Plane Couette-Poiseuille Flow Research Articles

Modal stability analysis of the density-stratified plane Couette–Poiseuille flow

Shear and density stratification strongly affect the flow mechanism of the different atmospheric and ocean flows. In this paper, we investigate the stability characteristics of plane Couette–Poiseuille (CP) flow with stable density stratification in the vertical direction. A modal stability analysis is carried out to examine the exponentially growing instability of stratified plane CP flow under different controlling parameters. The domain of the flow is periodic in streamwise and vertical directions. The stability problem is solved numerically using the spectral collocation method. The present analysis is carried out for Reynolds number Re=104 with different speeds of moving wall and different strengths of stratification. The results show that the mass diffusivity impact on the flow instability mechanism is almost negligible beyond the Schmidt number Sc≥20. The three-dimensional mode is generally the most unstable mode for density-stratified CP flow. However, the most unstable mode in unstratified CP flow is always two-dimensional. In contrast to unstratified CP flow, the density-stratified CP flow is unstable even moving wall velocity exceeds 70% of the center velocity of the plane Poiseuille flow component. It is also observed that the moving wall velocity and density stratification simultaneously affect the stability of the flow, which shows the importance of the shear and stratification in the flow.

Inertial enhancement of the polymer diffusive instability

Beneitez et al. (Phys. Rev. Fluids, vol. 8, 2023, L101901) have recently discovered a new linear ‘polymer diffusive instability’ (PDI) in inertialess rectilinear viscoelastic shear flow using the finitely extensible nonlinear elastic constitutive model of Peterlin (FENE-P) when polymer stress diffusion is present. Here, we examine the impact of inertia on the PDI for both plane Couette and plane Poiseuille flows under varying Weissenberg number ${W}$ , polymer stress diffusivity $\varepsilon$ , solvent-to-total viscosity ratio $\beta$ and Reynolds number ${Re}$ , considering the FENE-P and simpler Oldroyd-B constitutive relations. Both the prevalence of the instability in parameter space and the associated growth rates are found to significantly increase with ${Re}$ . For instance, as $Re$ increases with $\beta$ fixed, the instability emerges at progressively lower values of $W$ and $\varepsilon$ than in the inertialess limit, and the associated growth rates increase linearly with $Re$ when all other parameters are fixed. For finite $Re$ , it is also demonstrated that the Schmidt number $Sc=1/(\varepsilon Re)$ collapses curves of neutral stability obtained across various $Re$ and $\varepsilon$ . The observed strengthening of PDI with inertia and the fact that stress diffusion is always present in time-stepping algorithms, either implicitly as part of the scheme or explicitly as a stabilizer, implies that the instability is likely operative in computational work using the popular Oldroyd-B and FENE-P constitutive models. The fundamental question now is whether PDI is physical and observable in experiments, or is instead an artifact of the constitutive models that must be suppressed.

Predicting the energy stability limit of shear flows using weighted velocity components

Predicting the subcritical transition in fluid dynamic systems remains a challenging task, but recent advancements utilizing edge tracking methods, polynomial Lyapunov functions, and various energy norms have shown promise. In this study, we propose a novel approach by defining the general kinetic energy through weighted velocity components. The minimal Reynolds number is determined, where the derivative of this generalized energy with respect to time is zero. The procedure is similar to that of the well-known Reynolds–Orr equation. Unlike traditional methods, our approach does not necessitate the monotonic decay of the classic perturbation kinetic energy, resulting in a larger critical Reynolds number and reduced conservativeness of the Reynolds–Orr equation. However, the energy production of the pressure is not negligible, in contrast to the classical Reynolds–Orr equation. The pressure's implicit dependence on the velocity field complicates the variation process. To address this, a method is presented to handle the problem effectively. Our approach is then applied to analyze parallel flows, specifically the plane Couette and plane Poiseuille flows, wherein the problem can be further simplified using the complex Fourier transformation. The weights of velocity components are optimized to maximize the critical Reynolds number, resulting in a significant increase.

