Mathematical Equations of CFD

1 / 20

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

PowerPoint Slideshow about 'Mathematical Equations of CFD' - rolf

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

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
• 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

Conservation of Mass

Conservation of Momentum

Navier-Stokes Equations (2)

Conservation of Energy

Equation of State

Property Relations

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

Conservation of Mass

Conservation of Momentum

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.