Showing papers on "Stream function published in 1982"

Two‐dimensional convection in non‐constant shear: A model of mid‐latitude squall lines

Abstract: Numerical simulations of two-dimensional deep convection are analysed using analytical models extended to include shallow downdraughts and non-constant shear. The cumulonimbus are initiated by low-level convergence created by a finite amplitude downdraught. These experiments have constant low-level shear and differ only in the profile of mid-and upper-level winds. Quasi-steady convenction is produced if the mid- and upper-level flow has small shear and the low-level shear is large. The surface precipitation ismaximized for no intial relative relative flow aloft, if stationary, this storm (P(O)) can give prodigious locilized rainfall; P(O) is the two-dimentisonal equivalent of the supercell. These results are placed in context with previous two-dimensional simulations. Attention is drawn to the similiarity with previous two-dimensional simulations. Attention is drawn to the similarity with squall lines in central and eastern U.S.A. Storm P(O) is analysed by construction of time-averaged fields of streamfunction, vorticity, teperature, and height deviation. The smoothness of these fields suggests a conceptual model of the storm dynamics which involves cooperation between distinct charcteristic flows; an overturning updraught, a jump type updraught, a shallow downdraught, a low-level rotor, and a boundary layer. An idealized analytical model is described by solution of the equations for steady convection. These solutions, for the remote flow, are derived from energy conversation, mass continuity and a momentum budget, and they give relationships between the non-dimensional parameteres of the problem. It is apparent that the convection is a high Froude (or low Richardson) number flow demanding the existence of a cross-storm pressure gradient. Inherent in this idealized model is a vortex sheet between updraught and down-draught and it is considered that the dynamical instability of this sheet is related to complexities in the numerical simulation. Furthermore, these results show that in two-dimensions both non-constant shear and a shallow downdraught are necessary to maintain steady convection.

The velocity field induced by a helical vortex filament

Abstract: An exact analytical solution for the velocity field, both interior and exterior, induced by an infinite right‐handed helical vortex filament is derived. Due to the way the variables combine in this solution, the paper also shows that it is possible to derive a stream function for this nonaxisymmetric flow. Sample calculations of these expressions are included.

Long wavelength approximation to peristaltic motion of a power law fluid.

Abstract: Peristaltic motion of a power law fluid in a two-dimensional channel is studied. Assuming that the wavelength of the peristaltic wave is large in comparison to the mean half-width of the channel, a solution for the stream function is obtained as an asymptotic expansion in terms of slope parameter. Expressions for axial pressure gradient and shear stress are derived. The effect of flow behaviour indexn on the streamline pattern and shear stress is studied and the phenomenon of trapping is discussed.

Navier-Stokes Computations of Turbulent Compressible Two-Dimensional Impinging Jet Flowfields

Abstract: Planar mass-averaged compressible Navier-Stokes and energy equations in stream function/vorticity form are solved in conjunction with a two-equation (fc-e) turbulence model for impinging jet configurations relevant to VTOL aircraft design. The physical domain of the flow is mapped conformally into a rectangular computational region. An augmented central-difference scheme is used to preserve the diagonal dominance character of the difference equations at high Reynolds numbers. The resulting difference equations are solved by successive point relaxation. Excellent agreement with the experimental data is obtained for the airframe undersurface pressure, ground-plane pressure, and the centerline velocity decay along the jet axis. Computed turbulent kinetic energy along the jet axis, however, has a larger overshoot near the ground plane than indicated by the experimental data.

Marangoni-convection in a freely floating liquid sphere due to axial temperature field

Abstract: A freely floating liquid drop is subjected at its surface to an axial temperature field inducing a thermal Marangoni-convection due to the variation of the surface tension. Streamfunction and velocity distribution are analytically determined for steady and unsteady temperature fields, by solving the equation for the Streamfunction with the help of associated Legendre functions of the first kind. The special case of a steady linear axial temperature field is evaluated numerically and exhibits a single quadratic velocity profile, which in radial direction is proportional to cos ϑ and in ϑ-direction proportional to sin ϑ.

