Numerical Solutions of the Euler Equations for Steady Flow Problems (Albrecht Eberle, Arthur Rizzi, and Ernst Heinrich Hirschel)

Using a new approximate method, transient course of the local and mean diffusion fluxes following a step concentration change on the wall has been obtained for a broad class of steady flow problems.

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Transcritical two-layer flow over topography

Journal of Fluid Mechanics ◽

10.1017/s0022112087001101 ◽

1987 ◽

Vol 178 ◽

pp. 31-52 ◽

Cited By ~ 97

Author(s):

W. K. Melville ◽

Karl R. Helfrich

Keyword(s):

Steady Flow ◽

Solitary Waves ◽

Numerical Solutions ◽

Kdv Equation ◽

Single Layer ◽

Cubic Nonlinearity ◽

Arbitrary Length ◽

Weakly Nonlinear ◽

Flow Over Topography ◽

Layer Flow

The evolution of weakly-nonlinear two-layer flow over topography is considered. The governing equations are formulated to consider the effects of quadratic and cubic nonlinearity in the transcritical regime of the internal mode. In the absence of cubic nonlinearity an inhomogeneous Korteweg-de Vries equation describes the interfacial displacement. Numerical solutions of this equation exhibit undular bores or sequences of Boussinesq solitary waves upstream in a transcritical regime. For sufficiently large supercritical Froude numbers, a locally steady flow is attained over the topography. In that regime in which both quadratic and cubic nonlinearity are comparable, the evolution of the interface is described by an inhomogeneous extended Kortewegde Vries (EKdV) equation. This equation displays undular bores upstream in a subcritical regime, but monotonic bores in a transcritical regime. The monotonic bores are solitary wave solutions of the corresponding homogeneous EKdV equation. Again, locally steady flow is attained for sufficiently large supercritical Froude numbers. The predictions of the numerical solutions are compared with laboratory experiments which show good agreement with the solutions of the forced EKdV equation for some range of parameters. It is shown that a recent result of Miles (1986), which predicts an unsteady transcritical regime for single-layer flows, may readily be extended to two-layer flows (described by the forced KdV equation) and is in agreement with the results presented here.Numerical experiments exploiting the symmetry of the homogeneous EKdV equation show that solitary waves of fixed amplitude but arbitrary length may be generated in systems described by the inhomogeneous EKdV equation.

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Solution of incompressible steady flow problems in closed conduits using finite elements and a structural analogy

Engineering Computations ◽

10.1108/eb023735 ◽

1988 ◽

Vol 5 (2) ◽

pp. 165-168

Author(s):

Arturo O. Cifuentes

Keyword(s):

Finite Elements ◽

Steady Flow ◽

Flow Problems ◽

Structural Analogy

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Oscillatory flow past a circular cylinder in a rotating frame

Journal of Fluid Mechanics ◽

10.1017/s0022112097006204 ◽

1997 ◽

Vol 345 ◽

pp. 101-131

Author(s):

M. D. KUNKA ◽

M. R. FOSTER

Keyword(s):

Circular Cylinder ◽

Steady Flow ◽

Stagnation Point ◽

Oscillatory Flow ◽

Numerical Solutions ◽

Initial Conditions ◽

Temporal Integration ◽

Oncoming Flow ◽

Flow Past ◽

Rear Stagnation Point

Because of the importance of oscillatory components in the oncoming flow at certain oceanic topographic features, we investigate the oscillatory flow past a circular cylinder in an homogeneous rotating fluid. When the oncoming flow is non-reversing, and for relatively low-frequency oscillations, the modifications to the equivalent steady flow arise principally in the ‘quarter layer’ on the surface of the cylinder. An incipient-separation criterion is found as a limitation on the magnitude of the Rossby number, as in the steady-flow case. We present exact solutions for a number of asymptotic cases, at both large frequency and small nonlinearity. We also report numerical solutions of the nonlinear quarter-layer equation for a range of parameters, obtained by a temporal integration. Near the rear stagnation point of the cylinder, we find a generalized velocity ‘plateau’ similar to that of the steady-flow problem, in which all harmonics of the free-stream oscillation may be present. Further, we determine that, for certain initial conditions, the boundary-layer flow develops a finite-time singularity in the neighbourhood of the rear stagnation point.

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An Approximate Method for Transient Radial Flow

Society of Petroleum Engineers Journal ◽

10.2118/156-pa ◽

1962 ◽

Vol 2 (03) ◽

pp. 225-256 ◽

Cited By ~ 2

Author(s):

G. Rowan ◽

M.W. Clegg

Keyword(s):

Infinite Series ◽

Analytical Solutions ◽

Radial Flow ◽

Numerical Solutions ◽

Basic Theory ◽

Variable Pressure ◽

Flow Problems ◽

Disturbed Zone ◽

Approximate Analytical Solutions ◽

Wide Range

Abstract The basic equations for the flow of gases, compressible liquids and incompressible liquids are derived and the full implications of linearising then discussed. Approximate solutions of these equations are obtained by introducing the concept of a disturbed zone around the well, which expands outwards into the reservoir as fluid is produced. Many important and well-established results are deduced in terms of simple functions rather than the infinite series, or numerical solutions normally associated with these problems. The wide range of application of this approach to transient radial flow problems is illustrated with many examples including; gravity drainage of depletion-type reservoirs; multiple well systems; well interference. Introduction A large number of problems concerning the flow of fluids in oil reservoirs have been solved by both analytical and numerical methods but in almost all cases these solutions have some disadvantages - the analytical ones usually involve rather complex functions (infinite series or infinite integrals) which are difficult to handle, and the numerical ones tend to mask the physical principles underlying the problem. It would seem appropriate, therefore, to try to find approximate analytical solutions to these problems without introducing any further appreciable errors, so that the physical nature of the problem is retained and solutions of comparable accuracy are obtained. One class of problems will be considered in this paper, namely, transient radial flow problems, and it will be shown that approximate analytical solutions of the equations governing radial flow can be obtained, and that these solutions yield comparable results to those calculated numerically and those obtained from "exact" solutions. It will also be shown that the restrictions imposed upon the dependent variable (pressure) are just those which have to be assumed in deriving the usual diffusion-type equations. The method was originally suggested by Guseinov, whopostulated a disturbed zone in the reservoir, the radius of which increases with time, andreplaced the time derivatives in the basic differential equation by its mean value in the disturbed zone. In this paper it is proposed to review the basic theory leading to the equations governing the flow of homogeneous fluids in porous media and to consider the full implications of the approximation introduced in linearising them. The Guseinov-type approximation will then be applied to these equations and the solutions for the flow of compressible and incompressible fluids, and gases in bounded and infinite reservoirs obtained. As an example of the application of this type of approximation, solutions to such problems as production from stratified reservoirs, radial permeability discontinuities; multiple-well systems, and well interference will be given. These solutions agree with many other published results, and in some cases they may be extended to more complex problems without the computational difficulties experienced by other authors. THEORY In order to review the basic theory from a fairly general standpoint it is proposed to limit the idealising assumptions to the minimum necessary for analytical convenience. The assumptions to be made are the following:That the flow is irrotational.That the formation is of constant thickness.Darcy's Law is valid.The formation is saturated with a single homogeneous fluid. SPEJ P. 225^

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