Stability of plane Poiseuille flow to periodic disturbances of finite amplitude

1969 ◽  
Vol 39 (3) ◽  
pp. 611-627 ◽  
Author(s):  
C. L. Pekeris ◽  
B. Shkoller
Keyword(s):  
Poiseuille Flow ◽  
Subsequent Development ◽  
Critical Value ◽  
Finite Amplitude ◽  
Higher Order ◽  
Initial Disturbance ◽  
Plane Poiseuille Flow ◽  
Mean Flow ◽  
Linear Ordinary Differential Equations ◽  
Sommerfeld Equation

A disturbance of finite amplitude λ, which is periodic in the direction of the axis of the channel, is superimposed on plane Poiseuille flow, and the subsequent development of the disturbance is studied. The disturbance is represented by an expansion in the eigenfunctions of the Orr-Sommerfeld equation with coefficients which are functions of the time, and an accurate numerical solution of the truncated system of non-linear ordinary differential equations for the coefficients is obtained.It is found that even for Reynolds numbers R less than the critical value Rc, the flow breaks down when λ exceeds a critical value λc(R). This is shown in figure 11 for the case when the initial disturbance is represented by the first mode of the Orr-Sommerfeld equation. The development of this type of disturbance is illustrated in figures 1, 3 and 13 and, for the case of a higher-order mode initial disturbance, in figure 14. Near the time of breakdown, the curvature of the modified mean flow changes sign (figure 15), but a disturbance may die down even after a reversal in the sign of the curvature has taken place (see figure 2).The stability of plane Poiseuille flow to disturbances of finite amplitude is affected by the characteristics of the higher-order modes of the Orr-Sommerfeld equation. As shown in figures 4, 10, and 12, and in figures 5, 6, and 7, these modes are either of a ‘boundary type’, characteristic of the region near the wall, or of an ‘interior type’, characteristic of the centre of the channel. The modes in the transition zone, where the two types merge, are easily amplified through mutual constructive interference, even though individually they have high damping coefficients. It is these transition modes which are mainly responsible for the breakdown through finite amplitude effects.

The neutral curves for periodic perturbations of finite amplitude of plane Poiseuille flow

1969 ◽  
Vol 39 (3) ◽  
pp. 629-639 ◽  
Author(s):  
C. L. Pekeris ◽  
B. Shkoller
Keyword(s):  
Power Series ◽  
Perturbation Method ◽  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Neutral Curve ◽  
Plane Poiseuille Flow ◽  
Mean Flow ◽  
Periodic Perturbations ◽  
Neutral Curves ◽  
The Mean

It is shown that there exist undamped solutions for perturbations of finite amplitude of plane Poiseuille flow, which are periodic in the direction of the axis of the channel. The shift in the ‘neutral curve’ as a function of the amplitude λ* of the disturbance is shown in figure 2. The solution is obtained by a perturbation method in which the eigenfunctions and the eigenvalue c are expanded in power series of the amplitude λ, as shown in (14), (15), (16) and (17). Near the neutral curve for a finite amplitude disturbance, the curvature of the mean flow shows a tendency to become negative (figure 5).

On the stability of plane Poiseuille flow to finite-amplitude disturbances, considering the higher-order Landau coefficients

1983 ◽  
Vol 133 ◽  
pp. 179-206 ◽  
Author(s):  
P. K. Sen ◽  
D. Venkateswarlu
Keyword(s):  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Higher Order ◽  
Plane Poiseuille Flow ◽  
Supercritical Region ◽  
Subcritical Region ◽  
Poor Convergence ◽  
The Stability ◽  
Good Agreement ◽  
False Problem