Finite element, stream function–vorticity solution of steady laminar natural convection

Abstract: SUMMARY Stream function-vorticity finite element solution of two-dimensional incompressible viscous flow and natural convection is considered. Steady state solutions of the natural convection problem have been obtained for a wide range of the two independent parameters. Use of boundary vorticity formulae or iterative satisfaction of the no-slip boundary condition is avoided by application of the finite element discretization and a displacement of the appropriate discrete equations. Solution is obtained by Newton-Raphson iteration of all equations simultaneously. The method then appears to give a steady solution whenever the flow is physically steady, but it does not give a steady solution when the flow is physically unsteady. In particular, no form of asymmetric differencing is required. The method offers a degree of economy over primitive variable formulations. Physical results are given for the square cavity convection problem. The paper also reports on earlier work in which the most commonly used boundary vorticity formula was found not to satisfy the no-slip condition, and in which segregated solution procedures were attempted with very minimal success. Most finite difference studies of incompressible viscous flow and convection in two dimensions have used the stream function-vorticity formulation. This formulation has the advantage of satisfying mass conservation exactly and reducing the number of simultaneous partial differential equations by one (from three to two for uncoupled viscous flow alone). With the finite element method however the majority of studies have used the primitive variable approach. In one of the first finite element works Taylor and Hood',* examined both forms of approach, but dropped stream function-vorticity in favour of primitive variables. Primitive variables were also chosen by Oden and Wellf~rd,~ and then by Kawahara et aL4 and Gartling and Beckers amongst others. Stream function-vorticity was chosen in the early works of Cheng6 and Baker.7 The difficulty with the stream function-vorticity approach is that, of two second order partial differential equations, one, a Poisson equation for stream function, has two boundary conditions instead of the one required, and the other, the vorticity transport equation, has no boundary condition. Finite difference workers have for long met this difficulty by defining boundary vorticity from the variation along the boundary normal of the stream function field determined in a previous iteration. We shall call this the boundary vorticity formula method.

Velocity distribution due to marangoni effect for angular temperature field along an infinite liquid bridge under weightless condition

Abstract: Thermal Marangoni convection inside a liquid bridge is analytically determined for a purley angular temperature field at the free surface of the liquid. Radial- and angular velocity components as well as the stream function are presented for arbitrary stationary and time-periodic temperature fields. The case of a simple stationary sinusoidal temperature field has been evaluated numerically.

Numerical solution for the flow of highly viscous fluid in agitated vessel with anchor impeller

Abstract: Fluid motion in an agitated vessel with an anchor impeller is characterized by flow in a horizontal plane induced by the vertical arms of the impeller rotating near the vessel wall. A numerical algorithm of the two-dimensional flow in the horizontal plane is established using an iterative method for the determination of the boundary values of stream function. The computational results of the velocity profiles and agitation power are compared with those of the experiments, and it is shown that the numerical method used in this study is very useful to analyze the flow past the vertical arms of an anchor impeller.

Flow separation in shear-layer-driven cavities

Abstract: A study of the shear-layer flow over a range of open-top cavity configurations is reported. Emphasis is placed on the effect that altering the cavity's span length and aspect ratio has on the development of the shear layer. Computational results are obtained using an interactive method which adapts the compressible boundary-layer model for the flow above the cavity and incompressible Navier-Stokes equations within the enclosure. Interaction of this composite model with the outer, inviscid supersonic flow is also considered in one case. The results show that the location of the stagnation points is sensitive primarily to the variation of the span. When the span was fixed and the aspect ratio varied, the shear layer was nearly unaffected except at aspect ratios less than 0.5. Interaction with the outer flow had a smoothing effect on the shape of the dividing streamline, but did not significantly affect the location of the stagnation points. Nomenclature a = speed of sound = ^JyRT 6 = aspect ratio, -HlL cp = specific heat at constant pressure H = height of the cavity h = distance along the cavity's vertical wall measured from the convex corner k = thermal conductivity L = characteristic length M = Mach number, = u/a Pr =Prandtl number, -^cp/k p = pressure R = constant in perfect gas law Re = Reynolds number, =pUL/n T = temperature t = time U — freestream velocity vector u = velocity component in the x direction v = velocity component in the y direction x = Cartesian coordinate parallel to the plate y = Cartesian coordinate normal to the plate 7 = ratio of specific heats, =cp/cv A = small increment, difference <5 = boundary-layer thickness f = vorticity function fj, = viscosity coefficient in the boundary-layer equations p = density \l/ = stream function Subscripts BL = boundary-layer calculation c = cavity calculation e = local outer flow condition R = reference W '=wall

