Very large-scale motions in turbulent flows over streamwise traveling wavy boundaries

Turbulent flows over streamwise traveling wavy boundaries are investigated by large eddy simulations at a friction Reynolds number of ${\mathrm{Re}}_{\tau }=1000$. Four wave ages (i.e., the ratio between wave phase speed and bulk mean velocity) are considered, sequentially corresponding to the wave moving against the wind, stationary wavy wall, and intermediate and fast waves. A triple decomposition is performed to extract the mean, wave-induced, and turbulent components of the flow field. Very large-scale motions (VLSMs) of turbulent flow are identified by using the one-dimensional premultiplied energy spectra, instantaneous flow fields and conditionally averaged results. Compared with the flat-wall case, VLSMs in the negative wave age and stationary wavy wall cases are stronger, with larger length scales in both the streamwise and spanwise directions. The length scale and intensity of these motions decrease as the wave age increases. The conditionally averaged VLSMs involve the elongated low- and high-speed momentum regions and the roll cells in the streamwise direction. The transport equation of the two-point velocity correlation is investigated in different length scales by applying a spectral analysis. The wave-induced production that represents the interaction between the wave-induced and turbulent components of flow velocities provides extra input for the large-scale energy at low wave ages but play an opposite role at high wave ages.

Figure 1

Sketch of the turbulent open-channel flow over a traveling wavy boundary.

Figure 2

Illustration of the coordinate transformation: (a) boundary-fitted grid system of the irregular domain in the physical space; (b) mapped grid system of rectangular domain in the computational space.

Figure 3

Triple decomposition of the streamwise velocity: (a) instantaneous field; (b) mean profile; (c) wave-induced fluctuation; (d) turbulent fluctuation.

Figure 4

Profiles of the mean streamwise velocity for different wave ages as well as the flat-wall turbulence.

Figure 5

Profiles of the streamwise velocity fluctuations for different wave ages as well as the flat-wall turbulence: (a) turbulent component; (b) wave-induced component. The definitions of the lines are the same as those in Fig. 4.

Figure 6

Profiles of the vertical velocity fluctuations for different wave ages as well as the flat-wall turbulence: (a) turbulent component; (b) wave-induced component. The definitions of the lines are the same as those in Fig. 4.

Figure 7

Profiles of the shear stress for different wave ages as well as the flat-wall turbulence: (a) turbulent component; (b) wave-induced component. The definitions of the lines are the same as those in Fig. 4.

Figure 8

Contours of the wave-induced streamwise velocity fluctuation: (a) $c/U_b=-0.4$; (b) $c/U_b=0$; (c) $c/U_b=0.4$; (d) $c/U_b=1.2$.

Figure 9

Contours of the wave-induced vertical velocity fluctuation: (a) $c/\phantom{c{U}_b}{U}_b=-0.4$; (b) $c/\phantom{c{U}_b}{U}_b=0$; (c) $c/\phantom{c{U}_b}{U}_b=0.4$; (d) $c/\phantom{c{U}_b}{U}_b=1.2$.

Figure 10

Contours of the wave-induced shear stress: (a) $c/\phantom{c{U}_b}{U}_b=-0.4$; (b) $c/\phantom{c{U}_b}{U}_b=0$; (c) $c/\phantom{c{U}_b}{U}_b=0.4$; (d) $c/\phantom{c{U}_b}{U}_b=1.2$.

Figure 11

Instantaneous turbulent streamwise velocity in the x-z plane: (a), (f) flat wall; (b), (g) $c/\phantom{c{U}_b}{U}_b=-0.4$; (c), (h) $c/\phantom{c{U}_b}{U}_b=0$; (d), (i) $c/\phantom{c{U}_b}{U}_b=0.4$; (e), (j) $c/\phantom{c{U}_b}{U}_b=1.2$. Different vertical locations are chosen: (a)–(e) $\overline{y}/\phantom{\overline{y}\delta }\delta =0.03$; (f)–(j) $\overline{y}/\phantom{\overline{y}\delta }\delta =0.3$.

Figure 12

Conditionally averaged streamwise velocity ${\langle u\rangle }_c$ and velocity vector in the y-z plane: (a) the flat-wall case; (b) $c/\phantom{c{U}_b}{U}_b=0$; (c) $c/\phantom{c{U}_b}{U}_b=0.4$; (d) $c/\phantom{c{U}_b}{U}_b=1.2$.

