Effect of a bottom gap on the mean flow and turbulence structure past vertical solid and porous plates situated in the vicinity of a horizontal channel bed

The paper analyzes how the structure of the flow past solid (porosity $P=0\%$) and porous ($P=36\%$) plates containing parallel solid elements changes with increasing gap distance from the bottom surface. Fully three-dimensional large-eddy simulations are conducted in a straight channel with a horizontal smooth bottom surface and a vertical thin plate of height $H$ positioned at a distance $G$ from the channel bottom for $0\le G/H\le 1$. For small $G/H$ values, the main features of the wake flow are the formation of a bottom-attached recirculation region behind the plate and of corotating vortices originating from eddies shed inside the separated shear layer at the top edge of the plate. For sufficiently high $G/H$ values, the wake flow is dominated by the shedding of counter-rotating wake vortices behind the plate (antisymmetric vortex shedding). For $G/H\le 1$, the wake vortices originating in the bottom separated shear layer lose their coherence after they start interacting with the bottom surface. The effects of increasing $G/H$ on the turbulent kinetic energy, nondimensional streamwise drag force acting on the plate and friction velocity distributions on the bottom surface are analyzed. The paper also discusses how the bleeding flow present in the porous plate cases affects the dynamics of the large-scale coherent structures and its effect on the spectral content of the flow, streamwise drag force, and bed friction velocity distributions.

Figure 1

Sketch showing main variables characterizing a porous plate with a bottom gap (left) and computational domain containing a solid or a porous vertical plate placed in the vicinity of a horizontal surface (right).

Figure 2

Comparison of streamwise velocity, vertical velocity, streamwise normal Reynolds stress, vertical normal Reynolds stress, and primary shear Reynolds stress profiles for flow past a square solid cylinder at $\mathrm{Re}=U_0{H}_{\mathrm{s}}/\nu =21400$. 3D LES results, present study: solid line; experimental results of Lyn et al. [42]: red squares; LES results of Wang and Vanka [43]: green triangles; and LES results of Srinivas et al. [44]: blue circles. (a) $x=1H_{\mathrm{s}}$; (b) $x=5H_{\mathrm{s}}$ where the center of the square cylinder is situated at $x=0$ and its edge length is $H_{\mathrm{s}}$.

Figure 3

Mean flow structure visualized using 2D streamline patterns for the solid plate ($P=0\%$) cases. The blue and green lines indicate the end of the separation bubble at the back of the solid plate and the end of the main bottom-attached recirculation bubble for cases with low $G/H$, respectively. Also shown are the mean spanwise vorticity, ${\overline{\omega }}_z/(H/U_0)$, distributions.

Figure 4

Turbulent kinetic energy, $K/U_0^2$, in an $x-y$ plane for the solid plate ($P=0\%$) cases. Also shown are vertical profiles of $K/U_0^2$ at $x/H=8$ and $x/H=20$.

Figure 5

Instantaneous spanwise vorticity, ${\omega }_z/\left(H/U_0\right)$, and spanwise velocity, $w/U_0$, in an $x-y$ plane for the solid plate ($P=0\%$) cases. (a) $G=0H$; (b) $G=0.1H$; (c) $G=0.5H$; (d) $G=1.0H$.

Figure 6

Power spectra of the vertical velocity at point P1 and point P2 for the solid plate ($P=0\%$) cases. The positions of points P1 and P2 are shown in Fig. 5. The power spectra are compared for the $G=0H, G=0.1H, G=0.5H$, and $G=1.0H$ cases.

Figure 7

Spanwise- and time-averaged net pressure distribution on the plate for the solid plate ($P=0\%$) cases, where $y/H$ represents the vertical distance from the bottom edge of the plate.

Figure 8

Effect of varying the bottom gap, $G/H$, on the mean drag coefficient and the rms fluctuations of the drag coefficient for the solid plate ($P=0\%$) cases.

Figure 9

Effect of $G/H$ on the (a) nondimensional, spanwise-averaged bed friction velocity magnitude in the mean flow and (b) nondimensional rms fluctuations of the bed friction velocity, $u_{\tau \mathrm{SD}}$ for the solid plate ($P=0\%$) cases.

Figure 10

Instantaneous streamwise velocity, $u/U_0$, distribution on a horizontal plane situated at a distance of $0.0055H$ from the bottom surface for the solid plate ($P=0\%$) cases. (a) $G=0H$; (b) $G=0.1H$; (c) $G=0.2H$; (d) $G=0.5H$; (e) $G=1.0H$.

Figure 11

Mean flow structure visualized using 2D streamline patterns for the porous plate ($P=36\%$) cases. The blue lines indicate the end of the separation bubble at the back of the solid plate. Also shown are the mean spanwise vorticity, ${\overline{\omega }}_z/\left(H/U_0\right)$ distributions.

Figure 12

Turbulent kinetic energy, $K/U_0^2$, in an $x-y$ plane for the porous plate ($P=36\%$) cases. Also shown are vertical profiles of $K/U_0^2$ at $x/H=6$ and $x/H=20$.

Figure 13

Instantaneous spanwise vorticity, ${\omega }_z/\left(H/U_0\right)$, and spanwise velocity, $w/U_0$, in an $x-y$ plane for the porous plate ($P=36\%$) cases. (a) $G=0H$; (b) $G=0.1H$; (c) $G=0.5H$; (d) $G=1.0H$.

Figure 14

Power spectra of the vertical velocity at point P1 and point P2 for the porous plate ($P=36\%$) cases. The positions of points P1 and P2 are shown in Fig. 13. The power spectra are compared for the $G=0H, G=0.1H, G=0.5H$, and $G=1.0H$ cases.

Figure 15

Spanwise- and time-averaged net pressure distribution on the plate for the porous plate ($P=36\%$) cases, where $y/H$ represents the vertical distance from the bottom edge of the plate. The gray rectangles indicate the position of the openings between the solid elements.

Figure 16

Effect of varying the bottom gap, $G/H$, on the mean drag coefficient and the rms fluctuations of the drag coefficient for the porous plate ($P=36\%$) cases.

Figure 17

Instantaneous streamwise velocity, $u/U_0$, distribution on a horizontal plane situated at a distance of $0.0055H$ from the bottom surface for the porous plate ($P=36\%$) cases. (a) $G=0H$; (b) $G=0.1H$; (c) $G=0.2H$; (d) $G=0.5H$; (e) $G=1.0H$.

Figure 18

Effect of $G/H$ on the (a) nondimensional, spanwise-averaged bed friction velocity magnitude in the mean flow and (b) nondimensional rms fluctuations of the bed friction velocity, $u_{\tau \mathrm{SD}}$, for the porous plate ($P=36\%$) cases.