Showing papers on "Incompressible flow published in 1978"

An exact theory of nonlinear waves on a Lagrangian-mean flow

Abstract: An exact and very general Lagrangian-mean description of the back effect of oscillatory disturbances upon the mean state is given. The basic formalism applies to any problem whose governing equations are given in the usual Eulerian form, and irrespective of whether spatial, temporal, ensemble, or ‘two-timing’ averages are appropriate. The generalized Lagrangian-mean velocity cannot be defined exactly as the ‘mean following a single fluid particle’, but in cases where spatial averages are taken can easily be visualized, for instance, as the motion of the centre of mass of a tube of fluid particles which lay along the direction of averaging in a hypothetical initial state of no disturbance.The equations for the Lagrangian-mean flow are more useful than their Eulerian-mean counterparts in significant respects, for instance in explicitly representing the effect upon mean-flow evolution of wave dissipation or forcing. Applications to irrotational acoustic or water waves, and to astrogeophysical problems of waves on axisymmetric mean flows are discussed. In the latter context the equations embody generalizations of the Eliassen-Palm and Charney-Drazin theorems showing the effects on the mean flow of departures from steady, conservative waves, for arbitrary, finite-amplitude disturbances to a stratified, rotating fluid, with allowance for self-gravitation as well as for an external gravitational field.The equations show generally how the pseudomomentum (or wave ‘momentum’) enters problems of mean-flow evolution. They also indicate the extent to which the net effect of the waves on the mean flow can be described by a ‘radiation stress’, and provide a general framework for explaining the asymmetry of radiation-stress tensors along the lines proposed by Jones (1973).

The unsteady motion of a two-dimensional aerofoil in incompressible inviscid flow

Abstract: A numerical method has been developed for the calculation of the pressure distribution, forces and moments on a two-dimensional aerofoil undergoing an arbitrary unsteady motion in an inviscid incompressible flow. In a discussion of the appropriate Kutta condition(s) it is argued that two Kutta conditions are required to obtain a satisfactory solution. The method is applied to (i) a sudden change in aerofoil incidence, (ii) an aerofoil oscillating at high frequency and (iii) an aerofoil passing through a sharp-edged gust.

Approximate Equations for Long Nonlinear Waves on a Viscous Fluid

Abstract: A systematic perturbation method is applied to three-dimensional long waves on a viscous liquid film, and the nonlinear evolution equation incorporating the effects of dissipation and dispersion is derived. It is shown that both the fourth-order derivative term as well as the three-dimensionality have stabilizing effects.

Three‐dimensional magnetohydrodynamic turbulence in cylindrical geometry

Abstract: Ideal magnetohydrodynamic turbulence is treated using more realistic boundary conditions than rectangular periodic boundary conditions. The dynamical equations of incompressible magnetohydrodynamics and the associated fields are expanded in a set of vector eigenfunctions of the curl. The individual eigenfunctions represent force‐free fields, but superpositions of them do not. Three integral invariants have simple quadratic expressions in the expansion coefficients: the total energy, the magnetic helicity, and the cross helicity. The invariants remain temporally constant in the face of a truncation at a large but finite number of coefficients. Boundary conditions imposed are those for a rigid, perfectly‐conducting cylindrical boundary, with an arbitrary periodicity length parallel to the axis. Canonical distributions are constructed from the invariants. Mean‐square turbulent velocity fields 〈v2〉 have finite values for virtually all initial conditions, including quiescient ones. The stability problem can be reformulated as a search for values of the integral invariants which will minimize 〈v2〉. This leads to a principle of extremal helicity, which requires a magnetic configuration which will minimize 〈v2〉 for a given total energy. The development of helical macroscopic structures in the cylinder as a function of increasing ratio of axial current to axial magnetic flux is predicted.

Laser anemometer study of flow development in curved circular pipes

Abstract: An experimental investigation was carried out of the development of steady, laminar, incompressible flow of a Newtonian fluid in the entry region of a curved pipe for the entry condition of uniform motion. Two semicircular pipes of radius ratios 1/20 and 1/7 were investigated, covering a Dean number range from 138 to 679. The axial velocity and the component of secondary velocity parallel to the plane of curvature of the pipe were measured using laser anemometry. It was observed that, in the upstream region where the boundary layers are thin compared with the pipe radius, the axial velocity within the irrotational core first develops to form a vortex-like flow. In the downstream region, characterized by viscous layers of thickness comparable with the pipe radius, there appears to be three-dimensional separation at the inner wall. There is also an indication of an additional vortex structure embedded within the Dean-type secondary motion. The experimental axial velocity profiles are compared with those constructed from the theoretical analyses of Singh and Yao & Berger. The quantitative agreement between theory and experiment is found to be poor; however, some of the features observed in the experiment are in qualitative agreement with the theoretical solution of Yao & Berger.

