Streak generation in viscosity-stratified wall-bounded flows

TL;DR

This paper investigates how viscosity stratification affects streak generation in wall-bounded flows, revealing a new mechanism and explaining flow behavior differences in Poiseuille and Couette flows through theoretical and numerical analysis.

Contribution

It introduces a novel viscosity-stratified lift-up mechanism and analyzes its interaction with the classical lift-up effect in wall-bounded flows.

Findings

Viscosity stratification significantly alters the classical lift-up mechanism.

A new pathway for streak generation is identified.

Flow trends in Poiseuille and Couette flows are explained by the interaction of mechanisms.

Abstract

Streamwise elongated flow structures, or streaks, dominate wall-bounded flows. We show that the renowned lift-up mechanism, that generates streaks, is significantly altered by viscosity stratification. We additionally identify a novel pathway for streak generation: the viscosity-stratified lift-up mechanism. The competition and cooperation of the streaks generated by these mechanisms provides an explanation for trends in viscosity-stratified Poiseuille flows, and remarkably, the opposing trends in Couette flows as well. Our theoretical observations are substantiated by numerical experiments.

Equations

$$\begin{array}{rl}\frac{1}{Re}{\mathbf{L}}_{OS}\hat{w}& =-i{k}_y\frac{\partial }{\partial z}{\hat{f}}_y-k^2{\hat{f}}_z,\\ \frac{1}{Re}{\mathbf{L}}_{SQ}\hat{u}& =-U^{\prime }\hat{w}+\frac{1}{Re}{\mathbf{M}}_{u\theta }\hat{\theta }+{\hat{f}}_x,\\ \frac{1}{RePr}{\mathbf{L}}_{\Theta }\hat{\theta }& =-{\Theta }^{\prime }\hat{w}+{\hat{f}}_{\theta }.\end{array}$$

$$\begin{array}{rl}{\mathbf{L}}_{OS}& =i\omega \Delta -\overline{\mu }{\Delta }^2-2{\overline{\mu }}^{\prime }\mathcal{D}\Delta +{\overline{\mu }}^{\prime \prime }\left(\Delta -2{\mathcal{D}}^2\right),\\ {\mathbf{L}}_{SQ}& =i\omega -\overline{\mu }\Delta -{\overline{\mu }}^{\prime }\mathcal{D},\\ {\mathbf{M}}_{u\theta }& =\overline{\epsilon }\mathcal{D}+{\overline{\epsilon }}^{\prime }\text{mbox}and{\mathbf{L}}_{\Theta }=i\omega Pr-\Delta ,\end{array}$$

$$\begin{array}{rl}\hat{u}=& R{e}^2{\mathbf{R}}_1(z;k_y,\omega )\left[\begin{array}{c}{f}_y\\ f_z\end{array}\right]+R{e}^2{\mathbf{R}}_2(z;k_y,\omega )\left[\begin{array}{c}{f}_y\\ f_z\end{array}\right]+\\ & Re{\mathbf{R}}_3(z;k_y,\omega )\left[\begin{array}{c}{f}_x\end{array}\right]+Re{\mathbf{R}}_4(z;k_y,\omega )\left[\begin{array}{c}{f}_{\theta }\end{array}\right],\end{array}$$

$$\begin{array}{rl}{\mathbf{R}}_1& =-{\mathbf{L}}_{SQ}^{-1}{U}^{\prime }{\mathbf{L}}_{OS}^{-1}[-ik\mathcal{D},-k^2],\\ {\mathbf{R}}_2& =-Pr{\mathbf{L}}_{SQ}^{-1}{\mathbf{M}}_{u\theta }{\mathbf{L}}_{\Theta }^{-1}{\Theta }^{\prime }{\mathbf{L}}_{OS}^{-1}[-ik\mathcal{D},-k^2],\\ {\mathbf{R}}_3& =-{\mathbf{L}}_{SQ}^{-1},\text{mbox}and{\mathbf{R}}_4=-Pr{\mathbf{L}}_{SQ}^{-1}{\mathbf{M}}_{u\theta }{\mathbf{L}}_{\Theta }^{-1}.\end{array}$$

