柱坐标系和球坐标系下NS方程的直接推导VIP专享

Derivation of 3D Euler and Navier-Stokes Equations in Cylindrical Coordinates Dingxi Wang School of Engineering, Durham University Contents 1. Derivation of 3D Euler Equation in Cylindrical coordinates 2. Derivation of Euler Equation in Cylindrical coordinates moving at  in tangential direction 3. Derivation of 3D Navier-Stokes Equation in Cylindrical Coordinates 1. Derivation of 3D Euler Equation in Cylindrical coordinates Euler Equation in Cartesian coordinates 0zGyFxEtU (1.1) Where U  Conservative flow variables E  Inviscid/convective flux in x direction F  Inviscid/convective flux in y direction G  inviscid/convective flux in z direction And their specific definitions are as follows EwvuU,HuwuvupuuuE,HvwvpvvuvvF,HwpwwvwuwwG wwvvuuCvTE21 pEwwvvuuCpTH21 H  Total enthalpy Some relationship We w ant to perform the follow ing coordinates transformation rxzyx,,,, Because 1rzzrryyr 0zzryyr According to Cramer’s ruler, we have zJzyrzryzrzyr101 (1.2.1) yJzyrzryyryzr101 (1.2.2) Where zyrzryj Similar to the above 0rzzryy 1zzyy rzJzyrzryzrzy110 (1.2.3) ryJzyrzryyryz110 (1.2.4) In addition, the following relations hold between cylindrical coordinate and Cartesian coordinate cosry ,sinrz   cosry,sinrz,sinry,cosrz , (1...