Flow and heat transfer analysis of Couette and Poiseuille flow of a hybrid nanofluid with temperature-dependent viscosity and thermal conductivity

The purpose of this research is to quantify the influence of a number of independent parameters (−0.5≤α1,β1,β2≤0.5) on the heat transmission rate and the temperature of a nanofluid hybrid flowing between parallel plates. The linked nonlinear momentum and energy equations are solved using the regular perturbation method for two different cases resembling the plane Couette flow and Poiseuille flow. By combining engine oil with multiwalled carbon nanotubes (MWCNT) and aluminum oxide nanoparticles, a hybrid nanofluid is produced with wide-ranging lubricating and industrial applications. It has been found that the hybrid nanofluid can function as a coolant provided the relevant parameters are adjusted appropriately. It is demonstrated that these additives have a tendency to stabilise the thermal and the volume flow rate, hence decreasing randomization that is necessary for a good lubricant to function. The findings of the study indicate that β1 has a greater influence on temperature, while β2 has a stronger effect on velocity. In addition to the variable model, the results for the constant model have also been deduced, and the novel study demonstrates that the considered variable viscosity model tends to produce a more uniform temperature distribution.

Structured input–output analysis of oblique laminar–turbulent patterns in plane Couette–Poiseuille flow

In this work, we employ structured input–output analysis to study the flow patterns in transitional plane Couette–Poiseuille flow (CPF). First, we focus on the well-studied intermediate laminar profile, which balances the shear and pressure effects. We show that the highest structured gain corresponds to perturbations with wavelengths associated with the oblique turbulent bands observed in experiments. In addition, the inclination angles of these structures show a Reynolds number dependence consistent with experimentally observed trends. We then examine the Reynolds number scaling of the maximal structured frequency response as the velocity profile is varied from plane Couette to Poiseuille flow. Our results demonstrate that, as expected, the scaling exponent increases over this range, but this increase is not monotonic. We attribute the variation in the shape of the scaling exponent curve as a function of the velocity profile shape to changes in flow patterns that dominate each associated flow regime. We then focus on the particular case of plane Couette flow and compute the structured response modes. Their behavior and structural features are consistent with results obtained through direct numerical simulation (DNS) studies. Finally, we employ the structured analysis framework to examine the temporal evolution of the dominant structures. For the well-studied cases of plane Couette and plane Poiseuille flows, the computed advection speeds of the oblique structures are consistent with those predicted through DNS.

Electrohydrodynamic Couette–Poiseuille Flow Instability of Two Viscous Conducting and Dielectric Fluid Layers Streaming through Brinkman Porous Medium

The electrohydrodynamic plane Couette–Poiseuille flow instability of two superposed conducting and dielectric viscous incompressible fluids confined between two rigid horizontal planes under the action of a normal electric field and pressure gradient through Brinkman porous medium has been analytically investigated. The lower plane is stationary, while the upper one is moving with constant velocity. The details of the base state mathematical model and the linearized model equations for the perturbed state are introduced. Following the usual procedure of linear stability analysis for viscous fluids, we derived two non-dimensional modified Orr–Sommerfeld equations and obtained the associated boundary and interfacial conditions suitable for the problem. The eigenvalue problem has been solved using asymptotic analysis for wave numbers in the long-wavelength limit to obtain a very complicated novel dispersion relation for the wave velocity through lengthy calculations. The obtained dispersion equation has been solved using Mathematica software v12.1 to study graphically the effects of various parameters on the stability of the system. It is obvious from the figures that the system in the absence of a porous medium and/or electric field is more unstable than in their presence. It is found also that the velocity of the upper rigid boundary, medium permeability, and Reynolds number have dual roles on the stability on the system, stabilizing as well as destabilizing depending on the viscosity ratio value. The electric potential, dielectric constant and pressure gradient are found to have destabilizing influences on the system, while the porosity of the porous medium, density ratio and Froude number have stabilizing influences. A depth ratio of less than one has a dual role on the stability of the system, while it has a stabilizing influence for values greater than one. It is observed that the viscosity stratification brings about a stabilizing as well as a destabilizing effect on the flow system.