$$\mu (\theta )=\exp \left(A\left(\frac{B^{\ast }}{({\Theta }_r^{\ast }-C^{\ast })}\right)\frac{(1-\theta )}{(\theta -C^{\ast }/{\Theta }_r^{\ast })}\right),$$

$$\begin{array}{r}||({\mathbf{R}}_1+{\mathbf{R}}_2)|{|}_{\infty }(k_y)=\underset{\omega }{\max }\text{mbox}{\sigma }_1(k_y,\omega ).\end{array}$$

$${\psi }_{R_1}={\mathbf{R}}_1{\mathbf{\varphi }}_m=\sum _{i=1}^N{\psi }_i^S\text{mbox}{\sigma }_i^S\text{mbox}\underset{p_i}{\underbrace{({\varphi }_i^S\cdot U^{\prime }\mathbf{G}{\mathbf{\varphi }}_m)}}.$$

$$\begin{array}{rl}\frac{\partial u_i}{\partial t}& =-U_j\frac{\partial u_i}{\partial x_j}-u_j\frac{\partial U_i}{\partial x_j}-\frac{\partial p}{\partial x_i}+\\ & \frac{1}{Re}\frac{\partial }{\partial x_j}(\mu (\Theta )\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)\\ & +\frac{\partial \mu (\Theta )}{\partial \Theta }\left(\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i}\right)\theta )+f_i,\frac{\partial u_i}{\partial x_i}=0,\\ \frac{\partial \theta }{\partial t}& =-U_j\frac{\partial \theta }{\partial x_j}-u_j\frac{\partial \Theta }{\partial x_j}+\frac{1}{PrRe}\frac{{\partial }^2\theta }{\partial x_j\partial x_j}+f_{\theta }.\end{array}$$

$$\mu (\theta )={\theta }^{3/2}\frac{(1+C)}{(\theta +C)},$$

Full text

††preprint: APS/123-QED

Streaks, i.e., elongated flow structures, are characteristic of wall-bounded flows that are turbulent or transitioning to turbulence [1, 2]. These streaks play crucial roles in the statistics and dynamics of wall-bounded flows (see review [2]), and controlling these flows often involves controlling the streaks. Understanding these streaks is therefore crucial for a range of situations: from applications like drag reduction in wall-bounded flows [e.g. 3, 4] to atmospheric flows [e.g. 5] and hurricanes [e.g. 6].

In constant viscosity flows, the generation of these streaks is well characterized by a classical mechanism – the lift-up mechanism – where vortices in the flow extract energy from the mean shear to generate velocity streaks [7, 8]. However, in many practically relevant flows, viscosity stratification, i.e., a variation in the viscosity of the fluid with temperature or scalar concentration, is important: flows such as the flow in the Earth’s core [e.g. 9], crude oil flow in pipelines [e.g. 10] and polymer-laden flows [e.g. 11]. Fully compressible flows, like flows over reentry vehicles [e.g. 12], are also impacted by viscosity stratification. Importantly, in viscosity stratified flows, we find that the classical mechanism does not sufficiently explain the streaks. Here we present an entirely new mechanism and an altered lift-up mechanism that are now in operation. The interaction of these mechanisms leads to a fundamental revision of our understanding of streak generation in these flows, and also produces some interesting effects, including some counterintuitive ones.

To illustrate these mechanisms, let us consider incompressible channel flows, i.e. flows between two parallel walls. The bottom cold wall is at a temperature $\Theta_{C}^{*}$ and the top hot wall is at temperature $\Theta_{H}^{*}$, with temperature difference $\Delta\Theta^{*}=\Theta_{H}^{*}-\Theta_{C}^{*}$ (here $(\cdot)^{*}$ denotes dimensional quantities). We consider the Poiseuille flow, driven by a pressure gradient, and the Couette flow, driven by the velocity of the wall. Here we consider the case where the viscosity of the fluid $\mu(\theta)$ varies with temperature $\theta$. Note, the mechanisms described here will also impact flows where the viscosity is a function of scalar concentration. Mathematical models have shown that viscosity stratification alters wall-bounded flows [e.g. 13, 14, 15, 16, 17]. Direct numerical simulations (DNSs) have also demonstrated that viscosity stratification modifies turbulence [e.g. 18, 19]. For instance, while intuitively, higher viscosity would be expected to suppress turbulence, DNS studies have shown an enhancement of turbulence in the more-viscous regions of some flows, and a relative suppression in the less-viscous regions [e.g. 19, 18]. The mechanisms presented here explain this surprising trend.

