Showing papers on "Incompressible flow published in 1971"

A numerical study of laminar separation bubbles using the Navier-Stokes equations

Abstract: The flow in a two-dimensional laminar separation bubble is analyzed by means of finite-difference solutions to the Navier-Stokes equations for incompressible flow. The study was motivated by the need to analyze high-Reynolds-number flow fields having viscous regions in which the boundary-layer assumptions are questionable. The approach adopted in the present study is to analyze the flow in the immediate vicinity of the separation bubble using the Navier-Stokes equations. It is assumed that the resulting solutions can then be patched to the remainder of the flow field, which is analyzed using boundary-layer theory and inviscid-flow analysis. Some of the difficulties associated with patching the numerical solutions to the remainder of the flow field are discussed, and a suggestion for treating boundary conditions is made which would permit a separation bubble to be computed from the Navier-Stokes equations using boundary conditions from inviscid and boundary-layer solutions without accounting for interaction between individual flow regions. Numerical solutions are presented for separation bubbles having Reynolds numbers (based on momentum thickness) of the order of 50. In these numerical solutions, separation was found to occur without any evidence of the singular behaviour at separation found in solutions to the boundary-layer equations. The numerical solutions indicate that predictions of separation by boundary-layer theory are not reliable for this range of Reynolds number. The accuracy and validity of the numerical solutions are briefly examined. Included in this examination are comparisons between the Howarth solution of the boundary-layer equations for a linearly retarded freestream velocity and the corresponding numerical solutions of the Navier-Stokes equations for various Reynolds numbers.

Calculation of boundary-layer development using the turbulent energy equation: compressible flow on adiabatic walls

Abstract: The basic method described by Bradshaw, Ferriss & Atwell (1967) is extended to compressible flow in two-dimensional boundary layers in arbitrary pressure gradient (excluding shock waves and expansion fans) by invoking Morkovin's hypothesis (Favre 1964) that the turbulence structure is unaffected by compressibility Using the same empirical functions as in incompressible flow, skin friction in zero pressure gradient is predicted to within 3% of Spalding & Chi's (1964) correlation for free-stream Mach numbers less than 5 Comparisons with experiments in pressure gradient are restricted by the lack of data, but, since Morkovin's hypothesis does not depend on pressure gradient, methods which use it (of which the present method seems to be the first) can be checked fairly adequately by comparisons with data in zero pressure gradientNo ‘compressibility transformations’ are needed, although the Crocco relation is used, provisionally, for the temperature: since the calculations take only about 20% longer than in incompressible flow, Morkovin's hypothesis does as much as any transformation could do It is pointed out that, in supersonic flow, surface curvature which is large enough to induce a significant longitudinal pressure gradient is also large enough to have a very significant effect on the turbulence structure

Statistical investigation of pressure and velocity fields in the turbulent two-stream mixing layer

Abstract: Turbulent mixing between parallel incompressible air streams, using statistical investigation of pressure and velocity fields

On the scattering of aerodynamic noise

Abstract: According to the Lighthill acoustic analogy, the sound induced by a region of turbulence is the same as that due to an equivalent distribution of quadrupole sources within the fluid. It is known that the presence of scattering bodies situated near such multipoles can convert some of their intense near field energy into the form of sound waves whose amplitude is far greater than that of the incident field. Calculations are here presented to determine the extent of this conversion, for hard and soft bodies of various shapes, making use of the reciprocal theorem to recast the problem into one of finding the field, near the obstacle, induced by an incident plane wave. If the obstacle is small compared with a wavelength, then its presence is equivalent to an additional dipole (or source) whose greater efficiency as a sound radiator implies that the familiar intensity law I ∝ U8, for far field intensity I against typical turbulence velocity U for an unbounded flow, is replaced by I ∝ U6 (or I ∝ U4) for a hard (or soft) body. For the situation where the scatterer is large compared with wavelength, the prototype problem of a wedge of exterior angle (p/q)π is shown to yield an intensity law I ∝ U4+2q/p for both hard and soft surfaces. This result is shown to hold for the more general ‘wedge-like’ surfaces, whose dimensions are large scale and whose edges may be smoothed out on a small scale, compared with wavelength. The method used involves the matching of an incompressible flow, on the fine scales typical of the edge geometry, to an outer flow determined by the large scale features of the surface. Favourable comparisons are made with previous results pertaining to the two-dimensional semi-infinite duct and to the half-plate of finite thickness.

