APPLICATION OF THE METHOD OF WEIGHTED RESIDUALS TO THE BOUNDARY LAYER MOMENTUM AND TURBULENT STRESS TRANSPORT EQUATIONS
Abstract
A method is developed for the analysis of the turbulent boundary layer under the influence of a pressure gradient. The governing equations are the continuity equation, the boundary layer mean momentum equation, and the boundary layer turbulent kinetic energy equation. The kinetic energy equation is transformed into an equation which governs the transport of turbulent shear stress. This transformation is accomplished through the use of three empirical functions which relate the turbulent kinetic energy, the turbulent energy diffusion, and the turbulent energy dissipation to the Reynolds stress. These empirical functions are determined completely from zero pressure gradient boundary layers. The method of weighted residuals (modified Galerkin method) is then applied to the system consisting of the continuity, mean momentum, and Reynolds stress transport equations thereby reducing the system of partial differential equations to a system of ordinary differential equations. This technique involves the distribution of error which results when an exact solution is replaced with an approximate solution in the governing equations. The approximate solution is obtained by expanding the dependent variables (mean velocity derivative and Reynolds stress) in terms of complete sets of linearly independent functions of the velocity and adjustable coefficients which are functions of the streamwise coordinate. The approximate solutions are required to satisfy the boundary conditions on the dependent variables. The number of terms in the approximations (order of the solution) is specified as input, thus offering a rational method of successive approximations which is not generally available with integral techniques. The system of ordinary differential equations is then solved numerically. This procedure consists of first specifying appropriate initial conditions (skin friction coefficient, mean velocity profile, and Reynolds stress profile) and then integrating the system in the downstream direction. All of the calculated flow variables are given at the end of each specified integration interval. The method is applied to several sets of experimental data including flows with zero pressure gradient, favorable pressure gradient, and adverse pressure gradient. Results are given for comparisons of skin friction coefficient, thickness parameters, mean velocity profiles, and Reynolds stress profiles (where available). The agreement between the measured and the calculated values is generally good. The best agreement over all of the flow variables is obtained with the strong positive pressure gradient while the poorest agreement is obtained with the negative pressure gradient case. A comparison for the strong adverse pressure gradient flow of skin friction coefficient and momentum thickness determined using the present method and those variables determined using several other calculation methods is also given. This comparison indicates that the performance of the present method is generally equal to that of existing calculation methods. The present method offers several features which are not collectively available with current calculation methods. The skin friction coefficient is calculated directly from the solution for the mean velocity derivative thereby eliminating the need for an auxiliary skin friction relation. The technique offers the inherent numerical simplicity and reduced computation requirements of integral methods. The weighted residual method offers a rational method of successively reducing the error in approximating the dependent variables, and thus theoretically provides the opportunity to obtain the solution to any desired accuracy. Finally, the inclusion of a transport equation for turbulent stress offers the possibility of developing turbulence models which are more universal in nature, and thus applicable to a wider range of flow problems.
Degree
Ph.D.
Subject Area
Aerospace materials
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