The streamwise, spanwise and wall-normal directions of the channel are $x$, $y$ and $z$ and the corresponding velocity components are $u$, $v$ and $w$. The mean velocity $(U(z),0,0)$ and the mean temperature $\Theta(z)$ are obtained by averaging in $x$, $y$ and time. The Reynolds number is $Re\equiv\rho^{*}_{r}U_{b}^{*}h^{*}/\mu_{r}^{*}$, and the Prandtl number is $Pr\equiv\mu_{r}^{*}c_{p,r}^{*}/\lambda_{r}^{*}$. Here $U_{b}$ is the bulk velocity, $h$ is the channel half height, $\rho$ is density, $c_{p}$ is the specific heat at constant pressure and $\lambda$ is the thermal conductivity.

Streaks can be analyzed using linearized Navier–Stokes equation-based models [20, 21]. To obtain the model, we linearize the non-dimensional Navier–Stokes equations around the mean flow. For a majority of the discussion, we linearize around the laminar mean flow, except in figure 6 where we show that the conclusions drawn in this study are also more generally valid when the linearization is done around the turbulent mean flow. We Fourier transform the state variables in $x$, $y$ and time. Here $\widehat{\cdot}$ represents the Fourier transform. The streamwise and spanwise wavenumbers are ($k_{x},k_{y}$) and the temporal frequency is $\omega$ ($\omega$ is multiplied by $Re$ for ease of notation). Since streaks are elongated structures, we consider the response from streamwise-constant ($k_{x}=0$) structures. This streamwise-invariant model then becomes:

[TABLE]

Here $(f_{x},f_{y},f_{z})$ and $f_{\theta}$ represents any external forcing, and, for turbulent flows, the nonlinear terms of the equations. The boundary conditions imposed are $\widehat{w}(\pm 1)=\partial\widehat{w}/\partial z(\pm 1)=(ik_{y}\widehat{u}-ik_{x}\widehat{v})(\pm 1)=\widehat{\theta}(\pm 1)=0$. The linear operators in (1) are:

[TABLE]

where ′ and $\mathcal{D}$ represent derivatives in $z$, $\Delta=\nabla^{2}$ and $\overline{\varepsilon}=U^{\prime}(d\overline{\mu}/d\Theta)$. Figure 1 shows (1) as a block diagram.

To understand streak generation, we study how $\widehat{u}$ is generated by the inputs $\widehat{f}_{x}$, $\widehat{f}_{y}$, $\widehat{f}_{z}$ and $\widehat{f}_{\theta}$. Tracing out the different pathways in figure 1, $\widehat{u}$ is obtained as:

[TABLE]

where the linear operators are given by

[TABLE]

From (3), the contributions from $\bm{R}_{3}\widehat{f}_{x}$ and $\bm{R}_{4}\widehat{f}_{\theta}$ scale with $Re$, whereas the contributions from $(\widehat{f}_{y},\widehat{f}_{z})$, via $\bm{R}_{1}$ and $\bm{R}_{2}$, scale with $Re^{2}$. At moderate and high Reynolds numbers, the contributions of $\bm{R}_{3}$ and $\bm{R}_{4}$ can therefore be ignored. $\bm{R}_{1}$ corresponds to streak formation by the classical lift-up mechanism, albeit modulated by viscosity stratification. $\bm{R}_{2}$ corresponds to a completely new viscosity-stratified lift-up mechanism driven by the temperature gradient. Unlike in the case of constant viscosity flows, in viscosity stratified flows, the streaks generated by $\bm{R}_{1}$ and $\bm{R}_{2}$ interact. Crucially, the contribution of $\bm{R}_{2}$ to streak amplification increases with $Pr$, and therefore, at high enough $Pr$, it is likely that this novel viscosity-stratified lift-up becomes the dominant mechanism.

