Stability of plane Poiseuille flow of a dilute suspension of slender fibres

1978 ◽  
Vol 87 (2) ◽  
pp. 321-333 ◽  
Author(s):  
Fritz H. Bark ◽  
Hernán Tinoco
Keyword(s):  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Volume Fraction ◽  
Plane Poiseuille Flow ◽  
Mean Flow ◽  
Initial Value ◽  
Large Wave ◽  
Inclination Angles ◽  
Dilute Suspension ◽  
The Mean

The linear hydrodynamic stability problem for plane Poiseuille flow of a dilute suspension of rigid fibres is solved numerically. The constitutive equation given by Batchelor (1970a, b, 1971) is used to model the rheological properties of the suspension. The resulting eigenvalue problem is shown to be singular. The appropriate contour in the complex plane is determined by considering an initial-value problem. It is shown that, for a fixed, but not too large, inclination of the wave front to the mean flow, the fibres cause the critical Reynolds number to increase monotonically with the product of the volume fraction of the fibres and the square of their aspect ratio. The stabilizing influence of the fibres seems to vanish for large wave inclination angles.

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).

Instabilities of plane Poiseuille flow with a streamwise system rotation

2008 ◽  
Vol 603 ◽  
pp. 189-206 ◽  
Author(s):  
S. MASUDA ◽  
S. FUKUDA ◽  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Secondary Flow ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Plane Poiseuille Flow ◽  
Mean Flow ◽  
System Rotation ◽  
Spiral Vortex

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.

scholarly journals Quantifying the Effects of Wave—Current Interactions on Tidal Energy Resource at Sites in the English Channel Using Coupled Numerical Simulations

Energies ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 3625
Author(s):  
Jon Hardwick ◽  
Ed B. L. Mackay ◽  
Ian G. C. Ashton ◽  
Helen C. M. Smith ◽  
Philipp R. Thies
Keyword(s):  
Wave Model ◽  
Tidal Energy ◽  
English Channel ◽  
Flow Conditions ◽  
Mean Flow ◽  
Radiation Stress ◽  
Large Wave ◽  
Stress Gradients ◽  
The Mean ◽  
Flow Power

Numerical modeling of currents and waves is used throughout the marine energy industry for resource assessment. This study compared the output of numerical flow simulations run both as a standalone model and as a two-way coupled wave–current simulation. A regional coupled flow-wave model was established covering the English Channel using the Delft D-Flow 2D model coupled with a SWAN spectral wave model. Outputs were analyzed at three tidal energy sites: Alderney Race, Big Roussel (Guernsey), and PTEC (Isle of Wight). The difference in the power in the tidal flow between coupled and standalone model runs was strongly correlated to the relative direction of the waves and currents. The net difference between the coupled and standalone runs was less than 2.5%. However, when wave and current directions were aligned, the mean flow power was increased by up to 7%, whereas, when the directions were opposed, the mean flow power was reduced by as much as 9.6%. The D-Flow Flexible Mesh model incorporates the effects of waves into the flow calculations in three areas: Stokes drift, forcing by radiation stress gradients, and enhancement of the bed shear stress. Each of these mechanisms is discussed. Forcing from radiation stress gradients is shown to be the dominant mechanism affecting the flow conditions at the sites considered, primarily caused by dissipation of wave energy due to white-capping. Wave action is an important consideration at tidal energy sites. Although the net impact on the flow power was found to be small for the present sites, the effect is site specific and may be significant at sites with large wave exposure or strong asymmetry in the flow conditions and should thus be considered for detailed resource and engineering assessments.

Difference in Critical Reynolds Number Between Hagen-Poiseuille and Plane Poiseuille Flows

2001 ◽  
Author(s):  
Hidesada Kanda
Keyword(s):  
Experimental Data ◽  
Reynolds Number ◽  
Poiseuille Flow ◽  
Average Velocity ◽  
Critical Reynolds Number ◽  
Inlet Section ◽  
Parallel Plates ◽  
Plane Poiseuille Flow ◽  
Contraction Ratio ◽  
Significant Difference

Abstract For plane Poiseuille flow, results of previous investigations were studied, focusing on experimental data on the critical Reynolds number, the entrance length, and the transition length. Consequently, concerning the natural transition, it was confirmed from the experimental data that (i) the transition occurs in the entrance region, (ii) the critical Reynolds number increases as the contraction ratio in the inlet section increases, and (iii) the minimum critical Reynolds number is obtained when the contraction ratio is the smallest or one, and there is no-shaped entrance or straight parallel plates. Its value exists in the neighborhood of 1300, based on the channel height and the average velocity. Although, for Hagen-Poiseuille flow, the minimum critical Reynolds number is approximately 2000, based on the pipe diameter and the average velocity, there seems to be no significant difference in the transition from laminar to turbulent flow between Hagen-Poiseuille flow and plane Poiseuille flow.

