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ANSYS CFX-Solver

Theory Guide

ANSYS CFX Release 11.

December 2006

ANSYS, Inc.

Southpointe

275 Technology Drive

Canonsburg, PA 15317

[email protected]

http://www.ansys.com

(T) 724-746-

(F) 724-514-

ANSYS CFX-Solver Theory Guide Page v

Table of Contents

Copyright and Trademark Information

Disclaimer Notice

U.S. Government Rights

Third-Party Software

Basic Solver Capability Theory

Introduction........................................................................................... 1 Documentation Conventions........................................................................... 2 Dimensions........................................................................................ 2 List of Symbols..................................................................................... 2 Variable Definitions................................................................................. 6 Mathematical Notation............................................................................ Governing Equations.................................................................................. Transport Equations............................................................................... Equations of State................................................................................. Conjugate Heat Transfer........................................................................... Buoyancy............................................................................................. Full Buoyancy Model.............................................................................. Boussinesq Model................................................................................. Multicomponent Flow................................................................................. Multicomponent Notation......................................................................... Scalar Transport Equation......................................................................... Algebraic Equation for Components............................................................... Constraint Equation for Components.............................................................. Multicomponent Fluid Properties.................................................................. Energy Equation................................................................................... Multicomponent Energy Diffusion.................................................................