To illustrate how $\bm{R}_{1}$ and $\bm{R}_{2}$ operate, we will now consider a specific function for viscosity $\mu(\theta)$ as

[TABLE]

where $A=\log_{e}(10)$, $B^{*}=247.8K$, $C^{*}=140K$ and the reference temperature $\Theta_{r}^{*}=(\Theta_{H}^{*}+\Theta_{C}^{*})/2=323K$ [19]. The viscosity is here non-dimensionalized by $\mu_{r}$, where a subscript $(\cdot)_{r}$ denotes reference quantities defined at $\Theta_{r}^{*}$. This exponential law is an empirical fit frequently employed for liquids like water [19, 18]. The streak amplification mechanisms discussed here remain largely unaffected if the viscosity instead varies according to Sutherland’s law, commonly used for fluids like air.

$\bm{R}_{1}$, $\bm{R}_{2}$ and $(\bm{R}_{1}+\bm{R}_{2})$ are linear operators that take an input forcing $\widehat{\bm{f}}=(\widehat{f}_{y},\widehat{f}_{z})$ and give a response $\widehat{u}=(\bm{R}_{1}+\bm{R}_{2})\widehat{\bm{f}}$. To understand these operators, we can ask what the most sensitive forcing is, i.e. what is the $\widehat{\bm{f}}$ that gives the $\widehat{u}$ with maximum energy? For this we use singular value decomposition (SVD), which gives $(\bm{R}_{1}+\bm{R}_{2})=\sum_{i=0}^{N}\psi_{i}\sigma_{i}\bm{\phi}_{i}$, with $\sigma_{i}\geq\sigma_{i+1}$ and unit norm vectors $\bm{\phi}_{i}$ and $\psi_{i}$. In effect, for $i=1$, we have forcing $\widehat{\bm{f}}=\bm{\phi}_{1}$ that gives the maximum response $\widehat{u}=\sigma_{1}\psi_{1}=(\bm{R}_{1}+\bm{R}_{2})\bm{\phi}_{1}$.

Given these definitions at a temporal frequency $\omega$, we can now ask what the maximum possible response is across frequencies. For this, we use the infinity norm:

[TABLE]

Let the $\omega$ for the maximum response be $\omega_{m}$, the corresponding forcing $\bm{\phi}_{m}$ and the response $\psi_{m}$. Therefore, $\bm{\phi}_{m}$ is the optimum forcing $(\widehat{f}_{y},\widehat{f}_{z})$ that, across all frequencies, generates the maximum response $\widehat{u}=\sigma_{m}\psi_{m}$.

Now, we compute the separate contributions of $\bm{R}_{1}$ and $\bm{R}_{2}$ at $\omega_{m}$ as $\psi_{R_{1}}=\bm{R}_{1}\bm{\phi}_{m}$ and $\psi_{R_{2}}=\bm{R}_{2}\bm{\phi}_{m}$. We also consider the distinct contributions from the more-viscous and less-viscous channel halves. The energy of these responses numerically obtained on $N=101$ wall-normal Chebyshev grid points is considered.

Route R1: Classical lift-up mechanism – We now examine how the well-known lift-up mechanism in unstratified flows is modified by viscosity stratification. In lift-up, the wall-normal velocity ($\widehat{w}$) leverages the mean shear ($U^{\prime}$) to generate streamwise velocity streaks [e.g. 7, 8]. From figure 1 and (4), this corresponds to the transfer function $\bm{R}_{1}$. Viscosity stratification modifies $\bm{R}_{1}$ in two ways: through (i) additional terms in the linear operators due to viscosity and its gradients, and (ii) the modification of the mean shear. Figure 2 shows the contribution of $\bm{R}_{1}$ as a function of the spanwise wavenumber $k_{y}$.

When considering the channel halves separately, for Poiseuille flows in figure 22, viscosity stratification enhances the lift-up mechanism in the more-viscous channel half and diminishes it in the less-viscous half. Counterintuitively, therefore, higher viscosity is enhancing streaks rather than suppressing them. This observation is consistent with the DNS of turbulent viscosity-stratified Poiseuille flows of Zonta et al. [19]. Remarkably, this trend is absent for the Couette flow in figure 22, where streaks are more energetic in the less-viscous half. Additionally, for the Poiseuille flow, there is a mismatch between the optimal wavenumbers in the two channel halves, which is absent in the Couette flow.

