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Raghavarao, C. V.; Pramadavalli, K.
doi: 10.1002/fld.1650091102pmid: N/A
The Navier–Stokes equations, which are the governing equations for a steady, viscous, incompressible fluid rotating about the z‐axis with angular velocity ω, are linearized using the Oseen approximation. Two parameters, namely the Reynolds number Re = Ua/v and Reω = 2ωa2/v (the Reynolds number w.r.t. rotation), enter the linearized equations. These equations are solved by the Peaceman–Rachford ADI method and the resulting algebraic equations are solved by the SOR method. Streamlines are plotted and compared with the Oseen solution for the non‐rotating case. The magnitude of the vorticity vector with increasing θ is also plotted.
Raghavarao, C. V.; Pramadavalli, K.
doi: 10.1002/fld.1650091103pmid: N/A
The flow of steady incompressible viscous fluid rotating about the z‐axis with angular velocity ω and moving with velocity u past a sphere of radius a which is kept fixed at the origin is investigated by means of a numerical method for small values of the Reynolds number Reω. The Navier–Stokes equations governing the axisymmetric flow can be written as three coupled non‐linear partial differential equations for the streamfunction, vorticity and rotational velocity component. Central differences are applied to the partial differential equations for solution by the Peaceman–Rachford ADI method, and the resulting algebraic equations are solved by the ‘method of sweeps’.
doi: 10.1002/fld.1650091104pmid: N/A
A time‐marching method is presented for the calculation of two‐dimensional, high‐speed channel flow, including the usually neglected terms of slope and bottom friction. Time‐marching methods are potentially the most flexible means of calculating flow through geometrically complex channel passages, since they can readily deal with subcritical and supercritical flows. The adopted numerical scheme comes straight from gas flow computations in turbomachines. The flow is assumed to be fully mixed in the vertical direction, so that vertical variations may be neglected. Comparisons with other numerical solutions for various open channel configurations show that the proposed approach is a comparatively accurate, reliable and fast technique. It can be utilized for open channel designs.
Georgiou, Georgios C.; Olson, Lorraine G.; Schultz, William W.; Sagan, Susan
doi: 10.1002/fld.1650091105pmid: N/A
Abrupt changes in boundary conditions in viscous flow problems give rise to stress singularities. Ordinary finite element methods account effectively for the global solution but perform poorly near the singularity. In this paper we develop singular finite elements, similar in principle to the crack tip elements used in fracture mechanics, to improve the solution accuracy in the vicinity of the singular point and to speed up the rate of convergence. These special elements surround the singular point, and the corresponding field shape functions embody the form of the singularity. Because the pressure is singular, there is no pressure node at the singular point. The method performs well when applied to the stick–slip problem and gives more accurate results than those from refined ordinary finite element meshes.
doi: 10.1002/fld.1650091106pmid: N/A
The governing equations for depth‐averaged turbulent flow are presented in both the primitive variable and streamfunction–vorticity forms. Finite element formulations are presented, with special emphasis on the handling of bottom stress terms and spatially varying eddy viscosity. The primitive variable formulation is found to be preferable because of its flexibility in handling spatial variation in viscosity, variability in water surface elevations, and inflow and outflow boundaries. The substantial reduction in computational effort afforded by the streamfunction–vorticity formulation is found not to be sufficient to recommend its use for general depth‐averaged flows. For those flows in which the surface can be approximated as a fixed level surface, the streamfunction–vorticity form can produce results equivalent to the primitive variable form as long as turbulent viscosity can be estimated as a constant.
Chang, Yeon; Beris, Antony N.; Michaelides, Efstathios E.
doi: 10.1002/fld.1650091107pmid: N/A
A numerical scheme is developed to predict the heat transfer and pressure drop coefficients in flow through rigid tube bundles. The scheme uses the Galerkin finite element technique. The conservation equations for laminar steady‐state flow are cast in the form of streamfunction and vorticity equations. A Picard iteration method is used for the solution of the resulting system of non‐linear algebraic equations. Results for the heat transfer and pressure drop coefficients are obtained for tube arrays of pitch ratios of 1·5 and 2·0. Very good agreement of the present results and experimental data obtained in the past is observed up to Reynolds numbers of 1000. It is also observed that the results of the present method show better agreement with the experimental data and that they are applicable for higher Reynolds numbers than results of other studies.
doi: 10.1002/fld.1650091108pmid: N/A
The convergence properties of an iterative solution technique for the Reduced Navier–Stokes equations are examined for two‐dimensional steady subsonic flow over bump and trough geometries. Techniques for decreasing the sensitivity to the initial pressure approximation, for fine meshes in particular, are investigated. They are shown to improve the robustness of the relaxation process and to decrease the computational work required to obtain a converged solution. A semi‐coarsening multigrid technique that has previously been found to be particularly advantageous for high‐Reynolds‐number (Re) flows with flow separation and with highly stretched surface‐normal grids is applied herein to further accelerate convergence. Solutions are obtained for the laminar flow over a trough that is more severe than has been considered to date. Sufficient axial grid refinement in this case leads to a shock‐like reattachment and, for sufficiently large Re, to a local ‘divergence’ of the numerical computations. This ‘laminar flow breakdown’ appears to be related to an instability associated with high‐frequency fine‐grid modes that are not resolvable with the present modelling. This behaviour may be indicative of dynamic stall or of incipient transition. The breakdown or instability is shown to be controllable by suitable introduction of transition turbulence models or by laminar flow control, i.e. small amounts of wall suction. This lends further support to the hypothesis that the instability is of a physical rather than numerical character and suggests that full three‐dimensional analysis is required to properly capture the flow behaviour. Another inference drawn from this investigation is that there is a need for careful grid refinement studies in high‐Re flow computations, since coarser grids may yield oscillation‐free solutions that cannot be obtained on finer grids.
doi: 10.1002/fld.1650091109pmid: N/A
Two‐dimensional normal impinging jet flowfields, with or without an upper plate, were analysed by employing an implicit bidiagonal numerical method developed by Lavante and Thompkins Jr. The Jones–Launder K–ϵ two‐equation turbulent model was employed to study the turbulent effects of the impinging jet flowfield. The upper plate surface pressure, the ground plane pressure and other physical parameters of the momentum flowfield were calculated at various jet exit height and jet inlet Reynolds numbers. These results were compared with those of Beam and Warming's numerical method, Hsiao and Chuang, and others, along with experimental data. The potential core length of the impinging jet without an upper plate is longer than that of the free jet because of the effects of the ground plane, while the potential core length of the impinging jet with an upper plate is shorter than that of the free jet because of the effects of the upper plate. This phenomenon in the present analysis provides a fundamental numerical study of an impinging jet and a basis for further analysis of impinging jet flowfields on a variable angle plate.
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