Determining critical flow parameters for the Poiseuille, Couette and Taylor – Couette flows

2020 ◽  
Author(s):  
A.S. Kondratiev ◽  
K.F. Ogorodnikov
Keyword(s):  
Couette Flow ◽  
Critical Reynolds Number ◽  
Reynolds Numbers ◽  
Critical Parameters ◽  
Physical Conditions ◽  
Flow Parameters ◽  
Critical Reynolds Numbers ◽  
Taylor Couette ◽  
Mathematical Justification ◽  
Analytical Expressions

Based on a detailed analysis of the known Dou method for determining the critical parameters of a Newtonian fluid flow during the transition of a laminar flow regime to a turbulent one, an alternative to determining the critical Reynolds number for the Poiseuille and Couette flows in cylindrical, coaxial and flat channels, the article proposes a new mathematical justification of the Dou method leading to simpler calculation relations, while preserving the initial ideas about the physical conditions of the transition. New computing expressions for determining the critical Dou number for the generalized Poiseuille – Couette flow in a flat channel not considered by Dou are obtained. Analytical expressions for calculating critical parameters for the Taylor – Couette flow approximating experimental results for critical Reynolds numbers are given, as well as calcula-ted values of critical Dou numbers.

Modulated Taylor–Couette flow

1989 ◽  
Vol 208 ◽  
pp. 127-160 ◽  
Author(s):  
C. F. Barenghi ◽  
C. A. Jones
Keyword(s):  
Couette Flow ◽  
Bifurcation Theory ◽  
Modulation Frequency ◽  
Low Frequency ◽  
Reynolds Numbers ◽  
Narrow Gap ◽  
Initial Value ◽  
Critical Reynolds Numbers ◽  
Taylor Couette ◽  
Taylor Couette Flow

The onset of instability in temporally modulated Taylor-Couette flow is considered. Critical Reynolds numbers have been found by computing Floquet exponents. We find that frequency modulation of the inner cylinder introduces a small destabilization, in agreement with the narrow-gap theory of Hall and some recent experiments of Ahlers. We review the previous computational literature on this problem, and find a number of contradictory results: the source of these discrepancies is examined, and a satisfactory resolution is achieved. Nonlinear axisymmetric calculations on the modulated problem have been done with an initial value code using a spectral method with collocation. The results are compared satisfactorily with Ahlers' measurements.At low modulation frequency, a large destabilization has been observed in past experiments. We show that this cannot be explained on the basis of perfect bifurcation theory: an analysis of the modulated amplitude equation shows that very small imperfections can substantially affect the behaviour at low frequency by giving rise to ‘transient’ vortices at subcritical Reynolds number. We argue that these ‘transient’ vortices are the source of the large destabilization seen in some experiments. Modelling the imperfections in the initial-value code provides additional confirmation of this effect.

Non-axisymmetric subcritical bifurcations in viscoelastic Taylor–Couette flow

1994 ◽  
Vol 447 (1929) ◽  
pp. 135-153 ◽  
Keyword(s):  
Hopf Bifurcation ◽  
Couette Flow ◽  
Reynolds Numbers ◽  
Gap Size ◽  
Ginzburg Landau ◽  
Hopf Bifurcation Point ◽  
Stable Pattern ◽  
Flow Parameters ◽  
Taylor Couette ◽  
Taylor Couette Flow

Recently, an account of the linear and nonlinear analysis of the viscoelastic Taylor–Couette flow between independently rotating cylinders against axisymmetric disturbances was presented (Avgousti & Beris 1993 a ). However, more recent linear stability analysis has shown that for the range of geometric and kinematic parameters studied and for high enough values of flow elasticity, the critical instabilities are caused by non-axisymmetric, time-dependent disturbances (Avgousti & Beris 1993 b ). In this work, we calculate the bifurcating families corresponding to each one of the two possible non-axisymmetric patterns emerging at the point of criticality, namely the spirals and ribbons and determine their stability. It is shown that for a narrow gap size, for upper convected Maxwell and Oldroyd-B fluids, at least one of the non-axisymmetric families bifurcates subcritically. This result, in conjunction with the theoretical analysis of Hopf bifurcation in presence of symmetries (Golubitsky et al . 1988), implies that neither of the bifurcating families is stable. Consequently, there is a finite transition corresponding to infinitesimal changes of the flow parameters in the vicinity of the Hopf bifurcation point. Although a change in the ratio of the Deborah and Reynolds numbers has not been found to qualitatively affect this behaviour, calculations with a wider gap size have shown that both bifurcating families become supercritical. There, a Ginzburg–Landau analysis shows that the ribbons are the stable pattern. This behaviour is qualitatively similar to that seen for the newtonian fluid, but for counterrotating cylinders, albeit there, spirals have been found to be stable (Golubitsky & Langford 1988).

scholarly journals Optimal Taylor–Couette flow: direct numerical simulations

2013 ◽  
Vol 719 ◽  
pp. 14-46 ◽  
Author(s):  
Rodolfo Ostilla ◽  
Richard J. A. M. Stevens ◽  
Siegfried Grossmann ◽  
Roberto Verzicco ◽  
Detlef Lohse
Keyword(s):  
Angular Velocity ◽  
Couette Flow ◽  
Scaling Laws ◽  
Outer Cylinder ◽  
Reynolds Numbers ◽  
Velocity Profiles ◽  
Experimental Result ◽  
Counter Rotation ◽  
Taylor Couette ◽  
Taylor Couette Flow

