Centrifugal Body Force Research Articles

Three-Dimensional Free Convection in a Long Rotating Porous Box: Analytical Solution

A three-dimensional analytical solution to the steady-state free convection problem in a long rotating porous box is presented for large values of the porous media Ekman number. The convection results from differential heating of the horizontal walls leading to temperature gradients orthogonal to the centrifugal body force. The solution to the nonlinear set of partial differential equations was obtained through an asymptotic expansion of the dependent variables in terms of two small parameters representing the reciprocal Ekman number in porous media and the aspect ratio of the domain. The results are focused towards the Coriolis effect on the flow. Secondary circulation was obtained in a plane orthogonal to the leading free convection plane. The results show that the Coriolis effect on free convection is controlled by a combined dimensionless group representing the ratio of the centrifugal Rayleigh number to the porous media Ekman number.

Natural Convection in Porous Media Induced by the Centrifugal Body Force: The Solution for Small Aspect Ratio

The analytical solution to the natural convection problem in a rotating rectangular porous domain is presented for a small aspect ratio of the domain. The convection results from differential heating of the horizontal walls leading to temperature gradients orthogonal to the centrifugal body force. The solution to the nonlinear set of partial differential equations was obtained through an asymptotic expansion of the dependent variables in terms of a small parameter representing the aspect ratio of the domain. The convection regime is apparent in the results, although it has a weak effect on the mean heat flux.

Critical heat flux enhancement by means of liquid subcooling and centrifugal force induced by flow curvature

Experiments are performed to assess the feasibility of cooling high flux heat sources flush mounted on the inner wall of a cylindrical vessel by a dielectric liquid flowing at high velocity parallel to the vessel wall. Fluid motion is induced by rotating a stirrer having radial blades about the axis of the vessel. The study focusses on examining critical heat flux (CHF) enhancement attributed to liquid subcooling and the centrifugal body force induced by fluid rotation. This includes using test vessels with different inner radii, 4.19 and 7.62 cm, three heater lengths, 1.27, 2.54 and 5.08 cm, and stirrers having two or four blades. CHF is found to be independent of heater length and the number of stirrer blades. CHF enhancement is achieved by increasing the liquid fill volume, increasing the rotational speed and/or decreasing the radius of curvature. The system facilitates the removal of heat at nearly isothermal wall conditions over a wide range of heat fluxes. A CHF correlation is presented which accounts for the effects of subcooling and centrifugal force over all conditions tested.

BEM for general anisotropic 2D elasticity using particular integrals

AbstractIn the context of the direct boundary‐element method (DBEM) for most general two‐dimensional elastostatics, new particular integrals are presented in this paper to account for inertial and centrifugal body forces, as well as for those cases of three‐dimensional behaviour where the assumption of plane strain (or stress) may remain approximately valid despite the presence of non‐zero strain (or stress) in the out‐ofplane direction. These integrals have been implemented in a general‐purpose multi‐region boundary‐element code (namely, GPBEST) and tested via examples.

Analytic method of hydraulic calculation of eddy shaft spillways

1. The investigated analytic methods of hydraulic calculation of eddy shaft spillways are based on laws of conservation and, unlike the earlier known ones, take into account not only the size and shape of the water passageway but also contraction of the flow upon entering the spiral chamber from the inlet channel or conduit, hydraulic losses of energy at the entry to the vertical shaft, and the presence here of a zone of equalization of the flow. In calculating the characteristics of the swirled flow over the height of the vertical shaft, its potential energy related to the effect of centrifugal body forces is taken into account. 2. The investigated methods make it possible to obtain the calculated values of hydraulic characteristics of eddy shaft spillways and functions of the variation of the parameters of swirled flows in vertical shafts most closely coinciding with the experimental data. The deviations between experiment and calculation lie within the measurement accuracy. This enables recommending the given methods for practical use when designing eddy shaft spillways and for predicting and analyzing their operation.

Treatment of body forces in axisymmetric boundary element design sensitivity formulation

This paper deals with the treatment of uniform and non-uniform body forces in the implicit-differentiation formulation for axisymmetric boundary element design sensitivity analysis. The particular integral concept is extended to obtain the sensitivities due to gravitational and centrifugal body forces of uniform type. For thermal body forces of non-uniform type, the sensitivities are obtained through the implicit-differentiation of a surface integral. The efficiency and the accuracy of the formulation are determined for a wide range of problems which include different axisymmetric geometries under centrifugal, gravitational and thermal body forces.

Squeeze box in a centrifuge

A squeeze box which operates in a centrifuge is presented. In the experimental set-up the centrifugal body force mimics the role of gravity in nature and at the same time drives a hydraulic press which simulates lateral forces. The rate of lateral shortening is controlled by varying the loads placed on the hydraulic press. Utilization of a squeeze box in a centrifuge enhances the range of scaling and enables the use of coherent model materials without violating similitude requirements. In the traditional squeeze box which operates under normal gravity, similitude requirements severely limit the choice of model materials. As a test of the apparatus, simple experiments were carried out comparing models shortened in a centrifuge with similar models shortened under normal gravity. The realistic form of decollement structures e.g., folds, faults and diapirs, shortened in a centrifuge in these tests as well as in other tests which will be reported later, show the potential of this new technique for rigorous modelling of geodynamic systems, which evolve under the influence of both gravity and a push from the rear. The set-up can also be used as a simple shear box in a centrifuge, after a slight adjustment.

