Rotating Disk Flow Research Articles

On the nature of PSE approximation

The recently developed method of parabolized stability equations (PSE) offers a fast and efficient way of analyzing the spatial growth of linear and nonlinear (convective) disturbances in shear layers. For incompressible flows, the governing equations may be represented either in primitive variables or by using other formulations obtained by eliminating the pressure gradient (e.g., vorticity-streamfunction formulation). On the other hand, for compressible flows, primitive variables offer a natural and the only choice. We show that primitive-variable formulation is not well-posed due to the ellipticity introduced by the\(\partial \hat p/\partial x\) term and the marching solution eventually blows up for a sufficiently small step size. However, it is shown that this difficulty can be overcome if the minimum step size is greater than the inverse of the real part of the streamwise wave number, αr. An alternative is to drop the\(\partial \hat p/\partial x\) term, in which case the residual ellipticity is of no consequence for marching computations with much smaller step sizes. However, the ellipticity cannot be completely removed. Results obtained with streamfunction and vorticity-velocity formulations also show that the numerical difficulties arise for a sufficiently small marching step size. This step-size restriction can be overcome by dropping thedα/dx term from the governing equations. The effect of this term on solution accuracy is negligible for Blasius flow but not so for rotating-disk flow.

Simple cases of the streamline-curvature instability in three-dimensional boundary layers

A new instability of the centrifugal type due to the curvature of external streamlines was theoretically predicted in a recent study on boundary layers along a swept wing. It is, however, not clear how this instability relates to already-known instability phenomena in various three-dimensional flows. So the basic idea developed in the analysis of boundary layers is applied to the simpler problems of the flow on a rotating disk and along the leading edge of a yawed circular cylinder, and the resulting eigenvalue problems are numerically solved to show multiple stability characteristics of the flows. Computational results confirm that the streamline-curvature instability does appear in the rotating-disk flow and that it is in fact identical with the instability called the ‘parallel’ or ‘type 2’ mode in the atmospheric literature. This instability is also found to occur in the steady flow near the attachment line and to give the lowest values of the critical Reynolds number except for a very narrow region close to the attachment line, where the viscous and cross-flow instabilities are dominant. These facts provide evidence to show that the same mode of instability as the classical one observed in rotating flows can appear in general three-dimensional boundary layers without rotation.

An experimental study of absolute instability of the rotating-disk boundary-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.

Flows Generated by the Impingement of a Jet on a Rotating Surface: Part II—Detailed Flow Structure and Analysis

In Part I of this investigation, the basic flow patterns resulting from the interaction of a turbulent jet and a rotating disk flow were presented paying particular attention to mean and turbulent velocity fields around the jet impingement point on the disk and the flow patterns above the disk in the absence of the jet. The latter was used as a base flow pattern, on which the influence of the jet impingement could be assessed. In Part II, further details of the jet/disk flow interaction problem are discussed in the context of potential turbomachinery cooling applications and previous experimental results.

Experimental study of rotating disk instability. I. Natural flow

This article is devoted to the study of the rotating disk flow instability. This inflectional-type instability, called a cross-flow instability, exemplary of the transition to turbulence in three-dimensional boundary layers. We first present the experimental marginal stability curve of unstable waves obtained by hot-film probe measurements and compare it with the theoretical results available in the literature. The experiment is in accordance with different theoretical determinations of the linear threshold, but we note a difference between experimental and theoretical critical wave-number values. The unstable wave dynamics is then investigated by means of experimental dispersion curves (linking frequencies to the wave-number vector components) determined by two-probe measurements. The results show, in particular, the existence of traveling dispersive waves in the boundary layer of the rotating disk. Finally, we show that the emergence of nonlinear effects occurs very early in the system, far from the transition point.

THE KINETICS OF ACETIC ACID EXTRACTION STUDIED WITH THE ROTATING STABILIZED CELL TECHNIQUE.APPLICATION OF THE BARRIER MODEL.