AbstractWe numerically simulate turbulent Taylor–Couette flow for independently rotating inner and outer cylinders, focusing on the analogy with turbulent Rayleigh–Bénard flow. Reynolds numbers of $R{e}_{i} = 8\times 1{0}^{3} $ and $R{e}_{o} = \pm 4\times 1{0}^{3} $ of the inner and outer cylinders, respectively, are reached, corresponding to Taylor numbers $Ta$ up to $1{0}^{8} $. Effective scaling laws for the torque and other system responses are found. Recent experiments with the Twente Turbulent Taylor–Couette (${T}^{3} C$) setup and with a similar facility in Maryland at very high Reynolds numbers have revealed an optimum transport at a certain non-zero rotation rate ratio $a= - {\omega }_{o} / {\omega }_{i} $ of about ${a}_{\mathit{opt}} = 0. 33$. For large enough $Ta$ in the numerically accessible range we also find such an optimum transport at non-zero counter-rotation. The position of this maximum is found to shift with the driving, reaching a maximum of ${a}_{\mathit{opt}} = 0. 15$ for $Ta= 2. 5\times 1{0}^{7} $. An explanation for this shift is elucidated, consistent with the experimental result that ${a}_{\mathit{opt}} $ becomes approximately independent of the driving strength for large enough Reynolds numbers. We furthermore numerically calculate the angular velocity profiles and visualize the different flow structures for the various regimes. By writing the equations in a frame co-rotating with the outer cylinder a link is found between the local angular velocity profiles and the global transport quantities.

Effects of Low Uniform Relative Roughness on Single-Phase Friction Factors in Microchannels and Minichannels

2007 ◽  
Author(s):  
Timothy P. Brackbill ◽  
Satish G. Kandlikar
Keyword(s):  
Large Diameter ◽  
Reynolds Numbers ◽  
Transition To Turbulence ◽  
Roughness Effects ◽  
Relative Roughness ◽  
Flow Parameters ◽  
Friction Factors ◽  
Turbulent Flow Regime ◽  
Critical Reynolds Numbers ◽  
Large Diameter Pipes

Nikuradse’s [1] work on friction factors focused on the turbulent flow regime in addition to being performed in large diameter pipes. Laminar data was collected by Nikuradse, however only low relative roughness values were examined. A recent review by Kandlikar [2] showed that the uncertainties in the laminar region of Nikuradse’s experiments were very high, and his conclusion regarding no roughness effects in the laminar region is open to question. In order to conclusively resolve this discrepancy, we have experimentally determined the effects of relative roughness ranging from 0–5.18% in micro and minichannels on friction factor and critical Reynolds numbers. Reynolds numbers were varied from 30 to 7000 and hydraulic diameters ranged from 198μm to 1084μm. There is indeed a roughness effect seen in the laminar region, contrary to what is reported by Nikuradse. The resulting friction factors are well predicted using a set of constricted flow parameters. In addition to higher friction factors, transition to turbulence was observed at decreasing Reynolds numbers as relative roughness increased.

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.

Heat Transfer Enhancement in Axial Taylor-Couette Flow

2005 ◽  
Author(s):  
S. Gilchrist ◽  
C. Y. Ching ◽  
D. Ewing
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Nusselt Number ◽  
Couette Flow ◽  
Reynolds Numbers ◽  
Taylor Number ◽  
Number Range ◽  
Taylor Couette ◽  
Number Case ◽  
Taylor Couette Flow

An experimental investigation was performed to determine the effect that surface roughness has on the heat transfer in an axial Taylor-Couette flow. The experiments were performed using an inner rotating cylinder in a stationary water jacket for Taylor numbers of 106 to 5×107 and axial Reynolds numbers of 900 to 2100. Experiments were performed for a smooth inner cylinder, a cylinder with two-dimensional rib roughness and a cylinder with three-dimensional cubic protrusions. The heat transfer results for the smooth cylinder were in good agreement with existing experimental data. The change in the Nusselt number was relatively independent of the axial Reynolds number for the cylinder with rib roughness. This result was similar to the smooth wall case but the heat transfer was enhanced by 5% to 40% over the Taylor number range. The Nusselt number for the cylinder with cubic protrusions exhibited an axial Reynolds number dependence. For a low axial Reynolds number of 980, the Nusselt number increased with the Taylor number in a similar way to the other test cylinders. At higher axial Reynolds numbers, the heat transfer was initially independent of the Taylor number before increasing with Taylor number similar to the lower Reynolds number case. In this higher axial Reynolds number case the heat transfer was enhanced by up to 100% at the lowest Taylor number of 1×106 and by approximately 35% at the highest Taylor number of 5×107.

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.

Energy stability of modulated circular Couette flow

1977 ◽  
Vol 79 (3) ◽  
pp. 535-552 ◽  
Author(s):  
Peter J. Riley ◽  
Robert L. Laurence
Keyword(s):  
Couette Flow ◽  
Nonlinear Stability ◽  
Energy Analysis ◽  
Outer Cylinder ◽  
Reynolds Numbers ◽  
Initial Energy ◽  
Critical Parameters ◽  
Narrow Gap ◽  
Circular Couette Flow ◽  
The Stability

The stability of circular Couette flow when the outer cylinder is at rest and the inner is modulated both with and without a mean shear is examined in the narrow-gap limit. Disturbances are assumed to be axisymmetric. Two criteria are used to determine conditions for stability; the first requires that the motion be strongly stable, the second only that disturbances of arbitrary initial energy decay from cycle to cycle. The behaviour of critical parameters as a function of frequency is similar for the linear and the energy analysis. The range of Reynolds numbers bounded above by certain instability and below by conditional nonlinear stability is enlarged by modulation.

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