Numerical Solutions of a Viscous Uniform Approach Flow Past Square and Diamond Cylinders

2002 ◽  
Author(s):  
Charles Dalton ◽  
Wu Zheng
Keyword(s):  
Stream Function ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Elliptic Partial Differential Equation ◽  
Navier Stokes ◽  
Central Difference ◽  
Uniform Approach ◽  
Flow Past ◽  
Line Force ◽  
Vorticity Transport

Numerical results are presented for a uniform approach flow past square and diamond cylinders, with and without rounded corners, at Reynolds numbers of 250 and 1000. This unsteady viscous flow problem is formulated by the 2-D Navier-Stokes equations in vorticity and stream-function form on body-fitted coordinates and solved by a finite-difference method. Second-order Adams-Bashforth and central-difference schemes are used to discretize the vorticity transport equation while a third-order upwinding scheme is incorporated to represent the nonlinear convective terms. A grid generation technique is applied to provide an efficient mesh system for the flow. The elliptic partial differential equation for stream-function and vorticity in the transformed plane is solved by the multigrid iteration method. The Strouhal number and the average in-line force coefficients agree very well with the experimental and previous numerical values. The vortex structures and the evolution of vortex shedding are illustrated by vorticity contours. Rounding the corners of the square and diamond cylinders produced a noticeable decrease on the calculated drag and lift coefficients.

Optimized Accelerating Parameters of Convergence for the Navier-Stokes Equations

1992 ◽  
Author(s):  
Bashar S. AbdulNour
Keyword(s):  
Stream Function ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Relaxation Method ◽  
Reynolds Numbers ◽  
Computer Time ◽  
Navier Stokes ◽  
Computational Time ◽  
Navier Stokes Equations ◽  
Successive Over Relaxation

Abstract An over-relaxation procedure, that includes weighing factors, is applied to the steady, two-dimensional Navier-Stokes equations in order to reduce the computational time. The benefits obtained from this strategy are illustrated by the problem of viscous flow in the entrance region of an unconstricted and a constricted channel. The describing equations are expressed in terms of the stream function and vorticity. The convergence domain for the Successive Over-Relaxation method and the optimum values of the accelerating parameters, which consist of the over-relaxation and weighting factors for both the stream function and vorticity, are discussed. Numerical solutions are obtained for Reynolds numbers ranging from 20 to 2000. The computer time is reduced by as much as a factor of six using the optimum values of the accelerating parameters.

Finite Analytic Numerical Solution Axisymmetric Navier-Stokes and Energy Equations

1983 ◽  
Vol 105 (3) ◽  
pp. 639-645 ◽  
Author(s):  
Ching-Jen Chen ◽  
Young Hwan Yoon
Keyword(s):  
Heat Transfer ◽  
Numerical Solution ◽  
Numerical Method ◽  
Stream Function ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Inlet Temperature ◽  
Local Analytic Solution

Connective heat transfer for steady-state laminar flow in axisymmetric coordinates is considered. Numerical solutions for flow pattern and temperature distribution are obtained by the finite analytic numerical method applied to the Navier-Stokes equations expressed in terms of vorticity and stream function, and the energy equation. The finite analytic numerical method differs from other numerical methods in that it utilizes a local analytic solution in an element of the problem to construct the total numerical solution. Finite analytic solutions of vorticity, stream function, temperature, and heat transfer coefficients for flow with Reynolds numbers of 5, 100, 1000, and 2000, and Prandtl numbers of 0.1, 1.0, and 10.0 with uniform grid sizes, are reported for an axisymmetric pipe with a sudden expansion and contraction. The wall temperature is considered to be isothermal and differs from the inlet temperature. It is shown that the finite analytic is stable, converges rapidly, and simulates the convection of fluid flow accurately, since the local analytic solution is capable of simulating automatically the influence of skewed convection through the element boundary on the interior nodal values, thereby minimizing the false numerical diffusion.

Capsular flow in pipelines

1972 ◽  
Vol 56 (1) ◽  
pp. 49-59 ◽  
Author(s):  
A. E. Vardy ◽  
M. I. G. Bloor ◽  
J. A. Fox
Keyword(s):  
Reynolds Number ◽  
Pressure Gradient ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Steady Motion ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Hydraulic Pressure ◽  
Central Difference ◽  
Navier Stokes Equations

The problem considered is that of the steady motion of a series of neutrally buoyant, flat-faced, rigid, cylindrical capsules along the axis of a pipeline under the influence of a hydraulic pressure gradient. The Navier-Stokes equations are non-dimensionalized and expressed in central-difference form. Numerical solutions are found by the method of relaxation for Reynolds numbers up to 20 000 and a close agreement is obtained with readings from a laboratory apparatus for Reynolds numbers up to 2200.The flow is examined in detail and the existence of toroidal vortices between successive capsules is demonstrated. Their shape is shown to be increasingly influenced by inertial forces as the Reynolds number increases, but the overall pressure gradient is not greatly dependent on the Reynolds number.

