Two new versions of the k - w two-equation turbulence model will be presented. The new Baseline (BSL) model is designed to give results similar to those of the original k - w model of Wilcox. but without its strong dependency on arbitrary freestream values. The BSL model is identical to the Wilcox model in the inner SOC7£; of the boundary-layer but changes gradually to the standard k - f. model (in a k - w fonnulation) towards the boundary-layer edge. The new model is also virtually identical to the k - f. model for free shear layers. The second version of the model is called Shear-Stress Transport (SSn model. It is a variation of the BSL model with the additional ability to account for the transport of the principal turbulent shear stress in adverse pressure gradient boundary-layers. The model is based on Bradshaw's assumption that the principal shear-stress is pro­ portional to the turbulent kinetic energy, which is introduced into the definition of the eddy-viscosity. Both models are tested for a large number of different fiowfields. The results of the BSL model are similar to those of the original k - w model, but without the undesirable free stream dependency. The predictions of the SST model are also independent of the freestrearn values but show better agreement with exper­ imental data for adverse pressure gradient boundary-layer flows.

ZONAL TWO EQUATION k-w TURBULENCE MODELS FOR AERODYNAMIC FLOWS

2~5 However, the effects of the kinematic viscosity on the turbulence structure were ignored in many of these treatments. Consequently, the exact boundary conditions at the wall cannot be used when the turbulence Reynolds number is not high as, e.g., in flows with rapid expansions or near the transition/turbulence interface. The general goal of the present investigation was to develop a single transport model from the Navier-Stokes equation for accurate predictions of skin friction, heat transfer, and fluctuating kinetic energy distributions in transitional and turbulent flow regimes. As a first step toward this general goal, a new turbulence model valid down to the solid wall is formulated in this paper. Turbulence model equations which provide predictions of the flow within the viscous layer adjacent to the wall have been proposed by several investigators.3'4'6'7 Although the general approach of the present model is the same as that of Jones and Launder,3 the detailed proposals are substantially different. In the present study, the Taylor series expansion technique was used to systematically investigate the proper behavior of the turbulent shear stress and the kinetic energy and its rate of dissipation near a solid wall. The results were used in developing a new turbulence model which retains the proper physical behavior of the balance between the dissipation and the molecular diffusion of the turbulent kinetic energy at the solid wall. The model was applied to the problems of a fully developed turbulent channel flow and of a turbulent boundary-layer flow over a flat plate. Results on skin friction, the distribution of mean velocity, turbulent shear stress, and turbulent kinetic energy will be presented and compared with available experimental data and with the theory of Jones and Launder.

Predictions of Channel and Boundary-Layer Flows with a Low-Reynolds-Number Turbulence Model

Because of their ubiquitous presence in high-speed flight and their impact on vehicle and component performance, shock-wave/boundary-layer interactions have been studied for about 50 years. Despite truly remarkable progress in computational and measurement capabilities, there are still important quantities that cannot be predicted very accurately, that is, peak heating in strong interactions, or cannot be predicted at all, that is, unsteady pressure loads. There remain observations that cannot be satisfactorily explained and physical processes that are not well understood. Much work remains to be done. Based on the author's own views and those of colleagues, some suggestions are made as to where future efforts might be focused. Just as the first workers in the field could not have foreseen the capabilities generated by the computer/instrumentation revolution of the past 20 years, it is probably fair to assume that the extent of our vision and imagination in the year 2000 is equally limited. New simulation and measurement techniques will doubtlessly become available in the next 10 or 20 years, the results from which will render many of the current concerns moot

Fifty Years of Shock-Wave/Boundary-Layer Interaction Research: What Next?

