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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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Presentation Transcript
  • 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
introduction 2
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
Navier-Stokes Equations

Conservation of Mass

Conservation of Momentum

navier stokes equations 2
Navier-Stokes Equations (2)

Conservation of Energy

Equation of State

Property Relations

navier stokes equations 3
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 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
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
Incompressible Navier-Stokes Equations

Conservation of Mass

Conservation of Momentum

incompressible navier stokes equations 2
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
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).


= thermal expansion coefficient


= reference density and temperature

g = gravitational acceleration (assumed pointing in -y direction)

euler equations
Euler Equations
  • Neglecting all viscous terms in the Navier-Stokes equations yields the Euler equations:
euler equations 2
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
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
Discrete Phase Modeling (2)
  • Assuming the particle is spherical (diameter D), its trajectory is governed by
discrete phase modeling 3
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
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
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
  • 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.