Finite difference analysis of plane Poiseuille and Couette flow developments

Direct numerical solutions of the three-dimensional time-dependent Navier-Stokes equations are presented for the evolution of three-dimensional finite-amplitude disturbances of plane Poiseuille and plane Couette flows. Spectral methods using Fourier series and Chebyshev polynomial series are used. It is found that plane Poiseuille flow can sustain neutrally stable two-dimensional finite-amplitude disturbances at Reynolds numbers larger than about 2800. No neutrally stable two-dimensional finite-amplitude disturbances of plane Couette flow were found. Three-dimensional disturbances are shown to have a strongly destabilizing effect. It is shown that finite-amplitude disturbances can drive transition to turbulence in both plane Poiseuille flow and plane Couette flow at Reynolds numbers of order 1000. Details of the resulting flow fields are presented. It is also shown that plane Poiseuille flow cannot sustain turbulence at Reynolds numbers below about 500.

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.

Experiments and numerical simulations have shown that turbulence in transitional wall-bounded shear flows frequently takes the form of long oblique bands if the domains are sufficiently large to accommodate them. These turbulent bands have been observed in plane Couette flow, plane Poiseuille flow, counter-rotating Taylor–Couette flow, torsional Couette flow, and annular pipe flow. At their upper Reynolds number threshold, laminar regions carve out gaps in otherwise uniform turbulence, ultimately forming regular turbulent–laminar patterns with a large spatial wavelength. At the lower threshold, isolated turbulent bands sparsely populate otherwise laminar domains, and complete laminarization takes place via their disappearance. We review results for plane Couette flow, plane Poiseuille flow, and free-slip Waleffe flow, focusing on thresholds, wavelengths, and mean flows, with many of the results coming from numerical simulations in tilted rectangular domains that form the minimal flow unit for the turbulent–laminar bands.

We investigate the nonlinear phase-space dynamics of plane Couette flow and plane Poiseuille flow under the action of opposition control at low Reynolds numbers in domains close to the minimal unit. In Couette flow, the effect of the control is analysed by focussing on a pair of non-trivial equilibrium solutions. It is found that the control only slightly modifies the statistics, turbulent skin friction and phase-space projection of the lower-branch equilibrium solution, which, in this case, is in fact identical to the edge state. On the other hand, the upper-branch equilibrium solution and mean turbulent state are modified considerably when the control is applied. In phase space, they gradually approach the lower-branch equilibrium solution on increasing the control amplitude, and this results in an elevation of the critical Reynolds number at which the equilibrium solutions first occur via a saddle-node bifurcation. It is also found that the upper-branch equilibrium solution is stabilised by the control. In Poiseuille flow, we study an unstable periodic orbit on the edge state and find that it, too, is modified very little by opposition control. We again observe that the turbulent state gradually approaches the edge state in phase space as the control amplitude is increased. In both flows, we find that the control significantly reduces the fluctuating strength of the turbulent state in phase space. However, the reduced distance between the turbulent trajectory and the edge state yields a significant reduction in turbulence lifetimes for both Couette and Poiseuille flow. This demonstrates that opposition control greatly increases the probability of the trajectory escaping from the turbulent state, which takes the form of a chaotic saddle.

Reduced mass flow rates of a rarefied Couette and Poiseuille flow in a long rectangular channel are calculated in the whole range of the gas rarefaction and a wide range of the width to height ratio. Furthermore, walls may be made of different materials so that different tangential momentum accommodation coefficients (TMACs) may be applied. Analytical solutions are given for the slip regime, where all four surrounding walls may have a different TMAC. Due to a simplified modeling assumption, these solutions can be used to correct the well-known flow rates of a fully diffuse channel for different TMACs in the whole range of the gas rarefaction. If the slip solution and the diffuse solution are known, the procedure can principally be adapted for any channel shape. The results of the analytical model expressions are validated with simulation data of the plane Couette and Poiseuille flow and the Poiseuille flow through a pipe, which are found in the literature. In addition, the analytical solution is compared to results of the Direct Simulation Monte Carlo (DSMC) method of a Couette and a Poiseuille flow in a rectangular channel, which are provided as tabulated data for a variation of the gas rarefaction parameter at different aspect ratios and different combinations of TMACs. The procedure to calculate the mass flow rate of the certain flow as well as the application limits are discussed.

