Unsteady, viscous, circular flow Part 2. The cylinder of finite radius

The problem considered is that of the two-dimensional motion of the fluid in a cylinder of finite radius after the outer portion of the fluid is given an initial uniform velocity. The primary purpose of the investigation is the study of the changes in the energy distribution in the fluid as the initial motion decays. The appropriate flow equations are developed and then approximated by finite-difference equations. Numerical solutions of these equations are presented, and the energy-transfer processes are discussed in some detail. During the early stages of the flow, it is found that the spatial distribution of energy depends strongly on the Prandtl number. During the later stages, however, there is a net outward flow of energy for the case of a liquid and a net inward flow for a gas.

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A rapidly convergent procedure for solving the equations of subsonic potential flow. I. Numerical solutions

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1960.0056 ◽

1960 ◽

Vol 255 (1280) ◽

pp. 101-113 ◽

Cited By ~ 1

Keyword(s):

Potential Flow ◽

Numerical Solutions ◽

Initial Conditions ◽

Local Density ◽

Relaxation Method ◽

Two Dimensional ◽

Approximation Solution ◽

Flow Equations ◽

Bernoulli’S Equation ◽

First Approximation

The subsonic potential flow equations for a perfect gas are transformed by means of dependent variables s = ( ρ / ρ 0 ) n q/ a 0 and σ = 1/2 In ( ρ 0 / ρ ), where q is the local velocity, ρ and a the local density and speed of sound, and the suffix 0 indicates stagnation conditions, n is a parameter which is to be chosen to optimize the approximations. Bernoulli’s equation then becomes a relation between s 2 and σ which is independent of initial conditions. A family of first-approximation solutions in terms of the incompressible solution is obtained on linearizing. It is shown that for two-dimensional flow, the choice n = 0∙5 gives results as accurate as those obtained with the Karman—Tsien solution. The exact equations are then transformed into the plane of the incompressible velocity potential and stream function and the first-approximation results substituted in the non linear terms. The resulting second-approximation equations can then be solved by a relaxation method and the error in this approximation estimated by carrying out the third-approximation solution. Results are given for a circular cylinder at a free-stream Mach number, M ∞ = 0∙4, and a sphere at M ∞ = 0∙5. The error in the velocity distribution is shown to be less than ±1 % in the two-dimensional case. A rough and ready compressibility rule is formulated for axisymmetric bodies, dependent on their thickness ratios.

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On non-local energy transfer via zonal flow in the Dimits shift

Journal of Plasma Physics ◽

10.1017/s0022377817000708 ◽

2017 ◽

Vol 83 (5) ◽

Cited By ~ 13

Author(s):

Denis A. St-Onge

Keyword(s):

Energy Transfer ◽

Numerical Solutions ◽

Zonal Flow ◽

Two Dimensional ◽

Transfer Energy ◽

Wave Interactions ◽

Local Mechanism ◽

Non Local ◽

Quasilinear Approximation ◽

Mode Truncation

The two-dimensional Terry–Horton equation is shown to exhibit the Dimits shift when suitably modified to capture both the nonlinear enhancement of zonal/drift-wave interactions and the existence of residual Rosenbluth–Hinton states. This phenomenon persists through numerous simplifications of the equation, including a quasilinear approximation as well as a four-mode truncation. It is shown that the use of an appropriate adiabatic electron response, for which the electrons are not affected by the flux-averaged potential, results in an $\boldsymbol{E}\times \boldsymbol{B}$ nonlinearity that can efficiently transfer energy non-locally to length scales of the order of the sound radius. The size of the shift for the nonlinear system is heuristically calculated and found to be in excellent agreement with numerical solutions. The existence of the Dimits shift for this system is then understood as an ability of the unstable primary modes to efficiently couple to stable modes at smaller scales, and the shift ends when these stable modes eventually destabilize as the density gradient is increased. This non-local mechanism of energy transfer is argued to be generically important even for more physically complete systems.

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The Use of Von Karman’S Similarity Hypothesis in Turbulent Fluid Flow Computation

ASME 1992 International Computers in Engineering Conference: Volume 1 — Artificial Intelligence; Expert Systems; CAD/CAM/CAE; Computers in Fluid Mechanics/Thermal Systems ◽

10.1115/cie1992-0071 ◽

1992 ◽

Author(s):

David J. Foster

Keyword(s):

Turbulent Mixing ◽

Numerical Solutions ◽

Separated Flow ◽

Mixing Length ◽

Two Dimensional ◽

Turbulent Fluid Flow ◽

Flow Equations ◽

Similarity Hypothesis ◽

Step Flow ◽

Type Flow

Abstract Von Karman’s similarity hypothesis for the turbulent mixing length in boundary layer type flow is extrapolated to a plausable two dimensional expression. The corresponding incompressible turbulent flow equations are developed in terms of the transient vorticity transfer and stream function equations. Numerical solutions for the separated flow behind a step and in a rectangular cavity were obtained and the results are presented pictorally. The streamlines computed for the step flow solution compare favorably with those calculated from experimental measurements at a similar Reynolds Number.

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