Fourier analysis of the roll-up and merging of coherent structures in shallow mixing layers

Abstract

The current study investigates the role of nonlinearity in the development of two-dimensional coherent structures (2DCS) in shallow mixing layers. A nonlinear numerical model based on the depth-averaged shallow water equations is used to investigate temporal shallow mixing layers, where the mapping from temporal to spatial results is made using the velocity at the center of the mixing layer. The flow is periodic in the stream-wise direction and the transmissive boundary conditions are used in the cross-stream boundaries to prevent reflections. The numerical results are examined with the aid of Fourier decomposition. Results show that the previous success in applying local linear theory to shallow mixing layers does not imply that the flow is truly linear. Linear stability theory is confirmed to be only valid within a short distance from the inflow boundary. Downstream of this linear region, nonlinearity becomes important for the roll-up and merging of 2DCS. While the energy required for the merging of 2DCS is still largely provided by the velocity shear, the merging mechanism is one where nonlinear mode interaction changes the velocity field of the subharmonic mode and the gradient of the along-stream velocity profile which, in turn, changes the magnitude of the energy production of the subharmonic mode by the velocity shear implicitly. The nonlinear mode interaction is associated with energy up-scaling and is consistent with the inverse energy cascade which is expected to occur in shallow shear flows. Current results also show that such implicit nonlinear interaction is sensitive to the phase angle difference between the most unstable mode and its subharmonic. The bed friction effect on the 2DCS is relatively small initially and grows in tandem with the size of the 2DCS. The bed friction also causes a decrease in the velocity gradient as the flow develops downstream. The transition from unstable to stable flow occurs when the bed friction balances the energy production. Beyond this point, the bed friction is more dominant and the 2DCS are progressively damped and eventually get annihilated. The energy production by the velocity shear plays an important role from the upstream end all the way to the point of transition to stable flow. The fact that linear stability theory is valid only for a short distance from the inflow boundary suggests that some elements of nonlinearity is incorporated in the mean velocity profile in experiments by the averaging process. The implicit nature of nonlinear interaction in shallow mixing layers and the sensitivity of the nonlinear interaction to phase angle difference between the most unstable mode and its subharmonic allows local linear theory to be successful in reproducing features of the instability such as the dominant mode of the 2DCS and its amplitude.

Appendix: Derivation of the modal kinetic energy equations

The derivation of the modal kinetic energy equation is detailed here. The derivation starts with applying the rigid-lid assumption to the Galilean transformed shallow water equations (5)–(7). The assumption is to replace the free-surface with a rigid-lid, and to replace the hydrostatic pressure with the pressure under the rigid-lid, $p(x,y,t)$. This assumption is shown to be valid for shallow flows with small Froude numbers [14, 57]. Then, since the perturbations in shallow mixing layers are superpositions of interacting modes, a convenient way to investigate the growth and interaction of the modes is to perform Fourier decomposition of the flow variables. Substitute equations (12)–(14) into Eqs. (5)–(7). The mass equation for the mean flow vanishes. The mass equation for the perturbation is then:

$$\begin{aligned} i\left( \frac{2\pi k}{L_x}\right) \hat{u}_k+\frac{\partial \hat{v}_k}{\partial y}=0 \end{aligned}$$

(20)

Concerning the momentum equations, apply the normal flow condition at the inflexion point of the mixing layer:

$$\begin{aligned} 0=-\frac{c_f}{2} \overline{U}^2+g \overline{H}S_{0X} \end{aligned}$$

(21)

The momentum equation for the mean flow is then:

$$\begin{aligned} \frac{\partial (\overline{U}+\widetilde{u_m})}{\partial {t}} + 2 \mathfrak {R}\sum _{k=1}^N \frac{\partial \hat{u}_{-k} \hat{v}_k}{\partial y}&= -\frac{c_f}{\overline{H}}(\overline{U} \widetilde{u_m}) -\frac{c_f}{2\overline{H}}\widetilde{u_m}^2 -\frac{c_f}{\overline{H}}\sum _{k=1}^N \left( |\hat{u}_k|^2 +\frac{1}{2} |\hat{v}_k|^2\right) \nonumber \\&+\nu _{total}^t \frac{\partial ^2(\overline{U}+\widetilde{u_m})}{\partial y^2} \end{aligned}$$