ABSTRACT The kinetics of acetic acid extraction is studied with the use of an improved version of the rotating stabilized cell (RSG) technique at interfaces between an aqueous solution and an organic phase. The RSC is implemented as follows: the aqueous phase, containing the species of interest, is immobilized in a gel, placed in a rotating cylinder immersed in the organic phase. The transport processes in both phases are reliably controlled: pure diffusion in the gel, and rotating disc flow in the organic fluid. The amount of matter extracted from the gel is measured as a function of time; the data are analyzed at times much shorter than the mean diffusion time in the gel, in a transient regime. First, the reliability of the method is studied by examining the dependence of the aqueous-towards-organic rate constant upon the rotation speed; a critical value is found which corresponds to the maximum speed for a laminar flow. Then, the method is applied to the study of the kinetics of acetic acid extraction ...

Nonlinear effects on rotating disk flow in a cylindrical casing

The flow due to a finite disk rotating in an incompressible viscous fluid has been studied. A modified Newton-gradient finite difference scheme is used to obtain the solution of full Navier-Stokes equations numerically for different disk and cylinder sizes for a wide range of Reynolds numbers. The introduction of the aspect ratio and the disk-shroud gap, significantly alters the flow characteristics in the region under consideration, The frictional torque calculated from the flow data reveals that the contribution due to nonlinear terms is not negligible even at a low Reynolds number. For large Reynolds numbers, the flow structure reveals a strong boundary layer character.

Weakly non-linear development of rotating disk flow

The problem of primary stability of rotating disk boundary layer flow for stationary modes was studied by using the weakly non-linear theory where the Reynolds number R is close to its critical value R c as determined by linear theory. The system of equations for small amplitude disturbances is solved by a combination of theoretical and computational methods. A Landau type equation for the disturbance amplitude, A, is derived whose coefficients are determined numerically. It is found that for R < R c subcritical instability is exhibited while ¦A¦ becomes very large in a finite time for R > R c . These results suggest possible fast transition to turbulence in rotating-disk flow.

Secondary instability in rotating-disk flow

Primary instability of the three-dimensional boundary layer on a rotating disk introduces periodic modulation of the mean flow in the form of stationary crossflow vortices. Here we study the stability of this modulated mean flow with respect to secondary disturbances. These secondary disturbances are found to have quite large growth rates compared to primary disturbances. Both fundamental and subharmonic resonance cases are considered and their corresponding results indicate that the growth rate and the frequency of the secondary instability are insensitive to the exact nature of the resonance condition. The threshold primary stationary crossflow vortex amplitude for secondary instability found in this three-dimensional incompressible boundary layer is significantly larger than that for a two-dimensional boundary layer which is subjected to Tollmien–Schlichting instability. The secondary instability results in a pair of travelling counter-rotating vortices, tilted up and oriented at an angle to the primary stationary crossflow vortices. The computed velocity signals and flow visualization, evaluated based on this secondary disturbance structure, are compared with experimental results.

On a class of unsteady three-dimensional Navier—Stokes solutions relevant to rotating disc flows: threshold amplitudes and finite-time singularities

A class of ‘exact’ steady and unsteady solutions of the Navier—Stokes equations in cylindrical polar coordinates is given. The flows correspond to the motion induced by an infinite disc rotating in the (x, y)-plane with constant angular velocity about the z-axis in a fluid occupying a semi-infinite region which, at large distances from the disc, has velocity field proportional to (x, — y,O) with respect to a Cartesian coordinate system. It is shown that when the rate of rotation is large Kármán's exact solution for a disc rotating in an otherwise motionless fluid is recovered. In the limit of zero rotation rate a particular form of Howarth's exact solution for three-dimensional stagnation-point flow is obtained. The unsteady form of the partial differential system describing this class of flow may be generalized to time-periodic flows. In addition the unsteady equations are shown to describe a strongly nonlinear instability of Kármán's rotating disc flow. It is shown that sufficiently large perturbations lead to a finite-time breakdown of that flow whilst smaller disturbances decay to zero. If the stagnation point flow at infinity is sufficiently strong the steady basic states become linearly unstable. In fact there is then a continuous spectrum of unstable eigenvalues of the stability equations but, if the initial-value problem is considered, it is found that, at large values of time, the continuous spectrum leads to a velocity field growing exponentially in time with an amplitude decaying algebraically in time.

