Stability of film flow over inclined topography based on a long-wave nonlinear model

2013 ◽  
Vol 729 ◽  
pp. 638-671 ◽  
Author(s):  
D. Tseluiko ◽  
M. G. Blyth ◽  
D. T. Papageorgiou
Keyword(s):  
Reynolds Number ◽  
Model Equation ◽  
Critical Reynolds Number ◽  
Wave Model ◽  
Absolute Instability ◽  
Spatial Period ◽  
Integer Multiple ◽  
Long Wave ◽  
Critical Reynolds Numbers ◽  
Theoretical Finding

AbstractThe stability of a viscous liquid film flowing under gravity down an inclined wall with periodic corrugations is investigated. A long-wave model equation valid at near-critical Reynolds numbers is used to study the film dynamics, and calculations are performed for either sinusoidal or rectangular wall corrugations assuming either a fixed flow rate in the film or a fixed volume of fluid within each wall period. Under the two different flow assumptions, steady solution branches are delineated including subharmonic branches, for which the period of the free surface is an integer multiple of the wall period, and the existence of quasi-periodic branches is demonstrated. Floquet–Bloch theory is used to determine the linear stability of steady, periodic solutions and the nature of any instability is analysed using the method of exponentially weighted spaces. Under certain conditions, and depending on the wall period, the flow may be convectively unstable for small wall amplitudes but undergo transition to absolute instability as the wall amplitude increases, a novel theoretical finding for this class of flows; in other cases, the flow may be convectively unstable for small wall amplitudes but stable for larger wall amplitudes. Solutions with the same spatial period as the wall become unstable at a critical Reynolds number, which is strongly dependent on the period size. For sufficiently small wall periods, the corrugations have a destabilizing effect by lowering the critical Reynolds number above which instability occurs. For slightly larger wall periods, small-amplitude corrugations are destabilizing but sufficiently large-amplitude corrugations are stabilizing. For even larger wall periods, the opposite behaviour is found. For sufficiently large wall periods, the corrugations are destabilizing irrespective of their amplitude. The predictions of the linear theory are corroborated by time-dependent simulations of the model equation, and the presence of absolute instability under certain conditions is confirmed. Boundary element simulations on an inverted substrate reveal that wall corrugations can have a stabilizing effect at zero Reynolds number.

On the force fluctuations acting on a circular cylinder in crossflow from subcritical up to transcritical Reynolds numbers

1983 ◽  
Vol 133 ◽  
pp. 265-285 ◽  
Author(s):  
Günter Schewe
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Dynamic Range ◽  
Reynolds Numbers ◽  
Band Spectrum ◽  
Transitional Region ◽  
Force Measurements ◽  
Unsteady Forces ◽  
Number Range ◽  
Critical Reynolds Numbers

Force measurements were conducted in a pressurized wind tunnel from subcritical up to transcritical Reynolds numbers 2.3 × 104[les ]Re[les ] 7.1 × 106without changing the experimental arrangement. The steady and unsteady forces were measured by means of a piezobalance, which features a high natural frequency, low interferences and a large dynamic range. In the critical Reynolds-number range, two discontinuous transitions were observed, which can be interpreted as bifurcations at two critical Reynolds numbers. In both cases, these transitions are accompanied by critical fluctuations, symmetry breaking (the occurrence of a steady lift) and hysteresis. In addition, both transitions were coupled with a drop of theCDvalue and a jump of the Strouhal number. Similar phenomena were observed in the upper transitional region between the super- and the transcritical Reynolds-number ranges. The transcritical range begins at aboutRe≈ 5 × 106, where a narrow-band spectrum is formed withSr(Re= 7.1 × 106) = 0.29.

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.

