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## PowerPoint Slideshow about 'Mathematical Equations of CFD' - rolf

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Presentation Transcript

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)

- 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 (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 (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.

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