Steady Cavity Research Articles

Thermally driven shallow cavity flows in porous media: the intermediate régime

Steady cavity flows in porous media, driven by a horizontal temperature gradient, are examined in the intermediate limit when the Rayleigh number is comparable with the cavity aspect ratio, L (≫ 1). In this régime the flow departs from the single-cell Hadley structure, valid only for very shallow cavities, and separate circulations can develop at each end of the cavity. The existence of these closed cells is consistent with numerical solutions of the full cavity problem. Suitable heat-transfer laws are developed for conducting and for adiabatic horizontal boundaries.

Determination of the form of a thin axisymmetric cavity on the basis of an integrodifferential equation

With the principle of “the independence of expansion” proposed in [1] as a basis, simple differential equations are obtained which describe the form both of a steady and of an unsteady cavity [2–4], and solutions are obtained for some problems [1, 5] by the use of empirical constants and dependences. The mathematically closed formulation of the problem was proposed in [6–8]. At the present time, different variants of the equations for the form of a steady thin axisymmetric cavity are known together with their solutions (for example, [8–11]), and so are the numerical solutions of the steady problem in the exact formulation (without any assumption about the thinness of the cavity [12–14]). In the unsteady case all that is known is integrodifferential equations for the form of the cavity [3, 15] and the solutions to the equations of the first approximation [15]. These equations provide a good qualitative description of the form of steady and unsteady cavities [11, 15]. In order to achieve quantitative agreement, it is essential to solve the integrodifferential equations. In the present study, the solution of this type of equation is reduced to the integration of a chain of linear first-order differential equations. If one restricts oneself to the first two in this chain, the second approximation equations so obtained provide a high degree of accuracy and are fairly simple. In particular, analytical equations can be obtained in the steady case for the radius of the cavity during flow of a weightless, and of a heavy, fluid round an object. In the case of unsteady cavities, an analytical expression is obtained for the second derivative of the radius.

The motion of constant-volume air cavities in long horizontal tubes

Experimental results are presented for the instantaneous release of a constant volume of air into water in a long horizontal tube of square cross-section. The tube is closed at both ends and the volume of air is confined at one of the ends before it is released. The resulting motion, after the rapid formation of an air-cavity front, may be divided into three phases: initially the front of the air cavity moves at constant speed, later its speed decreases monotonically, and finally its speed executes a long series of erratic stops and starts before coming entirely to rest. The transition from the first to the second phase is observed to occur when a disturbance due to the tube end overtakes the cavity front. The final phase is dominated by surface-tension effects, complicated by surface contaminants. A simple model of the flow, based on Benjamin's (1968) theory of steady cavity flow and the classical theory of hydraulic jumps, is developed. With correction for surface tension, the model results compare well with the experimental results for the first two phases.

A finite difference galerkin formulation for the incompressible Navier-Stokes equations

The development of a new computational method for solving the incompressible Navier-Stokes equations in primitive variable form is presented. It is found that certain finite difference approximations for these equations can be transformed into an equivalent system which efficiently determines the discrete velocity field and which completely eliminates the pressure. Two such difference schemes for two dimensional problems are examined and some preliminary numerical results are discussed for the steady driven cavity problem.

Some problems of axisymmetric cavitation flows

The equations for the shape of a slender axisymmetric cavity [1–3] are used to consider problems relating to pulsations of the cavity shape, the drag of a slender cavity-forming body, and the influence of surface tension on the shape of a steady cavity.

Hydrodynamic Trends for Preliminary Design of Fully Cavitating Hydrofoil Sections

The object of this numerical study is to consider possible hydrodynamic trends for use in trade-off studies for the preliminary design of fully cavitating hydrofoil sections. Hydrodynamic data are obtained from inverse calculations which are based upon two-dimensional linearized cavity-flow theory. Supplementary data are also calculated from the direct problem of linearized cavity-flow theory in order to show off-design performance trends and to assess the effects of cavity-foil interference on the operating range of selected profiles. For the inverse calculations one specifies design values of the lift coefficient, cavitation number, and cavity thickness at the trailing edge, as well as the shape of the pressure distribution on the wetted surface of the hydrofoil section. In accordance with this specification, the ordinates of the profile wetted surface and upper-cavity contour are calculated, together with values of drag coefficient, moment coefficient, and attack angle at the design point. The paper summarizes the results of a parametric study of the effects of design cavitation number, lift coefficient, cavity thickness, and pressure distribution shape upon hydrofoil section performance and geometry. Three-dimensional wing effects, viscous drag, and the effects of structural design criteria are all outside the scope of the study. Results pertaining to steady two-dimensional cavity flows of an ideal incompressible fluid past a rigid hydrofoil section are presented.

