Rotationally symmetric flow over a rotating disk

The rotationally symmetric flow over a rotating disk in an incompressible viscous fluid is analyzed by a new method when the fluid at infinity is in a state of rigid rotation (in the same or in the opposite sense) about the same axis as that of the disk. Asymptotic expansions for the velocity field over the entire flow field are obtained for the general class of one-parameter rotationally symmetric flows. This method is further extended to the case when a uniform suction or injection is assumed at the rotating disk. Fluid motion induced by oscillatory suction of small amplitude at the rotating disk is also discussed.An initial-value analysis reveals that resonance is possible only when the angular velocity of the rotating fluid is greater than that of the rotating disk.     

Unsteady Flow of an Oscillating Porous Disk and a Fluid at Infinity

An exact analytic solution of the unsteady Navier–Stokes equations is obtained for the flow caused by the non-coaxial rotations of a porous disk and a fluid at infinity. The porous disk is executing oscillations in its own plane with superimposed injection or suction. An increasing or decreasing velocity amplitude of the oscillating porous disk is also discussed. Further, it is shown that a combination of suction/injection and decreasing/increasing velocity amplitude is possible as well. In addition, the flow due to porous oscillating disk and a fluid at infinity rotating about an axis parallel to the z-axis is attempted as a second problem. Sommario. Si studia il flusso non stazionario prodotto dall'oscillazione di un disco poroso in un fluido e si fornisce una soluzione analitica delle equazioni di Navier–Stokes. Si discute l'effetto di una suzione/iniezione e di una variazione sull'ampiezza della velocità' di oscillazione. Infine si studia il flusso dovuto alle oscillazioni non coassiali di un disco poroso e di un fluido all'infinito.     

A class of exact solutions for the incompressible viscous magnetohydrodynamic flow over a porous rotating disk

The present paper is concerned with a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous electrically conducting fluid flow due to a porous disk rotating with a constant angular speed.The three-dimensional hydromagnetic equations of motion are treated analytically to obtained exact solutions with the inclusion of suction and injection.The well-known thinning/thickening flow field effect of the suction/injection is better understood from the constructed closed form velocity equations.Making use of this solution,analytical formulas for the angular velocity components as well as for the permeable wall shear stresses are derived.Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation.The temperature field is shown to accord with the dissipation and the Joule heating.As a result,exact formulas are obtained for the temperature field which take different forms corresponding to the condition of suction or injection imposed on the wall.     

Exact solutions for the incompressible viscous fluid of a porous rotating disk flow

The present paper is concerned with a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous fluid flow motion due to a porous disk rotating with a constant angular speed. The three-dimensional equations of motion are treated analytically yielding derivation of exact solutions with suction and injection through the surface included. The well-known thinning/thickening flow field effect of the suction/injection is better understood from the exact velocity equations obtained. Making use of this solution, analytical formulas corresponding to the permeable wall shear stresses are extracted.Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. As a result, exact formulas are obtained for the temperature field which take different forms depending on whether suction or injection is imposed on the wall. The impacts of several quantities are investigated on the resulting temperature field. In accordance with the Fourier‘s heat law, a constant heat transfer from the porous disk to the fluid takes place. Although the influence of dissipation varies, suction enhances the heat transfer rate as opposed to the injection.     

Design, testing and two-dimensional flow modeling of a multiple-disk fan

A multiple-disk Tesla type fan has been designed, tested and analyzed two-dimensionally using the conservation of angular momentum principle. Experimental results showed that such multiple-disk fans exhibited exceptionally low performance characteristics, which could be attributed to the low viscosity, tangential nature of the flow, and large mechanical energy losses at both suction and discharge sections that are comparable to the total input power. By means of theoretical analysis, local and overall shearing stresses on the disk surfaces have been determined based on tangential and radial velocity distributions of the air flow of different volume flow rates at prescribed disk spaces and rotational speeds. Then the total power transmitted by rotating disks to air flow, and the power acquired by the air flow in the gap due to transfer of angular momentum have been obtained by numerically integrating shearing stresses over the disk surfaces. Using the measured shaft and hydraulic powers, these quantities were utilized to evaluate mechanical energy losses associated with the suction and discharge sections of the fan.     

Numerical study of the flow and heat transfer in a Reiner-Rivlin fluid on a rotating porous disk

An unsteady flow and heat transfer to an infinite porous disk rotating in a Reiner—Rivlin non-Newtonian fluid are considered. The effect of the non-Newtonian fluid characteristics and injection (suction) through the disk surface on velocity and temperature distributions and heat transfer is considered. Numerical solutions are obtained over the entire range of the governing parameters.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 85–95, January–February, 2005.     

Flow due to non-coaxial rotation of a porous disk and a second grade fluid rotating at infinity

An exact solution for the three-dimensional flow due to non-coaxial rotation of a porous disk and a second grade fluid at infinity is obtained. It is shown that for uniform suction or uniform blowing at the disk, an asymptotic profile exists for the velocity distribution. The velocity depends on two parameters: one of them is the suction parameter or blowing parameter and the other is the visco-elastic parameter. Furthermore, it is found that when the value of the visco-elastic parameter is fixed, the velocity decreases with an increase in the value of the suction parameter and when the value of the suction parameter is fixed, the velocity increases with an increase in the value of the visco-elastic parameter.     

