Stability Of Plane Poiseuille Flow Research Articles

Adjoint Analysis of Plane Poiseuille Flow Global and Convective Stability

Abstract The adjoint analysis of plane Poiseuille flow is conducted for both global and convective stability problems. Receptivity and structural sensitivity are analyzed using a stabilized finite element method. In the global stability problem at high Reynolds numbers, the leading adjoint modes exhibit a boundary-layer-like structure at the inlet and near the channel walls. For convective stability problem, receptivity is highest in regions close to the walls similar to the global adjoint modes. Structural separation of direct and adjoint modes indicates high non-normality of the linear operators. Structural sensitivity analysis shows that both the global and convective stability are most sensitive to momentum perturbation near the wall and become increasingly sensitive to near-wall perturbations as Reynolds number increases. Slight variation of wall shape tends to affect the linear stability of plane Poiseuille flow significantly at high Reynolds numbers.

The Stability of Plane Poiseuille Flow in a Finite-Length Channel

Abstract The linear stability of plane Poiseuille flow through a finite-length channel is studied. A weakly divergence-free basis finite element method with streamline upwind Petrov–Galerkin (SUPG) stabilization is used to formulate the weak form of the problem. The linear stability characteristics are studied under three possible inlet–outlet boundary conditions, and the corresponding perturbation kinetic energy transfer mechanisms are investigated. Active transfer of perturbation kinetic energy at the channel inlet and outlet, energy production due to convection and dissipation at the flow bulk provide a new perspective in understanding the distinct stability characteristics of plane Poiseuille flow under various boundary conditions.

Instability of vertically stratified horizontal plane Poiseuille flow

Abstract

Linear stability and energy stability of plane Poiseuille flow with isotropic and anisotropic slip boundary conditions

The linear stability and energy stability of the plane Poiseuille flow with the isotropic and anisotropic slip boundary conditions are theoretically analyzed and numerically calculated in this paper. The asymptotic expansions of the critical parameters for the linear stability and energy stability are derived from the eigenvalue equations characterizing the least stable modes. The critical Reynolds number for the linear stability based on 1.5 times of the bulk mean streamwise velocity is found to be Rl2D≈(1+2741l2)R02D when the non-dimensional isotropic slip length l ≪ 1, where R02D≈5772 is the critical Reynolds number under the no-slip boundary condition. The critical Reynolds numbers for the linear stability are calculated for a wide range of anisotropic slip lengths and are found to be no larger than their counterparts in the isotropic slip cases with the same streamwise slip lengths. In the energy stability analyses of the two-dimensional and three-dimensional plane Poiseuille flows with the isotropic slip boundary condition, the critical Reynolds numbers are found to be RlE2D≈(1+14.95l2)R0E2D and RlE3D≈(1+8.37l2)R0E3D, where R0E2D≈87.6 and R0E3D≈49.6 are their counterparts under the no-slip boundary condition. In the three-dimensional plane Poiseuille flow with the anisotropic slip boundary condition, the critical Reynolds number for the energy stability increases with the increase in streamwise slip length lx and with the decrease in spanwise slip length lz, and its first-order approximation is Rlx,lzE3D≈[1+2.41(lx−lz)]R0E3D.

Linear stability of plane Poiseuille flow of a Bingham fluid in a channel with the presence of wall slip

In the present paper, the effects of a channel symmetric wall slip are investigated on the stability of plane Poiseuille flow of a Bingham fluid, based on the modal and non-modal linear stability analysis approaches. Both streamwise and spanwise slip conditions are considered and their results are compared with those of the no-slip condition, in terms of the flow stability. The results are obtained for different dimensionless groups that govern the stability picture, for example the Bingham number, the Reynolds number and the streamwise/spanwise slip numbers. The linearized perturbation equations are obtained and the corresponding eigenvalue problem is solved based on the Chebyshev collocation method at the Gauss-Lobatto grid points, using the QZ algorithm. In the modal analysis, the focus is on the changes made by the channel wall slip on the eigenvalue spectra. The flow is shown to be linearly stable for no-slip and streamwise slip conditions. However, for the spanwise slip condition, an unstable eigenvalue is found for large values of the spanwise slip number. The non-modal stability analysis is based on the transient energy growth of the perturbations, which is calculated using the eigenfunctions of the eigenvalue problem representing the amplitude of the perturbations. Increasing the streamwise slip decreases the maximum energy growth, eventually leading to a flow for which the transient energy only decays in time. Thus, two distinct flows showing the energy growth and energy decay regimes can be identified, for a range of Bingham, Reynolds and streamwise slip numbers.

