The Stability of Pipe Entrance Flows Subjected to Axisymmetric Disturbances

1994 ◽  
Vol 116 (1) ◽  
pp. 61-65 ◽  
Author(s):  
D. F. da Silva ◽  
E. A. Moss
Keyword(s):  
Boundary Layer ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Axial Distance ◽  
Displacement Thickness ◽  
Critical Reynolds Numbers ◽  
Equilibrium Profile ◽  
Stability Data ◽  
The Stability ◽  
Theoretical Results

This paper reexamines an important unresolved problem in fluid mechanics—the discrepancy between measurements and predictions of stability in pipe entrance flows. Whereas measured critical Reynolds numbers are relatively insensitive to velocity profile shape in the streamwise direction, the theoretical results indicate a rapid increase, both as the equilibrium profile is approached, and toward the inlet. The current work uses the displacement thickness based Reynolds number as a rational basis on which to compare new stability predictions obtained by means of the Q-Z algorithm, with existing theoretical results. Although the present data are shown to be the only that are consistent with the classical parallel boundary layer limit towards the inlet, they still deviate increasingly with axial distance from the only available experimental results. By examining pipe inlet stability data in relation to boundary layer measurements and predictions, the work effectively questions the commonly held belief that streamwise variations of flow alone are responsible for these deviations, suggesting that the finite amplitude nature of the applied disturbances is the most likely cause.

The stability of free boundary layers between two uniform streams

1960 ◽  
Vol 7 (3) ◽  
pp. 433-441 ◽  
Author(s):  
T. Tatsumi ◽  
K. Gotoh
Keyword(s):  
Boundary Layer ◽  
Free Boundary ◽  
Reynolds Numbers ◽  
Velocity Profiles ◽  
Point Of View ◽  
Boundary Layer Flows ◽  
Type Curves ◽  
Neutral Curves ◽  
Critical Reynolds Numbers ◽  
The Stability

Hydrodynamic stability of free boundary-layer flows is treated in general. It is found that the situations at low Reynolds numbers are universal for all velocity profiles of free boundary-layer type. Curves of constant amplification are calculated as far as O(R3). In particular, the asymptotic form of the neutral curves for R [eDot ] 0 is found to be α = R/(4√3), so that the critical Reynolds numbers of these flows are identically zero. The phase velocity of the disturbance is also found to be zero, for all disturbances, up to the second approximation.A method of normalizing the velocity profiles is suggested, and existing results for the stability of various profiles at large Reynolds numbers are discussed from a new point of view.

The effect of external forcing on the stability of plane Poiseuille flow

1978 ◽  
Vol 359 (1699) ◽  
pp. 453-478 ◽  
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Stable Solution ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Plane Poiseuille Flow ◽  
Critical Reynolds Numbers ◽  
Threshold Amplitude ◽  
The Stability

The stability of plane Poiseuille flow in a channel forced by a wavelike motion on one of the channel walls is investigated. The amplitude Є of this forcing is taken to be small. The most dangerous modes of forcing are identified and it is found in general the critical Reynolds number is changed by O (Є) 2 . However, we identify two particular modes of forcing which give rise to decrements of order Є 2/3 and Є in the critical Reynolds number. Some types of forcing are found to generate sub critical stable finite amplitude perturbations to plane Poiseuille flow. This contrasts with the unforced case where the only stable solution is the zero amplitude solution. The forcing also deforms the unstable subcritical limit cycle solution from its usual circular shape into a more complicated shape. This has an effect on the threshold amplitude ideas suggested by, for example, Meksyn & Stuart (1951). It is found that the phase of disturbances must also be considered when finding the amplitude dependent critical Reynolds numbers.

scholarly journals Non-modal stability analysis of the boundary layer under solitary waves

2017 ◽  
Vol 836 ◽  
pp. 740-772 ◽  
Author(s):  
Joris C. G. Verschaeve ◽  
Geir K. Pedersen ◽  
Cameron Tropea
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Stability Analysis ◽  
Solitary Waves ◽  
Reynolds Numbers ◽  
Two Dimensional ◽  
Modal Theory ◽  
Critical Reynolds Numbers ◽  
Streamwise Streaks ◽  
The Stability

