# Calculation of the Governing Equations

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 July 25, 2008, 04:49 Calculation of the Governing Equations #1 Mihail Guest   Posts: n/a Good time! Help me please! I would like to understand how cfx Calculat Equations. So I understand cfx clculat it in Integration Point. And we have one Integration Point in one element. Is it write? Do cfx clculat value of variable in real nods of mesh? Thanks for any help. And I will be very gratitude if you give me more information. I am really need you help!!

 July 25, 2008, 17:27 Re: Calculation of the Governing Equations #2 Rogerio Fernandes Brito Guest   Posts: n/a CFx has in its tutorial a good explanation about it. Go to Help and check it! Transport Equations In this section, the instantaneous equation of mass, momentum, and energy conservation are presented. For turbulent flows, the instantaneous equations are averaged leading to additional terms. These terms, together with models for them, are discussed in Turbulence and Wall Function Theory. The instantaneous equations of mass, momentum and energy conservation can be written as follows in a stationary frame: The Continuity Equation Equation 77. The Momentum Equations Equation 78. Where the stress tensor, , is related to the strain rate by Equation 79. The Total Energy Equation Equation 80. Where is the total enthalpy, related to the static enthalpy by: Equation 81. The term represents the work due to viscous stresses and is called the viscous work term. The term represents the work due to external momentum sources and is currently neglected. The Thermal Energy Equation An alternative form of the energy equation, which is suitable for low-speed flows, is also available. To derive it, an equation is required for the mechanical energy . Equation 82. The mechanical energy equation is derived by taking the dot product of with the momentum equation Equation 78: Equation 83. Subtracting this equation from the total energy equation Equation 80 yields the thermal energy equation: Equation 84. The term is always negative and is called the viscous dissipation. Finally, the static enthalpy is related to the internal energy by: Equation 85. So Equation 84 can be simplified to: Equation 86. The term is currently neglected, although it may be non-zero for variable-density flows. This is the thermal energy equation solved by ANSYS CFX. Please note the following guidelines regarding use of the thermal energy equation: Although the thermal energy equation solves for , this variable is still called static enthalpy in ANSYS CFX-Post. The thermal energy equation is meant to be used for flows which are low speed and close to constant density The thermal energy equation is particularly suited for liquids, since compressibility effects are minor. In addition, the total energy equation may experience robustness problems due to the pressure transient and the contribution to enthalpy. For materials which have variable specific heats (e.g., set as a CEL expression or using an RGP table or Redlich Kwong equation of state) the solver includes the contribution in the enthalpy tables. This is inconsistent, because the variable is actually internal energy. For this reason, the thermal energy equation should not be used in this situation, particularly for subcooled liquids.

 July 25, 2008, 17:28 Re: Calculation of the Governing Equations #4 Rogerio Fernandes Brito Guest   Posts: n/a Numerical Discretization Analytical solutions to the Navier-Stokes equations exist for only the simplest of flows under ideal conditions. To obtain solutions for real flows, a numerical approach must be adopted whereby the equations are replaced by algebraic approximations which may be solved using a numerical method. ANSYS CFX-Solver Theory Guide | Discretization and Solution Theory | Numerical Discretization | Numerical Discretization Prev Up / Home The Coupled System of Equations Next Discretization of the Governing Equations This approach involves discretizing the spatial domain into finite control volumes using a mesh. The governing equations are integrated over each control volume, such that the relevant quantity (mass, momentum, energy, etc.) is conserved in a discrete sense for each control volume. The figure below shows a typical mesh with unit depth (so that it is two-dimensional), on which one surface of the control volume is represented by the shaded area. Figure 1. Control Volume Surface It is clear that each node is surrounded by a set of surfaces that define the control volume. All the solution variables and fluid properties are stored at the element nodes. Consider the mean form of the conservation equations for mass, momentum and a passive scalar, expressed in Cartesian coordinates: Equation 1. Equation 2. Equation 3. These equations are integrated over a control volume, and Gauss' Divergence Theorem is applied to convert some volume integrals to surface integrals. If control volumes do not deform in time, then the time derivatives can be moved outside of the volume integrals and the equations become: Equation 4. Equation 5. Equation 6. where V and s respectively denote volume and surface regions of integration, and dnj are the differential Cartesian components of the outward normal surface vector. The volume integrals represent source or accumulation terms, and the surface integrals represent the summation of the fluxes. Note that changes to these equations due to control volume deformation are presented below. For details, see Mesh Deformation. The first step in the numerical solution of these exact differential equations is to create a coupled system of linearized algebraic equations. This is done by converting each term into a discrete form. Consider, for example, an isolated mesh element like the one shown below. Figure 2. Mesh Element Volumetric (i.e., source or accumulation) terms are converted into their discrete form by approximating specific values in each sector and then integrating those values over all sectors that contribute to a control volume. Surface flow terms are converted into their discrete form by first approximating fluxes at integration points, ipn, which are located at the center of each surface segment in a 3D element surrounding the control volume. Flows are then evaluated by integrating the fluxes over the surface segments that contribute to a control volume. Many discrete approximations developed for CFD are based on series expansion approximations of continuous functions (such as the Taylor series). The order-accuracy of the approximation is determined by the exponent on the mesh spacing or timestep factor of the largest term in the truncated part of the series expansion. This is often the first term excluded from the approximation. Increasing the order-accuracy of an approximation generally implies that errors are reduced more quickly with mesh or timestep size refinement. Unfortunately, in addition to increasing the computational load, high-order approximations are also generally less robust (i.e., less numerically stable) than their low-order counterparts. The discrete form of the integral equations becomes:

 September 7, 2014, 04:08 #7 New Member   Julia Join Date: May 2011 Posts: 12 Rep Power: 6 Hi I use IAPWS IF97 in CFX for modelling the two phase flow (steam and water) in a nozzle, I test different table generation value , but this error is appeared ' Independent variables were clipped during table generation' for all temperature and pressure range, but solving process is continue, but when I change this bound the result is changed and also the convergence process is very slow. How can I choose proper bounds? I use twice more than max pressure in nozzle, and twice less than min pressure , and for temperature do same. Could you help me please?

 September 7, 2014, 06:27 #8 Super Moderator   Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 10,809 Rep Power: 85 The table needs to be defined to cover the highest to lowest value present in the simulation. Note that these limiting values may occur in the centre of the simulation rather than at a boundary and might not be known before hand (eg in a nozzle). A further thing to note is that as the simulation converges it is common for the flow to go through flaw states which result in the fluid properties going outside the steady state ranges. This means you should always give some margin over the steady state max and min values to allow for this. And finally: If your simulation is diverging then you probably have a weird flow occurring which will require wacky fluid properties. In this case you need to fix the divergence, not the table generation.

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