Internal flow is a flow for which the fluid is confined by a surface. Hence, the boundary layer is unable to develop without eventually being constrained. An internal flow is constrained by the bounding walls, and the viscous effects will grow and meet and permeate the entire flow. Some examples of internal flows are: flow between two parallel horizontal plates, Couette flow, plane Poiseuille flow, flow through pipes, etc. In the present study, a plane Couette flow has been analyzed by a classical method (exact solution of Navier-Stokes equation) as well as by an approximate method using central difference scheme (numerical solution of Navier-Stokes equation). The flow domain has been divided into various nodes and the velocities are obtained at different nodes for various time intervals. The stability condition for the convergence of solution has been determined and further the convergence of solution has been obtained using a MATLAB program for Couette flow using Crank-Nicolson scheme.

Two new families of exact coherent states are found in plane Poiseuille flow. They are obtained from the stationary and the travelling-wave mirror-symmetric solutions in plane Couette flow by a homotopy continuation. They are characterized by the mirror symmetry inherited from those continued solutions in plane Couette flow. The first family arises from a saddle-node bifurcation and the second family bifurcates by breaking the top–bottom symmetry of the first family. We find that both families exist below the minimum saddle-node-point Reynolds number known to date (Waleffe, Phys. Fluids, vol. 15, 2003, pp. 1517–1534).

Couette and Poiseuille flows in a low viscosity asthenosphere: Effects of internal heating rate, Rayleigh number, and plate representation

A linear stability analysis of plane Poiseuille flow of an upper-convected Maxwell (UCM) fluid, bounded between rigid plates separated by a distance $2L$, has been carried out to investigate the interplay of elasticity and inertia on flow stability. The stability is governed by the following dimensionless groups: the Reynolds number $Re=\unicode[STIX]{x1D70C}U_{max}L/\unicode[STIX]{x1D702}$ and the elasticity number $E\equiv W/Re=\unicode[STIX]{x1D706}\unicode[STIX]{x1D702}/(\unicode[STIX]{x1D70C}L^{2})$, where $W=\unicode[STIX]{x1D706}U_{max}/L$ is the Weissenberg number. Here, $\unicode[STIX]{x1D70C}$ is the fluid density, $\unicode[STIX]{x1D702}$ is the fluid viscosity, $\unicode[STIX]{x1D706}$ is the micro-structural relaxation time and $U_{max}$ is the maximum base-flow velocity. The stability is analysed for two-dimensional perturbations using both pseudo-spectral and shooting methods. We also analyse the linear stability of plane Couette flow which, along with the results for plane Poiseuille flow, yields insight into the structure of the complete elasto-inertial eigenspectrum. While the general features of the spectrum for both flows remain similar, plane Couette flow is found to be stable over the range of parameters examined ($Re\leqslant 10^{4},E\leqslant 0.01$). On the other hand, plane Poiseuille flow appears to be susceptible to an infinite hierarchy of elasto-inertial instabilities. Over the range of parameters examined, there are up to seven distinct neutral stability curves in the $Re$–$k$ plane (here $k$ is the perturbation wavenumber in the flow direction). Based on the symmetry of the eigenfunctions for the streamwise velocity about the centreline, four of these instabilities are antisymmetric, while the other three are symmetric. The neutral stability curve corresponding to the first antisymmetric mode is shown to be a continuation (to finite $E$) of the Tollmien–Schlichting (TS) instability already present for Newtonian channel flow. As $E$ is increased beyond $0.0016$, a new elastic mode appears at $Re\sim 10^{4}$, which coalesces with the continuation of the TS mode for a range of $Re$, thereby yielding a single unstable mode in this range. This trend persists until $E\sim 0.0021$, beyond which this neutral curve splits into two separate ones in the $Re$–$k$ plane. The new elastic mode which arises out of this splitting has been found to be the most unstable, with the lowest critical Reynolds number $Re_{c}\approx 1210.9$ for $E=0.0066$. The neutral curves for both the continuation of the original TS mode, and the new elastic antisymmetric mode, form closed loops upon further increase in $E$, which eventually vanish at sufficiently high $E$. For $E\ll 1$, the critical Reynolds number and wavenumber scale as $Re_{c}\sim E^{-1}$ and $k_{c}\sim E^{-1/2}$ for the first two of the symmetric modal families, and as $Re_{c}\sim E^{-5/4}$ for first two of the antisymmetric modal families; $k_{c}\sim E^{-1/4}$ for the third antisymmetric family. The critical wave speed for all of these unstable eigenmodes scales as $c_{r,c}\sim E^{1/2}$ for $E\ll 1$, implying that the modes belong to a class of ‘wall modes’ in viscoelastic flows with disturbances being confined in a thin region near the wall. The present study shows that, surprisingly, even in plane shear flows, elasticity acting along with inertia can drive novel instabilities absent in the Newtonian limit.