On the receptivity and nonparallel stability of traveling disturbances in rotating-disk flow

The generation and evolution of small amplitude long wavelength traveling disturbances in rotating-disk flow is the subject of this paper. The steady rotational speed of the disk is perturbed so as to introduce high-frequency oscillations in the flow field. Secondly, we introduce surface imperfections on the disk such as roughness elements. The interaction of these two disturbances will generate the instability waves whose evolution is governed by parabolic partial differential equations which are solved numerically. It is found that, for the class of disturbances considered here (wavelength on the order of Reynolds number), eigensolutions exist which decay or grow algebraically in the radial direction. However, these solutions grow only for frequencies larger than 4.58 times the steady rotational speed of the disk. The computed receptivity coefficient shows that there is an optimum size of roughness for which these modes are preferentially excited. The width of these roughness elements in the radial direction is about 0.1r 0 * where r 0 * is the radial location of the roughness. It is also found that the receptivity coefficient is larger for a negative spanwise wave number than for a positive one. The cumulative wave pattern produced from the roughness site shows that the typical wave angles for these disturbances are about −26° with about seven waves around the circumference. This is in contrast with the wave angles of 10°–14° found for the 30 or so inviscid cross-flow vortices.

Rotating stabilised cell: a new tool for the investigation of interfacial extraction kinetics between liquid phases

New experiments performed with an improved version of the rotating stabilised cell are presented. In this technique the aqueous phase is immobilised: it is found that a gel of polyacrylamide is a good stabilising medium for this purpose. The gel is placed in a cylindrical cell of small dimensions which can be set to rotate. A pure diffusion of the species to be extracted occurs in the aqueous phase. A flat interface is created for the contact with the organic extractant solution. A rotating-disc flow is promoted in this phase, which permits the control of the diffusive layer. The system investigated is the cobalt(II) ion at the interface water (+ buffer solution and supporting electrolyte)/DEHPA [bis(2-ethylhexyl)hydrogenphosphate] 0.2 mol dm–3 in dodecane, for two values of the pH of the aqueous phase: pH 3 and pH 5. A mathematical expression is derived for the amount of matter transferred, and the validity of the formula is examined mathematically. The influence of the rotation speed of the cell and the influence of its length are studied in the case where the pH is 3. The values of the aqueous-toward-organic rate constants found at pH 3 and pH 5 are in reasonable agreement with the results of other workers.

Traveling disturbances in rotating-disk flow

The stability curves for traveling disturbances in rotating-disk flow are computed using the sixth-order system of incompressible linear stability equations. It is found that the neutral curve has two minima for disturbances with positive frequencies as found earlier by Malik (1986) for stationary disturbances. The upper branch minimum occurs at ω=−2.9, R=283.6 while the lower branch minimum occurs at ω=7.9, R=64.46 where R is the Reynolds number. There exists a critical angle of approximately −35.34° (which is about 15° from the direction of maximum wall shear) below which all the waves are linearly damped. The results also show that at high frequencies the wave number for lower branch neutral disturbances varies with Reynolds number like R−1 while for stationary waves it behaves like R−1/2. The eigenfunction distribution suggests that the structure of the nonstationary high-frequency lower branch neutral disturbances are different from the structure of the viscous stationary disturbances.

