A global approach to error estimation and physical diagnostics in multidimensional computational fluid dynamics

Abstract

SUMMARYAn approach for simultaneously assessing numerical accuracy and extracting physical information from multidimensional calculations of complex (engineering) flows is proposed and demonstrated. The method is based on global balance equations, i.e. volume-integrated partial differential equations for primary or derived physical quantities of interest. Balances can be applied to the full computational domain or to any subdomain down to the single-cell level. Applications to in-cylinder flows in reciprocating engines are used for illustration. It is demonstrated that comparison of the relative magnitude of the terms in the balances provides insight into the physics of the flow being computed. Moreover, for quantities that are not conserved at the cell or control volume level in the construction of the numerical scheme, the imbalance allows a direct assessment of numerical accuracy in a single run using a single mesh. The mean kinetic energy imbalance is shown to be a particularly sensitive indicator of numerical accuracy. This simple and powerful diagnostic approach can be implemented for finite-difference, finite-volume or finite-element methods.

Adaptive Grid Refinement Using Cell-Level and Global Imbalances

“…3 In applications including incylinder flows, constraints such as the total number of cells or projected CPU time (cost), rather than the desired error level, are likely to limit the level of refinement that is practicable. Rapidly improving computing cost-to-performance ratios coupled with solution-adaptive mesh refinement and higher-order numerical schemes should eventually reduce numerical inaccuracy in three-dimensional time-dependent CFD to the point where the focus can return to physical modelling.…”

“…In the automotive industry, for example, CFD analysis of in-cylinder flow and combustion processes in reciprocating internal combustion (IC) engines represents an application at the frontier between research and practicable design tool. [1][2][3] Further progress in the modelling of such complex phenomena demands that numerical accuracy be isolated from physical submodel performance (e.g. turbulence, turbulent combustion and fuel spray models), quantified and controlled.…”

“…Thus measures of numerical accuracy and strategies for improved accuracy remain a major theme in the CFD literature. [3][4][5] Here the focus is on automated solution-adaptive local mesh refinement where mesh density is varied in space and (in time-dependent computations) in time to maintain a specified level of accuracy. Issues include (i) choice of numerical algorithm and data structure, (ii) formulation of local error estimators to establish the spatial distribution of grid points or cells and (iii) formulation of global error measures for determining when the solution is 'good enough'.…”

Assessing the credibility of a series of computational fluid dynamic simulations of open channel flow

“…The assumption of asymptotic convergence is especially important and applies to any sort of grid convergence study. Although the leading order term in a Taylor series analysis of a spatial discretization scheme provides a meaningful estimate of convergence rate, this only holds as the grid spacing approaches zero (Haworth et al, 1993). Extrapolation to zero mesh spacing of results computed on two or more finite resolution grids will be misleading, if the solutions are outside the convergence radius, and it is not possible to know a priori if a given mesh and numerical scheme lie within this limit.…”

Element residual error estimate for the finite volume method

“…Apart from several attempts aimed at measuring the numerical diffusion [16,17], the scope of single-mesh single-run error estimates aimed at the complete discretisation error is very limited. Some interesting attempts include the cell imbalance error estimate by Haworth et al [18], the method based on higher order face interpolation by Muzaferija [19] and the Taylor series and moment error estimate by Jasak [20,21], respectively based on the Taylor series truncation error analysis and the imbalance in the higher moments of the solution.…”