Hydrodynamic stability of plane Poiseuille flow in Maxwell fluid with cross-flow

This paper deals with the linear stability analysis of a plane Poiseuille in a Maxwell fluid in the presence of a uniform cross-flow with respect to two-dimensional wave disturbances. The physical problem is reduced to a modified Orr–Sommerfeld equation with nonlinear eigenvalues and solved numerically using the Chebyshev spectral collocation method. Attention is focused on the combined effects of uniform cross-flow and the relaxation time of the fluid (Deborah number) on the flow stability. Results obtained in this framework show that for a given Reynolds number the cross-flow can either delay or advances the stability of this system depending on the magnitude of the wave number. This result is observed in the case of a Newtonian fluid as well as in the case of a Maxwell one. An important feature concerns the existence of bicritical states that occur when two different instability modes with incommensurate wave numbers bifurcate simultaneously from the basis state is observed. In addition, it is attempted to determine the critical parameters of instability when one varies the cross-flow in the presence of fluid’s elasticity. It turns out that the cross-flow has a stabilizing effect which becomes less pronounced when the fluid’s elasticity is present.

Stability of plane Poiseuille flow of a Bingham fluid through a deformable neo-Hookean channel

Flow of Bingham fluids is unstable if channel walls are deformable. Laminar flows of Bingham fluids in rigid-walled channels are stable to tiny disturbances. A study finds that if the walls are made deformable, the flow becomes unstable at very low Reynolds number.

Calculation of the Orr-Sommerfeld stability equation for the plane Poiseuille flow

The stability of plane Poiseuille flow depends on eigenvalues and solutions which are generated by solving Orr-Sommerfeld equation with input parameters including real wavenumber and Reynolds number . In the reseach of this paper, the Orr-Sommerfeld equation for the plane Poiseuille flow was solved numerically by improving the Chebyshev collocation method so that the solution of the Orr-Sommerfeld equation could be approximated even and odd polynomial by relying on results of proposition 3.1 that is proved in detail in section 2. The results obtained by this method were more economical than the modified Chebyshev collocation if the comparison could be done in the same accuracy, the same collocation points to find the most unstable eigenvalue. Specifically, the present method needs 49 nodes and only takes 0.0011s to create eigenvalue while the modified Chebyshev collocation also uses 49 nodes but takes 0.0045s to generate eigenvalue with the same accuracy to eight digits after the decimal point in the comparison with , see [4], exact to eleven digits after the decimal point.

Instantaneous linear stability of plane Poiseuille flow forced by spanwise oscillations

In the present work, the stability of a plane Poiseuille flow forced by spanwise oscillations is studied via the instantaneous linear stability theory (LST). For streamwise Poiseuille flow and a spanwise Stokes layer, the superposition of these two linearly stable flows can lead to transient growth of perturbations. Periodic and aperiodic growth rates over time are found. Mode-crossing is observed in the aperiodic mode. Effects of oscillation amplitude W0 and frequency Ω are studied. The maximum instantaneous growth rate increases with increasing W0. At low W0, the stability of the flow is dominated by the corresponding Poiseuille flow or by the interactions between the two flows. At high W0, the stability characteristics of the Poiseuille–Stokes flow are very similar to those of the corresponding Stokes layer. It is demonstrated that oscillations have destabilizing effects in short-time intervals of one oscillation period. Oscillations of extremely low or high Ω have much weaker effects than those of medium Ω. The transient growth of the most unstable mode is traced by using direct numerical simulation (DNS). Instantaneous LST can capture the transient growth but fails to predict the accurate growth rate when the amplification ratio is higher than e9. The difference between LST and DNS is mainly due to the incorrect production of LST without nonlinear transportation. From the growing phase to the decay phase of the transient mode, the production term shifts from the destabilizing role to the stabilizing role because of the correlation-phase reversal of the perturbation velocities.

