亚/超声速楔状流层流边界层速度与温度相似解及拟合解

利用相似变换获得了楔状流层流边界层无量纲流函数的3阶非线性常微分方程,用Runge-Kutta法求解微分方程获得了不同楔形角楔状流层流边界层无量纲速度随相似变量的变化曲线;推导了亚声速和超声速楔状流层流边界层无量纲温度关于相似变量的2阶线性齐次和非齐次微分方程,获得了温度分布的通解,恒壁温条件下亚声速楔状流和绝热壁面条件下超声速楔状流层流边界层无量纲温度解析解及指数函数形式的拟合解.以楔形角为0为例利用相似变换研究了超声速条件下气体压缩性及黏度随温度变化等因素对层流边界层速度与温度的影响,得出不可压缩常物性与可压缩变物性条件下无量纲速度相对误差绝对值小于9.8%的结论.研究表明:Pr越大贴近壁面处无量纲温度变化越剧烈;超声速条件下壁温低于绝热壁温时黏性耗散作用可以使层流边界层气体温度从壁面到主流间出现先升高后降低的变化. 

Similarity solutions and approximation solutions of velocity and temperature in laminar boundary layer of subsonic and supersonic wedge flows

The variations in the dimensionless velocity due to changes in similarity variables of wedge flows with different wedge angles were obtained using the Runge-Kutta method by solving the third-order nonlinear ordinary differential equation of the dimensionless stream function that was obtained through similarity transformation, which described the wedge flow in the laminar boundary layer. The second-order linear homogeneous differential equation of dimensionless temperature based on the similarity variable in the laminar boundary layer of the subsonic wedge flow and the second-order linear non-homogeneous differential equation of dimensionless temperature on the similarity variable in the laminar boundary layer of the supersonic wedge flow were derived. The general temperature distribution solutions in the laminar boundary layers of subsonic and supersonic wedge flow, the similarity solutions and exponential form approximation solutions of dimensionless temperature in the laminar boundary layers under the condition of constant wall temperature for subsonic wedge flow and the condition of adiabatic wall for supersonic wedge flow were obtained by solving the above two differential equations. The effects of compressibility and viscosity of the supersonic gas in the laminar boundary layer on velocity and temperature were investigated using the wedge flow of wedge angle of 0 as a representative example. It is shown that the maximum absolute value of relative errors between the similarity solution obtained under the condition of incompressible and constant properties and the similarity solution obtained under the condition of compressible and variable properties is less than 9.8%. Results show that: the larger the Pr of the wedge flow, the more dramatic the change in dimensionless temperature in the area close to the wall; the viscous dissipation causes the temperature of the wedge flow from the wall to the main flow in the boundary layer to first increase and then decrease under supersonic flow conditions and with temperatures lower than the adiabatic wall temperature.