H. Mcdonald

Mixing Length and Kinematic Eddy Viscosity in a Compressible Boundary Layer

The effect of Mach number on the mixing lengths and kinematic eddy viscosities in the turbulent, flat-plate boundary layer has been evaluated. Using the generalized velocities suggested by Van Driest, it was possible to correlate measured boundary-layer profiles, for a range of Reynolds numbers and Mach numbers from 0 to 5, on a single curve. From this correlation, velocity profiles at desired Reynolds and Mach numbers were generated, and the boundary-layer equations of motion were then integrated, using these profiles, to obtain the local turbulent shear stress. The local mixing lengths and kinematic eddy viscosities were subsequently evaluated from the computed shear-stress distribution. The results of this study show that within the Mach number range of 0 to 5 the effect of compressibility on the normalized, turbulent shear stress and mixing length distribution is quite small, in keeping with Morkovins hypothesis concerning the structure of compressible turbulent shear flow. The kinematic eddy viscosity displays a more pronounced sensitivity to Mach number, but a normalizing length scale can be found such that this sensitivity is all but eliminated.

Practical calculations of transitional boundary layers

Abstract A general finite-difference procedure for computing the behavior of compressible two-dimensional boundary layers is presented together with a turbulence model which allows quantitative predictions of the location and extent of the transition region between laminar and turbulent flow as it is influenced by such disturbances as surface roughness and free-stream turbulence. Reverse transition, i.e. relaminarization, caused by large favorable streamwise accelerations, is also quantitatively predicted by this procedure. The solution procedure depends upon the calculation of the streamwise development of a turbulent mixing length whose magnitude is governed by the turbulence kinetic energy equation. A large number of comparisons between predictions and measurements have been made and in general very good agreement is obtained.

Three-dimensional supersonic flow of a viscous or inviscid gas

Abstract The initial-boundary value problem representing the supersonic flow of a viscous or inviscid gas is solved by a forward marching procedure which integrates a set of coupled nonlinear multidimensional equations. The numerical method is based upon an alternating-direction implicit scheme and sample calculations have been performed to demonstrate the capabilities of the procedure. The specific problem considered concerns the supersonic flow of a three-dimensional jet exhausting into a supersonic ambient stream. It is shown that stable and apparently accurate solutions can be obtained for axial steps considerably larger than those normally permissible with many conditionally stable procedures. The computational cost per grid point per axial step in the present problem was very approximately only a factor of 2 greater than that required with the conditionally stable methods.

Three-dimensional viscous flows with large secondary velocity

A new system of approximating equations is derived for three-dimensional steady viscous compressible flows in which a primary flow direction is present, but in which both transverse velocity components can be large. If the transverse velocity vector which corrects a given potential flow is first decomposed into ‘potential’ and ‘rotational’ vector components, then a re-examination of three-dimensional boundary-layer theory shows that both components ( v ϕ , w ϕ ) of the potential-velocity vector may be assumed small, whereas both components ( v ψ , w ψ ) of the rotational-velocity vector and hence of the composite secondary flow ( v, w ) can remain of order unity. An assumption of small scalar potential then leads to a system of governing equations whose characteristic polynomial has a non-elliptic form for arbitrary Mach number, without introducing any direct approximation of either streamwise or transverse pressure gradient terms. These non-elliptic equations can be solved very economically as a well-posed initial/boundary-value problem. Computed results for laminar subsonic flow in a curved square duct confirm the small scalar-potential approximation for both large ( R/d = 100) and small ( R/d = 2) radius of curvature. Other computations for R/d = 2.3 are in good agreement with the measurements of Taylor, Whitelaw & Yianneskis (1980).

Analysis and computation of viscous subsonic primary and secondary flows

A new approximate flow analysis, designed to enable numerical solution as an initial value problem, is developed for a wide class of viscous subsonic flows at high Reynolds number and in straight or smoothly curved three-dimensional flow geometries. The analysis is coordinate-independent and corrects an a priori known inviscid primary flow for viscous and thermal effects, secondary flows, total pressure distortion, internal flow blockage and pressure drop. Computed results include laminar solutions for three-dimensional boundary layer flow, fully viscous flow in circular arc ducts, and also flow in a curved duct shaped like a turbine blade passage.

