1. Fluent Lecture Dr. Thomas J. Barber www.engr.uconn.edu/~barbertj

2. Outline Background Issues
Codes, Flow Modeling, and Reduced Equation Forms
Numerical Methods: Discretize, Griding, Accuracy, Error
Data Structure, Grids
Turbulence
Fluent

3. CFD Connection to Other Solution Approaches
CFD (numerical) approach is most closely related to experimental approach, i.e.
can arbitrarily select physical parameters (tunnel conditions)
output is in form of discrete or point data
results have to be interpreted (corrected) for errors in simulation.

4. Background Limiting Factors - I
Computer size:
Moore’s law: First postulated by Intel CEO George Moore. Observation that logic density of silicon integrated circuits has closely followed curve: Bits per sq. in.(and MIPS) doubles power of computing (speed and reduced size), thereby quadrupling computing power every 24 months.
Calculations per second per year for $1000.

6. Outline Background Issues
Codes, Flow Modeling, and Reduced Equation Forms
Numerical Methods: Discretize, Griding, Accuracy, Error
Data Structure, Grids
Turbulence
Fluent

7. What is a CFD code?
Converts chosen physics into discretized forms and solves over chosen physical domain
Geometry
Definition
Computational Grid
and Domain
Definition
Boundary
Conditions
Preprocessing
Discretization
Approach
Solution
Approach
Computer
Usage
Strategy
Processing
Solution
Assessment
Solution
Display
Performance
Analysis
Postprocessing

8. Problem Formulation Equations of Motion
Conservation of mass (continuity) = particle identity
Conservation of linear momentum = Newton’s law
Conservation of energy = 1st law of thermo (E)
2nd law of thermo (S)
Any others?????
Most General Form: Navier-Stokes Equations
Written in differential or integral (control volume) form.
Dependent variables typically averaged over some time scale, shorter than the mean flow unsteadiness (Reynolds-averaged Navier-Stokes - RANS equations).

9. Reduced Forms of Governing Equations
Critical issue: modeling
viscous and turbulent
flow behavior

10. Complex Aircraft Analysis, Circa 1968 B747-100 with space shuttle Enterprise
What is different with these aircraft from normal operation?

11. Reduced Forms of Governing Equations
Euler Equations
Coupled system of 5 nonlinear first order PDE’s
Describes conservation mass, momentum, energy
Describes wave propagation (convective) phenomena
Full Potential Equation
Single nonlinear second order PDE
Describes conservation mass, energy
Conservation of momentum not fully satisfied in presence
of shocks
P:otential Flow Equation
Single linear second order PDE
Describes incompressible flow
Navier-Stokes Equations
Coupled system of 5 nonlinear second order PDE’s
Describes wave propagation phenomena damped
by viscosity
More Physics
(More complex equations)
Neglect viscosity &
heat conduction
Isentropic, irrotational flows
Neglect compressibility
More Geometry
(More complex grid generation)
(More grid points)

12. Outline Background Issues
Codes, Flow Modeling, and Reduced Equation Forms
Numerical Methods: Discretize, Griding, Accuracy, Error
Data Structure, Grids
Turbulence
Fluent

13. Finite Volume Finite Element All based on discretization approaches
Finite Difference
Finite Volume
Finite Element
All based on
discretization
approaches
P.D.E.
Lu=f
Discretize
System of Linear
Algebraic Eqns Up

14. Breakup Continuous Domain into a Finite Number of Locations
Boundary Condition
B. C.
B. C.
Boundary Condition

15. Discretization & Order of Accuracyx fi
fi+1
fi+2
fi+3
xi+1
xi+2
xi+3 xi x
Taylor Series Expansion
Polynomial Function [Power Series]
Accuracy Dependent on Mesh Size and Variable Gradients

16. Discretization Example
Derivative approximation proportional to polynomial order
Order of accuracy: mesh spacing, derivative magnitude
only reasonable if product is small

17. Numerical Error Sources - I
Truncation error
Finite polynomial effect
Diffusion: acts like artificial viscosity & damps out disturbances
Dispersion: introduces new frequencies to input disturbance
Effect is pronounced near shocks
Exact Diffusion Dispersion

18. Numerical Error Sources - II
at t=400
at t=0
Traveling linear wave model problem

19. Numerical Error Sources - III
at t=400

20. Numerical Error Sources - IV
at t=400

21. Time-Accurate vs. Time-Marching
Time-marching: steady-state solution from unsteady equations
Intermediate solution has no meaning
Time-accurate: time-dependent, valid at any time step

22. Numerical Properties of Method
Stability
Tendency of error in solution of algebraic equations to decay
Implies numerical solution goes to exact solution of discretized equations
Convergence
Solution of approximate equations approaches exact set of algebraic eqns.
Solutions of algebraic eqns. approaches exact solution of P.D.E.’s as x  t  0
Governing
P.D.E.’s
L(U)
Discretization
System of
Algebraic Equations
Consistency
Exact Solution U
Convergence
as x  t  0
Approximate Solution u

23. How good are the results?
Assess the calculation for
Grid independence
Convergence (mathematical): residuals as measure of how well the finite difference equation is satisfied.
Look for location of maximum errors
Look for non-monotonicity

24. How good are the results?
Convergence (physical): Check conserved properties: mass (for internal flows), atom balance (for chemistry), total enthalpy, e.g.