Numerical simulation of turbulent, plane parallel Couette–Poiseuille flow

We present numerical simulation and mean-flow modelling of statistically stationary plane Couette–Poiseuille flow in a parameter space $(Re,\theta )$ with $Re=\sqrt {Re_c^2+Re_M^2}$ and $\theta =\arctan (Re_M/Re_c)$ , where $Re_c,Re_M$ are independent Reynolds numbers based on the plate speed $U_c$ and the volume flow rate per unit span, respectively. The database comprises direct numerical simulations (DNS) at $Re=4000,6000$ , wall-resolved large-eddy simulations at $Re = 10\,000, 20\,000$ , and some wall-modelled large-eddy simulations (WMLES) up to $Re=10^{10}$ . Attention is focused on the transition (from Couette-type to Poiseuille-type flow), defined as where the mean skin-friction Reynolds number on the bottom wall $Re_{\tau,b}$ changes sign at $\theta =\theta _c(Re)$ . The mean flow in the $(Re,\theta )$ plane is modelled with combinations of patched classical log-wake profiles. Several model versions with different structures are constructed in both the Couette-type and Poiseuille-type flow regions. Model calculations of $Re_{\tau,b}(Re,\theta )$ , $Re_{\tau,t}(Re,\theta )$ (the skin-friction Reynolds number on the top wall) and $\theta _c$ show general agreement with both DNS and large-eddy simulations. Both model and simulation indicate that, as $\theta$ is increased at fixed $Re$ , $Re_{\tau,t}$ passes through a peak at approximately $\theta = 45^{\circ }$ , while $Re_{\tau,b}$ increases monotonically. Near the bottom wall, the flow laminarizes as $\theta$ passes through $\theta _c$ and then re-transitions to turbulence. As $Re$ increases, $\theta _c$ increases monotonically. The transition from Couette-type to Poiseuille-type flow is accompanied by the rapid attenuation of streamwise rolls observed in pure Couette flow. A subclass of flows with $Re_{\tau,b}=0$ is investigated. Combined WMLES with modelling for these flows enables exploration of the $Re\to \infty$ limit, giving $\theta _c \to 45^\circ$ as $Re\to \infty$ .

Understanding viscoelastic flow instabilities: Oldroyd-B and beyond

The Oldroyd-B model has been used extensively to predict a host of instabilities in shearing flows of viscoelastic fluids, often realized experimentally using polymer solutions. The present review, written on the occasion of the birth centenary of James Oldroyd, provides an overview of instabilities found across major classes of shearing flows. These comprise (i) the canonical rectilinear shearing flows including plane Couette, plane and pipe Poiseuille flows; (ii) viscometric shearing flows with curved streamlines such as those in the Taylor-Couette, cone-and-plate and parallel-plate geometries; (iii) non-viscometric shearing flows with an underlying extensional flow topology such as the flow in a cross-slot device; and (iv) multilayer shearing flows. While the underlying focus in all these cases is on results obtained using the Oldroyd-B model, we also discuss their relation to the actual instability, and as to how the shortcomings of the Oldroyd-B model may be overcome by the use of more realistic constitutive models. All the three commonly used tools of stability analysis, viz., modal linear stability, nonmodal stability, and weakly nonlinear stability analyses are discussed, with supporting evidence from experiments and numerical simulations as appropriate. Despite only accounting for a shear-rate-independent viscosity and first normal stress coefficient, the Oldroyd-B model is able to qualitatively predict the majority of instabilities in the aforementioned shearing flows. The review also highlights, where appropriate, open questions in the area of viscoelastic stability.

