A Numerical Procedure for Computing Fields of Stream Function and Velocity Potential

1967 ◽  
Vol 6 (6) ◽  
pp. 994-1001 ◽  
Author(s):  
Robert L. Mancuso
Keyword(s):  
Stream Function ◽  
Velocity Potential ◽  
Numerical Procedure

Contracting ducts of finite length

1951 ◽  
Vol 2 (4) ◽  
pp. 254-271 ◽  
Author(s):  
L. G. Whitehead ◽  
L. Y. Wu ◽  
M. H. L. Waters
Keyword(s):  
Stream Function ◽  
Finite Length ◽  
Velocity Potential ◽  
Three Dimensional ◽  
Relaxation Method ◽  
Two Dimensional ◽  
Dimensional Flow ◽  
Hodograph Plane ◽  
Two Dimensional Flow ◽  
Working Section

SummmaryA method of design is given for wind tunnel contractions for two-dimensional flow and for flow with axial symmetry. The two-dimensional designs are based on a boundary chosen in the hodograph plane for which the flow is found by the method of images. The three-dimensional method uses the velocity potential and the stream function of the two-dimensional flow as independent variables and the equation for the three-dimensional stream function is solved approximately. The accuracy of the approximate method is checked by comparison with a solution obtained by Southwell's relaxation method.In both the two and the three-dimensional designs the curved wall is of finite length with parallel sections upstream and downstream. The effects of the parallel parts of the channel on the rise of pressure near the wall at the start of the contraction and on the velocity distribution across the working section can therefore be estimated.

The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics

1978 ◽  
Vol 360 (1703) ◽  
pp. 471-488 ◽  
Keyword(s):  
Stream Function ◽  
Gravity Waves ◽  
Velocity Potential ◽  
Travelling Waves ◽  
Normal Modes ◽  
Phase Speed ◽  
Finite Amplitude ◽  
Fundamental Wave ◽  
Wave Steepness ◽  
Horizontal Scale

In this paper we embark on a calculation of all the normal-mode perturbations of nonlinear, irrotational gravity waves as a function of the wave steepness. The method is to use as coordinates the stream-function and velocity potential in the steady, unperturbed wave (seen in a reference frame moving with the phase speed) together with the time t. The dependent quantities are the cartesian displacements and the perturbed stream function at the free surface. To begin we investigate the ‘superharmonics’, i.e. those perturbations having the same horizontal scale as the fundamental wave, or less. When the steepness of the fundamental is small, the normal modes take the form of travelling waves superposed on the basic nonlinear wave. As the steepness increases the frequency of each perturbation tends generally to be diminished. At a steepness ak ≈ 0.436 it appears that the two lowest modes coalesce and an instability arises. There is evidence that this critical steepness corresponds precisely with the steepness at which the phase velocity is a maximum, considered as a function of ak. The calculations are facilitated by the discovery of some new identities between the coefficients in Stokes’s expansion for waves of finite amplitude.

scholarly journals A Viscous Velocity Potential/Stream Function

2022 ◽  
Author(s):  
Taofiq O Amoloye
Keyword(s):  
Stream Function ◽  
Velocity Potential ◽  
Three Dimensional ◽  
Panel Method ◽  
Theoretical Development ◽  
Cylinder Surface ◽  
Study Results ◽  
Classical Potential Theory ◽  
Reynolds Number Dependence ◽  
Mathematics And Physics

Abstract The motion of fluids presents interesting phenomena including flow separation, wakes, turbulence etc. The physics of these are enshrined in the continuity equation and the NSE. Therefore, their studies are important in mathematics and physics. They also have engineering applications. These studies can either be carried out experimentally, numerically, or theoretically. Theoretical studies using classical potential theory (CPT) have some gaps when compared to experiments. The present publication is part of a series introducing refined potential (RPT) that bridges these gaps. It leverages experimental observations, physical deductions and the match between CPT and experimentally observed flows in the theoretical development. It analytically imitates the numerical source/vortex panel method to describe how wall bounded eddies in a three-dimensional cylinder crossflow are linked to the detached wake eddies. Unlike discrete and arbitrary vortices/sources on the cylinder surface whose strengths are numerically determined in the panel method, the vortices/sources/sinks in RPT are mutually concentric and continuously distributed on the cylinder surface. Their strengths are analytically determined from CPT using physical deductions starting from Reynolds number dependence. This study results in the incompressible Kwasu function which is a Eulerian velocity potential/stream function that captures vorticity, boundary layer, shed wake vortices, three-dimensional effects, and turbulence. This Eulerian Kwasu function also theorizes streaklines. The Lagrangian form of the function is further exploited to obtain flow pathlines.

scholarly journals The line of action of the resultant pressure in discontinuous fluid motion

1922 ◽  
Vol 102 (716) ◽  
pp. 361-372
Keyword(s):  
Stream Function ◽  
Complex Variable ◽  
Velocity Potential ◽  
Fluid Motion ◽  
Two Dimensional ◽  
Fluid Stream

