Analytical solutions for algebraic growth of disturbances in a stably stratified shear flow

Jose, Sharath; Roy, Anubhab; Bale, Rahul; Govindarajan, Rama

doi:10.1098/rspa.2015.0267

Cited by 8 publications

(8 citation statements)

References 31 publications

(52 reference statements)

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“…Similarly to what observed in the present case, the authors found that the larger the initial density (or temperature) perturbation, the larger the kinetic energy growth, and the longer the amplification is sustained for. The present longtime energy growth mechanism, which is indeed optimal for some of the considered approximations of the energy, appears to be the same as that reported in [40].…”

Section: Long-time Energy Amplificationsupporting

confidence: 78%

“…In Ref. [40], an amplification mechanism has been studied in which density perturbations advected in a shear flow are able to force the wall-normal velocity perturbations, inducing an algebraic growth in nonstratified and weakly (stably) stratified shear flows, and a sublinear growth in strongly stratified ones. Similarly to what observed in the present case, the authors found that the larger the initial density (or temperature) perturbation, the larger the kinetic energy growth, and the longer the amplification is sustained for.…”

Section: Long-time Energy Amplificationmentioning

confidence: 99%

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Modal and nonmodal stability of a stably stratified boundary layer flow

et al. 2020

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Section: Long-time Energy Amplificationsupporting

confidence: 78%

Section: Long-time Energy Amplificationmentioning

confidence: 99%

Modal and nonmodal stability of a stably stratified boundary layer flow

et al. 2020

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“…1976; Jose et al. 2015). In this study, we will not consider such continuous spectrum solutions, for either the primary modes at frequency or the superharmonic modes at frequency .…”

Section: Resultsmentioning

confidence: 99%

Triadic resonances in internal wave modes with background shear

2021

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In this paper, we use asymptotic theory and numerical methods to study resonant triad interactions among discrete internal wave modes at a fixed frequency ( $\omega$ ) in a two-dimensional, uniformly stratified shear flow. Motivated by linear internal wave generation mechanisms in the ocean, we assume the primary wave field as a linear superposition of various horizontally propagating vertical modes at a fixed frequency $\omega$ . The weakly nonlinear solution associated with the primary wave field is shown to comprise superharmonic (frequency $2\omega$ ) and zero frequency wave fields, with the focus of this study being on the former. When two interacting primary modes $m$ and $n$ are in triadic resonance with a superharmonic mode $q$ , it results in the divergence of the corresponding superharmonic secondary wave amplitude. For a given modal interaction $(m, n)$ , the superharmonic wave amplitude is plotted on the plane of primary wave frequency $\omega$ and Richardson number $Ri$ , and the locus of divergence locations shows how the resonance locations are influenced by background shear. In the limit of weak background shear ( $Ri\to \infty$ ), using an asymptotic theory, we show that the horizontal wavenumber condition $k_m + k_n = k_q$ is sufficient for triadic resonance, in contrast to the requirement of an additional vertical mode number condition ( $q = |m-n|$ ) in the case of no shear. As a result, the number of resonances for an arbitrarily weak shear is significantly larger than that for no shear. The new resonances that occur in the presence of shear include the possibilities of resonance due to self-interaction and resonances that occur at the seemingly trivial limit of $\omega \approx 0$ , both of which are not possible in the no shear limit. Our weak shear limit conclusions are relevant for other inhomogeneities such as non-uniformity in stratification as well, thus providing an understanding of several recent studies that have highlighted superharmonic generation in non-uniform stratifications. Extending our study to finite shear (finite $Ri$ ) in an ocean-like exponential shear flow profile, we show that for cograde–cograde interactions, a significant number of divergence curves that start at $Ri\to \infty$ will not extend below a cutoff $Ri$ $\sim O(1)$ . In contrast, for retrograde–retrograde interactions, the divergence curves extend all the way from $Ri\to \infty$ to $Ri = 0.5$ . For mixed interactions, new divergence curves appear at $\omega = 0$ for $Ri\sim O(10)$ and extend to other primary wave frequencies for smaller $Ri$ . Consequently, the total ( $\text {cograde} + \text {retrograde} + \text {mixed}$ ) number of resonant triads is of the same order for small $Ri\approx 0.5$ as in the limit of weak shear ( $Ri\to \infty$ ), although it attains a maximum at $Ri\sim O(10)$ .

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“…This analysis framework has the advantage of computational tractability and is not subject to finite channel effects. Related analysis has shown promise in studying stratified flows, including inviscid stratified shear flow with constant shear (Farrell & Ioannou 1993b), stratified PCF (Jose et al 2015(Jose et al , 2018 and stratified turbulent channel flow (Ahmed et al 2021).…”

Section: Introductionmentioning

confidence: 99%

Structured input–output analysis of stably stratified plane Couette flow

2022

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We employ a recently introduced structured input–output analysis (SIOA) approach to analyse streamwise and spanwise wavelengths of flow structures in stably stratified plane Couette flow. In the low-Reynolds-number ( $Re$ ) low-bulk Richardson number ( $Ri_b$ ) spatially intermittent regime, we demonstrate that SIOA predicts high amplification associated with wavelengths corresponding to the characteristic oblique turbulent bands in this regime. SIOA also identifies quasi-horizontal flow structures resembling the turbulent–laminar layers commonly observed in the high- $Re$ high- $Ri_b$ intermittent regime. An SIOA across a range of $Ri_b$ and $Re$ values suggests that the classical Miles–Howard stability criterion ( $Ri_b\leq 1/4$ ) is associated with a change in the most amplified flow structures when the Prandtl number is close to one ( $Pr\approx 1$ ). However, for $Pr\ll 1$ , the most amplified flow structures are determined by the product $PrRi_b$ . For $Pr\gg 1$ , SIOA identifies another quasi-horizontal flow structure that we show is principally associated with density perturbations. We further demonstrate the dominance of this density-associated flow structure in the high $Pr$ limit by constructing analytical scaling arguments for the amplification in terms of $Re$ and $Pr$ under the assumptions of unstratified flow (with $Ri_b=0$ ) and streamwise invariance.

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Analytical solutions for algebraic growth of disturbances in a stably stratified shear flow

Cited by 8 publications

References 31 publications

Modal and nonmodal stability of a stably stratified boundary layer flow

Modal and nonmodal stability of a stably stratified boundary layer flow

Triadic resonances in internal wave modes with background shear

Structured input–output analysis of stably stratified plane Couette flow

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