Plane Couette Flow Research Articles

Transition to turbulence in rough plane Couette flow

This DNS study considers transition to turbulence in plane Couette flow (pCf) with a rough stationary wall and a smooth moving wall. The roughness elements are square ribs of height $k=0.2h$ (where $h$ is the half-channel height). Two different pitch separations, $\lambda =2k$ and $10k$ , are considered, i.e. d-type and k-type roughness, respectively. The transition in both rough pCf cases takes place through a stage of alternate laminar–turbulent bands aligned in an oblique fashion. However, roughness causes a shift in the transitional Reynolds number ( $Re$ ) range. In the k-type roughness, stable bands are observed in the range $Re \in [300, 325]$ , which is a downwards shift from the transitional $Re$ range for the smooth pCf ( $Re \in [325,400]$ ). The d-type roughness, on the other hand, surprisingly shifts the transitional $Re$ range upwards to $Re \in [350,425]$ . This peculiar behaviour is associated with the ability of the ribs to shed and regenerate vorticity. Large-scale components extracted using a filtering process relate to the transition bands and flow parallel to the oblique laminar–turbulent boundaries. Counter-rotating vortices are present in the turbulent regions of the flow field, which exist in tandem with the high- and low-velocity streaks. Another interesting observation is the secondary Reynolds shear stresses, $-\overline {v^{\prime }u^{\prime }}$ and $-\overline {w^{\prime }v^{\prime }}$ , which are non-zero in the transitional regime, in contrast to the turbulent regime where they are negligible.

Turbulence Structure and Mixing in Strongly Stable Couette Flows over Thermally Heterogeneous Surfaces: Effect of Heterogeneity Orientation

AbstractDirect numerical simulations (DNS) of plane Couette flows over thermally heterogeneous surfaces at bulk Reynolds number $$Re=10^4$$ R e = 10 4 and bulk Richardson number $$Ri=0.25$$ R i = 0.25 are performed. The focus of the present study (that extends previous work by the authors) is the effect of surface heterogeneity orientation on boundary-layer structure. The temperature of the upper and lower walls is either homogeneous or varies sinusoidally, where the temperature-wave crests are either normal or parallel to the mean flow (HETx and HETy cases, respectively). Importantly, the horizontal-mean surface temperature is the same in all simulations. The stratification is strong enough to quench turbulence over a homogeneous surface, but turbulence survives over heterogeneous surfaces. In all heterogeneous cases, both molecular diffusion and turbulence transfer momentum down the gradient of mean velocity. The total (turbulent plus diffusive) heat flux is down-gradient, but quasi-organized eddy motions generated by the surface thermal heterogeneity induce heat transfer up the gradient of the mean temperature. Comparative analysis of HETx and HETy cases shows that the configuration with the spanwise heterogeneity is more turbulent and more efficient in transporting momentum and heat vertically than its counterpart with the streamwise heterogeneity. Vertical profiles of mean fields and turbulence moments differ considerably between the HETx and HETy cases, e.g., the streamwise heat flux differs not only in magnitude but also in sign. A close examination of the second-order turbulence moments, vertical-velocity and temperature skewness, and the flow eddy structure helps explain the observed differences between the HETx and HETy cases. The implications of our DNS findings for modelling turbulence in stably-stratified environmental and industrial flows with surface heterogeneity are discussed.

Superposition of system response in modulated turbulent plane Couette flow

Superposition of system response in modulated turbulent plane Couette flow

Can isothermal plane Couette flow in fluid overlying porous layer be linearly unstable?

