Transient Starting Flow in a Cylinder With Counter-Rotating Endwall Disks

1985 ◽  
Vol 107 (1) ◽  
pp. 92-96 ◽  
Author(s):  
Jae Min Hyun
Keyword(s):  
Steady State ◽  
Stagnation Point ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Cell Structure ◽  
Radial Distance ◽  
Front Propagation ◽  
Rotating Disk ◽  
Azimuthal Velocity ◽  
Counter Rotation

Spin-up from rest in a cylinder with top and bottom endwall disks rotating in opposite directions (ΩT and ΩB are the respective rotation rate, but S[≡ ΩT/ΩB] < 0) is investigated. The sidewall is fixed to the faster-rotating disk. A finite-difference numerical model is adopted to integrate the unsteady Navier-Stokes equations. We consider a cylinder of aspect ratio 0(1) and minute Ekman numbers. Numerical solutions are presented to show the transient azimuthal flow structures, axial vorticity profiles, and meridional flow patterns. An azimuthal velocity front, which separates the rotating from the nonrotating fluid, propagates radially inward from the sidewall. The appearance of the front is similar to the front propagation in spin-up in a rigid cylinder. As S decreases from zero, the direction of rotation in the bulk of the interior fluid becomes the same as that of the faster-rotating disk. The azimuthal velocities are still vertically uniform in the bulk of the interior. The scaled time to reach the steady state decreases. The angular velocities of the interior fluid near the central axis become very small. Under counter-rotation, the meridional circulation forms a two-cell structure. A stagnation point appears on the slower-rotating disk. During spin-up, the stagnation point moves from the sidewall to its steady-state position. As counter-rotation increases, the radial distance traveled by the stagnation point decreases.

Flow Driven by a Finite, Shrouded Spinning Disk With Suction

1985 ◽  
Vol 52 (4) ◽  
pp. 766-770 ◽  
Author(s):  
J. M. Hyun
Keyword(s):  
Steady State ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Ekman Layer ◽  
Initial State ◽  
Disk Model ◽  
Stationary Disk ◽  
Spinning Disk ◽  
Real Flow ◽  
Finite Configuration

Numerical solutions are presented for the flow driven by a spinning disk which forms an endwall of a finite, closed cylinder. The effects of imposing a uniform suction (or blowing) through the spinning disk in finite configuration are investigated. The Reynolds number is large and the cylinder aspect ratio is 0(1). Finite-difference techniques are employed to integrate the time-dependent Navier-Stokes equations. The initial state is taken to be a uniform axial motion. Integration is performed until an approximate steady state is attained. When there is no suction, the infinite disk model is shown to provide a qualitatively representative approximation to the flow in the central core region. As a suction (blowing) is imposed, the core rotation rate in the case of finite configuration becomes smaller (larger) than that for the case of no suction, which is in disagreement with the predictions of the infinite disk model. These significant discrepancies point to a fundamental difficulty of the infinite disk model to adequately describe the real flow infinite geometry when there is a mass flux across the system boundary. Plots showing the meridional stream function at various times are constructed. Details of the flow structure in the approximate steady state are analyzed. When there is a suction, a strong Ekman layer is present on the spinning disk but the Ekman layer on the stationary disk fades. When there is a blowing, a strong Ekman layer forms on the stationary disk. It is shown that the dynamic effects influencing the character of the flow are confined to these Ekman layers.

On the Scaling of Impulsively Started Incompressible Turbulent Round Jets

1982 ◽  
Vol 104 (2) ◽  
pp. 191-197 ◽  
Author(s):  
T.-W. Kuo ◽  
F. V. Bracco
Keyword(s):  
Steady State ◽  
Kinetic Energy ◽  
Time Scales ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Scaling Law ◽  
Navier Stokes ◽  
Centerline Velocity ◽  
Arrival Times ◽  
Round Jets