In this work a study has been made of the Stuart (1960)–Watson (1960) formalism as applied to plane Poiseuille flow. In particular, the higher-order Landau coefficients have been calculated for the Reynolds & Potter (1967) method and for the Watson (1960) method. The results have been used to study the convergence of the Stuart–Landau series. A convergence curve in the (α, R)-plane has been obtained by using suitable Domb–Sykes plots. In the region of poor convergence of the series, and also in a part of the divergent region of the series, it has been found that the Shanks (1955) method, using the em1 transformation, serves as a very effective way of finding the proper sum of the series, or of finding the proper antilimit of the series. The results for the velocity calculations at R = 5000 are in very good agreement with Herbert's (1977) Fourier-truncation method using N = 4. The Watson method and the Reynolds & Potter method have also been compared inthe subcritical and supercritical regions. It is found in the supercritical region that there is not much difference in the results by the ‘true problem’ of Watson and the ‘false problem’ of Reynolds & Potter when the respective series in both methods are summed by the Shanks method. This fact could possibly be capitalized upon in the subcritical region, where the Watson method is difficult to apply.

On the instability of the flow in a squeeze lubrication film

1990 ◽  
Vol 430 (1879) ◽  
pp. 347-375 ◽  
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Flow Instability ◽  
Initial Distribution ◽  
Neutral Curve ◽  
Time Dependent ◽  
Order Partial Differential Equation ◽  
Plane Poiseuille Flow ◽  
Disturbance Wave ◽  
Sommerfeld Equation

When two parallel plates move normal to each other with a slow time-dependent speed, the velocity field developed in the intervening film of fluid is approximately that of plane Poiseuille flow, except that the magnitude of the velocity is dependent on time and on the coordinate parallel to the planes. This fact is intrinsic to Reynolds’ lubrication theory, and can be shown to follow from the Navier-Stokes equations when both the modified Reynolds number ( Re M ) and an aspect ratio ( δ ) are small. The modified Reynolds number is the product of δ and an actual Reynolds number ( Re ), which is based on the gap between the planes and on a characteristic velocity. The occurrence of flow instability and of turbulence in the film depend on Re . Typical values of Re , which are known to be required for the linear instability of plane Poiseuille flow, are of order 6000. This condition can be achieved, even if Re M is of order 1, provided that δ is of order 10 -4 . Such parameter values are typical of lubrication problems. The Orr-Sommerfeld equation governing flow instability is derived in this paper by use of the WKBJ technique, δ being the approximate small parameter to represent the small length-scale of the disturbance oscillations compared with the larger scale of the basic laminar flow. However, the coefficients in the Orr-Sommerfeld equation depend on slow space and time variables. Consequently the eigenrelation, derivable from the Orr-Sommerfeld equation and the associated boundary conditions, constitutes a nonlinear first-order partial differential equation for a phase function. This equation is solved by use of Charpit’s method for certain special forms of the time-dependent gap between the planes, followed by detailed numerical calculations. The relation between time-dependence and flow instability is delineated by the calculated results. In detail the nature of the instability can be described as follows. We consider a disturbance wave at or near a particular station, the initial distribution of amplitude being gaussian in the slow coordinate parallel to the planes. In the context of the Orr-Sommerfeld equation and its eigenrelation, the particular station implies an equivalent Reynolds number, while the initial distribution of the disturbance wave implies an equivalent wavenumber. As time increases, the disturbance wave can be considered to move in the instability diagram of equivalent wavenumber against Reynolds number, in the sense that these parameters are time- and space-dependent for the evolution of the disturbance-wave system. For our detailed calculations we use a quadratic approximation to the eigenrelation, an approximation which is quite accurate. If the initial distribution implies a point within the neutral curve, when the plates are squeezed together the equivalent wavenumber falls while the equivalent Reynolds number rises, and amplification takes place until the lower branch of the neutral curve is nearly crossed. If the plates are pulled apart (dilatation) the equivalent wavenumber rises, while the Reynolds number drops, and amplification takes place until the upper branch of the neutral curve has been just crossed. In the case of dilatation the transition from amplification to damping takes place more quickly than for the case of squeezing, in part due to the geometry of the neutral curve.

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.