The present study aims to examine the temporal linear stability analysis of isothermal plane Couette flow over a porous layer using the two-domain approach. The flow in the porous layer is described by the unsteady Darcy–Brinkman equations, whereas it is characterised by the Navier–Stokes equations in the fluid layer. In contrast to the Darcy model, it is observed that the isothermal plane Couette flow becomes unstable for such a superposed system on the inclusion of the Brinkman term. From the stability analysis, the two-dimensional mode is found to be least stable, and two modes of instability, namely porous mode and mixed mode are obtained under the consideration of the Darcy–Brinkman model along with advection term (DBA model). For Darcy number $(\delta )=0.01$ , depending on the value of the stress-jump coefficient, mixed mode controls the instability of the system at small values of depth ratio $(\hat {d})$ , and it disappears for relatively high values of $\hat {d}$ , where the porous mode dominates. In addition, it has been observed that when $\hat {d}=0.1$ , the critical mode of instability is found to be mixed for $\delta >0.02$ and porous for $\delta \le 0.02$ . The stress-jump coefficient destabilises the flow in terms of energy production through perturbed stresses at the interface. As observed in the case of isothermal plane Poiseuille flow studied by Chang, Chen & Straughan (J. Fluid Mech., vol. 564, 2006, pp. 287–303), here also depth ratio (Darcy number) stabilises (destabilises) the flow. However, this characteristic does not remain valid when the advection term is eliminated from the considered momentum equation. For a certain range of $\hat {d} (\delta )$ , the destabilising (stabilising) characteristic of the respective parameters are encountered when the fluid mode of instability prevails.

Revisit compressible plane Couette and Poiseuille instability: Spectrum features of disturbances and dominant instability

Understanding the instability of compressible shear flows is of both academic interest and practical importance for predicting transition in such flows. In this paper, the linear stability of compressible plane Couette flow and Poiseuille flow in temporal mode is revisited, with emphasis on spectrum features of disturbances and dominant instability at different Mach numbers. The disturbance spectra of different discrete modes are presented, including acoustic modes (i.e., even modes II, IV, etc. and odd modes I, III, etc.) and viscous modes. The results show that as wavenumber increases, the even modes are first synchronized with the viscous branch when their phase velocities are a bit larger than zero, and subsequently synchronized with odd modes when their phase velocities coincide with each other. The synchronizations cause branching of the dispersion curves, and the even modes have several maxima of growth rate. The low-wavenumber maximum (with phase velocity close to zero) is associated with viscous instability and the higher-wavenumber maxima correspond to inviscid instability. Depending on the flow parameters, the branching topology may involve multimodes, which is different from the spectrum branching observed in hypersonic boundary layer. In compressible Couette flow, the mode-II viscous and inviscid instabilities dominate the instability at different Mach numbers. In compressible plane Poiseuille flow, multiple instability modes coexist, including the viscous Tollmien-Schlichting (TS) mode, the acoustic viscous instability modes II and IV and acoustic inviscid instability modes I and III. As Mach number increases (i.e., from incompressible to hypersonic), the mode TS, II, IV, and III would successively dominate the Poiseuille-flow instability. These results yield a comprehensive understanding of the disturbance spectrum and modal instability in compressible plane Couette and Poiseuille flows.

Asymptotic behavior of the two‐phase flow around the planar Couette flow in three‐dimensional space

This paper is concerned with the nonlinear stability of planar Couette flow for the three‐dimensional NS‐NS equations, which are used to model the motion of two‐phase flow. Our result shows that the planar Couette flow is asymptotically stable for the initial perturbations sufficiently small in some Sobolev space if the background velocity and the Reynolds number are sufficiently small. Moreover, it is proved that the solution converges to the stationary state at an algebraic time decay rate, and we also give the decay rate of the relative velocity.

Numerical Solution of the Newtonian Plane Couette Flow with Linear Dynamic Wall Slip

An efficient numerical approach based on weighted-average finite differences is used to solve the Newtonian plane Couette flow with wall slip, obeying a dynamic slip law that generalizes the Navier slip law with the inclusion of a relaxation term. Slip is exhibited only along the fixed lower plate, and the motion is triggered by the motion of the upper plate. Three different cases are considered for the motion of the moving plate, i.e., constant speed, oscillating speed, and a single-period sinusoidal speed. The velocity and the volumetric flow rate are calculated in all cases and comparisons are made with the results of other methods and available results in the literature. The numerical outcomes confirm the damping with time and the lagging effects arising from the Navier and dynamic wall slip conditions and demonstrate the hysteretic behavior of the slip velocity in following the harmonic boundary motion.

