fluent lecture dr. thomas j. barber engr.uconn/~barbertj
DESCRIPTION
Fluent Lecture Dr. Thomas J. Barber www.engr.uconn.edu/~barbertj. Outline. Background Issues Codes, Flow Modeling, and Reduced Equation Forms Numerical Methods: Discretize, Griding, Accuracy, Error Data Structure, Grids Turbulence Fluent. CFD Connection to Other Solution Approaches. - PowerPoint PPT PresentationTRANSCRIPT
Fluent Lecture
Dr. Thomas J. Barberwww.engr.uconn.edu/~barbertj
Outline
• Background Issues• Codes, Flow Modeling, and Reduced
Equation Forms• Numerical Methods: Discretize, Griding,
Accuracy, Error• Data Structure, Grids• Turbulence• Fluent
CFD Connection to Other Solution Approaches
Approach Advantages Disadvantages Experimental Capable of being most realistic Costly Long lead times Scaling problems Tunnel corrections Measurement difficulties Theoretical Analytical forms Restricted geometry and physics Usually for linear problems Numerical Complicated geoms. can be treated Ability to describe physics Unsteady flows can be treated Truncation errors B.C. problems Computer costs
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.
BackgroundLimiting 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.
Date Device Add Time Calculations cost cost CPS/$1000(sec) per sec then $'s 1998 $'s
1946 ENIAC 2.00E-04 5.00E+03 $750,000 $6,265,000 7.98E-01
1951 Univac I 1.20E-04 8.33E+03 $930,000 $5,827,000 1.43E+00
1960 IBM 1620 6.00E-04 1.67E+03 $200,000 $1,101,000 1.51E+00
1966 IBM 360 Model 75 8.00E-07 1.25E+06 $5,000,000 $25,139,000 4.97E+01
1976 DEC PDP-11Model 70 3.00E-06 333000 $150,000 $429,000 7.77E+02
1977 Cray I 1.00E-08 1.00E+08 $10,000,000 $26,881,000 3.72E+03
1977 Apple II 1.00E-05 1.00E+05 $1,300 $3,722 2.69E+04
1979 DEC VAX I IModel 780 2.00E-06 5.00E+05 $200,000 $449,000 1.11E+03
1980 Sun-i 3.00E-06 3.33E+05 $30,000 $59,300 5.62E+03
1982 IBM PC 1.56E-06 6.41E+05 $3,000 $5,064 1.27E+05
1993 Pentium PC 1.00E-07 1.00E+07 $2,500 $2,818 3.55E+06
1996 Pentium PC 1.00E-08 1.00E+08 $2,000 $2,080 4.81E+07
1998 Pentium II PC 5.00E-09 2.00E+08 $1,500 $1,500 1.33E+08
Outline
• Background Issues• Codes, Flow Modeling, and Reduced
Equation Forms• Numerical Methods: Discretize, Griding,
Accuracy, Error• Data Structure, Grids• Turbulence• Fluent
What is a CFD code?
GeometryDefinition
Computational Grid and Domain
Definition
Boundary Conditions
DiscretizationApproach
SolutionApproach
Performance Analysis
SolutionDisplay
Preprocessing
Processing
Postprocessing
Converts chosen physics into discretized forms and solves over chosen physical domain
Computer Usage
Strategy
SolutionAssessment
Problem FormulationEquations of Motion
Conservation of mass (continuity) = particle identityConservation of linear momentum = Newton’s lawConservation 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).
Reduced Forms of Governing Equations
v vE FQ E F
t x y x y
2
2
0 0 0
u v
uvu puQ E F
uv v pv
e e p u e p v
00
xx
vxy
xy
xx xy xxx xy x
u
yE
u v qu v q
Critical issue: modelingviscous and turbulentflow behavior
Complex Aircraft Analysis, Circa 1968B747-100 with space shuttle Enterprise
What is different with these aircraft from normal operation?
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 conservation mass, energy• Describes incompressible flow• Conservation of momentum not fully satisfied in presence of shocks
Navier-Stokes Equations• Coupled system of 5 nonlinear second order PDE’s• Describes conservation mass, momentum, energy• Describes wave propagation phenomena damped by viscosity
Neglect viscosity &heat conduction
Isentropic, irrotational flows
Neglect compressibility
More Physics(More complex equations)
More Geometry(More complex grid generation)
(More grid points)
Outline
• Background Issues• Codes, Flow Modeling, and Reduced
Equation Forms• Numerical Methods: Discretize, Griding,
Accuracy, Error• Data Structure, Grids• Turbulence• Fluent
• Finite Difference
• Finite Volume
• Finite Element
All based ondiscretizationapproaches
P.D.E.Lu=f
DiscretizeSystem of LinearAlgebraic Eqns Up
Breakup Continuous Domain into a Finite Number of Locations
Boundary Condition
Boundary Condition
B. C.
B. C.
Discretization & Order of Accuracy
• Taylor Series Expansion
• Polynomial Function [Power Series]
• Accuracy Dependent on Mesh Size and Variable Gradients
2 2
2
( )( ) ( ) ...
