Showing papers on "Stream function published in 1969"

Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow

Abstract: A method of numerically integrating the Navier-Stokes equations for certain three-dimensional incompressible flows is described. The technique is presented through application to the particular problem of describing thermal convection in a rotating annulus. The equations, in cylindrical polar co-ordinate form, are integrated with respect to time by a marching process, together with the solving of a Poisson equation for the pressure. A suitable form of the finite difference equations gives a computationally-stable long-term integration with reasonably faithful representation of the spatial and temporal characteristics of the flow.Trigonometric interpolation techniques provide accurate (discretely exact) solutions to the Poisson equation. By using an auxiliary algorithm for rapid evaluation of trigonometric transforms, the proportion of computation needed to solve the Poisson equation can be reduced to less than 25% of the total time needed to’ advance one time step. Computing on a UNIVAC 1108 machine, the flow can be advanced one time-step in 2 sec for a 14 × 14 × 14 grid upward to 96 sec for a 60 × 34 × 34 grid.As an example of the method, some features of a solution for steady wave flow in annulus convection are presented. The resemblance of this flow to the classical Eady wave is noted.

Numerical Studies of Viscous Flow around Circular Cylinders

Abstract: Numerical solutions have been obtained for incompressible Newtonian flow around a circular cylinder for Reynolds numbers of 1, 2, 4, 10, 15, 30, 50, 100, and 500 The results are presented in the form of vorticity and stream function distributions The solutions allow detailed characterization of the vortex rings Drag coefficients, pressure distributions, and vortex dimensions are compared with experimental data and with available theoretical predictions for Reynolds numbers up to 50 The agreement with both experimental data and previous theoretical solutions is considered to be excellent

The generation of continental shelf waves

Abstract: Assuming zero divergence, the equations of forced long waves in a uniformly rotating, homogeneous ocean are reduced to a single partial differential equation for the stream function. A shelf of exponential slope between a rigid continent and a sea of uniform depth is taken as a model, and certain other assumptions are made which appear physically reasonable. Calculations made on the basis of this simplified theory are in good qualitative agreement with observations of shelf waves, indicating that these waves are generated by the stress of the longshore component of the geostrophic wind.

Plane Subcritical Flow Past a Lifting Aerofoil

Abstract: A method for solving the equations of steady two-dimensional inviscid isentropic irrotational flow past an obstacle is presented. The exterior of the obstacle is mapped on the interior of a circle, any sharp edge being taken into consideration, so that the far flow with circulation becomes an oriented dipole and vortex. The stream function is introduced and these two singularities are removed to leave a modified stream function which is finite everywhere; the differential equation which it satisfies is represented by a difference scheme on an annular mesh inside the circle. Provided that the flow is everywhere subsonic (subcritical problem), the iterative method of solution of this and Bernoulli’s equation is convergent. Results illustrating the method are presented for the 20% ellipse and the 10% R. A. E. 101 section, at zero and non-zero incidence.

Buoyancy Effects in Forced-Convection Flow and Heat Transfer

Abstract: An analysis is made for the effects of buoyancy forces in laminar forced-convection on a vertical flat plate. In general, since it is difficult to obtain an exact solution, an approximate similarity solution is given by introducing the assumption that a role of Gr/Re2 in stream function, [numerical formula] is negligibly small as compared with η. The validity of this assumption is examined by comparing with the experimental data of Kliegel. Numerical calculations are performed for both the constant fluid properties and the variable fluid properties, and the results are compared with Szewczyk's solution previously obtained by perturbation analysis. The velocity and temperature distribution, the friction factor and heat transfer coefficient are given for various values of the parameter Gr/Re2.

Numerical Study of Natural Convection in a Vertical Rectangular Enclosure

Abstract: The steady‐state natural convection of a fluid in a vertical rectangular cavity with isothermal side walls and adiabatic top and bottom walls is considered for 6 × 104 ≤ Ra ≤ 3.6 × 105 with Pr = 1, 6, 2000, and at an aspect ratio of 5.0. The governing nonlinear fourth‐order equation in the stream function and the coupled second‐order energy equation are solved numerically by a stable and rapidly converging iteration scheme. The computed flow distributions, including the appearance of multicellular flows, the temperature profiles, and the heat transfer predictions compare favorably with experimental results and with other numerical studies.

