Linear instability of tilted parallel shear flow in a strongly stratified and viscous medium

Abstract

A linear stability analysis is performed on a tilted parallel wake in a strongly stratified fluid at low Reynolds numbers. A particular emphasis of the present study is given to the understanding of the low-Froude-number mode observed by the recent experiment (Meunier in J Fluid Mech 699:174–197, 2012). In the limit of low Froude number, the linearised equations of motion can be reduced to the Orr–Sommerfeld equation on the horizontal plane, except the viscous term that contains vertical dissipation. Based on this equation, it is proposed that the low-Froude-number mode would be a horizontal inflectional instability and should remain two-dimensional at small tilting angles as long as the Reynolds number is sufficiently low. To support this claim, the asymptotic regime where this analysis is strictly valid is subsequently discussed in relation to previous work on the proper vertical length scale. The absolute and convective instability analysis of parallel wake is further performed, showing qualitatively good agreement with the experimental result. The low-Froude-number mode is found to be stabilised on increasing Froude number, as in the experiment. It is shown that the emergence of small vertical velocity at finite Froude number, the size of which is proportional to the square of Froude number, plays the key role in the stabilisation by modifying the inflectional instability and paradoxically creating stabilising buoyancy effect with the increase of Froude number.

Graphic abstract

Appendices

Derivation of energy budget analysis

Now, (8a)–(8e) can be rewritten as:

$$\begin{aligned} i\omega E_{total}= \int _{-\infty }^{\infty } \left( \begin{array}{c} \bar{{\tilde{u}}}\\ \bar{{\tilde{v}}}\\ \bar{{\tilde{w}}}\\ \bar{{\tilde{b}}}\\ \bar{{\tilde{p}}} \end{array}\right) ^T \left( \begin{array}{ccccc} {\mathcal {L}} &{} DU &{} 0 &{} 0 &{} i\alpha \\ 0 &{} {\mathcal {L}} &{} 0 &{} -\sin \theta &{} D\\ 0 &{} 0 &{} {\mathcal {L}} &{} \cos \theta &{} 0\\ 0 &{} \sin \theta &{} -\cos \theta &{} {\mathcal {L}}_{\rho } {\textit{Fr}}^2 &{} 0\\ i\alpha &{} D &{} 0 &{} 0 &{} 0 \end{array}\right) \left( \begin{array}{c} {\tilde{u}}\\ {\tilde{v}}\\ {\tilde{w}}\\ {\tilde{b}}\\ {\tilde{p}} \end{array}\right) \mathrm{d}y, \end{aligned}$$

(34)

where

$$\begin{aligned} E_{total}=\int _{-\infty }^{\infty } |{\tilde{u}}|^2 + |{\tilde{v}}|^2 + |{\tilde{w}}|^2 + {\textit{Fr}}^2 |{\tilde{b}}|^2 \mathrm{d}y. \end{aligned}$$

(35)

We note that we have deliberately scaled (8d) by ${\textit{Fr}}^2$ to recover an energy budget that has physical significance. Therefore, the first three terms in the integrand of (35) form

$$\begin{aligned} E_{u}=\int _{-\infty }^{\infty } |{\tilde{u}}|^2 + |{\tilde{v}}|^2 + |{\tilde{w}}|^2 \mathrm{d}y, \end{aligned}$$

(36)

representing the kinetic energy, and the last term

$$\begin{aligned} E_{b}=\int _{-\infty }^{\infty } {\textit{Fr}}^2 |{\tilde{b}}|^2 \mathrm{d}y \end{aligned}$$

(37)

being the potential energy.

Now, since we are only interested in the growth rate, the imaginary part of $\omega$, we can therefore define the following terms:

$$\begin{aligned} P_{uu}=&-{\mathrm {Re}}\left( \int _{-\infty }^{\infty }\bar{{\tilde{u}}}{\mathcal {L}}{\tilde{u}}\mathrm{d}y\right) , \end{aligned}$$

(38)

$$\begin{aligned} P_{uv}=&-{\mathrm {Re}}\left( \int _{-\infty }^{\infty }\bar{{\tilde{u}}}DU{\tilde{v}}\mathrm{d}y\right) ,\end{aligned}$$

(39)

$$\begin{aligned} P_{up}=&-{\mathrm {Re}}\left( i\alpha \int _{-\infty }^{\infty }\bar{{\tilde{u}}}{\tilde{p}}\mathrm{d}y\right) ,\end{aligned}$$