Unsteady state fluid structure of two-sided nonfacing lid-driven cavity induced by a semicircle at different radii sizes and velocity ratios

2019 ◽  
Vol 30 (08) ◽  
pp. 1950060 ◽  
Author(s):  
Basma Souayeh ◽  
Fayçal Hammami ◽  
Najib Hdhiri ◽  
Huda Alfannakh
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Velocity Ratio ◽  
Time History ◽  
Standard Case ◽  
Driven Cavity ◽  
Opposite Case ◽  
Critical Reynolds Numbers ◽  
Volume Method ◽  
Radius Size

This paper aims in analyzing the effect of velocity ratio [Formula: see text] and Radius size of an inner semicircle inserted at the bottom wall of two-sided nonfacing lid-driven cavity on the bifurcation occurrence phenomena. The study has been performed by using finite volume method (FVM) and multigrid acceleration for certain pertinent parameters; Reynolds number, velocity ratios ([Formula: see text]) by step of 0.25 and Radius size of the inner semicircle ([Formula: see text]) by step of 0.05. An analysis of the flow evolution shows that, when increasing Re beyond a certain critical value, the flow becomes unstable then bifurcates for various velocity ratios and radius size of the semicircle. Therefore, critical Reynolds numbers are determined for each case. It is worth to mention that the transition to unsteadiness follows the classical scheme of a Hopf bifurcation. Results show also that in the standard case of a single lid-driven cavity ([Formula: see text]), the highest critical Reynolds number corresponds to the lowest radius of the semicircle and the same for ([Formula: see text]). Conversely, from ([Formula: see text]) where the left moving lid take effect, the opposite phenomenon occurs. In harmony with this, it has been found that elongating the cylinder radius accelerates the appearance of the unsteady regime and delays it in the opposite case. Flow periodicity has been verified through time history plots for the velocity component and phase-space trajectories as a function of Reynolds number. The numerical results are correlated in a sophisticated correlation of the critical Reynolds number with other parameters.

Influence of the Reynolds Number on Spar Vortex Induced Motions (VIM): Multiple Scale Model Test Comparisons

2009 ◽  
Author(s):  
Dominique Roddier ◽  
Tim Finnigan ◽  
Stergios Liapis
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Multiple Scale ◽  
Scale Model ◽  
Taylor Model ◽  
Testing Procedures ◽  
Critical Reynolds Numbers ◽  
Number Of Publications ◽  
Six Degree Of Freedom ◽  
First Time

There have been a number of publications on spar Vortex-Induced-Motions (VIM) model testing procedures and results over the past few years. All tests allowing full 6 DOF response to date have been done under sub-critical Reynolds Number conditions. Prior to 2006 tests under super-Critical Reynolds Number conditions had only been done with a fully submerged 1 DOF rig. Early in 2006, a series of Spar VIM experiments was undertaken in three different facilities: Force Technology in Denmark, the David Taylor Model Basin in Bethesda Maryland and UC Berkeley in California. The motivation of this work was to investigate the effect of Reynolds Number and hull appurtenances on spar vortex induced motions (VIM) for a vertically moored 6DOF truss spar hull model with strakes. The three series of tests were done at both sub and super-critical Reynolds Numbers, with matching Froude Numbers. In order to assess the importance of appurtenances (chains, pipes and anodes) and current heading on strake effectiveness, tests were done with several sets of appurtenances, and at various headings and reduced velocities. These experiments were unique and groundbreaking in many ways: • For the first time the issue of scalability of Spar VIM experiments has been addressed and tested in a systematic way. • For the first time the effect of appurtenances (pipes, chains and anodes) was systematically tested. • The model tested at the David Taylor Model Basin (DTMB) had a diameter of 5.8′ and a weight of 15,600 lbs. It is the largest spar model ever tested. Furthermore the DTMB tests series is the only supercritical spar VIM performed with a six degree of freedom (6DOF) rig. This paper describes the three model tests campaigns, focusing on the efforts made to ensure three complete geo-similar programs, and on the significant findings of these tests, effectively that the influence of Re is to add some conservativeness in the results as the testing scale is smaller.

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.

