The Mathematical Underpinnings of Stratified Turbulence

Studying at Cambridge

Projects

# Stratified shear flow in an inclined duct

This project examines the flow fields and mixing properties of a stratified shear flow generated in an inclined duct. The duct connects two reservoirs of fluid with different densities, and contains a counterflow with a dense layer flowing beneath a less-dense layer moving in the opposite direction. So far we have studied two systems: a large stratified inclined duct (SID) measuring 3m long and 10cm by 10cm in cross section, and a smaller duct (miniSID) measuring 1.35m long and 2.25cm by 2.25cm in cross section. The large scale SID experiments were mainly concerned with flow visualisation, while miniSID has been used to obtain detailed measurements using PIV and PLIF to obtain the velocity and density fields, respectively.

## Stratified Inclined Duct (SID)

We identify four flow states in this experiment, depending on the fractional density differences, characterised by the dimensionless Atwood number, and the angle of inclination $\theta$, which is defined to be positive (negative) when the along-duct component of gravity reinforces (opposes) the buoyancy-induced pressure differences across the ends of the duct. For sufficiently negative angles and small fractional density differences the flow is observed to be laminar ($\mathsf{L}$ state) with an undisturbed density interface separating the two layers. For positive angles and/or high fractional density differences three other states are observed. For small angles of inclination the flow is wave-dominated and exhibits Holmboe modes ($\mathsf{H}$ state) on the interface with characteristic cusp-like wave breaking. At the highest positive angles and density differences there is a turbulent ($\mathsf{T}$ state) high-dissipation interfacial region typically containing Kelvin-Helmholtz (KH)-like structures sheared in the direction of the mean shear and connecting both layers. For intermediate angles and density differences an intermittent state ($\mathsf{I}$ state) is found, which exhibits a rich range of spatio-temporal behaviour and an interfacial region that contains features of KH-like structures and of the other two lower-dissipation states: thin interfaces and Holmboe-like structures. We map the state diagram of these flows in the Atwood number -- $\theta$ plane and examine the force balances that determine each of these states. We find that the $\mathsf{L}$ and $\mathsf{H}$ states are hydraulically controlled at the ends of the duct and the flow is determined by the pressure difference associated with the density difference between the reservoirs. As the inclination increases, the along-slope component of the buoyancy force becomes more significant and the $\mathsf{I}$ and $\mathsf{T}$ states are associated with increasing dissipation within the duct. We replot the state-space in the Grashof number -- $\theta$ phase plane and find the transition to the $\mathsf{T}$-state is governed by a critical Grashof number. We find that the corresponding buoyancy Reynolds number of the transition to the $\mathsf{T}$-state is of order 100, and that this state is also found to be hydraulically controlled at the ends of the duct. In this state the dissipation balances the force associated with the along-slope component of buoyancy and the counterflow has a critical composite Froude number.

## Mini Stratified Inclined Duct (mSID)

We have carried out matched pairs of experiments in which we separately measure the velocity and density fields, over the range of states observed in SID. Examples can be seen in the movies. The analysis of these data is on going.

# Stratified Plane Couette (SPC)

Although turbulence is generally suppressed in very statically stable conditions, intermittent bursts of turbulence are still seen when the Reynolds number is sufficiently large. We study stratified turbulence in plane Couette flow using direct numerical simulations, focusing on the complexity arising from the spatio-temporal intermittency of the flow as the stabilizing stratification increases. Two external dimensionless parameters control the dynamics: the Reynolds number $Re$ and the bulk Richardson number $Ri_b$. We trace the boundary between laminar and turbulent states in the $Re$ , $Ri_b$ plane and discuss the relevant dynamical quantities involved in the relaminarization process. The aim is on analyzing the structures populating the intermittent regime and the coexistence between laminar and turbulent patches, focusing on similarities and differences between small-$Re$ small $Ri_b$ and large $Re$ large $Ri_b$ intermittent dynamics. We also aim at investigating the applicability and breakdown of existing stratified turbulence theories, including the Monin-Obukhov self-similarity theory and stratified variation of TKE models.

## Coherent structures in stratified plane Couette flows

Wall-bounded shear flows typically follow a subcritical transition scenario where finite amplitude solutions unconnected to the basic flow play a key role. Edge tracking has been very useful in finding some of these by following the laminar-turbulent boundary in phase space for many canonical shear flows. However, it has yet to be used to probe stratified flows. In this project we investigate the results of edge tracking in stably-stratified plane Couette flow over large and small domains (click here for a movie).

