International Journal for Numerical Methods in Fluids

doi: 10.1002/fld.1650040603pmid: N/A

doi: 10.1002/fld.1650040602pmid: N/A

doi: 10.1002/fld.1650040601pmid: N/A

O'Carroll, M. J.; Toro, E. F.

doi: 10.1002/fld.1650040604pmid: N/A

Computing critical flows in hydraulics involves three problems in one: the internal flow problem, the location of the free surface and the determination of the critical flow rate. The subject can involve such difficulties as non‐uniqueness, non‐existence, ill‐conditioning and catastrophes. This paper discusses the difficulties relating to computing critical flows over weirs. A new rapidly convergent method of determining the critical flow rate is presented and various results are shown using it with finite element discretization and with a new streamline shifting method. Numerical results are in good agreement with published data, both numerical and experimental.

Doctor, H. D.; Bulsari, A. B.; Kalthia, N. L.

doi: 10.1002/fld.1650040605pmid: N/A

A well‐known Stokes problem is discussed by a cubic spline collocation method. Two consecutive cubic splines are obtained for the problem. The results by this method are compared with those of an orthogonal collocation method. The selection of the length of the subintervals of the range of the boundary value problem is also justified. The results obtained by these two methods are compared with the analytic solution. The methods involve simple algebra, and hence the calculations do not require the help of a computer. Necessary error analysis has been carried out.

Maliska, C. R.; Raithby, G. D.

doi: 10.1002/fld.1650040606pmid: N/A

For three‐dimensional fluid flows in complex geometries, it is convenient to make predictions using a non‐orthogonal boundary‐fitted mesh. The present paper describes an economical method of solving the equations of motion for two and three dimensional problems using such meshes. The locations on the mesh at which the depenent variables are calculated, and the methods used to solve the equations, are key issues in the development of a successful algorithm; these are discussed in the present paper. Results obtained when the proposed method is applied to several problems are also described. The method is intended for flows in which compressibility effects do not dominate.

Fuchs, L.; Zhao, H.‐S.

doi: 10.1002/fld.1650040607pmid: N/A

The three‐dimensional Navier‐Stokes equations for viscous incompressible fluids are discretized on staggered or non‐staggered grids. The system of finite‐difference equations is solved by a multi‐grid method. The method and some possible sources of difficulties and their remedies are described. The numerical algorithm has been applied to the computations of flows in ducts for a range of Reynolds numbers up to 2000. As outflow boundary conditions, either the fully developed flow profile (Dirichlet condition) or parabolic conditions have been applied. The multi‐grid method has a fast rate of convergence (with both types of boundary conditions), and it is not sensitive to the number of mesh points and the Reynolds number. The numerical solution, using parabolic boundary conditions, is insensitive to the location of the outflow boundary, even for large Reynolds numbers, in contrast to the solution with Dirichlet boundary conditions.

Gresho, Philip M.; Chan, Stevens T.; Lee, Robert L.; Upson, Craig D.

doi: 10.1002/fld.1650040608pmid: N/A

Beginning with the Galerkin finite element method and the simplest appropriate isoparametric element for modelling the Navier‐Stokes equations, the spatial approximation is modified in two ways in the interest of cost‐effectiveness: the mass matrix is ‘lumped’ and all coefficient matrices are generated via 1‐point quadrature. After appending an hour‐glass correction term to the diffusion matrices, the modified semi‐discretized equations are integrated in time using the forward (explicit) Euler method in a special way to compensate for that portion of the time truncation error which is intolerable for advection‐dominated flows. The scheme is completed by the introduction of a subcycling strategy that permits less frequent updates of the pressure field with little loss of accuracy. These techniques are described and analysed in some detail, and in Part 2 (Applications), the resulting code is demonstrated on three sample problems: steady flow in a lid‐driven cavity at Re ≤ 10,000, flow past a circular cylinder at Re ≤ 400, and the simulation of a heavy gas release over complex topography.