On the stability of Poiseuille-Couette flow: a bifurcation from infinity (original) (raw)
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Nonlinear stability results for plane Couette and Poiseuille flows
Physical Review E
In this article we prove, choosing an appropriately weighted L 2energy equivalent to the classical energy, that the plane Couette and Poiseuille flows are nonlinearly stable with respect to streamwise perturbations for any Reynolds number. In this case the coefficient of time-decay of the energy is π 2 /(2Re), and it is a bound from above of the time-decay of streamwise perturbations of linearized equations. We also prove that the plane Couette and Poiseuille flows are nonlinearly stable if the Reynolds number is less then Re Orr / sin ϕ when the perturbation is a tilted perturbation, i.e. 2D perturbations with wave vector which forms an angle ϕ ∈ [0, π/2] with the direction i of the motion. Re Orr is the Orr (1907) critical Reynolds number for spanwise perturbations which, for the Couette flow is Re Orr = 177.22 and for the Poiseuille flow is Re Orr = 175.31.
Stability of plane Couette–Poiseuille flow
Journal of Fluid Mechanics, 1966
The stability of a two-dimensional Couette-Poiseuille flow is investigated. The primary unidirectional flow is between two infinite parallel plates, one of which moves relative to the other. The results for the case of Poiseuille flow agree with Lin's results and all flows for which the plate ...
Linear Instability of Plane Couette and Poiseuille Flows
—It is shown that linear instability of plane Couette flow can take place even at finite Reynolds numbers Re > Re th ≈ 139, which agrees with the experimental value of Re th ≈ 150 ± 5 [16, 17]. This new result of the linear theory of hydrodynamic stability is obtained by abandoning traditional assumption of the longitudinal periodicity of disturbances in the flow direction. It is established that previous notions about linear stability of this flow at arbitrarily large Reynolds numbers relied directly upon the assumed separation of spatial variables of the field of disturbances and their longitudinal periodicity in the linear theory. By also abandoning these assumptions for plane Poiseuille flow, a new threshold Reynolds number Re th ≈ 1035 is obtained, which agrees to within 4% with experiment—in contrast to 500% discrepancy for the previous estimate of Re th ≈ 5772 obtained in the framework of the linear theory under assumption of the " normal " shape of disturbances [2].
The instability of oscillatory plane Poiseuille flow
Journal of Fluid Mechanics, 1982
The instability of oscillatory plane Poiseuille flow, in which the pressure gradient is time-periodically modulated, is investigated by a perturbation technique. The Floquet exponents (i.e. the complex growth rates of the disturbances to the oscillatory flow) are computed by series expansions, in powers of the oscillatory to steady flow velocity amplitude ratio, about the values of the growth rates of the disturbances of the steady flow. It is shown that the oscillatory flow is more stable than the steady flow for values of Reynolds number and disturbance wave number in the vicinity of the steady flow critical point and for values of frequencies of imposed oscillation greater than about one tenth of the frequency of the steady flow neutral disturbance. At very high and low values of imposed oscillation frequency, the unsteady flow is slightly less stable than the steady flow. These results hold for the values of the velocity amplitude ratio at least up to 0·25.
Modal and non-modal linear stability of the plane Bingham Poiseuille flow
Journal of Fluid Mechanics, 2007
The receptivity problem of plane Bingham-Poiseuille flow with respect to weak perturbations is addressed. The relevance of this study is highlighted by the linear stability analysis results (spectra and pseudospectra). The first part of the present paper thus deals with the classical normal-mode approach in which the resulting eigenvalue problem is solved using the Chebychev collocation method. Within the range of parameters considered, the Poiseuille flow of Bingham fluid is found to be linearly stable. The second part investigates the most amplified perturbations using the non-modal approach. At a very low Bingham number (B 1), the optimal disturbance consists of almost streamwise vortices, whereas at moderate or large B the optimal disturbance becomes oblique. The evolution of the obliqueness as function of B is determined. The linear analysis presented also indicates, as a first stage of a theoretical investigation, the principal challenges of a more complete nonlinear study.
Physics of Fluids A: Fluid Dynamics, 1991
The linear stability of pressure-driven flow between a sliding inner cylinder and a stationary outer cylinder is studied numerically. Attention is restricted to axisymmetric disturbances (n = 0), and asymmetric disturbances with azimuthal wave numbers n = 1,2, and 3. Neutral stability curves in the Reynolds number versus the wave-number plane are presented as a function of the sliding velocity of the inner cylinder for select values of the radius ratio K. Overall, the sliding velocity of the inner cylinder has a net stabilizing effect on all modes studied. Results presented for K = 2 show that individual disturbance modes can be completely stabilized by increasing the sliding velocity. In particular, when the sliding velocity is approximately 25% of the maximum Poiseuille velocity, the neutral curve for the n = 2 mode vanishes; at 36% of the maximum Poiseuille velocity, the neutral curve for the n = 0 mode vanishes, and at 65%, the neutral curve for the n = 1 mode vanishes. For a stationary inner cylinder the asymmetric modes are generally the least stable, though this conclusion does depend on the magnitude of K. As K+ 1 the axisymmetric mode is found to be the most dangerous.
Stability of plane Couette flow and pipe Poiseuille flow
2007
The theoretical derivations were done in close cooperation between the authors, both of them contributing in an equal amount. The author of this thesis had the main responsibility for the computer implementations and wrote section 5 in the report. Malin Siklosi had the main responsibility for the literature studies and wrote sections 1-4 in the report. This paper is also part of the licentiate thesis [1].
Physics of Fluids
The bifurcation and instability of nonisothermal annular Poiseuille flow (NAPF) of air as well as water is studied. We have emphasized the impact of a gap between cylinders in terms of curvature parameter (C) for axisymmetric as well as nonaxisymmetric disturbances. The results from the linear stability analysis reveal that the first azimuthal mode acts as a least stable mode of the NAPF of air for relatively small values of C. In this situation, even though for some values of C, the NAPF has supercritical bifurcation, but the same flow may experience subcritical bifurcation under zero azimuthal mode. It has also been observed that for relatively larger values of the Reynolds number (Re) and lower values of C, the NAPF under axisymmetric disturbance always exhibits subcritical bifurcation. However, for small values of Re, the NAPF exhibits only supercritical bifurcation. The finite amplitude analysis predicts only supercritical bifurcation of NAPF of water. The influence of nonlinear interaction of different harmonics on the amplitude profile as well as kinetic energy spectrum is investigated. The amplitude profile possesses a jump in the vicinity of a point where the type of bifurcation is changed. In the subcritical regime, the induced shear production due to modification of the gradient production acts as a main destabilizing factor balanced by the gradient production of kinetic energy.
High Speed Couette - Poiseuille Flow Stability in R everse Flow Conditions
The linear stability of reverse high speed-viscous plane Couette - Poiseuille flow is investigated numerically. The conservation equations along with Sutherland's viscosity law are studied using a second order finite difference scheme. Basic velocity and temperature distributions are perturbed by a small amplitude normal-mode disturbance. Small amplitude disturbance equations are solved numerically using a global method to find all the eigenvalues at finite Reynolds numbers. The results indicate that instabilities occur, although the corresponding growth rates are often small. The aim of the study is to see the effect of the reverse flow on the stability compared to the direct flow. In the combined plane Couette - Poiseuille flow, the new mode, Mode 0, which seems to be a member of even modes such as Mode II, is the most unstable mode.