Hydrodynamic stability in plane Poiseuille flow with finite amplitude disturbances

1972 ◽  
Vol 51 (4) ◽  
pp. 687-704 ◽  
Author(s):  
W. D. George ◽  
J. D. Hellums
Keyword(s):  
Reynolds Number ◽  
Linear Theory ◽  
Hydrodynamic Stability ◽  
Poiseuille Flow ◽  
Classical Problem ◽  
Finite Amplitude ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Navier Stokes Equation ◽  
Numerical Process

A general method for studying two-dimensional problems in hydrodynamic stability is presented and applied to the classical problem of predicting instability in plane Poiseuille flow. The disturbance stream function is expanded in a Fourier series in the axial space dimension which, on substitution into the Navier-Stokes equation, leads to a system of parabolic partial differential equations in the coefficient functions. An efficient, stable and accurate numerical method is presented for solving these equations. It is demonstrated that the numerical process is capable of accurate reproduction of known results from the linear theory of hydrodynamic stability.Disturbances that are stable according to linear theory are shown to become unstable with the addition of finite amplitude effects. This seems to be the first work of quantitative value for disturbances of moderate and larger amplitudes. A relationship between critical amplitude and Reynolds number is reported, the form of which indicates the existence of an absolute critical Reynolds number below which an arbitrary disturbance cannot be made unstable, no matter how large its initial amplitude. The critical curve shows significantly less effect of amplitude than do those obtained by earlier workers.

Finite-amplitude neutral disturbances in plane Poiseuille flow

1974 ◽  
Vol 63 (4) ◽  
pp. 765-771 ◽  
Author(s):  
W. D. George ◽  
J. D. Hellums ◽  
B. Martin
Keyword(s):  
Reynolds Number ◽  
Asymptotic Stability ◽  
Poiseuille Flow ◽  
Velocity Fluctuation ◽  
Fourier Expansion ◽  
Finite Amplitude ◽  
Neutral Stability ◽  
Plane Poiseuille Flow ◽  
Number Of Solutions ◽  
Degree Of Confidence

Finite-amplitude disturbances in plane Poiseuille flow are studied by a method involving Fourier expansion with numerical solution of the resulting partial differential equations in the coefficient functions. A number of solutions are developed which extend to relatively long times so that asymptotic stability or instability can be established with a degree of confidence. The amplitude for neutral stability is established for a fixed wavenumber for two values of the Reynolds number. Details of the neutral velocity fluctuation are presented. These and earlier results are expressed in terms of the asymptotic amplitude and compared with results obtained by prior workers. The results indicate that the expansion methods used by prior workers may be valid only for amplitudes considerably smaller than 0·01.

Non-linear theory of unstable plane Poiseuille flow

1969 ◽  
Vol 38 (2) ◽  
pp. 401-414 ◽  
Author(s):  
E. H. Dowell
Keyword(s):  
Linear Theory ◽  
Poiseuille Flow ◽  
Theoretical Study ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Mathematical Technique ◽  
Navier Stokes Equations ◽  
Fourier Decomposition ◽  
Non Linear

A theoretical study of plane Poiseuille flow is made using the full non-linear Navier-Stokes equations. The mathematical technique employed is to use a Fourier decomposition in the streamwise spatial variable, a Galerkin expansion in the lateral variable and numerical integration with respect to time. By retaining the non-linear terms, the limit cycle oscillations of an unstable (in a linear sense) flow are obtained. A brief investigation of the possibility of instability due to large (non-linear) disturbances is also made. The results are negative for the cases examined. Comparisons with results previously obtained by others from linear theory illustrate the accuracy and efficacy of the method.

scholarly journals Study of the instability of the Poiseuille flow using a thermodynamic formalism

2015 ◽  
Vol 112 (31) ◽  
pp. 9518-9523 ◽  
Author(s):  
Jianchun Wang ◽  
Qianxiao Li ◽  
Weinan E
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Thermodynamic Formalism ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Action Function ◽  
Continuous Transition ◽  
Wide Range ◽  
The Stability

