Unconditional Linear Stability of Plane Couette-Poiseuille Flow in Presence of Cross Flow

2010 ◽  
Author(s):  
Anirban Guha ◽  
Ian A. Frigaard
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Poiseuille Flow ◽  
Energy Production ◽  
Energy Analysis ◽  
Cross Flow ◽  
Base Flow ◽  
Lower Range ◽  
Long Wavelength ◽  
Flow Reynolds Number

We have investigated the linear stability of plane Couette-Poiseuille flow in the presence of a cross-flow. The base flow is characterised by the cross flow Reynolds number, Ri and the dimensionless wall velocity, k. Corresponding to each k ∈ [0,1], we have observed two ranges of Ri for which the flow is unconditionally linearly stable. In the lower range, we have a stabilisation of long wavelengths leading to a cut-off Ri. In this range, cross-flow stabilisation and Couette stabilisation appear to act via very similar mechanisms in this range, leading to the potential for robust compensatory design of flow stabilisation using either mechanism. As Ri is increased, we see first destabilisation and then stabilisation at very large Ri. The instability is again a long wavelength mechanism. A linear energy analysis reveals that in this range the Reynolds stress becomes amplified, the critical layer is irrelevant and viscous dissipation is completely dominated by the energy production/negation, which approximately balances at criticality.

scholarly journals On the stability of plane Couette–Poiseuille flow with uniform crossflow

2010 ◽  
Vol 656 ◽  
pp. 417-447 ◽  
Author(s):  
ANIRBAN GUHA ◽  
IAN A. FRIGAARD
Keyword(s):  
Linear Stability ◽  
Poiseuille Flow ◽  
Energy Production ◽  
Base Flow ◽  
Two Dimensional ◽  
Wall Velocity ◽  
Resonant Interactions ◽  
Flow Stabilization ◽  
Tollmien Schlichting ◽  
The Stability

We present a detailed study of the linear stability of the plane Couette–Poiseuille flow in the presence of a crossflow. The base flow is characterized by the crossflow Reynolds number Rinj and the dimensionless wall velocity k. Squire's transformation may be applied to the linear stability equations and we therefore consider two-dimensional (spanwise-independent) perturbations. Corresponding to each dimensionless wall velocity, k ∈ [0, 1], two ranges of Rinj exist where unconditional stability is observed. In the lower range of Rinj, for modest k we have a stabilization of long wavelengths leading to a cutoff Rinj. This lower cutoff results from skewing of the velocity profile away from a Poiseuille profile, shifting of the critical layers and the gradual decrease of energy production. Crossflow stabilization and Couette stabilization appear to act via very similar mechanisms in this range, leading to the potential for a robust compensatory design of flow stabilization using either mechanism. As Rinj is increased, we see first destabilization and then stabilization at very large Rinj. The instability is again a long-wavelength mechanism. An analysis of the eigenspectrum suggests the cause of instability is due to resonant interactions of Tollmien–Schlichting waves. A linear energy analysis reveals that in this range the Reynolds stress becomes amplified, the critical layer is irrelevant and viscous dissipation is completely dominated by the energy production/negation, which approximately balances at criticality. The stabilization at very large Rinj appears to be due to decay in energy production, which diminishes like Rinj−1. Our study is limited to two-dimensional, spanwise-independent perturbations.

scholarly journals Linear Stability of a Steady Convective Flow between Permeable Cylinders

Fluids ◽  
2021 ◽  
Vol 6 (10) ◽  
pp. 342
Author(s):  
Maksims Zigunovs ◽  
Andrei Kolyshkin ◽  
Ilmars Iltins
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Chemical Reaction ◽  
Linear Stability Analysis ◽  
Convective Flow ◽  
Base Flow ◽  
Marginal Stability ◽  
Heat Sources ◽  
Rate Of Increase

Linear stability analysis of a steady convective flow in a tall vertical annulus caused by nonlinear heat sources is conducted in the paper. Heat sources are generated as a result of a chemical reaction. The effect of radial cross-flow through permeable porous walls of the annulus is analyzed. The problem is relevant to biomass thermal conversion. The base flow solution is obtained by solving nonlinear boundary value problem. Linear stability analysis is performed, using collocation method. The calculations show that radial inward or outward flow has a stabilizing effect on the flow, while the increase in the Frank–Kamenetskii parameter (proportional to the intensity of the chemical reaction) destabilizes the flow. The increase in the Reynolds number based on the radial velocity leads to the appearance of the second minimum on the marginal stability curves. The rate of increase in the critical Grashof number with respect to the Reynolds number is different for inward and outward radial flows.