Figure 13

Contours of the one-dimensional pre-multiplied spanwise energy spectra of turbulent velocity fluctuations: (i) the flat-wall case; (ii) $c/\phantom{c{U}_b}{U}_b=-0.4$; (iii) $c/\phantom{c{U}_b=0}{U}_b=0$; (iv) $c/\phantom{c{U}_b}{U}_b=0.4$; (v) $c/\phantom{c{U}_b}{U}_b=1.2$. Different components, i.e., (a) $\overline{u^{\prime }{u}^{\prime }}$, (b) $\overline{v^{\prime }{v}^{\prime }}$, (c) $\overline{w^{\prime }{w}^{\prime }}$, and (d) $\overline{u^{\prime }{v}^{\prime }}$ are presented.

Figure 14

Normalized one-dimensional premultiplied spanwise energy spectra of turbulent velocity fluctuations for various wave ages at different vertical locations: (a) streamwise component $u^{\prime },\overline{y}/\phantom{\overline{y}\delta }\delta =0.03$; (b) streamwise component $u^{\prime },\overline{y}/\phantom{\overline{y}\delta }\delta =0.3$; (c) vertical component $v^{\prime },\overline{y}/\phantom{\overline{y}\delta }\delta =0.5$; (d) spanwise component $w^{\prime },\overline{y}/\phantom{\overline{y}\delta }\delta =0.08$. The different lines represent the different wave ages, the same as those in Fig. 4.

Figure 15

Instantaneous turbulent streamwise velocity in the x-z plane: (a) $\overline{y}/\phantom{\overline{y}\delta }\delta =0.03$; (b) $\overline{y}/\phantom{\overline{y}\delta }\delta =0.3$; the contours are same as those in Fig. 11. (c) Conditionally averaged streamwise velocity and velocity vector in the y-z plane; the contours are same as those in Fig. 12.

Figure 16

Contours of the one-dimensional pre-multiplied spanwise energy spectra of the production and pressure-strain terms: (i) the flat-wall case; (ii) $c/\phantom{c{U}_b}{U}_b=-0.4$; (iii) $c/\phantom{c{U}_b=0}{U}_b=0$; (iv) $c/\phantom{c{U}_b}{U}_b=0.4$; and (v) $c/\phantom{c{U}_b}{U}_b=1.2$. Different terms, i.e., (a) $P_{11}$, (b) ${\mathrm{\Pi }}_{s22}$, (c) ${\mathrm{\Pi }}_{s33}$, and (d) $P_{12}$ are presented.

Figure 17

Contours of the one-dimensional pre-multiplied spanwise energy spectra of wave-induced production terms for different wave-ages: (i) $c/\phantom{c{U}_b}{U}_b=-0.4$; (ii) $c/\phantom{c{U}_b=0}{U}_b=0$; (iii) $c/\phantom{c{U}_b}{U}_b=0.4$; (iv) $c/\phantom{c{U}_b}{U}_b=1.2$. Different terms, i.e., (a) $W{P}_{11,uy}$, (b) $W{P}_{11,ux}$, (c) $W{P}_{22,vx}$, and (d) $W{P}_{12,vy}$ are presented.

Figure 18

Sketch of the paths through which the large-scale turbulent fluctuations and momentum flux gain/lose energy.

Figure 19

One-dimensional premultiplied spanwise energy spectra of turbulent velocity fluctuation for different wave steepness: (a) streamwise component $u^{\prime },\overline{y}/\phantom{\overline{y}\delta }\delta =0.03$; (b) streamwise component $u^{\prime },\overline{y}/\phantom{\overline{y}\delta }\delta =0.3$; (c) vertical component $v^{\prime },\overline{y}/\phantom{\overline{y}\delta }\delta =0.5$; (d) spanwise component $w^{\prime },\overline{y}/\phantom{\overline{y}\delta }\delta =0.08$.

Figure 20

One-dimensional premultiplied spanwise energy spectra of turbulent velocity fluctuation for different Reynolds numbers: (a) streamwise component $u^{\prime },\overline{y}/\phantom{\overline{y}\delta }\delta =0.03$; (b) streamwise component $u^{\prime },\overline{y}/\phantom{\overline{y}\delta }\delta =0.3$; (c) vertical component $v^{\prime },\overline{y}/\phantom{\overline{y}\delta }\delta =0.5$; (d) spanwise component $w^{\prime },\overline{y}/\phantom{\overline{y}\delta }\delta =0.08$.