Spectral Method Solutions for Some Laminar Channel Flows with Separation

Abstract: Spectral method solutions to the plane slender-channel equations are investigated for some laminar, incompressible, high Reynolds number internal flows with separation. The equations are formulated in fourthorder stream-function form with the stream function taken as a Fourier series expansion plus a cubic polynomial in the transverse coordinate. Solutions are obtained for a sudden channel expansion, base in a channel, channel entry flow, and slowly diverging channel which agree well with the results of previous investigations.

A numerical treatment of two-dimensional flow in a branching channel

Abstract: A numerical method for treating the steady two-dimensional flow of a viscous incompressible fluid in a branching channel is given. The upstream and downstream boundary conditions are discussed and a logarithmic transformation is applied to the coordinate measuring distance downstream in order to extend the numerical solution far enough downstream. Two methods are presented for dealing with the singularity in the vorticity at the sharp corners associated with the geometrical division of the flow. The Navier-Stokes equations are written in terms of the stream function and vorticity giving the usual two coupled nonlinear partial differential equations. These equations are solved using the method of Dennis and Hudson (1978). The effect of the relative widths of the channels upstream and downstream of the branch on the separation of the flow is discussed using results obtained from three separate grid sizes.

Rotational Plane Flows of a Viscous Fluid

Abstract: Navier–Stokes equations for steady plane flows are recast in the hodograph plane by using the Legendre transform of the stream function. As applications, various flows and the corresponding geometries are investigated.

Finite-Amplitude Waves in Inviscid Shear Flows

Abstract: This paper examines the existence and properties of steady finite-amplitude waves of cats-eye form superposed on a unidirectional inviscid, incompressible shear flow. The problem is formulated as the solution of nonlinear Poisson equations for the stream function with boundary conditions on the unknown edges of the cats-eyes. The dependence of vorticity on stream function is assumed outside the cats-eyes to be as in the undisturbed flow, and uniform unknown vorticity is assumed inside. It is argued on the basis of a finite difference discretization that the problem is determinate, and numerical solutions are obtained for Couette-Poiseuille channel flow. These are compared with the predictions of a weakly nonlinear theory based on the approach of Benney & Bergeron (1969) and Davis (1969). The phase speed of the waves is found to be linear in the wave amplitude.

A boundary integral method for eddy current flow around cracks in thin plates

Abstract: A boundary element method which employs a Green's function for a crack has been developed to calculate the induced eddy current flow around cracks in thin conducting plates. The theoretical equations employ a stream function for the current density vector and is equivalent to the electric field vector potential method. A low frequency or large skin depth approximation leads to a Poisson equation for steady harmonic inductor fields. Induced currents around a crack in a square plate due to a uniform inductor field for various crack positions and sites have been calculated in this paper. The effect of the relative position and length of the crack, with respect to the plate width, on the eddy current density near the tips of the crack is given special attention. These results may be useful to simulate eddy current flow detection phenomena.

Buoyant Convection Computed in a Vorticity, Stream-Function Formulation

Abstract: Model equations describing large scale buoyant convection in an enclosure are formulated with the vorticity and stream function as dependent variables. The mathematical model, based on earlier work of the authors, is unique in two respects. First, it neglects viscous and thermal conductivity effects. Second the fluid is taken to be thermally expandable: large density variations are allowed while acoustic waves are filtered out. A volumetric heat source of specified spatial and temporal variation drives the flow in a two-dimensional rectangular enclosure. An algorithm for solution of the equations in this voticity, stream-function formulation is presented. Results of computations using this algorithm are presented. Comparison of these results with those obtained earlier by the authors using a finite difference code to integrate the primitive equations show excelent agreement. A method for periodically smoothing the computational results during a calculation, using Lanczos smoothing, is also presented. Computations with smoothing at different time intervals are presented and discussed.