Convergence of vortex methods for Euler’s equations

Abstract: A numerical method for approximating the flow of a two dimensional incompressible, inviscid fluid is examined. It is proved that for a short time interval Chorin's vortex method converges superlinearly toward the solution of Euler's equations, which govern the flow. The length of the time interval depends upon the smoothness of the flow and of the particular cutoff. The theory is supported by numerical experiments. These suggest that the vortex method may even be a second order method.

Motion of a curved vortex filament with decaying vortical core and axial velocity

Abstract: The motion and decay of a vortex filament with large axial and circumferential velocity components in a three-dimensional stream are studied. Solutions are constructed to the Navier–Stokes equations by use of matched asymptotic expansions; the small parameter used is a measure of the ratio of the viscous effects to the vortex strength. The outer flow, which corresponds to the classical Biot–Savart type analysis is matched to the solution in an inner viscous region. The radius of the viscous core is assumed to be much smaller than the radius of curvature. The present viscous analysis yields the classical inviscid theory as a limiting case for the leading term in the outer region and thus can be used to correct various deficiencies in the latter. We show in particular, that the inner solution yields a finite velocity at all points in the filament and we determine how the components of both vorticity and velocity diffuse due to the viscous forces. The matching conditions guarantee the continuity of velocity ...

Spatial Decay Estimates for the Navier–Stokes Equations with Application to the Problem of Entry Flow

Abstract: The development of velocity profiles in the inlet region of channels or pipes is a classic problem of laminar flow theory which has given rise to an extensive literature. Most previous work on this entry flow problem has involved some degree of simplification either in flow geometry or in the governing equations. Here we treat the flow development within the general framework of the Navier–Stokes equations governing the steady laminar flow of an incompressible viscous fluid in a cylindrical pipe of arbitrary cross-section.The problem to be treated is that of an “end effect” involving comparison between two distinct solutions of the Navier–Stokes equations. Thus we consider the spatial evolution of the difference between the base flow and the fully developed solution. The corresponding velocity difference clearly satisfies a condition of zero net inflow. In this way, we draw an analogy between the issue of concern here and the celebrated “Saint-Venant’s Principle” of elasticity theory involving the effect ...

Free convective heat transfer in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall

Abstract: Analyses of fluid flow over a wavy wall are of interest because of their applications to the physical problems mentioned in § 1. The authors have therefore devoted their attention to the effect of waviness of one of the walls on the flow and heat-transfer characteristics of an incompressible viscous fluid confined between two long vertical walls and set in motion by a difference in the wall temperatures. The equations governing the fluid flow and heat transfer have been solved subject to the relevant boundary conditions by assuming that the solution consists of two parts: a mean part and a disturbance or perturbed part. To obtain the perturbed part of the solution use has been made of the long-wave approximation. The mean (zeroth-order) part of the solution has been found to be in good agreement with that of Ostrach (1952) after certain modifications resulting from the different non-dimensionalizations employed by Ostrach and the present authors respectively. The perturbed part of the solution is the contribution from the waviness of the wall. The zeroth-order, the first-order and the total solution of the problem have been evaluated numerically for several sets of values of the various parameters entering the problem. Certain qualitatively interesting properties of the flow and heat transfer, along with the changes in the fluid pressure on the wavy and flat wall, are recorded in §§ 5 and 6.

Uniqueness and drag for fluids of second grade in steady motion

Abstract: For incompressible fluids of second grade that are compatible with the Clausius-Duhem inequality, non-uniqueness of steady flows with small Reynolds number (i.e. creeping flows) is possible provided the ‘absorption number’ is also small. We discuss this uniqueness question, generalize a well-known theorem of Tanner concerning how solutions of the Stokes equations may be used to generate solutions of the creeping flow equations for fluids of second grade, and give a new uniqueness theorem appropriate to a class of problems for the steady creeping flow of fluids of second grade. Under the conditions for uniqueness, we obtain a simple formula for the drag force on a fixed body which is immersed in a fluid of second grade which is undergoing uniform creeping flow. For bodies with certain geometric symmetries, the non-Newtonian nature of the fluid has no effect upon the drag.

Fully Multidimensional Flux-Corrected Transport.