The stability of the plane Poiseuille flow is analyzed using a thermodynamic formalism by considering the deterministic Navier–Stokes equation with Gaussian random initial data. A unique critical Reynolds number, Rec≈2,332, at which the probability of observing puffs in the solution changes from 0 to 1, is numerically demonstrated to exist in the thermodynamic limit and is found to be independent of the noise amplitude. Using the puff density as the macrostate variable, the free energy of such a system is computed and analyzed. The puff density approaches zero as the critical Reynolds number is approached from above, signaling a continuous transition despite the fact that the bifurcation is subcritical for a finite-sized system. An action function is found for the probability of observing puffs in a small subregion of the flow, and this action function depends only on the Reynolds number. The strategy used here should be applicable to a wide range of other problems exhibiting subcritical instabilities.

The effect of external forcing on the stability of plane Poiseuille flow

1978 ◽  
Vol 359 (1699) ◽  
pp. 453-478 ◽  
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Stable Solution ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Plane Poiseuille Flow ◽  
Critical Reynolds Numbers ◽  
Threshold Amplitude ◽  
The Stability

The stability of plane Poiseuille flow in a channel forced by a wavelike motion on one of the channel walls is investigated. The amplitude Є of this forcing is taken to be small. The most dangerous modes of forcing are identified and it is found in general the critical Reynolds number is changed by O (Є) 2 . However, we identify two particular modes of forcing which give rise to decrements of order Є 2/3 and Є in the critical Reynolds number. Some types of forcing are found to generate sub critical stable finite amplitude perturbations to plane Poiseuille flow. This contrasts with the unforced case where the only stable solution is the zero amplitude solution. The forcing also deforms the unstable subcritical limit cycle solution from its usual circular shape into a more complicated shape. This has an effect on the threshold amplitude ideas suggested by, for example, Meksyn & Stuart (1951). It is found that the phase of disturbances must also be considered when finding the amplitude dependent critical Reynolds numbers.

Instability due to viscosity stratification

1967 ◽  
Vol 27 (2) ◽  
pp. 337-352 ◽  
Author(s):  
Chia-Shun Yih
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Hydrodynamic Stability ◽  
Poiseuille Flow ◽  
Plane Couette Flow ◽  
Plane Poiseuille Flow ◽  
Neutral Mode ◽  
Usual Theory

The principal aim of this paper is to show that the variation of viscosity in a fluid can cause instability. Plane Couette-Poiseuille flow of two superposed layers of fluids of different viscosities between two horizontal plates is considered, and it is found that both plane Poiseuille flow and plane Couette flow can be unstable, however small the Reynolds number is. The unstable modes are in the neighbourhood of a hidden neutral mode for the case of a single fluid, which is entirely ignored in the usual theory of hydrodynamic stability, and are brought out by the viscosity stratification.

scholarly journals Localized vortex/Tollmien–Schlichting wave interaction states in plane Poiseuille flow

2016 ◽  
Vol 791 ◽  
pp. 97-121 ◽  
Author(s):  
L. J. Dempsey ◽  
K. Deguchi ◽  
P. Hall ◽  
A. G. Walton
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Wave Interaction ◽  
Travelling Wave ◽  
Navier Stokes ◽  
Spanwise Direction ◽  
Plane Poiseuille Flow ◽  
Tollmien Schlichting

Strongly nonlinear three-dimensional interactions between a roll–streak structure and a Tollmien–Schlichting wave in plane Poiseuille flow are considered in this study. Equations governing the interaction at high Reynolds number originally derived by Bennett et al. (J. Fluid Mech., vol. 223, 1991, pp. 475–495) are solved numerically. Travelling wave states bifurcating from the lower branch linear neutral point are tracked to finite amplitudes, where they are observed to localize in the spanwise direction. The nature of the localization is analysed in detail near the relevant spanwise locations, revealing the presence of a singularity which slowly develops in the governing interaction equations as the amplitude of the motion is increased. Comparisons with the full Navier–Stokes equations demonstrate that the finite Reynolds number solutions gradually approach the numerical asymptotic solutions with increasing Reynolds number.

