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Introduction to Computational Fluid Dynamics Lecture 1: Review. Introductions. Instructor Bio Ph.D in ME (CFD applications in Materials Processing) Post Doctoral Fellowship at Stanford (Hydrodynamic instabilities in cavity flows) 7 years of cfd development at FDI, Fluent

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  • Instructor Bio
    • Ph.D in ME (CFD applications in Materials Processing)
    • Post Doctoral Fellowship at Stanford (Hydrodynamic instabilities in cavity flows)
    • 7 years of cfd development at FDI, Fluent
    • 6 years as Member of Technical Staff and R&D Manager at FSI Intl.
    • 2 years as a Principal Consultant at Applied Thermal Technologies
  • Student introductions
  • 1. Review of basic numerical analysis [1 week]
  • 2. System-level solution of cfd problems [2 weeks]
  • 3. CFD analysis using commercial codes, applicability and relevance [1 week] (Handout Project 1)
  • 4. Grid generation [2 weeks] (Handout Projects 2, 3)
  • 5. Incompressible flows with heat transfer [4 weeks] (Final project selection)
  • 6. Importance of boundary conditions [2 weeks]
  • 7. (Based on student feedback, some of these topics will be discussed) Special topics: Turbulent flows, Free surfaces, Melt interfaces, Porous Media, User functions [2 weeks]
  • Though the presentations and notes includes references or materials from some CFD vendors, the instructor is not a representative or an advocate of these companies’ products or services.
  • These materials are used only for educational purposes and it should be clear that the choice of suitable CFD software is entirely up to the user.
  • For the purposes of instruction and evaluation of projects macroflow and fluent softwares will be used.
web sites for cfd
Web-sites for CFD
  • Etc..
  • Computational fluid dynamics by Roache, P.J.
    • Classic book, arguably the earliest text book, still referred to by many
  • Computational fluid mechanics and heat transfer by Anderson, Tannehill and Pletcher
    • Good general descriptions for finite difference methods
  • Numerical heat transfer and fluid mechanics by Patankar
    • Written by the Professor who came up with SIMPLE algorithms
  • Introduction to finite element method by Zienkiewicz
    • THE text book for FEM (stress analysis, heat transfer)
  • Fluent online resources
    • Downloadable examples and tutorial problems
  • Computational methods for fluid dynamics by Ferziger & Peric
    • Good explanation of cfd codes, Text book for this class
  • resources
  • 1910 – Richardson, 50 page paper to Royal Society,
    • Laplace’s eqn, Biharmonic eqn
    • Numerical boundary conditions at sharp corners, at infinity
    • Finite difference equations, iterative solution
    • Grid convergence, extrapolation to zero grid size
    • (Hand calculations, n/18 of pence for each pt (n digits), used human computers)
    • Discounts for wrong answers, 2000 operations per week !)
  • 1918 – Liebermann (Continuous substitution)
  • 1928 – Courant, Friedrichs and Levy (Existence and uniqueness for elliptic, parabolic and hyperbolic systems, CFL stability limit)
  • 1933 – Thom (first viscous fluid dynamics problem)
  • 1946 – Southwell (residual relaxation method)
  • 1955 – Allen & Southwell (Coordinate transformation, Flow past a cylinder)
  • 1950, 54 – Frankel, Young (Successive over-relaxation method, Optimum relaxation factor)
  • 1950 – Von Neuman (Stability of parabolic difference equations)
  • 1955, 56 – Peaceman & Rachford, Douglas & Rachford (ADI, larger time steps)
  • 1965 – Scientific American article on CFD, Harlow & Fromm

“In fluid mechanics, we obtain exact solutions of approximate equations or approximate solutions of exact equations”

transport equations
Transport Equations
  • Mass conservation

The integral form of mass conservation equation is

where ρ is the density in domain Ω , v the velocity of the fluid and n the unit normal to the boundary, S.

  • Marker and Cell methods – Harlow & Welch (> 1965)
  • Finite difference methods for Navier Stokes (> 1970)
  • Finite element methods for stress analysis (> 1970)
  • Finite volume methods (>1980)
  • Finite element and Spectral element methods for CFD (>1980)
  • Lattice-gas methods (> 1990)
transport equations11
Transport Equations
  • Momentum Conservation

T = Stress tensor, n = normal to the boundary

b = body force (gravity, centrifugal, Coriolis, Lorentz etc..)

transport equations12
Transport Equations
  • Energy transport

T = temperature, k = thermal conductivity, c = specific heat at constant pressure, Q = heat flux

(Species transport is similar – no specific heat term)

navier stokes equations
Navier-Stokes Equations

Conservation of Mass

Conservation of Momentum

Courtesy: Fluent, Inc.

navier stokes equations 2
Navier-Stokes Equations (2)

Conservation of Energy

Equation of State

Property Relations

Courtesy: Fluent, Inc.

  • Incompressibility - Ma < 0.3
  • Boussinesq approximation – Linear variation of density with temperature ρ = ρ0 (1 - β(T-T0))
  • Turbulence – models (k-e, RNG, LES etc.)
  • Viscoelasticity (generalized second-order fluid model)
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

Courtesy: Fluent, Inc.

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.

Courtesy: Fluent, Inc.

incompressible navier stokes equations
Incompressible Navier-Stokes Equations

Conservation of Mass

Conservation of Momentum

Courtesy: Fluent, Inc.

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.

Courtesy: Fluent, Inc.

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)

Courtesy: Fluent, Inc.

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

Courtesy: Fluent, Inc.

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).

Courtesy: Fluent, Inc.

constitutive equations
Constitutive Equations
  • Newtonian, non-Newtonian fluids (stress-strain relationship)
  • Fourier Law (flux vs. temperature gradient)
  • Fick’s law (species flux vs. species gradient)
  • Material properties – density, viscosity, thermal conductivity, species diffusivity, coefficient of thermal expansion etc.
  • Equations of State (ex. ideal gas law)
boundary conditions
Boundary Conditions
  • Dirichlet – constant or function of time
  • Neuman – gradient = constant or function of time
  • Robin – mixed type
non dimensionalization
  • Re = (u0 L0)/υ
  • Ra = Gr Pr = gβΔTL03/υα (natural convection)
  • Ma = u0 /a (compressibility)
  • Ca = μu0/γ (free-surfaces)
  • Fr = u0/sqrt(gL0) (hydrodynamic flows)
  • St = L0/(u0 t0) (shedding frequency)
  • Proper length scales
  • Importance of various terms
  • Wider applicability of solutions
  • Numerically stable solutions
gauss divergence theorem
Gauss’ Divergence Theorem
  • Convert a volume integral to a surface integral and reduce the order of equations by one
  • Used extensively in finite element and finite volume methods

V = vector (velocity ex.)

T = Stress tensor

Ω [Volume]

Γ (all surfaces)

classification of flows
Classification of Flows
  • Hyperbolic flows – Unsteady, inviscid compressible flow (b^2-4ac>0)
  • Parabolic flows – Boundary-layer equations, unsteady conduction eqn
  • Elliptic flows – steady, incompressible flows (b^2-4ac<0)
  • Unsteady, incompressible flows – elliptic in space, parabolic in time