Showing papers on "Incompressible flow published in 1968"

Laminar separation bubbles in two and three dimensional incompressible flow

Abstract: A theoretical and experimental study is made of the closed 'bubbles` of separated flow formed when a laminar boundary layer separates from an aerofoil surface and, after undergoing transition to turbulence, subsequently re-attaches. Attention is mainly confined to the so-called 'short' type of bubble, which is distinguished from the 'long' type by its relatively s .i ht overall effect upon the pressure distribution. In Part I, a semi-empirical theory for the prediction of the growth and bursting of two-dimensional short bubbles is developed. The existinF, data concerning short bubbles are re-examined, with particular emphasis upon the conditions governing re-attachment. A criterion for the determination of turbulent re-attachment is proposed, and approximate nuadrature methods developed for the calculation of the momentum thickness in the separated region. These results, together k". 'ith am empirical formula for the dotermi-n tion of the position of transition, are comb nod i: ith a simplified model of the pre -r ure di_-trtbu1 ion in the buhl>le region to predict the re-attachment position. ? 't in fr. und that, for a given imposed pressure dic, trihution, there exi=; ts a Reynolds number at separation below which re-attachment

Integral Theory for Supersonic Turbulent Base Flows

Abstract: The integral near- wake analysis of Reeves and Lees developed for supersonic laminar base flows with viscous-inviscid interaction is extended to the case of fully turbulent separated adiabatic flow behind a rearward-facin g step at supersonic speeds. A turbulent eddy viscosity model is formulated for the shear stress scaling of the dissipation integral in the mechanical energy equation. It is shown that the eddy viscosity can be described simply by one incompressible constant (valid for both shear layers and wakes) and one reference density pr. Using a compressibility transformation, theoretical solutions for the spreading rates of free shear layers are found to agree with experiment when the reference density is chosen to be the centerline density for the wake flow. The wake flow solution, uniquely determined by the wake critical point, is joined to the body through a turbulent free shear layer mixing solution. A simple conservation model is presented to relate the viscous sublayer after expansion to the initial boundary layer ahead of the step. For freestream Mach numbers MI < 2.3, where lip shock effects may be neglected, the integral theory is found to give good estimates for the length scales and centerline pressure variations measured experimentally for both wake flows and step flows (where reattachment is to a solid surface). b C f f'M h

Influence of an Axial Magnetic Field on the Steady Linear Ekman Boundary Layer

Abstract: A hydromagnetic version of the Ekman boundary layer is developed in a simple form in order to study how the geophysically important Ekman suction velocity is affected by magnetic fields. The problem treated consists of a viscous, incompressible, conducting fluid in the presence of an infinite, flat, insulating boundary which rotates at speed Ω0. Outside the boundary layer, the fluid rotates uniformly with speed Ω1 = Ω0 (1 + e), and there is a uniform magnetic field aligned with the rotation axis. An expansion in powers of e, the Rossby number, together with von Karman similarity, leads to an exact solution which to first order in e, describes a continuous transition between pure Ekman flow and a rotating analog of Hartmann flow. The magnetic field is found to inhibit Ekman suction; yet, such a boundary layer may still exert a strong influence on the outer flow because of a new feature that replaces the suction, namely an induced axial current outside the boundary layer. This “Hartmann current,” not presen...

Solution of a three-dimensional boundary-layer flow with separation.

Abstract: A method of calculating accurate solutions of the three-dimensional laminar boundarylayer equations has been developed. The method is applied to a problem that exhibits interesting crossflow phenomena, such as a flow reversal driven by crossflow convective terms and crossflow influences with zero freestream crossflow velocity. The method of solution used is an implicit finite difference scheme, and the stability and convergence properties of this scheme seemed to be good. Also, the important question of initial conditions for threedimensional boundary-layer flow is discussed, and a method of obtaining initial conditions is derived.