## Minimal seeds in stratified Plane Couette flow

The stability properties of shear flows have received wide attention due to the important engineering applications of understanding how and when turbulence might emerge in a given flow geometry. Research has recently focused on identifying "minimal seeds," i.e. the initial perturbations to a laminar state with the smallest initial perturbation energy E0 = Ec that ultimately trigger the transition to turbulence. In unstratified plane Couette flow, Rabin, Caulfield & Kerswell (J. Fluid Mech. 2012 712) identified both such a minimal seed, and other "nonlinear optimal perturbations"(NLOPS) with E0 < Ec which maximised the gain in kinetic energy over some finite time while the flow still remained laminar. We use the same variational method of "nonlinear adjoint looping" to identify NLOPS and minimal seeds in stably stratified plane Couette flow, where a constant (stabilising) density difference is maintained across the flow. We also identify the mechanisms through which such perturbations may transiently gain both kinetic and potential energy as the bulk Richardson number is varied, identifying how stratification changes the qualitative characteristics of the optimal perturbations.

For the linear problem in which transient growth of perturbations can occur due to the nonnormality of the governing time evolution operator, we find that the possible energy gain of perturbations is rapidly reduced as the bulk Richardson number is increased. There is also a transition in the form of perturbation that is optimal for energy growth from two-dimensional, streamwise aligned vortices that primarily use the lift-up mechanism for energy growth, to three-dimensional oblique structures that primarily use the Orr mechanism for energy growth. We also find that the possibility of non-trivial transient energy growth remains for bulk Richardson numbers exceeding 0.25.

## Multiple instabilities in layered stratified plane Couette flow

We consider the linear stability and nonlinear evolution of a Boussinesq fluid consisting of three layers with density $\rho_a + \Delta \rho/2$, $\rho_a$ and $\rho_a - \Delta \rho/2$ of equal depth $d/3$ in a 2D channel where the horizontal boundaries are driven at a constant relative velocity $\Delta U$. Unlike unstratified flow, we demonstrate that for all $Ri_b = g \Delta \rho d /(\rho_a \Delta U)^2$, and for sufficiently large $Re= \Delta U d/(4 \nu)$, this flow is linearly unstable to normal mode disturbances of the form first considered by Taylor (1931). These instabilities, associated with a coupling between Doppler-shifted internal waves on the density interfaces, have a growth rate (maximised across wavenumber and $Ri_b$) which is a non-monotonic function of $Re$. Through 2D simulation, we explore the nonlinear evolution of these primary instabilities at various $Re$, demonstrating that the primary instabilities grow to finite amplitude as vortices in the intermediate fluid layer before rapidly breaking down, modifying the mean flow to become susceptible to strong and long-lived secondary instabilities of Holmboe (1962) type, associated with vortices now localised in the top and bottom layers.

## Turbulence and mixing from optimal perturbations to a stratified shear layer

The stability and mixing of stratified shear layers is a canonical problem in fluid dynamics with relevance to flows in the ocean and atmosphere. The Miles-Howard theorem states that a necessary condition for normal-mode instability in parallel, inviscid, steady stratified shear flows is that the gradient Richardson number, $Ri_g$ is less than 1/4 somewhere in the flow. However, substantial transient growth of non-normal modes may be possible at finite times even when $Ri_g>1/4$ everywhere in the flow. We have calculated the `optimal perturbations ' associated with maximum perturbation energy gain for a stably-stratified shear layer. These optimal perturbations are then used to initialize direct numerical simulations. For small but finite perturbation amplitudes, the optimal perturbations grow at the predicted linear rate initially, but then experience sufficient transient growth to become nonlinear and susceptible to secondary instabilities, which then break down into turbulence. Remarkably, this occurs even in flows for which $Ri_g>1/4$ everywhere. We will describe the nonlinear evolution of the optimal perturbations and characterize the resulting turbulence and mixing.

