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

Some new developments of explicit algebraic Reynolds stress turbulence models (EARSM) are presented. The new developments include a new near-wall treatment ensuring realizability for the individual stress components, a formulation for compressible flows, and a suggestion for a possible approximation of diffusion terms in the anisotropy transport equation. Recent developments in this area are assessed and collected into a model for both incompressible and compressible three-dimensional wall-bounded turbulent flows. This model represents a solution of the implicit ARSM equations, where the production to dissipation ratio is obtained as a solution to a nonlinear algebraic relation. Three-dimensionality is fully accounted for in the mean flow description of the stress anisotropy. The resulting EARSM has been found to be well suited to integration to the wall and all individual Reynolds stresses can be well predicted by introducing wall damping functions derived from the van Driest damping function. The platform for the model consists of the transport equations for the kinetic energy and an auxiliary quantity. The proposed model can be used with any such platform, and examples are shown for two different choices of the auxiliary quantity.

An explicit algebraic Reynolds stress model for incompressible and compressible turbulent 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

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

This paper reports a comprehensive set of hot-wire measurements of a round buoyant plume which was generated by forcing a jet of hot air vertically up into quiescent environment. The boundary conditions of the experiment were measured, and are documented in the present paper in an attempt to sort out the contradictory mean flow results from the earlier studies. The ambient temperature was monitored to insure that the facility was not stratified and that the experiment was conducted in a neutral environment. The axisymmetry of the flow was checked by using a planar array of sixteen thermocouples and the mean temperature measurements from these are used to supplement the hot-wire measurements. The source flow conditions were measured so as to ascertain the rate at which the buoyancy was added to the flow. The measurements conserve buoyancy within 10 percent. The results are used to carry out the balances of the mean energy and momentum differential equations. In the mean energy equation it is found that the vertical advection of the energy is primarily balanced by the radial turbulent transport. In the mean momentum equation the vertical advection of momentum and the buoyancy force balance the radial turbulent transport. The buoyancy force is the second largest term in this balance and is responsible for the wider (and higher) velocity profiles in plumes as compared to jets. Budgets of the temperature variance and turbulence kinetic energy are also carried out in which thermal and mechanical dissipation rates are obtained as the closing terms. Similarities and differences between the two balances are discussed. It is found that even though the direct affect of buoyancy on turbulence, as evidenced by the buoyancy production term, is substantial, most of the turbulence is produced by shear. This is in contrast to the mean velocity field where the affect of buoyancy force is quite strong. Therefore, it is concluded that in a buoyant plume the primary affect of buoyancy on turbulence is indirect, and enters through the mean velocity field (giving larger shear production).

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Experiments on a round turbulent buoyant plume

A computational study is carried out to understand the physical mechanism responsible for the improvement in stall margin of an axial flow rotor due to the circumferential casing grooves. Computational fluid dynamics simulations show an increase in operating range of the low speed rotor in the presence of casing grooves. A budget of the axial momentum equation is carried out at the rotor casing in the tip gap in order to understand the physical process behind this stall margin improvement. It is shown that for the smooth casing the net axial pressure force at the rotor casing in the tip gap is balanced by the net axial shear stress force. However, for the grooved casing the net axial shear stress force acting at the casing is augmented by the axial force due to the radial transport of axial momentum, which occurs across the grooves and power stream interface. This additional force adds to the net axial viscous shear force and thus leads to an increase in the stall margin of the rotor.

Flow Mechanism for Stall Margin Improvement due to Circumferential Casing Grooves on Axial Compressors

In developing turbulence models, different authors have proposed various model constraints in an attempt to make the model equations more general (or universal). Most recent of these are the realizability principle (Lumley 1978, Schumann 1977), linearity principle (Pope 1983), rapid distortion theory (Reynolds 1987) and material indifference principle (Speziale 1983). In this paper we will discuss several issues concerning these principles and will pay special attention to the realizability principle raised by Lumley (1978). Realizability (defined as the requirement of non-negative energy and Schwarz’ inequality between any fluctuating quantities) is the basic physical and mathematical principle that any modeled equation should obey. Hence, it is the most universal, important and also the minimal requirement for a model equation to prevent it from producing unphysical results. In this paper we will describe in detail the principle of realizability, derive the realizability conditions for various turbulence models, and propose the model forms for the pressure correlation terms in the second moment equations. Detailed comparisons of various turbulence models (Launder et al 1975, Craft et al 1989, Zeman and Lumley 1976, Shih and Lumley 1985 and the one proposed here) with experiments and direct numerical simulations will be presented. As a special case of turbulence, we will also discuss the two-dimensional, two-component turbulence modeling.

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Advances in modeling the pressure correlation terms in the second moment equations

Wall functions are used in CFD to provide skin friction values and hence the boundary conditions for flow variables along solid surfaces. In this paper the effect of surface roughness on skin friction is incorporated in a wall function approach which uses Spalding’s formula. The use of Spalding’s formula extends the method to a wider range of wall distance than the logarithmic friction law. For CFD applications the results are then re-formulated for explicit calculation throught the use of an additional variable — the ratio of the surface roughness height to the distance from the solid surface. For a given computational grid this information is readily available in a CFD calculation. This methodology is applied to a linear compressor cascade that has been experimentally measured for different roughness values. The comparison of the simulation followed the same trends as the experiment, but with less overall effect.Copyright © 2004 by ASME

Aamir Shabbir

Papers

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

Experiments on a round turbulent buoyant plume

Flow Mechanism for Stall Margin Improvement due to Circumferential Casing Grooves on Axial Compressors

Advances in modeling the pressure correlation terms in the second moment equations

A Wall Function for Calculating the Skin Friction With Surface Roughness