The instability of oscillatory plane Poiseuille flow (original) (raw)
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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].
On absolute linear instability analysis of plane Poiseuille flow by a semi-analytical treatment
Journal of the Brazilian Society of Mechanical Sciences and Engineering, 37 (2): 495–505 (2015), DOI: 10.1007/s40430-014-0187-2, 2015
"The absolute linear hydrodynamic instability of the plane Poiseuille flow is investigated by solving the Orr– Sommerfeld equation using the semi-analytical treatment of the Adomian decomposition method (ADM). In order to use the ADM, a new zero-order ADM approximation is defined. The results for the spectrum of eigenvalues are obtained using various orders of the ADM approximations and discussed. A comparative study of the results for the first, second and third eigenvalues with the ones from a previously published work is also presented. A monotonic trend of approach of decreasing relative error with the increase of the orders of ADM approximation is indicated. The results for the first, second and third eigenvalues show that they are in good agreement within 1.5 % error with the ones obtained by a previously published work using the Chebyshev spectral method. The results also show that the first eigenvalue is positioned in the unstable zone of the spectrum, while the second and third eigenvalues are located in the stable zone."
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.
Asymptotic stability and transient growth in pulsatile Poiseuille flow through a compliant channel
Journal of Fluid Mechanics, 2017
The time-asymptotic linear stability of pulsatile flow in a channel with compliant walls is studied together with the evaluation of modal transient growth within the pulsation period of the basic flow as well as non-modal transient growth. Both one (vertical-displacement) and two (vertical and axial) degrees-of-freedom compliant-wall models are implemented. Two approaches are developed to study the dynamics of the coupled fluid–structure system, the first being a Floquet analysis in which disturbances are decomposed into a product of exponential growth and a sum of harmonics, while the second is a time-stepping technique for the evolution of the fundamental solution (monodromy) matrix. A parametric study of stability in the non-dimensional parameter space, principally defined by Reynolds number ($Re$), Womersley number ($Wo$) and amplitude of the applied pressure modulation ($\unicode[STIX]{x1D6EC}$), is then conducted for compliant walls of fixed geometric and material properties. ...
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.
On the stability of Poiseuille-Couette flow: a bifurcation from infinity
Journal of Fluid Mechanics, 1985
The linear and weakly nonlinear stability of Poiseuilldouette flow is considered for various values of the relative wall velocity 2u,. An account is given first of the asymptotic upper and lower branches of the linear neutral curve(s), followed by their disappearance, as u, is increased. Two main (and one minor) neutral curves are found to exist for smaller O(1) (or lesser) values of u,, then one for moderate O(1) values of u,, and none for larger 0 ( 1 ) values of u,. The cut-off velocity at which each main neutral curve disappears is determined, and in each case the whole neutral curve for u, just below the cut-off value is determined in closed form. Secondly, weakly nonlinear solutions are found to bifurcate subcritically from the neutral curve for u, just below cut-off, but to 'bifurcate from infinity ' just above cut-off. This identifies a minimum threshold amplitude at the entry to the regime where no linear neutral curve exists.
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].
Stability bounds on turbulent Poiseuille flow
Journal of Fluid Mechanics, 1988
For steady-state turbulent flows with unique mean properties, we determine a sense in which the mean velocity is linearly supercritical. The shear-turbulence literature on this point is ambiguous. As an example, we reassess the stability of mean profiles in turbulent Poiseuille flow. The Reynolds & Tiederman (1967) numerical study is used as a starting point. They had constructed a class of one-dimensional flows which included, within experimental error, the observed profile. Their numerical solutions of the resulting Orr-Sommerfeld problems led them to conclude that the Reynolds number for neutral infinitesimal disturbances was twenty-five times the Reynolds number characterizing the observed mean flow. They found also that the first nonlinear corrections were stabilizing. In the realized flow, this latter conclusion appears incompatible with the former. Hence, we have sought a more complete set of velocity profiles which could exhibit linear instability, retaining the requirement that the observed velocity profile is included in the set. We have added two dynamically generated modifications of the mean. The first addition is a fluctuation in the curvature of the mean flow generated by a Reynolds stress whose form is determined by the neutrally stable Orr-Sommerfeld solution. We find that this can reduce the stability of the observed flow by as much as a factor of two. The second addition is the zero-average downstream wave associated with the above Reynolds stress. The three-dimensional linear instability of this modification can even render the observed flow unstable. Those wave amplitudes that just barely will ensure instability of the observed flow are determined. The relation of these particular amplitudes t o the limiting conditions admitted by an absolute stability criterion for disturbances on the mean flow is found. These quantitative results from stability theory lie in the observationally determined Reynolds-Tiederman similarity scheme, and hence are insensitive to changes in Reynolds number.
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.
Quarterly of Applied Mathematics, 2001
This paper presents an instability theory in which a mean flow and multiple wave interactions in the Poiseuille flow transition process are studied. It is shown that not only can this mean flow term come as the result of the Fourier decomposition of a general disturbance, it can also come as an exact solution to the unsteady Navier-Stokes equations. The presence of this term, though small, can produce totally different linear and nonlinear stability behavior for the flow at subcritical Reynolds numbers. In the linear stability case, with the presence of this mean flow perturbation term, the instabilities are obtained well below the critical value of 5772. When this mean flow term is introduced into the interactions with other harmonic perturbation waves, for the plane Poiseuille flow case with Reynolds numbers around 1200, the nonlinear interactions rapidly modify the total mean flow profile toward the mean flow profile observed in turbulence while the other two-and three-dimensional waves remain small. The initial energies needed to trigger the instabilities are much smaller than those reported by previous investigators. The intermittent character of the disturbance observed in transition experiments is also captured.