Energy equation in cylindrical coordinates

This currently is, to the authors knowledge, the only scheme available that is demonstrably capable of conserving mass, momentum and kinetic energy (in the absence of viscosity) on both uniform and non-uniform grids. We confirm the importance of conservation of total energy for correctly modeling shock waves. We present a finite difference discretization of the incompressible Navier-Stokes equations in cylindrical coordinates. We present numerical examples that demonstrate the robustness of the new ALE method on a variety of grids and test problems including polygonal meshes and some realistic experiments. The principal contributions of this part of the paper are the extension of this algorithm to general polygonal meshes and 2D rz geometry with requirement of symmetry preservation on special meshes. 1 The ALE algorithm conserves total energy on arbitrary meshes and preserves symmetry when remapping from one equiangular polar mesh to another.

I know it is a very lengthy and relatively complex derivation, but come on. They just threw the end result in cylindrical coordinated in the appendix and call it a day. In this paper we give a systematic and complete description of the algorithm for the axisymmetric case and provide justification for our approach. 'Derive the control volume three dimensional energy equation in cylindrical coordinates.' It's as if the publisher refused to print the cylindrical derivation. We describe a distribution-based algorithm for the definition of remapped subzonal densities and a local constrained-optimization-based approach for each zone to find the subzonal mass fluxes. The general idea is based on the following papers, ,, , where it was described for Cartesian coordinates. In the second part of the paper we describe the remapping stage of the arbitrary LagrangianвЂ“Eulerian algorithm. SPHERICAL COORDINATES 497 F.2 CYLINDRICAL COORDINATES The following form of the thermal energy equation in cylindrical coordinates allows for nonconstant physical properties, energy generation, and conversion of mechan-ical to internal energy using viscous dissipation, which is expressed in terms of unspecied viscous stresstensor. In particular, we demonstrate the importance of conservation of total energy for correctly modeling shock waves. We present numerical examples that demonstrate the robustness of the new method for Lagrangian equations on a variety of grids and test problems including polygonal meshes. The principal contributions of this part of the paper are as follows: a definition of вЂњplanarвЂќ subzonal mass for nodes on the z axis ( r = 0) that does not require a special procedure for movement of these nodes proof of conservation of the total energy formulated for general polygonal meshes. We complete our analysis of eccentric cylindrical coordinates by using the new equations to solve one momentum, one energy, and one mass transport problem. The momentum equation is discretized in its вЂњCartesianвЂќ form with a special definition of вЂњplanarвЂќ masses (area-weighted). The method for discretization of the Lagrangian equations is based on ideas presented in, ,, where the authors use a special procedure to distribute zonal mass to corners of the zone (subzonal masses).

It has a consistent definition of kinetic energy in the zone that is exact for a velocity field with constant magnitude. The discretization conserves total energy exactly up to machine round-off on any mesh. In particular, spherical symmetry is preserved on polar equiangular meshes. It exactly preserves planar, cylindrical and spherical symmetry of the flow on meshes aligned with the flow. The foregoing equations (1), (2), and (3) represent the continuity, NavierStokes, and energy. In the first part of the paper, we describe the discretization of the basic Lagrangian hydrodynamics equations in axisymmetric 2D rz geometry on general polygonal meshes. coordinates, and cylindrical coordinates are provided. We present a new discretization for 2D arbitrary LagrangianвЂ“Eulerian hydrodynamics in rz geometry (cylindrical coordinates) that is compatible, total energy conserving and symmetry preserving.

Energy equation in cylindrical coordinates