Some Steady MHD Flows of the Second Order Fluid

The exact analytic solutions of two problems of a second order fluid in presence of a uniform transverse magnetic field are investigated. The governing equation is of fourth order ordinary differential equation and is solved using perturbation method. In the first problem we discuss the flow of a second order fluid due to non-coaxial rotations of a porous disk and a fluid at infinity. In second problem the flow of a second order conducting fluid between two infinite plates rotating about the same axis is investigated, with suction or blowing along the axial direction. For second order conducting fluid it is observed that asymptotic solution exists for the velocity both in the case of suction and blowing.     

Magnetohydrodynamic flow past a porous rotating disk in a circular magnetic field

This paper studies the effects of a circular magnetic field on the flow of a conducting fluid about a porous rotating disk. Using modern quasi-Newton and globally convergent homotopy methods, numerical solutions are obtained for a wide range of magnetic field strengths, suction and injection velocities and Alfven and disk speeds. Results are presented graphically in terms of three non-dimensional parameters. There is excellent agreement with previous work and asymptotic formulae.     

高速转动圆盘流动数值模拟研究

重点研究高速离心压气机叶轮与机匣间的间隙流动及其温度分布。研究将离心压气机简化为高速转动圆盘,搭建了相关实验平台,并开展了相应的数值模拟研究。通过改变转动圆盘的转速和轴向进入的冷却流的流量,研究了转速和流量对于间隙内温度和速度分布的影响。结果显示,转速是影响温度变化的最主要因素,转速越大,温度越高;同等幅度的流量变化对温度的影响则较小。研究发现,在实验和模拟对应的大雷诺数条件下,无量纲的速度分布基本不受到圆盘转速、冷却流量和温度场的影响。     

Flow between a porous rotating disk and a plane

It is shown that when a viscous incompressible fluid is sucked through a stationary porous disk spontaneous rotation of the fluid sets in at a certain Reynolds number. This is consistent with the results of a specially designed experiment. Another unusual result is the existence of multicell regimes, corresponding to suction, when the force acting on the porous, rapidly rotating disk is a lift force and, moreover, anomalously large. Charts of the possible steady-state flow regimes, stable and unstable, have been constructed. In the case of fairly intense suction and rotation a stable self-oscillating regime is observed. In the limit of vanishingly small viscosity unusual boundary layer properties associated with suction are noted.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 53–65, November–December, 1989.     

Investigation on convective heat transfer over a rotating disk with discrete pins

A three-dimensional numerical study on the flow and heat transfer characteristics over a rotating disk surface with discrete pins was conducted by the use of RNG k–ε turbulent model. And some experiments were also made for validation. The effects of rotating angular speed and pin configuration on the temperature maps and convective heat transfer characteristics on the rotating surface were analyzed. As the increase of rotating velocity, the impingement of pumping jet on the centre of rotating disk becomes stronger and the transition from laminar to turbulent occurs at the outer radius of rotating disk, which resulting in heat transfer enhancement. The pins on the disk make the pumping action of a rotating disk weaker. Simultaneously, they also act as perturbing elements to the cyclone flow near the rotating disk surface, making the overall heat transfer to be enhanced. The needle pins have higher convective heat transfer capacity than the discrete ring pins with the same extend pin areas.     

MHD flow of a power-law fluid over a rotating disk

Magnetohydrodynamic flow of an electrically conducting power-law fluid in the vicinity of a constantly rotating infinite disk in the presence of a uniform magnetic field is considered. The steady, laminar and axi-symmetric flow is driven solely by the rotating disk, and the incompressible fluid obeys the inelastic Ostwald de Waele power-law model. The three-dimensional boundary layer equations transform exactly into a set of ordinary differential equations in a generalized similarity variable. These ODEs are solved numerically for values of the magnetic parameter m up to 4.0. The effect of the magnetic field is to reduce, and eventually suppress, the radially directed outflow. An accompanying reduction of the axial flow towards the disk is observed, together with a thinning of the boundary layer adjacent to the disk, thereby increasing the torque required to maintain rotation of the disk at the prescribed angular velocity. The influence of the magnetic field is more pronounced for shear-thinning than for shear-thickening fluids.     

Some features of the flow around rapidly rotating bodies made of cellular-porous materials

Two types of gas flows arising near a rapidly rotating cellular-porous disk are studied numerically and experimentally. Steady-state limits for the flow around a disk rotating in free space and the type and scenario of the loss of stability are determined. Transitional flows are characterized by formation of a vortex sheet at the boundary of the exhausting jet. Numerical simulations of the flow around a cellular-porous disk rotating near a flat screen show that it is possible to form a closed swirl flow responsible for redistribution of swirl in the gap between the disk and the flat screen. The computed results offer an explanation for the experimentally observed excess of tangential velocity of the flow in the gap over the velocity of disk rotation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 86–96, January–February, 2007.     

