this article, we emphasize practical issues; an earlier survey by Bern and Eppstein [24] emphasized theoretical results. Although there is inevitably some overlap between these two surveys, we intend them to be complementary.

Many problems in fluid dynamics require the representation of complicated internal or external boundaries of the flow. Here we present a method for calculating time-dependent incompressible inviscid flow which combines a projection method with a "Cartesian grid" approach for representing geometry. In this approach, the body is represented as an interface embedded in a regular Cartesian mesh. The advection step is based on a Cartesian grid algorithm for compressible flow, in which the discretization of the body near the flow uses a volume-of-fluid representation. A redistribution procedure is used to eliminate time-step restrictions due to small cells where the boundary intersects the mesh. The projection step uses an approximate projection based on a Cartesian grid method for potential flow. The method incorporates knowledge of the body through volume and area fractions along with certain other integrals over the mixed cells. Convergence results are given for the projection itself and ...

An adaptive mesh projection method for the time-dependent incompressible Euler equations is presented. The domain is spatially discretised using quad/octrees and a multilevel Poisson solver is used to obtain the pressure. Complex solid boundaries are represented using a volume-of-fluid approach. Second-order convergence in space and time is demonstrated on regular, statically and dynamically refined grids. The quad/octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features. The source code of the method implementation is freely available.

this paper were performed with a scale factor of f Dt =6. Thus, for a prescribed setting of CFL=150, the actual CFL number will be linearly scaled between 25 for the thinnest tetrahedral cell to 150 for the most isotropic cell. The ultimate benefit of this procedure was a factor-of-two reduction in required solution cycles and, hence, computer time.

Cut Cell fs 0 1 fs 0 0 fs 5 2 fs 5 1 fs 5 0 T 1 T 2 T 0 T 3 tp 0 tp 1 tp 2 tp 3 fp 0 1 fp 3 0 fp 5 1 F [0-5] fp [face#] [poly#] fs [face#][seg#] T [0-n] tp[0-n] fp 0 0 x x fp 5 0 1 0 x 2 Figure 3-5: Anatomy of an abstract cut-cell.

this paper were performed with double precision, but the results have been truncated for brevity. The table shows a very good agreement between the actual L 2 error and E 2(I) for which we used CE defined by (A.21) and f(t) is computed by using the central difference formula with a step size 1.0E-05 which was found to give the identical result as that obtained by using the exact derivative of (A.28). In the last column of the table, the relative errors, defined by fi fi fi fi 1 \Gamma E 2(I) kx\Gammauk L 2 (I) fi fi fi fi , are shown. Starting with 1:15% error for NE = 5, the relative error goes down as the grid is refined. The rate of convergence is found to be 1:98 which verifies the result in the last section. As mentioned earlier, the local L 2 error 139 NE kx \Gamma uk L 2 (I) E 2(I) Relative Error 5 1:947E \Gamma 01 1:722E \Gamma 01 1:152E \Gamma 01 10 4:657E \Gamma 02 4:513E \Gamma 02 3:100E \Gamma 02 20 1:173E \Gamma 02 1:164E \Gamma 02 7:740E \Gamma 03 40 2:941E \Gamma 03 2:935E \Gamma 03 1:933E \Gamma 03 80 7:359E \Gamma 04 7:356E \Gamma 04 4:833E \Gamma 04 Table A.1: The results for (A.28) cannot be predicted accurately near inflection points. To see this, we computed the relative error locally for each element with NE =160. The plots are shown in Figure A.4. Note that the ordinates of the lower three plots are logarithmically scaled. On the top, the first derivative f(t) is plotted so that the inflection points can be easily located, and the upper-middle one is the plot of the elementwise relative error j1 \Gamma E 2(E) kx\Gammauk L 2 (E) j plotted in the middle of each element. It is seen that the relative error is significantly larger near inflection points than in the other region. However, as seen in the plot in the lower-middle, the actual ...

A parallel, finite-volume algorithm has been developed for large-eddy simulation (LES) of compressible turbulent flows. This algorithm includes piecewise linear least-square reconstruction, trilinear finite-element interpolation, Roe flux-difference splitting, and second-order MacCormack time marching. Parallel implementation is done using the message-passing programming model. In this paper, the numerical algorithm is described. To validate the numerical method for turbulence simulation, LES of fully developed turbulent flow in a square duct is performed for a Reynolds number of 320 based on the average friction velocity and the hydraulic diameter of the duct. Direct numerical simulation (DNS) results are available for this test case, and the accuracy of this algorithm for turbulence simulations can be ascertained by comparing the LES solutions with the DNS results. The effects of grid resolution, upwind numerical dissipation, and subgrid-scale dissipation on the accuracy of the LES are examined. Comparison with DNS results shows that the standard Roe flux-difference splitting dissipation adversely affects the accuracy of the turbulence simulation. For accurate turbulence simulations, only 3--5 percent of the standard Roe fluxdifference splitting dissipation is needed.