25. Outline Background Issues
Codes, Flow Modeling, and Reduced Equation Forms
Numerical Methods: Discretize, Griding, Accuracy, Error
Data Structure, Grids
Turbulence
Fluent

26. 2-D Problem Setup Structured Grid / Data
i,j+1
i-1,j
i,j
i,j-1
i+1,j
X , i
Y, j
Ui,j
Structured Grid / Data
Unstructured Data / Structured Grid 61 35 36 11 37 X Y
U36 60 10 12 62

27. 2-D Problem Setup Semi -Structured Grid / Unstructured Data61 35 36 11 37 X Y
U36 60 10 12 62
Semi -Structured Grid / Unstructured Data
Unstructured Data / Unstructured Grid 61 35 36 11 37 X Y
U36 60 10 12 62

28. Grid Generation Transformation to a new coordinate system
Transformation to a stretched grid

29. Grid Generation - Generic Topologies
More complicated grids can be constructed by combining the basic grid
topologies - cylinder in a duct
Block-structured
O + H
Overset or Chimera
Cartesian + Polar
Both take advantage of natural symmetries of the geometry

30. Grid Generation - Generic Topologies
More complicated grids can be constructed taking advantage of simple elements
Cartesian-stepwise
Unstructured-hybrid
Dimension Unstructured Structured
2D triangular quadrilateral
3D tetrahedra hexahedra

31. Outline Background Issues
Codes, Flow Modeling, and Reduced Equation Forms
Numerical Methods: Discretize, Griding, Accuracy, Error
Data Structure, Grids
Turbulence
Fluent

32. Viscosity and Turbulence

33. Viscosity and Turbulence
Laminar
Steady Unsteady
Turbulent
Steady Unsteady

34. Viscosity and Turbulence Properties Averaged Over Time Scale Much Smaller Than Global Unsteadiness

35. Viscosity and Turbulence
Laminar viscosity modeled by algebraic law: Sutherland
Turbulent viscosity modeled by 1 or 2 Eqn. Models
Realizable k- model is most reliable
k=turbulence kinetic energy
 = turbulence dissipation
Model near wall behavior by:
Wall integration; more mesh near wall, y+  1-2
Wall functions: less mesh, algebraic wall model, y+  30-50

36. Outline Background Issues
Codes, Flow Modeling, and Reduced Equation Forms
Numerical Methods: Discretize, Griding, Accuracy, Error
Data Structure, Grids
Turbulence
Fluent

37. Finite Volume Cell centered Corner centered
Basic conservation laws of fluid dynamics are expressed in terms of mass, momentum and energy in control volume form.
F.V. method: on each cell, conservation laws are applied at a discrete point of the cell [node].
Cell centered
Corner centered
Piecewise constant
interpolation
Piecewise linear
Interpolation

38. 2D Steady Flux Equation Finite-difference: centered in space schemeW E N S
 X
i,j+1
i-1,j
i,j-1

39. Steady Governing Equations
 = transport coeff.
 /  = diffusivity
Start with generalized RANS equations

40. Fluent Solution Method Simple Scheme
SIMPLE: Semi-Implicit Method for Pressure Linked Equations

41. Fluent Solution Method Simple Scheme
Solution algorithm:
Staggered grid; convected on different grid from pressure.
Avoids wavy velocity solutions

42. Fluent Solution Method Simple Scheme
CV for u-eqn.
Two sets of indices or one and one staggered at half-cell

43. Fluent Solution Method Simple Scheme
CV for v-eqn.

44. Fluent Solution Method Simple Scheme
CV for p-eqn.

45. Fluent Solution Method Simple Scheme
5-point computational molecules for linearized system
using geographical not index notation

46. Fluent Solution Method Simple Scheme – Multidimensional Model
2-D and 3-D computational molecules using geographical not index notation

47. Fluent Operational Procedures
Generate Geometry
Generate Computational Grid
Set Boundary Conditions
Set Flow Models: Equation of State, Laminar or Turbulent, etc.
Set Convergence Criteria or Number of Iterations
Set Solver Method and Solve
Check Solution Quality Parameters: Residuals, etc.
Post-process: Line Plots, Contour Plots
Export Data for Further Post-processing

48. Suggested Fluent Development Path
Read FlowLab FAQ notes [Barber Web site]
Run FlowLab to familiarize yourself with GUI, solution process and post-processing
Read Cornell University training notes [Handout]
Develop a relevant validation-qualification process, i.e. compare with known analyses or data
Developing laminar flow in straight pipe
Developing turbulent flow in a straight pipe [if appropriate]
Convection process
Convergent-divergent nozzle flow ….