The question that follows is what causes this contrasting trend in Poiseuille and Couette flows? To highlight the effect of shear in $\bm{R}_{1}$, let us write $\bm{R}_{1}=-\bm{S}\cdot U^{\prime}\cdot\bm{G}$. The operators $\bm{S}$ and $\bm{G}$ follow from (4) and are identical for Poiseuille and Couette flows. Using SVD $\bm{S}=(\sum_{i=1}^{N}\psi^{S}_{i}\sigma^{S}_{i}\phi^{S}_{i}$), and the response to $\bm{R}_{1}$ becomes:

[TABLE]

The contribution of a mode $i$ in the summation is determined by $p_{i}$ that is the projection of $U^{\prime}\bm{G}\bm{\phi}_{m}$ onto the $\phi_{i}^{S}$. Since $\sigma^{S}_{1}>\sigma^{S}_{i},i=2,3,....N$, the lift-up represented by $\bm{R}_{1}$ is most efficient when $p_{1}$ is the maximum.

Let us first consider unstratified flows as in Illingworth [22]. The wall-normal profile of $\phi_{1}^{S}$ is symmetric across $z=0$, and $\phi_{2}^{S}$ is antisymmetric (figures 3(a)). $\bm{G}\bm{\phi}_{m}$ is symmetric (figures 3(b)). These profiles are identical for Poiseuille and Couette flows. In Couette flows, $U^{\prime}$ is symmetric, making $U^{\prime}\bm{G}\bm{\phi}_{m}$ symmetric. The symmetric $U^{\prime}\bm{G}\bm{\phi}_{m}$ projects well onto the symmetric $\phi_{1}^{S}$, thereby producing a large $p_{1}$. Lift-up is therefore efficient in Couette flows [22]. In Poiseuille flows however, $U^{\prime}$ is antisymmetric, and therefore so is $U^{\prime}\bm{G}\bm{\phi}_{m}$ (figures 3(b)). The antisymmetric $U^{\prime}\bm{G}\bm{\phi}_{m}$ does not project onto the symmetric $\phi_{1}^{S}$ and instead projects onto the antisymmetric $\phi_{2}^{S}$. The projection $p_{1}=0$, and the contribution of the first mode $p_{1}\sigma_{1}\psi_{1}=0$ (figures 3(c)). Lift-up in Poiseuille flows is therefore inefficient compared to Couette flows [22].

Now let us follow the same arguments for viscosity-stratified flows. Viscosity stratification introduces asymmetry. Therefore, in Poiseuille flows, $U^{\prime}\bm{G}\bm{\phi}_{m}$ is nearly antisymmetric, but not perfectly. The projection of $U^{\prime}\bm{G}\bm{\phi}_{m}$ onto the symmetric $\phi_{1}^{S}$, although small, is now not zero, i.e. $p_{1}>0$. Since $\sigma^{S}_{1}\gg\sigma^{S}_{2}$, this small $p_{1}$ creates a significant contribution from the first mode $p_{1}\sigma_{1}\psi_{1}$ (figures 3(c)). This now creates the scenario where the nearly-symmetric $p_{1}\sigma_{1}\psi_{1}$ and nearly anti-symmetric $p_{2}\sigma_{2}\psi_{2}$ interact, as illustrated in figure 3(c). We see that the two modes cooperate (have same sign) in the more-viscous bottom half, thereby enhancing streaks, and the modes compete (have opposite signs) in the less-viscous top half, thereby attenuating streaks. Therefore, the interaction of the symmetric and antisymmetric modes of $\bm{R}_{1}$ causes the enhancement of classical lift-up in the more-viscous region of Poiseuille flows. Since classical lift-up is already optimal for unstratified Couette flows, this enhancement of lift-up due to viscosity-stratification-induced asymmetry is not significant in these flows.

This comparison of Poiseuille and Couette flows therefore demonstrates that the classical lift-up mechanism is significantly modified by viscosity stratification. More generally, any asymmetry in the flow can strongly influence this key mechanism in shear flows. The observations also illustrate how the impact of this modification of classical lift-up varies across flow regimes.

