# 3.2. Math Models

Fire simulation requires the solution of variable property, high Grashof number, turbulent, low Mach number flow including the effects of species and soot transport, radiation, and buoyancy.

Conservation laws include mass of the mixture, momentum, mass of the individual species, and energy. Length scales vary from molecular to convection dominated. For purposes of discussion, length scales are also categorized by the method of resolution.

The transport equations used to describe fire physics are based on two sets of approximations to the fundamental equations of fluid dynamics. Fast acoustic time scales are removed from the equations using low Mach number asymptotics, described in Low Mach Number Equations.

Turbulent transport at high Grashof numbers is modeled using a Reynolds averaging approach, described in RANS Temporal Filtering.

In what follows, we note that unless specifically stated otherwise all units in the equations and submodel expressions are cgs. For a more extensive treatment of units and unit conversions in Fuego, please see the “Units and Unit Conversions” section in the User’s Manual. The numerical methods we use to solve the transport equations are of the finite volume class. Therefore, we generally write the transport equations in the integral form.

- 3.2.1. Low Mach Number Equations
  - 3.2.1.1. Asymptotic Expansion
  - 3.2.1.2. Variable Thermodynamic Pressure
- 3.2.2. Laminar Flow Equations
  - 3.2.2.1. Conservation of Mass
  - 3.2.2.2. Conservation of Momentum
  - 3.2.2.3. Conservation of Energy
  - 3.2.2.4. Conservation of Species
  - 3.2.2.5. Conservation of Momentum, Axisymmetric with Swirl
  - 3.2.2.6. Laminar Flow Boundary Conditions
- 3.2.3. Radiation Transport Equation
  - 3.2.3.1. Boltzmann Transport Equation
  - 3.2.3.2. Radiation Intensity Boundary Condition
- 3.2.4. Turbulence Modeling Overview
  - 3.2.4.1. RANS Temporal Filtering
  - 3.2.4.2. LES Spatial Filtering
- 3.2.5. Turbulent Flow Equations, Favre-Averaged
  - 3.2.5.1. Conservation of Mass
  - 3.2.5.2. Conservation of Momentum
  - 3.2.5.3. Conservation of Energy
  - 3.2.5.4. Conservation of Species
  - 3.2.5.5. Radiation Transport
  - 3.2.5.6. Comparison Between Time-Averaging and Favre-Averaging
- 3.2.6. Turbulence Closure Models
  - 3.2.6.1. Standard RANS Model
  - 3.2.6.2. Low Reynolds Number RANS Model
  - 3.2.6.3. RNG RANS Model
  - 3.2.6.4. RANS Model
  - 3.2.6.5. RANS Model
  - 3.2.6.6. Shear Stress Transport (SST)
  - 3.2.6.7. Standard Smagorinsky LES Model
  - 3.2.6.8. Dynamic Smagorinsky LES Model
  - 3.2.6.9. Subgrid-Scale Kinetic Energy One-Equation LES Model
  - 3.2.6.10. Dynamic Subgrid-Scale Kinetic Energy One-Equation LES Model
  - 3.2.6.11. Buoyancy Models for the Production Rate
  - 3.2.6.12. Turbulence closure model constants
- 3.2.7. Wall Boundary Conditions for Turbulence Models
  - 3.2.7.1. Resolution of Boundary Layer; Momentum
  - 3.2.7.2. Resolution of Boundary Layer; Turbulence Quantities
  - 3.2.7.3. Resolution of Boundary Layer; Enthalpy
  - 3.2.7.4. Wall Functions for Turbulent Flow Boundary Conditions
  - 3.2.7.5. Wall Functions; Momentum
  - 3.2.7.6. Wall Functions; Turbulent Kinetic Energy
  - 3.2.7.7. Wall Functions; Turbulent Kinetic Energy
  - 3.2.7.8. Wall Functions; Turbulence Dissipation Transport
  - 3.2.7.9. Wall Functions; Turbulent Frequency Transport
  - 3.2.7.10. Wall Functions; Enthalpy Transport
  - 3.2.7.11. Wall Functions; Scalar Transport
- 3.2.8. Inlet Conditions for Turbulence Quantities
  - 3.2.8.1. Turbulent Kinetic Energy
  - 3.2.8.2. Turbulence Dissipation Rate
- 3.2.9. Mass Injection Boundary Condition
  - 3.2.9.1. Mass Blowing Correction for Turbulent Applications
- 3.2.10. EDC Turbulent Combustion Model
  - 3.2.10.1. Model Characteristics
  - 3.2.10.2. Physical Interpretation
  - 3.2.10.3. Thermochemistry
  - 3.2.10.4. Chemical Mechanism
  - 3.2.10.5. Species Consumption/Production Limits
  - 3.2.10.6. Conservation Laws
  - 3.2.10.7. Effect Of Turbulence On Combustion Rates
  - 3.2.10.8. Average Control Volume Properties
  - 3.2.10.9. Limits Testing
  - 3.2.10.10. Cell Value Information Used By Model
  - 3.2.10.11. Model Outputs
  - 3.2.10.12. Combustion Products Transport Equation
  - 3.2.10.13. Chemical Equilibrium Models
- 3.2.11. Laminar Flamelet Turbulent Combustion Model
  - 3.2.11.1. Adiabatic Property Table Generation
  - 3.2.11.2. Nonadiabatic Property Table Generation
  - 3.2.11.3. Filtered Scalar Dissipation Rate
  - 3.2.11.4. Filtered Mixture Fraction
  - 3.2.11.5. Filtered Scalar Variance
- 3.2.12. Turbulent Reacting Mixing Models
  - 3.2.12.1. Modified EDC Model (PARENTE)
- 3.2.13. Soot Generation Model for Multicomponent Combustion
  - 3.2.13.1. EDC Soot Model
  - 3.2.13.2. Transport Equations and Source Terms
- 3.2.14. Absorptivity Model
  - 3.2.14.1. Theory
  - 3.2.14.2. Emittance Model
- 3.2.15. Fuel Boundary Condition Submodel
  - 3.2.15.1. Option 1: Constant, Specified Mass Flux
  - 3.2.15.2. Option 2: Mass Flux as a Function of Incident Heat Flux
- 3.2.16. Fuel Spreading Submodel
- 3.2.17. One-Dimensional Composite Fire Boundary Condition
  - 3.2.17.1. Conceptual Overview
  - 3.2.17.2. Model Formulation
- 3.2.18. Non-Conformal DG Boundary Condition
  - 3.2.18.1. Conceptual Overview
  - 3.2.18.2. Performance Considerations
  - 3.2.18.3. Non-Conformal Moving Walls
- 3.2.19. Porous-Fluid Coupling Algorithm
  - 3.2.19.1. Fluid Flow
  - 3.2.19.2. Enthalpy Transport
  - 3.2.19.3. Species Transport
- 3.2.20. Volume of Fluid Model
  - 3.2.20.1. Governing Equation
  - 3.2.20.2. Interface Reconstruction
  - 3.2.20.3. Continuity Equation
  - 3.2.20.4. Property Evaluation