Viscous incompressible flow between concentric rotating spheres. Part 2. Hydrodynamic stability

1971 ◽  
Vol 49 (2) ◽  
pp. 305-318 ◽  
Author(s):  
B. R. Munson ◽  
D. D. Joseph
Keyword(s):  
Reynolds Number ◽  
Incompressible Flow ◽  
Hydrodynamic Stability ◽  
Stability Problem ◽  
Critical Reynolds Number ◽  
Viscous Incompressible Flow ◽  
Concentric Spheres ◽  
Angular Velocities ◽  
Energy Theory ◽  
The Stability

The energy theory of hydrodynamic stability is applied to the viscous incompressible flow of a fluid contained between two concentric spheres which rotate about a common axis with prescribed angular velocities. The critical Reynolds number is calculated for various radius and angular velocity ratios such that it is certain the basic laminar motion is stable to any disturbances. The stability problem is solved by means of a toroidal–poloidal representation of the disturbance flow and numerical integration of the resulting eigenvalue problem.

Stability analysis of a two-phase suspension between rotating porous cylinder with the radial flow

2015 ◽  
Vol 29 (05) ◽  
pp. 1550014
Author(s):  
Feng-Hui Wang ◽  
Yu-Chuan Zhu ◽  
Zhan-Hong Wan ◽  
Song He
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Hydrodynamic Stability ◽  
Stability Problem ◽  
Radial Flow ◽  
Wave Number ◽  
Porous Cylinder ◽  
Two Phase ◽  
Stabilizing Effect ◽  
The Stability

The hydrodynamic stability analysis of viscous flow between rotating porous cylinder has been researched for a long time by many researchers. But little works have been carried out about the linear stability analysis of the two-phase suspension. When the radial flow is present, the linear hydrodynamic stability analysis of suspension has been carried out between rotating porous cylinder. We know that the continuous and Stokes equations cannot only solve the stability problem of the continuous fluid phase, but also solving the stability problem of the discontinuous particle phase. The stability equations from an eigenvalue problem that was solved by a numerical technique based on Wan's method. The results reveal that the radial Reynolds number have a great effect on the critical Taylor number in the suspension. In this paper, we also researched on how the critical Taylor number changes as the radius ratio η, the axial wave number k, the particle concentration and the circumferential direction wave number happen to change with the radial Reynolds number increasing range from -5 to 5. Thus, our research discovered the radial inflow and outflow have a stabilizing effect on the two-phase suspension and the circumferential direction wave number also has a stabilizing effect.

scholarly journals Non-normal origin of modal instabilities in rotating plane shear flows

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190550
Author(s):  
Sharath Jose ◽  
Rama Govindarajan
Keyword(s):  
Reynolds Number ◽  
Shear Flows ◽  
Critical Reynolds Number ◽  
Flow System ◽  
Neutral Line ◽  
Plane Couette Flow ◽  
Plane Shear ◽  
Perturbation Amplitude ◽  
Fundamental Reason ◽  
The Stability

Small variations introduced in shear flows are known to affect stability dramatically. Rotation of the flow system is one example, where the critical Reynolds number for exponential instabilities falls steeply with a small increase in rotation rate. We ask whether there is a fundamental reason for this sensitivity to rotation. We answer in the affirmative, showing that it is the non-normality of the stability operator in the absence of rotation which triggers this sensitivity. We treat the flow in the presence of rotation as a perturbation on the non-rotating case, and show that the rotating case is a special element of the pseudospectrum of the non-rotating case. Thus, while the non-rotating flow is always modally stable to streamwise-independent perturbations, rotating flows with the smallest rotation are unstable at zero streamwise wavenumber, with the spanwise wavenumbers close to that of disturbances with the highest transient growth in the non-rotating case. The instability critical rotation number scales inversely as the square of the Reynolds number, which we demonstrate is the same as the scaling obeyed by the minimum perturbation amplitude in non-rotating shear flow needed for the pseudospectrum to cross the neutral line. Plane Poiseuille flow and plane Couette flow are shown to behave similarly in this context.

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.

Instability of Viscoelastic Fluids in Blasius Flow

2010 ◽  
Author(s):  
Amin Doostmohammadi ◽  
Seyyedeh Negin Mortazavi
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Orthogonal Polynomials ◽  
Hydrodynamic Stability ◽  
Analytical Approach ◽  
Critical Reynolds Number ◽  
Blasius Flow ◽  
Linearized Stability ◽  
De Moivre’S Formula ◽  
Elasticity Number

In this paper, we study the hydrodynamic stability of a viscoelastic Walters B liquid in the Blasius flow. A linearized stability analysis is used and orthogonal polynomials which are related to de Moivre’s formula are implemented to solve Orr–Sommerfeld eigenvalue equation. An analytical approach is used in order to find the conditions of instability for Blasius flow and Critical Reynolds number is found for various combinations of the elasticity number. Based on the results, the destabilizing effect of elasticity on Blasius flow is determined and interpreted.

Linear Spatial Stability of Developing Flow in a Parallel Plate Channel

1981 ◽  
Vol 48 (1) ◽  
pp. 192-194 ◽  
Author(s):  
S. C. Gupta ◽  
V. K. Garg
Keyword(s):  
Reynolds Number ◽  
Velocity Profile ◽  
Parallel Plate ◽  
Critical Reynolds Number ◽  
Two Dimensional ◽  
Spatial Stability ◽  
Percent Change ◽  
Developing Flow ◽  
The Stability ◽  
Plate Channel

It is found that even a 5 percent change in the velocity profile produces a 100 percent change in the critical Reynolds number for the stability of developing flow very close to the entrance of a two-dimensional channel.