Viscous incompressible flow between concentric rotating spheres. Part 1. Basic flow

Abstract: The steady motion of a viscous fluid contained between two concentric spheres which rotate about a common axis with different angular velocities is considered. A high-order analytic perturbation solution, through terms of order Re7, is obtained for low Reynolds numbers. For larger Reynolds numbers an approximate Legendre polynomial series representation is used to reduce the governing system of equations to a non-linear ordinary differential equation boundary-value problem which is solved numerically. The resulting flow pattern and the torque required to rotate the spheres are presented for various cases considered.

Gust Loading on a Thin Aerofoil

Abstract: A closed-form expression is derived which gives an approximate solution to the lift generated on a two-dimensional thin aerofoil in incompressible flow with a normal velocity component of the form exp [ i ( ωt – x x + y y )]. The inaccuracy of the solution when compared with other published work is compensated by the simplicity of the final expression, particularly if the result is required for the calculation of the sound power radiated by an aerofoil in a turbulent flow.

A Lifting-Surface Theory for the Rectangular Wing in Non-Stationary Flow

Abstract: A method of obtaining the load distribution on thin rectangular wings in non-stationary incompressible flow is presented, with particular reference to the problem of gust-induced loading. The method utilises solutions to a Fourier transform of the downwash integral equation, enabling the problem to be expressed in the form of a set of dual integral equations having a series solution. Some values of the overall lift coefficient are computed and compared with values of lift coefficient obtained by collocation methods. The method is easily extendable to subsonic compressible flows.

Magnetohydrodynamic duct flow in a uniform transverse magnetic field of arbitrary orientation

Abstract: The paper presents an approximate analysis for high Hartmann number of the flow of an electrically conducting, incompressible fluid in a duct of square crosssection, having one pair of opposite walls insulating, and the other pair perfectly conducting and inclined at arbitrary orientation to a uniform transverse magnetic field. The flow is considered to be either pressure-driven with the two perfectly conducting electrodes short-circuited together or electrically driven by a potential difference applied between these electrodes in the absence of axial pressure gradient. The paper describes experiments on the pressure-driven, short circuited case using mercury in copper ducts to investigate the variation of the streamwise pressure gradient and of the potential distribution along one insulating wall with orientation, magnetic field and flow rate.At general orientations the analysis suggests and the experiments confirm the existence of regions of stationary fluid in the corners of the duct, together with viscous shear layers parallel to the magnetic field. For the case in which the electrodes are parallel to the magnetic field the experimental results for the pressure gradient, but not those for the potential distribution, agree reasonably well with Hunt & Stewartson's (1965) asymptotic solution. Both pressure gradient and potential results agree closely with the analysis by Hunt (1965) of the case in which the electrodes are perpendicular to the magnetic field.

Difference methods for the solution of the time-dependent semiconductor flow equations

Abstract: The letter describes the solution of the instationary semiconductor transport equations by means of finite-difference methods. This new approach is based on the formal similarity with the equations of incompressible flow. Implicit schemes have proven useful for the computation of both transient and steady-state solutions. Comparisons with experimental results are extremely favourable for f.e.t.s.

Aerodynamics of Finned Missiles at High Angle of Attack

Abstract: A theoretical and experimental investigation of the aerodynamics of finned missiles at high angle-of-attack is presented. The interaction of the body vortex wake region with the missile fins aiid the resulting effect on the forces and moments produced by the fins is studied. An aerodynamic flow model is derived for a circular cylinder body at high angle-of-attack in subsonic or supersonic flow. The forces and moments produced by the fins in subsonic flow are calculated by utilizing the present flow model and present lifting theory. The experimental investigation consists primarily of strain gage measurements on a finned body in subsonic flow. The present aerodynamic flow model is compared with experimental crossflow data and good agreement between theory and experiment is obtained. Extensive comparison is made between the present force and moment predictions and available experimental data. Generally good agreement is obtained between theory and experiment for all of the forces and moments. Results given particular attention are nonlinear rolling phenomena, induced side force, and fin Magnus.

