Critical Reynolds number of the Orr-Sommerfeld equation

1973 ◽  
Vol 16 (2) ◽  
pp. 329 ◽  
Author(s):  
D. P. Chock
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Sommerfeld Equation

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.

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.

Primary instability of a shear-thinning film flow down an incline: experimental study

2017 ◽  
Vol 821 ◽  
Author(s):  
M. H. Allouche ◽  
V. Botton ◽  
S. Millet ◽  
D. Henry ◽  
S. Dagois-Bohy ◽  
...  
Keyword(s):  
Experimental Study ◽  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Primary Stability ◽  
Shear Thinning ◽  
Driving Frequency ◽  
Neutral Curves ◽  
Theoretical Predictions ◽  
Sommerfeld Equation ◽  
Primary Instability

The main objective of this work is to study experimentally the primary instability of non-Newtonian film flows down an inclined plane. We focus on low-concentration shear-thinning aqueous solutions obeying the Carreau law. The experimental study essentially consists of measuring wavelengths in marginal conditions, which yields the primary stability threshold for a given slope. The experimental results for neutral curves presented in the $(Re,f_{c})$ and $(Re,k)$ planes (where $f_{c}$ is the driving frequency, $k$ is the wavenumber and $Re$ is the Reynolds number) are in good agreement with the numerical results obtained by a resolution of the generalized Orr–Sommerfeld equation. The long-wave asymptotic extension of our results is consistent with former theoretical predictions of the critical Reynolds number. This is the first experimental evidence of the destabilizing effect of the shear-thinning behaviour in comparison with the Newtonian case: the critical Reynolds number is smaller, and the ratio between the critical wave celerity and the flow velocity at the free surface is larger.

On hydrodynamic stability in unlimited fields of viscous flow

1957 ◽  
Vol 238 (1215) ◽  
pp. 489-501 ◽  
Keyword(s):  
Reynolds Number ◽  
Hydrodynamic Stability ◽  
Critical Reynolds Number ◽  
Integral Form ◽  
Neutral Curve ◽  
Wave Number ◽  
Neutral Stability ◽  
Laminar Jet ◽  
Fourth Derivative ◽  
Sommerfeld Equation

This paper considers the hydrodynamic stability of flows in which there are no solid boundaries in the field of flow. The method used is an extension of that initiated by McKoen (1957), in which the fourth derivative, 0 iv , is assumed to be significant only near to the singular layer, but otherwise the complete fourth-order Orr—Sommerfeld equation is considered. An alternative derivation is given for McKoen’s integral form of the boundary condition for an antisymmetrical perturbation. In this integral it is necessary to approximate for (j) but not for any of its derivatives. It is shown that the present method will always lead to a neutral stability curve of wave number against Reynolds number, having two branches as R ->oo and hence a least critical R . The case of the plane laminar jet is considered, and a critical Reynolds number of 4 is obtained, which does not compare unreasonably with experiment in which unsteadiness is first detected at a Reynolds number of about 10. The lower branch of the neutral curve is found to be almost coincident with the R -axis.

Influence of viscosity temperature dependence on the spectral characteristics of the thermoviscous liquids flow stability equation

2019 ◽  
Vol 14 (1) ◽  
pp. 52-58 ◽  
Author(s):  
A.D. Nizamova ◽  
V.N. Kireev ◽  
S.F. Urmancheev
Keyword(s):  
Reynolds Number ◽  
Spectral Characteristics ◽  
Wave Number ◽  
Critical Parameters ◽  
Fluid Viscosity ◽  
Flat Channel ◽  
Stability Equation ◽  
The Stability ◽  
Thermoviscous Fluid ◽  
Sommerfeld Equation

The flow of a viscous model fluid in a flat channel with a non-uniform temperature field is considered. The problem of the stability of a thermoviscous fluid is solved on the basis of the derived generalized Orr-Sommerfeld equation by the spectral decomposition method in Chebyshev polynomials. The effect of taking into account the linear and exponential dependences of the fluid viscosity on temperature on the spectral characteristics of the hydrodynamic stability equation for an incompressible fluid in a flat channel with given different wall temperatures is investigated. Analytically obtained profiles of the flow rate of a thermovisible fluid. The spectral pictures of the eigenvalues of the generalized Orr-Sommerfeld equation are constructed. It is shown that the structure of the spectra largely depends on the properties of the liquid, which are determined by the viscosity functional dependence index. It has been established that for small values of the thermoviscosity parameter the spectrum compares the spectrum for isothermal fluid flow, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has a nontrivial solution. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of eigenvalues found with fixed Reynolds number and wavenumber parameters. It is shown that with a fixed Reynolds number and a wave number with an increase in the thermoviscosity parameter, the flow becomes unstable. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermally viscous fluid. The eigenfunctions constructed in the subsequent works show the behavior of transverse-velocity perturbations, their possible growth or decay over time.