The Stability of Water Flow Over Heated and Cooled Flat Plates

1968 ◽  
Vol 90 (1) ◽  
pp. 109-114 ◽  
Author(s):  
Ahmed R. Wazzan ◽  
T. Okamura ◽  
A. M. O. Smith
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Free Stream ◽  
Reynolds Numbers ◽  
Stream Temperature ◽  
Heated Wall ◽  
Flat Plates ◽  
Critical Reynolds Numbers ◽  
The Stability ◽  
Sommerfeld Equation

The theory of two-dimensional instability of laminar flow of water over solid surfaces is extended to include the effects of heat transfer. The equation that governs the stability of these flows to Tollmien-Schlichting disturbances is the Orr-Sommerfeld equation “modified” to include the effect of viscosity variation with temperature. Numerical solutions to this equation at high Reynolds numbers are obtained using a new method of integration. The method makes use of the Gram-Schmidt orthogonalization technique to obtain linearly independent solutions upon numerically integrating the “modified Orr-Sommerfeld” equation using single precision arithmetic. The method leads to satisfactory answers for Reynolds numbers as high as Rδ* = 100,000. The analysis is applied to the case of flow over both heated and cooled flat plates. The results indicate that heating and cooling of the wall have a large influence on the stability of boundary-layer flow in water. At a free-stream temperature of 60 deg F and wall temperatures of 60, 90, 120, 135, 150, 200, and 300deg F, the critical Reynolds numbers Rδ* are 520, 7200, 15200, 15600, 14800, 10250, and 4600, respectively. At a free-stream temperature of 200F and wall temperature of 60 deg F (cooled case), the critical Reynolds number is 151. Therefore, it is evident that a heated wall has a stabilizing effect, whereas a cooled wall has a destabilizing effect. These stability calculations show that heating increases the critical Reynolds number to a maximum value (Rδ* max = 15,700 at a temperature of TW = 130 deg F) but that further heating decreases the critical Reynolds number. In order to determine the influence of the viscosity derivatives upon the results, the critical Reynolds number for the heated case of T∞ = 40 and TW = 130 deg F was determined using (a) the Orr-Sommerfeld equation and (b) the present governing equation. The resulting critical Reynolds numbers are Rδ* = 140,000 and 16,200, respectively. Therefore, it is concluded that the terms pertaining to the first and second derivatives of the viscosity have a considerable destabilizing influence.

The stability of the flow of an elastic-viscous liquid in a curves channel

1963 ◽  
Vol 274 (1358) ◽  
pp. 371-385 ◽  
Keyword(s):  
Reynolds Number ◽  
Pressure Gradient ◽  
Viscous Liquid ◽  
Critical Reynolds Number ◽  
Curved Channel ◽  
Main Effect ◽  
The Stability

Consideration is given to the stability of the flow of an idealized elastico-viscous liquid in a narrow curved channel, the motion being due to a pressure gradient acting around the channel. It is shown that the main effect of the elasticity of the liquid is to lower the value of the critical Reynolds number at which instability occurs.

Accurate solution of the Orr–Sommerfeld stability equation

1971 ◽  
Vol 50 (4) ◽  
pp. 689-703 ◽  
Author(s):  
Steven A. Orszag
Keyword(s):  
Reynolds Number ◽  
Chebyshev Polynomials ◽  
Critical Reynolds Number ◽  
Great Accuracy ◽  
Accurate Solution ◽  
Plane Poiseuille Flow ◽  
Stability Equation ◽  
Orthogonal Functions ◽  
The Stability ◽  
Sommerfeld Equation

The Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the QR matrix eigenvalue algorithm. It is shown that results of great accuracy are obtained very economically. The method is applied to the stability of plane Poiseuille flow; it is found that the critical Reynolds number is 5772·22. It is explained why expansions in Chebyshev polynomials are better suited to the solution of hydrodynamic stability problems than expansions in other, seemingly more relevant, sets of orthogonal functions.

Experimental Study on the Stability of the Flow Behind an Accelerating Circular Cylinder

2007 ◽  
Author(s):  
A. Inasawa ◽  
K. Toda ◽  
M. Asai
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Cylinder Wake ◽  
Global Instability ◽  
The Stability ◽  
Dimensional Instability ◽  
Wake Oscillation

Disturbance growth in the wake of a circular cylinder moving at a constant acceleration is examined experimentally. The cylinder is installed on a carriage moving in the still air. The results show that the critical Reynolds number for the onset of the global instability leading to a self-sustained wake oscillation increases with the magnitude of acceleration, while the Strouhal number of the growing disturbance at the critical Reynolds number is not strongly dependent on the magnitude of acceleration. It is also found that with increasing the acceleration, the Ka´rma´n vortex street remains two-dimensional even at the Reynolds numbers around 200 where the three-dimensional instability occurs to lead to the vortex dislocation in the case of cylinder moving at constant velocity or in the case of cylinder wake in the steady oncoming flow.

Torque Characteristics for Spherical Annulus Flow

1979 ◽  
Vol 101 (2) ◽  
pp. 284-286 ◽  
Author(s):  
A. M. Waked ◽  
B. R. Munson
Keyword(s):  
Reynolds Number ◽  
Viscous Fluid ◽  
Secondary Flow ◽  
Reynolds Numbers ◽  
Flow Effect ◽  
Spherical Annulus ◽  
Concentric Spheres ◽  
Annulus Flow ◽  
Angular Velocities

The torque needed to rotate concentric spheres when the spherical annulus gap between them is filled with a viscous fluid depends on the Reynolds number and the ratio of the angular velocities of the two spheres. Experimental torque results for low to moderate Reynolds numbers are presented. The secondary flow effect is evident.

Sign in / Sign up

Export Citation Format

Share Document