Linear stability of rotating Hagen-Poiseuille flow

1981 ◽  
Vol 108 ◽  
pp. 101-125 ◽  
Author(s):  
Fredrick W. Cotton ◽  
Harold Salwen
Keyword(s):  
Reynolds Number ◽  
Angular Velocity ◽  
Linear Stability ◽  
Poiseuille Flow ◽  
Neutral Stability ◽  
Unstable Region ◽  
Azimuthal Index ◽  
Expansion Technique ◽  
Simple Scaling

Linear stability of rotating Hagen-Poiseuille flow has been investigated by an orthonormal expansion technique, confirming results by Pedley and Mackrodt and extending those results to higher values of the wavenumber |α|, the Reynolds number R, and the azimuthal index n. For |α| [gsim ] 2, the unstable region is pushed to considerably higher values of R and the angular velocity, Ω. In this region, the neutral stability curves obey a simple scaling, consistent with the unstable modes being centre modes. For n = 1, individual neutral stability curves have been calculated for several of the low-lying eigenmodes, revealing a complicated coupling between modes which manifests itself in kinks, cusps and loops in the neutral stability curves; points of degeneracy in the R, Ω plane; and branching behaviour on curves which circle a point of degeneracy.

Turbulent Wake of a Stack and the Influence of Velocity Ratio

2006 ◽  
Author(s):  
M. S. Adaramola ◽  
D. Sumner ◽  
D. J. Bergstrom
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Reynolds Shear Stress ◽  
Velocity Ratio ◽  
Cross Flow ◽  
Turbulent Wake ◽  
Mean Velocity ◽  
Distinct Structure ◽  
Flow Reynolds Number ◽  
Cross Flow Velocity

The effect of the jet-to-cross-flow velocity ratio, R, on the turbulent wake of a cylindrical stack of AR = 9 was investigated with two-component thermal anemometry. The cross-flow Reynolds number was ReD = 2.3×104, the jet Reynolds number ranged from Red = 7×103 to 4.6×104, and R was varied from 0 to 3. The stack was partially immersed in a flat-plate turbulent boundary layer, with a boundary layer thickness-to-height ratio of δ/H = 0.5 at the location of the stack. The flow around the stack was broadly classified into three flow regimes depending on the value of R, which were the downwash (R < 0.5), cross-wind dominated (0.5 < R < 1.5), and jet-dominated (R > 1.5) regimes. Each flow regime had a distinct structure to the mean velocity (streamwise and wall-normal directions), turbulence intensity (streamwise and wall-normal directions), and Reynolds shear stress fields.

scholarly journals Linear stability of confined flow around a 180-degree sharp bend

2017 ◽  
Vol 822 ◽  
pp. 813-847 ◽  
Author(s):  
Azan M. Sapardi ◽  
Wisam K. Hussam ◽  
Alban Pothérat ◽  
Gregory J. Sheard
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Two Dimensional ◽  
Two Dimensional Flow ◽  
The Stability

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

VORTEX INTERFERENCE IN THE WAKE OF TWO TANDEM CIRCULAR CYLINDERS IN A CROSS FLOW

2005 ◽  
Vol 19 (28n29) ◽  
pp. 1595-1598 ◽  
Author(s):  
KAZUO OHMI ◽  
SUXIA LI ◽  
SEUNGHEE JEON ◽  
LINGYUN CHEN
Keyword(s):  
Reynolds Number ◽  
Image Analysis ◽  
Cross Correlation ◽  
Laser Sheet ◽  
Cross Flow ◽  
Circular Cylinders ◽  
Water Tank ◽  
Cylinder Wake ◽  
Flow Reynolds Number ◽  
Spacing Ratio

The wake of two circular cylinders in tandem arrangement is investigated by flow visualization and PIV experiments in a towing water tank. The two cylinders are spaced at L/d (spacing ratio) = 2.0 to 15.0 and the cross flow Reynolds number ranges from 60 to 120. The flow is seeded with fine Rilsan particles and illuminated by a 2 mm thick laser sheet. The PIV image analysis is done by a standard cross correlation scheme with a powerful validation algorithm followed by multi-pass adaptive cross correlation iterations. The main objective of the study is to investigate the characteristics of the downstream cylinder wake changing considerably with the spacing ratio of the two cylinders.