Study of Third-Grade Fluid under the Fuzzy Environment with Couette and Poiseuille Flows

In this work, fundamental flow problems, namely, Couette flow, fully developed plane Poiseuille flow, and plane Couette–Poiseuille flow of a third-grade non-Newtonian fluid between two horizontal parallel plates separated by a finite distance in a fuzzy environment are considered. The governing nonlinear differential equations (DEs) are converted into fuzzy differential equations (FDEs) and explain our approach with the help of the membership function (MF) of triangular fuzzy numbers (TFNs). Adomian decomposition method (ADM) is used to solve fundamental flow problems based on FDEs. In a crisp environment, the current findings are in good accord with their previous numerical and analytical results. Finally, the effect of the α -cut α ∈ 0,1 and other engineering constants on fuzzy velocity profile are invested in graphically and tabular forms. Also, the variability of the uncertainty is studied through the triangular MF.

Unsteady MHD Plane Couette-Poiseuille Flow of Fourth-Grade Fluid with Thermal Radiation, Chemical Reaction and Suction Effects

This study investigates the unsteady MHD flow of a fourth-grade fluid in a horizontal parallel plates channel. The upper plate is oscillating and moving while the bottom plate is stationary. Solutions for momentum, energy and concentration equations are obtained by the He-Laplace scheme. This method was also used by Idowu and Sani [12] and there is agreement with our results. The effect of various flow parameters controlling the physical situation is discussed with the aid of graphs. Significant results from this study show that velocity and temperature fields increase with the increase in the thermal radiation parameter, while velocity and concentric fields decrease with an increase in the chemical reaction parameter. Furthermore, velocity, temperature and concentric fields decrease with an increase in the suction parameter. It is also interesting to note that when <i>S4</i> = <i>0</i>, our results will be in complete agreement with Idowu and Sani [12] results. The results of this work are applicable to industrial processes such as polymer extrusion of dye, draining of plastic films etc.

Purely elastic linear instabilities in parallel shear flows with free-slip boundary conditions

We perform a linear stability analysis of viscoelastic plane Couette and plane Poiseuille flows with free-slip boundary conditions. The fluid is described by the Oldroyd-B constitutive model, and the flows are driven by a suitable body force. We find that both types of flow become linearly unstable, and we characterise the spatial structure of the unstable modes. By performing a boundary condition homotopy from the free-slip to no-slip boundaries, we demonstrate that the unstable modes are directly related to the least stable modes of the no-slip problem, destabilised under the free-slip situation. We discuss how our observations can be used to study recently discovered purely elastic turbulence in parallel shear flows.

Structured input–output analysis of transitional wall-bounded flows

Input–output analysis of transitional channel flows has proven to be a valuable analytical tool for identifying important flow structures and energetic motions. The traditional approach abstracts the nonlinear terms as forcing that is unstructured, in the sense that this forcing is not directly tied to the underlying nonlinearity in the dynamics. This paper instead employs a structured-singular-value-based approach that preserves certain input–output properties of the nonlinear forcing function in an effort to recover the larger range of key flow features identified through nonlinear analysis, experiments and direct numerical simulation (DNS) of transitional channel flows. Application of this method to transitional plane Couette and plane Poiseuille flows leads to not only the identification of the streamwise coherent structures predicted through traditional input–output approaches, but also the characterization of the oblique flow structures as those requiring the least energy to induce transition, in agreement with DNS studies, and nonlinear optimal perturbation analysis. The proposed approach also captures the recently observed oblique turbulent bands that have been linked to transition in experiments and DNS with very large channel size. The ability to identify the larger amplification of the streamwise varying structures predicted from DNS and nonlinear analysis in both flow regimes suggests that the structured approach allows one to maintain the nonlinear effects associated with weakening of the lift-up mechanism, which is known to dominate the linear operator. Capturing this key nonlinear effect enables the prediction of a wider range of known transitional flow structures within the analytical input–output modelling paradigm.

