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Luettich, Richard A.; Westerink, Joannes J.
doi: 10.1002/fld.1650121002pmid: N/A
A simple technique is presented that allows a numerical solution to be sought for the vertical variation of shear stress as a substitute for the vertical variation of velocity in a three‐dimensional hydrodynamic model. In its most general form the direct stress solution (DSS) method depends only upon the validity of an eddy viscosity relation between the shear stress and the vertical gradient of velocity. The rationale for preferring a numerical solution for shear stress to one for velocity is that shear stress tends to vary more slowly over the vertical than velocity, particularly near boundaries. Consequently, a numerical solution can be obtained much more efficiently for shear stress than for velocity. When needed, the velocity profile can be recovered from the stress profile by solving a one‐dimensional integral equation over the vertical. For most practical problems this equation can be solved in closed form. Comparisons are presented between the DSS technique, the standard velocity solution technique and analytical solutions for wind‐driven circulation in an unstratified, closed, rectangular channel governed by the linear equations of motion. In no case was the computational effort required by the velocity solution competitive with the DSS when a physically realistic boundary layer was included. The DSS technique should be particularly beneficial in numerical models of relatively shallow water bodies in which the bottom and surface boundary layers occupy a significant portion of the water column.
Carriere, Philippe; Jeandel, Denis
doi: 10.1002/fld.1650121003pmid: N/A
A three‐dimensional finite element method for the simulation of thermoconvective flows is presented. Vector‐parallel performances of some preconditioned conjugate gradient methods are compared for solving both large linear systems and the Stokes problem. As significant examples, numerical experiments on the steady two‐ and three‐dimensional Rayleigh‐Bénard convection at high Prandtl number are reported.
doi: 10.1002/fld.1650121004pmid: N/A
One of the main factors limiting the widespread use of computational fluid dynamics codes for engineering design is their very large requirements both in terms of computer memory and CPU time. Distributed memory parallel computers offer both the potential for a dramatic improvement in cost/performance over conventional supercomputers and the scalability to large numbers of processors that is required if performance beyond that of current supercomputers is to be achieved. As part of an evaluation to explore the potential of such machines for computational fluid mechanics applications, a concurrent algorithm for the solution of the Navier‐Stokes equations has been developed and demonstrated on a hypercube parallel computer. The algorithm is based on a domain decomposition of a well‐established serial pressure correction algorithm. The algorithm is demonstrated on both a 32‐node scalar and eight‐node vector Intel iPSC/2 for complicated two‐dimensional laminar and turbulent flow problems with different grid sizes and numbers of processors. Speed‐ups relative to a single processor of 12.9 with 16 processors and 20.2 with 32 processors are achieved on a scalar iPSC/2, demonstrating the parallel efficiency of the algorithm. Measured performance on a 32‐node scalar iPSC/2 exceeds one‐sixth that of a Cray X‐MP running the original serial algorithm. The performance of the algorithm on an eight‐node vector iPSC/2 exceeds that of the larger scalar hypercube and is about one‐fifth that of the Cray X‐MP. With cost/performance more than 10 times better than the Cray, these results dramatically show the cost effectiveness of vector hypercubes for this class of fluid mechanics algorithm.
doi: 10.1002/fld.1650121005pmid: N/A
An algorithm, called the Algebraic Continuity Equations Solver (ACES), is developed based on the concept that two algebraic equations (three for 3D problems) can be generated from rearranging the discretized continuity equations. These rearranged equations are used to re‐compute the two velocity components (three for 3D problems), whose values are already obtained from solving the momentum equations. When written in a Navier‐Stokes computer code, this algorithm is equivalent to a fairly concise set of statements and can be implemented immediately after the computation of the continuity equation. In our analysis, ACES is used in conjunction with a grid having nodal velocity components at the vertices and the nodal pressure at the centre of each computational cell. With the aid of ACES, correction of velocity components during the iteration can be inexpensively made, leading to faster convergence rates or rendering otherwise divergent computations convergent. Test problems include benchmark problems such as lid‐driven cavity flows and buoyancy‐driven cavity flows of various parametric values and grid sizes. A 3D time‐dependent flow in an irregular geometry is also investigated. Discussions are presented to clarify some relevant issues. A possible reason why we think ACES is capable of improving the convergence rates is also given.
doi: 10.1002/fld.1650121006pmid: N/A
A semi‐implicit pseudo‐spectral collocation method using a third‐order Runge‐Kutta numerical scheme for the full Navier‐Stokes equations is described. The Courant‐Friedrichs‐Lewy condition is overcome by the implicit handling of a diffusive term, as suggested by Harned and Kerner. All such terms are solved with an iterative scheme in the Fourier space. Simulation of thermal convection in 2D compressible fluids is made by expanding variables on a Fourier‐Chebyshev basis. We give some examples of sub‐ and supersonic steady solutions in the case where the heat flux at the upper boundary is governed by a black body.
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