Kelvin–Helmholtz instability in the presence of variable viscosity for mudflow resuspension in estuaries

Abstract

The temporal stability of a parallel shear flow of miscible fluid layers of different density and viscosity is investigated through a linear stability analysis and direct numerical simulations. The geometry and rheology of this Newtonian fluid mixing can be viewed as a simplified model of the behavior of mudflow at the bottom of estuaries for suspension studies. In this study, focus is on the stability and transition to turbulence of an initially laminar configuration. A parametric analysis is performed by varying the values of three control parameters, namely the viscosity ratio, the Richardson and Reynolds numbers, in the case of initially identical thickness of the velocity, density and viscosity profiles. The range of parameters has been chosen so as to mimic a wide variety of real configurations. This study shows that the Kelvin–Helmholtz instability is controlled by the local Reynolds and Richardson numbers of the inflection point. In addition, at moderate Reynolds number, viscosity stratification has a strong influence on the onset of instability, the latter being enhanced at high viscosity ratio, while at high Reynolds number, the influence is less pronounced. In all cases, we show that the thickness of the mixing layer (and thus resuspension) is increased by high viscosity stratification, in particular during the non-linear development of the instability and especially pairing processes. This study suggests that mud viscosity has to be taken into account for resuspension parameterizations because of its impact on the inflection point Reynolds number and the viscosity ratio, which are key parameters for shear instabilities.

Appendix

In this appendix, we describe the method that we have developed to identify the discretised K–H modes among spurious modes tracing back the continuous spectrum of the linear problem.

1.1 Characterization of the spurious modes

Spurious modes are characterized by a Chebyshev decomposition of the $z$-profiles that is not converged. The associated spurious eigenvalues stand close to the marginal curve of null growth rate $\sigma ^*=0$. We interpret them as a continuous spectrum of internal waves trapped in critical layers. A physical hint of these modes can be caught by considering the refraction of plane internal waves by density stratification and shear. The WKB method describes the evolution of wave packets of the wave vector ${k,m(z)}$ in the two-dimensional plane $(x,z)$. They form a solution of the Eikonale equation $\Omega [k,m(z),z]=\Omega _0$ with $\Omega (k,m,z)=\sqrt{\frac{g}{\rho _r}\frac{\partial \overline{\rho }}{\partial z}}\frac{k}{\sqrt{m^2+k^2}}+k\, \overline{u}(z)$ and $\Omega _0$ constant, where $\rho _r$ is the reference density used in the framework of a Boussinesq approximation of the Navier–Stokes equations.

Figure 19 presents isovalues of the frequency for fixed wave number, representing rays of internals waves, in the $(m,z)$ plane. Divergent trajectories are observed and characterized by horizontals asymptotes (as dotted line for $\omega =0.9$) with $m \rightarrow \infty$ and $z \rightarrow z_c$. These asymptotes correspond to absorption critical layers of internal waves. Due to viscosity, equation does not degenerate as Rayleigh equation for $\overline{u}=c$. Yet, very high resolution is needed since these layers present a thickness of $(k^*Re)^{-1/3}$ (see [12]). Non-converged but physical modes with positive growth rate have their origin in these critical layers. To validate this hypothesis, the relation of dispersion $\omega \sim k\, \overline{u}(z_c)$ have been verified. Frequency of modes are compared to $k\, \overline{u}(z_c)$, by obtaining $z_c$ from vertical profiles of the mode. Figure 21 shows that the growth rate of these modes decreases when resolution grows: they are under-resolved. Figure 20 shows that these modes also decrease around $Ri=0$. This evolution corroborate the relation between modes and internal waves.

Fig. 19

Isovalues of $\Omega [k,m(z),z]$ in the $(m,z)$ plane for $Ri=0,15$, $k^*=0,8$, zoomed on $z \in [0.9,2.3]$

Fig. 20

Growth rate of the two most unstable eigenmodes as a function of the Richardson number for $W=0$, $k^*=1.1$

Fig. 21

Growth rate of the two most unstable eigenmode, for $W=3$, $k^*=0.95$, $Re=100$, $Ri=0.15$. a Variation with the resolution. b Variation with $\alpha$

1.2 Complex mapping function

The elimination of these spurious modes at the profit of the searched K–H modes is achieved through the use of a complex mapping in the LiSa code. This stability code is based on a spectral decomposition using Chebyshev polynomials with a collocation method using a mapping transforming the spectral variable $s \in [-1, 1]$ (Gauss–Lobatto collocation points) into the physical variable $z$. A first real mapping $z=\varphi _0(s)$ reads:

$$\begin{aligned} \varphi _0(z)=C_1 \left[ 1- e^{ C_2\, \mathrm {arctanh}\left( C_3 s\right) +C_4 } \right] , \end{aligned}$$

where the constants $C_1$, $C_2$, $C_3$ and $C_4$ are adjusted to match the positions of the interface and the two boundaries as well as to adjust the refinement of the collocation grid near the interface. Based on an idea of Fabre et al. [14], we have extended it to the complex mapping $\varphi _{\alpha }(z) = \varphi _0(s)\, (1- i\, \alpha )$ with $\alpha =0.010369-0.035621\, \mathrm {Ri}$. As presented on Fig. 21b, $\alpha$ can been determined through a sensitivity analysis of the eigenvalues with respect to its value. We choose the lowest value of $\alpha$ for which growth rate of spurious eigenvalues pass horizontal neutral axes (corresponding to their supposed position).

Figure 22 presents the eigenvalues spectrum for the two mappings. With the real mapping $\varphi _0$, the KH mode is “hidden” in a continuous spectrum of non-converged mode corresponding to internal waves trapped in absorption critical layers as presented earlier. With the complex mapping $\varphi _\alpha$, the continuous spectrum is pushed below while the unstable KH-mode subsists and exhibit a converged Chebyshev decomposition. The use of the complex mapping is equivalent to a deformation of the integration path in the complex plane, which avoids the singularities at the $z$ real points corresponding to the critical layers. The error done with a constant $\alpha$ mapping, which does not bring back the integration path towards the real axes, appears to be negligible for the eigenvalue precision looked at in the present work.

Fig. 22

Eigenvalues spectra for$Ri=0.15$, $W=3$, $Re=10^2$, $k^*=0.95$. Comparison of the real and complex mappings. Eigenvalues with red circles correspond to converged modes