Analysis of Turbulent Flow Past a Class of Semi-Infinite Bodies

1986 ◽  
Vol 108 (2) ◽  
pp. 157-165
Author(s):  
A. M. Abdelhalim ◽  
U. Ghia ◽  
K. N. Ghia
Keyword(s):  
Flat Plate ◽  
Stream Function ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Analytical Data ◽  
Flow Region ◽  
Two Dimensional ◽  
Infinite Body ◽  
Similarity Type ◽  
Flow Past

This study was undertaken with the primary purpose of developing an analysis for flow past a class of two-dimensional and axisymmetric semi-infinite bodies. The time-averaged Navier-Stokes equations for these flows are derived in surface-oriented conformal coordinates (ξ, η) in terms of similarity-type vorticity and stream-function variables. Turbulence closure is achieved by means of a two-equation turbulence model utilizing the kinetic energy k and its dissipation rate ε which enable determination of the isotropic eddy viscosity. The coupled vorticity and stream-function equations are solved simultaneously using an incremental formulation of the factored alternating-direction implicit scheme. The turbulence equations for k and ε are solved by the standard ADI method. Numerical solutions are obtained for the thin flat plate and compared with available experimental and analytical data. Also, results are obtained for flow over a parabola and compared with the flat-plate results in order to assess the effects of longitudinal curvature on the flow results. Finally, solutions are obtained for flow past a two-dimensional semi-infinite body with a shoulder, at Red = 24,000. The computed results have the same general trend as the experimental data; possible causes for the differences within the separated-flow region are cited.

Flow in a channel with a moving indentation

1988 ◽  
Vol 190 ◽  
pp. 87-112 ◽  
Author(s):  
M. E. Ralph ◽  
T. J. Pedley
Keyword(s):  
Stream Function ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Maximum Magnitude ◽  
Computational Domain ◽  
Generation Process ◽  
Navier Stokes Equations ◽  
Moving Wall ◽  
Experimental Findings ◽  
Vorticity Transport

The unsteady flow of a viscous, incompressible fluid in a channel with a moving indentation in one wall has been studied by numerical solution of the Navier-Stokes equations. The solution was obtained in stream-function-vorticity form using finite differences. Leapfrog time-differencing and the Dufort-Frankel substitution were used in the vorticity transport equation, and the Poisson equation for the stream function was solved by multigrid methods. In order to resolve the boundary-condition difficulties arising from the presence of the moving wall, a time-dependent transformation was applied, complicating the equations but ensuring that the computational domain remained a fixed rectangle.Downstream of the moving indentation, the flow in the centre of the channel becomes wavy, and eddies are formed between the wave crests/troughs and the walls. Subsequently, certain of these eddies ‘double’, that is a second vortex centre appears upstream of the first. These observations are qualitatively similar to previous experimental findings (Stephanoff et al. 1983, and Pedley & Stephanoff 1985), and quantitative comparisons are also shown to be favourable. Plots of vorticity contours confirm that the wave generation process is essentially inviscid and reveal the vorticity dynamics of eddy doubling, in which viscous diffusion and advection are important at different stages. The maximum magnitude of wall vorticity is found to be much larger than in quasi-steady flow, with possibly important biomedical implications.

Laminar Incompressible Recirculating Flow in a Driven Cavity of Polar Cross Section

1977 ◽  
Vol 99 (4) ◽  
pp. 774-777 ◽  
Author(s):  
U. Ghia ◽  
R. K. Goyal
Keyword(s):  
Stream Function ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Cavity Flow ◽  
Rectangular Cavity ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Driven Cavity ◽  
Recirculating Flow ◽  
Alternating Direction

The driven flow in a polar cavity has been analyzed using the complete Navier-Stokes equations formulated in terms of a stream function and vorticity. An alternating-direction implicit method, with careful treatment of the convective terms in the equations, is used to obtain the numerical solutions. Results are obtained for the stream function, vorticity, velocities and pressure for various values of the two characteristic parameters of the problem, namely, the flow Reynolds number Re and the aspect ratio of the cavity. The formulation is general and produces results for the driven rectangular cavity-flow problem as a special case. Good agreement is obtained between the present solutions for this case and available corresponding results. The overall features of the driven polar-cavity flow are found to be generally similar to those for the rectangular cavity.

A numerical study of the interaction between unsteay free-stream disturbances and localized variations in surface geometry

1989 ◽  
Vol 209 ◽  
pp. 285-308 ◽  
Author(s):  
R. J. Bodonyi ◽  
W. J. C. Welch ◽  
P. W. Duck ◽  
M. Tadjfar
Keyword(s):  
Numerical Solutions ◽  
Free Stream ◽  
Stokes Equations ◽  
Numerical Study ◽  
Navier Stokes ◽  
Surface Geometry ◽  
Navier Stokes Equations ◽  
S Waves ◽  
Tollmien Schlichting

A numerical study of the generation of Tollmien-Schlichting (T–S) waves due to the interaction between a small free-stream disturbance and a small localized variation of the surface geometry has been carried out using both finite–difference and spectral methods. The nonlinear steady flow is of the viscous–inviscid interactive type while the unsteady disturbed flow is assumed to be governed by the Navier–Stokes equations linearized about this flow. Numerical solutions illustrate the growth or decay of the T–S waves generated by the interaction between the free-stream disturbance and the surface distortion, depending on the value of the scaled Strouhal number. An important result of this receptivity problem is the numerical determination of the amplitude of the T–S waves.

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