Stability of thermocapillary flows in non-cylindrical liquid bridges

2002 ◽  
Vol 458 ◽  
pp. 35-73 ◽  
Author(s):  
CH. NIENHÜSER ◽  
H. C. KUHLMANN
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Critical Reynolds Number ◽  
Volume Fraction ◽  
Thermocapillary Flow ◽  
Surface Shape ◽  
Liquid Bridges ◽  
Gravity Level ◽  
Neutral Curves ◽  
Prandtl Numbers

The thermocapillary flow in liquid bridges is investigated numerically. In the limit of large mean surface tension the free-surface shape is independent of the flow and temperature fields and depends only on the volume of liquid and the hydrostatic pressure difference. When gravity acts parallel to the axis of the liquid bridge the shape is axisymmetric. A differential heating of the bounding circular disks then causes a steady two-dimensional thermocapillary flow which is calculated by a finite-difference method on body-fitted coordinates. The linear-stability problem for the basic flow is solved using azimuthal normal modes computed with the same discretization method. The dependence of the critical Reynolds number on the volume fraction, gravity level, Prandtl number, and aspect ratio is explained by analysing the energy budgets of the neutral modes. For small Prandtl numbers (Pr = 0.02) the critical Reynolds number exhibits a smooth minimum near volume fractions which approximately correspond to the volume of a cylindrical bridge. When the Prandtl number is large (Pr = 4) the intersection of two neutral curves results in a sharp peak of the critical Reynolds number. Since the instabilities for low and high Prandtl numbers are markedly different, the influence of gravity leads to a distinctly different behaviour. While the hydrostatic shape of the bridge is the most important effect of gravity on the critical point for low-Prandtl-number flows, buoyancy is the dominating factor for the stability of the flow in a gravity field when the Prandtl number is high.

Bifurcation Characteristics of Flows in Rectangular Sudden Expansion Channels

2005 ◽  
Author(s):  
Francine Battaglia ◽  
George Papadopoulos
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Expansion Ratio ◽  
Sudden Expansion ◽  
Two Dimensional ◽  
Effective Expansion ◽  
Three Dimensional Simulations

The effect of three-dimensionality on low Reynolds number flows past a symmetric sudden expansion in a channel was investigated. The geometric expansion ratio of in the current study was 2:1 and the aspect ratio was 6:1. Both experimental velocity measurements and two- and three-dimensional simulations for the flow along the centerplane of the rectangular duct are presented for Reynolds numbers in the range of 150 to 600. Comparison of the two-dimensional simulations with the experiments revealed that the simulations fail to capture completely the total expansion effect on the flow, which couples both geometric and hydrodynamic effects. To properly do so requires the definition of an effective expansion ratio, which is the ratio of the downstream and upstream hydraulic diameters and is therefore a function of both the expansion and aspect ratios. When the two-dimensional geometry was consistent with the effective expansion ratio, the new results agreed well with the three-dimensional simulations and the experiments. Furthermore, in the range of Reynolds numbers investigated, the laminar flow through the expansion underwent a symmetry-breaking bifurcation. The critical Reynolds number evaluated from the experiments and the simulations was compared to other values reported in the literature. Overall, side-wall proximity was found to enhance flow stability, helping to sustain laminar flow symmetry to higher Reynolds numbers in comparison to nominally two-dimensional double-expansion geometries. Lastly, and most importantly, when the logarithm of the critical Reynolds number from all these studies was plotted against the reciprocal of the effective expansion ratio, a linear trend emerged that uniquely captured the bifurcation dynamics of all symmetric double-sided planar expansions.

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.

Loss Characteristics of Orifices and Nozzles

1978 ◽  
Vol 100 (3) ◽  
pp. 299-307 ◽  
Author(s):  
S. H. Alvi ◽  
K. Sridharan ◽  
N. S. Lakshmana Rao
Keyword(s):  
Reynolds Number ◽  
Turbulent Flow ◽  
Flow Regime ◽  
Discharge Coefficient ◽  
Critical Reynolds Number ◽  
Reynolds Numbers ◽  
Center Line ◽  
Laminar Regime ◽  
Variation Of Parameters ◽  
Turbulent Flow Regime

Loss characteristics of sharp-edged orifices, quadrant-edged orifices for varying edge radii, and nozzles are studied for Reynolds numbers less than 10,000 for β ratios from 0.2 to 0.8. The results may be reliably extrapolated to higher Reynolds numbers. Presentation of losses as a percentage of meter pressure differential shows that the flow can be identified into fully laminar regime, critical Reynolds number regime, relaminarization regime, and turbulent flow regime. An integrated picture of variation of parameters such as discharge coefficient, loss coefficient, settling length, pressure recovery length, and center line velocity confirms this classification.

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