Tim Craft

Development and application of a cubic eddy-viscosity model of turbulence

Abstract Many quadratic stress-strain relations have been proposed in recent years to extend the applicability of linear eddy-viscosity models at modest computational cost. However, comparison shows that none achieves much greater width of applicability. This paper, therefore, proposes a cubic relation between the strain and vorticity tensor and the stress tensor, which does much better than a conventional eddy-viscosity scheme in capturing effects of streamline curvature over a range of flows. The flows considered range from simple shear at high strain rates and pipe flow, to flows involving strong streamline curvature and stagnation.

Impinging jet studies for turbulence model assessment—II. An examination of the performance of four turbulence models

Abstract Four turbulence models are applied to the numerical prediction of the turbulent impinging jets discharged from a circular pipe measured by Cooper el al. [ Int. J. Heat Mass Transfer 36, 2675–2684 (1993)], Baughn and Shimizu [ ASME J. Heat Transfer 111, 1096–1098 (1986)] and Baughn el al. [ASME Winter Annual Meeting, November 1992]. They comprise one k-e eddy viscosity model and three second-moment closures. In the test cases selected, the jet discharge was two and six diameters above a plane surface orthogonal to the jets axis. The Reynolds numbers were 2.3 × 10 4 and 7 × 10 4 the flow being fully developed at the discharge plane. The numerical predictions, obtained with an extended version of the finite-volume TEAM code, indicate that the k-e model and one of the Reynolds stress models lead to far too large levels of turbulence near the stagnation point. This excessive energy in turn induces much too high heat transfer coefficients and turbulent mixing with the ambient fluid. The other two second-moment closures, adopting new schemes for accounting for the walls effect on pressure fluctuations, do much better though one of them is clearly superior in accounting for the effects of the height of the jet discharge above the plate. None of the schemes is entirely successful in predicting the effects of Reynolds number. It is our view, however, that the main cause of this failure is the two-equation eddy viscosity scheme adopted in all cases to span the near-wall sublayer rather than the outer layer models on which the present study has focused.

A Reynolds stress closure designed for complex geometries

Abstract The paper describes steps in the development of a low Reynolds number second-moment closure for general flow geometries. This requirement means that the model cannot contain geometry-specific quantities, such as the wall-normal vector or wall distance. In their place, invariant dimensionless “gradient indicators” are introduced. New models are also devised for stress dissipation to capture the very diverse behaviour of the different components of eij in the walls vicinity with and without shear. A novel decomposition of the fluctuating pressure terms is also proposed. Applications are shown for shear-free boundary regions, plane channel, and stagnation flows.

Progress in the generalization of wall-function treatments

Abstract This paper describes progress in developing an analytical representation of the variation of the dynamic variables and temperature across the near-wall sublayer of a turbulent flow. The aim is to enable the effective “resistance” of the viscous sublayer to the transport of heat and momentum to be packaged in the form of a “wall function”, thus enabling CFD predictions of convective heat transfer to be made without incurring the cost of the very fine near-wall grid that would otherwise have to be adopted. While the general idea is not new, the detailed strategy contains many new features, which have led to a scheme capable of accounting for the effects of buoyancy, pressure gradient and of variations in molecular transport properties. The scheme is applied to the problem of forced and mixed convection in a vertical pipe and to the opposed wall jet with encouraging results.

Prediction of turbulent transitional phenomena with a nonlinear eddy-viscosity model

Abstract This paper describes a new nonlinear eddy-viscosity model of turbulence designed with a view to predicting flow far from equilibrium, including transition. The scheme follows earlier UMIST practice in adopting a cubic relation between the stress and the strain/ vorticity tensors but broadens the range of flows to which the model applies by including a third transport equation for an anisotropy parameter of the stress field. Applications are shown for transition on a flat plate at different levels of free-stream turbulence, for the normal impingement of a turbulent jet on a flat plate, and for the flow around a turbine blade. The model is shown to generate much more realistic predictions than what is said to be the best of the linear eddy-viscosity schemes.