The initial-value problem for three-dimensional disturbances in plane Poiseuille flow of helium II

2008 ◽  
Vol 598 ◽  
pp. 227-244 ◽  
Author(s):  
LARS B. BERGSTRÖM
Keyword(s):  
Poiseuille Flow ◽  
Three Dimensional ◽  
Peak Level ◽  
Minor Effect ◽  
Transient Growth ◽  
Plane Poiseuille Flow ◽  
Mean Flow ◽  
Normal Fluid ◽  
Helium Ii ◽  
Fluid Field

The time development of small three-dimensional disturbances in plane Poiseuille flow of helium II is considered. The study is conducted by considering the interaction of a normal fluid field and a superfluid field. The interaction is caused by a mutual friction forcing between the two flow fields. Specifically, the stability of the normal fluid affected by the mutual forcing is considered. Compared to the ordinary fluid case where the mutual forcing is not present, the presence of the mutual forcing implies a substantial increase of the transient growth of the disturbances. The increase of the transient growth occurs because the mutual forcing reduces the damping of the disturbances. The phase of transient growth becomes thereby more prolonged and higher levels of amplification are reached. There is also a minor effect on the transient growth caused by the modification of the mean flow owing to the mutual forcing. The strongest transient growth occurs for streamwise elongated disturbances, i.e. disturbances with streamwise wavenumber α = 0. When α increases beyond zero, the transient amplification quickly becomes reduced. Striking differences compared to the ordinary fluid case are that the largest transient amplification does not occur when the spanwise wavenumber (β) is close to two and that the peak level of the disturbance energy density amplification does not depend on the square of the Reynolds number.

Numerical Computation of the Stability of Plane Poiseuille Flow

1978 ◽  
Vol 45 (1) ◽  
pp. 13-18 ◽  
Author(s):  
L. Wolf ◽  
Z. Lavan ◽  
H. J. Nielsen
Keyword(s):  
Stream Function ◽  
Poiseuille Flow ◽  
Numerical Technique ◽  
Finite Amplitude ◽  
Wave Number ◽  
Plane Poiseuille Flow ◽  
Governing Equations ◽  
The Stability ◽  
Successive Over Relaxation ◽  
Small Disturbances

The hydrodynamic stability of plane Poiseuille flow to infinitesimal and finite amplitude disturbances is investigated using a direct numerical technique. The governing equations are cast in terms of vorticity and stream function using second-order central differences in space. The vorticity equation is used to advance the vorticity values in time and successive over-relaxation is used to solve the stream function equation. Two programs were prepared, one for the linearized and the other for the complete disturbance equations. Results obtained by solving the linearized equations agree well with existing solutions for small disturbances. The nonlinear calculations reveal that the behavior of a disturbance depends on the amplitude and on the wave number. The behavior at wave numbers below and above the linear critical wave number is drastically different.

Drop fall-off from pendent rivulets

1997 ◽  
Vol 338 ◽  
pp. 173-201 ◽  
Author(s):  
ALEXANDRA INDEIKINA ◽  
IGOR VERETENNIKOV ◽  
HSUEH-CHIA CHANG
Keyword(s):  
Critical Value ◽  
Finite Amplitude ◽  
Mean Flow ◽  
Capillary Length ◽  
Lubrication Equation ◽  
Long Wave ◽  
Static Contact ◽  
Axial Curvature ◽  
Good Agreement ◽  
Matched Asymptotic Analysis

Drops fall off a viscous pendent rivulet on the underside of a plane when the inclination angle θ, measured with respect to the horizontal, is below a critical value θc. We estimate this θc by studying the existence of finite-amplitude drop solutions to a long-wave lubrication equation. Through a partial matched asymptotic analysis, we establish that fall-off occurs by two distinct mechanisms. For θ>ϕ, where ϕ is the static contact angle, a jet mechanism results when a mean-flow steepening effect cannot provide sufficient axial curvature to counter gravity. This fall-off mechanism occurs if the rivulet width B, which is normalized with respect to the capillary length H=(σ/ρg cosθ)1/2, exceeds a critical value defined by β=−cosB>1/4. For θ<ϕ, the normal azimuthal curvature is the dominant force against fall-off and the azimuthal capillary force. The corresponding critical condition is found to be 1.5β1/6>tanθ/tanϕ. Both criteria are in good agreement with our experimental data.

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