(22)

where $\mathfrak {R}$ mean taking the real part. The momentum equation for mode $k$ is:

$$\begin{aligned}&\frac{\partial {\hat{u}_k}}{\partial {t}} + i\left( \frac{2 \pi k}{L_x}\right) \widetilde{u_m}\hat{u}_k + \hat{v}_k\frac{\partial \widetilde{u_m}}{\partial {y}}+ i\left( \frac{2 \pi k}{L_x}\right) \sum _{n+s=k}\hat{u}_n\hat{u}_s+\sum _{n+s=k} \frac{\partial \hat{u}_n \hat{v}_s}{\partial y}\nonumber \\&\quad = -i\left( \frac{2 \pi k}{L_x}\right) \hat{p}_k-\frac{c_f}{\overline{H}}(\overline{U}+\widetilde{u_m})\hat{u}_k\nonumber \\&\quad \quad -\frac{c_f}{2\overline{H}}\sum _{n+s=k}\left[ \left( \hat{u}_n \hat{u}_s+\frac{1}{2} \hat{v}_n \hat{v}_s\right) \right] +\nu _{total}^t \left[ -\left( \frac{2 \pi k}{L_x}\right) ^2 \hat{u}_k+\frac{\partial {^2 \hat{u}_k}}{\partial {y^2}}\right] \end{aligned}$$

(23)

$$\begin{aligned}&\frac{\partial {\hat{v}_k}}{\partial {t}} + i\left( \frac{2 \pi k}{L_x}\right) \widetilde{u_m}\hat{v}_k + i\left( \frac{2 \pi k}{L_x}\right) \sum _{n+s=k}\hat{u}_n\hat{v}_s+\sum _{n+s=k} \frac{\partial \hat{v}_n \hat{v}_s}{\partial y} \nonumber \\&\quad = -\frac{\partial \hat{p}_k}{\partial y}-\frac{c_f}{2\overline{H}}(\overline{U}+\widetilde{u_m})\hat{v}_k -\frac{c_f}{2\overline{H}}\sum _{n+s=k}(\hat{u}_n \hat{v}_s)+\nu _{total}^t \left[ -\left( \frac{2 \pi k}{L_x}\right) ^2 \hat{v}_k+\frac{\partial {^2 \hat{v}_k}}{\partial {y^2}}\right] \nonumber \\ \end{aligned}$$

(24)

where $n$ and $s$ goes from $-N$ to $N$, and $n \ne 0$ and $s \ne 0$. $\nu _{total}^t$ is assumed to be a constant, corresponding to the eddy viscosity at the center of the mixing layer where the 2DCS are situated. It is a simplifying assumption to prevent from including additional nonlinearity stemming from the eddy viscosity in the current work.

To obtain the modal kinetic energy equation for mode $k$, multiply equation (23) by $\hat{u}_{-k}$ and multiply the conjugate equation of (23) by $\hat{u}_k$. The pair of conjugate equations are added together. The pair of equations become:

