The stability of a family of Jeffery–Hamel solutions for divergent channel flow

1966 ◽  
Vol 24 (1) ◽  
pp. 191-207 ◽  
Author(s):  
P. M. Eagles
Keyword(s):  
Reynolds Numbers ◽  
Neutral Curve ◽  
Wave Number ◽  
Finite Limit ◽  
Lower Branch ◽  
Negative Wave ◽  
Wave Velocities ◽  
Critical Reynolds Numbers ◽  
Divergent Channel ◽  
The Stability

A set of Jeffery–Hamel profiles (for radial, viscous, incompressible flow) have been shown by Fraenkel (1962, 1963) to approximate to profiles in certain two-dimensional divergent channels. The stability of a family of these profiles is investigated by a numerical solution of the Orr-Sommerfeld problem. Neutralstability curves are calculated in the (R,k)-planes (where R is the Reynolds number of the basic flow and k is the wave-number of the disturbance), and fairly low critical Reynolds numbers are found. For those profiles that have regions of reversed flow, negative wave velocities are found on the lower branch of the neutral curve, and also it is found that Rk tends to a finite limit as R → ∞ on the lower branch. These unexpected results are further discussed and verified by independent methods. The relation of the calculations to some experiments of Patterson (1934, 1935) is discussed.

scholarly journals Linear stability of boundary layers under solitary waves

2014 ◽  
Vol 761 ◽  
pp. 62-104 ◽  
Author(s):  
Joris C. G. Verschaeve ◽  
Geir K. Pedersen
Keyword(s):  
Linear Stability ◽  
Solitary Waves ◽  
Reynolds Numbers ◽  
Quantitative Agreement ◽  
Linear Instability ◽  
Navier Stokes ◽  
Acceleration Region ◽  
Critical Reynolds Numbers ◽  
Tollmien Schlichting ◽  
The Stability

AbstractIn the present treatise, the stability of the boundary layer under solitary waves is analysed by means of the parabolized stability equation. We investigate both surface solitary waves and internal solitary waves. The main result is that the stability of the flow is not of parametric nature as has been assumed in the literature so far. Not only does linear stability analysis highlight this misunderstanding, it also gives an explanation why Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207–231), Vittori & Blondeaux (Coastal Engng, vol. 58, 2011, pp. 206–213) and Ozdemir et al. (J. Fluid Mech., vol. 731, 2013, pp. 545–578) each obtained different critical Reynolds numbers in their experiments and simulations. We find that linear instability is possible in the acceleration region of the flow, leading to the question of how this relates to the observation of transition in the acceleration region in the experiments by Sumer et al. or to the conjecture of a nonlinear instability mechanism in this region by Ozdemir et al. The key concept for assessment of instabilities is the integrated amplification which has not been employed for this kind of flow before. In addition, the present analysis is not based on a uniformization of the flow but instead uses a fully nonlinear description including non-parallel effects, weakly or fully. This allows for an analysis of the sensitivity with respect to these effects. Thanks to this thorough analysis, quantitative agreement between model results and direct numerical simulation has been obtained for the problem in question. The use of a high-order accurate Navier–Stokes solver is primordial in order to obtain agreement for the accumulated amplifications of the Tollmien–Schlichting waves as revealed in this analysis. An elaborate discussion on the effects of amplitudes and water depths on the stability of the flow is presented.

The stability of a stratified fluid

1967 ◽  
Vol 29 (2) ◽  
pp. 233-240
Author(s):  
J. B. Hinwood
Keyword(s):  
Turbulent Flow ◽  
Froude Number ◽  
Reynolds Numbers ◽  
Stratified Fluid ◽  
Stratified Flows ◽  
Rectangular Duct ◽  
Interfacial Waves ◽  
Homogeneous Flow ◽  
Critical Reynolds Numbers ◽  
The Stability

For the flow of a stably-stratified fluid in the inlet region of a rectangular duct, it is shown experimentally that the upper and lower critical Reynolds numbers are functions of the interfacial Froude number F, and that if F is large they are lower than for a homogeneous flow. In stratified flows the disturbances leading to turbulent flow sometimes arise at the interface and lead to interfacial waves, whose wavelength at breaking is equal to the conduit depth.

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.