Abstract: : An alternative formula is proposed for the flux limiting phase of the flux-corrected transport (fct) algorithms of Boris and Book. The advantages of the proposed new formula are three: trivial generalization to multidimensions without resort to time-step splitting; elimination of the clipping phenomenon for vanishing velocity; and reduction of the clipping phenomenon in a finite velocity field. the new method makes possible for the first time multidimensional FCT calculations for problems not amenable to time splitting, such as those involving incompressible or nearly incompressible flow. (Author)

Large eddy simulation of incompressible turbulent channel flow

Abstract: The three-dimensional, time-dependent primitive equations of motion were numerically integrated for the case of turbulent channel flow A partially implicit numerical method was developed An important feature of this scheme is that the equation of continuity is solved directly The residual field motions were simulated through an eddy viscosity model, while the large-scale field was obtained directly from the solution of the governing equations An important portion of the initial velocity field was obtained from the solution of the linearized Navier-Stokes equations The pseudospectral method was used for numerical differentiation in the horizontal directions, and second-order finite-difference schemes were used in the direction normal to the walls The large eddy simulation technique is capable of reproducing some of the important features of wall-bounded turbulent flows The resolvable portions of the root-mean square wall pressure fluctuations, pressure velocity-gradient correlations, and velocity pressure-gradient correlations are documented

Rimming flow: numerical simulation of steady, viscous, free-surface flow with surface tension

Abstract: Flow in a partly liquid-filled, rotating, horizontal cylinder is analysed by means of finite-element numerical simulation. Of alternative methods for locating the free surface, a boundary collocation scheme with Newton-Raphson iteration converges. This method forces the residual in the normal-stress boundary condition to zero at a finite set of points on the liquid meniscus. Solutions of the steady, two-dimensional, incompressible flow problem show circumferential variation of the liquid-film thickness and corresponding pressure and velocity fields, including recirculation zones. The complications of an unknown meniscus location and a nonlinear normal-stress condition when surface tension is significant are illustrated. The finite-element method proves an effective and convenient tool for such flows, in which inertial, gravitational, pressure, viscous and capillary forces are all important.

Numerical study of pressure and velocity fluctuations in nearly isotropic turbulence

Abstract: The spectral method of Orszag and Patterson has been extended to calculate the static pressure fluctuations in incompressible homogeneous decaying turbulence at Reynolds numbers Reλ [less, similar] 35. In real space 323 points are treated. Several cases starting from different isotropic initial conditions have been studied. Some departure from isotropy exists owing to the small number of modes at small wavenumbers. Root-mean-square pressure fluctuations, pressure gradients and integral length scales have been evaluated. The results agree rather well with predictions based on velocity statistics and on the assumption of normality. The normality assumption has been tested extensively for the simulated fields and found to be approximately valid as far as fourth-order velocity correlations are concerned. In addition, a model for the dissipation tensor has been proposed. The application of the present method to the study of the return of axisymmetric turbulence to isotropy is described in the companion paper.

Finite-element solution procedures for solving the incompressible, Navier-Stokes equations using equal order variable interpolation

Abstract: The conventional finite-element formulation of the equations of motion (written in pressure-velocity variables) requires that the order of interpolation for pressure be one less than that used for the velocity components. This constraint is inconvenient and can be argued to be physically inconsistent when inertial effects are dominant. The origins of the constraint are discussed and three new finite-element formulations are advanced that permit equal order representation of pressure and velocity. Of these, the velocity correction scheme, similar to that commonly used in finite-difference procedures, offers superior performance for the examples examined in this paper.

The dilatational properties of suspensions of gas bubbles in incompressible newtonian and non-newtonian fluids

Abstract: We derive expressions for the dilatational properties of suspensions of gas bubbles in incompressible fluids, using a cell model for the suspension. A cell, consisting of a gas bubble centered in a spherical shell of incompressible fluid, is subjected to a purely dilatational boundary motion and the resulting stress at the cell boundary is obtained. The same dilatational boundary motion is prescribed at the boundary of an “equivalent” cell composed of a one-phase, uniformly compressible fluid with unknown dilatational properties. By specifying that the stress at the boundary of the one-phase cell is equal to the stress at the boundary of the two-phase suspension cell, we obtain expressions for the unknown dilatational properties as a function of observable properties of the suspension. The dilatational viscosity of a suspension with a Newtonian continuous phase and the analogous properties for suspensions with non-Newtonian continuous phases are obtained as functions of the boundary motion, volume fraction of gas, and properties of the incompressible continuous phase. Results are presented for continuous phases which are Newtonian fluids, second-order fluids, and Goddard—Miller model fluids.

Turbulent diffusion of magnetic fields.

Abstract: We use the exact Eulerian formulation of the problem of the diffusion of passive scalar and magnetic fields by a turbulent velocity field to obtain sufficient conditions for the validity of the mean-field approximation. We present a general derivation of the dynamo equation in a homogeneous, stationary, incompressible, perfectly conducting turbulent fluid, and discuss the conditions for its validity. We show that the renormalized transport coefficients are formally equivalent to those obtained in the exact Lagrangian analysis.