A solid-liquid phase-transfer catalysis in rotating-disk flow

A solid-liquid phase-transfer catalysis in rotating-disk flow

Stability of Boundary Layers along a Rotating Disk with Uniform Suction

AbstractA theory is developed for the linear stability analysis on the effect of uniform suction through the disk due to three‐dimensional rotating disk flow, in which the effect of Coriolis forces and streamline curvature are included. The characteristics of the laminar boundary layer with uniform suction along a rotating disk are calculated, and then the perturbation equation describing the spatial evolution of disturbances is solved numerically according to the Adams‐Bashforth method for the velocity profiles obtained above. Variation of neutral stability curves and also of the critical Reynolds number are given for the values of the suction parameter. The results show that the boundary layer flow with uniform suction along a rotating disk becomes stable.

Similarity solutions for stratified rotating-disk flow

This work treats the vertically stratified system of two homogeneous fluid layers confined between horizontal infinite rotating disks which rotate steadily about a common vertical axis. Allowance is made for uniform injection of fluid at either disk. With appropriate restrictions on disk rotational speeds and injection rates a flat interface is possible, and the problem admits similarity solutions to the Navier-Stokes equations of the Karman-Bodewadt-Batchelor variety. This type of flow allows for a uniformly accessible surface of interphase mass and heat transfer at the two-fluid interface, and, with that as the primary motivation, the present work provides exploratory numerical solutions of the above equations for both corotation and counter-rotation combined with injection.A linearized theory is given for the case of nearly rigid rotation, with explicit analytical results for the large-Reynolds-number boundary-layer limit. Also, we offer a theoretical discussion of the inviscid limit for arbitrary rotation and injection rates. Based on the type of Euler-cell solutions identified in previous work, we derive the remarkably simple formula \[ \rho_1\omega_1^2\cot^2\frac{\omega_1d_1}{V_1} = \rho_2\omega^2_2\cot^2\frac{\omega_2d_2}{V_2} \] connecting densities ρ, depths d, rotation speeds ω and injection velocities V.Sample calculations and comparisons are given for property ratios typical of water-kerosene layers. In this case, the linearized theory works exceedingly well for corotation with small injectional Rossby numbers V/ωd. The simple inviscid theory cited above shows excellent agreement with the numerical computations for Reynolds numbers greater than 500 and for Rossby numbers > 1/π, corresponding to strong blowing in the inviscid regime. The larger-wavelength inviscid cell structure appears to provide the kind of stagnation-flow pattern essential to the application envisaged.

The neutral curve for stationary disturbances in rotating-disk flow

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.

Mass-transfer enhancement and reaction-rate limitations in solid-liquid phase-transfer catalysis

We present an analysis of mass transfer in solid-liquid phase-transfer catalysis (SLPTC), obtaining analytical solutions for the stagnant-film model and numerical solutions to the complete transport equations for the von Kármán rotating-disk flow. The latter is viewed as a prime candidate for experimental studies on SLPTC. A catalyst-effectiveness plot is developed, which displays both mass-transfer enhancement and reaction-rate limitation associated with the two distinct Damköhler parameters for the bulk and surface reactions. Solutions are obtained from the film model for pseudo-first-order reaction and for rapid equilibrium reaction, including irreversible reaction with flame-like reaction fronts. Comparison of stagnant-film model against the numerical solutions for the rotating disk shows generally good agreement, as one would expect for uniformly accessible surfaces.

Instability and transition in rotating disk flow

The stability of three dimensional rotating disk flow and the effects of Coriolis forces and streamline curvature were investigated. It was shown that this analysis gives better growth rates than Orr-Sommerfeld equation. Results support the numerical prediction that the number of stationary vortices varies directly with the Reynolds number.

Analytical and numerical results for the non-stationary rotating disk flow

This paper deals with the time-dependent flow due to an infinite rotating disk. The Navier-Stokes equations are transformed by Von Karman's similarity approach. The resulting equations have been studied both numerically and analytically for two cases: (1) the flow due to a disk whose angular velocity abruptly changes sign, and (2) the oscillating disk flow. Some numerical results are compared with the outcome of the analysis.