An experimental study of absolute instability of the rotating-disk boundary-layer flow

1996 ◽  
Vol 314 ◽  
pp. 373-405 ◽  
Author(s):  
R. J. Lingwood
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Wave Packet ◽  
Boundary Layer Flow ◽  
Critical Reynolds Number ◽  
Linear Stability Theory ◽  
Absolute Instability ◽  
Unstable Flow ◽  
Average Value ◽  
Layer Flow

In this paper, the results of experiments on unsteady disturbances in the boundary-layer flow over a disk rotating in otherwise still air are presented. The flow was perturbed impulsively at a point corresponding to a Reynolds numberRbelow the value at which transition from laminar to turbulent flow is observed. Among the frequencies excited are convectively unstable modes, which form a three-dimensional wave packet that initially convects away from the source. The wave packet consists of two families of travelling convectively unstable waves that propagate together as one packet. These two families are predicted by linear-stability theory: branch-2 modes dominate close to the source but, as the packet moves outwards into regions with higher Reynolds numbers, branch-1 modes grow preferentially and this behaviour was found in the experiment. However, the radial propagation of the trailing edge of the wave packet was observed to tend towards zero as it approaches the critical Reynolds number (about 510) for the onset of radial absolute instability. The wave packet remains convectively unstable in the circumferential direction up to this critical Reynolds number, but it is suggested that the accumulation of energy at a well-defined radius, due to the flow becoming radially absolutely unstable, causes the onset of laminar–turbulent transition. The onset of transition has been consistently observed by previous authors at an average value of 513, with only a small scatter around this value. Here, transition is also observed at about this average value, with and without artificial excitation of the boundary layer. This lack of sensitivity to the exact form of the disturbance environment is characteristic of an absolutely unstable flow, because absolute growth of disturbances can start from either noise or artificial sources to reach the same final state, which is determined by nonlinear effects.

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 Interfacial Electrokinetic Effect on the Microchannel Flow Linear Stability

2004 ◽  
Vol 126 (1) ◽  
pp. 10-13 ◽  
Author(s):  
Sedat Tardu
Keyword(s):  
Reynolds Number ◽  
Hydrodynamic Stability ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Reynolds Numbers ◽  
Microchannel Flow ◽  
Electrokinetic Effect ◽  
Electrostatic Double Layer ◽  
Microchannel Flows ◽  
Critical Reynolds Numbers

The electrostatic double layer (EDL) effect on the linear hydrodynamic stability of microchannel flows is investigated. It is shown that the EDL destabilizes the Poiseuille flow considerably. The critical Reynolds number decreases by a factor five when the non-dimensional Debye-Huckel parameter κ is around ten. Thus, the transition may be quite rapid for microchannels of a couple of microns heights in particular when the liquid contains a very small number of ions. The EDL effect disappears quickly for κ⩾150 corresponding typically to channels of heights 400 μm or larger. These results may explain why significantly low critical Reynolds numbers have been encountered in some experiments dealing with microchannel flows.

Temporal stability of Jeffery–Hamel flow

1994 ◽  
Vol 268 ◽  
pp. 71-88 ◽  
Author(s):  
Mahmoud Hamadiche ◽  
Julian Scott ◽  
Denis Jeandel
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Temporal Stability ◽  
Reynolds Numbers ◽  
Critical Value ◽  
Volume Flux ◽  
Short Channel ◽  
Oscillatory Solutions ◽  
Half Angle ◽  
Critical Reynolds Numbers

In this study of the temporal stability of Jeffery–Hamel flow, the critical Reynolds number based on the volume flux, Rc, and that based on the axial velocity, Rec, are computed. It is found that both critical Reynolds numbers decrease very rapidly when the half-angle of the channel, α, increases, such that the quantity αRc remains very nearly constant and αRecis a nearly linear function of α. For a short channel there can be more than one value of the critical Reynolds number. A fully nonlinear analysis, for Re close to the critical value, indicates that the loss of stability is supercritical. The resulting asymmetric oscillatory solutions show staggered arrays of vortices positioned along the channel.

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.

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