KPP fronts in shear flows with cutoff reaction rates

AbstractWe consider the effect of a shear flow which has, without loss of generality, a zero mean flow rate, on a Kolmogorov–Petrovskii–Piscounov (KPP)‐type model in the presence of a discontinuous cutoff at concentration . In the long‐time limit, a permanent‐form traveling wave solution is established which, for fixed , is unique. Its structure and speed of propagation depends on (the strength of the flow relative to the propagation speed in the absence of advection) and (the square of the front thickness relative to the channel width). We use matched asymptotic expansions to approximate the propagation speed in the three natural cases , , and , with particular associated orderings on , while remains fixed. In all the cases that we consider, the shear flow enhances the speed of propagation in a manner that is similar to the case without cutoff (). We illustrate the theory by evaluating expressions (either directly or through numerical integration) for the particular cases of the plane Couette and Poiseuille flows.

Stability of plane Couette flow under anisotropic superhydrophobic effects

We study the linear stability of plane Couette flow subject to an anisotropic slip boundary condition that models the slip effect of parallel microgrooves with a misalignment about the direction of the wall motion. This boundary condition has been reported to be able to destabilize channel flow far below the critical Reynolds number of the no-slip case. Unlike channel flow, the no-slip plane Couette flow is known to be linearly stable at arbitrary Reynolds numbers. Nevertheless, the results show that the slip can cause linear instability at finite Reynolds numbers also. The misalignment angle of the microgrooves that maximizes the destabilizing effect is nearly π/4, and the unstable modes are of small streamwise wavenumbers and relatively large spanwise wavenumbers. The flow is always more destabilized by two slippery walls compared to a single slippery wall. These observations are in qualitative agreement with the slippery channel flow with the same boundary condition, indicating that such an anisotropic superhydrophobic effect has a rather general destabilizing effect in shear flows regardless of the profile of the base flow. The absence of the Tollmien–Schlichting instability allows us to reveal the inverse relationship between the critical Reynolds number and the slip length as well as the misalignment in the small-parameter regime. The results suggest that arbitrary nonvanishing slip length and misalignment, with arbitrarily weak anisotropy, may suffice to destabilize plane Couette flow.

Phase Transition to Turbulence via Moving Fronts.

Directed percolation (DP), a universality class of continuous phase transitions, has recently been established as a possible route to turbulence in subcritical wall-bounded flows. In canonical straight pipe or planar flows, the transition occurs via discrete large-scale turbulent structures, known as puffs in pipe flow or bands in planar flows, which either self-replicate or laminarize. However, these processes might not be universal to all subcritical shear flows. Here, we design a numerical experiment that eliminates discrete structures in plane Couette flow and show that it follows a different, simpler transition scenario: turbulence proliferates via expanding fronts and decays via spontaneous creation of laminar zones. We map this phase transition onto a stochastic one-variable system. The level of turbulent fluctuations dictates whether moving-front transition is discontinuous, or continuous and within the DP universality class, with profound implications for other hydrodynamic systems.

Sensitivity analysis of the Cercignani - Lampis accommodation coefficients in prototype rarefied gas flow and heat transfer problems via the Monte Carlo method

In rarefied gas dynamics, the Cercignani-Lampis (CL) scattering kernel, containing two accommodation coefficients (ACs), namely the tangential momentum and normal energy ones, is widely employed to characterize gas-surface interaction, particularly in non-isothermal setups, where both momentum and energy may simultaneously be exchanged. Here, a formal and detailed sensitivity analysis of the effect of the CL ACs on the main output quantities of several prototype problems, namely the cylindrical Poiseuille, thermal creep and thermomolecular pressure difference (TPD) flows, as well as the plane Couette flow and heat transfer (Fourier flow), is performed. In each problem, some uncertainties are randomly introduced in the ACs (input parameters) and via a Monte Carlo propagation analysis, the deduced uncertainty of the corresponding main output quantity is computed. The output uncertainties are compared to each other to determine the flow configuration and the gas rarefaction range, where a high sensitivity of the output quantities with respect to the CL ACs is observed. The flow setups and rarefaction regimes with high sensitivities are the most suitable ones for the estimations of the ACs, since larger modeling and experimental errors may be acceptable. In the Poiseuille and Couette flows, the uncertainties of the flow rate and shear stress respectively are several times larger than the input uncertainty in the tangential momentum AC and much smaller than the uncertainty in the normal energy AC in a wide range of gas rarefaction. In the thermal creep flow, the uncertainty of the flow rate depends on the input ones of both ACs, but, in general, it remains smaller than the input uncertainties. A similar behavior with the thermal creep flow is obtained in the TPD flow. On the contrary, in the Fourier flow, the uncertainty of the heat flux may be about the same or even larger than the input ones of both ACs in a wide range of gas rarefaction. It is deduced that in order to characterize the gas-surface interaction via the CL ACs by matching computations with measurements, it is more suitable to combine the Poiseuille (or Couette) and Fourier configurations, rather than, as it is commonly done, the Poiseuille and thermal creep ones. For example, in order to estimate the normal energy AC within an accuracy of 10 %, experimental uncertainties should be less than 4 % in the thermal creep or TPD flows, while may be about 10 % in the Fourier flow.

