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# Mathematical Equations of CFD - PowerPoint PPT Presentation

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.

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### Mathematical Equations of CFD

• Introduction

• Navier-Stokes equations

• Turbulence modeling

• Incompressible Navier-Stokes equations

• Buoyancy-driven flows

• Euler equations

• Discrete phase modeling

• Multiple species modeling

• Combustion modeling

• Summary

• 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

• 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

• Conservation of Mass

Conservation of Momentum

Conservation of Energy

Equation of State

Property Relations

• 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 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

• 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.

• Conservation of Mass

Conservation of Momentum

• 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.

• 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)

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

• 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).

• 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

• Assuming the particle is spherical (diameter D), its trajectory is governed by

• 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)

• 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.).

• 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

• 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.