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Page 378
APPLICATION OF COMPUTATIONAL FLUID DYNAMICS TO VENTILATION SYSTEM DESIGN
formed the basis for most of the initial work done in computational fluid dynamics. Subsequently the method of weighted residuals provided an alternative. This approach starts by equating an integral of the product of a weighting function and the residual (difference between the true solution and the approximation) to zero. The different weighting functions define the particular method.
Method of Weighted Residuals (MWR). Let L(u) represent a differential operator, and f, a scalar valued real function then for the equation L(u) f the residual ε is defined by L(û) f ε
(14.1)
where û is an approximation of u. The MWR multiplies this residual by an orthogonal weighting function w; thus
冕w [L(û) f ] dΩ 0
(14.2)
where Ω is the domain of integration. This form allows a host of numerical approximations, including collocation, subdomain (finite-volume), and Galerkin methods. The weighting and interpolation functions are of compact local support and zero elsewhere, thus generating an N N sparse system of simultaneous equations for the discrete problem. In the Galerkin approach the functions used to approximate u are the same as the weighting functions. Finite-element approximations for the Galerkin method follow directly from this approach. Petrov–Galerkin methods employ different functions for the weights and approximations. In the finite-volume method the weighting function is one over the cell volume. 14.2.2
Grid-Free Methods
All the CFD ventilation studies to date employing commercially available software packages use a fixed grid with either finite-element or finite-volume discretizations. A significant amount of time and effort go into creating the mesh for these problems, and one limitation is the ability to generate adequate, efficient meshes. In addition, all of these codes employ either a RANS (Reynold’s averaged Navier–Stokes) approach with scalar eddy viscosity turbulence models, such as k-epsilon; or a large eddy simulation (LES) approach with a subgrid turbulence model. Both approaches require assumptions and empirically determined fit factors to complete the model. A different approach is possible, one that employs no assumptions about turbulence but solves the Navier–Stokes equations directly. This direct numerical simulation (DNS) approach can be implemented without a mesh via the discrete-vortex method (DVM) developed by Chorin (1973). This method has several appealing advantages and some drawbacks. Here a time-dependent calculation is used to solve the vorticity transport form of the incompressible Navier–Stokes equations, and discretize the vorticity into little “blobs.” The method works well for certain types of high-Reynolds-number flows with boundary-layer separation, and has been applied to