Stability analysis of viscous multi-layer shear flows with interfacial slip

Abstract One of the most fundamental interfacial instabilities in ideal, immiscible, incompressible multifluid flows is the celebrated Kelvin–Helmholtz (KH) instability. It predicts short-wave instabilities that, in the absence of other mollifying physical mechanisms (e.g. surface tension, viscosity), render the nonlinear problem ill-posed and lead to finite-time singularities. The crucial driving mechanism is the jump in tangential velocity across the liquid–liquid interface, i.e. interfacial slip, that can occur since viscosity is absent. The purpose of the present work is to analyse analogous instabilities for viscous flows at small or moderate Reynolds numbers as opposed to the infinite Reynolds numbers that underpin KH instabilities. The problem is physically motivated by both experiments and simulations. The fundamental model considered consists of two superposed viscous, incompressible, immiscible fluid layers sheared in a plane Couette flow configuration, with slip present at the deforming liquid–liquid interface. The origin of slip in viscous flows has been observed in experiments and molecular dynamics simulations, and can be modelled by employing a Navier-slip boundary condition at the liquid–liquid interface. The emerging novel instabilities are studied in detail here. The linear stability of the system is addressed asymptotically for long- and short-waves, and for arbitrary wavenumbers using a combination of analytical and numerical calculations. Slip is found to be capable of destabilising perturbations of all wavelengths. In regimes where the flow is stable to perturbations of all wavelengths in the absence of slip, its presence can induce a Turing-type instability by destabilization of a small band of finite wavenumber perturbations. In the case where the underlying layer is asymptotically thin, the results are found to agree with the linear properties of a weakly non-linear asymptotic model that is also derived here. The weakly nonlinear model extends previous work by the authors that had a thin overlying layer that produces a different evolution equation.

Stability of plane Couette flow with constant wall transpiration

Plane Couette flow with constant wall transpiration, i.e., constant blowing from below and constant suction from above, is a solution of the Navier-Stokes equations in terms of exponentials. It is characterized by two Reynolds numbers Re and ReV, where Re parametrizes the moving wall and ReV describes the influence of the transpiration. For this flow the modified Orr-Sommerfeld equation admits one of the very few exact solutions in terms of hypergeometric functions. Based on this, we solve the stability problem, though for the main part focus on temporal stability. For small wall-transpiration rates up to ReV≃6.71, the flow remains unconditionally stable, analogous to the classical Couette flow for arbitrary Reynolds numbers Re. By further increasing ReV an instability sets in and a minimum critical Reynolds number of Re=668350.491 is reached at ReV=9.799. Hence, around this point, the destabilizing effect of blowing outweighs the stabilizing effect of suction. By further increasing the transpiration rate beyond this point, the corresponding critical Reynolds number Re increases again and continues to grow. This limiting case is accompanied by the development of a strong boundary-layer-like velocity profile near the upper wall and the flow transitions to the asymptotic suction boundary layer (ASBL). Thus, the present analysis comprises the whole range from classical Couette flows extended by transpiration to the ASBL, which is known to have a strongly stabilizing effect. Published by the American Physical Society 2024

Turbulent spot formation in three-dimensional yukawa liquids using large-scale molecular dynamics simulation—effect of system size