Route R2: viscosity-stratified lift-up mechanism - Viscosity stratification also introduces an entirely new pathway for streak generation, separate from the classical lift-up mechanism. In this viscosity-stratified lift-up mechanism, the wall-normal velocity $(\widehat{w})$ leverages the mean temperature gradient $(\Theta^{\prime})$ to generate a temperature streak $(\widehat{\theta})$, or equivalently, a viscosity streak. This in turn generates a streamwise velocity streak. From figure 1 and (4), this corresponds to the transfer function $\bm{R}_{2}$. From (4), we note that, unlike $\bm{R}_{1}$, the contribution of $\bm{R}_{2}$ increases with $Pr$ and $\Delta\Theta^{*}$ ($\Theta^{\prime}$ increases with $\Delta\Theta^{*}$).

This prompts us to ask how the separate contributions from R1 and R2 interact to give the full response. Figure 4 shows the $\widehat{u}$ from R1, $\psi_{R_{1}}=\bm{R}_{1}\bm{\phi}_{m}$, and the $\widehat{u}$ from R2, $\psi_{R_{2}}=\bm{R}_{2}\bm{\phi}_{m}$, alongside the full response $\psi_{m}=(\psi_{R_{1}}+\psi_{R_{2}})=(\bm{R}_{1}+\bm{R}_{2})\bm{\phi}_{m}$. Crucially, for the Poiseuille flow in figure 44, the streaks from R1 and R2 cooperate (have same sign) at the more-viscous bottom channel half, thereby enhancing streaks, and they compete (have opposite signs) at the less-viscous top half, hence attenuating streaks. This interaction can cause streaks to be more energetic than in the unstratified case, with the major contribution to this energy coming from the more-viscous half of the channel, as seen in the trends of $||(\bm{R}_{1}+\bm{R}_{2})||_{\infty}^{2}$ in figure 59. The opposite trend is true for the Couette flow in figure 44, with streaks from R1 and R2 cooperating at the less-viscous region of the channel, and therefore the less-viscous half showing higher energy in figure 59. Additionally, since R1 is more efficient in Couette flows, in figure 44, at a given $Pr$ and $\Delta\Theta^{*}$, the contribution of R1 is higher than for the Poiseuille flow. The competition and cooperation of R1 and R2 can therefore produce different, sometimes counterintuitive, trends in viscosity-stratified shear flows.

From the model-based observations, we conclude that the asymmetry induced by viscosity stratification alters the classical lift-up mechanism for streak generation in shear flows. Additionally, an entirely new mechanism for streak generation, a viscosity stratified lift-up, gains dominance over the classical mechanism with increasing stratification. Illustrating the significance of this new and modified mechanisms, we found that the interaction of the streaks from the viscosity stratified lift-up with the altered classical lift-up produces contrasting trends in Poiseuille and Couette flows. In Poiseuille flows, the model predicts higher streak energy in the more-viscous regions; this trend is absent in Couette flows.

The final question we address here is, can this particular prediction from the model, i.e. the contrasting trends between Poiseuille and Couette flows, be demonstrated numerically? For this, we use the fully nonlinear minimal flow unit simulations that resolve the minimum set of flow structures that sustain turbulence [23]. First, we confirm if the trend obtained from the laminar flow-based model in figure 5 is valid for a turbulent flow-based model. Figure 6 shows the predictions from the model (8) that is now constructed using linearization around the turbulent mean profiles obtained from minimal flow unit simulations. Consistent with the laminar case, figure 6 shows that the model predicts streak enhancement at the more-viscous half of the Poiseuille flow; a trend that is not present for the Couette flow. At the low $Pr$ considered here, the asymmetry-induced modification of the classical lift-up mechanism drives this trend.

Now, to verify if this trend is present in the fully nonlinear flow, figure 77 shows the variance of streamwise velocity obtained from the simulation of a viscosity-stratified Poiseuille flow. In line with the theoretical observations here, and also consistent with the DNS of Zonta et al. [19], higher variance is obtained in the more-viscous half than in the less-viscous half. Turbulence is therefore enhanced in the more-viscous half. Now considering the Couette flow in figure 77, contrary to the Poiseuille flow, higher variance is observed in the less-viscous half, i.e. turbulence is enhanced in the less-viscous half. The numerical experiments therefore support the theoretical findings presented here. It is also worth noting that, in figure 4, the model predicts a stratification-induced overall shift of $\widehat{u}$ towards the less-viscous region of the channel (i.e. shift towards top wall). In DNS, this is observed as a shift in the wall-normal location at which the variance peaks – away from the wall in the more-viscous half and closer to the wall in the less-viscous half.