Flow properties for the continuum viewpoint of a non-equilibrium gas-particle mixture

Abstract: Flow properties for the non-equilibrium two-phase flow of a gas-particle mixture are formulated from the theoretical standpoint. A quasi-one-dimensional flow containing an arbitrary volume of particles is considered, and mass transfer between the phases is allowed. It is shown that meaningful definitions of the flow properties of each phase can be constructed as area-averages of (time-averaged local flow-field properties). Special definitions of averages overcome the difficulties introduced by the fact that one phase does not occupy the entire region at all times. Conservation equations for the newly defined properties are given and criteria for their validity determined. The results give fresh interpretation to several aspects of two-phase flow: the particle-phase pressure is associated with the internal particle pressure, whereas Reynolds-stress terms are introduced by fluctuations in particle velocity. Reynolds stresses for both phases are important in laminar as well as turbulent flow and provide a significant particlephase viscous effect. The interphase momentum transfer because of condensation or vaporization is shown to be characterized by the particle-phase velocity irrespective of the direction of the mass transfer.

Finite Element Analysis of Fluid Flows

Abstract: : The finite element method is applied to several simple cases of steady flow of a perfect, incompressible fluid. It is shown that the finite element representation accurately reflects the behavior of the classical flow equations. Finite elements form the basis for a versatile analysis procedure applicable to problems in several different fields. The earliest applications were to problems in structural mechanics. In recent years, nonstructural problems also have been treated by this method. The finite element method represents an approximate procedure for satisfying the problem in terms of its variational formulation. In structural mechanics this is generally accomplished by determining displacement fields based on satisfying the minimum potential energy theorem. Consequently, finite elements furnish a useful alternative scheme for applying the well-known Ritz method. For nonstructural problems, it is essential that the appropriate variational expressions be known beforehand. For the flow problems taken up in this paper, such expressions are well known. The governing matrix equation for the assemblage of elements is based on the properties derived for a single typical element. These properties, in turn, depend on assuming a mathematical form for the primary unknown of the problem and then satisfying the variational principle. For the elasticity problem, the unknowns are the displacements, while for the perfect incompressible fluid, either the velocity potential or the stream function may be used. Of great interest is that structural and nonstructural elements may often be identical in shape and may be represented by similar mathematical expressions. By way of illustration, the problems taken up in this paper were solved using the new ASTRA structural program developed at The Boeing Company. Finally, it should be pointed out that the major difference between the elasticity and fluid flow problems lies in the boundary conditions to be satisfied.

The prediction of turbulent boundary layer development in compressible flow

Abstract: Starting from Head's semi-empirical method for incompressible flow, two approaches to the prediction of turbulent boundary-layer development in compressible flow are explored. The first uses Head's incompressible method in conjunction with a compressibility transformation similar to Stewartson's transformation for laminar flow; the second carries over Head's physical arguments to treat the compressible flow directly. Measurements in supersonic flow, both on flat plates and downstream of an abrupt pressure rise, show broad agreement with the predictions of the second method but do not support the compressibility transformation. In particular, measurements on flat plates reveal that as Mach number increases the entrainment rate decreases to a lesser extent than the skin-friction coefficient. Whilst this result is consistent with the second treatment in this paper, it is difficult to reconcile with any of the compressibility transformations discussed, and the validity of these transformations in turbulent flow is therefore questioned.

Attenuation of circumferential inlet distortion in multistage axial compressors.

Abstract: Circumferential inlet distortion in a high hub-tip ratio multistage machine is treated by analyzing the compressor as a region in which a large number of small stages produce a pressure rise that is a function of the local mass flow rate. The resistance to circumferential flow due to the blading is included through an empirical factor. It is found that the over-all attenuation of both total pressure distortion and axial velocity distortion is mainly dependent on the slope of the compressor pressure rise vs flow rate characteristic. The attenuation increases when the slope of the characteristic is made more negative. In addition, considerable flow redistribution is found to occur upstream of the compressor. The theory has been compared with interstage data obtained on a three-stage, low-speed compressor with axial clearances that are 26% of the total length and a hub-tip ratio of 0.675. It is found that the approximation of zero axial clearance (infinite resistance to circumferential flow) gives excellent results. In consequence, it appears that for the normal range of axial clearances, the circumferential flow within the compressor can be neglected in a first-order analysis of the effects of inlet distortion.

Compressible slow viscous flow past a vaporizing droplet.