The effect of pressure gradient on the law of the wall in turbulent flow

The effect of a streamwise pressure gradient on the velocity profile in the viscous sublayer of a turbulent flow along a smooth wall in two-dimensional flow is estimated. In the analysis, a similarity argument is used and the necessary empirical information obtained from a constant pressure flow. An allowance is made for the departure from the wall value of the gradient of total shear stress normal to the wall. The results of analysis were used to generate new additive constants for use with Townsends modified law of the wall velocity profile and subsequently Townsends profile is found to be in good agreement with the measured velocity profiles in an adverse pressure gradient.

Combustion modeling in two and three dimensions—Some numerical considerations

Abstract The wide disparity in claims for combusting flow prediction models which are based upon numerical solution of a set of governing partial differential equations is examined in the light of potential and observed problem areas encountered in the equation solving process. These numerical difficulties are attributed to such case dependent items as large flow gradients requiring dense meshes, the presence of singular points such as re-entrant corners, and equation stiffness. In simple flows where none of these difficulties is encountered or is mitigated, experience is good and an encouraging level of predictive agreement is observed. Thus there exists considerable motivation to continue to develop the equation solving process so that these difficulties are either minimized or eliminated when encountered in the more practical systems. Techniques which treat some of these problem areas are discussed and their current status reviewed.

An overview and generalization of implicit Navier–Stokes algorithms and approximate factorization

Abstract A theme of linearization and approximate factorization provides the context for a retrospective overview of the development and evolution of implicit numerical methods for the compressible and incompressible Euler and Navier–Stokes algorithms. This topic was chosen for this special volume commemorating the recent retirements of R.M. Beam and R.F. Warming. A generalized treatment of approximate factorization schemes is given, based on an operator notation for the spatial approximation. The generalization focuses on the implicit structure of Euler and Navier–Stokes algorithms as nonlinear systems of partial differential equations, with details of the spatial approximation left to operator definitions. This provides a unified context for discussing noniterative and iterative time-linearized schemes, and Newton iteration for unsteady nonlinear schemes. The factorizations include alternating direction implicit, LU and line relaxation schemes with either upwind or centered spatial approximations for both compressible and incompressible flows. The noniterative schemes are best suited for steady flows, while the iterative schemes are well suited for either steady or unsteady flows. This generalization serves to unify a large number of schemes developed over the past 30 years.

General Three-Dimensional Viscous Primary/Secondary Flow Analysis

Generalized primary/secondary flow equations are derived as an approximation to the Navier-Stokes equations for three-dimensional viscous flows with a dominant flow direction. The primary/secondary flow equations are well posed for solution by spatial marching and can be solved one to two orders of magnitude faster than the Navier-Stokes equations. A key element in the approximations, which is a distinguishing feature of the present approach, is that accuracy is related to curvature terms obtained from a local primary flow direction rather than the coordinate system used to describe geometry of the flowfield. Potential flow streamlines for the flow geometry under consideration are a suitable choice for the primary flow direction. A sequentially decoupled implicit algorithm has been developed that exploits the form of the primary/secondary flow equations to obtain decoupled subsets of equations through choices for dependent variables, the sequence of equations, and the linearization scheme. Each of the equation subsets is solved by efficient and appropriate implicit numerical procedures. Computed solutions for flow in 90-deg bends agree very well with experimental data and NavierStokes solutions. The combined efficiency and accuracy of the approximate equations and solution algorithm make this approach attractive for problems in which a suitable primary flow direction can be identified.

Transonic Flows with Viscous Effects

Publisher Summary The transonic flow regime is of critical practical interest in view of the desire to be able to design vehicles that can cruise efficiently at these speeds. Frequently, the transonic regime is also critical in regard to loading, stability and control, and flutter for vehicles that have to traverse this regime to achieve some desired higher flight speed. Viscous effects are important as these effects contribute to the vehicle drag and modify the lift. In addition, at transonic speeds, the shock wave location is both important and particularly sensitive to the boundary layer. This chapter focuses on the aspects of the transonic flow problem, which either cannot be treated or cannot be treated efficiently, by interacting a wake and boundary layer with an inviscid outer flow. The chapter highlights problems involving steady and unsteady flow separation in two and three dimensions.