We have performed classical ‘first principles’ 3D Molecular Dynamics (MD) simulation of plane-Couette flow (PCF) in a 3D Yukawa liquid. The main aim of the present investigation is to study the effect of stream-wise (flow-direction) and span-wise (transverse direction to the flow) width on the subcritical transition to turbulence in plane Couette flow (PCF), separately. In the past, taking very large-aspect ratio systems, subcritical transition to turbulence in PCF has been studied in great detail. However, the effect of stream-wise and span-wise width on the turbulent dynamics separately, have not been studied in detail. In the present work, we have investigated the effect of stream-wise and span-wise width or length and found that the stream-wise length enhances the large-scale energy, which gives rise to strong large-scale flow, and span-wise width enhances the small-scale energy, which gives rise to strong small-scale structures. In other words, topology of the turbulent spot is observed to change with the change in system size. The spectral separation between the modes governing the large and small-scale dynamics improves with the increase in system sizes. The connection between the stream-wise vortices and the stream-wise velocity streaks is investigated in details. We have found that the number stream-wise velocity streaks is one unit larger than the number of stream-wise vortices. A qualitative comparison between the results obtained from the MD simulation and the hydrodynamics experiment is presented in this article. The behavior of the system was observed to vary with the range of interactions among the dust grains.

Modal-based generalised quasilinear approximations for turbulent plane Couette flow

Modal-based generalised quasilinear approximations for turbulent plane Couette flow

Instability in strongly stratified plane Couette flow, with application to supercritical fluids.

This paper addresses the stability of plane Couette flow in the presence of strong density and viscosity stratifications. It demonstrates the existence of a generalised inflection point that satisfies the generalised Fjørtoft's criterion of instability when a minimum of kinematic viscosity is present in the base flow. The characteristic scales associated with this minimum are identified as the primary controlling parameters of the associated instability, regardless of the type of stratification. To support this finding, analytical stability models are derived in the long wave approximation using piecewise linear base flows. Numerical stability calculations are carried out to validate these models and to provide further information on the production of disturbance vorticity. All instabilities are interpreted as arising from the interaction between two vorticity waves. Depending on the type of stratification, these two waves are produced by different physical mechanisms. When both strong density and viscosity stratifications are present, we show that they result from the concurrent action of shear and inertial baroclinic effects. The stability models developed for simple fluid models ultimately shed light on a recently observed unstable mode in supercritical fluids (Ren et al., J. Fluid Mech., vol. 871, 2019, pp. 831-864), providing a quantitative prediction of the stability diagram and identifying the dominant mechanisms at play. Furthermore, our study suggests that the minimum of kinematic viscosity reached at the Widom line in these fluids is the leading cause of their instability. The existence of similar instabilities in different fluids and flows (e.g., miscible fluids) is finally discussed.

Transition Threshold for the 3D Couette Flow in a Finite Channel

In this paper, we study nonlinear stability of the 3D plane Couette flow ( y , 0 , 0 ) (y,0,0) at high Reynolds number R e {Re} in a finite channel T × [ − 1 , 1 ] × T \mathbb {T}\times [-1,1]\times \mathbb {T} . It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. This work shows that if the initial velocity v 0 v_0 satisfies ‖ v 0 − ( y , 0 , 0 ) ‖ H 2 ≤ c 0 R e − 1 \|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1} for some c 0 > 0 c_0>0 independent of R e Re , then the solution of the 3D Navier-Stokes equations is global in time and does not transit away from the Couette flow in the L ∞ L^\infty sense, and rapidly converges to a streak solution for t ≫ R e 1 3 t\gg Re^{\frac 13} due to the mixing-enhanced dissipation effect. This result confirms the threshold result obtained by Chapman via an asymptotic analysis(JFM 2002). The most key ingredient of the proof is the resolvent estimates for the full linearized 3D Navier-Stokes system around the flow ( V ( y , z ) , 0 , 0 ) (V(y,z), 0,0) , where V ( y , z ) V(y,z) is a small perturbation(but independent of R e Re ) of y y .