The flow regimes considered so far span low $Pr$ where the modified lift-up is active, or moderate $Pr$ where the interaction of the classical and viscosity-stratified lift-up dictates the trends. The model predicts that, at higher values of $Pr$, the well-known classical lift-up mechanism will cease to impact the flow, and viscosity stratified lift-up will dominate. In these regimes, the length and time scales of the turbulent flow structures will therefore likely change, and this will impact practical applications like estimation and control. Additionally, viscosity stratification is one of several factors distinguishing streaks in compressible from incompressible flows. Even considering this single factor, the observations here suggest that the streaks in compressible flows are mechanistically distinct from their incompressible counterparts, therefore contributing to the ongoing debate on the equivalence of the streaks in these two flow regimes [e.g. 24, 25, 26, 27, 28]. The observations here therefore highlight the importance of re-characterizing the physical mechanisms that generate the dominant features in flows where practically relevant factors like stratification are active.

Acknowledgments – AM and RG would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program ‘Mathematical aspects of turbulence: where do we stand?’ where part of the work on this paper was undertaken. This was partly supported by EPSRC grant number EP/R014604/1. AM gratefully acknowledges the financial support provided by the Department of Science and Technology (DST), Government of India, under the INSPIRE Faculty Fellowships, Grant number SP/DSTO-23-0192. RG thanks the Department of Atomic Energy, Government of India, for support under Project No. RTI4001.

END MATTER

Governing equations: For the linear model in (1), velocities are non-dimensionalized by the bulk velocity $U_{b}^{*}=0.5\int_{-1}^{1}U^{*}(z)dz$, temperature by $\Theta_{ref}^{*}=(\Theta_{H}^{*}+\Theta_{C}^{*})/2$, the spatial variables by the channel half-height $h^{*}$, and the viscosity by the viscosity at the reference temperature $\mu_{ref}^{*}$. The non-dimensional Navier–Stokes equations linearized around the mean flow are:

[TABLE]

Here $(x_{1},x_{2},x_{3})=(x,y,z)$ and $(u_{1},u_{2},u_{3})=(u,v,w)$ and $p$ is pressure. Viscosity is linearized as $\mu(\Theta+\theta)=\mu(\Theta)+(d\mu(\Theta)/d\Theta)\theta$, where $\mu(\Theta)$ is the mean viscosity and $(d\mu(\Theta)/d\Theta)\theta$ are the fluctuations of viscosity.

The laminar mean profiles that are provided as input to (8) are obtained by first assuming a mean temperature that varies linearly with $z$. To obtain the mean velocities for the Poiseuille flow, the mean momentum equations are solved assuming the mean imposed pressure gradient $d\overline{P}/dx$ to be constant for the unstratified and stratified flows. Similarly, the mean velocity profiles for the Couette flow are obtained by assuming a constant velocity for the top wall. The laminar mean profiles do not change with $Re$ or $Pr$ and are shown in figure 8. Fourier transforming in $x$, $y$ and time and considering the $k_{x}=0$ structures alone in (8) gives (1). The wavenumbers ($k_{x},k_{y}$) are non-dimesionalized by $1/h^{*}$.

Minimal channel simulations: The minimal channel simulations reported in this study are carried out using the open source partial differential equation solver Dedalus [29]. Fourier decompositions are used in the $x$ and $y$ directions and a Chebyshev decomposition is employed in the $z$ direction. Viscosity varies with temperature, as in (5). The simulations are performed at a constant friction Reynolds number $Re_{\tau}=\rho_{ref}^{*}u_{\tau}^{*}h^{*}/\mu_{ref}^{*}$, where $u_{\tau}^{*}=(\tau_{w}^{*}/\rho_{ref}^{*})$ is the friction velocity defined using the wall-shear stress $\tau_{w}^{*}$. In viscosity-stratified channel flows, the wall-shear stress at the hot and the cold walls are different. We therefore use $\tau_{w}^{*}=(|\tau_{w,C}^{*}|+|\tau_{w,H}^{*}|)/2$ [19]. The spatial variables are normalized by $h^{*}$ and the velocities by $u_{\tau}^{*}$. The Prandtl number is kept at $Pr=1$ for low computational cost. We consider $\Delta\Theta^{*}=40K$.

