Cfx Theroy Guide

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Guide to CFD theory for use with Ansys Fluent 14.0 Computational. Ansoft, AUTODYN, EKM, Engineering Knowledge Manager, CFX, FLUENT, HFSS and any and all ANSYS.

Cfx Theory Guide

(Parte 3 de 5) If you again assume an ideal gas equation of state with variable specific heat capacity you can compute relative total temperature, total temperature and stationary frame total temperature using: (Eqn. 5) and: (Eqn. 56) and: (Eqn. 57) whereallthetotaltemperaturequantitiesareobtainedbyinvertingtheenthalpytable.If is constant, then these values are obtained directly from these definitions: (Eqn.

60) At this point, given, and you can compute relative total pressure,totalpressureorstationaryframetotalpressureusingtherelationshipgiveninthe section describing total pressure. For details, seeTotal Pressure (p.14). The names of the various total enthalpies, temperatures, and pressures when visualizing results in ANSYS CFX-Post or for use in CEL expressions is as follows. Hstat htot htot href– cp T Td Ih ref– cp T Td htot,stn href– cp T Td Ttot,rel Tstat Urel Urel⋅ Ttot Tstat Urel Urel⋅ω R×ω R×⋅– Ttot,stn Tstat Ustn Ustn⋅ Ttot,rel Ttot Ttot,stn pstat Tstat Basic Solver Capability Theory: Documentation Conventions. ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 1.0. © 1996-2006 ANSYS Europe, Ltd.

All rights reserved.Page 19 Contains proprietary and confidential information of ANSYS, Inc. And its subsidiaries and affiliates. Table 1Variable naming: Total Enthalpies, temperatures, and pressures The Mach Number and stationary frame Mach numbers are defined as: (Eqn.

62) where is the local speed of sound. Material with Variable Density and Specific Heat Rotating and stationary frame total temperature and pressure are calculated the same way as described inTotal Temperature andTotal Pressure.

The only changes in the recipes are that rotating frame total pressure and temperature require rothalpy, as the starting point and stationary frame total pressure and temperature require stationary frame total enthalpy,. Courant numberThe Courant number is of fundamental importance for transient flows. For a one-dimensional grid, it is defined by: (Eqn. 63) where isthefluidspeed, isthetimestepand isthemeshsize.TheCourantnumber calculated in ANSYS CFX is a multidimensional generalization of this expression where the velocity and length scale are based on the mass flow into the control volume and the dimension of the control volume. VariableLong Variable NameShort Variable Name Total Enthalpyhtot Rothalpy rothalpy Total Enthalpy in StnFrame htotstn Total Temperature inRel Frame Ttotrel Total Temperature Ttot Total Temperature inStn Frame Ttotstn Total Pressure in RelFrame ptotrel Total Pressureptot Total Pressure in StnFrame ptotstn htot I htot,stn Ttot,rel Ttot Ttot,stn Ptot,rel Ptot Ptot,stn Mstn Ustn I htot,stn Basic Solver Capability Theory: Documentation Conventions. Page 20ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 1.0.

Cfx Theory Guide

And its subsidiaries and affiliates. For explicit CFD methods, the timestep must be chosen such that the Courant number is sufficiently small. The details depend on the particular scheme, but it is usually of order unity. As an implicit code, ANSYS CFX does not require the Courant number to be small for stability. However, for some transient calculations (e.g., LES), one may need the Courant number to be small in order to accurately resolve transient details. ANSYS CFX uses the Courant number in a number of ways: 1.

The timestep may be chosen adaptively based on a Courant number condition (e.g., to reach RMS or Courant number of 5). 2.For transient runs, the maximum and RMS Courant numbers are written to the output file every timestep. 3.The Courant number field is written to the results file. Mathematical Notation This section describes the basic notation which is used throughout the ANSYS CFX-Solver documentation. The vectoroperators Assume a Cartesian coordinate system in which,and are unit vectors in the three coordinate directions.is defined such that: (Eqn.

64) Gradient operator For a general scalar function, the gradient of is defined by: (Eqn. 65) Divergence operator For a vector function where: (Eqn. 6) the divergence of is defined by: (Eqn.