Direct numerical simulation of a turbulent plane Couette-Poiseuille flow with zero-mean shear

Direct numerical simulations of a turbulent Couette-Poiseuille flow with zero-mean-shear at the moving wall (SL-flow) is performed to examine flow features compared to those for a turbulent pure Poiseuille flow (P-flow). Profiles of the streamwise mean velocity, indicator function and ratio of production to dissipation show that the logarithmic region is significantly elongated for the SL-flow compared to that for the P-flow at a similar Reynolds number. In addition, the magnitudes of the Reynolds stresses are found to be larger in both inner and outer layers for the SL-flow than those for the P-flow. The spanwise spectra of the production term in the turbulent kinetic energy equation are examined to provide a structural basis for explaining the statistical behaviors. In addition, because the growth of the energy-containing motions extends to the outer layer further for the SL-flow due to the presence of a positive mean shear throughout the entire wall layer, the self-similar behavior of the energy balance between the production and transport terms with respect to the self-similar wavenumber is found far from the wall. We also find the increase in the number of uniform momentum zones in the SL-flow, revealing the hierarchical distribution of the energy-containing eddies which are composed of multiple uniform momentum zones. These coherent motions lead to the elongation of the logarithmic region for the SL-flow. Finally, investigation of the turbulent energy transfer process in a spectral domain for the SL-flow demonstrates importance of outer layer very-long structures, and these structures attribute to the energy transport process in an entire flow field.

Decay of streaks and rolls in plane Couette–Poiseuille flow

We report the results of an experimental investigation into the decay of turbulence in plane Couette–Poiseuille flow using ‘quench’ experiments where the flow laminarises after a sudden reduction in Reynolds number $Re$ . Specifically, we study the velocity field in the streamwise–spanwise plane. We show that the spanwise velocity containing rolls decays faster than the streamwise velocity, which displays elongated regions of higher or lower velocity called streaks. At final Reynolds numbers above $425$ , the decay of streaks displays two stages: first a slow decay when rolls are present and secondly a more rapid decay of streaks alone. The difference in behaviour results from the regeneration of streaks by rolls, called the lift-up effect. We define the turbulent fraction as the portion of the flow containing turbulence and this is estimated by thresholding the spanwise velocity component. It decreases linearly with time in the whole range of final $Re$ . The corresponding decay slope increases linearly with final $Re$ . The extrapolated value at which this decay slope vanishes is $Re_{a_z}\approx 656\pm 10$ , close to $Re_g\approx 670$ at which turbulence is self-sustained. The decay of the energy computed from the spanwise velocity component is found to be exponential. The corresponding decay rate increases linearly with $Re$ , with an extrapolated vanishing value at $Re_{A_z}\approx 688\pm 10$ . This value is also close to the value at which the turbulence is self-sustained, showing that valuable information on the transition can be obtained over a wide range of $Re$ .

Experimental measurements in plane Couette–Poiseuille flow: dynamics of the large- and small-scale flow

Abstract

Power law fluid model on wave mitigation, 2D simulation using smoothed particle hydrodynamics

This article is focused on numerical modelling to describe influence of coastal vegetation in wave mitigation. The model based on Navier-Stokes equations with stress tensor written in power law model. Numerical approach used to solve the problem is SPH (Smoothed Particle Hydrodynamics). Three numerical simulation are conducted; plane Couette-Poiseuille flow, wave mitigation on flat bottom, and wave mitigation on incline bottom. The first simulation shows that our numerical results are in good agreement with analytic solution provided in [13]. The second simulation shows that existence of the coastal vegetation reduce the wave amplitude. Whereas the last simulation shows that the vegetation reduce the run up height. Further, the power law constant influences how high the run up.

