Variable grid scheme applied to turbulent boundary layers
ABSTRACT A Crank-Nicolson type finite-difference scheme with a nonuniform grid spacing has been interpreted in terms of a coordinate stretching approach to show that it is second-order accurate. The variable grid scheme is applied to a flat plate laminar to turbulent boundary layer flow with a rapidly changing grid interval across the layer. The accuracy of the solution is determined for a different number of intervals and compared to results obtained with the Keller box scheme. The influence of changing the grid spacing on the accuracy of the solutions is determined for one coordinate stretching or grid spacing relation. The use of Richardson extrapolation is also investigated.
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- "If the mesh is uniform, the finite-difference scheme provides the second-order approximation with respect to both x and y. The two momentum equations are solved by the Blottner-type technique , which is well suited for marching problems. The induction equation is solved at each time step using the well-known ADI method. "
ABSTRACT: The proposed use of a flowing liquid metal layers as virtual first-walls for magnetic fusion energy reactors has prompted the development of numerical models capable of predicting the motion of such free surface liquid-metal flows within complex geometry boundaries and in the presence of strong magnetic fields. Several model variants were developed that utilize the assumption of toroidal axisymmetry to simplify the governing Navier–Stokes and Maxwell's equations to a 2D form. Typically an induced magnetic field formulation has been used to model eddy current formation and various numerical methods and free surface tracking techniques (including height function and volume-of-fluid) have been employed. These axisymmetric models predict a variety of interesting behavior including the effect of toroidal field gradients on the velocity profiles and stability, and the effect of surface-normal magnetic field components on toroidal motion and flow thickness. However, axisymmetric models cannot be used to simulate the true 3D geometry and magnetic field configuration of a magnetic fusion reactor. And so, a 3D, flexible geometry, multiple material, free surface magnetohydrodynamic (MHD) solver called HIMAG has been developed over the past several years. The HIMAG formulation is described in detail along with the results of several initial benchmark problems. Preliminary data from the application of HIMAG to several fusion relevant liquid wall problems including: (1) motion of lithium in a new sample holder for the Diverter Materials Evaluation System (DiMES) experiment on the DIII-D tokamak facility; (2) motion of gallium alloy in a quasi-2D film flow test section in the MTOR facility; (3) motion of gallium alloy in a 3D field film flow test section in the MTOR facility; are also presented and discussed. Finally, future plans for the HIMAG code, including application to the simulation of the effect of insulator coating cracks on closed channel MHD flows, are described.Fusion Engineering and Design 11/2004; 72(1-3-72):3-34. DOI:10.1016/j.fusengdes.2004.07.013 · 1.15 Impact Factor
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ABSTRACT: In the APEX study, one of the tasks focuses on the exploration and identification of the attractive options and issues for flowing liquid lithium walls in the NSTX device. In addition to constraints imposed by the machine, the operating conditions of the flowing liquid walls along the center stack and divertor areas are guided by MHD and heat removal requirements. In this paper, we present important MHD and heat removal issues and analysis for the proposed free surface lithium flows under NSTX conditions. It is shown that of all MHD effects, the one caused by the normal magnetic field is the most important. The flow over the center stack area is not affected by MHD interaction significantly, whereas flow over the inboard divertor undergoes strong MHD drag resulting in flow thickening by several times. The flow over the outboard divertor is essentially stopped. The analysis shows that a flow with an inlet velocity of 2 m/s and film thickness of about 4 mm can be established to provide surface temperature less than 400 ° C for the center stack under a projected NSTX total heating power of 10 MW operation.
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ABSTRACT: I0. I Introduction. This paper is a review of recent developments and problem areas in computational fluid dynamics, wherein "recent" is defined in relation to the publication of my book (Roache, 1972). (For brevity, that book will be referred to as "CFD" in this paper). Some of these topics were mentioned in CFD but not emphasized sufficiently; others were not in existence at the time of writing. CFD is not a truly comprehensive review of all the work done in the area up to mid-1972, and this review article is much less so for the work done up to the time of this writing, which is the fall of 1974. It is hoped that this review will still be of some utility, especially to younger people just getting into the field. 10.2 Significant General Publications. The literature in computational fluid dynamics has become very extensive, so that general, wide-ranging publications are very valuable. Three sets of conference proceedings are recommended: the AIAA Computational Fluid Dynamics Conference of July 1973 at Palm Springs (AIAA, 1973), and the Third (Paris) and Fourth (Boulder) International Conferences on Numerical Methods in Fluid Dynamics (Cabannes and Temann, 1972; Riehtmyer, 1974). These two biannual affairs alternate, so that we may expect another addition of a conference proceedings to the literature every year. In addition, three survey works are to be recommended. The AGARDograph by Taylor (1973) covers inviscid and viscous flows throughout the Mach number range from incompressible, transonic, through supersonic flows. It contains 261 references and covers well the background for "conventional" methods. The article by Orszag and Israeli (1974) in the Annual Review of Fluid Mechanics series is recommended for an introduction to higher-order methods and to spectral methods, and for general discussions on viscous flow simulation. The monograph by Kreiss and Oliger (1973) presents a detailed analysis of higher-order finite difference methods for inviscid equations, and will provide the nonmathematician with the background vocabulary for mathematical papers.