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4.3. Numerical methods Since the NS equations are non-linear, no general analytic solution exists; in- stead, numerical techniques can be used. By using a discretization practice, the NS equations are transferred into a system of non-linear algebraic equa- tions which than can be solved numerically. There are three main discretization methods that are used for fluid problems; the finite difference (FD) method, the finite element (FE) method and the finite volume method (FV). In all nu- merical studies presented in this thesis, the commercial general CFD solver STAR-CD ver. 3.26 has been used. This code is based on the FV methods. For the temporal discretization, two different schemes have been used, a first order implicit scheme and the Crank-Nicholson scheme. The implicit scheme has in principle no restrictions on the time step δt, but it must be small enough so as to resolve the fluctuations and additionally to limit the temporal approximation errors. The Crank-Nicholson scheme is of formal sec- ond order accuracy, but as for all central second-orders schemes, it is liable to introduce non-physical oscillations when the viscous smallest scales are not fully resolved. The Crank-Nicholson scheme can be blended with an implicit scheme. This will then lead to a reduction in the formal order of accuracy of the numerical scheme but with the gain of enhanced stability. When deter- mining the size of the time step it is important to consider the coupling to the spatial discretization and the speed which the information is propagating with (i.e. the local physical properties of the flow). This means that time step must be small enough to ensure that information can not propagate no more than over a computational cell δx during a time step δt. This condition allows the numerical scheme to track the propagation of physical information and it is ex- pressed in numerical analysis in form of the Courant condition (CFL) number that must be below 1: CFL= |U|δt ≤1 (4.7) δx where U = u ± c is the physical propagation speed, with c being the speed of sound and u the convection velocity. For the approximation of the spatial discretization of the convection terms, different schemes are available in the used code. Three of them have been used, the first order Upwind Differencing (UD) scheme, the formal second or- der Monotone Advection and Reconstruction Scheme (MARS) and a blended Central Differencing (CD) scheme. The UD scheme is known to preserve the physical bounds of the fluxes, but can in many cases lead to numerical dif- fusion. A second order scheme will better preserve steep gradients, but can also introduce non-physical oscillations, known as numerical dispersion. The MARS scheme employs a Total Variation Diminishing (TVD) scheme, making 4.3. NUMERICAL METHODS 27PDF Image | Numerical computations of the unsteady flow in a radial turbine
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