Table of Contents: GGI and MFR Theory

Table of Contents: Particle Transport Theory

Table of Contents: Radiation Theory

• Additional Variables. Page vi ANSYS CFX-Solver Theory Guide
  • Transport Equation.
  • Diffusive Transport Equation..
  • Poisson Equation.
  • Algebraic Equation.
• Rotational Forces..
  • Alternate Rotation Model.
• Sources.
  • Momentum Sources.
  • General Sources.
  • Mass (Continuity) Sources.
  • Bulk Sources.
  • Radiation Sources.
  • Boundary Sources.
• Boundary Conditions.
  • Inlet (subsonic).
  • Inlet (supersonic).
  • Outlet (subsonic).
  • Outlet (supersonic).
  • Opening.
  • Wall..
  • Symmetry Plane.
• Automatic Time Scale Calculation..
  • Fluid Time Scale Estimate.
  • Solid Time Scale Estimate.
• Mesh Adaption..
  • Adaption Criteria.
  • Mesh Refinement Implementation in ANSYS CFX.
  • Mesh Adaption Limitations.
• Flow in Porous Media..
  • Darcy Model.
  • Directional Loss Model.
• Introduction. Turbulence and Wall Function Theory
• Turbulence Models.
  • Statistical Turbulence Models and the Closure Problem..
• Eddy Viscosity Turbulence Models.
  • The Zero Equation Model in ANSYS CFX.
  • Two Equation Turbulence Models..
  • The Eddy Viscosity Transport Model..
• Reynolds Stress Turbulence Models.
  • The Reynolds Stress Model.
  • Omega-Based Reynolds Stress Models.
  • Rotating Frame of Reference for Reynolds Stress Models..
• ANSYS CFX Transition Model Formulation.
• Large Eddy Simulation Theory.
  • Smagorinsky Model.
  • Wall Damping.
• Detached Eddy Simulation Theory ANSYS CFX-Solver Theory Guide Page vii
  • SST-DES Formulation Strelets et al.
  • Zonal SST-DES Formulation in ANSYS CFX
  • Discretization of the Advection Terms.
  • Boundary Conditions
• Scale-Adaptive Simulation Theory
  • SAS-SST Model Formulation
• Modeling Flow Near the Wall
  • Mathematical Formulation.
• Wall Distance Formulation.
  • 1D Illustration of Concept.
  • Concept Generalized to 3D
• Introduction GGI and MFR Theory
• Interface Characteristics
• Numerics
• Introduction Multiphase Flow Theory
• Multiphase Notation
  • Multiphase Total Pressure
• The Homogeneous and Inhomogeneous Models
  • The Inhomogeneous Model
  • The Homogeneous Model
• Hydrodynamic Equations
  • Inhomogeneous Hydrodynamic Equations
  • Homogeneous Hydrodynamic Equations.
• Multicomponent Multiphase Flow
• Interphase Momentum Transfer Models.
  • Interphase Drag
  • Interphase Drag for the Particle Model
  • Interphase Drag for the Mixture Model
  • Interphase Drag for the Free Surface Model
  • Lift Force
  • Virtual Mass Force
  • Wall Lubrication Force
  • Interphase Turbulent Dispersion Force
• Solid Particle Collision Models
  • Solids Stress Tensor
  • Solids Pressure
  • Solids Bulk Viscosity
  • Solids Shear Viscosity
  • Granular Temperature
• Interphase Heat Transfer
  • Phasic Equations
  • Inhomogeneous Interphase Heat Transfer Models
  • Homogeneous Heat Transfer in Multiphase Flow
• Multiple Size Group (MUSIG) Model Page viii ANSYS CFX-Solver Theory Guide
  • Model Derivation.
  • Size Group Discretization
  • Breakup Models
  • Coalescence Models
• The Algebraic Slip Model
  • Phasic Equations
  • Bulk Equations
  • Drift and Slip Relations
  • Derivation of the Algebraic Slip Equation.
  • Turbulence Effects.
  • Energy Equation.
  • Wall Deposition
• Turbulence Modeling in Multiphase Flow
  • Phase-Dependent Turbulence Models
  • Turbulence Enhancement
  • Homogeneous Turbulence for Multiphase Flow
• Additional Variables in Multiphase Flow.
  • Additional Variable Interphase Transfer Models
  • Homogeneous Additional Variables in Multiphase Flow
• Sources in Multiphase Flow
  • Fluid-specific Sources.
  • Bulk Sources
• Interphase Mass Transfer
  • Secondary Fluxes.
  • User Defined Interphase Mass Transfer
  • General Species Mass Transfer
  • The Thermal Phase Change Model
  • The Cavitation Model
  • The Droplet Condensation Model
• Free Surface Flow
  • Implementation
  • Surface Tension
• Introduction Particle Transport Theory
• Lagrangian Tracking Implementation
  • Integration
  • Interphase Transfer Through Source Terms.
• Momentum Transfer
  • Drag Force
  • Buoyancy Force
  • Rotation Force
  • Virtual or Added Mass Force
  • Pressure Gradient Force
  • Turbulence in Particle Tracking
  • Turbulent Dispersion
• Heat and Mass Transfer ANSYS CFX-Solver Theory Guide Page ix
  • Heat Transfer.
  • Simple Mass Transfer
  • Liquid Evaporation Model
  • Oil Evaporation/Combustion.
  • Reactions
  • Coal Combustion
  • Hydrocarbon Fuel Analysis Model.
• Basic Erosion Model
  • Model of Finnie
  • Model of Tabakoff and Grant.
  • Overall Erosion Rate and Erosion Output
• Spray Breakup Models
  • Primary Breakup/Atomization Models.
  • Secondary Breakup Models
  • Dynamic Drag Models
  • Dynamic Drag Law Control
  • Penetration Depth and Spray Angle.
• Introduction Combustion Theory
• Transport Equations
• Chemical Reaction Rate.
• Fluid Time Scale for Extinction Model
• The Eddy Dissipation Model
  • Reactants Limiter
  • Products Limiter.
  • Maximum Flame Temperature Limiter
• The Finite Rate Chemistry Model.
  • Third Body Terms.
• The Combined Eddy Dissipation/Finite Rate Chemistry Model
• Combustion Source Term Linearization
• The Flamelet Model
  • Laminar Flamelet Model for Non Premixed Combustion
  • Coupling of Laminar Flamelet with the Turbulent Flow Field
  • Flamelet Libraries
• Burning Velocity Model (Premixed or Partially Premixed)
  • Reaction Progress
  • Weighted Reaction Progress
• Burning Velocity Model (BVM)
  • Equivalence Ratio, Stoichiometric Mixture Fraction
• Laminar Burning Velocity
  • Value
  • Equivalence Ratio Correlation.
• Turbulent Burning Velocity
  • Value
  • Zimont Correlation
  • Peters Correlation
  • Mueller Correlation
• Spark Ignition Model Page x ANSYS CFX-Solver Theory Guide
• Phasic Combustion
• NO Formation Model
  • Formation Mechanisms.
• Chemistry Post-Processing
• Soot Model
  • Soot Formation.
  • Soot Combustion.
  • Turbulence Effects.
• Introduction Radiation Theory
• Radiation Transport
  • Blackbody Emission
  • Quantities of Interest
  • Radiation Through Domain Interfaces
• Rosseland Model
  • Wall Treatment
• The P1 Model
  • Wall Treatment
• Discrete Transfer Model
• Monte Carlo Model
• Spectral Models
  • Gray
  • Multiband Model
  • Multigray Model.
• Introduction Discretization and Solution Theory
• Numerical Discretization.
  • Discretization of the Governing Equations.
  • The Coupled System of Equations.
• Solution Strategy - The Coupled Solver.
  • General Solution
  • Linear Equation Solution.
  • Residual Normalization Procedure
• Discretization Errors
  • Controlling Error Sources
  • Controlling Error Propagation.