Boundary-fitted coordinate systems for numerical solution of partial differential equations—A review

Progress in shock wave/boundary layer interactions

Several multiequation eddy viscosity models of turbulence are used with the Navier-Stokes equations to compute three classes of experimentally documented shock-separated turbulent boundary-layer flows. The types of flow studied are: (1) a normal shock at transonic speeds in both a circular duct and a two-dimensional channel; (2) an incident oblique shock at supersonic speeds on a flat surface; and (3) a two-dimensional compression corner at supersonic speeds. Established zero-equation (algebraic), one-equation (kinetic energy), and two-equation (kinetic energy plus length scale) turbulence models are each utilized to describe the Reynolds shear stress for the three classes of flows. These models are assessed by comparing the calculated values of skin friction, wall pressure distribution, velocity, Mach number, and turbulent kinetic energy profiles with experimental measurements. Of the models tested the two-equation model results gave the best overall agreement with the data.

Comparison of Multiequation Turbulence Models for Several Shock Boundary-Layer Interaction Flows

Extensive boundary-layer measurements have been made on a cone-ogive-cylinder model at a free-stream Mach number of 7.0 and momentum-thickness Reynolds number of 8500. Mean flow transformations and calculated turbulence correlations are presented which are in good agreement with previous incompressible results. New quantitative turbulence measurements including measurements of the first higher moment and probability density of fluctuations in mass flow and total temperature in hypersonic flow are also presented. The higher moment and probability density data show that the characters of the fluctuation modes of the mass flow and total temperature are significantly different in the wall region and in the outer part of the boundary layer. These differences together with data on the turbulence scale and lifetime obtained from autocorrelation and space-time correlation measurements are discussed.

Mean and fluctuating flow measurements of a fully-developed, non-adiabatic, hypersonic boundary layer

An experiment is described that tests and guides computations of a shock-wave turbulent boundary-layer interaction flow over a 20° compression corner at Mach 2.85. Numerical solutions of the time-averaged NavierStokes equations for the entire flow field, employing various turbulence models, are compared with the data. Each model is evaluated critically by comparisons with the details of the experimental data. Experimental results for the extent of upstream pressure influence and separation location are compared with numerical predictions for a wide range of Reynolds numbers and shock-wave strengths.

Reynolds Number Effects on Shock-Wave Turbulent Boundary-Layer Interactions

Solutions of the time-dependent, mass-averaged Navier-Stokes equations are compared In detail with experimental results obtained on an axisymmetric "bump" model at a transonic Mach number that produced an extensive separated now region. In addition, an inverse boundary method is evaluated for this type of flow. The Cebeci-Smith algebraic and the Wilcox-Rubesin two-equation turbulence models used in the Navier-Stokes calculations both predict the maximum boundary-layer displacement thickness generated by the interaction reasonably well, with the details of the now best described with the two-equation formulation. However, both models predict a shock location substantially farther aft on the bump than observed experimentally. This error in shock location was slightly less with the two-equation model (0.12 chord compared with 0.16 chord). In the vicinity of the shock, the calculations predict a more rapid increase in turbulent shear stress than observed in the experimental results; this more rapid increase is believed to be the cause or the poor predictions in shock position.

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Comparison Between Experiment and Prediction for a Transonic Turbulent Separated Flow

Turbulent shear-stress, eddy-viscosity, mixing-length, heat-flux and Prandtl number distributions across a hypersonic turbulent boundary layer have been determined from the 'time-averaged' conservation equations using experimental mean profile data obtained at several streamwise locations in a fully developed turbulent boundary layer with negligible pressure gradient. The eddy-viscosity, mixing-length and Prandtl number results show general agreement with previous incompressible and adiabatic compressible correlations. However, when the turbulent Prandtl number is defined using total enthalpy as opposed to static enthalpy no clear correlation of the results in the outer portion of the boundary layer could be obtained.

C. C. Horstman

Papers

Comparison of Multiequation Turbulence Models for Several Shock Boundary-Layer Interaction Flows

Mean and fluctuating flow measurements of a fully-developed, non-adiabatic, hypersonic boundary layer

Reynolds Number Effects on Shock-Wave Turbulent Boundary-Layer Interactions

Comparison Between Experiment and Prediction for a Transonic Turbulent Separated Flow

Turbulent properties of a compressible boundary layer.