Mathematical Equations of CFD

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) • 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.