Taylor vortices between two concentric rotating spheres

1982 ◽  
Vol 119 ◽  
pp. 1-25 ◽  
Author(s):  
Fritz Bartels
Keyword(s):  
Reynolds Number ◽  
Finite Difference ◽  
Flow Field ◽  
Critical Reynolds Number ◽  
Stokes Equations ◽  
Initial Conditions ◽  
Finite Difference Methods ◽  
Taylor Vortices ◽  
Concentric Spheres ◽  
Inner Sphere

The laminar viscous flow in the gap between two concentric spheres is investigated for a rotating inner sphere. The solution is obtained by solving the Navier-Stokes equations by means of finite-difference techniques, where the equations are restricted to axially symmetric flows. The flow field is hydrodynamically unstable above a critical Reynolds number. This investigation indicates that the critical Reynolds number beyond which Taylor vortices appear is slightly higher in a spherical gap than for the flow between concentric cylinders. The formation of Taylor vortices could be observed only for small gap widths s ≤ 0·17. The final state of the flow field depends on the initial conditions and the acceleration of the inner sphere. Steady and unsteady flow modes are predicted for various Reynolds numbers and gap widths. The results are in agreement with experiment if certain accuracy conditions of the finite-difference methods are satisfied. It is seen that the equatorial symmetry is of great importance for the development of the Taylor vortices in the gap.

A Parametric Study of Turbulent Flow Past a Circular Cylinder Using Large Eddy Simulation

2015 ◽  
Vol 137 (9) ◽  
Author(s):  
W. Sidebottom ◽  
A. Ooi ◽  
D. Jones
Keyword(s):  
Reynolds Number ◽  
Finite Difference ◽  
Circular Cylinder ◽  
Stokes Equations ◽  
Finite Difference Methods ◽  
Large Eddy Simulations ◽  
Near Wake ◽  
Flow Past ◽  
Large Eddy ◽  
Difference Methods

Flow over a circular cylinder at a Reynolds number of 3900 is investigated using large eddy simulations (LES) to assess the affect of four numerical parameters on the resulting flow-field. These parameters are subgrid scale (SGS) turbulence models, wall models, discretization of the advective terms in the governing equations, and grid resolution. A finite volume method is employed to solve the incompressible Navier–Stokes equations (NSE) on a structured grid. Results are compared to the experiments of Ong and Wallace (1996, “The Velocity Field of the Turbulent Very Near Wake of a Circular Cylinder,” Exp. Fluids, 20(6), pp. 441–453) and Lourenco and Shih (1993, “Characteristics of the Plane Turbulent Near Wake of a Circular Cylinder: A Particle Image Velocimetry Study,” private communication (taken from Ref. [2]); and the numerical results of Beaudan and Moin (1994, “Numerical Experiments on the Flow Past a Circular Cylinder at Sub-Critical Reynolds Number,” Technical Report No. TF-62), Kravchenko and Moin (2000, “Numerical Studies of Flow Over a Circular Cylinder at ReD = 3900,” Phys. Fluids, 12(2), pp. 403–417), and Breuer (1998, “Numerical and Modelling Influences on Large Eddy Simulations for the Flow Past a Circular Cylinder,” Int. J. Heat Fluid Flow, 19(5), pp. 512–521). It is concluded that the effect of the SGS models is not significant; results with and without a wall model are inconsistent; nondissipative discretization schemes, such as central finite difference methods, are preferred over dissipative methods, such as upwind finite difference methods; and it is necessary to properly resolve the boundary layer in the vicinity of the cylinder in order to accurately model the complex flow phenomena in the cylinder wake. These conclusions are based on the analysis of bulk flow parameters and the distribution of mean and fluctuating quantities throughout the domain. In general, results show good agreement with the experimental and numerical data used for comparison.