$$\begin{aligned}&\frac{\partial {|\hat{u}_k|^2}}{\partial {t}} + (\hat{u}_{-k}\hat{v}_k + \hat{v}_{-k}\hat{u}_k) \frac{\partial \widetilde{u_m}}{\partial {y}}\nonumber \\&+ i\left( \frac{2 \pi k}{L_x}\right) \sum _{n+s=k}(\hat{u}_n\hat{u}_s\hat{u}_{-k} - \hat{u}_{-n}\hat{u}_{-s}\hat{u}_{k}) +\sum _{n+s=k} \left( \hat{u}_{-k} \frac{\partial \hat{u}_n \hat{v}_s}{\partial y} + \hat{u}_{k} \frac{\partial \hat{u}_{-n} \hat{v}_{-s}}{\partial y}\right) \nonumber \\&\quad = -i\left( \frac{2 \pi k}{L_x}\right) \hat{p}_k\hat{u}_{-k} +i\left( \frac{2 \pi k}{L_x}\right) \hat{p}_{-k}\hat{u}_k - \frac{2c_f}{\overline{H}}(\overline{U}+\widetilde{u_m}) |\hat{u}_k|^2\nonumber \\&\quad \quad -\frac{c_f}{2\overline{H}}\sum _{n+s=k}(\hat{u}_n \hat{u}_s \hat{u}_{-k} + \hat{u}_{-n} \hat{u}_{-s} \hat{u}_{k}) - \frac{c_f}{4\overline{H}}\sum _{n+s=k} (\hat{v}_n \hat{v}_s \hat{u}_{-k} + \hat{v}_{-n} \hat{v}_{-s} \hat{u}_k))\nonumber \\&\quad \quad +\nu _{total}^t \left[ -2\left( \frac{2 \pi k}{L_x}\right) ^2 |\hat{u}_k|^2+ \hat{u}_{-k}\frac{\partial {^2 \hat{u}_k}}{\partial {y^2}} + \hat{u}_k\frac{\partial {^2 \hat{u}_{-k}}}{\partial {y^2}}\right] \end{aligned}$$

(25)

Also, multiply equation (24) by $\hat{v}_{-k}$ and multiply the conjugate equation of (24) by $\hat{v}_k$. The pair of conjugate equations are added together. The pair of equations become:

$$\begin{aligned}&\frac{\partial {|\hat{v}_k|^2}}{\partial {t}} + i\left( \frac{2 \pi k}{L_x}\right) \sum _{n+s=k}(\hat{u}_n\hat{v}_s\hat{v}_{-k} - \hat{u}_{-n}\hat{v}_{-s}\hat{v}_{k}) +\sum _{n+s=k} \left( \hat{v}_{-k}\frac{\partial \hat{v}_n \hat{v}_s}{\partial y} + \hat{v}_{k}\frac{\partial \hat{v}_{-n} \hat{v}_{-s}}{\partial y}\right) \nonumber \\&=-\hat{v}_{-k}\frac{\partial \hat{p}_k}{\partial y}-\hat{v}_{k}\frac{\partial \hat{p}_{-k}}{\partial y} -\frac{c_f}{\overline{H}}(\overline{U}+\widetilde{u_m})|\hat{v}_k|^2 -\frac{c_f}{2\overline{H}}\sum _{n+s=k}(\hat{u}_n \hat{v}_s\hat{v}_{-k}+\hat{u}_{-n} \hat{v}_{-s}\hat{v}_{k})\nonumber \\&\quad +\nu _{total}^t \left[ -2\left( \frac{2 \pi k}{L_x}\right) ^2 |\hat{v}_k|^2 + \hat{v}_{-k}\frac{\partial {^2 \hat{v}_k}}{\partial {y^2}}+ \hat{v}_{k}\frac{\partial {^2 \hat{v}_{-k}}}{\partial {y^2}}\right] \end{aligned}$$

(26)

Equations (25) and (26) are added together and then take domain average along $y$-direction (i.e. $\frac{1}{L_y} \int _{-L_y/2}^{L_y/2}\ldots dy$). The mean flow is further decomposed into a base flow and a mean flow correction, $\widetilde{u_m}(y,t) = \widetilde{u_b}(y,t)+\hat{u}_0(y,t)$. Use the mass equation (20) and take $L_y$ to be much larger than the coherent structures so that $\hat{u}_k=0$, $\hat{v}_k=0$, $\partial (\widetilde{u_b}+\hat{u}_0) /\partial y=0$, $\partial \hat{u}_k/\partial y=0$ and $\partial \hat{v}_k/\partial y=0$ at $y=\pm \frac{L_y}{2}$. This implies the turbulent energy fluxes by turbulent convection, the viscous diffusion of turbulent energy and the turbulent energy flux by turbulent pressure across $y= \frac{L_y}{2}$ and $y= -\frac{L_y}{2}$ are zero. The modal kinetic energy equation for mode $k$ is derived as:

