Brian Launder

Progress in the development of a Reynolds-stress turbulence closure

The paper develops proposals for a model of turbulence in which the Reynolds stresses are determined from the solution of transport equations for these variables and for the turbulence energy dissipation rate E. Particular attention is given to the approximation of the pressure-strain correlations; the forms adopted appear to give reasonably satisfactory partitioning of the stresses both near walls and in free shear flows. Numerical solutions of the model equations are presented for a selection of strained homogeneous shear flows and for two-dimensional inhomogeneous shear flows including the jet, the wake, the mixing layer and plane channel flow. In addition, it is shown that the closure does predict a very strong influence of secondary strain terms for flow over curved surfaces.

The prediction of laminarization with a two-equation model of turbulence

Abstract The paper presents a new model of turbulence in which the local turbulent viscosity is determined from the solution of transport equations for the turbulence kinetic energy and the energy dissipation rate. The major component of this work has been the provision of a suitable form of the model for regions where the turbulence Reynolds number is low. The model has been applied to the prediction of wall boundary-layer flows in which streamwise accelerations are so severe that the boundary layer reverts partially towards laminar. In all cases, the predicted hydrodynamic and heat-transfer development of the boundary layers is in close agreement with the measured behaviour.

Ground effects on pressure fluctuations in the atmospheric boundary layer

Proposals are made for modelling the pressure-containing correlations which appear in the transport equations for Reynolds stress and heat flux in a simple way which accounts for gravitational effects and the modification of the fluctuating pressure field by the presence of a wall. The predicted changes in structure are shown to agree with Youngs (1975) measurements in a free stratified shear flow and with the Kansas data on the atmospheric surface layer.

The calculation of low-Reynolds-number phenomena with a two-equation model of turbulence

Abstract The paper presents numerical predictions of various turbulent shear flows in which the structure of the viscous sublayer exerts appreciable influence on the flow. The model of turbulence employed is one where the turbulence energy and its dissipation rate are calculated by way of transport equations which are solved simultaneously with the conservation equations for the mean flow. The flows considered include isothermal low Reynolds number pipe flows, and wall boundary layers with streamwise pressure gradient and wall injection; the predictions span both natural transition and laminarisation. Although complete agreement with experiment is not yet achieved in every case, it is argued that only a turbulence model of (at least) this level of complexity will permit a universal modelling of the near-wall turbulence structures commonly found in thermal power equipment.

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.

Second-moment closure: present… and future?☆

Abstract The paper summarizes the present position of second-moment closure and outlines possible directions for future development. It is first argued that a simple form of second-moment treatment that has been widely used for computing industrial flows gives demonstrably superior predictive accuracy that any eddy—viscosity model. More complex schemes, now in the final phases of development, that exactly satisfy various limiting constraints, give a further marked improvement in our ability to mimic the response of turbulence to external inputs. The inclusion of such models into commercial software over the next few years is quite feasible. The desirability of introducing a further second-rank tensor into the closure is considered; the conclusion reached is that, for most applications, the likely benefits would not justify the additional effort. The split-spectrum approach may, however, be attractive for certain flows with unusual spectral distributions of energy. The introduction of a second scale-related equation is arguably a more sensible approach since the extra computational cost is small while the added flexibility could bring significant benefits in modeling turbulence far from equilibrium.

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.

Contribution towards a Reynolds-stress closure for low-Reynolds-number turbulence

The problem of closing the Reynolds-stress and dissipation-rate equations at low Reynolds numbers is considered, specific forms being suggested for the direct effects of viscosity on the various transport processes. By noting that the correlation coefficient uv2/u2v2 is nearly constant over a considerable portion of the lowReynolds-number region adjacent to a wall the closure is simplified to one requiring the solution of approximated transport equations for only the turbulent shear stress, the turbulent kinetic energy and the energy dissipation rate. Numerical solutions are presented for turbulent channel flow and sink flows at low Reynolds number as well as a case of a severely accelerated boundary layer in which the turbulent shear stress becomes negligible compared with the viscous stresses. Agreement with experiment is generally encouraging.

Impinging jet studies for turbulence model assessment—I. Flow-field experiments

Abstract The paper reports an extensive set of measurements of a turbulent jet impinging orthogonally onto a large plane surface. Two Reynolds numbers have been considered, 2.3 × 10 4 and 7 × 10 4 while the height of the jet discharge above the plate ranges from two to ten diameters, with particular attention focused on two and six diameters. The experiment has been designed so that it provides hydrodynamic data for conditions the same as those for which Baughn and Shimizu [ ASME J. Heal Transfer 111, 1096 (1989)] have recently reported Nusselt number data (at Re = 23 000). In both experiments, before discharge, the air passed along a smooth pipe sufficiently long to give fully developed flow at the exit plane of the jet—a feature that is helpful in using the data for turbulence-model evaluation. Hot-wire measurements have been made with pipes of nominally one-inch (26 mm) and four inches (101.6 mm) diameter. Data are reported of the mean velocity profile in the vicinity of the plate surface and also of the three Reynolds-stress components lying in the x-r plane. Computational results reported in a companion paper [ Inl. J. Heal Mass Transfer 36, 2685–2697 (1993)] indicate a good degree of internal consistency between the mean and turbulent field data in that models predicting the mean flow poorly (or well) also predict the turbulence data poorly (well).

On the effects of a gravitational field on the turbulent transport of heat and momentum

This paper suggests a simple way of including gravitational effects in the pres-sure-containing correlations that appear in the equations for the transport of Reynolds stress and heat flux. The predicted changes in structure due to the gravitational field are shown to agree closely with Websters (1964) measurements in a stably stratified shear flow.