Turbulent momentum and kinetic energy transfer of channel flow over three-dimensional wavy walls

In this paper, we conduct a numerical investigation to analyze the turbulent flow characteristics over three-dimensional wavy walls with varying amplitudes and wavelengths, with a specific focus on the transfer of momentum and kinetic energy. Detailed three-directional momentum statistics are presented to elucidate the influence of shape parameters on flow behavior. The temporal-spatial averaging at relative height is employed to highlight the impact of the wall on mean, time-averaged, and dispersive momentum flux as well as kinetic energy production. Our findings indicate that the current wall amplifies the effects of spanwise momentum flux, which is determined by the transverse flow around the crest. The dispersive shear stress (DSS) demonstrates a notable correlation with vorticity enhancement, while the near-wall vertical momentum flux is jointly governed by the counteraction of Reynolds shear stress and DSS. Through an analysis of kinetic energy conservation, we observe the transfer of kinetic energy among time-averaged, mean, dispersive, and turbulent motions. Overall, kinetic energy is transferred from dispersion to the mean flow and subsequently to turbulence downstream of the crest. On the windward side, both turbulent and dispersive energy are injected into the mean flow, suggesting the possible growth of internal boundary layer. Additionally, the exchange between dispersion and turbulence significantly contributes to turbulent kinetic energy production, particularly in cases with high amplitudes or spanwise wavelengths.

Figure 1

Elevation of the three-dimensional wavy wall.

Figure 2

Comparison of (a) subgrid-scale kinetic energy and (b) resolved turbulent kinetic energy (TKE) of turbulent flow over a three-dimensional wavy wall. The results are shown on the longitudinal section where $y/{\lambda }_y=0$. The subgrid-scale kinetic energy is approximately two orders of magnitude less than the resolved TKE.

Figure 3

The turbulence statistics on three-directional sections for G1-2. (a)–(c) Time-averaged velocity, (d)–(f) Reynolds normal stress (RNS), and (g)–(i) Reynolds shear stress (RSS) components.

Figure 4

The turbulence statistics on three-directional sections for G1-6. (a)–(c) Time-averaged velocity, (d)–(f) Reynolds normal stress (RNS), and (g)–(i) Reynolds shear stress (RSS) components.

Figure 5

The turbulence statistics on three-directional sections for G2-6. (a)–(c) Time-averaged velocity, (d)–(f) Reynolds normal stress (RNS), and (g)–(i) Reynolds shear stress (RSS) components.

Figure 6

The schematic of temporal and spatial averaging used in this paper on (a) longitudinal section with $y/{\lambda }_y=0$ and (b) cross-section with $x/{\lambda }_x=0.75$. (a) The instantaneous streamwise velocity is decomposed into time-averaged and turbulent fluctuating velocities; a spatial averaging along the streamwise direction divides the time-averaged velocity into temporal-spatial averaged velocity and dispersive velocity in the streamwise direction. (b) The decomposition along the spanwise direction. The subscript $x$ or $y$ means the spatial averaging along the $x$ or $y$ direction.

Figure 7

(a) The transformation from Cartesian to curvilinear coordinate. The time-averaged (b) and (d) streamwise and (c) and (e) vertical velocities shown in (b) and (c) Cartesian and (d) and (e) curvilinear coordinates for case G1-2.

Figure 8

(a) and (b) Reynolds shear stress and (c) and (d) dispersive shear stress under (a) and (c) Cartesian and (b) and (d) curvilinear coordinates for case G1-2.

Figure 9

The profiles of mean streamwise velocity at the (a) and (c) logarithmic and (b) and (d) semilogarithmic plots. (a) and (b) Group 1; (c) and (d) group 2.

Figure 10

Roughness function.

Figure 11

The Reynolds shear stress (RSS), dispersive shear stress (DSS), viscous shear stress (VSS), and total shear stress (TSS) on the cross-section for cases (a) G1-2, (b) G1-6, and (c) G2-6.

Figure 12

The dimensionless spatial-averaged Reynolds shear stress (RSS), dispersive shear stress (DSS), and viscous shear stress (VSS) profiles.

Figure 13

The contours of dispersive shear stress (DSS) and isolines of dispersive velocity components for G1-3 on (a) YS1, (b) XS1, and (c) ZS1 sections. The solid and dashed lines are the dispersive streamwise and vertical velocities, with blue (red) denoting less (larger) than zero.

Figure 14

The contours of dispersive shear stress (DSS) and isolines of dispersive velocity components for G2-6 on (a) YS1, (b) XS1, and (c) ZS1 sections. The solid and dashed lines are the dispersive streamwise and vertical velocities, with blue (red) denoting less (larger) than zero.

Figure 15

Comparison of the dispersive shear stress (DSS) for (a) and (b) case G1-2 and (c) and (d) two-dimensional wavy wall case from our previous work [38]. The blue solid lines denote $\tilde{u}<0$ or $\tilde{w}<0$, whereas the red solid lines represent $\tilde{u}>0$ or $\tilde{w}>0$.

Figure 16

Local-averaged dispersive shear stress (DSS) as a function of the local-averaged (a) and (d) streamwise, (b) and (e) spanwise, and (c) and (f) vertical vorticity for (a)–(c) group 1 and (d)–(f) group 2.

Figure 17

(a)–(c) Turbulent kinetic energy (TKE) production, (d)–(f) mean TKE production, and (g)–(i) TKE-dispersive kinetic energy (DKE) exchange on longitudinal and vertical sections for cases (a), (d), and (g) G1-2, (b), (e), and (h) G1-6, and (c), (f), and (i) G2-6.

Figure 18

(a)–(c) Dispersive kinetic energy (DKE) production, (d)–(f) mean DKE production, and (g)–(i) DKE transportation on longitudinal and vertical sections for cases (a), (d), and (g) G1-2, (b), (e), and (h) G1-6, and (c), (f), and (i) G2-6.