New Axisymmetric BEM Formulation for Body Forces Using Particular Integrals

A new boundary element formulation is introduced for the treatment of gravitational and centrifugal body forces in axisymmetric elastic media. This new formulation is based on the method of particular integrals, which require neither volume nor surface integration for the treatment of body forces. Particular integrals for both gravitational and centrifugal body forces are derived and all kernel functions are defined in an appendix. The formulation is implemented into a multiregion system that utilizes quadratic isoparametric shape functions to model the geometry and field variables on the boundary of the body. A number of examples of practical interest have been included to demonstrate the power and versatility of the present formulation.

Asymptotic realization of the method of integral relations for turbulent jets to compute wall-jets curtain and their control

The protection of the surface of bodies from high-temperature fluid flows is a real problem in modern technology. The aerodynamic protection is effected with the help of gas curtain. One of the principal means of organizing such a curtain is to blow cold fluid through slots at the initial section of the surface being protected [i]. The computational problem consists in the determination of the surface-temperature distribution in the region of the gas curtain. The existing methods to compute wall-jet curtains are in three principal directions depending on the physical model being used [2]: free turbulent jet model; twolayer scheme with laws on semibounded turbulent jets; flow in a boundary layer with wall-jet curtains determined by wall turbulence. The simplest method appeared to be based on the use of integral relations for the boundary layer and asymptotic conditions when equalization of temperature inside the boundary layer [3] takes place at x § ~ due to turbulent mixing. Under these conditions, there is a limiting relation between the momentum and energy thicknesses which cannot fully reflect the influence of initial conditions and the previous history of the flow. Expressions for the effect of cooling are developed using interpolations of the type q = (i + na )b where a and b are constants, is the value of the efficiency UX_~ , ~X_~O o , obtained on the basis of the laws of boundary-layer growth away from the location of blowing. The model for the wall jet takes into consideration the behavior of mixing processes near the lip of the nozzle [4]. Studies on turbulent boundary layer in wall jets have been carried out in many theoretical and experimental works [5-7]. The majority of theoretical studies is based on the simultaneous solution of the equations for turbulent jets and boundary layer growing on a flat plate, the difference being in the manner of specifying velocity profiles and skin-friction. In studying flows near curvilinear surfaces, disagreement has been noticed between experimental data and theoretical results computed from Karman integral relations which do not take into account the surface curvature in an explicit form. In [8, 9], it has been observed that the effect of surface curvature on the semibounded jet and the efficiency of wall-jet curtain mainly depend on active or conservative role of centrifugal body forces. In the present work, a method is developed to compute the flow and heat transfer on an adiabatic curvilinear surface, based on the asymptotic continuation of the method of integral relations for turbulent wall jets away from the location of blowing. The above-mentioned limiting cases correspond to conditions for the reorganization of nonmonotonous velocity profile which is present near the lip of the nozzle, in the flow that is characteristic of a developed boundary layer [7]. Such an approach makes it possible to develop a computational method for the wall-jet curtain with initial conditions for outflow while computational expressions for the efficiency of film cooling are developed on the basis of interpolation formulas. i. Problem Formulation. Adiabatic Curvilinear Surface with Gas Curtain. Consider a quasiisothermal turbulent wall jet. It is assumed that the physical characteristics of the fluid are constant in the given temperature range. The basic flow has a velocity u 0 and temperature T o . The same fluid is blown through a slot of height s with a mean velocity u s and temperature T s at the exit section. In the initial segment 0 < x < x s the curvilinear surface is in contact only with the fluid curtain and, hence, has a temperature T w = T s (Fig. i) at all points. The thermal boundary layer begins from the section x = x s as a result of mixing of the fluid curtain with the basic flow. Equations of quasisteady turbulent, incompressible boundary layer on the curvilinear surface with a constant curvature (neglecting normal turbulent stress components) take the form [i0]

Dynamic models of continental rift valley systems

In experimental scale models simulating the major continental rift valley systems rifting is caused by diapiric uprise of viscous material induced by a centrifugal body force; this force plays the same role as gravity in nature. The scale models show that bending alone as a mechanism of rifting is unrealistic. Rather, the rift structures form as a result of the combined effect of gentle bending and lateral stretching of the crust. The models also show that graben structures could form at any stage of development of an ascending and laterally spreading diapiric body depending upon the density and viscosity contrast of the body and its surroundings and the strength of the model crust. The close resemblance in structure of the models with natural rift systems suggests a close similarity of dynamics and driving forces.