Vortex computation by the method of weighted residuals using exponentials

Abstract: A method of weighted residuals for the computation of rotationally symmetric quasicylindrical viscous incompressible vortex flow is presented. The method approximates the axial velocity and circulation profiles by series of exponentials having (N + 1) and TV free parameters, respectively. Exponentials are also used as weighting functions. Formal integration results in a set of (2N + 1) ordinary differential equations for the free parameters. The governing equations are shown to have an infinite number of discrete singularities. Sample solutions for different swirl parameters and three typical vortex flows (initially uniform axial flow, leading edge vortex, and trailing vortex) are presented, and the effects of external axial velocity and circulation gradients are investigated. The computations point to the controlling influence of the inner core flow on vortex behavior. They also confirm the existence of two particular critical swirl parameter values: So, which separates vortex flow which decays smoothly from vortex flows which eventually "breaks down," and Si, the first singularity of the quasi-cylindrical system, at which point physical vortex breakdown is thought to occur. The results are close to the inviscid values for S0 and Si[(2)lf2 and 3.8317/2 for initially uniform axial flow].

Approximate transfer functions for large aspect ratio wings in turbulent flow

Abstract: The fundamental aerodynamic problem associated with flight through atmospheric turbulence is calculation of the response of a wing flying through a single sinusoidal wave of upwash with lines of constant phase arbitrarily inclined to the flight path. Approximate closed form expressions for the gust transfer functions" relating the lift and moments to the upwash in such a wave are derived for large aspect ratio rectangular wings in incompressible flow. The lift transfer function is expressed as the usual two-dimensional Sears function multiplied by a factor to correct for finite span and a further factor to account for the span wise gust wave number. Multiplying this expression by the chordwise (or spanwise) center of pressure leads to the pitching (or rolling) moment transfer function. Some simple scaling laws, based on these results, are then suggested for flight through large-scale turbulence.

The development of an incompressible boundary-layer theory valid to second order

Abstract: A set of incompressible boundary-layer equations is derived which include second-order curvature effects. These equations are used to analyze flows past spheres and ellipsoids without surface suction or injection to determine the effect of the curvature terms on boundary-layer properties near separation. Finally the suction velocity necessary to maintain a particular value of constant displacement thickness on a sphere is determined. It appears that infinite suction is required as the rear stagnation point of the sphere is approached.

The hydromagnetic Kelvin—Helmholtz problem in a Hall plasma

Abstract: The hydromagnetic analogue of the Kelvin–Helmholtz problem is extended to include the effects of the Hall term. In contrast to other results in the literature it is shown that, in the case of incompressible fluids, the stability of a shear plane is unaffected by the introduction of the Hall term. The special case of a hot, uninagnetized fluid on one side of the interface and a cold, magnetized fluid on the other is studied in some detail. In this case it is shown that the presence of the Hall term can have either a stabilizing or a destabilizing effect, depending upon whether the sound speed in the hot fluid is very much greater than the Alfven speed in the cold fluid or vice versa.

Magnetohydrodynamic laminar flow between porous disks

Abstract: The steady two-dimensional laminar flow of an incompressible conducting fluid between two parallel circular disks in the presence of a transverse magnetic field is investigated. A solution is obtained by perturbing the creeping flow solution and it is valid only for small suction or injection Reynolds numbers. Expressions for velocity, induced magnetic field, pressure, and shear stress distribution are determined and are compared with the creeping flow and hydrodynamic solutions. It is found that the overall effect of the magnetic field on the flow is the same as that in the Hartmann flow.