A scaling law for transient, turbulent, incompressible, round jets is reported. Numerical solutions of the Navier-Stokes equations were obtained using a k-ε model for turbulence. The constants of the k-ε model were optimized by comparing computed centerline velocity, mean radial velocity distribution, longitudinal kinetic energy distributions with those measured by other authors in steady round jets. The resulting constants are those also used in computations of steady planar jets except for the one that multiplies the source term in the ε-equation. After optimization, the agreement is satisfactory for all mean quantities but is still rather poor for the kinetic energy distribution. Parameteric studies of the transient were performed for 9•103 ≤ ReD ≤ 105. Then the definition was adopted that a jet reaches steady state between the nozzle and an axial location when, at that location, the centerline velocity achieves 70 percent of its steady state value, and characteristic steadying length and time scales (D•ReD0.053 and D•ReD0.053/u cL,0 respectively) were determined as well as a unique function that relates dimensionless steadying time to dimensionless steadying length. This function changes in a predictable way if a percent other than 70 is selected but the characteristic length and time scales do not. It is found that the 70 percent threshold is reached within the head vortex of the transient jet. Thus a transient jet, practically, is a steady jet except within its head vortex. This, in part, justifies our use of steady state k-ε constants in our transient computations. The computed jet tip arrival times are shown to compare favorably with measured ones.

Analytical solutions for the unsteady MHD rotating flow over a rotating sphere near the equator

Open Physics ◽  
2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Erik Sweet ◽  
Kuppalapalle Vajravelu ◽  
Robert Gorder
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Nonlinear Partial Differential Equations ◽  
Coupled System ◽  
Polar Coordinates ◽  
Rotating Flow ◽  
Rotating Sphere

AbstractIn this paper we investigate the three-dimensional magnetohydrodynamic (MHD) rotating flow of a viscous fluid over a rotating sphere near the equator. The Navier-Stokes equations in spherical polar coordinates are reduced to a coupled system of nonlinear partial differential equations. Self-similar solutions are obtained for the steady state system, resulting from a coupled system of nonlinear ordinary differential equations. Analytical solutions are obtained and are used to study the effects of the magnetic field and the suction/injection parameter on the flow characteristics. The analytical solutions agree well with the numerical solutions of Chamkha et al. [31]. Moreover, the obtained analytical solutions for the steady state are used to obtain the unsteady state results. Furthermore, for various values of the temporal variable, we obtain analytical solutions for the flow field and present through figures.

MHD mixed convection oblique stagnation-point flow on a vertical plate

2017 ◽  
Vol 27 (12) ◽  
pp. 2744-2767
Author(s):  
Giulia Giantesio ◽  
Anna Verna ◽  
Natalia C. Roşca ◽  
Alin V. Rosca ◽  
Ioan Pop
Keyword(s):  
Mixed Convection ◽  
Stagnation Point ◽  
Newtonian Fluid ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Boussinesq Approximation ◽  
Stagnation Point Flow ◽  
Content Type ◽  
Boundary Layer Equations ◽  
Similarity Transformations

Purpose This paper aims to study the problem of the steady plane oblique stagnation-point flow of an electrically conducting Newtonian fluid impinging on a heated vertical sheet. The temperature of the plate varies linearly with the distance from the stagnation point. Design/methodology/approach The governing boundary layer equations are transformed into a system of ordinary differential equations using the similarity transformations. The system is then solved numerically using the “bvp4c” function in MATLAB. Findings An exact similarity solution of the magnetohydrodynamic (MHD) Navier–Stokes equations under the Boussinesq approximation is obtained. Numerical solutions of the relevant functions and the structure of the flow field are presented and discussed for several values of the parameters which influence the motion: the Hartmann number, the parameter describing the oblique part of the motion, the Prandtl number (Pr) and the Richardson numbers. Dual solutions exist for several values of the parameters. Originality value The present results are original and new for the problem of MHD mixed convection oblique stagnation-point flow of a Newtonian fluid over a vertical flat plate, with the effect of induced magnetic field and temperature.

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.

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.

