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 0zGyFxEtU (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 wwvvuuCvTE21 pEwwvvuuCpTH21 H Total enthalpy Some relationship We w ant to perform the follow ing coordinates transformation rxzyx,,,, Because 1rzzrryyr 0zzryyr According to Cramer’s ruler, we have zJzyrzryzrzyr101 (1.2.1) yJzyrzryyryzr101 (1.2.2) Where zyrzryj Similar to the above 0rzzryy 1zzyy rzJzyrzryzrzy110 (1.2.3) ryJzyrzryyryz110 (1.2.4) In addition, the following relations hold between cylindrical coordinate and Cartesian coordinate cosry ,sinrz cosry,sinrz,sinry,cosrz , (1...