Marine Cloud Brightening

The idea behind the marine cloud-brightening (MCB) geoengineering technique is that seeding marine stratocumulus clouds with copious quantities of roughly monodisperse sub-micrometre sea water particles might significantly enhance the cloud droplet number concentration, and thereby the cloud albedo and possibly longevity. This would produce a cooling, which general circulation model (GCM) computations suggest could—subject to satisfactory resolution of technical and scientific problems identified herein—have the capacity to balance global warming up to the carbon dioxide-doubling point. We describe herein an account of our recent research on a number of critical issues associated with MCB. This involves (i) GCM studies, which are our primary tools for evaluating globally the effectiveness of MCB, and assessing its climate impacts on rainfall amounts and distribution, and also polar sea-ice cover and thickness; (ii) high-resolution modelling of the effects of seeding on marine stratocumulus, which are required to understand the complex array of interacting processes involved in cloud brightening; (iii) microphysical modelling sensitivity studies, examining the influence of seeding amount, seed-particle salt-mass, air-mass characteristics, updraught speed and other parameters on cloud–albedo change; (iv) sea water spray-production techniques; (v) computational fluid dynamics studies of possible large-scale periodicities in Flettner rotors; and (vi) the planning of a three-stage limited-area field research experiment, with the primary objectives of technology testing and determining to what extent, if any, cloud albedo might be enhanced by seeding marine stratocumulus clouds on a spatial scale of around 100×100 km. We stress that there would be no justification for deployment of MCB unless it was clearly established that no significant adverse consequences would result. There would also need to be an international agreement firmly in favour of such action.

On the spreading mechanism of the three-dimensional turbulent wall jet

The paper explores, using different levels of turbulence closure, the computed behaviour of the three-dimensional turbulent wall jet in order to determine the cause of the remarkably high lateral rates of spread observed in experiments. Initially, to ensure accurate numerical solution, the equations are cast into the form appropriate to a self-similar shear flow thereby reducing the problem to one of two independent variables. Our computations confirm that the strong lateral spreading arises from the creation of streamwise vorticity, rather than from anisotropic diffusion. The predicted ratio of the normal to lateral spreading rates is, however, very sensitive to the approximation made for the pressure–strain correlation. The version that, in other flows, has led to the best agreement with experiments again comes closest in calculating the wall jet, although the computed rate of spread is still some 50% greater than in most of the measurements. Our subsequent calculations, using a forward-marching scheme show that, because of the strong coupling between axial and secondary flow, the flow takes much longer to reach its self-preserving state than in a two-dimensional wall jet. Thus, it appears very probable that none of the experimental data are fully developed.

A new wall-function strategy for complex turbulent flows

Wall functions are widely used and offer significant computational savings compared with low-Reynolds-number formulations. However, existing schemes are based on assumed near-wall profiles of velocity, turbulence parameters, and temperature which are inapplicable in complex, nonequilibrium flows. A new wall function has therefore been developed which solves boundary-layer-type transport equations across a locally defined subgrid. This approach has been applied to a plane channel flow, an axisymmetric impinging jet, and flow near a spinning disc using linear and nonlinear k–ϵ turbulence models. Computational costs are an order of magnitude less than low-Reynolds-number calculations, while a clear improvement is shown in reproducing low-Re predictions over standard wall functions.

Progress in the Use of Non-Linear Two-Equation Models in the Computation of Convective Heat-Transfer in Impinging and Separated Flows

The paper considers the application of the Craft et al. [6]non-linear eddy-viscosity model to separating and impinging flows. The original formulation was found to lead to numerical instabilities when applied to flow separating from a sharp corner. An alternative formulation for the variation of the turbulent viscosity parametercμ with strain rate is proposed which, together with a proposed improvement in the implementation of the non-linear model, removes this weakness. It does, however, lead to worse predictions in an impinging jet, and a further modification in the expression for cμ is proposed, which both retains the stability enhancements and improves the prediction of the stagnating flow. The Yap [24] algebraic length-scale correction term, included in the original model, is replaced with a differential form, developed from that proposed by Iacovides and Raisee [10]. This removes the need to prescribe the wall-distance, and is shown to lead to superior heat-transfer predictions in both an abrupt pipe flow and the axisymmetric impinging jet. One predictive weakness still, however, remains. The proposed model, in common with other near-wall models tested for the abrupt pipe expansion, returns a stronger dependence of Nusselt number on the Reynolds number than that indicated by the experimental data.

Recent developments in second-moment closure for buoyancy-affected flows

Abstract The paper summarizes a new type of second-moment closure, more elaborate in form than earlier versions but designed to satisfy the two-component limit to which turbulence reduces at a wall or at a sharp density interface. Because they are intrinsically realizable, closures of this type are believed to offer the prospects of a wider range of applicability than earlier schemes. They may also be expected to display better numerical stability. Several illustrative applications are provided including the downward directed warm jet, the stratified mixing layer and buoyancy affected grid-turbulence decay. Extension of the scheme to near wall flows appears possible without introducing empirical ‘wall-reflection’ terms, at least in flows parallel to walls.