Derivation of 3D Euler and Navier-Stokes Equations in Cylindrical CoordinatesContents1. Derivation of 3D Euler Equation in Cylindrical coordinates2. Derivation of Euler Equation in Cylindrical coordinates moving at in tangential direction3. Derivation of 3D Navier-Stokes Equation in Cylindrical Coordinates1. Derivation of 3D Euler Equation in Cylindrical coordinatesEuler Equation in Cartesian coordinates0zGyFxEtU (1.1)Where U Conservative flow variablesE Inviscid/convective flux in x directionF Inviscid/convective flux in y directionG inviscid/convective flux in z directionAnd their specific definitions are as followsEwvuU,HuwuvupuuuE,HvwvpvvuvvF,HwpwwvwuwwGwwvvuuCvTE21pEwwvvuuCpTH21H Total enthalpySome relationshipWe want to perform the following coordinates transformationrxzyx,,,,Because1rzzrryyr0zzryyrAccording to Cramer’s ruler, we havezJzyrzryzrzyr101 (1.2.1)yJzyrzryyryzr101 (1.2.2)WherezyrzryjSimilar to the above0rzzryy1zzyyrzJzyrzryzrzy110 (1.2.3) ryJzyrzryyryz110 (1.2.4)In addition, the following relations hold between cylindrical coordinate and Cartesian coordinatecosry,sinrzcosry,sinrz,sinry,cosrz, (1.3)rrrzyrzryJcossinsincossincosFFrrrzFzFrrzFzrFyFyrrFJyFJ (1.4.1)cossinGGrrryGyGrryGyrGzGzrrGJzGJ (1.4.2)DerivationMultiplying the both side of equation (1.1) by J and applying equalities (1.4.1) and (1.4.2) gives,0sincossincoscossinsincosFGGrFrrxErtUrGGrrFFrrxErtUrzGJyFJxEJtUJzGyFxEtUJ (1.5)Differentiating the following w.r.t. time givescosry,sinrzdtdrdtdrdtdysincos,dtdrdtdrdtdzcossinwdtdzvdtdrvdtdrvdtdyr,,,rvwvsincos (1.6.1) vwvcossin (1.6.2)Expanding the term sincosGrFr and applying the relationships (1.6) yields,rrrrrrrGHvpwvpvvuvvrwvHpwvwpwvvwvuwvrrHwpwwvwuwwrHvwvpvvuvvGrFrsincossincossinsincoscossincossincossincossincossincos (1.7.1)Expanding the term sincosFG and applying the relationships (1.6) y...
