Stability of a viscous liquid flowing down a flexible boundary

1968 ◽  
Vol 46 (18) ◽  
pp. 2059-2063
Author(s):  
A. S. Gupta
Keyword(s):  
Reynolds Number ◽  
Shear Stress ◽  
Phase Shift ◽  
Viscous Liquid ◽  
Critical Reynolds Number ◽  
Elastic Parameter ◽  
Viscous Damping ◽  
Wave Disturbances ◽  
Long Wave ◽  
The Stability

The stability characteristics of a viscous liquid flowing down a flexible boundary are investigated for long-wave disturbances. In the case of a normally compliant boundary which resembles a thin stretched membrane resting on an elastic foundation with viscous damping, it is found that the critical Reynolds number Rc increases with the elastic parameter. For a boundary showing purely tangential compliance, Rc is found to depend on the phase shift between the oscillation of the shear stress and tangential deformation.

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.

On stability of parallel flow of an incompressible fluid of variable density and viscosity

1962 ◽  
Vol 58 (4) ◽  
pp. 646-661 ◽  
Author(s):  
P. G. Drazin
Keyword(s):  
Incompressible Fluid ◽  
Water Waves ◽  
Salt Water ◽  
Reynolds Numbers ◽  
Variable Density ◽  
Stability Equation ◽  
Wave Disturbances ◽  
Long Wave ◽  
Vertical Variations ◽  
The Stability

ABSTRACTSome aspects of generation of water waves by wind and of turbulence in a heterogeneous fluid may be described by the theory of hydrodynamic stability. The technical difficulties of these problems of instability have led to obscurities in the literature, some of which are elucidated in this paper. The stability equation for a basic steady parallel horizontal flow under the influence of gravity is derived carefully, the undisturbed fluid having vertical variations of density and viscosity. Methods of solution of the equation for large Reynolds numbers and for long-wave disturbances are described. These methods are applied to simple models of wind blowing over water and of fresh water flowing over salt water.

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.

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.

Stability of spiral flow between concentric circular cylinders at low axial Reynolds number

1965 ◽  
Vol 21 (4) ◽  
pp. 635-640 ◽  
Author(s):  
Subhendu K. Datta
Keyword(s):  
Reynolds Number ◽  
Perturbation Theory ◽  
Viscous Liquid ◽  
Axial Flow ◽  
Circular Cylinders ◽  
Rotating Cylinders ◽  
Spiral Flow ◽  
The Stability

The stability of a viscous liquid between two concentric rotating cylinders with an axial flow has been investigated. Attention has been confined to the case when the cylinders are rotating in the same direction, the gap between the cylinders is small and the axial flow is small. A perturbation theory valid in the limit when the axial Reynolds number R → 0 has been developed and corrections have been obtained for Chandrasekhar's earlier results.

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.

Effect of surfactants on the long-wave stability of two-layer oscillatory film flow

2021 ◽  
Vol 928 ◽  
Author(s):  
Cheng-Cheng Wang ◽  
Haibo Huang ◽  
Peng Gao ◽  
Xi-Yun Lu
Keyword(s):  
Film Flow ◽  
Viscosity Ratio ◽  
Density Ratio ◽  
Thickness Ratio ◽  
High Frequency Oscillation ◽  
Key Factors ◽  
Layer Film ◽  
Wave Disturbances ◽  
Long Wave ◽  
The Stability

The stability of the two-layer film flow driven by an oscillatory plate under long-wave disturbances is studied. The influence of key factors, such as thickness ratio ( $n$ ), viscosity ratio ( $m$ ), density ratio ( $r$ ), oscillatory frequency ( $\beta$ ) and insoluble surfactants on the stability behaviours is studied systematically. Four special Floquet patterns are identified, and the corresponding growth rates are obtained by solving the eigenvalue problem of the fourth-order matrix. A small viscosity ratio ( $m\le 1$ ) may stabilize the flow but it depends on the thickness ratio. If the viscosity ratio is not very small ( $m>0.1$ ), in the $(\beta ,n)$ -plane, stable and unstable curved stripes appear alternately. In other words, under the circumstances, if the two-layer film flow is unstable, slightly adjusting the thickness of the upper film may make it stable. In particular, if the upper film is thin enough, even under high-frequency oscillation, the flow is always stable. The influence of density ratio is similar, i.e. there are curved stable and unstable stripes in the $(\beta ,r)$ -planes. Surface surfactants generally stabilize the flow of the two-layer oscillatory membrane, while interfacial surfactants may stabilize or destabilize the flow but the effect is mild. It is also found that gravity can generally stabilize the flow because it narrows the bandwidth of unstable frequencies.

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