Creeping flow past a fluid globule when a trace of surfactant is present

Abstract: The effects of a trace quantity of a surface-active agent on creeping flow past a bubble or droplet are investigated. The equations describing mass and momentum transfer are simultaneously solved by a perturbation technique, consistent with the jump mass and momentum balances at the phase interface. The stream function for the velocity distribution is evaluated as an infinite series of spherical harmonics. Galerkin's method, which reduces the partial differential equation of continuity to a set of ordinary differential equations, is used to evaluate the concentration distribution of surfactant. A sample calculation is carried out for relative motion between an air bubble and an infinite body of water which contains a trace of isoamyl alcohol. The relative velocity of the water at an infinite distance from the bubble is found to be highly sensitive to small changes in surfactant concentration from zero, although the bubble varies imperceptibly from a spherical shape.

Numerical Integration of the Navier‐Stokes Equations for Steady Two‐Dimensional Flow

Abstract: A summary is given of a numerical method of solving steady‐flow problems for the symmetrical motion of a viscous fluid past a fixed cylinder in two dimensions. The stream function and vorticity are used as dependent variables. Two features which have not previously been mentioned are considered. The first is concerned with the convergence of the iterative procedure which is used to obtain successive approximations to the stream function and vorticity. The second is the method of approximating the equation which governs the vorticity by finite‐difference schemes. The latter is discussed in the light of recent suggestions that approximations whose accuracy is demonstrably lower than that of the usual central‐difference scheme should be used in order to secure a matrix which is diagonally dominant. It is shown that the diagonally dominant system can be used while still retaining the higher accuracy of central differences. The results of some illustrative calculations are given for flow past a finite flat plate for Reynolds numbers up to 200, and for flow past elliptic cylinders of various ratios of major to minor axes at Reynolds number 40.

Two and Three-Dimensional Flow using Finite Elements *

Abstract: The method of finite elements is in certain cases advantageous when dealing with flow problems in a finite domain. This is particularly so when attempting to include subcritical compressibility effects. In the present note we first consider the two-dimensional flow using TRIC and TRIM-like triangular elements in conjunction with the concept of the stream function, which is assigned to the nodal points of the elements. The application of the stream function allows a direct and exact satisfaction of the boundary conditions. Strictly the elements in question could also be used in conjunction with the potential function but the observance of the boundary conditions is then cumbersome.

The lee-wave régime for a slender body in a rotating flow

Abstract: The uniform motion of a closed, axisymmetric body along the axis of an unbounded, rotating, inviscid, incompressible fluid is considered on Long's hypotheses that: the flow is steady; the flow is uniform far upstream of the body; the inertial waves excited by the body cannot propagate upstream. The appropriate similarity parameters are k, an inverse Rossby number based on the body length, and δ, the slenderness ratio of the body. It is conjectured that an upper bound to the parametric regime in which the solution implied by Long's hypotheses remains valid, say kδ ≡ k < kc, is determined by the first occurrence, with increasing k, of a local reversal of the flow.A general solution for the stream function is established in terms of an assumed distribution of dipoles along the axis of the body. The disturbance upstream of the body is found to be proportional to the product of k2 and the dipole moment (total dipole strength) and to fall off only as the inverse distance, as compared with the inverse cube of the distance for a potential flow. The corresponding wave drag is found to depend on the power spectrum of the dipole distribution in the axial wave-number interval (0, k) and to be a monotonically decreasing function of the axial velocity for fixed angular velocity. Asymptotic solutions for prescribed bodies are established in the following limits: (i) K → 0 with δ fixed; (ii) δ → 0 with k fixed; (iii) k → ∞ with kδ fixed. Both the upstream disturbance and the wave drag in the limit (i) depend essentially on the dipole moment of the body with respect to a uniform, potential flow. The limit (ii) is analogous to conventional slender-body theory and yields a dipole density that is proportional to the cross-sectional area of the body. The limit (iii) leads to a singular integral equation that is solved to determine kc and the dipole moment for a slender body.The results are applied to a sphere and a slender ellipsoid. The upstream axial velocity and the drag coefficient based on Stewartson's results for a sphere are found to differ significantly from Maxworthy's (1969) measurements, presumably in consequence of viscous separation effects. Maxworthy's measured values of upstream axial velocity are found to agree with the theoretical values for an equivalent ellipsoid, based on the sphere plus its upstream wake, for k [lsim ] kc.