ANSYS CFX Release 11.0. © 1996-2006 ANSYS Europe, Ltd. All rights reserved. Page 1 Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

ANSYS CFX-Solver Theory Guide

Basic Solver Capability Theory

Introduction

This chapter describes:

• Documentation Conventions (p. 2)
• Governing Equations (p. 22)
• Buoyancy (p. 33)
• Multicomponent Flow (p. 35)
• Additional Variables (p. 40)
• Rotational Forces (p. 42)
• Sources (p. 43)
• Boundary Conditions (p. 46)
• Automatic Time Scale Calculation (p. 58)
• Mesh Adaption (p. 61)
• Flow in Porous Media (p. 65)

This chapter describes the mathematical equations used to model fluid flow, heat, and mass transfer in ANSYS CFX for single-phase, single and multi-component flow without combustion or radiation. It is designed to be a reference for those users who desire a more detailed understanding of the mathematics underpinning the ANSYS CFX-Solver, and is therefore not essential reading. It is not an exhaustive text on CFD mathematics; a reference section is provided should you wish to follow up this chapter in more detail. Information on dealing with multiphase flow:

• Multiphase Flow Theory (p. 123)
• Particle Transport Theory (p. 187) Information on combustion and radiation theory:
• Combustion Theory (p. 227)
• Radiation Theory (p. 263)

Recommended books for further reading on CFD and related subjects:

• Further Background Reading (p. 6 in "ANSYS CFX Introduction")

Basic Solver Capability Theory: Documentation Conventions

ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 11.0. © 1996-2006 ANSYS Europe, Ltd. All rights reserved. Page 3 Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

concentration of components A and B i.e. mass per unit volume of components A and B (single-phase flow) Reynolds Stress model constant

specific heat capacity at constant pressure specific heat capacity at constant volume Reynolds Stress model constant

Reynolds Stress model constant

binary diffusivity of component A in component B kinematic diffusivity of an additional variable,

distance or length

constant used for near-wall modeling

Zero Equation turbulence model constant RNG- - turbulence model coefficient

gravity vector

specific static (thermodynamic) enthalpy

For details, see Static Enthalpy (p. 7). heat transfer coefficient

specific total enthalpy For details, see Total Enthalpy (p. 8). turbulence kinetic energy per unit mass

local Mach number, mass flow rate

shear production of turbulence

static (thermodynamic) pressure For details, see Static Pressure (p. 6). reference pressure For details, see Reference Pressure (p. 6). total pressure For details, see Total Pressure (p. 14). modified pressure For details, see Modified Pressure (p. 6). universal gas constant

Symbol Description Dimensions Value cA , cB M L –^3

cS 1 0.

c (^) p (^) L 2 T –^2 Θ –^1

c (^) v L 2 T –^2 Θ –^1

cε 1 1 1.

cε 2 1 1.

D (^) AB (^) L 2 T –^1

DΦ

Γ (^) Φ ⁄ ρ

L

2 T

• 1

d L

E 1 9.

f (^) μ 1 0.

f (^) h k ε 1

g (^) L T –^2

h h, (^) stat L 2 T –^2

hc M T –^3 Θ –^1

htot (^) L 2 T –^2

k (^) L 2 T –^2

M U ⁄ c 1

m˙ (^) M T –^1

P (^) k (^) M L –^1 T –^3

p p, (^) stat M L –^1 T –^2

pref (^) M L –^1 T –^2

ptot (^) M L –^1 T –^2

p' (^) M L –^1 T –^2

R 0 L 2 T –^2 Θ –^1 8314.

Basic Solver Capability Theory: Documentation Conventions

Page 4 ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 11.0. © 1996-2006 ANSYS Europe, Ltd. All rights reserved. Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