In the past, several stability theories have been developed to describe the mechanism of flow instability. These are: (1) The linear stability theory, which can be traced back to Rayleigh (1880) is a widely used method and has been applied to some problems [1]; (2) The energy method (Orr, 1907) based on Orr-Reynolds equations; (3) The weak nonlinear stability theory developed by Stuart and others; (4) The secondary instability theory proposed by Herbert et al. (1988) and his colleagues. However, there are still significant discrepancies between the predictions obtained using these methods and experimental data; particularly at transition. Dou [2,3] proposed a mechanism with the aim to clarify the phenomenon of transition from laminar flow to turbulence for wall-bounded shear flows. In this mechanism, the whole flow field is treated as an energy field. By rigorous derivations based on physics, it is found that that the energy gradient in the transverse direction is able to amplify a velocity disturbance, while the viscous friction loss in the streamwise direction can resist and absorb this disturbance. The flow instability or the transition to turbulence depends on the relative magnitude of these two roles of energy gradient amplification and viscous friction damping of the initial disturbance. The analysis has obtained very good consistent agreement for the plane Poiseuille flow and the pipe Poiseuille flow. This theory is named as “energy gradient theory”. In this theory, the flow instability is characterized by a function K of coordinates and the disturbance amplitude. In this paper, using the energy analysis, the equation for calculating K in plane Couette flow is derived. It is demonstrated that the critical value of K at the critical condition which is observed from experiments [4~6] is about 370 for plane Couette flow. This value is about the same as for plane Poiseuille flow and pipe Poiseuille flow (385~389). These results demonstrate that the critical value of Kmax for wall-bounded parallel flows including both pressure driven and shear driven flows is about 370~389. This consistency also suggests that the mechanisms of instabilities in wall-bounded parallel shear flow are perhaps the same. They are all dominated by the transverse energy gradient and the streamwise flow energy loss. The results obtained in this study provide further basis for better understanding of the mechanism of instability and transition to turbulence in parallel shear flows, and provide a basis for the modelling and prediction of the transition process.

Rarefied gas flows in micro-electro-mechanical systems (MEMS) devices, calculated from the linearized Bhatnagar-Gross-Krook model equation [P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. 94, 511 (1954)], are studied in a wide range of Knudsen numbers. Both plane Poiseuille and Couette flows are investigated numerically by extending a finite difference technique first introduced by Cercignani and Daneri [J. Appl. Phys. 34, 3509 (1963)]. Moreover, a variational approach, applied to the integrodifferential form of the linearized Boltzmann equation [C. Cercignani, J. Stat. Phys. 1, 297 (1969)], is used to solve in a unified manner the plane Poiseuille-Couette problem by means of the computation of only one functional. General boundary conditions of Maxwell’s type have been considered, assuming both symmetric and nonsymmetric molecular interaction between gas-solid interfaces, in order to take into account possible differences in the accommodation coefficients on the walls of MEMS devices. Based on the analysis presented in this paper, an accurate database valid in the entire Knudsen regime can be created for the Poiseuille-Couette problem, to be used in micromechanical applications.

