# Linear Non-Modal Growth of Planar Perturbations in a Layered Couette Flow

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Linear Equations for the Evolution of Small Perturbations

## 3. Modal Characteristics of the TCI

## 4. Non-Modal Growth of Perturbations

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

TCI | Taylor–Caulfield Instability |

## References

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**Figure 1.**(

**a**) The non-dimensional mean flow velocity $U\left(z\right)=z$. (

**b**) The non-dimensional mean flow density ${\rho}_{0}\left(z\right)-{\rho}_{m}/\mathrm{\Delta}\rho $ with $R=30$.

**Figure 2.**(

**a**) Growth rate of the most unstable modes as a function of wavenumber k and bulk Richardson number J. Also shown is the line $J=2k/9$ for which resonance in the long-wave limit between the gravity edge-waves supported at the interfaces of sharp density gradient is achieved. (

**b**) Phase speed of the most unstable modes with $k=2$ as a function of J (solid line). Also shown are the phase speeds ${c}_{t}^{+}$ (dashed line) and ${c}_{t}^{-}$ (dash-dotted line) of the neutral modes in the $R\to \infty $ limit.

**Figure 3.**Streamfunction (

**a**) and density (

**b**) for the most unstable mode in the TCI branch with ${k}_{max}=1.9$ and ${J}_{max}=0.5$. Streamfunction (

**c**) and density (

**d**) for the most unstable mode in the ‘overtone’ branch with $k=2.6$ and $J=1.7$. Also shown in all panels are the interfaces of large density gradients (dashed lines) and the critical layers of the two modes (dashed-dotted) lines.

**Figure 4.**(

**a**) Optimal growth ${G}_{0}$ at ${T}_{opt}=5$ for an unstratified flow as a function of wavenumber. Also shown is the corresponding growth for an unbounded shear flow $1+{T}_{opt}^{2}$. (

**b**) Optimal growth G at ${T}_{opt}=5$ as a function of wavenumber and bulk Richardson number. (

**c**) Ratio of the optimal growth G at ${T}_{opt}=5$ over the corresponding growth for an unstratified flow (${G}_{0}$) as a function of wavenumber and bulk Richardson number.

**Figure 5.**Finite time Lyapunov exponent $ln\left(G\left({T}_{opt}\right)\right)/2{T}_{opt}$ as a function of wavenumber and bulk Richardson number for (

**a**) ${T}_{opt}=10$, (

**b**) ${T}_{opt}=50$ and (

**c**) ${T}_{opt}=100$.

**Figure 6.**(

**a**) Excitation amplitude ${p}_{i}$ for the unstable modes in the TCI branch by their bi-orthogonals. Also shown is the cut-off scale for the TCI branch (dashed line). (

**b**) Energy evolution for the bi-orthogonal as an initial perturbation (solid) and for the most unstable mode as an initial perturbation (dashed).

**Figure 7.**(

**a**) Streamfunction of the bi-orthogonal to the most unstable mode for ${k}_{max}$ and ${J}_{max}$. (

**b**,

**c**) Snapshots of the streamfunction at (

**b**) $t=4$ and (

**c**) $t=20$ showing the evolution of the bi-orthogonal. In all panels the interfaces of large density gradients are also shown (dashed lines).

**Figure 8.**(

**a**) The evolution of the energy of the optimal perturbations with $k=3$ (solid line), $k=2.5$ (dashed line) and $k=2.475$ (dash-dotted line). The bulk Richardson number is $J=0.5$ and ${T}_{opt}=100$. (

**b**) Hovmöller diagram of the streamfunction of the optimal perturbation for $k=3$. That is, contours of $\Re \left[\widehat{\psi}(z=1/3,t){e}^{ikx}\right]$ as a function of x and t for the optimal perturbation. Also shown (dashed line) is the phase speed of the mode onto which the optimal perturbation projects (see panel (

**c**)). (

**c**) Projection coefficients of the optimal perturbation for $k=3$ on the normal modes and the continuous spectrum modes. Also shown (dashed lines) are the ${c}_{t}^{\pm}$ phase speeds of the neutral normal modes in the $R\to \infty $ limit.

**Figure 9.**The evolution of the optimal perturbation for $k=3$, $J=0.5$ and ${T}_{opt}=100$. (

**a**) The streamfunction of the optimal perturbation. (

**b**) The absolute value of the Fourier amplitude of the streamfunction obtained through a discrete Fourier transform. Note that the minimum wavenumber is $\pi $ leading to the jagged appearance of the line. (

**c**–

**e**) Snapshots of the streamfunction at (

**c**) $t=10$, (

**d**) $t=23$ and (

**e**) $t=29$. (

**f**) The absolute value of the Fourier amplitude of the streamfunction at $t=10$ (solid line) and $t=29$ (dashed line).

**Figure 10.**Late stage of the evolution of the optimal perturbation for $k=3$, $J=0.5$ and ${T}_{opt}=100$. Shown are contours of the streamfunction at (

**a**) $t=100$, (

**b**) $t=105$, (

**c**) $t=111$ and (

**d**) $t=114$.

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**MDPI and ACS Style**

Iliakis, E.G.; Bakas, N.A.
Linear Non-Modal Growth of Planar Perturbations in a Layered Couette Flow. *Fluids* **2021**, *6*, 442.
https://doi.org/10.3390/fluids6120442

**AMA Style**

Iliakis EG, Bakas NA.
Linear Non-Modal Growth of Planar Perturbations in a Layered Couette Flow. *Fluids*. 2021; 6(12):442.
https://doi.org/10.3390/fluids6120442

**Chicago/Turabian Style**

Iliakis, Emmanouil G., and Nikolaos A. Bakas.
2021. "Linear Non-Modal Growth of Planar Perturbations in a Layered Couette Flow" *Fluids* 6, no. 12: 442.
https://doi.org/10.3390/fluids6120442