Reynolds number, location vector

volume fraction of phase

energy source

momentum source

mass source

turbulent Schmidt number,

mass flow rate from phase to phase . static (thermodynamic) temperature For details, see Static Temperature (p. 9). domain temperature For details, see Domain Temperature (p. 9). buoyancy reference temperature used in the Boussinesq approximation saturation temperature

total temperature For details, see Total Temperature (p. 10). vector of velocity

velocity magnitude

fluctuating velocity component in turbulent flow fluid viscous and body force work term

molecular weight (Ideal Gas fluid model) mass fraction of component A in the fluid used as a subscript to indicate that the quantity applies to phase used as a subscript to indicate that the quantity applies to phase coefficient of thermal expansion (for the Boussinesq approximation) RNG - turbulence model constant

diffusivity

molecular diffusion coefficient of component

Symbol Description Dimensions Value Re rU d ⁄ m 1

r L

rα α 1

S (^) E (^) M L –^1 T –^3

S M (^) M L –^2 T –^2

S (^) MS (^) M L –^3 T –^1

Sct μt /Γt 1

sαβ α β

M T

• 1

T T, (^) stat Θ

Tdom Θ

Tref Θ

Tsat Θ

Ttot Θ

U U x y z, , (^) L T –^1

U (^) L T –^1

u (^) L T –^1

W (^) f (^) M L –^1 T –^3

w 1

Y A

α α

β β

β (^) Θ –^1

βRNG k ε 1 0.

Γ (^) M L –^1 T –^1

ΓA A

M L

• 1 T - 1

Basic Solver Capability Theory: Documentation Conventions

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Such quantities are only used in the chapters describing multicomponent and multiphase flows.

• Multicomponent Flow (p. 35)
• Multiphase Flow Theory (p. 123)

Variable Definitions

Isothermal Compressibility

The isothermal compressibility defines the rate of change of the system volume with pressure. For details, see Variables Relevant for Compressible Flow (p. 62 in "ANSYS CFX-Solver Manager User's Guide").

(Eqn. 1)

Isentropic Compressibility

Isentropic compressibility is the extent to which a material reduces its volume when it is subjected to compressive stresses at a constant value of entropy. For details, see Variables Relevant for Compressible Flow (p. 62 in "ANSYS CFX-Solver Manager User's Guide").

(Eqn. 2)

Reference Pressure

The Reference Pressure (Eqn. 3) is the absolute pressure datum from which all other pressure values are taken. All relative pressure specifications in ANSYS CFX are relative to the Reference Pressure. For details, see Setting a Reference Pressure (p. 10 in "ANSYS CFX-Solver Modeling Guide").

(Eqn. 3)

Static Pressure ANSYS CFX solves for the relative Static Pressure (thermodynamic pressure) (Eqn. 4) in

the flow field, and is related to Absolute Pressure (Eqn. 5).

(Eqn. 4)

(Eqn. 5)

Modified Pressure

When the - turbulence model is used, the fluctuating velocity components give rise to

an additional pressure term to give the modified pressure (Eqn. 6), where is the turbulent kinetic energy. In this case, ANSYS CFX solves for the modified pressure. This variable is named Pressure in ANSYS CFX.

(Eqn. 6)

ρ

⎝⎛– ---⎠⎞^

dρ dp

T

ρ

⎝⎛– ---⎠⎞^

dρ dp

S

Pref

pstat

pabs = pstat +pref

k ε

k

p' pstat^2 ρk 3

Basic Solver Capability Theory: Documentation Conventions

ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 11.0. © 1996-2006 ANSYS Europe, Ltd. All rights reserved. Page 7 Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

Static Enthalpy Specific static enthalpy (Eqn. 7) is a measure of the energy contained in a fluid per unit mass.

Static enthalpy is defined in terms of the internal energy of a fluid and the fluid state:

(Eqn. 7)

When you use the thermal energy model, the ANSYS CFX-Solver directly computes the static enthalpy. General changes in enthalpy are also used by the solver to calculate thermodynamic properties such as temperature. To compute these quantities, you need to know how enthalpy varies with changes in both temperature and pressure. These changes are given by the general differential relationship (Eqn. 8):

(Eqn. 8)

which can be rewritten as (Eqn. 9)

(Eqn. 9)

where is specific heat at constant pressure and is density. For most materials the first

term always has an effect on enthalpy, and, in some cases, the second term drops out or is not included. For example, the second term is zero for materials which use the Ideal Gas equation of state or materials in a solid thermodynamic state. In addition, the second term is also dropped for liquids or gases with constant specific heat when you run the thermal energy equation model.

Material with Variable Density and Specific Heat In order to support general properties, which are a function of both temperature and

pressure, a table for is generated by integrating Equation 9 using the functions

supplied for and. The enthalpy table is constructed between the upper and lower

bounds of temperature and pressure (using flow solver internal defaults or those supplied

by the user). For any general change in conditions from to , the change

in enthalpy, , is calculated in two steps: first at constant pressure, and then at constant temperature using Equation 10.