All rights reserved.Page 21 Contains proprietary and confidential information of ANSYS, Inc. And its subsidiaries and affiliates. Dyadic operator The dyadic operator (or tensor product) of two vectors, and, is defined as: (Eqn. 68) By using specific tensor notation, the equations relating to each dimension can be combined into a single equation.

Thus, in the specific tensor notation: (Eqn. 69) Matrixtransposition The transpose of a matrix is defined by the operator. For example, if the matrix is defined by: (Eqn.

70) then: (Eqn. 71) The Identity Matrix (Kronecker Delta function) The Identity matrix is defined by: (Eqn.

72) Index notationAlthough index notation is not generally used in this documentation, the following may help you if you are used to index notation. UV⊗ UxVx UxVy UxVz UyVx UyVy UyVz UzVx UzVy UzVz Basic Solver Capability Theory: Governing Equations. Page 22ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 1.0.

© 1996-2006 ANSYS Europe, Ltd. All rights reserved. Contains proprietary and confidential information of ANSYS, Inc. And its subsidiaries and affiliates. In index notation, the divergence operator can be written: (Eqn. 73) where the summation convention is followed, i.e., the index is summed over the three components. The quantity can be represented by (when and are vectors), or by (when is a vector and is a matrix), and so on.

Hence, the quantitycan be represented by: (Eqn. 74) Note the convention that the derivatives arising from the divergence operator are derivativeswithrespecttothesamecoordinateasthefirstlistedvector.Thatis,thequantity is represented by: (Eqn. 75) and not: (Eqn. 76) The quantity (when and are matrices) can be written. Governing Equations ThesetofequationssolvedbyANSYSCFXaretheunsteadyNavier-Stokesequationsintheir conservation form. If you are new to CFD, review the introduction. For details, seeComputational Fluid Dynamics (p.1 in 'ANSYS CFX Introduction').

A list of recommended books on CFD and related subjects is available. For details, see Further Background Reading (p.6 in 'ANSYS CFX Introduction'). For all the following equations, static (thermodynamic) quantities are given unless otherwise stated. UV⊗ UiVj UV UiVjk UV ρU⊗∇. xi∂∂ ρUiU j xj∂∂ ρUiU j ab. ab aijbij Basic Solver Capability Theory: Governing Equations.

ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 1.0. © 1996-2006 ANSYS Europe, Ltd. All rights reserved.Page 23 Contains proprietary and confidential information of ANSYS, Inc. And its subsidiaries and affiliates. 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(p.69). The instantaneous equations of mass, momentum and energy conservation can be written as follows in a stationary frame: The ContinuityEquation (Eqn. 7) The MomentumEquations (Eqn. 78) Where the stress tensor, is related to the strain rate by (Eqn. 79) The Total EnergyEquation (Eqn. 80) Where is the total enthalpy, related to the static enthalpy by: (Eqn.

81) The term represents the work due to viscous stresses and is called the viscous work term. Theterm representstheworkduetoexternalmomentumsourcesandiscurrently neglected. The Thermal EnergyEquationAn 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. 82) The mechanical energy equation is derived by taking the dot product of with the momentum equation(Eqn.

83) Basic Solver Capability Theory: Governing Equations. Page 24ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 1.0.

And its subsidiaries and affiliates. Subtractingthisequationfromthetotalenergyequation(Eqn.80)yieldsthethermalenergy equation: (Eqn. 84) The term is always negative and is called the viscous dissipation.

Finally, the static enthalpy is related to the internal energy by: (Eqn. 84) can be simplified to: (Eqn.

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. Formaterialswhichhavevariablespecificheats(e.g.,setasaCELexpressionorusingan RGPtableorRedlichKwongequationofstate)thesolverincludesthe 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. Equations of State In ANSYS CFX, the flow solver calculates pressure and static enthalpy. Finding density requiresthatyouselectthethermalequationofstateandfindingtemperaturerequiresthat you select the constitutive relation. The selection of these two relationships is not necessarily independent and is also a modeling choice. The thermal equation of state is described as a function of both temperature and pressure: (Eqn.

87) Basic Solver Capability Theory: Governing Equations. ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 1.0. © 1996-2006 ANSYS Europe, Ltd. All rights reserved.Page 25 Contains proprietary and confidential information of ANSYS, Inc.