Programs for Computation of Velocities and Streamlines on a Blade-to-Blade Surface of a Turbomachine

Abstract: This paper describes Fortran programs that give the solution to the two-dimensional, subsonic, nonviscous flow problem on a blade-to-blade surface of revolution of a turbomachine. Flow may be axial, radial, or mixed. There may be a change in stream channel thickness in the through-flow direction. Either single, tandem, or slotted blades may be handled as well as blade rows with splitter vanes. Also, small regions may be magnified to give more detail where desired, such as around a leading or trailing edge or through a slot. The method is based on a finite difference solution of the stream function equations. Numerical examples are shown to illustrate the type of blades which can be analyzed, and to show results which can be obtained. Results are compared with experimental data.Copyright © 1969 by ASME

Free-Surface Flow Through Heterogeneous Porous Media

Abstract: Methods are developed for obtaining finite difference solutions to steady-state, free-surface saturated seepage flows through nonhomogeneous porous media. The formulation considers the total fluid head and the stream function as the independent variables and the coordinates x and y as the dependent variables. In the plane defined by the total fluid head and the stream function free surfaces are straight lines and the region of seepage is rectangular if no surfaces of seepage are present. The variation in permeability of the porous media can be specified by continuous functions. The methods are applied to obtain solutions to problems of seepage from canals through nonhomogeneous porous media to a drained layer at some specified depth. Several representative flownets from these solutions are presented.

Numerical Studies of a Viscous Liquid past a Circular Cylinder

Abstract: In the study of the steady two‐dimensional flows past a circular cylinder in an infinite extent, the two sets of boundary conditions at infinity for the perturbed stream function and the vorticity, which are generally used, are examined numerically on the basis of Navier‐Stokes equations. It is confirmed that, for sufficiently small Reynolds number, if points far enough from the cylinder are taken as inner mesh points, then the boundary conditions have little influence on the results, and the drag coefficients approach the values calculated by Imai's formula rather than those calculated based on Oseen's theory.

Paper 27: Calculation of the Flow past Turbomachine Blades

Abstract: This paper considers the problem of determining the inviscid, subsonic compressible (or incompressible) blade-to-blade flow field through a turbomachine blade row, and presents two computer solutions, known as the ‘matrix’ and ‘streamline curvature’ methods, for solving this elliptic problem. In the former method the equations of motion are formulated in terms of a stream function and the numerical solution is based on finite difference approximations; in the latter, the equations of motion are solved in conjunction with the equation of continuity by an iterative process starting from a guess for the streamline pattern. Solutions for several blade profiles, giving flow patterns and blade surface velocity distributions, are included.

Time‐Dependent Viscous Incompressible Flow past a Finite Flat Plate

Abstract: : The flow is started suddenly with a constant velocity and is parallel with the plate. This simple flow configuration is used to compare the accuracy and the speed of various numerical methods and to examine the effect of the far field boundary conditions and the variable mesh size on the flow. The governing Navier-Stokes equations are expressed in terms of vorticity and stream function. The (parabolic) vorticity equation is approximated by two finite difference methods. The (elliptic) stream function equation is approximated by two relaxation schemes. The details of the flow field around the flat plate are computed for Reynolds number equal to 4 and 993. One of the interesting results is that the velocity at the outer edge of the boundary layer exceeds that of the free stream and the maximum velocity overshoot is 9.5% at Re = 4 and 5.4% at Re - 993. The implicit alternating direction method is two to three times faster than the explicit method. (Author)

A Theoretical and Experimental Study of Confined Vortex Flow

Abstract: The interaction of a vortex with a stationary surface was studied both theoretically and experimentally. The flow field examined was that produced by radially inward flow through a pair of concentric rotating porous cylinders that were perpendicular to, and in contact with, a stationary flat plane. The complete Navier-Stokes equations were solved over a range of tangential Reynolds numbers from 0–300 and a range of radial Reynolds numbers from 0 to −13, the minus sign indicating radially inward flow. In order to facilitate the solution, the original equations were recast in terms of a dimensionless stream function, vorticity, and third variable related to the tangential velocity. The general validity of the numerical technique was demonstrated by the agreement between the theoretical and experimental results. Examination of the numerical results over a wide range of parameters showed that the entire flow field is very sensitive to the amount of radial flow, especially at the transition from zero radial flow to some finite value.

Viscous Flow in a Bifurcate Channel

Abstract: The technique presented by Dean is applied to find the solution for a slow two-dimensional steady motion of liquid in an infinite channel. The channel is composed of two parallel infinite plates and a semi-infinite partition plate which is in the middle of the infinite plates. The fluid velosity is assumed to have a parabolic distribution at infinity and the direction perpendicular to the edge line of the partition plate. The stream function is first assumed and two sets of constants contained in it are then adjusted so that slip of the velosity on the boundaries becomes sufficiently small. From the approximate solution obtained, the properties of the solution in the neighbourhood of the leading edge of the partition plate and the pressure drop along the channel due to the partition are discussed.