2!
df x d ff x x f x x
dx dx
f
x
fi fi+1 fi+2
fi+3
xi+1xi+2 xi+3xi
x
2( ) ( ) ( ) ...f x A B x C x
Discretization Example
• Derivative approximation proportional to polynomial order• Order of accuracy: mesh spacing, derivative magnitude
– only reasonable if product is small
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
Numerical Error Sources - II
0 100 200 300 400Time
-0.1
0
0.1
0.2
0.3
0.4
Am
plit
ud
e
ux
0 5 23
2. exp (ln )( )
at t=0
at t=400
Traveling linear wave model problem
0u u
t x
380 390 400 410 420Time
-0.1
0
0.1
0.2
0.3
0.4A
mp
litu
de
'1st''2nd''3rd''3tvd''exact'
Numerical Error Sources - III
at t=400
380 390 400 410 420Time
-0.1
0
0.1
0.2
0.3
0.4
Am
plit
ud
e
'1st''2nd''3rd''3tvd''exact'
Numerical Error Sources - IV
at t=400
380 390 400 410 420Time
-0.1
0
0.1
0.2
0.3
0.4
Am
plit
ud
e
'5th''7th''9th''exact'
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
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
Exact SolutionU
GoverningP.D.E.’s
L(U)
System of Algebraic Equations
Approximate Solutionu
Discretization
Consistency
Convergence
as x t 0
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
2
2,
max
. . . 0
. . . ( , ) 0
1. . . ( )
,
n ni j ij
nij
i j
nij i j
P D E Lu
F D E Lu x y
R M S error L normN
Max error at x y
How good are the results?
• Convergence (physical): Check conserved properties: mass (for internal flows), atom balance (for chemistry), total enthalpy, e.g.
2 2, ,H H O OH H constant
O constant
Outline
• Background Issues• Codes, Flow Modeling, and Reduced
Equation Forms• Numerical Methods: Discretize, Griding,
Accuracy, Error• Data Structure, Grids• Turbulence• Fluent
2-D Problem Setup
• Structured Grid / Data
• Unstructured Data / Structured Grid
i,j+1
i-1,j i,j
i,j-1
i+1,j
X , i
Y, j
Ui,j
61
35 36
11
37
X
Y
U3660
10 12
62
2-D Problem Setup
• Semi -Structured Grid / Unstructured Data
• Unstructured Data / Unstructured Grid
61
35 36
11
37
X
Y
U3660
10 12
62
61
3536
11
37
X
Y
U3660
1012
62
Grid Generation Transformation to a new coordinate system Transformation to a stretched grid
Grid Generation - Generic Topologies
Block-structured O + H
• More complicated grids can be constructed by combining the basic grid topologies - cylinder in a duct
Overset or Chimera Cartesian + Polar
Both take advantage of natural symmetries of the geometry
Grid Generation - Generic Topologies
Cartesian-stepwise
• More complicated grids can be constructed taking advantage of simple elements
Unstructured-hybrid
Dimension Unstructured Structured 2D triangular quadrilateral 3D tetrahedra hexahedra
Outline
• Background Issues• Codes, Flow Modeling, and Reduced
Equation Forms• Numerical Methods: Discretize, Griding,
Accuracy, Error• Data Structure, Grids• Turbulence• Fluent
Viscosity and Turbulence
0.99e
Boundary Layer
uy
u
Viscosity and Turbulence
Laminar
Turbulent
Steady Unsteady
Steady Unsteady
Viscosity and TurbulenceProperties Averaged Over Time Scale Much Smaller Than Global
Unsteadiness
/ / / /
*
*
1/ 2
*
; 0;
1 2ln
w
u u u u uv uv u v
u yuu F F y
u
u
yu y B f
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
Outline
• Background Issues• Codes, Flow Modeling, and Reduced
Equation Forms• Numerical Methods: Discretize, Griding,
Accuracy, Error• Data Structure, Grids• Turbulence• Fluent
Finite Volume• 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
0QdV H n dAt
����������������������������
2D Steady Flux Equation
1, 1, 2
,
, 1, , 1, 1, 1,2 2
,
0
( )2
( ) ( )2 2 2
i j i j
i j
i j i j i j i j i j i jE W
i j
F G
x y
Classical view
F FFO x
x x
Alternative view
F F F F F FF FFO x O x
x x x x x
Finite-difference: centered in space scheme
W E
N
S
X
i-1,j
i,j+1
i,j-1
Steady Governing Equations
( )
1, , ,
. . .'
jj i i
j j j
Tj i v
P P i iE W N S
u S Sx x x
where u H f
Coupled system of nonlinear P D E s
Discretizing over control volume yields
A A B
Start with generalized RANS equations = transport coeff. / = diffusivity
Fluent Solution MethodSimple Scheme
SIMPLE: Semi-Implicit Method for Pressure Linked Equations
Fluent Solution MethodSimple Scheme
( )
/ 2
...
...
...
P P i i
W W W
E
S
N
P E W N S
A A B S
A D F
A
A
A
A A A A A
Solution algorithm: • Staggered grid; convected on different grid from pressure. • Avoids wavy velocity solutions
Fluent Solution MethodSimple Scheme
CV for u-eqn.
Two sets of indices or one and one staggered at half-cell
Fluent Solution MethodSimple Scheme
CV for v-eqn.
Fluent Solution MethodSimple Scheme
CV for p-eqn.
Fluent Solution MethodSimple Scheme
5-point computational molecules for linearized system using geographical not index notation
Fluent Solution MethodSimple Scheme – Multidimensional Model
2-D and 3-D computational molecules using geographical not index notation
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
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– ….