Theoretical and Experimental Investigations of Coolant Flow in Inlet Channels of the BTA and Ejector Drills

1995 ◽  
Vol 209 (3) ◽  
pp. 211-220 ◽  
Author(s):  
V P Astakhov ◽  
P S Subramanya ◽  
M O M Osman
Keyword(s):  
Axial Flow ◽  
Annular Channel ◽  
Reynolds Numbers ◽  
Coolant Flow ◽  
Experimental Investigations ◽  
Annular Channels ◽  
Critical Reynolds Numbers ◽  
Flow Through ◽  
The Stability ◽  
The Mathematical Model

The coolant flow through inlet annular channels in BTA and ejector drills is investigated. The study was conducted in order to understand the influence of the channel's parameters (the channel's clearance variation along its length and eccentricity) on the coolant pressure distribution and hydraulic resistance. A new design of the ejector drill with the eccentrical location on the inner tube is proposed. A study is made of the stability in the coolant flow in the inlet annular channels. The appearance of instability is explained by the presence of Taylor macrovortices in these channels under certain combinations of boring bar rotating velocity and axial flow velocity. In order to define the unstable regimes (the critical Reynolds numbers), the mathematical model for non-isothermal flow through the annular channel is solved. The heat transfer from the swarf to the incoming coolant is investigated under different flow conditions.

On the neutral curve of the flat-plate boundary layer: comparison between experiment, Orr–Sommerfeld theory and asymptotic theory

1995 ◽  
Vol 288 ◽  
pp. 59-73 ◽  
Author(s):  
J. J. Healey
Keyword(s):  
Boundary Layer ◽  
Branch Point ◽  
Plate Boundary ◽  
Reynolds Numbers ◽  
Wall Layer ◽  
Neutral Curve ◽  
Critical Layer ◽  
Lower Branch ◽  
Neutral Curves ◽  
Flat Plate Boundary Layer

The neutral stability curve for the flat-plate boundary layer has been calculated using the Orr–Sommerfeld equation and compared to those obtained using upper- and lower-branch scalings. The Orr–Sommerfeld results agree well with the lower-branch scaling at Reynolds numbers relevant to experiment, but agree well with the upper-branch scaling only for Rδ > 105. It is shown that the critical layer only emerges from the viscous wall layer when Rδ > 105. This suggests that for Rδ < 105, when the critical layer lies within the viscous wall layer, the disturbance has a triple-deck structure, even for the upper branch of the neutral curve (which can be reached if the phase jump across the critical layer is retained).The transition from a triple-deck to a five-deck structure with increasing Reynolds number on the upper branch occurs relatively abruptly and can be associated with a square-root branch point in the Tietjens function. Essentially, the lower- and upper-branch scalings pertain to two different modes, the first possessing a triple-deck structure, the second a five-deck structure. The modes are connected at the branch point, and the neutral curves of each mode join to give a single curve close to this branch point. The asymptotic expansions for the upper- and lower-branch neutral curves depend upon the analyticity of the dispersion relationship, and so the proximity of the branch point indicates where these expansions will be liable to inaccuracies. This explains the poor neutral-curve predictions made by five-deck analyses at the Reynolds numbers where transition occurs.

The stability of pipe flow Part 1. Asymptotic analysis for small wave-numbers

1970 ◽  
Vol 43 (2) ◽  
pp. 279-290 ◽  
Author(s):  
W. P. Graebel
Keyword(s):  
Asymptotic Analysis ◽  
Poiseuille Flow ◽  
Reynolds Numbers ◽  
Transverse Component ◽  
Wave Number ◽  
Radial Velocity Component ◽  
Wave Numbers ◽  
Flow Part ◽  
The Stability ◽  
High Degree

The instability of Poiseuille flow in a pipe is considered for small disturbances. An asymptotic analysis is used which is similar to that found successful in plane Poiseuille flow. The disturbance is taken to travel in a spiral fashion, and comparison of the radial velocity component with the transverse component in the plane case shows a high degree of similarity, particularly near the critical point where the disturbance and primary flow travel with the same speed. Instability is found for azimuthal wave-numbers of 2 or greater, although the corresponding minimum Reynolds numbers are too small to compare favourably with either experiments or the initial restrictions on the magnitude of the wave-number.