Linear stability of plane Poiseuille flow in an infinite elastic medium and volcanic tremors

The linear stability of a plane compressible laminar (Poiseuille) flow sandwiched between two semi-infinite elastic media was investigated with the aim of explaining the excitation of volcanic tremors. Our results show that there are several regimes of instability, and the nature of stability significantly depends on the symmetry of oscillatory fluid and solid motion. It has been shown that long-wave symmetric instability occurs at a very small value of the Reynolds number, but it is unlikely that this is the cause of volcanic tremors. We show that antisymmetric (flexural) instability also occurs, involving two parallel Rayleigh waves traveling against the Poiseuille flow, but the critical flow speed is faster than that of symmetric instability. However, if the basic flow profile is nonparabolic because of a nonuniform driving force or nonuniform viscosity, the critical flow speed of antisymmetric instability can be considerably slower than that of symmetric instability. Based on numerical calculations and analytical consideration, we conclude that this anomalous antisymmetric instability is possibly produced by a basaltic magma flow of a few meters per second through a dike with thickness of 1 m and extending for several kilometers; this origin can explain some of the characteristics of volcanic tremors.

Hydrodynamic Stability of Plane Poiseuille Flow of Non-Newtonian Fluids in the Presence of a Transverse Magnetic Field

Hydrodynamic Stability of Plane Poiseuille Flow of Non-Newtonian Fluids in the Presence of a Transverse Magnetic Field

Effect of vortex line distribution in superfluid plane Poiseuille flow instability

AbstractThe hydrodynamic stability of plane Poiseuille flow of superfluid is studied using modal and non-modal analysis. Two modes of instability are predicted, in normal mode stability analysis of the normal fluid, one caused by viscosity similar to the classical mode and another due to mutual friction between superfluid and normal fluid. The mutual friction mode occurs at high wavenumbers, which are stable wavenumbers in classical plane Poiseuille flow. A high superfluid vortex line density alone is not enough to induce instability in normal fluid; a localization of vortex lines is shown to play a major role. The extent of vortex line concentration required to cause instability depends on the density itself. Non-modal instability analysis shows that oblique waves are stronger than streamwise waves, unlike the scenario in classical plane Poiseuille flow.

Stability of plane Poiseuille flow of viscoelastic fluids in the presence of a transverse magnetic field

The linear stability of plan Poiseuille flow of an electrically conducting viscoelastic fluid in the presence of a transverse magnetic field is investigated numerically. The fourth-order Sommerfeld equation gov- erning the stability analysis is solved by spectral method with expansions in lagrange's polynomials, based on collocation points of Gauss-Lobatto. The critical values of Reynolds number, wave number and wave speed are computed. The results are shown through the neutral curve. The main purpose of this work is to check the com- bined effect of magnetic field and fluid's elasticity on the stability of the plane Poiseuille flow. Based on the results obtained in this work, the magnetic field is predicted to have a stabilizing effect on the Poiseuille flow of viscoelastic fluids. Hence, it will be shown that for second-order fluids (K 0) is that the critical Reynolds numbers Rec increase when the Hartman number M increases for certain value of elasticity number K and decrease for others. The latter result is in contrast to previous studies.

Linear stability of plane Poiseuille flow over a generalized Stokes layer

Linear stability of plane Poiseuille flow subject to spanwise velocity forcing applied at the wall is studied. The forcing is stationary and sinusoidally distributed along the streamwise direction. The long-term aim of the study is to explore a possible relationship between the modification induced by the wall forcing to the stability characteristic of the unforced Poiseuille flow and the signifcant capabilities demonstrated by the same forcing in reducing turbulent friction drag. We present in this paper the statement of the mathematical problem, which is considerably more complex that the classic Orr-Sommerfeld-Squire approach, owing to the streamwise-varying boundary condition. We also report some preliminary results which, although not yet conclusive, describe the effects of the wall forcing on modal and non-modal characteristics of the flow stability.

The linear stability of plane Poiseuille flow of Burgers fluid at very low Reynolds numbers

The linear stability of plane Poiseuille flow of Burgers fluid at very low Reynolds numbers is studied. In the previous works, it was shown that plane Poiseuille flows of UCM and Oldroyd-B fluids at very low Reynolds numbers are linearly stable. However, in the present work, unstable modes for Burges fluid are detected with some special value of parameters. In critical stable condition, there are two kinds of perturbation waves. One travels downstream with the name of forward wave, while the other travels upstream with the name of reverse wave. The mean value of the two wave speeds is independent of the characteristic velocity of flow, while the difference between the wave speeds reflects the equivalent velocity of flow. By comparing the perturbation modes for Newtonian fluid and Burgers fluid under critical stable conditions, it can be found that although the shapes of their amplitudes are basically the same, their phase diagrams are significantly different. The neutral curve of Burgers fluid is also different from that of Newtonian fluid. Unstable modes can still be found at high wave numbers. The flow instability is induced due to the phase difference between the perturbation strain and stress. The perturbation energy caused by the forward wave, which is dissipated by the effects of the component of perturbation normal stress in the streamwise direction, is transported by the effects of both the perturbation normal stress in the wall-normal direction and the perturbation shear stress. The converse is true for the reverse wave.

