Mathematical Equations of CFD

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Mathematical Equations of CFD
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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

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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

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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

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

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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

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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.

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

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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)
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Euler Equations

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

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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).

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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 (lt 10-12 by
  volume)
• Model individual particles by constructing a
  force balance on the moving particle

Drag Force
Particle path
Body Force
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Discrete Phase Modeling (2)

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

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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)

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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.).

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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

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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.