Lateral Migration of Large and Small Droplets Suspended in Channel Flow

2013 ◽  
Author(s):  
Masato Makino ◽  
Masako Sugihara-Seki
Keyword(s):  
Numerical Simulation ◽  
Channel Flow ◽  
Poiseuille Flow ◽  
Size Difference ◽  
Two Dimensional ◽  
Lateral Migration ◽  
Plane Poiseuille Flow ◽  
Surface Tensions ◽  
Segregation Behavior ◽  
The Mean

In order to investigate the effect of the size differences between suspended particles on the segregation behavior in channel flow of multicomponent suspensions, we conduct a two-dimensional numerical simulation for suspensions of fluid droplets of two different sizes subjected to a plane Poiseuille flow. The large and small droplets are assumed to have equal surface tensions and equal internal viscosities. The temporal evolutions of the lateral positions of the large and small droplets relative to the channel centerline are computed for various size ratios and area ratios of the large and small droplets. It is found that the small droplets tend to migrate toward the channel walls with increasing fraction of the large droplets and that the mean lateral positions of the large droplets are always closer to the channel centerline compared to the mean lateral positions of the small droplets, which represent the margination of the small droplets and the segregation of the droplets caused by the size difference. These trends are enhanced as the size ratio of large and small droplets is increased.

Effect of Number of Roughing Passes on the Dynamic Transformation of Austenite during Simulated Plate Rolling

2018 ◽  
Vol 941 ◽  
pp. 717-722
Author(s):  
Samuel F. Rodrigues ◽  
Fulvio Siciliano ◽  
Clodualdo Aranas Jr. ◽  
Gedeon S. Reis ◽  
Brian J. Allen ◽  
...  
Keyword(s):  
Grain Size ◽  
Volume Fraction ◽  
Austenite Phase ◽  
Mean Flow ◽  
Grain Sizes ◽  
Dynamic Transformation ◽  
Plate Rolling ◽  
Forward Transformation ◽  
The Mean ◽  
Rough Rolling

When austenite is deformed within the austenite phase field, it partially transforms dynamically into ferrite. Here, plate rolling simulations were carried out on an X70 steel using rough rolling passes of 0.4 strain each. The influence of the number of roughing passes on the grain size and volume fraction of induced ferrite was determined. Up to three roughing passes applied at 1100 °C followed by 5 finishing passes at 900 °C were employed. The sample microstructures were analysed by means of metallographic techniques. Both the critical strain to the onset of dynamic transformation as well as the grain size decreased with pass number during the roughing simulations. For the finishing passes, the mean flow stresses (MFS`s) applicable to each schedule decreased when a higher number of roughing passes was applied. The volume fraction of dynamically formed ferrite retained after simulated rolling increased with the roughing pass number. This is ascribed to the increased amount of ferrite retransformed into austenite and the finer grain sizes produced during roughing. The forward transformation is considered to occur displacively while the retransformation into austenite during holding takes place by a diffusional mechanism. This indicates that both dynamic transformation (DT) and dynamic recrystallization were taking place during straining.

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 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.

Energy growth of three-dimensional disturbances in plane Poiseuille flow

1991 ◽  
Vol 224 ◽  
pp. 241-260 ◽  
Author(s):  
L. Hårkan Gustavsson
Keyword(s):  
Energy Density ◽  
Poiseuille Flow ◽  
Initial Conditions ◽  
Formal Solution ◽  
Three Dimensional ◽  
Propagation Speed ◽  
Inhomogeneous Equation ◽  
Normal Velocity ◽  
Plane Poiseuille Flow ◽  
Initial Value

The development of a small three-dimensional disturbance in plane Poiseuille flow is considered. Its kinetic energy is expressed in terms of the velocity and vorticity components normal to the wall. The normal vorticity develops according to the mechanism of vortex stretching and is described by an inhomogeneous equation, where the spanwise variation of the normal velocity acts as forcing. To study specifically the effect of the forcing, the initial normal vorticity is set to zero and the energy density in the wavenumber plane, induced by the normal velocity, is determined. In particular, the response from individual (and damped) Orr–Sommerfeld modes is calculated, on the basis of a formal solution to the initial-value problem. The relevant timescale for the development of the perturbation is identified as a viscous one. Even so, the induced energy density can greatly exceed that associated with the initial normal velocity, before decay sets in. Initial conditions corresponding to the least-damped Orr–Sommerfeld mode induce the largest energy density and a maximum is obtained for structures infinitely elongated in the streamwise direction. In this limit, the asymptotic solution is derived and it shows that the spanwise wavenumbers at which the largest amplification occurs are 2.60 and 1.98, for symmetric and antisymmetric normal vorticity, respectively. The asymptotic analysis also shows that the propagation speed for induced symmetric vorticity is confined to a narrower range than that for antisymmetric vorticity. From a consideration of the neglected nonlinear terms it is found that the normal velocity component cannot be nonlinearly affected by the normal vorticity growth for structures with no streamwise dependence.

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