The full Navier-Stokes equations are solved [19]. For the Poiseuille flow, we impose a pressure gradient of $-1$ and use Dirichlet velocity boundary conditions at both the walls, and for the Couette flow, we set a constant streamwise velocity $U_{w}$ at the top wall, and use Dirichlet boundary conditions for all other velocity components at the walls. The temperature boundary conditions are $\Theta_{C}$ and $\Theta_{H}$ at the cold and hot walls, respectively. The time integration is performed using a 2nd-order semi-implicit BDF scheme (SBDF2 in Dedalus). A ‘+’ superscript will indicate the normalization of the spatial variables by the viscous length scale $\mu_{ref}^{*}/(\rho_{ref}^{*}u_{\tau}^{*})$. The statistics reported in figure 7 are obtained by averaging across $T^{+}=500$ time units in intervals of $dt^{+}=0.5$, obtained after transients.

The Poiseuille flow is considered at a fixed Reynolds number of $Re_{\tau}=186$. The streamwise, spanwise and wall-normal domain sizes are $L_{x}^{+}=350$, $L_{y}^{+}=350$ and $L_{z}^{+}=372$, respectively, with $N_{x}=32$, $N_{y}=64$ and $N_{z}=80$ grid points in each direction. This gives grid spacings of $\Delta x^{+}=10.9$, $\Delta y^{+}=5.4$ and $0.14\leq\Delta z^{+}\leq 7.3$ [30]. For the Couette flow, we take $Re_{\tau}=100$. The lower $Re_{\tau}$ for the Couette flow (in comparison to the Poiseuille flow) does not impact the discussions here, but makes the simulation, with the recommended grid spacings in the literature [31], more computationally manageable. The streamwise, spanwise and wall-normal domain sizes are $L_{x}^{+}=549$, $L_{y}^{+}=376$ and $L_{z}^{+}=200$, respectively, and we use $N_{x}=48$, $N_{y}=48$ and $N_{z}=80$ grid points in each direction. This gives grid spacings of $\Delta x^{+}=11.4$, $\Delta y^{+}=7.8$ and $0.077\leq\Delta z^{+}\leq 3.92$.

It should be emphasized that minimal flow unit simulations do not represent flows that can be realized in a laboratory. They instead serve as computationally tractable models that can be used to study some features of turbulent flows [23].

Route R2: Figure 9 shows the amplification of $\bm{R}_{2}$ as a function of $k_{y}$. We see that $\bm{R}_{2}$ for the Poiseuille flow in figure 9(a) shows higher amplification at the less-viscous wall, and therefore does not exhibit the counterintuitive trend of higher viscosity enhancing turbulence.

Comparison with Sutherland’s law of viscosity:

The numerical results in this study have been obtaining using the exponential variation of viscosity with temperature (5) that is an empirical relation frequently employed for liquids like water. Figures 10 and 11 show that the conclusions drawn are also valid when considering the Sutherland’s law of viscosity:

[TABLE]

where $C=110.4K/\Theta^{*}_{r}$. The key difference between the two functions for viscosity is that, while with the exponential law, the cold channel half is more-viscous than the hot half, for the Sutherland’s law the opposite trend holds, with the cold half being less-viscous than the hot half. Here we choose $\Delta\Theta^{*}=20K$ for the exponential law and $\Delta\Theta^{*}=160K$ for the Sutherland’s law, such that the ratio of the minimum to the maximum viscosity remains similar between the two cases.

Figures 10 shows $||\bm{R}_{1}+\bm{R}_{2}||_{\infty}$ for the Poiseuille flow at $Pr=3$ for the exponential law and the Sutherland’s law. We see that the trends are strikingly similar with the more-viscous channel half showing higher streak amplification. Now considering the streamwise variance obtained from minimal channel simulations in figure 11, we again see that the trends are similar, with the more-viscous half showing higher variance. This illustrates that the discussions here remain valid for the exponential law used for liquids, and the Sutherland’s law used for gases.