribution by this term is small (less than 1%), as expected.

Figure 5

Weighted conditional averages of $F_Q$ (a) and $F_{\nu }$ (b) based on four events: (i) $\varphi >0$ and $Q>0$ (vortex-dominated region, upstream of the cube elements), (ii) $\varphi >0$ and $Q<0$ (strain-dominated, upstream), (iii) $\varphi <0$ and $Q>0$ (vortex-dominated, downstream), and (iv) $\varphi <0$ and $Q<0$ (strain-dominated, downstream). Please note we plot the weighted conditional averages $E_s$ to accurately quantify the contribution of each event to the total drag. Here $E_s\left(\text{event}\right)=E\left(\text{event}\right)\times P\left(\text{event}\right)$, where $E$ and $P$ are the true (intrinsic) conditional averages and probability of an event. Also note $E_s\left(f\right|\varphi >0)=E_s\left[f\right|(\varphi >0,Q>0)]+E_s\left[f\right|(\varphi >0,Q<0)]$, shows contribution of the upstream events to the respective force, and $E_s\left(f\right|\varphi <0)=E_s\left[f\right|(\varphi <0,Q>0)]+E_s\left[f\right|(\varphi <0,Q<0)]$ shows contribution of the downstream events. The total drag per volume is shown by the solid black lines $E_s\left(f\right)$.

Figure 6

Contribution to drag by $\overline{Q}=-0.5\overline{u_{i,j}{u}_{j,i}}$ (note $u_{i,j}=\frac{\partial u_i}{\partial x_j}$), ${\overline{Q}}_{\text{uavg}}=-0.5{\overline{u}}_{i,j}{\overline{u}}_{j,i}$ and ${\overline{Q}}_{\text{urms}}=-0.5\overline{u_{i,j}^{\prime }{u}_{j,i}^{\prime }}$. Their respective forces are $f_{\overline{Q}}, f_{Q_{\text{uavg}}}$, and $f_{Q_{\text{urms}}}$, intrinsically averaged in both time and $x-z$ plane. Note $F_Q=\int A_f{f}_{\overline{Q}}dy=22.0, F_{Q_{\text{uavg}}}=\int A_f{f}_{Q_{\text{uavg}}}dy=12.3$ ($=0.56F_Q$) and $F_{Q_{\text{urms}}}=\int A_f{f}_{Q_{\text{urms}}}dy=9.7$ ($=0.44F_Q$), all normalized by $\delta$ and $u_{\tau }$. Also $A_f$ is the planar fluid occupied area.

Figure 7

Contours of $\overline{Q}=-0.5\overline{u_{i,j}{u}_{j,i}}, {\overline{Q}}_{\text{uavg}}=-0.5{\overline{u}}_{i,j}{\overline{u}}_{j,i}$ and ${\overline{Q}}_{\text{urms}}=-0.5\overline{u_{i,j}^{\prime }{u}_{j,i}^{\prime }}$ with their respective forces $f_{\overline{Q}}, f_{Q_{\text{uavg}}}$, and $f_{Q_{\text{urms}}}$. All quantities are time averaged and phase averaged over each cube element. Possible signatures of U-shape structures (occurring around $y=0.5k_c$) and roller structures or turbulent motions generated in the thin shear layer above the cubes (occurring at $y=k_c$) are encircled in the contour of $Q_{\text{uavg}}$.

Figure 8

Sand-grain roughness elements (a), and isosurfaces of $\varphi$ (b), $\varphi =+0.02$ (red surfaces) and $\varphi =-0.02$ (blue surfaces).

Figure 9

Contours of instantaneous $u, Q, \varphi$ (constant in time), $f_Q$ and $f_{\nu }$.

Figure 10

Weighted conditional averages of $F_Q$ (a) and $F_{\nu }$ (b) for sand grain. Legends same as Fig. 5.

Figure 11

Roughness database to predict $k_s$, reproduced with permission from Ref. [34].

Figure 12

Calculated $\varphi$ field for four sample cases from Ref. [34]. Only a portion of the domain is shown on a representative $x-y$ plane.

Figure 13

Scatter plots of the parameter ${\psi }_1={\int }_0^{\infty }\langle \left|\varphi \right|\rangle dy/k_r^2$ against surface geometric parameters: surface porosity $P_o$ (a) and spanwise effective slope $E_z$ (b).

Figure 14

Normalized sand-grain roughness length as function of the three parameters ${\psi }_1$ (a), ${\psi }_2$ (b), and ${\psi }_3$ (c), and results from three-parameter curve-fitting for which ${\text{Err}}_{\text{avg}}=10.5\%$ and ${\text{Err}}_{\text{max}}=26\%$ (d).