A Reduced Order Model of the Mean Properties of a Turbulent Wall Boundary Layer at a Zero Pressure Gradient

Abstract

A novel twoequations model for computing the flow properties of a spatiallydeveloping, incompressible, zeropressuregradient, turbulent boundary layer over a smooth, flat wall is developed. The mean streamwise velocity component inside the boundary layer is described by the Reynoldsaveraged Navier–Stokes equation where the Reynolds shear stress is given by an extended mixinglength model. The nondimensional form of the mixing length is described by a polynomial function in terms of the nondimensional wall normal coordinate. Moreover, a stream function approach is applied with a leadingorder term described by a similarity function. Two ordinary differential equations are derived for the solution of the similarity function along the wall normal coordinate and for its streamwise location. A numerical integration scheme of the model equations is developed and enables the solution of flow properties. The coefficients of the mixinglength polynomial function are modified at each streamwise location as part of solution iterations to satisfy the wall and farfield boundary conditions and adjust the local boundary layer thickness, خ´99.4, to a location where streamwise speed is 99.4% of the farfield streamwise velocity. The elegance of the present approach is established through the successful solution of the various flow properties across the boundary layer (i.e., mean streamwise velocity, viscous stress, Reynolds shear stress, skin friction coefficient, and growth rate of boundary layer among others) from the laminar regime all the way to the fully turbulent regime. It is found that results agree with much available experimental data and direct numerical simulations for a wide range of Reخ¸ based on the momentum thickness (Reخ¸) from 15 up to 106, except for the transition region from laminar to turbulent flow. Furthermore, results shed light on the von Kأ،rmأ،n constant as a function of Reخ¸, the possible fourlayer nature of the mean streamwise velocity profile, the universal profiles of the streamwise velocity and the Reynolds shear stress at high Reخ¸, and the scaling laws at the outer region.