Linear stability of a plane Couette–Poiseuille flow overlying a porous layer

The modal and non-modal linear stability analyses of a three-dimensional plane Couette–Poiseuille flow through a porous channel are studied based on the two-domain approach, where fluid and porous layers are treated as distinct layers separated by an interface. The unsteady Darcy–Brinkman equations are used to describe the flow in the porous layer rather than the unsteady Darcy’s equations. In fact, the Brinkman viscous diffusion terms are necessary to capture the momentum boundary layer developed close to the fluid-porous interface. The modal stability analysis is performed under the framework of the Orr–Sommerfeld boundary value problem. On the other hand, the non-modal stability analysis is performed under the framework of the time-dependent initial value problem in terms of normal velocity and normal vorticity components. The Chebyshev spectral collocation method along with the QZ algorithm is implemented to solve the boundary value problem numerically for disturbances of arbitrary wavenumbers. The convergence test of spectrum demonstrates that more Chebyshev polynomials are required to arrest the flow dynamics in the momentum boundary layer once the Couette flow component is turned on. Two different types of modes, so-called the fluid layer mode and the porous layer mode are identified in the modal stability analysis. The most unstable fluid layer mode intensifies while the most unstable porous layer mode attenuates in the presence of the Couette flow component. Further, the mechanism of modal instability is deciphered by using the method of the energy budget. It is found that the energy production term supplies energy from the base flow to the disturbance via the Reynolds stress, and boosts the disturbance kinetic energies for the fluid layer and the porous layer. Moreover, the non-modal stability analysis demonstrates that short time energy growth exists in the parameter space and becomes significant in the presence of the Couette flow component and the permeability of the porous medium.

Heat Transfer of Power-Law Fluids in Plane Couette–Poiseuille Flows with Viscous Dissipation

Analytical expressions for the velocity and temperature profiles, bulk temperature and Nusselt numbers, in a fully-developed laminar Couette–Poiseuille flow between parallel plates of a power-law fluid with constant, and distinct, wall heat fluxes, in the presence of viscous dissipation are deduced and presented. Both favorable and adverse pressure gradient cases were analyzed. The walls’ shear stresses ratio, which arises naturally when the dimensionless hydrodynamic solution is obtained, together with the fluid power-law index Brinkman number and the walls’ heat fluxes ratio are the independent variables in the heat transfer solutions. With the exception of Newtonian fluids, there are in general two distinct analytical solutions, one for positive and another for negative values of the walls’ shear stresses ratio. The existence of singular points are also observed, where for a given value of the power-law index, there are values of the walls’ shear stresses ratio for which the Nusselt number becomes independent of the Brinkman number. It was also found that in a Couette–Poiseuille flow, for each value of the power-law index there exists a certain negative value of the walls’ shear stresses ratio that makes the Nusselt numbers at both walls identically zero.

Self-sustaining dual critical layer states in plane Poiseuille–Couette flow at large Reynolds number

The nonlinear stability of plane Poiseuille–Couette flow subjected to three-dimensional disturbances is studied asymptotically at large Reynolds number R . By analysing the nature of the instability for increasing disturbance size Δ, the scaling Δ = O ( R −1/3 ) is identified at which a strongly nonlinear neutral wave structure emerges, involving the interaction of two inviscid critical layers. The striking feature of this structure is that the travelling wave disturbances have both streamwise and spanwise wavelengths comparable to the channel width, with an associated phase speed of O (1). An alternative method to the classical balancing of phase shifts is proposed, involving vorticity jumps, that uses a global property of the flow-field and enables the amplitude-dependence of the neutral modes to be determined in terms of the wavenumbers and the properties of the basic flow. Numerical computation of the Rayleigh equation which governs the flow outside of the critical layers shows that neutral solutions exist for non-dimensional wall sliding speeds in the range 0 ≤ V < 2. It transpires that the critical layers merge and the asymptotic structure referred to above breaks down both in the large-amplitude limit and the limit V → 2 when the maximum of the basic flow becomes located at the upper wall.

Dissimilarity between heat and momentum transfer in turbulent plane Couette-Poiseuille flow

Dissimilarity between heat and momentum transfer in turbulent plane Couette-Poiseuille flow