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4.4. NUMERICAL ACCURACY AND UNCERTAINTY 33 • Errors in computing the discrete solution (i.e. convergence errors in the solution process). • Errors due to round-off error due to a limited number of significant digits in the computations. The influence of these different errors can be examined by performing nu- merical accuracy studies (using successively refined grids) and comparing the results from the computations against measured or computed data. In the latter case, data from DNS is often used. The purpose of numerical accuracy studies is to state the order of accuracy of the used schemes. The numerical accuracy studies can also be used to compute the uncertainty of the results. To investigate the numerical accuracy and the uncertainty, the GCI-method proposed by Celik (2005) was used. The results from these studies show that the order of accuracy for the used code is between 1 and 2, depending on the flow case and the used turbulence modelling approach, see Paper 1 and Paper 2. This reflect the order of accuracy of the used schemes, where the formal second order MARS scheme is used for spatial discretization of the convection terms and a blended central difference scheme for the viscous terms. A blended Crank-Nicholson scheme was used for temporal discretization of the LES computations for the pipe cases and the first order Euler implicit temporal discretization and was used for the turbine computations and for the RANS computations. The GCI-method can also be used to estimate the uncertainty of the results. For the pipe-flow cases, the uncertainty of the computed velocity field at the finest grid was below 4% for the LES computations for the single bend case, and for the RANS computations on the double bend case the uncertainty was below 9.6% for the finest grid. For pulsatile flow, the evaluated quantity must be phase averaged. The total number of samples can be computed with a formula given in Johansson & Alfredsson (1988) for a given a given sampling error ε: ε(X) = √1 xrms (4.13) N Xm where N is the total number statistically independent samples, Xm is the mean value and xrms is the root mean square value. Since the inflow conditions for the double bend pipe was pulsating, the velocities must be phase averaged in the LES computations, which implies that 400 cycles had to be computed if the sampling error should be below 1%, and hence no uncertainty has been computed for the pulsatile flow cases. For the turbine computations, the numerical accuracy study was performed on simplified turbine geometry, where the complete wheel and only parts of the volute and the diffuser were modelled to reduce the computational time. ThePDF Image | Numerical computations of the unsteady flow in a radial turbine
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