Unsteady MHD Flow on a Rotating Cone in a Rotating Fluid

Unsteady flow over an infinite permeable rotating cone in a rotating fluid in the presence of an applied magnetic field has been investigated. The unsteadiness is induced by the time-dependent angular velocity of the body, as well as that of the fluid. The partial differential equations governing the flow have been solved numerically by using an implicit finite-difference scheme in combination with the quasi-linearization technique. For large values of the magnetic parameter, analytical solutions have also been obtained for the steady-state case. It is observed that the magnetic field, surface velocity, and suction and injection strongly affect the local skin friction coefficients in the tangential and azimuthal directions. The local skin friction coefficients increase when the angular velocity of the fluid or body increases with time, but these decrease with decreasing angular velocity. The skin friction coefficients in the tangential and azimuthal directions vanish when the angular velocities of fluid and the body are equal but this does not imply separation. When the angular velocity of the fluid is greater than that of the body, the velocity profiles reach their asymptotic values at the edge of the boundary layer in an oscillatory manner, but the magnetic field or suction reduces or suppresses these oscillations.     

Dynamic response of a rotating disk submerged and confined. Influence of the axial gap

In this paper, the natural frequencies and mode shapes of a rotating disk submerged and totally confined inside a rigid casing, have been obtained. These have been calculated analytically, numerically and experimentally for different axial gaps disk-casing. A simplified analytical model to analyse the dynamic response of a rotating disk submerged and confined, that has been used and validated in previous researches, is used in this case, generalised for arbitrary axial gaps disk-casing. To use this model, it is necessary to know the averaged rotating speed of the flow with respect to the disk. This parameter is obtained after an analytical discussion of the motion of the flow inside the casing where the disk rotates, and by means of CFD simulations for different axial positions of the disk. The natural frequencies of the rotating disk for the different axial confinements can be calculated following this method. A Finite Element Model has been built up to obtain the natural frequencies by means of computational simulation. The relative velocity of the flow with respect to the disk is also introduced in the simulation model in order to estimate the natural frequencies of the rotating disk. Experimental tests have been performed with a rotating disk test rig. A thin stainless steel disk (thickness of 8 mm, (h/r<5%) and mass of 7.6 kg) rotates inside a rigid casing. The position of the disk can be adjusted at several axial gaps disk-casing. A piezoelectric patch (PZT) attached on the rotating disk is used to excite the structure. Several miniature and submergible accelerometers have measured the response from the rotating frame. Excitation and measured signals are transmitted from the rotating to the stationary frame through a slip ring system. Experimental results are contrasted with the results obtained by the analytical and numerical model. Thereby, the influence of the axial gap disk-casing on the natural frequencies of a rotating disk totally confined and surrounded by a heavy fluid is determined.     

Flow of a third grade fluid due to an accelerated disk

The magnetohydrodynamic (MHD) flow induced by non‐coaxial rotation of porous disk and a third grade fluid at infinity is investigated. The disk is moving with uniform acceleration and rotating with a uniform angular velocity. Numerical solution of the governing nonlinear initial and boundary value problem is obtained. The effects of physical parameters on the velocity profiles are examined in detail. The present study shows that the constant acceleration part has a greater influence than the time part of the assumed variable velocity of the disk. Copyright © 2009 John Wiley & Sons, Ltd.     

Three-dimensional rotational MHD flows in bounded volumes

Three-dimensional laminar flows of a viscous conducting gas in the neighborhood of a rotating disk are considered. The simultaneous impact of an external magnetic field, suction from the disk surface, and the axial temperature gradient as well as the action of the external axial magnetic field on three-dimensional flows in the neighborhood of rigid permeable surfaces are first studied. An exact analytic solution of the system of the boundary layer equations is obtained. It is found that the direction of the radial flow initiated in the boundary layer can be varied by changing the temperature ratio in the external flow and on the disk for various Prandtl numbers Pr. An approximate solution of the problem of flow in the rotating cylinder in the presence of a retarding cover is constructed on the basis of the approach developed for extended disks.     

Spiral and circular waves in the flow between a rotating and a stationary disk

 Circular and spiral waves are observed in the flow between a rotating and a stationary disk. These waves are generated by instabilities of the stationary disk boundary layer. This experimental work is devoted to their study by means of flow visualization and measurements of the associated velocity fields. In particular, instantaneous velocity profiles are measured by ultrasonic Doppler anemometry. The spatio-temporal characteristics of the waves are studied with the help of Fourier transforms of these velocity signals. Received: 21 April 1997/Accepted: 2 February 1998     

A note on similarity-type flows of elastico-viscous fluids

A study is undertaken to ascertain non-Newtonian effects in steady flows of elastic fluids due to an infinite rotating disk when there is suction across its surface. The fluids considered are of a class for which the similarity-type solution of von Kármán is an exact solution. It is shown that the presence of elasticity (of the type considered) does not result in flow reversal, the disk acting as a centrifugal fan as in Newtonian flow.