A-posteriori error estimation using higher moments in computational fluid dynamics

• Main Author: Russant, Stuart
• Published: University of Manchester 2015
• Subjects: 621.4
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.647412

In industrial situations time is expensive and simulation accuracy is not always investigated because it requires grid refinement studies or other time consuming methods. With this in mind the goal of this research is to develop a method to assess the errors and uncertainties on computational fluid dynamics (CFD) simulations that can be adopted by industry to meet their requirements and time constraints. In a CFD calculation there are a number of sources of errors and uncertainties. An uncertainty is a potential deficiency that is due to a lack of knowledge of an activity of the modelling process, for example turbulence modelling. An error is defined as a recognisable deficiency that is not due to a lack of knowledge, for example numerical discretisation error. The process of determining the level of errors and uncertainties is termed verification and validation. The work aims to define an error estimation method for verification of numerical errors that can be produced during one simulation on a single grid. The second moment solution error estimate for scalar and vector quantities was proposed to meet these requirements. Where the governing equations of CFD, termed the first moments, represent the transport of primary variables such as the velocity, the second moments represents the transport of the primary variables squared such as the total kinetic energy. The second moments are formed by a rearrangement of the first moments. Based on a mathematical justification, an error estimate for vector or scalar quantities was defined from combinations of the solutions to the first and second moments. The error estimate was highly successful when applied to six test cases using laminar flow and scalar transport. These test cases used either central differencing with Gaussian elimination, or the finite volume method with the CFD solver Code_Saturne to conduct the simulations, demonstrating the applicability of the error estimate across solution methods. Comparisons were made to the numerical simulation errors, which were found using either the analytical or refined solutions. The comparisons were aided by the normalised cross correlation coefficient, which compared the similarity of the shape prediction, and the averaged summation coefficients, which compared the scale prediction. When using the first order upwind scheme the method consistently produced good predictions of the locations of error. When using the second order centred or second order linear upwind schemes there was similar success, but limited by influences from solution unboundedness, non-resolution of the boundary layer, the near-wall gradient approximation, and numerical pressure error. At high Reynolds numbers these caused the prediction of the location of error to degrade. This effect was made worse when using the second order schemes in conjunction with the constant value boundary condition. This was the case for the scalar or velocity simulations, and is caused by the unavoidable drop to first order accuracy during the near-wall gradient approximation that is required for the second moment source term approximation. The prediction of the scale demonstrated a dependence on the cell Peclet number. Below cell Peclet number 4 the increase of the estimate scale was linearly related to the increase of the error scale. The estimate scale consistently over-predicts by up to a factor of 3. This allows confidence that the true error level is below that which is predicted by the error estimate. At cell Peclet numbers greater than 4 the relationship between the scales remained linear, however, the estimate begins to under-predict the estimate. The exact relation becomes case dependent, and the highest under-prediction was by a factor of 10. In such circumstances a computationally inexpensive calibration can be done.