Upwelling of a stratified fluid in a rotating annulus: steady state. Part 2. Numerical solutions

1973 ◽  
Vol 59 (2) ◽  
pp. 337-368 ◽  
Author(s):  
J. S. Allen
Keyword(s):  
Steady State ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Rotational Frequency ◽  
Navier Stokes ◽  
Rossby Number ◽  
Navier Stokes Equations ◽  
Finite Difference Approximations ◽  
Side Walls ◽  
Steady State Solutions

Numerical solutions of finite-difference approximations to the Navier–Stokes equations have been obtained for the axisymmetric motion of a Boussinesq liquid in a rigidly bounded rotating annulus. For most of the cases studied, a temperature difference is maintained between the top and bottom surfaces such that essentially a basic stable density stratification is imposed on the fluid. The side walls are thermally insulated and the motion is driven by a differential rotation of the top surface. Approximate steady-state solutions are obtained for various values of the Rossby number ε and the stratification parameter S = N2/Ω2, where N is the Brunt–Väisälä frequency and Ω is the rotational frequency. The changes in the flow field with the variation of these parameters is studied. Particular attention is given to an investigation of the meridional, or up welling, circulation and its dependence on the stratification parameter. The effects on the flow of different boundary conditions, such as an applied stress driving, specified temperature at the side walls and an applied heat flux at the top and bottom surfaces, are also investigated.

Complex dynamics in a stratified lid-driven square cavity flow

2018 ◽  
Vol 855 ◽  
pp. 43-66 ◽  
Author(s):  
Ke Wu ◽  
Bruno D. Welfert ◽  
Juan M. Lopez
Keyword(s):  
Stokes Equations ◽  
Complex Dynamics ◽  
Boussinesq Approximation ◽  
Navier Stokes ◽  
Square Cavity ◽  
Navier Stokes Equations ◽  
Stable Temperature ◽  
Wide Range ◽  
Side Walls ◽  
Steady State Solutions

The dynamic response to shear of a fluid-filled square cavity with stable temperature stratification is investigated numerically. The shear is imposed by the constant translation of the top lid, and is quantified by the associated Reynolds number. The stratification, quantified by a Richardson number, is imposed by maintaining the temperature of the top lid at a higher constant temperature than that of the bottom, and the side walls are insulating. The Navier–Stokes equations under the Boussinesq approximation are solved, using a pseudospectral approximation, over a wide range of Reynolds and Richardson numbers. Particular attention is paid to the dynamical mechanisms associated with the onset of instability of steady state solutions, and to the complex and rich dynamics occurring beyond.

The motion generated by a rising particle in a rotating fluid – numerical solutions. Part 2. The long container case

2002 ◽  
Vol 454 ◽  
pp. 345-364 ◽  
Author(s):  
E. MINKOV ◽  
M. UNGARISH ◽  
M. ISRAELI
Keyword(s):  
Steady State ◽  
Flow Field ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Rotating Fluid ◽  
Order Unity ◽  
Navier Stokes Equations ◽  
Theory And Experiments ◽  
Taylor Column

Numerical finite-difference results from the full axisymmetric incompressible Navier–Stokes equations are presented for the problem of the slow axial motion of a disk particle in an incompressible, rotating fluid in a long cylindrical container. The governing parameters are the Ekman number, E = ν*/(Ω*a*2), Rossby number, Ro = W*/(Ω*a*), and the dimensionless height of the container, 2H (the scaling length is the radius of the particle, a*; Ω* is the container angular velocity, W* is the particle axial velocity and ν* the kinematic viscosity). The study concerns the flow field for small values of E and Ro while HE is of order unity, and hence the appearance of a free Taylor column (slug) of fluid ‘trapped’ at the particle is expected. The numerical results are compared with predictions of previous analytical approximate studies. First, developed (quasi-steady-state) cases are considered. Excellent agreement with the exact linear (Ro = 0) solution of Ungarish & Vedensky (1995) is obtained when the computational Ro = 10−4. Next, the time-development for both an impulsive start and a start under a constant axial force is considered. A novel unexpected behaviour has been detected: the flow field first attains and maintains for a while the steady-state values of the unbounded configuration, and only afterwards adjusts to the bounded container steady state. Finally, the effects of the nonlinear momentum advection terms are investigated. It is shown that when Ro increases then the dimensionless drag (scaled by μ*a*W*) decreases, and the Taylor column becomes shorter, this effect being more pronounced in the rear region (μ* is the dynamic viscosity). The present results strengthen and extend the validity of the classical drag force predictions and therefore the issue of the large discrepancy between theory and experiments (Maxworthy 1970) concerning this force becomes more acute.

Fluid motion inside a spinning nutating cylinder

1985 ◽  
Vol 150 ◽  
pp. 121-138 ◽  
Author(s):  
Harold R. Vaughn ◽  
William L. Oberkampf ◽  
Walter P. Wolfe
Keyword(s):  
Steady State ◽  
Finite Difference ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Fluid Motion ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Experimental Measurements ◽  
Navier Stokes Equations ◽  
Internal Fluid

The incompressible three-dimensional Navier–Stokes equations are solved numerically for a fluid-filled cylindrical cannister that is spinning and nutating. The motion of the cannister is characteristic of that experienced by spin-stabilized artillery projectiles. Equations for the internal fluid motion are derived in a non-inertial aeroballistic coordinate system. Steady-state numerical solutions are obtained by an iterative finite-difference procedure. Flow fields and liquid induced moments have been calculated for viscosities in the range of 0.9 × 104−1 × 109 cSt. The nature of the three-dimensional fluid motion inside the cylinder is discussed, and the moments generated by the fluid are explained. The calculated moments generally agree with experimental measurements.

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