The accuracy of numerical simulation algorithms is one of main concerns in modern Computational Fluid Dynamics. Development of new and more accurate mathematical models requires an insight into the problem of numerical errors. In order to construct an estimate of the solution error in Finite Volume calculations, it is first necessary to examine its sources. Discretisation errors can be divided into two groups: errors caused by the discretisation of the solution domain and equation discretisation errors. The first group includes insufficient mesh resolution, mesh skewness and non-orthogonality. In the case of the second order Finite Volume method, equation discretisation errors are represented through numerical diffusion. Numerical diffusion coefficients from the discretisation of the convection term and the temporal derivative are derived. In an attempt to reduce numerical diffusion from the convection term, a new stabilised and bounded second-order differencing scheme is proposed. Three new methods of error estimation are presented. The Direct Taylor Series Error estimate is based on the Taylor series truncation error analysis. It is set up to enable single-mesh single-run error estimation. The Moment Error estimate derives the solution error from the cell imbalance in higher moments of the solution. A suitable normalisation is used to estimate the error magnitude. The Residual Error estimate is based on the local inconsistency between face interpolation and volume integration. Extensions of the method to transient flows and the Local Residual Problem error estimate are also given. Finally, an automatic error-controlled adaptive mesh refinement algorithm is set up in order to automatically produce a solution of pre-determined accuracy. It uses mesh refinement and unrefinement to control the local error magnitude. The method is tested on several characteristic flow situations, ranging from incompressible to supersonic flows, for both steady-state and transient problems.

Error analysis and estimation for the finite volume method with applications to fluid flows

Publisher Summary This chapter presents a discussion on jet impingement heat transfer. The chapter describes the applications and physics of the flow and heat transfer phenomena, available empirical correlations and values they predict, and numerical simulation techniques and results of impinging jet devices for heat transfer. The relative strengths and drawbacks of the Reynolds stress model, algebraic stress models, shear stress transport, and v 2 f turbulence models for impinging jet flow and heat transfer are compared in the chapter. The chapter provides select model equations as well as quantitative assessments of model errors and judgments of model suitability. The review of recent impinging jet research publications identified a series of engineering research tasks that are important for improving the design and resulting performance of impinging jets: (1) clearly resolve the physical mechanisms by which multiple peaks occur in the transfer coefficient profiles, and clarify which mechanism(s) dominate in various geometries and Reynolds number regimes, (2) develop a turbulence model, and associated wall treatment if necessary, that reliably and efficiently provides time-averaged transfer coefficients, (3) develop alternate nozzle and installation geometries that provide higher efficiency, meaning improved Nu profiles at either a set flow or set blower power, and (4) further explore the effects of jet interference in jet array geometries, both experimentally and numerically. This includes improved design of exit pathways for spent flow in array installations.

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Jet Impingement Heat Transfer: Physics, Correlations, and Numerical Modeling

Computational Fluid Dynamics for urban physics: Importance, scales, possibilities, limitations and ten tips and tricks towards accurate and reliable simulations

Rayleigh–Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing. II

50 years of Computational Wind Engineering: Past, present and future

A new k-ϵ eddy viscosity model for high reynolds number turbulent flows

A new model dissipation rate equation is proposed based on the dynamic equation of the mean-square vorticity fluctuation for large Reynolds number turbulence. The advantage of working with the vorticity fluctuation equation is that the physical meanings of the terms in this equation are more clear than those in the dissipation rate equation. Hence, the model development based on the vorticity fluctuation equation is more straightforward. The resulting form of the model equation is consistent with the spectral energy cascade analysis introduced by Lumley. The proposed model dissipation rate equation is numerically well behaved and can be applied to any level of turbulence modeling. It is applied to a realizable eddy viscosity model. Flows that are examined include: rotating homogeneous shear flows; free shear flows; a channel flow and flat plate boundary layers with and without pressure gradients; and backward facing step separated flows. In most cases, the present model predictions show considerable improvement over the standard kappa-epsilon model.

Zhigang Yang

Papers

A new k-ϵ eddy viscosity model for high reynolds number turbulent flows

A vorticity dynamics based model for the turbulent dissipation: Model development and validation