$$\begin{aligned} \frac{dE_k}{dt}&= - \left[ \frac{1}{L_y} \int _{-L_y/2}^{L_y/2}{2 \mathfrak {R}(\hat{u}_{-k} \hat{v}_k) \frac{\partial \widetilde{u_b}}{\partial y} dy}\right] - \left[ \frac{1}{L_y} \int _{-L_y/2}^{L_y/2}{2 \mathfrak {R}(\hat{u}_{-k} \hat{v}_k) \frac{\partial \hat{u}_0}{\partial y} dy}\right] \nonumber \\&- \frac{1}{L_y} \int _{-L_y/2}^{L_y/2}2 \sum _{n+s=k} \mathfrak {R}\left[ i\left( \frac{2 \pi k}{L_x}\right) \hat{u}_n \hat{u}_s \hat{u}_{-k}\!-\!\hat{u}_n \hat{v}_s \frac{\partial \hat{u}_{-k}}{\partial y} \!+\! i\left( \frac{2 \pi k}{L_x}\right) \hat{u}_n \hat{v}_s \hat{v}_{-k} \! -\!\hat{v}_n \hat{v}_s \frac{\partial \hat{v}_{-k}}{\partial y}\right] dy\nonumber \\&- \frac{1}{L_y} \frac{c_f}{\overline{H}} \int _{-L_y/2}^{L_y/2}{2 (\overline{U}+\widetilde{u_b}) \vert \hat{u}_k \vert ^2 + (\overline{U}+\widetilde{u_b}) \vert \hat{v}_k \vert ^2 dy} - \frac{1}{L_y} \frac{c_f}{\overline{H}} \int _{-L_y/2}^{L_y/2} 2 \hat{u}_0 \vert \hat{u}_k \vert ^2 + \hat{u}_0 \vert \hat{v}_k \vert ^2 dy\nonumber \\&- \frac{1}{L_y}\frac{c_f}{\overline{H}} \int _{-L_y/2}^{L_y/2}{\sum _{n+s=k} {\mathfrak {R}\left[ \hat{u}_n \hat{u}_s \hat{u}_{-k} + \hat{u}_n \hat{v}_s \hat{v}_{-k} + \frac{1}{2} \hat{v}_n \hat{v}_s \hat{u}_{-k}\right] }dy}\nonumber \\&+ \nu _{total}^t \frac{1}{L_y} \int _{-L_y/2}^{L_y/2}{-2 (2 \pi k/L_x)^2 (\vert \hat{u}_k \vert ^2 + \vert \hat{v}_k \vert ^2) - 2\left( \vert \frac{\partial \hat{u}_k}{\partial y} \vert ^2 + \vert \frac{\partial \hat{v}_k}{\partial y} \vert ^2\right) dy} \end{aligned}$$

(27)

Where:

$$\begin{aligned} E_k=\frac{1}{L_y}\int _{-L_y/2}^{L_y/2}{\vert \hat{u}_k \vert ^2 + \vert \hat{v}_k \vert ^2 dy} \end{aligned}$$

(28)

There are $N$ modal kinetic energy equations $k=1,2,3,\ldots ,N$, because the modal kinetic energy equation for mode $k$ is derived from both the momentum equation for mode $k$ and the momentum equation for mode $-k$ for all $k \le N$. $n$ and $s$ goes from $-N$ to $N$, with $n \ne 0$ and $s \ne 0$.

The full set of modal kinetic energy equations should in fact include the mean flow kinetic energy equation to be complete, but the derivation of the mean flow kinetic energy equation is omitted here for brevity.