Asymptotic realization of the method of integral relations for turbulent jets to compute wall-jets curtain and their control

Abstract

The protection of the surface of bodies from high-temperature fluid flows is a real problem in modern technology. The aerodynamic protection is effected with the help of gas curtain. One of the principal means of organizing such a curtain is to blow cold fluid through slots at the initial section of the surface being protected [i]. The computational problem consists in the determination of the surface-temperature distribution in the region of the gas curtain. The existing methods to compute wall-jet curtains are in three principal directions depending on the physical model being used [2]: free turbulent jet model; twolayer scheme with laws on semibounded turbulent jets; flow in a boundary layer with wall-jet curtains determined by wall turbulence. The simplest method appeared to be based on the use of integral relations for the boundary layer and asymptotic conditions when equalization of temperature inside the boundary layer [3] takes place at x § ~ due to turbulent mixing. Under these conditions, there is a limiting relation between the momentum and energy thicknesses which cannot fully reflect the influence of initial conditions and the previous history of the flow. Expressions for the effect of cooling are developed using interpolations of the type q = (i + na )b where a and b are constants, is the value of the efficiency UX_~ , ~X_~O o , obtained on the basis of the laws of boundary-layer growth away from the location of blowing. The model for the wall jet takes into consideration the behavior of mixing processes near the lip of the nozzle [4]. Studies on turbulent boundary layer in wall jets have been carried out in many theoretical and experimental works [5-7]. The majority of theoretical studies is based on the simultaneous solution of the equations for turbulent jets and boundary layer growing on a flat plate, the difference being in the manner of specifying velocity profiles and skin-friction. In studying flows near curvilinear surfaces, disagreement has been noticed between experimental data and theoretical results computed from Karman integral relations which do not take into account the surface curvature in an explicit form. In [8, 9], it has been observed that the effect of surface curvature on the semibounded jet and the efficiency of wall-jet curtain mainly depend on active or conservative role of centrifugal body forces. In the present work, a method is developed to compute the flow and heat transfer on an adiabatic curvilinear surface, based on the asymptotic continuation of the method of integral relations for turbulent wall jets away from the location of blowing. The above-mentioned limiting cases correspond to conditions for the reorganization of nonmonotonous velocity profile which is present near the lip of the nozzle, in the flow that is characteristic of a developed boundary layer [7]. Such an approach makes it possible to develop a computational method for the wall-jet curtain with initial conditions for outflow while computational expressions for the efficiency of film cooling are developed on the basis of interpolation formulas. i. Problem Formulation. Adiabatic Curvilinear Surface with Gas Curtain. Consider a quasiisothermal turbulent wall jet. It is assumed that the physical characteristics of the fluid are constant in the given temperature range. The basic flow has a velocity u 0 and temperature T o . The same fluid is blown through a slot of height s with a mean velocity u s and temperature T s at the exit section. In the initial segment 0 < x < x s the curvilinear surface is in contact only with the fluid curtain and, hence, has a temperature T w = T s (Fig. i) at all points. The thermal boundary layer begins from the section x = x s as a result of mixing of the fluid curtain with the basic flow. Equations of quasisteady turbulent, incompressible boundary layer on the curvilinear surface with a constant curvature (neglecting normal turbulent stress components) take the form [i0]