1. The problem of any barrier in a fluid stream is best attacked by the method due to Levi-Civita, of which useful accounts, with extensions, are given by Cisotti and Brillouin. The resultant pressure for any barrier has been given in terms of the constants defining the barrier; but the calculations required to find the line of action of this pressure have not been carried out. It is our object to supply this deficiency here. The motion is two-dimensional. Let the complex variable z (≡ x + iy ) define position in any plane perpendicular to the generators of the barrier, the x axis being parallel to the direction of the stream at infinity. We define u = ∂ϕ/∂ x = ∂ψ/∂ y , v = ∂ϕ/∂ y = -∂ψ/∂ x , where u , v are the velocity components, and ϕ, ψ are the velocity potential and stream function respectively. Let w ≡ ϕ + iψ and define ζ, Ω, r, θ so that ζ ≡ re θ = dz / dw ; Ω = log ζ ≡ log r + iθ . (1)

Method of Fundamental Solutions for Stokes Problems by the Pressure-Stream Function Formulation

2008 ◽  
Vol 24 (2) ◽  
pp. 137-144 ◽  
Author(s):  
D. L. Young ◽  
J. T. Wu ◽  
C. L. Chiu
Keyword(s):  
Stream Function ◽  
Numerical Scheme ◽  
Numerical Procedure ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Circular Cavity ◽  
Stokes Flows ◽  
Function Formulation ◽  
Stream Function Formulation ◽  
Harmonic Equation

ABSTRACTThe main purpose of this paper is to investigate the pressure-stream function formulation to solve 2D and 3D Stokes flows by the meshless numerical scheme of the method of fundamental solutions (MFS). The MFS can be regarded as a truly scattered, grid-free (or meshless) and non-singular numerical scheme. By the proposed algorithm, the stream function is governed by the bi-harmonic equation while the pressure is governed by the Laplace equation. The velocity field is then obtained by the curl of the stream function for 2D flows and curl of the vector stream function for 3D flows. We can simultaneously solve the pressure, velocity, vorticity, stream function and traction forces fields. Furthermore during the present numerical procedure no pressure boundary condition is needed which is a tedious and forbidden task. The developed algorithm is used to test several numerical experiments for the benchmark examples, including (1) the driven circular cavity, (2) the circular cavity with eccentric rotating cylinder, (3) the square cavity with traction boundary conditions and (4) the uniform flow past a sphere. The results compare very well with the solutions obtained by analytical or other numerical methods such as finite element method (FEM). It is found that the meshless MFS will give a simpler and more efficient and accurate solutions to the Stokes flows investigated in this study.

A note on Stokes's stream function for motion with a spherical boundary

1953 ◽  
Vol 49 (1) ◽  
pp. 169-174 ◽  
Author(s):  
S. F. J. Butler
Keyword(s):  
Circular Cylinder ◽  
Stream Function ◽  
Complex Potential ◽  
Velocity Potential ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Irrotational Flow ◽  
Circle Theorem ◽  
Spherical Boundary

The circle theorem of Milne-Thomson(1) connecting the complex potential in a two-dimensional irrotational flow about a circular cylinder with that of the flow when the cylinder is absent has a three-dimensional counterpart in the result due to Weiss (3) for the perturbed velocity potential in an unlimited irrotational flow when the rigid spherical boundary r = a is inserted.

scholarly journals A THEORY FOR WAVES OF FINITE HEIGHT

2011 ◽  
Vol 1 (7) ◽  
pp. 9
Author(s):  
Charles L. Bretschneider
Keyword(s):  
Stream Function ◽  
Particle Velocity ◽  
Velocity Potential ◽  
Wave Theory ◽  
Stokes Wave ◽  
Particle Position ◽  
Exact Theory ◽  
Bernoulli’S Equation ◽  
Finite Height ◽  
Expanded Form

A theory for waves of finite height, presented in this paper, is an exact theory, to any order for which it is extended. The theory is represented by a summation liarmdnic series, each term of which is in an unexpanded form. The terms of the series when expanded result in an approximation of the exact theory, and this approximation is identical to Stokes' wave theory extended to the same order. The theory represents irrotational - divergenceless flow. The procedure is to select the form of equations for the coordinates of the particles in anticipation of later operations to be performed in the evaluation of the coefficients of the series. The horizontal and vertical components of these coordinates are given respectively by the following; (equations given). From the above equations it is possible to deduce the expressions for velocity potential and stream function. The horizontal and vertical components of particle velocity are obtained by differentiating £ and ^with respect to time. Along the free surface z -1?!a 0 and z = Vs and all expressions reduce to simple forms, which in turn saves considerable work in the evaluation of the coefficients. The coefficients are evaluated by use of Bernoulli's equation. The final form of the solution is given by two sets of equations. One set of equations (same as above) is used to compute the particle position and the second set (the first time derivatives of the above) is used to compute the components of particle velocity at the particle position. That is, the particles and velocities are referenced to the lines of the stream function and the velocity potential. Expanding the two sets of equations, by approximation methods, results in one set of equation for computing particle velocity and no equations are required for the particle position.The unexpanded form requiring two sets of equations, being an exact solution, is more accurate theoretically, than the Stokes or the expanded form to the same order. Coefficients have been formulated for all terms of the order one to five for both the unexpanded and the expanded form of the theory, and are presented in tabular form as functions of d/L, as consecutive equations.

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