The Stewartson layer of a rotating disk of finite radius

It is shown that if a disk of finite radius and the surrounding medium rotate coaxially with slightly different angular velocities, an axial layer in the form of a cylindrical shell exists at the edge of the disk. This shell of thickness O(E ^1/3) has length O(E ^–1) in axial direction, where E is the Ekman number. Its most characteristic element is the axial velocity of O(E ^1/6) which is larger than everywhere else in the field. We calculate the velocity components and the pressure in this layer.     

Fluid flow between a rotating disk and a fixed wall

The results of a theoretical investigation of the fluid flow in the gap between a rotating disk and a fixed wall are presented. The variations of the rate of twisting of the fluid and the statistical pressure along the radius are analyzed for the case of centripetal flow.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 26, No. 4, pp. 611–617, April, 1974.     

Fluctuating flow induced by periodic suction through a rotating disk

Summary This paper is concerned with the time-harmonic and time-mean flow induced by a normal periodic suction through a disk rotating in contact with an incompressible viscous fluid. Two separate analytical-numerical solutions are developed for the extreme cases of low and high frequency oscillations, employing the Fettis-Benton method. General formulae for the skin-friction components and the far-field suction velocity are obtained. The deduced quasi-steady results are in agreement with the available numerical results.

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Die durch periodisches Saugen durch eine rotierende Scheibe erzeugte schwankende Strömung

Zusammenfassung Die vorliegende Arbeit behandelt die zeitlich harmonische und die zeitlich gemittelte Strömung, verursacht durch eine periodische Saugwirkung normal durch eine Scheibe, die in einer inkompressiblen zähen Flüssigkeit rotiert. Zwei getrennte analytisch-numerische Lösungen werden mit Hilfe der Fettis-Benton-Methode für die beiden Sonderfälle niedriger bzw. hoher Oszillationsfrequenz hergeleitet. Allgemeine Beziehungen ergeben sich für die Komponenten der Oberflächenreibungskraft und die Sauggeschwindigkeit. Die Ergebnisse für den stationären Fall sind in Übereinstimmung mit numerisch gewonnenen Resultaten.

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On leave from Indian Institute of Technology Kharagpur India     

The flow between a rotating disk and a stationary porous disk of finite thickness

The flow of a viscous incompressible fluid between a rotating disk and a stationary disk is studied. The two disks are impermeable. There is a layer of porous medium over the stationary disk. The generalized Darcy law is used to represent the fluid motion in the porous medium and the Navier-Stokes equations in the pure fluid medium. The nonlinear equations representing the motion, together with the boundary conditions and the matching conditions at the interface, constitute a two point boundary value problem. In view of the sensitive nature of the problem to the choice of missing initial conditions required for the forward integration, a shooting method in conjunction with the continuation method is used to produce the numerical solution. The response of the velocity profiles and stresses on the boundaries to the increase in the thickness of the porous layer is studied.     

Transitional flow conditions on a rotating disk

Transitional flow conditions on the surface of a freely rotating disk were investigated by cine-recording, acoustic, and visual methods.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 31, No. 6, pp. 1060–1067, December, 1976.     

Rotationally symmetric fluctuating flow about a rotating disk

Superposed mean flow due to fluid performing torsional oscillations at large distances from a rotating disk is analysed. For low frequency of oscillations an analytical-numerical solution is obtained for small values of the amplitude parameter, which agrees fairly well with the available numerical results in the limit of zero frequency. Method of multiple scales is employed to derive a solution for high frequency range and for finite values of the amplitude parameter.     

Nonlinear effects on rotating disk flow in a cylindrical casing

Summary 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.     

Method of calculation of a turbulent flow in an axial gap with a variable radius between a rotating disk and an axisymmetric casing

An integral method for calculation of a turbulent flow in an axial gap between a rotating disk and an axisymmetric casing is developed with account for the small flow through the gap, the variation of the gap width over the radius, and the interaction with outer flows. Limitations of the mathematical models used by most researchers and ways of surmounting them are revealed. The method is confirmed by a comparison with known experimental data. The obtained computational integral parameters were used repeatedly to improve economy and reliability of industrial pumps, turbines, and compressors. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 71, No. 6, pp. 1107–1115, November–December, 1998.     

Flow induced by an asymmetrically placed disk rotating coaxially inside a cylindrical casing

Summary In this paper, the flow due to a rotating disk non-symmetrically placed with respect to the height of the enclosing stationary cylinder is analyzed numerically. The full Navier-Stokes equations expressed in terms of stream function and vorticity are solved by successive over-relaxation for different disk radii, its distance from the bottom casing and rotational Reynolds numbers. It is observed that the flow pattern is strongly influenced by the size and the position of the disk. When the disk is very close to the top casing and small in radius, there are two regions of different scales and the vortices in the region of small scale are trapped between the disk and the top casing. Further, the variation of the moment coefficient is determined for different positions and sizes of the rotating disk. The calculations shows that the frictional torque increases rapidly, when the disk approaches the top casing. This finding is of importance for the design of vertical rotating disk reactors applied in chemical vapor deposition.     