(Eqn. 10)

hstat ustat

pstat ρstat

dh

∂h ∂ T

p

dT

∂h ∂ p

T

= + dp

dh c (^) p dT

ρ

T

ρ

∂ρ ∂ T

p

= + + dp

c (^) p ρ

h T( ,p)

ρ c (^) p

( p 1 ,T 1 ) ( p 2 ,T 2 )

d h

h 2 – h 1 c (^) p T 1

T 2

∫ dT^

ρ

T

ρ

∂ρ ∂ T

p

p 1

p 2 = + ∫ dp

Basic Solver Capability Theory: Documentation Conventions

ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 11.0. © 1996-2006 ANSYS Europe, Ltd. All rights reserved. Page 9 Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

where is the flow velocity. When you use the total energy model the ANSYS CFX-Solver directly computes total enthalpy, and static enthalpy is derived from this expression. In rotating frames of reference the total enthalpy includes the relative frame kinetic energy. For details, see Rotating Frame Quantities (p. 17).

Domain Temperature

The domain temperature, , is the absolute temperature at which an isothermal

simulation is performed. For details, see Isothermal (p. 8 in "ANSYS CFX-Solver Modeling Guide").

Static Temperature

The static temperature, , is the thermodynamic temperature, and depends on the

internal energy of the fluid. In ANSYS CFX, depending on the heat transfer model you select, the flow solver calculates either total or static enthalpy (corresponding to the total or thermal energy equations). The static temperature is calculated using static enthalpy and the constitutive relationship for the material under consideration. The constitutive relation simply tells us how enthalpy varies with changes in both temperature and pressure.

Material with Constant Density and Specific Heat In the simplified case where a material has constant and , temperatures can be

calculated by integrating a simplified form of the general differential relationship for enthalpy:

(Eqn. 12)

which is derived from the full differential form for changes in static enthalpy. The default reference state in the ANSYS CFX-Solver is and.

Ideal Gas or Solid with cp=f(T) The enthalpy change for an ideal gas or CHT solid with specific heat as a function of temperature is defined by:

(Eqn. 13)

When the solver calculates static enthalpy, either directly or from total enthalpy, you can

back static temperature out of this relationship. When varies with temperature, the

ANSYS CFX-Solver builds an enthalpy table and static temperature is backed out by inverting the table.

Material with Variable Density and Specific Heat To properly handle materials with an equation of state and specific heat that vary as functions of temperature and pressure, the ANSYS CFX-Solver needs to know enthalpy as a function of temperature and pressure,.

U

Tdom

Tstat

ρ c (^) p

hstat – href=c (^) p ( Tstat – Tref)

Tref = 0 [ K] href = 0 [ J ⁄( kg)]

hstat – href c (^) p ( T) dT Tref

Tstat = ∫

c (^) p

h T( ,p)

Basic Solver Capability Theory: Documentation Conventions

Page 10 ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 11.0. © 1996-2006 ANSYS Europe, Ltd. All rights reserved. Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

can be provided as a table using, for example, an RGP file. If a table is not pre-supplied, and the equation of state and specific heat are given by CEL expressions or CEL user functions, the ANSYS CFX-Solver will calculate by integrating the full differential definition of enthalpy change.

Given the knowledge of and that the ANSYS CFX-Solver calculates both static enthalpy and static pressure from the flow solution, you can calculate static temperature by inverting the enthalpy table:

(Eqn. 14)

In this case, you know , from solving the flow and you calculate by table

inversion.

Total Temperature

The total temperature is derived from the concept of total enthalpy and is computed exactly the same way as static temperature, except that total enthalpy is used in the property relationships.

Material with Constant Density and Specific Heat If and are constant, then the total temperature and static temperature are equal

because incompressible fluids undergo no temperature change due to addition of kinetic energy. This can be illustrated by starting with the total enthalpy form of the constitutive relation:

(Eqn. 15)

and substituting expressions for Static Enthalpy and Total Pressure for an incompressible fluid:

(Eqn. 16)

(Eqn. 17)

some rearrangement gives the result that:

(Eqn. 18)

for this case.

h T( ,p)

h T( ,p)

h T( ,p)

h (^) stat – href=h T( (^) stat ,p (^) stat) – h T( (^) ref ,pref)

hstat pstat Tstat

ρ c (^) p

htot – href=c (^) p ( Ttot – Tref)

htot hstat

= +^ --^ ( U ⋅ U )

ptot pstat

= + --ρ ( U ⋅ U )

Ttot =Tstat