^{1,a)}and Rama Govindarajan

^{2}

### Abstract

Absolute instabilities in shear flows can cause a catastrophic breakdown into a new unsteady state, or even into turbulence. We demonstrate that in a double-diffusive channel flow with a viscosity stratification across the channel, rapidly growing absolute instability may be obtained at Reynolds numbers of a few hundreds. The instability is much weaker in an equivalent single solute fluid with the same viscosity contrast, or even in one which is made up only of the more dangerous of the two diffusing species. This is a novel characteristic of double-diffusive systems driven by viscosity, rather than density variations. Convective instabilities too are stronger in the double-diffusive case.

K.S. gratefully acknowledges the support from the Indian Institute of Technology Hyderabad, India.

I. INTRODUCTION

II. FORMULATION

A. Base state

B. Linear stability analysis

III. RESULTS AND DISCUSSION

IV. SUMMARY

### Key Topics

- Flow instabilities
- 63.0
- Viscosity
- 40.0
- Viscous flow instabilities
- 14.0
- Channel flows
- 10.0
- Reynolds stress modeling
- 9.0

## Figures

Schematic of the three-layer base state flow. The fluids “1” and “2” occupy the channel core and the region adjacent to the channel walls, respectively. The two fluids are separated by a mixed layer of uniform thickness *q*, with fluid “1” located in the region −*h* ⩽ *y* ⩽ *h*.

Schematic of the three-layer base state flow. The fluids “1” and “2” occupy the channel core and the region adjacent to the channel walls, respectively. The two fluids are separated by a mixed layer of uniform thickness *q*, with fluid “1” located in the region −*h* ⩽ *y* ⩽ *h*.

Comparison of DD with SC convective instability for *h* = 0.6 and *q* = 0.1: (a) dispersion curve (ω_{ i, max } versus α_{ r }) for Re = 200, (b) neutral stability curve. Curves *C* _{DD}, with Sc = 50, *R* _{ s } = 5, *R* _{ f } = −0.5, and δ = 10 are for a DD system. The SC cases, denoted by *C* _{SC}, are obtained by setting one of the stratification rates, *R* _{ f } in this case, to zero. To make the two systems equivalent, for the SC case we set Sc = 9.091 and *R* _{ s } = 4.5. To obtain the maximum convective temporal growth rate, we have set the imaginary part of the disturbance wavenumber, α_{ i } = 0.

Comparison of DD with SC convective instability for *h* = 0.6 and *q* = 0.1: (a) dispersion curve (ω_{ i, max } versus α_{ r }) for Re = 200, (b) neutral stability curve. Curves *C* _{DD}, with Sc = 50, *R* _{ s } = 5, *R* _{ f } = −0.5, and δ = 10 are for a DD system. The SC cases, denoted by *C* _{SC}, are obtained by setting one of the stratification rates, *R* _{ f } in this case, to zero. To make the two systems equivalent, for the SC case we set Sc = 9.091 and *R* _{ s } = 4.5. To obtain the maximum convective temporal growth rate, we have set the imaginary part of the disturbance wavenumber, α_{ i } = 0.

Comparison of DD with SC stability for the case where the slower diffusing species is stabilizing, with Re = 200, *h* = 0.6, and *q* = 0.1. Curve *C* _{DD}, with Sc = 50, *R* _{ s } = −0.5, *R* _{ f } = 4, and δ = 10 is for a DD system, while *C* _{SC} with Sc = 9.091, *R* _{ s } = 3.5, and *R* _{ f } = 0 is for an SC fluid. Here, α_{ i } = 0.

Comparison of DD with SC stability for the case where the slower diffusing species is stabilizing, with Re = 200, *h* = 0.6, and *q* = 0.1. Curve *C* _{DD}, with Sc = 50, *R* _{ s } = −0.5, *R* _{ f } = 4, and δ = 10 is for a DD system, while *C* _{SC} with Sc = 9.091, *R* _{ s } = 3.5, and *R* _{ f } = 0 is for an SC fluid. Here, α_{ i } = 0.

Isocontours of (a) ω_{ r } and (b) ω_{ i } in the complex wavenumber plane for a DD system at Re = 200. The rest of the parameters are Sc = 50, δ = 10, *h* = 0.6, *q* = 0.1, *R* _{ s } = 4, and *R* _{ f } = −0.5. The absolute frequency ω_{0}, i.e., ω at the saddle point, is 1.156 + 0.041**i**, showing absolute instability.