Stability of plane Poiseuille flow to periodic disturbances of finite amplitude in the vicinity of the neutral curve

1967 ◽  
Vol 29 (1) ◽  
pp. 31-38 ◽  
Author(s):  
Chaim L. Pekeris ◽  
Boris Shkoller
Keyword(s):  
Periodic Solutions ◽  
Linear Theory ◽  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Neutral Curve ◽  
Lower Branch ◽  
Plane Poiseuille Flow ◽  
Periodic Disturbances ◽  
Existence Of Periodic Solutions ◽  
The Stability

Stuart (1960) has developed a theory of the stability of plane Poiseuille flow to periodic disturbances of finite amplitude which, in the neighbourhood of the neutral curve, leads to an equation of the Landau (1944) type for the amplitude A of the disturbance: \[ d|A|^2/dt = k_1|A|^2 - k_2|A|^4. \] If k2 is positive in the supercritical region (R > RC) where k1 is positive, then, according to Stuart, there is a possibility of the existence of periodic solutions of finite amplitude which asymptotically approach a constant value of (k1/k2)½. We have evaluated the coefficient k2 and found that there indeed exists a zone in the (α, R)-plane where it is positive. This is the zone inside the dashed curve shown in figure 1, with the region of instability predicted by the linear theory included inside the ‘neutral curve’. Stuart's theory and Eckhaus's generalization thereof could apply in the overlapping zone just above the lower branch of the neutral curve.

Analysis of Burnett Stresses and Entropy Generation for Pressure-Driven Plane Poiseuille Flow

2020 ◽  
Author(s):  
Upendra Yadav ◽  
Amit Agrawal
Keyword(s):  
Shear Stress ◽  
Entropy Generation ◽  
Knudsen Number ◽  
Poiseuille Flow ◽  
Gas Flow ◽  
Constitutive Expression ◽  
Navier Stokes ◽  
Normal Stresses ◽  
Plane Poiseuille Flow ◽  
Navier Stokes Equation

Abstract In this paper, we undertake an analytical study of stresses (augmented and Onsager-Burnett) and entropy generation for the plane Poiseuille flow problem, and their variation with Knudsen number. The gas flow is assumed to be 2D laminar, fully developed, compressible, and isothermal; these assumptions make the problem amenable to analytical treatment. The variation of stresses and entropy generation have been analyzed over a large range of Knudsen number. It is found that the augmented and OBurnett normal stresses are of opposite signs to the corresponding Navier-Stokes stresses, while the magnitude of the net normal stress increases with Knudsen number. The magnitude of the augmented Burnett shear stress is insignificant as compared to the augmented Burnett normal stresses. A minimum in the variation of normalized entropy generation against the Knudsen number (Kn) is observed at Kn close to unity, and is being reported for the first time. The magnitude of net entropy generation from the summation of Navier-Stokes and augmented Burnett stresses is found to be positive, even in the transition regime of gas flow. Further, an appearance of minimum or maximum in normalized net shear stress versus Knudsen number, depending upon the lateral position in the micro-channel has also been observed. Altogether, this analysis supports the validity of the Navier-Stokes equation with modified constitutive expression, even for higher Knudsen numbers. Moreover, the significant terms of Burnett stress are pointed out by the analysis, which can help in developing reduced-order model for these equations.

Finite amplitude instability of second-order fluids in plane Poiseuille flow

1972 ◽  
Vol 52 (2) ◽  
pp. 273-285 ◽  
Author(s):  
Larry V. McIntire ◽  
C. H. Lin
Keyword(s):  
Hydrodynamic Stability ◽  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Second Order ◽  
Equilibrium States ◽  
Newtonian Fluids ◽  
Plane Poiseuille Flow ◽  
Entire Spectrum ◽  
Second Order Fluids

The hydrodynamic stability of plane Poiseuille flow of second—order fluids to finite amplitude disturbances is examined using the method of Stuart, and Watson as extended by Reynolds & Potter. For slightly non-Newtonian fluids subcritical instabilities are predicted. No supercritical equilibrium states are expected if the entire spectrum of disturbance wavelengths is present. Possible implications with respect to the Toms phenomenon are discussed.

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