## External and internal mixing in stratified plane Couette flows

We perform direct numerical simulations (DNS) of stratified plane Couette (SPC) flow at a wide range of Reynolds, Richardson and Prandtl numbers The primary aim is to investigate the turbulent mixing characteristics in this canonical flow set-up; in particular, by varying $Pr$ from 0.7, 7 and 70, we examine the effects of $Pr$ which have not been examined extensively in the literature. The fluid between the oppositely moving walls is either continuous stratified or initialized with a two-layer density profile, which enables us to investigate the external (boundary-driven) and internal (local-shear-driven) mixing scenarios as described by Turner (1973).

# Stratified Taylor-Couette (STC)

## Mitigating end-effects with stratification in quasi-Keplerian Taylor–Couette flow

Efforts to model accretion disks in the laboratory using Taylor–Couette flow are plagued with problems due to the substantial impact the end-plates have on the flow (Burin et al. 2006; Paoletti & Lathrop, 2011; Balbus, 2011). We explored the possibility of mitigating the influence of these end-plates (meridional circulation causing transition to turbulence; Avila et al., 2008; Avila, 2012) by imposing linearly stratified layers in their vicinity. We performed numerical computations of axisymmetric base flows, with end-plates, and checked the linear stability of steady solutions. We successfully identified a sweet-spot in parameter space for suppressing end-effects without triggering additional instabilities. These results were confirmed experimentally and no subcritical transition to turbulence was observed in the unstratified core of the flow, up to $Re = O(10^4)$.

## The connection between centrifugal and stratorotational instabilities in viscous Taylor–Couette flow

In 2001, Molemaker et al. (2001) and Yavneh et al. (2001) showed that axially linearly stratified Taylor–Couette flow was prone to non-axisymmetric instabilities beyond the classical limit of centrifugal instability (CI): Ωo/Ωi < (ri/ro)2 (Ω: angular velocity; r: radius; subscript i/o for inner/outer cylinder), known as the Rayleigh criterion. This new instability was later called ‘stratorotational’ (SRI; Dubrulle et al. 2005). We investigated the competition between the two instability mechanisms, CI and SRI, when they coexist. This work has been done in collaboration with an intern from ENS Paris (Master 1), Florian Nguyen.

## Dynamical layering and mass transport in stratified Taylor–Couette flow

Direct numerical simulations (using code from Shi et al. 2015 and Lopez et al. 2013) of axially linearly stratified Taylor–Couette flow are carried out in order to elucidate the mechanisms responsible for mixing in this configuration, with fixed outer cylinder. Key questions are the mechanisms a) setting up the series of well-mixed layers and sharp interfaces observed experimentally and b) driving the buoyancy flux measured by Oglethorpe et al. (2013).

# Shear orientation in stably stratified Kolmogorov flow

Turbulence driven by a large scale shear in periodic geometries is considered to investigate the effect of shear orientation on the properties of the turbulence and its transition. Stable stratification is known to suppress vertical motion and damp turbulent fluctuations more when a uniform imposed mean shear is vertical rather than horizontal (Jacobitz & Sarkar 1998). In the body-forced scenario we discover that this stabilising mechanism gives rise to more highly energetic mean flow due to energy injection across the domain. A surprising result is thus that vertically sheared body-forced stratified turbulence is typically more vigorous than it's horizontal counterpart, with other inclination angles intermediate in energetics. Further in the horizontally sheared case we observe a new coherent state arising, seemingly subcritical and disconnected from the base flow. This state exhibits streamwise independence and structure in the vertical which is observed to organise the turbulence at higher Reynolds numbers via the introduction of vertical shear. A generic feature observed is the recurrent bursting of turbulence when the stratification is large, again due to stratification stabilising the shear in isolated layers until such energy is accumulated in the mean to overcome a local gradient Richardson number criterion. We introduce a throttling method to maintain a mean dissipation rate by modulating the forcing strength to enable a comparison between shear directions when the energetics are similarly statistically stationary. Movies here show total density and streamwise velocity and show the spontaneous appearance of layers via the inclination of the background shear.

This true "zig-zag" pattern is found as the nonlinear saturated state arising from a sequence of instabilities of the base flow. We converge this unstable steady state using a Newton-GMRES-hookstep algorithm from a guess extracted directly from the chaotic simulation. This is state (C) in the figure below. This solution can be traced in parameter space to construct a bifurcation diagram and we find that the state originates in two separate instabilities, the first of which is a new stratified instability of the base flow, giving rise to state (A).