Some Characteristics of Cavity Flow Past Cylindrical Inducers in a Venturi

The characteristic dimensions of the steady cavity and the shedding frequency of vortices behind six circular cylinders in a two-dimensional venturi have been studied. The normalized length and maximum width of cavity for cavitation sources of different sizes indicated unified trends with a modified cavitation number km. The angles of detachment θ increased with cavitation number k and decreased with increasing Reynolds number R. The Strouhal number Sd reached minimum values for all cavitation sources at small values of k. The possible role of wall effects on the investigations are discussed.

Linearized Theory of Finite Cavity Flow around a Thin Jet-Flapped Hydrofoil

A theoretical analysis of the steady finite cavity flow around a thin plane hydrofoil with a thin jet flap is presented. Distributions of sources and vorticies are used to obtain a governing equation of this cavity flow. The resulting equation is solved by the method of matched asymptotic expansions for small value of the jetmomentum coefficient. Using the present method, there are three cavity models around conventional hydrofoils: open model; closed; and semiclosed models. It is found that the open cavity model is reasonable from a theoretical viewpoint and other cavity models are unreasonable.

Unsteady Separation Phenomena in a Two-Dimensional Cavity

Introduction I is generally recognized that the problem of separation is very important in practical aerodynamics and in many other fluid flowfields. Although a great deal of theoretical work has been done on steady separation, mostly through the use of the boundary-layer equations, the unsteady situation has received little attention in spite of its significance. Because the boundarylayer approximation is incomplete, recourse has been made to the full momentum equation. A two-dimensional oscillatory cavity flow presents a tractable model for approximate finite-difference solution of the equation to study the unsteady separation process. Although other workers have used steady cavity flow models, this is the first time an oscillatory cavity.flow solution has been published. The results are first compared to the steady case where there is only one fixed separation streamline along the line of symmetry. Then a dynamic pattern of separation and reattachment is illustrated.

The dynamic balances of dissolved air and heat in natural cavity flows

In steady, fully developed and unventilated cavity flows occurring in practice, air (originally dissolved in the water) and heat are diffused through the fluid towards the interface providing a continuous supply of air and vapour to the cavity. This must be balanced by the rate of entrainment of volume of air and vapour away from the cavity in the wake. These equilibria which determine respectively the partial pressure of air within the cavity and the temperature differences involved in the flow are studied in this paper. The particular case of the cavitating flow past a spherical headform has been investigated in detail. Measurements indicate a near-linear relation between the partial pressure of air in the cavity and the total air content of the water. From a second set of experiments, designed to estimate the volume rates of entrainment under various conditions by employing artificial ventilation, it appears that this is a function only of tunnel speed and cavity size within the range of the experiments. A simplified theoretical approach involving the turbulent boundary layer on the surface of the cavity is then used to estimate the rates of diffusion into the cavity. The resulting air balance yields a partial pressure of air/air content relation compatible with experiment. The water vapour or heat balance suggests that the temperature differences involved are likely to be virtually undetectable experimentally.

General Formulation of a Perturbation Theory for Unsteady Cavity Flows

The problem of a two-dimensional cavity flow of an ideal fluid with small unsteady disturbances in a gravity free field is considered. By regarding the unsteady motion as a small-perturbation of an established steady cavity flow, a fundamental formulation of the problem is presented. It is shown that the unsteady disturbance generates a surface wave propagating downstream along the free cavity boundary, much in the same way as the classical gravity waves in water, only with the centrifugal acceleration owing to the curvature of the streamlines in the basic flow playing the role of an equivalent gravity effect. As a particularly simple example, the surface waves in a hollow potential vortex flow is calculated using the present theory.

The Linearized Theory of a Supercavitating Hydrofoil With a Jet Flap

A theoretical analysis is carried out for the steady cavity flow about thin hydrofoil sections at small incidence α, at the trailing edge of which a thin jet emerges at a small deflection τ. The flow is assumed to be inviscid and incompressible and the cavitation number is taken to be zero. The jet is assumed so thin that it can be considered as a vortex sheet across which the velocity is discontinuous. For the case of a flat plate, expressions have been obtained for lift, drag, pressure distribution, pitching moment, the jet shape, and the cavity shape. Numerical calculations are made for a number of jet momentum-flux coefficients CJ lying between 0.01 and 5.

Small-time behavior of unsteady cavity flows

A perturbation theory is applied to investigate the small-time behavior of unsteady cavity flows in which the time-dependent part of the flow may be taken as a small-time expansion superimposed on an established steady cavity flow of an ideal fluid. One purpose of this paper is to study the effect of the initial cavity size on the resulting flow due to a given disturbance. Various existing steady cavity-flow models have been employed for this purpose to evaluate the initial reaction of a cavitated body in an unsteady motion. Furthermore, a physical model is proposed here to give a proper representation of the mechanism by which the cavity volume may be changed with time; the initial hydrodynamic force resulting from such change is calculated based on this model.