This paper presents a reformulated version of the author'sk-ω model of turbulence. Revisions include the addition of just one new closure coefficient and an adjustment to the dependence of eddy viscosity on turbulence properties. The result is a significantly improved model that applies to both boundary layers and free shear flows and that has very little sensitivity to finite freestream boundary conditions on turbulence properties. The improvements to the k-ω model facilitate a significant expansion of its range of applicability. The new model, like preceding versions, provides accurate solutions for mildly separated flows and simple geometries such as that of a backward-facing step. The model's improvement over earlier versions lies in its accuracy for even more complicated separated flows. This paper demonstrates the enhanced capability for supersonic flow into compression corners and a hypersonic shock-wave/ boundary-layer interaction. The excellent agreement is achieved without introducing any compressibility modifications to the turbulence model.

Formulation of the k-w Turbulence Model Revisited

The incompressible zero-pressure-gradient turbulent boundary layer: An assessment of the data

It is generally accepted that the direct effects of density fluctuations on turbulence are small if the root-mean-square density fluctuation is small compared with the absolute density: this is Morkovin's hypothesis (Favre 1964, p. 367). This means that the turbulence structure of boundary layers and wakes at free-stream Mach numbers Me less than about 5, and of jets at Mach numbers less than about 1.5, is closely the same as in the corrcsponding constant-density flow. By "turbulence structure" we mean dimensionless properties like correlation coefficients and spectrum shapes: the skin-friction coefficient cf == Tw/tPeU; and other ratios of turbulence quantities to mean flow quantities are greatly affected by the influence of mean density changes on the mean motion. The effect of mean density variations in x or y on the turbulence structure is not covered by Morkovin's hypothesis, but is often negligible at the lower Mach numbers if stream wise pressure gradients are small. Therefore assumptions about turbulence structure that give good results in calculation methods for constant-density /low will, if properly scaled, give good results in compressible boundary layers or wakes for Me 5, say. Basic equations for compressible shear layers are given by Howarth (1953) and Lin (1959). More recent treatments (FavJe 1971, Cebeci & Smith 1974, Rubesin & Rose 1973, Bilger 1975) use "mass-weighted" variables, which remove density iluctuations from the time-averaged equations of motion but not from the turbulence or from the response of measuring instruments (although Laufer, in Birch et aI1972, p. 462, suggests that pitot tubes probably yield mass-averaged velocities). It seems probable that the difference between conventional and mass-weighted averages rises more slowly with Mach number than current errors in measuring either. Of problems 1 The author is grateful for a number of helpful comments or contributions, especialIy

Compressible Turbulent Shear Layers

An outline of the techniques available for the measurement of skin friction in turbulent boundary layers

Interactions between shock waves and turbulent boundary layers

Measurements of the turbulent boundary layer on the sidewall of the RAE 8ft x 8ft wind tunnel are described. The measurements consisted of surface shearing stress, using a large strain-gauge balance, together with velocity and temperature profiles. Maximum Reynolds numbers, based on an effective streamwise run, range from about 200 million at M = 0.2 to 60 million at M = 2.8. The analysis of the results shows that accepted descriptions of turbulent boundary layers in incompressible flow, derived from the concept of a universal velocity defect law, are preserved for compressible flows if the boundary-layer parameters are expressed in kinematic form. This leads to a simple method of calculating skin friction and velocity profiles for constant pressure flows without heat transfer.

Turbulent boundary-layer studies at high Reynolds numbers at mach numbers between 0.2 and 2.8

Experiments have been made, in supersonic turbulent boundary layers, on a simple type of surface pitot tube, formed by cementing a piece of razor blade over a statie pressure hole. This type of surface pitot tube has great practical advantages, as surface shear measurements may be made on any model designed for pressure measurements.
A calibration curve, which is sensibly independent of Mach nurnber, has been obtained for the dependenee of the pressure rise at the pitot tube on the surface shear, both terms being made non-dimensional. Examples are included of the use of surface pitot tubes on two models. The models were a delta wing and a biconvex cone and the tests covered a range of Mach numbers, Reynolds numbers and model incidence.

The use of surface pilot tubes as skin friction meters at supersonic speeds

The measurements were made in the R.A.E. 8 ft x 8 ft wind tunnel, using a self-aligning yawmeter, over a plane normal to the model longitudinal axis at 82.5% root chord from the apex. They show a region of large total pressure loss and large flow direction variation about the leading edge vortex and another region of total pressure loss attributed to the bow shockwave. Between the regions affected by the leading edge vortices and the region affected by the bow shock wave there is an area of fairly small total pressure loss and flow direction deviation. The vortex is located well inboard of its position for similar incidence at low speeds. The streamwise Mach numbers near the core are of doubtful accuracy but, as measured, are smaller than freestream in contrast to the high velocities found there in incompressible flow.

L. Gaudet

Papers

Turbulent boundary-layer studies at high Reynolds numbers at mach numbers between 0.2 and 2.8

The use of surface pilot tubes as skin friction meters at supersonic speeds

Preliminary measurements of the flow field on the leeside of a delta wing unit aspect ratio at a Mach number of 2.6 and an incidence of 15º