And its subsidiaries and affiliates. The specific heat capacity, may also be described as a function of temperature and pressure: (Eqn. 8) For an Ideal Gas, the density is defined by the Ideal Gas Law and, in this case, can be a function of only temperature: (Eqn. 89) Important:When or are also functions of an algebraic additional variable, in addition to temperature and pressure, then changes of that additional variable are neglected in the enthalpyandentropyfunctions.However,ifthatadditionalvariableisitselfonlydependent on pressure and temperature, then the effects will be correctly accounted for. Ideal Gas Equation of state For an Ideal Gas, the relationship is described by the Ideal Gas Law: (Eqn. 90) where is the molecular weight of the gas, and is the universal gas constant.

Real Gas and Liquid Equations of State In the current version of ANSYS CFX the Redlich Kwong equation of state is available as a built-in option for simulating real gases. It is also available through several pre-supplied CFX-TASCflowRGPfiles.TheVukalovichVirialequationofstateisalsoavailablebutcurrently only by using CFX-TASCflow RGP tables. Redlich Kwong Gas Properties The Redlich-Kwong equation of state was first published in 1949 and is considered one of the most accurate two parameter corresponding states equations of state.

This equation of state is quite useful from an engineering standpoint because it only requires that the user know the fluid critical temperature and pressure. More recently, Aungier (1995)96 has modifiedtheRedlich-Kwongequationofstatesothatitprovidesmuchbetteraccuracynear the critical point. The Aungier form of this equation of state is used by ANSYS CFX and is given by: (Eqn.

91) cp cp pT,= cp cp cp T= ρ cp wR 0 p RT ν b– c+ Basic Solver Capability Theory: Governing Equations. Page 26ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 1.0. © 1996-2006 ANSYS Europe, Ltd. All rights reserved.

Contains proprietary and confidential information of ANSYS, Inc. And its subsidiaries and affiliates. Thisformdiffersfromtheoriginalbytheadditionalparameter,whichisaddedtoimprove the behavior of isotherms near the critical point, as well as the form of. The parameters, and in(Eqn.

91) are given by: (Eqn. 95) In the expression for, the standard Redlich Kwong exponent of has been replaced by a general exponent. Optimum values of depend on the pure substance. Aungier (1995)96 presented values for twelve experimental data sets to which he provided a best fit polynomial for the temperature exponent in terms of the acentric factor,: (Eqn. 96) The acentric factor must be supplied when running the Redlich Kwong model and is tabulatedformanycommonfluidsinPolingetal84.Ifyoudonotknowtheacentricfactor, or it is not printed in a common reference, it can be estimated using knowledge of the critical point and the vapor pressure curve with this formula: (Eqn. 97) where the vapor pressure, is calculated. In addition to the critical point pressure, this formula requires knowledge of the vapor pressure as a function of temperature.

Inordertoprovideafulldescriptionofthegasproperties,theflowsolvermustalsocalculate enthalpy and entropy. These are evaluated using slight variations on the general relationships for enthalpy and entropy which were presented in the previous section on variable definitions. The variations depend on the zero pressure, ideal gas, specific heat c aT ab c b 0.08664RTc c RTc pc a0 n ω pv T 0.7Tc= Basic Solver Capability Theory: Governing Equations. ANSYS CFX-Solver Theory Guide.

And its subsidiaries and affiliates. Capacity and derivatives of the equation of state. The zero pressure specific heat capacity must be supplied to ANSYS CFX while the derivatives are analytically evaluated from(Eqn. Internal energy is calculated as a function of temperature and volume (, ) by integrating from the reference state (, ) along path 'amnc' (see diagram below) to the required state (,) using the following differential relationship: (Eqn. 98) Firsttheenergychangeiscalculatedatconstanttemperaturefromthereferencevolumeto infinite volume (ideal gas state), then the energy change is evaluated at constant volume using the ideal gas.

The final integration, also at constant temperature, subtracts the energy change from infinite volume to the required volume. In integral form, the energy change along this path is: (Eqn. 9) Once the internal energy is known, then enthalpy is evaluated from internal energy: Tref vref Tv udc v TdT pd T pd vref ∫ cv0 Td Tref ∫ T pd Basic Solver Capability Theory: Governing Equations Page 28ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 1.0. © 1996-2006 ANSYS Europe, Ltd. All rights reserved.

Contains proprietary and confidential information of ANSYS, Inc. And its subsidiaries and affiliates. (Parte 3 de 5).

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