The aggregation of red blood cells (RBCs) is a reversible phenomenon in which RBCs form a pile or network at low shear rate via the interactions of electrostatic and aggregating forces. In previous experimental results under both Couette and Poiseuille flows, cyclic variations in blood echogenicity were observed but their cyclic patterns were different. In this paper, a two-dimensional particle model capable of mimicking the main characteristics of RBC aggregation kinetics was proposed to elucidate the different effects of hemodynamics under Couette and Poiseuille flows on RBC aggregation. In simulation results, cyclic variation of RBC aggregation was observed but its magnitude of mean aggregate size (MAS) was not changed by variations of velocity and stroke rate under Couette flow. These results are in agreement with the experiment results. In contrast, the simulation results under Poiseuille flow revealed that cyclic variation of RBC aggregation and its MAS magnitude were changed by variations of velocity and stroke rate. As stroke rate increased from 20 to 120 beats/min, the phase of MAS variation compared with velocity profile was shifted. These simulation results may provide the theoretical explanation of the different experimental results of cyclic variation of RBC aggregation under Couette and Poiseuille flows. [Work supported by NIPA-2013-H0401-13-1007 and 2013R1A1A2043478.]

The method of moments is employed to extend the validity of continuum-hydrodynamic models into the transition-flow regime. An evaluation of the regularized 13 moment equations for two confined flow problems, planar Couette and Poiseuille flows, indicates some important limitations. For planar Couette flow at a Knudsen number of 0.25, they fail to reproduce the Knudsen-layer velocity profile observed using a direct simulation Monte Carlo approach, and the higher-order moments are not captured particularly well. Moreover, for Poiseuille flow, this system of equations creates a large slip velocity leading to significant overprediction of the mass flow rate for Knudsen numbers above 0.4. To overcome some of these difficulties, the theory of regularized moment equations is extended to 26 moment equations. This new set of equations highlights the importance of both gradient and non-gradient transport mechanisms and is shown to overcome many of the limitations observed in the regularized 13 moment equations. In particular, for planar Couette flow, they can successfully capture the observed Knudsen-layer velocity profile well into the transition regime. Moreover, this new set of equations can correctly predict the Knudsen layer, the velocity profile and the mass flow rate of pressure-driven Poiseuille flow for Knudsen numbers up to 1.0 and captures the bimodal temperature profile in force-driven Poiseuille flow. Above this value, the 26 moment equations are not able to accurately capture the velocity profile in the centre of the channel. However, they are able to capture the basic trends and successfully predict a Knudsen minimum at the correct value of the Knudsen number.

Analytical solutions for the plane Couette flow and the plane Poiseuille flow of the one-mode Giesekus fluid without any retardation time have been obtained by considering the domain of definition for each of the two branch solutions which arise due to the presence of the quadratic stress terms in the constitutive equations. For each fixed value of the mobility parametera, the limiting value of the Weissenberg number for the upper branch solution, i.e., the physically realistic solution is determined in terms of the corresponding dimensionless shear stress for the plane Couette flow and in terms of the corresponding dimensionless pressure gradient for the plane Poiseuille flow. In the case of the plane Couette flow, it is shown that fora falling in the range 0≤a≤1/2 only the physically realistic solution exists while for 1/2<a ≤1 a nonphysical solution coexists with the realistic one. In the case of the plane Poiseuille flow, it is shown that the non-physical solution cannot even exist around the center plane of the channel, and the effects of the mobility parameter and the dimensionless pressure gradient on the flow variables are investigated. Possible extensions of the present approach to other steady simple shear flows with and without the introduction of the retardation time are also discussed.

A proof by Petrov [10] that plane Couette flow and plane Poiseuille flow are always stable with respect to infinitesimal disturbances is examined and found to be based on a set of incomplete functions, thus rendering the proof invalid.