Initial value problem for Rayleigh–Taylor instability of viscous fluids

Abstract: The initial value problem associated with the development of small amplitude disturbances in Rayleigh–Taylor unstable, viscous, incompressible fluids is studied. Solutions to the linearized equations of motion which satisfy general initial conditions are obtained in terms of Fourier–Laplace transforms of the hydrodynamic variables, without restriction on the density or viscosity of either fluid. When the two fluids have equal kinematic viscosities, these transforms can be inverted explicitly to express the fluid variables as integrals of Green’s functions multiplied by initial data. In addition to normal modes, a set of continuum modes, not treated explicitly in the literature, makes an important contribution to the development of the fluid motion.

Radial source flow between parallel disks

Abstract: The vorticity transport equation is solved for radial incompressible flow between disks by a finite-difference method with discretization based on the method of Allen & Southwell. The solution permits detailed characterization of the flow for the Reynolds number range investigated, 1 ≤ Re ≤ 300. Above Re = 60 separation is observed with the bubble size increasing rapidly with Re. The streamwise and transverse pressure and velocity gradients are examined to interpret the observed phenomena.

Nonlinear analysis of laminar boundary layer flow over a periodic wavy surface

Abstract: The problem of laminar, incompressible flow over a periodic wavy surface is treated as a first‐order perturbation to the boundary layer flow on a flat surface. The analysis demonstrates that some nonlinear terms in the disturbance boundary‐layer equations are first order if the wave amplitude and disturbance sublayer thickness are comparable in magnitude. Further, the theory predicts that the nonlinear effects are confined to the thin sublayer adjacent to the wavy surface. Computer‐generated, nonlinear solutions are presented for sinusoidal waves with a range of wave amplitudes, including cases with local separated flow regions.

Merging of magnetic fields with field-aligned plasma flow components

Abstract: The Sonnerup merging model for an incompressible plasma is extended to allow a flow component along the field lines in the inflow regions. Solutions are found to exist as long as the difference between the quantities B. V for the two inflow regions does not exceed a critical magnitude dependent on the inflow field magnitudes and plasma densities. All such solutions satisfy Vasyliunas' definition of merging, but some classes of solution have radically altered geometries, i.e. geometries in which the inflow regions are much smaller than the outflow regions. The necessary but not sufficient condition for these unusual geometries is that the field-aligned flow component in at least one inflow region be super Alfvenic. A solution for the case of a vacuum field in one inflow region is obtained in which any flow velocity is allowed in the non-vacuum inflow region, although super-Alfvenic flow can still result in an unusual geometry. For symmetric configurations, the usual geometry, that of Petschek and Sonnerup, is retained as long as both field-aligned flow components in the inflow regions are less than twice the inflow Alfven speed. For the case of a vacuum field on one side and fields approximating the boundary between the solar wind and the earth's dayside magnetosphere, the usual geometry is retained for flow less than about 2·5 times the local Alfven speed.

Lifting-line theory of oblique wings

Abstract: The study extends Prandtl's lifting-line theory to planar wings involving swept and curved centerlines. Attention is focused on distinct features of such a theory, with special emphasis on an oblique wing in a steady incompressible potential flow. The analysis presented is compared with exact solutions derived from an inverse method and with results from a panel method.

A Basic Study of the VTOL Ground Effect Problem for Planar Flow

Abstract: The force interaction between airframe undersurfaces and the ground in the presence of lift jets is an important consideration for VTOL aircraft design. As a first step toward prediction of this phenomenon, a combined theoretical and experimental investigation of planar turbulent jet impingement flowfields has been undertaken. Unvectored jets in close ground effect have been modeled using the incompressible Reynolds equations with a one-equation turbulence model. Distributions of the flow properties are computed as functions of undersurface shape, length scales, and jet exit height above ground. Computed flowfield properties are presented and comparisons are made with experimental measurements.

The Computation of Optimum Pressure Recovery in Two-Dimensional Diffusers

Abstract: A method is presented for computation of optimum recovery at fixed length (Cp *) in two-dimensional diffusers with incompressible flow and turbulent inlet boundary layers. Since Cp * lies in the zone of transitory stall, the method involves computation of not only attached but also detaching and detached turbulent boundary layers. The results agree with available data to the level of the uncertainty in the data. The model is zonal in character. Results suggest that the most important feature in computing detaching flows is the treatment of the interaction between the outer (inviscid) flow and the boundary layer; the use of velocity-profile forms that represent average back-flows adequately is also important.