Abstract: A small rigid spherical droplet is taken to undergo quasi-steady adiabatic vaporization. The droplet, near its boiling temperature, lies in an unbounded expanse of hot ideal gas, which flows slowly past the droplet. Whenever even a slight ambient gaseous flow exists relative to the droplet (that is, for small but finite Reynolds number based on freestream conditions), piecewise-valid asymptotic expansions must be adopted to restore the convective transport to proper importance far from the body. Incipient forced convection thus represents a singular perturbation to the classical limit of sphericosymme tric, diffusion-dominated vaporization. Inner-and-outer (matched asymptotic) expansions here yield the solution in general and two gross flow properties in particular: the modifications to both the Stokes drag and also the Maxwell-Stefan vaporization rate (Sherwood number) owing to ablationlike mass transfer, fluid compressibility, and variability of fluid properties. The dependence of the solution on all problem parameters is studied within the restrictions that the Mach number is taken as vanishingly small, and the Schmidt and Lewis numbers are constant and of order unity.

Rayleigh Step Journal Bearing, Part II—Incompressible Fluid

Abstract: Rayleigh step journal bearings, considering pressure distribution, load capacity and attitude angle and optimal film thickness ratio for incompressible fluid lubrication

Magnetohydrodynamic flows over a rotating disk

Abstract: Magnetohydrodynamic flow of incompressible viscous electrically conducting fluid over rotating disk

Numerical solutions of compressible and incompressible laminar separated flows.

Abstract: Numerical techniques and solutions for compressible and incompressible laminar separated flows using time dependent finite difference equations

Some Remarks on Vortex Drag and its Spanwise Distribution in Incompressible Flow

Abstract: Summary Theoretical data from lifting-surface theory are presented to illustrate (i) that the vortex drag factor is closely related to the half-wing spanwise centre of pressure on simple planforms without camber or twist, (ii) that lifting-line theory is useless for predicting the spanwise distribution of vortex drag on swept wings, (iii) that recent numerical improvements in lifting-surface theory help to reconcile the concepts of wake energy and leading-edge suction in relation to vortex drag.

Axially-symmetric eddies embedded in a rotational stream

Abstract: Eddies structure embedded in swirling incompressible fluid assuming laminar motion and flow field axial symmetry

Wall shear stress diagnostics in three-dimensional turbulent boundary layers.

Abstract: : Several of the more common means of inferring local wall shear values in two-dimensional turbulent boundary layer flows are reviewed and the suitability of these means for use in three-dimensional turbulent flows is discussed A simple, floating element, direct force sensing device, corrected for pressure gradient effects, was used to measure the local wall shear stress in a strongly skewed, low-speed, incompressible three-dimensional turbulent boundary layer flow These pressures corrected, direct force measurements are compared to wall shear stress values inferred by variations or adaptations of the two-dimensional techniques and/or devices reviewed (Author)

The Incompressible Laminar Axisymmetric Far Wake

Abstract: : The memorandum considers the asymptotic development of the axisymmetric incompressible wake far behind a long, thin cylinder. This is the axisymmetric analogue of the classical two-dimensional flat-plate problem; however, the resulting solution is applicable to a wider class of axisymmetric bodies. In Section II an asymptotic expansion in inverse powers of the axial distance downstream of the cylinder is attempted. In Section III the axisymmetric problem is reformulated, and it is shown that as in the two- dimensional problem the cause of the breakdown of the solution given in Section II is the existence of a logarithmic term in the second-order approximation.

On compressible turbulent-plane Couette flow.

Abstract: : Equations describing compressible turbulent plane Couette flow for the case of equal heat transfer at both walls were developed for a perfect gas based on the von Karman mixing length model. This model was selected because of its good agreement with available turbulent plane Couette flow measurements of incompressible flow and furthermore, lacking compressible Couette flow data, it is shown by Spalding and Chi that those theories based on the von Karman model give best agreement with compressible turbulent boundary layer measurements. A sample calculation is given for the heat rate to the inner surface of a rail- guided slipper (representative of slippers supporting rocket boosted sleds) based on the assumption of a small air gap between both surfaces.