In the present work, a stability analysis of the bottom boundary layer under solitary waves based on energy bounds and non-modal theory is performed. The instability mechanism of this flow consists of a competition between streamwise streaks and two-dimensional perturbations. For lower Reynolds numbers and early times, streamwise streaks display larger amplification due to their quadratic dependence on the Reynolds number, whereas two-dimensional perturbations become dominant for larger Reynolds numbers and later times in the deceleration region of this flow, as the maximum amplification of two-dimensional perturbations grows exponentially with the Reynolds number. By means of the present findings, we can give some indications on the physical mechanism and on the interpretation of the results by direct numerical simulation in Vittori & Blondeaux (J. Fluid Mech., vol. 615, 2008, pp. 433–443) and Özdemir et al. (J. Fluid Mech., vol. 731, 2013, pp. 545–578) and by experiments in Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207–231). In addition, three critical Reynolds numbers can be defined for which the stability properties of the flow change. In particular, it is shown that this boundary layer changes from a monotonically stable to a non-monotonically stable flow at a Reynolds number of $Re_{\unicode[STIX]{x1D6FF}}=18$.

The Effect of Finite Amplitude Disturbance Magnitude on Departures From Laminar Conditions in Impulsively Started and Steady Pipe Entrance Flows

2001 ◽  
Vol 124 (1) ◽  
pp. 235-240 ◽  
Author(s):  
E. A. Moss ◽  
A. H. Abbot
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Shape Parameter ◽  
Critical Reynolds Number ◽  
Flow Instability ◽  
Finite Amplitude ◽  
Pipe Flows ◽  
Displacement Thickness ◽  
Layer Flow ◽  
The Stability

The aim of this study was to investigate first departures from laminar conditions in both impulsively started and steady pipe entrance flows. Wall shear stress measurements were conducted of transition in impulsively started pipe flows with large disturbances. These results were reconciled in a framework of displacement thickness Reynolds number and a velocity profile shape parameter, with existing measurements of pipe entrance flow instability, pipe-Poiseuille and boundary layer flow responses to large disturbances, and linear stability predictions. Limiting critical Reynolds number variations for each type of flow were thus inferred, corresponding to the small and gross disturbance limits respectively. Consequently, insights have been provided regarding the effect of disturbance levels on the stability of both steady and unsteady pipe flows.

scholarly journals Finite-amplitude steady waves in plane viscous shear flows

1985 ◽  
Vol 160 ◽  
pp. 281-295 ◽  
Author(s):  
F. A. Milinazzo ◽  
P. G. Saffman
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Navier Stokes ◽  
Unidirectional Flow ◽  
Viscous Shear ◽  
Finite Amplitude Waves

Computations of two-dimensional solutions of the Navier–Stokes equations are carried out for finite-amplitude waves on steady unidirectional flow. Several cases are considered. The numerical method employs pseudospectral techniques in the streamwise direction and finite differences on a stretched grid in the transverse direction, with matching to asymptotic solutions when unbounded. Earlier results for Poiseuille flow in a channel are re-obtained, except that attention is drawn to the dependence of the minimum Reynolds number on the physical constraint of constant flux or constant pressure gradient. Attempts to calculate waves in Couette flow by continuation in the velocity of a channel wall fail. The asymptotic suction boundary layer is shown to possess finite-amplitude waves at Reynolds numbers orders of magnitude less than the critical Reynolds number for linear instability. Waves in the Blasius boundary layer and unsteady Rayleigh profile are calculated by employing the artifice of adding a body force to cancel the spatial or temporal growth. The results are verified by comparison with perturbation analysis in the vicinity of the linear-instability critical Reynolds numbers.

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.

Similarity and stability of the laminar boundary layer in a streamwise corner

1981 ◽  
Vol 377 (1770) ◽  
pp. 269-288 ◽  
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Conclusive Evidence ◽  
Experimental Results ◽  
Solid Surfaces ◽  
Boundary Layer Problem ◽  
Streamwise Pressure Gradient ◽  
The Stability ◽  
Limiting Case ◽  
Theoretical Results

The study of laminar viscous flow along the line of intersection of two solid surfaces at right angles is examined in its present state, and out­standing differences between various experimental and theoretical results are analysed. New experimental results are presented in which the stability of the corner boundary layer is examined in terms of the degree of streamwise similarity of its velocity profiles. Conclusive evidence is found that the layer does not exist in stable laminar form when the streamwise pressure gradient is zero and the Reynolds number much above about 10 4 . The new results also help to explain the differences between various experimental results, and between theory and experi­ment, which have characterized the corner boundary layer problem for several years. By extrapolation, an approximate prediction is obtained of what the velocity profile of the corner boundary layer would be in the limiting case of zero pressure gradient, if the layer were stable in that state. The predicted profile is compared with the results of current theories.

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.

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