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doi: 10.1002/fld.1650080402pmid: N/A
A numerical method for the solution of the vector potential/vorticity vector formulation of the transient, fully three‐dimensional Navier‐Stokes energy and continuity equations has been applied to simulate the development of natural convective flow within a ‘box’ after a sudden temperature change on a vertical portion of the wall. Only one cavity size has been considered, this having a vertical height of three times its width and a horizontal length of six times its width. A single heated rectangular hot spot or ‘element’ on an otherwise adiabatic wall is centred between the vertical end walls. The opposite vertical wall is held at the intial fluid temperature, and all other walls are assumed to be adiabatic. Fluid properties have been assumed constant except for the density change with temperature that gives rise to the buoyancy force. The numerical method is an underrelaxation Gauss‐Seidel method using finite differencing at each time step. Solutions have been obtained for a Prandtl number of 0.71, for Rayleigh numbers, based on the width, of between 0 and 100000 and for a number of heated element locations and sizes.
doi: 10.1002/fld.1650080403pmid: N/A
The incompressible flow through a two‐dimensional cascade is computed using the SIMPLE algorithm in a boundary‐fitted co‐ordinate system. With the standard staggered grid arrangement the numerical solution was found to allow localized pressure oscillations to persist adjacent to the periodic boundaries. These oscillations were found to be a consequence of the extended momentum control volumes which are required in this region of the cascade. Such control volumes may be removed by the use of appropriately non‐staggered velocity storage locations, which are also desirable in the boundary‐fitted system since the Cartesian velocity components are no longer related to the grid line orientations. However, this storage permits the propagation of global pressure oscillations, which were previously suppressed by the staggered grid arrangement. This paper attempts to define a solution procedure which uses non‐staggered velocity locations and is able to eliminate the consequent global pressure oscillations. To achieve this aim, two forms of pressure correction scheme were considered. The first implemented the scheme proposed by Vanka et al. but was found to be inadequate in the open part of the cascade, whereas the second employed a modification of the scheme proposed by Rhie and Chow and was found to be successful in all regions of the flow. The results computed using this scheme were compared with the available experiment results.
doi: 10.1002/fld.1650080404pmid: N/A
A numerical method for computing high‐Re laminar steady flows is presented. The incompressible Navier‐Stokes equations are expressed in terms of vorticity‐velocity variables, discretized in space by finite differences on a staggered grid and advanced in time by a scalar alternating direction implicit (ADI) procedure, which allows a fully vectorized computer code. The accuracy and efficiency of the present formulation are discussed in comparison with the standard ω‐ψ and u, v, P forms. Numerical results are presented for two test cases: the driven cavity at Re up to 5000 and the backward‐facing step at Re up to 800.
Sivaloganathan, S.; Shaw, G. J.
doi: 10.1002/fld.1650080405pmid: N/A
The use of multigrid methods in complex fluid flow problems is recent and still under development. In this paper we present a multigrid method for the incompressible Navier‐Stokes equations. The distinctive features of the method are the use of a pressure‐correction method as a smoother and a novel continuity‐preserving manner of grid coarsening. The shear‐driven cavity problem is used as a test case to demonstrate the efficiency of the method.
Shaw, G. J.; Sivaloganathan, S.
doi: 10.1002/fld.1650080406pmid: N/A
A local mode Fourier analysis is used to assess the suitability of the SIMPLE pressure‐correction algorithm to act as a smoother in a multigrid method. The necessary ellipticity of the Navier‐Stokes equations and their discrete representation are established. The theoretical analysis is compared with practical results.
doi: 10.1002/fld.1650080407pmid: N/A
Three‐dimensional, compressible, internal flow solutions obtained using a thin‐layer Navier‐Stokes code are presented. The code, formulated by P.D. Thomas, is based on the Beam‐Warming implicit factorization scheme; the boundary conditions also are formulated implicitly. Turbulent flow is treated through the use of the Baldwin‐Lomax two‐layer, algebraic eddy viscosity model. Steady‐state solutions are obtained by solving numerically the time‐dependent equations from given initial conditions until the time‐dependent terms become negligible. The configuration considered is a rectangular cross‐section, S‐shaped centreline diffuser duct with an exit/inlet area ratio of 2.25. The Mach number at the duct entrance is 0.9, with a Reynolds number of 5.82 × 105. Convergence to the final results required about 2700 time steps or 11 hours of CPU time on our CRAY‐1M computer. The averaged residuals were reduced by about two orders of magnitude during the computations. Several regions of separated flow exist within the diffuser. The separated flow region on the upper wall, downstream of the second bend, is by far the largest and extends to the exit plane.
doi: 10.1002/fld.1650080408pmid: N/A
Recently the concept of adaptive grid computation has received much attention in the computational fluid dynamics research community. This paper continues the previous efforts of multiple one‐dimensional procedures in developing and asessing the ideas of adaptive grid computation. The focus points here are the issue of numerical stability induced by the grid distribution and the accuracy comparison with previously reported work. Two two‐dimensional problems with complicated characteristics—namely, flow in a channel with a sudden expansion and natural convection in an enclosed square cavity—are used to demonstrate some salient features of the adaptive grid method. For the channel flow, by appropriate distribution of the grid points the numerical algorithm can more effectively dampen out the instabilities, especially those related to artificial boundary treatments, and hence can converge to a steady‐state solution more rapidly. For a more accurate finite difference operator, which contains less undesirable numerical diffusion, the present adaptive grid method can yield a steady‐state and convergent solution, while uniform grids produce non‐convergent and numerically oscillating solutions. Furthermore, the grid distribution resulting from the adaptive procedure is very responsive to the different characteristics of laminar and turbulent flows. For the problem of natural convection, a combination of a multiple one‐dimensional adaptive procedure and a variational formulation is found very useful. Comparisons of the solutions on uniform and adaptive grids with the reported benchmark calculations demonstrate the important role that the adaptive grid computation can play in resolving complicated flow characteristics.
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