Three dimensional numerical simulation of polymer flooding in homogeneous and heterogeneous systems

Abstract: The 3-dimensional simultaneous flow of oil, water, and polymer solution is treated as 2-phase incompressible flow, with the oil being the nonwetting phase and water and polymer together being the wetting phase--the latter 2 flow miscibly with each other. The 2-phase flow takes into account effects of gravity, capillary pressure, and sand heterogeneity, and appropriate finite difference analogs are solved by standard iterative ADI technique. Propagation of polymer within the wetting phase is accomplished purely by mass transport with the assumption of negligible dispersion, and is evaluated explicity using a moving reference technique. Results show that injection of polymer at a watered-out stage in the flood history has no effect on the 5-spot recoveries for the homogeneous cases but has a slightly favorable effect in the layered cases with thief zones. When injected earlier, however, incremental oil recoveries of 5% to 26% over the base floods are obtained at the time of oil-bank production. At polymer flood termination at WOR of 25:1, the homogeneous cases show slight increase in ultimate recoveries.

Calculation of mean and fluctuating properties of the incompressible turbulent boundary layer

Abstract: Mathematical model for calculation of mean and fluctuating properties of incompressible turbulent boundary layers

Spheroidal particle flow in a cylindrical tube.

Abstract: : Viscous flow in a circular cylindrical tube containing an infinite line of rigid spheroidal particles equally spaced along the axis of the tube is considered for (a) uniform axial translation of the spheroids (b) flow past a line of stationary spheroids and (c) flow of the suspending fluid and spheroids under an imposed pressure gradient. The fluid is assumed to incompressible and Newtonian. The Reynolds number is assumed to be small and the equations of creeping flow are used. Two types of solutions are developed for each case (i) an exact solution in the form of infinite series which is valid for ratios of the spheroid diameter to the tube diameter up to 0.80. (ii) an approximate solution using lubrication theory which is valid for spheroids which nearly fill the tube. The drag on each spheroid and the pressure drop are computed for all cases. Both prolate and oblate spheroids are considered. The results show that the drag and pressure drop depend on the spheroidal diameter perpendicular to the axis of tube primarily and the effects of the spheroidal thickness and spacing are secondary. The results are of interest in connection with mechanics of capillary blood flow, sedimentation, fluidized beds, and fluid-solid transport. (Author)

Linearized analysis of constant-property duct flows

Abstract: The conservation equations describing steady, incompressible flow in a variable-area duct with mass transfer occurring at its walls are simplified by linearizing the inertial and convective terms. Solutions to a large class of problems can be obtained by means of the general method which is presented. Particular examples considered are entrance flow with heat transfer to an isothermal wall in the presence of mass addition, constant rate of injection of a foreign gas through the wall, vaporization and sublimation of a volatile wall material, and gas-phase combustion of a fuel which enters the duct from its wall. Comparison of the present results with previous work and with new experimental results is discussed for the first of these applications. It is concluded that the present results for velocity and pressure fields are likely to be accurate for small values of the Reynolds number based on wall injection velocity, that the present results for temperature and composition fields are likely to be accurate in the absence of wall mass transfer, and that in the absence of wall mass transfer the linearization technique is likely to exhibit its highest accuracy for flows with uniform entrance conditions.

Study of swirling fluid flows.

Abstract: : Three distinct aspects of swirling flow were studied both analytically and experimentally: (1) incompressible flow in a rotating tube, (2) turbulent, swirling flow in a stationary tube, and (3) compressible, turbulent swirling flow in vaneless radial diffusers (Author)

Hydromagnetic flow in a rotating straight pipe.

Abstract: A theoretical analysis is made of the flow of a conducting viscous and incompressible fluid through a straight pipe of circular cross-section flowing under a constant pressure gradient. The pipe is rotated about an axis perpendicular to it and also there is imposed a uniform magnetic field transverse to the motion. It is assumed for the purpose of mathematical analysis, the angular velocity about the axis of rotation, is small. A solution is developed by successive approximations in ascending powers of the Hartmann number, the first approximation corresponds to the non-magnetic case, formulated and discussed by Barua. The stream lines in the central plane and the projection of the stream lines on the cross-section of the pipe are compared with those in the non-magnetic case. An expression for the induced electric potential difference and sensitivity has been obtained.