(40)

$$\begin{aligned} P_{vv}=&-{\mathrm {Re}}\left( \int _{-\infty }^{\infty }\bar{{\tilde{v}}}{\mathcal {L}}{\tilde{v}}\mathrm{d}y\right) ,\end{aligned}$$

(41)

$$\begin{aligned} P_{vb}=&-{\mathrm {Re}}\left( -\sin \theta \int _{-\infty }^{\infty }\bar{{\tilde{v}}}{\tilde{b}}\mathrm{d}y\right) ,\end{aligned}$$

(42)

$$\begin{aligned} P_{vp}=&-{\mathrm {Re}}\left( \int _{-\infty }^{\infty }\bar{{\tilde{v}}}D{\tilde{p}}\mathrm{d}y\right) ,\end{aligned}$$

(43)

$$\begin{aligned} P_{ww}=&-{\mathrm {Re}}\left( \int _{-\infty }^{\infty }\bar{{\tilde{w}}}{\mathcal {L}}{\tilde{w}}\mathrm{d}y\right) ,\end{aligned}$$

(44)

$$\begin{aligned} P_{wb}=&-{\mathrm {Re}}\left( \cos \theta \int _{-\infty }^{\infty }\bar{{\tilde{w}}}{\tilde{b}}\mathrm{d}y\right) ,\end{aligned}$$

(45)

$$\begin{aligned} P_{bv}=&-{\mathrm {Re}}\left( \sin \theta \int _{-\infty }^{\infty }\bar{{\tilde{b}}}{\tilde{v}}\mathrm{d}y\right) ,\end{aligned}$$

(46)

$$\begin{aligned} P_{bw}=&-{\mathrm {Re}}\left( -\cos \theta \int _{-\infty }^{\infty }\bar{{\tilde{b}}}{\tilde{w}}\mathrm{d}y\right) ,\end{aligned}$$

(47)

$$\begin{aligned} P_{bb}=&-{\mathrm {Re}}\left( \int _{-\infty }^{\infty }{\textit{Fr}}^2 \bar{{\tilde{b}}}{\mathcal {L}}_{\rho }{\tilde{b}}\mathrm{d}y\right) ,\end{aligned}$$

(48)

$$\begin{aligned} P_{pu}=&-{\mathrm {Re}}\left( i\alpha \int _{-\infty }^{\infty }\bar{{\tilde{p}}}{\tilde{u}}\mathrm{d}y\right) ,\end{aligned}$$

(49)

$$\begin{aligned} P_{pv}=&-{\mathrm {Re}}\left( \int _{-\infty }^{\infty }\bar{{\tilde{p}}}D{\tilde{v}}\mathrm{d}y\right) , \end{aligned}$$

(50)

where $\omega _i E_{total}$ is equal to the sum of the above terms. Here, we note that the sum of pressure terms ($P_{up}$+ $P_{vp}$+ $P_{wp}$) and the continuity terms ($P_{pu}+P_{pv}+P_{pw}$) cancel out each other. Also, in the inviscid limit, the Doppler shift terms ($P_{uu}$, $P_{vv}$, $P_{ww}$, $P_{bb}$) are purely imaginary and do not contribute to the growth rate either. Finally, the buoyancy terms $P_{wb}$ and $P_{vb}$ play a role only in exchanging kinetic energy with potential energy in conjunction with the terms $P_{bv}$ and $P_{bw}$ from the density equation. Therefore, the only term that would contribute to the total energy becomes the production by base flow shear, $P_{uv}$.

Froude number effect on three-dimensional temporal instability

Fig. 10

Temporal growth rate of the most unstable mode with respect to Froude number for several $\beta$ at a$\theta =30^{\circ }, {\textit{Re}}=25, \alpha =1$ and b$\theta =60^{\circ }, {\textit{Re}}=50, \alpha =0.4$

A three-dimensional temporal stability analysis is performed here. Here, we consider several sets of $\alpha \ne 0$ and $\beta \ne 0$ over a range of ${\textit{Fr}}$. The result is shown in Fig. 10. As expected from the analysis in Sect. 4.1, the behaviour of three-dimensional instability mode with respect to ${\textit{Fr}}$ is qualitatively the same as that of absolute instability for $\beta =0$.