Introduction to Computational Fluid Dynamics Lecture 1: Review

1. Introduction to Computational Fluid DynamicsLecture 1: Review

2. 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 • 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

3. Outline • 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]

4. Disclaimer • 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.

5. Web-sites for CFD • www.cfd-online.com • www.fluent.com • www.exa.com • www.starcd.com • www.cfdrc.com • www.ansys.com • www.inres.com • Etc..

6. References • 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 • Cfd-online.com resources

7. History • 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)

8. History • 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”

9. 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.

10. History • 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)

11. Transport Equations • Momentum Conservation T = Stress tensor, n = normal to the boundary b = body force (gravity, centrifugal, Coriolis, Lorentz etc..)

12. 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)

13. Navier-Stokes Equations Conservation of Mass Conservation of Momentum Courtesy: Fluent, Inc.

14. Navier-Stokes Equations (2) Conservation of Energy Equation of State Property Relations Courtesy: Fluent, Inc.

15. Simplifications • 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)

16. 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.

17. 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.

18. Incompressible Navier-Stokes Equations Conservation of Mass Conservation of Momentum Courtesy: Fluent, Inc.

19. 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.

20. 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) Courtesy: Fluent, Inc.

21. Euler Equations • Neglecting all viscous terms in the Navier-Stokes equations yields the Euler equations: Courtesy: Fluent, Inc.

22. 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.

23. 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)

24. Boundary Conditions • Dirichlet – constant or function of time • Neuman – gradient = constant or function of time • Robin – mixed type

25. 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

26. 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)

27. 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