Development of a two-layer fluid flow near a rotating disk upon acceleration

We consider the problem of the development of boundary-layer flow on a suddenly accelerated disk in a two-layer fluid when at the initial instant of time the interface between the fluids is perpendicular to the plane of the disk. Kazan State Technological University, Kazan, Russia. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 69, No. 1, pp. 129–132, January–February, 1996     

Exact flow/heat solutions for the non-axisymmetric flow over a rotating disk

The present note is concerned with the exact non-axisymmetric solutions for the flow over a rotating disk. The governing non-axisymmetric flow equations of motion generate exact flow solutions from which analytic expressions for the vorticity, shear stresses, flow/thermal layer thicknesses and rate of heat transfer are obtained. The effects of Brinkman number, heat generation/absorption as well as thermal radiation on the temperature field can be better pursued from the extracted formulae.     

Characteristics of flow between a rotating and a static disk in the presence of radial flow

An improved method is proposed for the calculation of the flow in the gap between a rotating and a static disk in the presence of radial flow. The algorithm of the solution is realized on a Nairi-2 computer.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 32, No. 2, pp. 234–241, February, 1977.     

Computation of the flow between a rotating and a stationary naturally permeable disk

A numerical exploration for the flow between a rotating and a stationary disk made up of porous material having depth equal to the distance between the two disks, is carried out by continuation method. The flow field has been studied by solving Navier-Stokes equations in the free fluid region and using Darcy's equations in the porous region.     

Fluid flow induced by a small amplitude harmonically oscillating cascade

Summary Fluid flow induced by a harmonically oscillating cascade of normal flat plates and square cylinders has been investigated. We assume that the amplitude of the oscillation is small in comparison with the typical length of the body and that the Reynolds number, based on the typical length and speed being the size of the body and the frequency of the oscillation, respectively, is small. First-order solutions have been obtained using both a numerical technique and the series truncation method and the results from these two methods show good agreement. The streaming flow induced by the oscillation has also been investigated by solving the set of coupled perturbation equations numerically.     

A mixed-mode crack analysis of rotating disk using finite element method

This paper presents a rigorous elastodynamic hybrid-displacement finite element procedure for a safety analysis of fast rotating disks with mixed-mode cracks. Based on a modified Hamilton's principle, the finite element model is derived such that the proper crack-tip singularities are taken into consideration and the interelement displacement compatibility conditions are still satisfied. Thus, the specimen can be represented by a finite element assemblage in which “singular” elements are used around the crack-tip and high-order isoparametric “regular” elements are taken elsewhere.To determine the mixed-mode stress intensity factors, the modified $\text{J}\text{?}{}_{\mathrm{k}}$ integrals for rotating cracked disks have been established taking into account the effect of centrifugal force. Using the “strain-energy-density factor” concept, the direction of crack growth of a rotating disk with an arbitrary internal crack is predicted. To provide a method of non-destructive testing in evaluating the integrity of structures, natural vibrations of cracked disk are then studied. Lastly, the influence of inertia effects due to rotating speed changes in determining the dynamic stress intensity factors is examined.For verification purposes, the simple case of a rotating disk with radial cracks is first solved. Excellent correlations between the computed results and available referenced solutions are drawn. New solutions for the circular disk with circumferential or arbitrarily-oriented cracks are also presented.     

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

Summary This paper deals with the time-dependent flow due to an infinite rotating disk. The Navier-Stokes equations are transformed by Von Kármán'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.     

Periodic flow of a second-grade fluid induced by non-torsional oscillations of eccentric rotating disks

This paper deals with the periodic flow of a second-grade fluid caused by non-torsional oscillations of two disks rotating about non-coincident axes. While the two parallel disks are initially rotating with the same angular velocity about distinct axes, they start to execute non-torsional oscillations in their own planes and in the opposite directions. An exact solution is obtained for the components of the horizontal force per unit area exerted by the top and bottom disks on the fluid in the periodic state. The results are graphically displayed and the influence of the second-grade fluid parameter, the ratio of the frequency of oscillation to the angular velocity of the disks, the Reynolds number and the dimensionless velocity amplitudes of oscillation is discussed. It is observed that the change in the \( x \)-component of the mentioned force gets larger when the second-grade fluid parameter increases. However, an opposite effect is seen for the change in the \( y \)-component.