Isocontours of (a) ω_{ r } and (b) ω_{ i } in the complex wavenumber plane for a DD system at Re = 200. The rest of the parameters are Sc = 50, δ = 10, *h* = 0.6, *q* = 0.1, *R* _{ s } = 4, and *R* _{ f } = −0.5. The absolute frequency ω_{0}, i.e., ω at the saddle point, is 1.156 + 0.041**i**, showing absolute instability.

Isocontours of (a) ω_{ r } and (b) ω_{ i } in the complex wavenumber plane for a single component system equivalent to the double-diffusive system of Figure 4. The parameters are Re = 200, Sc = 9.091, *h* = 0.6, *q* = 0.1, *R* _{ s } = 3.5, and *R* _{ f } = 0. At the saddle point, ω_{0} is 1.265 − 0.266**i**, i.e., the flow is not absolutely unstable. In fact, the zero group velocity mode is damped out rapidly.

Isocontours of (a) ω_{ r } and (b) ω_{ i } in the complex wavenumber plane for a single component system equivalent to the double-diffusive system of Figure 4. The parameters are Re = 200, Sc = 9.091, *h* = 0.6, *q* = 0.1, *R* _{ s } = 3.5, and *R* _{ f } = 0. At the saddle point, ω_{0} is 1.265 − 0.266**i**, i.e., the flow is not absolutely unstable. In fact, the zero group velocity mode is damped out rapidly.

Isocontours of (a) ω_{ r } and (b) ω_{ i } in the complex wavenumber plane. The parameters are Re = 200, Sc = 50, δ = 10, *h* = 0.6, *q* = 0.1, *R* _{ s } = 4, and *R* _{ f } = 0. At the saddle point, ω_{0} is 0.787 + 0.007**i**.

Isocontours of (a) ω_{ r } and (b) ω_{ i } in the complex wavenumber plane. The parameters are Re = 200, Sc = 50, δ = 10, *h* = 0.6, *q* = 0.1, *R* _{ s } = 4, and *R* _{ f } = 0. At the saddle point, ω_{0} is 0.787 + 0.007**i**.

Isocontours of (a) ω_{ r } and (b) ω_{ i } in the complex wavenumber plane. The parameters are the same as those used to generate the *C* _{DD} curve of Figure 3. The absolute frequency ω_{0}, i.e., ω at the saddle point, is 1.366 − 0.485**i**, showing that the flow is not absolutely unstable.

Isocontours of (a) ω_{ r } and (b) ω_{ i } in the complex wavenumber plane. The parameters are the same as those used to generate the *C* _{DD} curve of Figure 3. The absolute frequency ω_{0}, i.e., ω at the saddle point, is 1.366 − 0.485**i**, showing that the flow is not absolutely unstable.

Variation of the absolute growth rate with *R* _{ s } for different δ values for (a) *q* = 0.1 and (b) *q* = 0.05, respectively. The rest of the parameter values are *R* _{ f } = −1, Re = 200, Sc = 50, and *h* = 0.6. The absolute instability is stronger for a DD case than for an SC one.

Variation of the absolute growth rate with *R* _{ s } for different δ values for (a) *q* = 0.1 and (b) *q* = 0.05, respectively. The rest of the parameter values are *R* _{ f } = −1, Re = 200, Sc = 50, and *h* = 0.6. The absolute instability is stronger for a DD case than for an SC one.

Stability diagrams showing the regions of convective and absolute instabilities in *R* _{ s }-*Re* space for (a) *q* = 0.1 and (b) *q* = 0.05, respectively. The rest of the parameter values are Sc = 50, δ = 10, *h* = 0.6, and *q* = 0.1. The horizontal lines in each case show the location where *R* _{ s } + *R* _{ f } = 0. Above this line, the average viscosity increases as we move from the centerline of the channel towards the wall.

Stability diagrams showing the regions of convective and absolute instabilities in *R* _{ s }-*Re* space for (a) *q* = 0.1 and (b) *q* = 0.05, respectively. The rest of the parameter values are Sc = 50, δ = 10, *h* = 0.6, and *q* = 0.1. The horizontal lines in each case show the location where *R* _{ s } + *R* _{ f } = 0. Above this line, the average viscosity increases as we move from the centerline of the channel towards the wall.

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