数学
布拉修斯边界层
偏微分方程
常微分方程
层流
边界层
数学分析
微分方程
边界层厚度
边界(拓扑)
物理
机械
作者
Yeunwoo Cho,Angelica Aessopos
摘要
The system of equations in a steady, compressible, laminar boundary layer is composed of four fundamental equations. Those are: the continuity equation, the momentum equation, the energy equation, and the equation of state. The solutions of these equations, when solved simultaneously for a 2-dimensional boundary layer, are: the velocity in the x and y direction ( u and v ), the pressure (p) and the density (ρ ). The system of equations is a system of partial differential equations (PDE) and is usually difficult to solve. Therefore, sophisticated transformation methods, called similarity transformations are introduced to convert the original partial differential equation set to a simplified ordinary differential equation (ODE) set. The solutions of this ordinary differential equation set are usually nondimensionalized velocities and temperature. By principle, these ordinary equations are coupled mathematically and usually can be solved by numerical methods. However, with further appropriate assumptions related to the transport properties (e.g. Prandtl number), and flow conditions (e.g. Mach number, geometry around flow), these ODEs can be uncoupled mathematically or can have simpler forms, almost similar to the forms obtained from the incompressible boundary layer analysis. (e.g. Blasius solution, Falkner-Skan equation). Hence, the simplified ODE set makes it possible to get the solution from the already existing solutions of the incompressible analysis and also reduces the computing time in the numerical analysis. In this paper, three different transformation methods will be described. A detailed derivation of the generalized (Levy-Ilingworth) transformation method and the appropriate assumptions made during the derivation will be explained. The Howarth transformation and the Illingworth-Stewartson transformation will be described briefly.
科研通智能强力驱动
Strongly Powered by AbleSci AI