Mathematical Equations of CFD

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

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

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

Navier-Stokes Equations Conservation of Mass Conservation of Momentum

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

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.

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

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.

Incompressible Navier-Stokes Equations Conservation of Mass Conservation of Momentum

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.

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)

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

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).

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

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

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)

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.).

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

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.