An Analysis of Laminar Forced Convection Between Concentric Spheres

1976 ◽  
Vol 98 (4) ◽  
pp. 601-608 ◽  
Author(s):  
K. N. Astill
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Temperature Distribution ◽  
Prandtl Number ◽  
Stokes Equations ◽  
Initial Conditions ◽  
Radius Ratio ◽  
Flow And Heat Transfer ◽  
Concentric Spheres ◽  
Inner Surface

A numerical solution for predicting the behavior of laminar flow and heat transfer between concentric spheres is developed. Axial symmetry is assumed. The Navier-Stokes equations and energy equation are simplified to parabolic form and solved using finite-difference methods. Hydrodynamic and energy equations are uncoupled, which allows the hydrodynamic problem to be solved independently of the heat-transfer problem. Velocity and temperature are calculated in terms of the two spatial coordinates. Solutions depend on radius ratio of the concentric spheres, Reynolds number of the flow, Prandtl number, initial conditions of temperature and velocity, temperature distribution along the spherical surfaces, and azimuthal position of the start of the flow. The effect on flow and heat transfer of these variables, except surface temperature distribution, is evaluated. While the computer solution is not restricted to isothermal spheres, this is the only case treated. Velocity profiles, pressure distribution, flow losses, and heat-transfer coefficients are determined for a variety of situations. Local and average Nusselt numbers are computed, and a correlation is developed for mean Nusselt number on the inner surface as a function of Reynolds number, Prandtl number, and radius ratio. Flow separation is predicted by the analysis. Separation is a function of Reynolds number, radius ratio, and azimuthal location of the initial state. Separation was observed at the outer surface as well as from the inner surface under some conditions. In cases where separation occurred, the solution was valid only to the point of separation.

scholarly journals Calculating Rotordynamic Coefficients of Seals by Finite-Difference Techniques

1987 ◽  
Vol 109 (3) ◽  
pp. 388-394 ◽  
Author(s):  
F. J. Dietzen ◽  
R. Nordmann
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Flow Field ◽  
Pressure Distribution ◽  
Difference Method ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Rotordynamic Coefficients ◽  
Finite Difference Techniques

For modelling the turbulent flow in a seal the Navier-Stokes equations in connection with a turbulence model (k-ε-model) are solved by a finite-difference method. A motion of the shaft around the centered position is assumed. After calculating the corresponding flow field and the pressure distribution, the rotordynamic coefficients of the seal can be determined. These coefficients are compared with results obtained by using the bulk flow theory of Childs [1] and with experimental results.

ON THE RATE OF CONVERGENCE OF APPROXIMATION SCHEMES FOR BELLMAN EQUATIONS ASSOCIATED WITH OPTIMAL STOPPING TIME PROBLEMS

2003 ◽  
Vol 13 (05) ◽  
pp. 613-644 ◽  
Author(s):  
ESPEN ROBSTAD JAKOBSEN
Keyword(s):  
Finite Difference ◽  
Rate Of Convergence ◽  
Optimal Stopping ◽  
Initial Conditions ◽  
Finite Difference Methods ◽  
Approximation Schemes ◽  
Dynamic Programming Principle ◽  
Bellman Equations ◽  
Difficult Time ◽  
Difference Methods

We provide estimates on the rate of convergence for approximation schemes for Bellman equations associated with optimal stopping of controlled diffusion processes. These results extend (and slightly improve) the recent results by Barles & Jakobsen to the more difficult time-dependent case. The added difficulties are due to the presence of boundary conditions (initial conditions!) and the new structure of the equation which is now a parabolic variational inequality. The method presented is purely analytic and rather general and is based on earlier work by Krylov and Barles & Jakobsen. As applications we consider so-called control schemes based on the dynamic programming principle and finite difference methods (though not in the most general case). In the optimal stopping case these methods are similar to the Brennan & Schwartz scheme. A simple observation allows us to obtain the optimal rate 1/2 for the finite difference methods, and this is an improvement over previous results by Krylov and Barles & Jakobsen. Finally, we present an idea that allows us to improve all the above-mentioned results in the linear case. In particular, we are able to handle finite difference methods with variable diffusion coefficients without the reduction of order of convergence observed by Krylov in the nonlinear case.

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