Waves in a shear flow: transition between the KH, Holmboe and Miles instability

TL;DR

This study analyzes shear-driven interfacial waves between two immiscible fluids using an upper-layer exponential velocity profile and a sharp density jump, uncovering a novel transition in the fastest-growing mode from Kelvin-Helmholtz to Holmboe and then to Miles instability as the density ratio $\delta$ decreases from near unity to air–water values, at high Bond number and moderate Fr. The authors derive and solve the inviscid Rayleigh equations with a Gauss hypergeometric representation to obtain a dispersion relation, and validate linear predictions against nonlinear simulations in Basilisk, demonstrating distinct nonlinear states from capillary ripples to cusped Holmboe-like waves and KH spirals. They also compare the exponential profile to a piecewise-linear background, quantify the role of background curvature and the critical-layer dynamics (including the Reynolds-stress signature at $z_c$), and extend the analysis to a self-similar double-exponential background to show robustness of the transitions. The findings unify three canonical instabilities within a single background state, reveal that the Miles mode persists up to $\delta\approx 0.01$ (ten times air–water), and illuminate how the energy extraction mechanism shifts from a localized critical-layer process to a shear-wide interaction as $\delta$ grows, with implications for geophysical and astrophysical flows and possible laboratory validation.

Abstract

We investigate shear driven wave generation at the interface between two immiscible fluids, using an exponential velocity profile with a sharp density interface representing stable stratification. At low Froude and high Bond numbers, conditions relevant to geophysical and astrophysical flows, we identify a novel transition in the fastest growing mode: from Kelvin Helmholtz (KH) instability at high density ratio (delta = 0.9), to Holmboe (H) instability as delta approaches 0.5, and ultimately to the Miles (1957) critical layer instability as delta approaches 0.001, representative of the air water system. Remarkably, the Miles mode, characterized by a sharp jump in inviscid Reynolds stress (tau) at the critical layer, persists up to delta = 0.01, i.e., ten times the air water value. As delta increases, the vertical variation of tau undergoes a qualitative change, from a sharp jump at the critical layer for delta much less than 1 to a smooth transition through it for delta greater than or equal to 0.5. A theoretical explanation is provided. In the moderate to high density ratio regime, comparison with a piecewise-linear (PL) velocity profile confirms the presence of both H and KH instabilities in the exponential profile. Nonlinear simulations of the incompressible Euler equations with gravity and surface tension show excellent agreement with linear theory for delta = 0.01 up to five wave periods. At delta = 0.1, waves saturate into finite-amplitude structures with capillary ripples, while at delta = 0.5, the waves develop sheared cusps and emit spume, resembling asymmetric Holmboe waves observed experimentally. At delta = 0.9, the waves rapidly evolve into classic KH spirals. Comparisons with the PL profile highlight the role of background curvature and the critical layer. This work presents, for the first time, all three canonical instabilities within a single background state.

Figures (31)

• Figure 1: Exponential velocity profile with quiescent fluid below, in the base (background) state with upper fluid velocity varying as $U_u(z)=U_{\infty}\left[1-\exp(-z/\Delta)\right]$. A sharp interface separates two fluids of differing densities in a statically stable configuration: $\rho(z)=\rho_l,\; z< 0$ and $\rho(z)=\rho_u,\; z >0$ with $\rho_l > \rho_u$.
• Figure 2: Panel (a) The piecewise linear (PL) profile with a shear layer of thickness $\Delta$. The upper fluid background velocity is given by $U_u(z)=U_\infty\left( \dfrac{z}{\Delta}\right),\; 0\leq z \leq \Delta$ and $U_u(z)=U_{\infty},\; z \geq \Delta$. Panel (b) The classic KH discontinuous profile, obtained when $\Delta\rightarrow0$ in the PL profile on the left panel. In both panels, the lower fluid is quiescent in the base-state i.e. $U_l(z)=0,\; z<0$. The statically stable, base-state density profile is $\rho(z)=\rho_l,\; z< 0$ and $\rho(z)=\rho_u,\; z >0$ with $\rho_l > \rho_u$.
• Figure 3: $Bo\rightarrow\infty$, low (air-water) density ratio ($\delta=0.001$) and high Froude ($Fr=3194.38$). (Panel (a)) Growth rate ($\kappa \tilde{c}_i$) and (panel (b)) phase speed ($c_r/c_{_{KH}}$) as a function of the non-dimensional wavenumber $\kappa$ for the Kelvin-Helmholtz (KH) dispersion relation from eqn. \ref{['eq3.1']} (pink dashed curve), for the exponential profile (solid blue curve) from eqn. \ref{['eq2.12']} and for the piecewise-linear (PL) profile from the dispersion relation in eqn. \ref{['eq3.2']} ('$\cdot\times\cdot$' symbols) for the base-states of figures \ref{['fig1']} and \ref{['fig2']}. We see that the growth rate and phase speed are very similar for $\kappa\rightarrow0$. The KH dispersion relation of eqn. \ref{['eq3.1']} does not contain the Froude number as a parameter and represents the $Fr\rightarrow\infty$ limit. The red and green dots in both panels depict a small wavenumber and the fastest growing wavenumber respectively.
• Figure 4: $(Bo\rightarrow\infty,Fr=3194.8)$ Vertical ($z$) profiles of the (Panel (a)) Reynolds stress and (Panel (b)) perturbation kinetic energy in the upper fluid, normalised by their values at the location of the undisturbed interface at $z=0$, for the exponential profile. In panels (a) and (b), the red and green curves correspond to the low wavenumber and fastest growing modes from figure \ref{['fig3']}, respectively. The vertical coordinate $z$ is scaled here by the coordinate $z_c$ of the critical layer where the base velocity $U_{u}\left(z_{c}\right)$ matches the real part of the phase speed of an unstable mode i.e. $c_{r}\left(k\right)$. The blue curve is for the fastest growing mode with $\delta=0.5$.
• Figure 5: $Bo\rightarrow\infty$, high density ratio ($\delta=0.5$) and high Froude ($Fr=3194.38$). (Panel (a)) Growth rate ($\kappa \tilde{c}_i$) and (Panel (b)) phase speed ($c_r/c_{_{KH}}$) as a function of the non-dimensional wavenumber $\kappa$ for the Kelvin-Helmholtz (KH) dispersion relation from eqn. \ref{['eq3.1']} (pink dashed curve), for the exponential profile (solid blue curve) from eqn. \ref{['eq2.12']} and for the piecewise-linear (PL) profile from the dispersion relation in eqn. \ref{['eq3.2']} ('+' symbols) for the base-states of figures \ref{['fig1']} and \ref{['fig2']}. Green dot indicates the fastest growing mode in both panels. The phase speeds of the PL and the exponential profile are markedly different.