All Speed and High-Resolution Scheme Applied to Three-Dimensional Multi-Block Complex Flowfield System

2004 ◽  
Vol 20 (1) ◽  
pp. 13-25 ◽  
Author(s):  
Uzu- Kuei Hsu ◽  
Chang- Hsien Tai ◽  
Chien- Hsiung Tsai
Keyword(s):  
Steady State ◽  
Incompressible Flows ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Diffusion Flux ◽  
Time Derivative ◽  
Splitting Method ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Complex Flow

ABSTRACTThe improved numerical approach is implemented with preconditioned Navier-Stokes solver on arbitrary three-dimensional (3-D) structured multi-block complex flowfield. With the successful application of time-derivative preconditioning, present hybrid finite volume solver is performed to obtain the steady state solutions in compressible and incompressible flows. This solver which combined the adjective upwind splitting method (AUSM) family of low-diffusion flux-splitting scheme with an optimally smoothing multistage scheme and the time-derivative preconditioning is used to solve both the compressible and incompressible Euler and Navier-Stokes equations. In addition, a smoothing procedure is used to provide a mechanism for controlling the numerical implementation to avoid the instability at stagnation and sonic region. The effects of preconditioning on accuracy and convergence to the steady state of the numerical solutions are presented. There are two validation cases and three complex cases simulated as shown in this study. The numerical results obtained for inviscid and viscous two-dimensional flows over a NACA0012 airfoil at free stream Mach number ranging from 0.1 to 1.0E-7 indicates that efficient computations of flows with very low Mach numbers are now possible, without losing accuracy. And it is effectively to simulate 3-D complex flow phenomenon from compressible flow to incompressible by using the advanced numerical methods.

High Reynolds number flow between two inflnite rotating disks

1971 ◽  
Vol 12 (4) ◽  
pp. 483-501 ◽  
Author(s):  
H. Rasmussen
Keyword(s):  
Stokes Equations ◽  
Axisymmetric Flow ◽  
Radial Distance ◽  
Rotating Disk ◽  
Navier Stokes ◽  
Rotating Disks ◽  
Axial Velocity Component ◽  
Navier Stokes Equations ◽  
Reynolds Number Flow ◽  
High Reynolds

In 1921 von Karman [1] showed that the Navier-Stokes equations for steady viscous axisymmetric flow can be reduced to a set of ordinary differential equations if it is assumed that the axial velocity component is independent of the radial distance from the axis of symmetry. He used these similarity equations to obtain a solution for the flow near an infinite rotating disk. Later Batchelor [2] and Stewartson [3] applied these equations to the problem of steady flow between two infinite disks rotating in parallel planes a finite distance apart.

Magnetohydrodynamic flow between rotating coaxial disks

1969 ◽  
Vol 38 (2) ◽  
pp. 335-352 ◽  
Author(s):  
C. J. Stephenson
Keyword(s):  
Numerical Solutions ◽  
Hartmann Number ◽  
Axial Magnetic Field ◽  
Radial Distance ◽  
Rotating Disk ◽  
Magnetohydrodynamic Flow ◽  
The Other ◽  
Experimental Measurements ◽  
Velocity Increase ◽  
Axis Of Rotation

This is a study of the magnetohydrodynamic flow of an incompressible viscous fluid between coaxial disks, with a uniform axial magnetic field B. The fluid has density ρ, viseosity η and electrical conductivity σ. The flow is assumed to be steady, and to be similar in the sense that the radial and tangential components of velocity increase linearly with radial distance from the axis of rotation. Most of the work is concerned with disks which are electrical insulators, one of which rotates while the other remains stationary. The imposed conditions can then be represented by the Reynolds number R = ρΩad2/η and the Hartmann number M2 = σB2d2/η, where Ωa is the angular velocity of the rotating disk and d is the gap between the disks. Asymptotic solutions are given for R [Lt ] M2, and numerical solutions are obtained for values of R and M2 up to 512. Experimental measurements are presented which are in general agreement with the theoretical flows, and the results for small values of the Hartmann number provide the first known experimental support for the purely hydrodynamic solutions in the range 100 < R < 800.

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