scholarly journals Modal and non-modal stability analysis of electrohydrodynamic flow with and without cross-flow

2015 ◽  
Vol 770 ◽  
pp. 319-349 ◽  
Author(s):  
Mengqi Zhang ◽  
Fulvio Martinelli ◽  
Jian Wu ◽  
Peter J. Schmid ◽  
Maurizio Quadrio
Keyword(s):  
Stability Analysis ◽  
Electric Field ◽  
Linear Stability ◽  
Poiseuille Flow ◽  
Cross Flow ◽  
Transient Growth ◽  
Unstable Flow ◽  
Electrohydrodynamic Flow ◽  
Stability Threshold ◽  
Energy Growth

We report the results of a complete modal and non-modal linear stability analysis of the electrohydrodynamic flow for the problem of electroconvection in the strong-injection region. Convective cells are formed by the Coulomb force in an insulating liquid residing between two plane electrodes subject to unipolar injection. Besides pure electroconvection, we also consider the case where a cross-flow is present, generated by a streamwise pressure gradient, in the form of a laminar Poiseuille flow. The effect of charge diffusion, often neglected in previous linear stability analyses, is included in the present study and a transient growth analysis, rarely considered in electrohydrodynamics, is carried out. In the case without cross-flow, a non-zero charge diffusion leads to a lower linear stability threshold and thus to a more unstable flow. The transient growth, though enhanced by increasing charge diffusion, remains small and hence cannot fully account for the discrepancy of the linear stability threshold between theoretical and experimental results. When a cross-flow is present, increasing the strength of the electric field in the high-$\mathit{Re}$Poiseuille flow yields a more unstable flow in both modal and non-modal stability analyses. Even though the energy analysis and the input–output analysis both indicate that the energy growth directly related to the electric field is small, the electric effect enhances the lift-up mechanism. The symmetry of channel flow with respect to the centreline is broken due to the additional electric field acting in the wall-normal direction. As a result, the centres of the streamwise rolls are shifted towards the injector electrode, and the optimal spanwise wavenumber achieving maximum transient energy growth increases with the strength of the electric field.

scholarly journals Linear stability of Hunt's flow

2010 ◽  
Vol 649 ◽  
pp. 115-134 ◽  
Author(s):  
JĀNIS PRIEDE ◽  
SVETLANA ALEKSANDROVA ◽  
SERGEI MOLOKOV
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Linear Stability ◽  
Critical Reynolds Number ◽  
Transverse Magnetic ◽  
Base Flow ◽  
Maximal Velocity ◽  
Square Duct ◽  
Stream Function Formulation ◽  
The Magnetic Field

We analyse numerically the linear stability of the fully developed flow of a liquid metal in a square duct subject to a transverse magnetic field. The walls of the duct perpendicular to the magnetic field are perfectly conducting whereas the parallel ones are insulating. In a sufficiently strong magnetic field, the flow consists of two jets at the insulating walls and a near-stagnant core. We use a vector stream function formulation and Chebyshev collocation method to solve the eigenvalue problem for small-amplitude perturbations. Due to the two-fold reflection symmetry of the base flow the disturbances with four different parity combinations over the duct cross-section decouple from each other. Magnetic field renders the flow in a square duct linearly unstable at the Hartmann number Ha ≈ 5.7 with respect to a disturbance whose vorticity component along the magnetic field is even across the field and odd along it. For this mode, the minimum of the critical Reynolds number Rec ≈ 2018, based on the maximal velocity, is attained at Ha ≈ 10. Further increase of the magnetic field stabilizes this mode with Rec growing approximately as Ha. For Ha > 40, the spanwise parity of the most dangerous disturbance reverses across the magnetic field. At Ha ≈ 46 a new pair of most dangerous disturbances appears with the parity along the magnetic field being opposite to that of the previous two modes. The critical Reynolds number, which is very close for both of these modes, attains a minimum, Rec ≈ 1130, at Ha ≈ 70 and increases as Rec ≈ 91Ha1/2 for Ha ≫ 1. The asymptotics of the critical wavenumber is kc ≈ 0.525Ha1/2 while the critical phase velocity approaches 0.475 of the maximum jet velocity.

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