The stability of quasi-geostrophic fields induced by potential vorticity sources

Abstract: A determination of the conditions leading to instabilities which grow in place in quasi-geostrophic fields induced by localized potential vorticity sources is presented. The analysis considers a quasi-geostrophic barotropic flow in a horizontally open domain, and a nondimensional quasi-geostrophic vorticity equation is defined which governs the deviation stream function. Weak forcing is considered, and it is shown that the field induced by the topography is stable while the field induced by the potential vorticity source can be unstable. The growth rate is exponential and a function of nonlinearity and friction, which, if absent, indicates an unstable flowfield. The instability is shown to be capable of changing the flowfield from one quasi-steady state to another, with energy extraction occurring at the source

Numerical Solution of Flow Near the Rotor of a Wind Turbine

Abstract: A numerical procedure based on the full Navier-Stokes equations as applied to the flow near a wind turbine rotor is developed. The flow is assumed axisymmetric, and the unsteady equations of motion are cast in terms of a stream function, one vorticity component, and the peripheral velocity. The vorticity equation and the peripheral momentum equation are solved by an alternating difference implicit technique, and the Poisson equation for the stream function is solved by direct matrix reduction. The rotor is modeled as an actuator disk, and the direct simulation of a given actual wind turbine rotor is considered in detail. Turbulent transport is modeled by an integrated turbulent kinetic energy equation with a simple extension to represent the effects of swirl. Comparison with field measurements on a horizontal-axis, three-bladed Elektro 10-kW machine at a station right behind the rotor shows that a good prediction of the radial distribution of the axial velocity was obtained.

Generalized stream function representation for current flow in semiconducting media

Abstract: Stream functions comprised of stream vectors and stream potentials are introduced to represent electron and hole current densities in a semiconductor. These stream functions satisfy electron and hole current continuity equations individually for general cases involving, for example, transient behavior and recombination‐generation processes. The terminal currents are expressed in terms of the stream functions. The nonuniqueness property of the stream functions is discussed briefly.

A numerical method to solve the steady-state Navier-Stokes equations for natural convection in enclosures

Abstract: A numerical method is presented for thermally driven laminar flow in two-dimensional enclosures. The procedure solves the Navier-Stokes equations in terms of the stream function by using the finite-difference method of Hermitian type. For the solution of the complex system of finite-difference equations a direct solver is developed based on the LU decomposition of the matrix. In order to linearize the nonlinear stream function equation two methods, Biharmonic Driver and Newton-Raphson, are considered. Results with respect to accuracy and rate of convergence demonstrate the reliability of the procedure for low Rayleigh number flows.

Flow aerodynamics modeling of an MHD swirl combustor - Calculations and experimental verification

Abstract: This paper describes a computer code for calculating the flow dynamics of constant density flow in the second stage trumpet shaped nozzle section of a two stage MHD swirl combustor for application to a disk generator. The primitive pressure-velocity variable, finite difference computer code has been developed to allow the computation of inert nonreacting turbulent swirling flows in an axisymmetric MHD model swirl combustor. The method and program involve a staggered grid system for axial and radial velocities, and a line relaxation technique for efficient solution of the equations. Turbulence simulation is by way of a two-equation Kappa-epsilon model. The code produces as output the flowfield map of the nondimensional stream function, axial, and swirl velocity. Good agreement was obtained between the theoretical predictions and the qualitative experimental results. The best seed injector location for uniform seed distribution at combustor exit is with injector located centrally on the combustor axis at entrance to the second stage combustor.

On the use of several compact methods for the study of unsteady incompressible viscous flow for outer problems (II)

Abstract: The main features of the methods are as follows. The Poisson equation is solved by an optimized A.D.I. method. The "Mehrstellen" discretization is defer-corrected so that sixth order accuracy is attainable on 4. The vorticity equation can be time dif-ferenced either with the help of a Beam & Warming two-step, quasi one leg method which is At2-accurate and A-stable or with a more common A.D.I. method ( Peaceman-Rachford). Both methods allow the splitting of the space operator so that unidimensional problems are solved sequentially.