1. Mathematical Equations of CFD

2. Outline Introduction Navier-Stokes equations Turbulence modeling
Incompressible Navier-Stokes equations
Buoyancy-driven flows
Euler equations
Discrete phase modeling
Multiple species modeling
Combustion modeling
Summary

3. Introduction
In CFD we wish to solve mathematical equations which govern fluid flow, heat transfer, and related phenomena for a given physical problem.
What equations are used in CFD?
Navier-Stokes equations
most general
can handle wide range of physics
Incompressible Navier-Stokes equations
assumes density is constant
energy equation is decoupled from continuity and momentum if properties are constant

4. Introduction (2) Other equations and models
Euler equations
neglect all viscous terms
reasonable approximation for high speed flows (thin boundary layers)
can use boundary layer equations to determine viscous effects
Other equations and models
Thermodynamics relations and equations of state
Turbulence modeling equations
Discrete phase equations for particles
Multiple species modeling
Chemical reaction equations (finite rate, PDF)
We will examine these equations in this lecture

5. Navier-Stokes Equations
Conservation of Mass
Conservation of Momentum

6. Navier-Stokes Equations (2)
Conservation of Energy
Equation of State
Property Relations

7. Navier-Stokes Equations (3)
Navier-Stokes equations provide the most general model of single-phase fluid flow/heat transfer phenomena.
Five equations for five unknowns: r, p, u, v, w.
Most costly to use because it contains the most terms.
Requires a turbulence model in order to solve turbulent flows for practical engineering geometries.

8. Turbulence Modeling
Turbulence is a state of flow characterized by chaotic, tangled fluid motion.
Turbulence is an inherently unsteady phenomenon.
The Navier-Stokes equations can be used to predict turbulent flows but…
the time and space scales of turbulence are very tiny as compared to the flow domain!
scale of smallest turbulent eddies are about a thousand times smaller than the scale of the flow domain.
if 10 points are needed to resolve a turbulent eddy, then about 100,000 points are need to resolve just one cubic centimeter of space!
solving unsteady flows with large numbers of grid points is a time-consuming task

9. Turbulence Modeling (2)
Conclusion: Direct simulation of turbulence using the Navier-Stokes equations is impractical at the present time.
Q: How do we deal with turbulence in CFD?
A: Turbulence Modeling
Time-average the Navier-Stokes equations to remove the high-frequency unsteady component of the turbulent fluid motion.
Model the “extra” terms resulting from the time-averaging process using empirically-based turbulence models.
The topic of turbulence modeling will be dealt with in a subsequent lecture.

10. Incompressible Navier-Stokes Equations
Conservation of Mass
Conservation of Momentum

11. Incompressible Navier-Stokes Equations (2)
Simplied form of the Navier-Stokes equations which assume
incompressible flow
constant properties
For isothermal flows, we have four unknowns: p, u, v, w.
Energy equation is decoupled from the flow equations in this case.
Can be solved separately from the flow equations.
Can be used for flows of liquids and gases at low Mach number.
Still require a turbulence model for turbulent flows.

12. Buoyancy-Driven Flows
A useful model of buoyancy-driven (natural convection) flows employs the incompressible Navier-Stokes equations with the following body force term added to the y momentum equation:
This is known as the Boussinesq model.
It assumes that the temperature variations are only significant in the buoyancy term in the momentum equation (density is essentially constant). b
= thermal expansion coefficient
roTo
= reference density and temperature
g = gravitational acceleration (assumed pointing in -y direction)

13. Euler Equations
Neglecting all viscous terms in the Navier-Stokes equations yields the Euler equations:

14. Euler Equations (2)
No transport properties (viscosity or thermal conductivity) are needed.
Momentum and energy equations are greatly simplified.
But we still have five unknowns: r, p, u, v, w.
The Euler equations provide a reasonable model of compressible fluid flows at high speeds (where viscous effects are confined to narrow zones near wall boundaries).

15. Discrete Phase Modeling
We can simulate secondary phases in the flows (either liquid or solid) using a discrete phase model.
This model is applicable to relatively low particle volume fractions (< %10-12 by volume)
Model individual particles by constructing a force balance on the moving particle
Drag Force
Particle path
Body Force

16. Discrete Phase Modeling (2)
Assuming the particle is spherical (diameter D), its trajectory is governed by

17. Discrete Phase Modeling (3)
Can incorporate other effects in discrete phase model
droplet vaporization
droplet boiling
particle heating/cooling and combustion
devolatilization
Applications of discrete phase modeling
sprays
coal and liquid fuel combustion
particle laden flows (sand particles in an air stream)

18. Multiple Species Modeling
If more than one species is present in the flow, we must solve species conservation equations of the following form
Species can be inert or reacting
Has many applications (combustion modeling, fluid mixing, etc.).

19. Combustion modeling
If chemical reactions are occurring, we can predict the creation/depletion of species mass and the associated energy transfers using a combustion model.
Some common models include
Finite rate kinetics model
applicable to non-premixed, partially, and premixed combustion
relatively simple and intuitive and is widely used
requires knowledge of reaction mechanisms, rate constants (introduces uncertainty)
PDF model
solves transport equation for mixture fraction of fuel/oxidizer system
rigorously accounts for turbulence-chemistry interactions
can only include single fuel/single oxidizer
not applicable to premixed systems

20. Summary
General purpose solvers (such as those marketed by Fluent Inc.) solve the Navier-Stokes equations.
Simplified forms of the governing equations can be employed in a general purpose solver by simply removing appropriate terms
Example: The Euler equations can be used in a general purpose solver by simply zeroing out the viscous terms in the Navier-Stokes equations
Other equations can be solved to supplement the Navier-Stokes equations (discrete phase model, multiple species, combustion, etc.).
Factors determining which equation form to use:
Modeling - are the simpler forms appropriate for the physical situation?
Cost - Euler equations are much cheaper to solver than the Navier-Stoke equations
Time - Simpler flow models can be solved much more rapidly than more complex ones.