On hydromagnetic fluctuating flow past an infinite porous plate in slip flow regime

Abstract: The effect of a uniform external magnetic field on the laminar, incompressible rarefied gas flow along an infinite porous flat plate is studied under the following conditions: 1) there is uniform suction, 2) the external flow velocity varies periodically with time in magnitude but not in direction, 3) the magnetic Reynolds number is small and 4) the current occurs under slip flow boundary conditions. Expressions for the velocity and temperature fields in the boundary layer are obtained. The response of skin friction, and heat transfer to the fluctuating stream is studied for variations in the rarefaction parameter h 1, the magnetic field parameter M, and the frequency of the fluctuating stream.

Kinematic-dynamo action under incompressible, isotropic velocity turbulence.

Abstract: Regenerative kinematic-dynamo action under incompressible isotropic velocity turbulence, noting turbulent Lorentz force role

Aerodynamic Loading Induced on a Two-Dimensional Wing by a Free Vortex in Incompressible Flow

Abstract: Practical problems often arise in which the aerodynamic loadings on wings and fins, due to nearby free vortices, are required. For example, aerodynamic loads on helicopter blades are affected by the tip vortices which are continuously being shed and which lie beneath the rotor plane. And vortices shed from the nose of an aircraft fuselage can induce significant side loads on a fin. The estimation of the aerodynamic load distribution on a moving wing in the neighbourhood of a free vortex (whose axis is primarily in the free stream direction) is an interesting problem which, as far as the author is aware, has not been discussed previously in the literature. Standard techniques for conventional linearised wing theory cannot be applied directly to this problem because the trailing vorticity shed from the wing trailing edge is affected by the flow field of the free vortex and so the mathematical application of the Kutta trailing edge condition is more complex.

Magnetohydrodynamics of Conical Flows

Abstract: The effect of diverging or converging electric currents on a stationary, viscous, incompressible, conducting body of fluid, confined in a cone of infinite extent, is investigated. The fluid moves down from upstream infinity near the walls and as it approaches the apex of the cone, it changes its direction in the core of the cone and moves toward upstream infinity near the axis. With increasing currents, the velocity near the axis rises very rapidly. Similar phenomenon have been observed in the laboratory electric arcs and in thermal plasmas of the solar atmosphere. Next, when a fixed amount of fluid is drained from the apex of the cone, the flow divides into two regions far from the apex. The drained fluid reaches the apex only through a narrow region near the walls whereas fluid in the core of the cone executes a secondary motion, similar to the first case of confined flow. There exists a stagnation point on the axis and its location is primarily controlled by current density, fluid drainage rate, and the cone angle of the configuration.

Laminar Incompressible Flow in the Entrance Region of Ducts of Arbitrary Cross Section

Abstract: A combination of the numerical technique of Chorin for the solution of the Navier-Stokes equations and a transformation of the initial value problem to a boundary value problem is shown to allow calculation of the laminar hydrodynamic entrance region of ducts of arbitrary cross section. Numerical examples consisting of the solution for ducts of square and triangular cross sections are presented along with the associated friction factors.

An Implicit Numerical Solution of the Turbulent Three-Dimensional Incompressible Boundary Layer Equations.

Abstract: A method of solving the three-dimensional, incompressible turbulent boundary-layer equations was developed using a Crank-Nicholson implicit finite-difference technique, with the turbulent stress terms modeled with an eddy-viscosity model obtained from mixing length theory. The method was applied to two three-dimensional flow geometries for which experimental data exists and a comparison with this data showed excellent agreement. A complete computer program was sufficiently generalized for application to two-dimensional laminar and turbulent flows with arbitrary pressure gradients. The method was applied to several such test cases and the solutions agreed well with both theory and experiment. An analysis was presented to determine the conditions for which the finite difference equations were stable and convergent. (Author)

The Turbulent Boundary Layer With Mass Transfer and Pressure Gradient

Abstract: : An analysis of the incompressible, turbulent boundary layer, including the combined effects of mass transfer and pressure gradient is presented. An integral method employing the integral mechanical energy equation forms the basis of the analysis. Stevenson's velocity profiles are used to obtain the functional dependence of the integral properties and also obtain a skin friction law. Comparisons of the numerical results with a wide variety of experimental data, including cases where the blowing rate and pressure are varying simultaneously, show good agreement. In addition, several problems with discontinuities in blowing or suction are solved and also seen to be in good agreement with the data. (Author)