Multi-scale invariant solutions in plane Couette flow: a reduced-order model approach

Plane Couette flow at Reynolds number $Re=1200$ (based on the channel half-height and half the velocity difference between the top and bottom plates) is investigated with a spatial domain designed to retain only two spanwise integral length scales. In this system, the computation of invariant solutions that are physically representative of the turbulent state has been understood to be challenging. To address this challenge, our approach is to employ an accurate reduced-order model with 600 degrees of freedom (Cavalieri & Nogueira, Phys. Rev. Fluids, vol. 7, 2022, L102601). Using the two-scale energy budget and the temporal cross-correlation of key observables, it is first demonstrated that the model contains most of the multi-scale physical processes identified recently (Doohan et al., J. Fluid Mech., vol. 913, 2021, A8); i.e. the large- and small-scale self-sustaining processes, the energy cascade for turbulent dissipation, and an energy-cascade mediated small-scale production mechanism. Invariant solutions of the reduced-order model are subsequently computed, including 96 equilibria and 43 periodic orbits. It is found that none of the computed equilibrium solutions are able to reproduce an accurate energy balance associated with the multi-scale dynamics of the turbulent state. Incorporation of unsteadiness into invariant solutions is seen to be essential for a sensible description of the multi-scale turbulent dynamics and the related energetics, at least in this type of flow, as periodic orbits with a sufficiently long period are mainly able to describe the complex spatio-temporal dynamics associated with the known multi-scale phenomena.

Non-Isothermal Plane Couette Flow and Its Stability in an Anisotropic and Inhomogeneous Porous Layer Underlying a Fluid Layer Saturated by Water

Abstract In this article, the linear stability of nonisothermal plane Couette flow (NPCF) in an anisotropic and inhomogeneous porous layer underlying a fluid layer is investigated. The Darcy model is utilized to describe the flow in the porous layer. The stability analysis indicates that the introduction of media-anisotropy (K*) and media-inhomogeneity (in terms of inhomogeneity parameter A) still renders the isothermal plane Couette flow (IPCF) in such superposed fluid-porous systems unconditionally stable. For NPCF, three different modes, unimodal (porous or fluid mode), bimodal (porous and fluid mode) and trimodal (porous, fluid and porous mode), are observed along the neutral stability curves and characterized by the secondary flow patterns. It has been found that the instability of the fluid-porous system increases on increasing the media permeability and inhomogeneity along the vertical direction. Contrary to natural convection, at d̂=0.2 (d̂=depth of fluid layer/depth of porous layer) and K*=1, in which the critical wavelength shows both increasing and decreasing characteristics with increasing values of A (0≤A≤5), here in the present study, the same continuously decreases with increasing values of A. Finally, scale analysis indicates that the onset of natural convection requires a relatively higher temperature difference (ΔT) between lower and upper plates in the presence of Couette flow. However, by including media anisotropy and inhomogeneity in the porous media, the system becomes unstable even for a small critical temperature difference of about 2 °C.

Invariant scaling laws for plane Couette flow with wall-transpiration

The effect of the wall-transpiration Reynolds number ReV0 on the large-scale rolls of the plane Couette flow with wall-transpiration is investigated using resolvent analysis. We observe streamwise-elongated large-scale rolls, moving closer to the suction wall and becoming narrower in the wall-normal and spanwise direction as well as shortened in the streamwise direction for growing ReV0 due to the decreasing characteristic length of the sheared region. Invariant scaling laws describing the growth of the amplification gain over various parameters and determining the optimal parameters yielding most amplification are found enhancing our understanding of the influence on the coherent structures, which can be used to predict their exact size. In a potential second step, this allows to manipulate and efficiently modify these flow characteristics increasing the functionality for various engineering applications as for mass, momentum and heat transfer, which is dominated by turbulent superstructures. For large enough wall-transpiration, the most amplified structures are oblique at optimal streamwise wavenumbers αmax≠0 growing linearly with ReV0. The optimal spanwise wavenumber βmax yielding most amplified structures shows a polynomial dependence of second order on ReV0. For a constant ratio of the wall-transpiration and streamwise Reynolds numbers γ=ReV0Re, the most amplified structures occur at optimal streamwise Reynolds numbers represented by Remax·γa=C. The amplification gain changes its trend from growing with Re2 to decreasing with Re−1.368 after Remax is reached. The resulting streamwise structures are identical for each set of Remax and γ within this invariant scaling law.