Numerical Model for Wind‐Driven Circulation of the Oceans of the World

Abstract: A two‐dimensional linear steady‐state model of the ocean circulation of the world which includes the effects of the rotation of the earth, lateral and bottom friction, and a steady‐state wind stress is considered. The two‐dimensional equations of motion for mass transports are obtained by integration of the equations of motion with respect of depth and linearization. A stream function ψ for the mass transports is then introduced and a fourth‐order partial differential equation for ψ is obtained. The boundary conditions ψ = const.and ∂ψ/∂n = 0 are implemented at the land‐sea boundaries of the model, which correspond closely to actual boundaries except that small isolated islands are omitted and larger islands such as Australia are joined to the mainland. The numerical solution of the partial differential equation is achieved by approximating it by finite difference equations on the Mercator projection of the earth, and solving these equations by a combination of iterative and direct techniques. The finite difference equations involve values of ψ at certain land grid points near the boundaries, and at each stage of the iteration, values of ψ are assigned to these points in such a way that the boundary conditions are satisfied for the current approximation to ψ The results so far produced qualitatively agree very well with the known patterns of ocean circulation. Western boundary currents such as the Gulf Stream are prominent and smaller circulations such as the eddies off the coast of Antarctica are also present.

The Iterated Equation of Generalized Axially Symmetric Potential Theory

Abstract: Milne-Thomson's well-known circle theorem [1] gives the stream function for steady two-dimensional irrotational flow of a perfect fluid past a circular cylinder when the flow in the absense of the cylinder is known. Butler's sphere theorem [2] gives the corresponding result for axially symmetric irrotational flow of a perfect fluid past a sphere. Collins [3] has obtained a sphere theorem for axially symmetric Stokes flow of a viscous liquid which gives a stream function satisfying the appropriate viscous boundary conditions on the surface of a sphere when the stream function for irrotational flow in the absence of the sphere is known.

Numerical solutions of the time-dependent incompressible viscous flow over a disk or a sphere,

Abstract: : The Navier-Stokes equations for incompressible axisymmetric flow over a disk or a sphere are solved by means of finite difference approximation. The Reynolds number range covered is 1-1000. Time-dependent Stream function-Vorticity formulation in cylindrical and spherical polar coordinate systems is adopted. The surface of the disk and the sphere is conveniently represented by coordinate lines. Explicit central Dufort-Frankel differencing in time is employed. Second order accuracy conservative differencing is used for the spatial variables. For the disk case, vorticity and stream function are defined on mesh points, while the velocity components are defined at the midpoints of the mesh cells. The latter leads to difficulties with the implementation of boundary conditions. Therefore, in the sphere case all the dependent variables are defined on mesh points. Separate linearized criterions for diffusion and convective terms were used to guide the selection of an empirical relation between time and space increments of the entire Navier-Stokes equations. It was found that in the initial phase of the solution, especially at low Reynolds numbers, time steps considerably smaller than those suggested by the stability criterion must be used. The condition to be observed for 'stable' solutions appears to be that the fractional change of the vorticity at any point over a time step is small. (Author)

FORTRAN 4 program calculates velocities and streamlines in a tandem blade turbomachine

Abstract: Computer program gives blade-to-blade solution of the two-dimensional, subsonic, compressible, nonviscous flow problem for a circular or straight infinite cascade of tandem or slotted turbomachine blades. The method of solution is based on the stream function using iterative solution of nonlinear finite-difference equations.

On Axially Symmetric Magnetohydrostatic Fields

Abstract: 1 Introductiot Solutions of the magnetohydrostatic equations for a medium of large conductivity are of interest in connection with problems in cosmic physics In this paper a general class of axially symmetric solutions is found to satisfy a single partial differential equation of the second order containing two arbitrary functions From this equation a family of solutions depending on two parameters a, ,B and belonging to the force-free field class is constructed in which the toroidal field is in general a nonlinear function of the magnetic stream function, but the equation satisfied by the magnetic stream function can be reduced to the solution of a linear partial differential equation For ,B = 0, the solution reduces to the wellknown force-free field solution suitable for simply connected domains discussed by Chandrasekhar [1], [2] and Woltjer [3] The equipartition of energy theorem between the poloidal -and toroidal energies is shown to be valid for any simply connected region for which the normal component of the magnetic field vanishes on the boundary This result was established for the case of the sphere without the assumption of axial symmetry by Chandrasekhar and Kendall [4] For a = 0, a representation is found suitable for malultiply connected domains in which there is a tripartition of the total energy, the toroidal energy being twice the poloidal An explicit solution is determined for the case in which the meridian curve is a rectangle, and finally it is shown how the new class of solutions may be characterized by a variational principle

The approximate solution of the equation for the supersonic flow of a gas

Abstract: A new approximation of the Chaplygin function is proposed in this paper, one that is suitable at 1 < M < 3, making it possible to find analytical solutions for the equations of supersonic gas flow An explicit form of the Riemann function has been derived for the potential equation and the stream function, and the limit transitions to the Tricomi equation or to the Euler-Poisson-Darboux equation are examined