A study of separation and reattachment of a turbulent flow behind a backward-facing step. (1st report, Effect of streamline curvature on a turbulent reattaching flow)

The paper presents the behavior of a separating shear layer and its reattachment when an additional centrifugal body force due to streamline curvature is imposed on the shear layer. A curved channel is attached to a backward-facing step to obtain such a flow situation. In addition to he inherent stability within the conventional straight channel, the stabilizing and destabilizing effects on the reattachment process are given according to the direction of curvature. Three different surface configurations are used (flat, convex, and concave); with weak, mild, and strong surface curvature, the measurements were performed for seven separated cases. The reattachments point was determined by a thermal tuft probe. A hot-wire anemometer was used for the measurements of mean velocity and turbulence. Flow visualization of the shear layer was made by means of a smoke-wire method. Reynolds number based on step height was 8×104, curvature paramenter being in the rage -0.04<δ/R<0.05.

Fir tree fastening of turbomachinery blades—II: Step load analysis

Blade-root fastening of the fir-tree shape, used in securing turbomachinery blades to the rotor has been analysed by considering it as an assembly of the step-segments, the stiffness characteristics of which are already obtained in Part-I. The general equation of equilibrium as well as the compatability of deformation between the steps of the blade and the disc give the necessary ‘ n’ linear algebraic equations for the evaluation of the step loads in the n- step root. The method can accommodate the centrifugal body forces of the root itself, the friction between the blade and disc-steps and the difference in the material of the blade and the disc.

Rotating Elasto-Plastic Interference Fits

By an elasto-plastic design the transferable torque of an interference fit can be increased significantly. In high-speed applications the initial contact pressure is reduced by the centrifugal body force. A fit having a full shaft and equal elastic constants of shaft and hub material is considered. Using Tresca’s yield criterion an analytical solution determining the dependence of contact pressure on angular velocity is derived. All formulae necessary for design of fits are presented. Numerical results are given and conclusions for practical applications of rotating elasto-plastic interference fits are drawn.

The design of orthotropic materials for stress optimization

The availability of materials such as fibrous composites now makes it feasible to approach problems of the theory of elasticity from a different perspective. Rather than determining the stresses within a homogeneous body subjected to known loads, it may be possible to design a nonhomogeneous material to yield an optimum internal stress field. The approach is demonstrated for two dimensional problems in polar coordinates, with detailed results given for the problems of the pressurized, thick-walled cylinder and the rotating disk subjected to centrifugal body forces.

The Analysis of Radial Flow Impellers by the Matrix Finite Element Method

The stress analysis of an axisymmetrical body under the influence of a field of centrifugal body force is of considerable importance to the design of rotating machinery. The classical problem of the rotating disc of uniform thickness was considered by Maxwell as long ago as 1850. For this, the simplifying assumption that the stresses are functions of the distance from the centre only, with no change across the thickness, was made. Since then, the same basic analysis has been extended to discs of conical and hyperbolical cross-section profile, with or without a central hole, references to which may be found in a number of text books. The analysis of a disc with an elliptical cross-section was published by Chree in 1895. The solution for a disc of constant thickness has been refined to allow stress variation across the thickness. Although these solutions, in common with many analytical solutions in the theory of elasticity, do not satisfy all the boundary conditions exactly, they serve as very good approximations in practice.

Non-isothermal flow between rotating disks

Consideration is given to laminar axially symmetric flow and heat transfer between two horizontal disks which rotate around a common axis with a common angular velocity. The temperatures of both disks are constant with that of the upper disk being hotter. The centrifugal body force drives a tangential velocity through its coupling with the Coriolis force in a thermal layer near the lower disk. Ekman layers are produced on each disk and heat is convected by the secondary axial flow. Significant convective transport occurs for σβϵ − 1 2 ⪢ 1, where σ is the Prandtl number, ϵ is the inverse of rotational Reynolds number, and β is the product of a temperature difference and the thermal coefficient of expansion. A perturbation solution is carried out for σβϵ − 1 2 ⪢ 1. Some numerical solutions are given for lower rotational rates.

Magnetohydrodynamic Flows in Closed Regions with Complete Symmetry about an Axis

A formal perturbational technique was developed to obtain solutions for magnetohydrodynamic flow problems in closed regions with symmetric properties of flow and boundary conditions about an axis. The system of highly nonlinear partial differential equations, governing the problems of this class, was found to reduce to systems of simultaneous linear partial differential equations, thus creating a possibility of obtaining solutions by classical mathematical methods. The technique used was based on a well-known power series type of perturbational expansion in terms of a small parameter, which in the limit was assumed to become the solution of the original system. The general technique was applied to the magnetohydrodynamic flow in a toroid under the influence of a uniform applied magnetic field along the direction of the main axis of the toroid. It was found that the secondary flow caused by the centrifugal body force on the fluid element due to the curvature created induced electromagnetic components and a pressure variation along the radial direction. The solutions were carried out to include completely the second perturbational equations and the electromagnetic and flow fields were discussed for different values of characteristic parameters appearing in the solutions.