Linear stability analysis for plane-Poiseuille flow of an elastoviscoplastic fluid with internal microstructure for large Reynolds numbers

We study the linear stability of Plane Poiseuille flow of an elastoviscoplastic fluid using a revised version of the model proposed by Putz and Burghelea (A.M.V. Putz, T.I. Burghelea, Rheol. Acta 48 (2009) 673–689). The evolution of the microstructure upon a gradual increase of the external forcing is governed by a structural variable (the concentration of solid material elements) which decays smoothly from unity to zero as the stresses are gradually increased beyond the yield point. Stability results are in close conformity with the ones of a pseudo-plastic fluid. Destabilizing effects are related to the presence of an intermediate transition zone where elastic solid elements coexist with fluid elements. This region brings an elastic contribution which does modify the stability of the flow.

Instabilities of plane Poiseuille flow with a streamwise system rotation

We analyse the stability of plane Poiseuille flow with a streamwise system rotation. It is found that the instability due to two-dimensional perturbations, which sets in at the well-known critical Reynolds number, Rc = 5772.2, for the non-rotating case, is delayed as the rotation is increased from zero, showing a stabilizing effect of rotation. As the rotation is increased further, however, the laminar flow becomes most unstable to perturbations which are three-dimensional. The critical Reynolds number due to three-dimensional perturbations at this higher rotation case is many orders of magnitude less than the corresponding value due to two-dimensional perturbations. We also perform a nonlinear analysis on a bifurcating three-dimensional secondary flow. The secondary flow exhibits a spiral vortex structure propagating in the streamwise direction. It is confirmed that an antisymmetric mean flow in the spanwise direction is generated in the secondary flow.

Dual Role of Wall Slip on Linear Stability of Plane Poiseuille Flow

The linear stability of plane Poiseuille flow is extended to the cases of two wall boundaries maintaining different slip coefficients β1 and β2. We determine the slip coefficient pairs, β1c and β2c, which yield the same critical Reynolds number as the classical no-slip case. It is found that the wall slip may stabilize the flow for β1 > β1c and β2 > β2c, whereas slightly destabilize the flow for β1 < β1c and β2 < β2c.

Some Global Stability Results for Shear Flows of Viscoelastic Fluids

We consider plane shear flows of viscoelastic fluids. For a number of constitutive models, we prove stability of the rest state for perturbations of arbitrary size. We also consider stability of plane Poiseuille flow in a few special cases.

Influence of small imperfections on the stability of plane Poiseuille flow and the limitation of Squire’s theorem

In a recent paper [Luo and Wu, Phys. Fluids 16, 2852 (2004)], a theoretical model was proposed to illustrate the key role played by small imperfections in the subcritical transition of plane Poiseuille flow. In that model, the imperfections are represented by a steady but spatially periodic two-dimensional surface roughness, and a small oscillatory pressure gradient representing the inevitable background fluctuations. The present paper extends the previous work by considering an improved model, where the wall roughness is three-dimensional. A steady perturbation in the form of spatially periodic suction is also considered. In both cases, the resulting steady and unsteady components interact to produce a forcing that is in near resonance with a pair of oblique Tollmien-Schlichting (T-S) waves. The latter are generated as a result, and grow in proportion to the time during the initial stage. This excitation process and the subsequent nonlinear development of the T-S waves are studied by a weakly nonlinear analysis. A forced nonlinear (and nonlocal) amplitude equation is derived to provide a simple mathematical framework to link the critical Reynolds number for nonlinear instability to the level of imperfections. The theoretical result demonstrates that the onset critical Reynolds number is lower for three-dimensional than for two-dimensional disturbances if the imperfection exceeds a critical level, which turns out to be remarkably small. This offers a possible explanation as to why Squire’s theorem is applicable only when external perturbations are suppressed to an extremely low level. Further analysis of the nonlinear interactions indicates that the much reduced critical Reynolds number for three-dimensional external perturbations can be attributed to the fact that the excited pair of oblique T-S waves interact to generate extremely strong streaks, which then result in a very strong nonlinear destabilizing effect.