The stability of a two-dimensional laminar jet

1958 ◽  
Vol 4 (3) ◽  
pp. 261-275 ◽  
Author(s):  
T. Tatsumi ◽  
T. Kakutani
Keyword(s):  
Reynolds Number ◽  
Basic Assumption ◽  
Wave Number ◽  
Neutral Stability ◽  
Two Dimensional ◽  
Lower Branch ◽  
Laminar Jet ◽  
Numerical Result ◽  
The Stability ◽  
Different Parts

This paper deals with the stability of a two-dimensional laminar jet against the infinitesimal antisymmetric disturbance. The curve of the neutral stability in the (α, R)-plane (α, the wave-number; R, Reynolds number) is calculated using two different methods for the different parts of the curve; the solution is developed in powers of (αR)−1 for obtaining the upper branch of the curve and in powers of αR for the lower branch.The asymptotic behaviour of these branches is that for branch I,$\alpha \rightarrow 2, \;\; c \rightarrow \frac{2}{3}$ for $R \rightarrow \infty$; and for branch II, $R \sim 1\cdot12\alpha^{-1|2},\; c \sim 1\cdot 20 \alpha^2$ for α → 0. Some discussion is given on the validity of the basic assumption of the stability theory in relation to the numerical result obtained here.

scholarly journals Global stability of swept flow around a parabolic body: the neutral curve

2011 ◽  
Vol 678 ◽  
pp. 589-599 ◽  
Author(s):  
CHRISTOPH J. MACK ◽  
PETER J. SCHMID
Keyword(s):  
Boundary Layer ◽  
Leading Edge ◽  
Neutral Curve ◽  
Parameter Study ◽  
Global Approach ◽  
Complex Flow ◽  
Unstable Boundary Layer ◽  
Critical Reynolds Numbers ◽  
Acoustic Instabilities ◽  
The Stability

The onset of transition in the leading-edge region of a swept blunt body depends crucially on the stability characteristics of the flow. Modelling this flow configuration by swept compressible flow around a parabolic body, a global approach is taken to extract pertinent stability information via a DNS-based iterative eigenvalue solver. Global modes combining features from boundary-layer and acoustic instabilities are presented. A parameter study, varying the spanwise disturbance wavenumber and the sweep Reynolds number, showed the existence of unstable boundary-layer and acoustic modes. The corresponding neutral curve displays two overlapping regions of exponential growth and two critical Reynolds numbers, one for boundary-layer instabilities and one for acoustic instabilities. The employed global approach establishes a first neutral curve, delineating stable from unstable parameter configurations, for the complex flow about a swept parabolic body with corresponding implications for swept leading-edge flow.

The neutral curve for stationary disturbances in rotating-disk flow

1986 ◽  
Vol 164 ◽  
pp. 275-287 ◽  
Author(s):  
Mujeeb R. Malik
Keyword(s):  
Rotating Disk ◽  
Reynolds Numbers ◽  
Neutral Curve ◽  
Order System ◽  
Lower Branch ◽  
Rotating Disk Flow ◽  
Streamline Curvature ◽  
Stationary Vortex ◽  
Wave Angle ◽  
Vortex Disturbances

The neutral curve for stationary vortex disturbances in rotating-disk flow is computed up to a Reynolds number of 107 using the sixth-order system of linear stability equations which includes the effects of streamline curvature and Coriolis force. It is found that the neutral curve has two minima: one at R = 285.36 (upper branch) and the other at R = 440.88 (lower branch). At large Reynolds numbers, the upper branch tends to Stuart's asymptotic solution while the lower branch tends to a solution that is associated with the wave angle corresponding to the direction of zero mean wall shear.

On the stability of flow in an elliptic pipe which is nearly circular

1978 ◽  
Vol 87 (2) ◽  
pp. 233-241 ◽  
Author(s):  
A. Davey
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Cross Section ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Major Axis ◽  
Reynolds Numbers ◽  
Minor Axis ◽  
Critical Reynolds Numbers ◽  
The Stability

The linear stability of Poiseuille flow in an elliptic pipe which is nearly circular is examined by regarding the flow as a perturbation of Poiseuille flow in a circular pipe. We show that the temporal damping rates of non-axisymmetric infinitesimal disturbances which are concentrated near the wall of the pipe are decreased by the ellipticity. In particular we estimate that if the length of the minor axis of the cross-section of the pipe is less than about 96 ½% of that of the major axis then the flow will be unstable and a critical Reynolds number will exist. Also we calculate estimates of the ellipticities which will produce critical Reynolds numbers ranging from 1000 upwards.

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