柱坐标系和球坐标系下N-S方程的直接推导

Derivation of 3D Euler and Navier-Stokes Equationsin 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 coordinates (1.1)Where Conservative flow variables Inviscid/convective flux in x direction Inviscid/convective flux in y direction inviscid/convective flux in z directionAnd their specific definitions are as follows, Total enthalpySome relationshipWe want to perform the following coordinates transformationBecauseAccording to Cramers ruler, we have (1.2.1) (1.2.2)WhereSimilar to the above (1.2.3) (1.2.4)In addition, the following relations hold between cylindrical coordinate and Cartesian coordinate, (1.3) (1.4.1) (1.4.2)DerivationMultiplying the both side of equation (1.1) by and applying equalities (1.4.1) and (1.4.2) gives, (1.5)Differentiating the following w.r.t. time gives, (1.6.1) (1.6.2)Expanding the term and applying the relationships (1.6) yields, (1.7.1)Expanding the term and applying the relationships (1.6) yields, (1.7.2)Substituting relationships (1.7) into equation (1.5) and rearranging gives,(1.8)As we can see from expressions (1.7), the momentum equations in radial and tangential directions contain velocities in Cartesian coordinate; we need to replace them with corresponding variables in cylindrical coordinate. Writing down the momentum equations in radial and tangential directions as follows, (1.9.1) (1.9.2)Multiplying (1.9.1) by and (1.9.2) by, then summing up and applying expressions (1.6) and rearranging yields (1.10.1)Multiplying (a) by and (b) by, then summing up and applying expressions (1.6) yields, (1.10.2)Replacing (1.10) with (1.9) and rearranging equation (1.8) gives (1.11)Where,Note: different from Euler equation in Cartesian coordinates, the Euler equation in cylindrical coordinates contains source terms from momentum equations in radial and tangential equations.2. Derivation of Euler Equation in Cylindrical coordinates moving at in tangential directionWhere, , ,Then equation (1.11) can be written as follows (2.1)Where Equation (2.1) adopts rotating coordinates but the variables are measured in absolute cylindrical coordinates.3. Derivation of 3D Navier-Stokes Equation in Cylindrical Coordinates3D Navier-Stokes Equations in Cartesian coordinates (3.1)Where,，,In the following derivation, only viscous terms will be derived from Cartesian coordinates to cylindrical coordinates, those inviscid terms having been derived in section 1 will be not repeated. Replacing with gives (3.2.1)Replacing with gives (3.2.2)Multiplying equation (3.1) by , the viscous terms are gives as follows (omitting the negative sign before it from simplicity),(3.3) (3.4.1), (3.4.2), (3.4.3),(3.4.4) (3.4.5) (3.4.6)Expanding expression (3.3) gives,(3.5)= (3.6.1)= (3.6.2)(3.6.3)(3.7.1)Divergence in Cartesian Coordinates (3.7.2)Divergence in cylindrical coordinates (3.7.3)(3.8.1)(3.8.2)(3.9.1)(3.9.2)As we can see from the above that viscous terms in expression (3.5) for the momentum equation in axial/x direction and energy equation can be expressed in variables in cylindrical coordinates, while the viscous terms in (3.5) for momentum equations in radial and tangential directions still contain variables in Cartesian coordinates. Similar manipulation to (1.10) will be adopted in the following.Writing out the viscous terms for momentum equations in radial and tangential coordinates as follows, (3.10.1) (3.10.2)Multiplying (3.10.1) by and multiplying (3.10.2) by , then summing up and rearranging gives,(3.11.1)Multiplying (3.10.1) by and multiplying (3.10.2) by , then summing up and rearranging gives,(3.11.2)(3.12.1)(3.12.2)(3.12.3) (3.12.4)Substituting (3.6.1), (3.6.2) and (3.12) into expressions (3.11) and rearranging yields, (3.13.1) (3.13.2) Making use of expressions (3.4.1), (3.6.1), (3.6.2), (3.8.1), (3.8.2), (3.9.1), (3.9.2),(3.13.1) and (3.13.2), we can get the final expression of 3D Navier-Stokes Equation in cylindrical coordinates as follows,3D Navier-Stokes Equation in cylindrical coordinates，,，,， If the moment of momentum equation is adopted to replace the tangential momentum equation, its expression will be s