Mathematical equations of cfd
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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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  • 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.