# SC07 Education Program - CFD desktop to cluster by wuyunqing

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SC08 Engineering Track: Introducing
Modeling Skills – CFD
Jim Giuliani
Client and Technology Support Manager
jimg@osc.edu
Agenda
The CFD Process
•Solid Modeling
•Mesh Generation
•Physical Properties
•Initial and Boundary Conditions
•Numerical Solution
•Data Visualization

Computational to Fluid Dynamics
•Partial Differential Equations
•Governing Equations
•Discretization
•Numerical Solutions

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Computational Fluid Dynamics (CFD)
• In CFD the equations that govern a process of interest are
solved numerically
• Once we have classified our problem, we can derive
equations that describe the fluid system
• Approximations can be made that allow the governing
equations to be reduced in
complexity
• For complex flows or
geometries we can
approximate the
governing equations in a
form that can be solved
numerically.
• CFD encompasses the
entire process of solving
the governing equations
numerically
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The CFD Process

Solid Modeling
Mesh Generation
Physical Properties
Initial and Boundary Conditions
Numerical Solution
Data Visualization

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The CFD Process
• Based on the classification of the problem to be analyzed,
select the appropriate solver
– Based on flow classification, simplifications are made to the
governing equations
– Simplified governing equations need to be solved using different
numerical techniques
• Describe the physical problem
– Geometry
– Physical properties
• Set solution and solver settings
• Start the solution and monitor progress
– Solution parameters, residuals, Courant number, are monitored to
determine if the solution is converging

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Solid Modeling
The first step in creating a numerical model is to describe the
geometry of the problem to be analyzed

• Most grid
generation software
have basic solid
modeling capabilities

• Parts/geometries
can be imported from
packages

6
Mesh Generation
The physical domain defined by the
solid model is discretized into small
elemental surfaces/volumes
A solution to the governing equations
can now be approximated

7
Grid/Mesh Limitations

• Overall flowfield model is on order of hundreds of feet
• Only 1.5 inch gap between belly and ground
• To properly represent flow through any gap, should have at least 5
cells between wall surfaces
• Turbulence modeling required first layer of cells on road and body to
be ~0.5”
– This leaves only ~0.5” for remaining 3 layers; results in very high
aspect ratio cells and misrepresented ground/vehicle boundary
layer interaction
Boundary layer cells   0.5”
1.5”
Boundary layer cells   0.5”

8
Physical Properties
Temperature
Pressure
Density
Viscosity is a measure of the
resistance of a fluid which is being
deformed.
– Dynamic
– Kinematic (dynamic viscosity/density)

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Initial and Boundary Conditions
For the solution to be well posed
– A solution must exist
– The solution must be unique
– The solution must depend on initial or boundary
conditions
There are 3 types of Boundary Conditions
– Dirchelt                   u  f (x)

– Neumann                    u
 g (x)
n
– Robins or mixed                            u
a( x)u b( x)       h( x )
n

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Initial and Boundary Conditions
Initial Conditions
Specify properties at the beginning of the analysis
– Properties may change during simulation
– Choice of initial conditions can shorten solution time
Boundary Conditions
Describe how the simulation will behave at the edges
of the computational domain
– Properties often are constant throughout simulation
Common properties that are specified
– Velocity, Pressure and Temperature
Special types of boundaries
– Symmetry, Extrapolated

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Boundary Conditions
For the example of simulating the flow around the
body of a car (U > 200 m.p.h.), boundary conditions
must be specified on all surfaces and edges of the
computational domain
Symmetry on top and sides

Solution driven by fixed inlet
velocities

Flow exits domain through
pressure outlet
Zero velocity (no slip) at car surface
Solid, moving boundary for ground (speed equal to
free stream velocity)
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Numerical Solution
•Solution to the discretization equations are approximated
numerically
•Choose solver settings (relaxation factor, tolerances)
•Set solution control parameters (start time, end time, time step)

Residuals
•Errors of the
discretized equations
•Monitored as a
means to determine
when solution has
converged

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Data Visualization
Grids that contain millions of
elements can provide very high
resolution information on flow field
Colorized contour plots, particle
traces and time lapse animations
allow subtle flow patterns and
overall flow characteristics to be
examined
User must specify at what interval
data is written out
More steps = more data
(gigabytes per simulation)
(terabytes per design problem)
Solutions can be run remotely, but
visualizations perform best locally

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Exercise – 2D Heat Conduction

• Cooling_channel_descripton.doc contains a detailed description of the
problem
• We have used Java applet to solve for a solution, but adiabatic
boundary condition was not able to be modeled accurately
• Open Heat_transfer_CFD_portal.doc for instructions on how to solve
this problem with the OpenFOAM CFD solver
• Compare results to solution obtained with Java Applet

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Steady flow denotes a system where the flow does
not change with time
When the fundamental equations are discussed, we
will see that steady flow denotes that all time
derivatives are zero
When the stability of a numerical solution is
discussed, we will see that selection of B.C. for
unstable flow strongly impacts stability

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Fluid Flow Streamlines
Streamlines
– A moving fluid element is seen to trace out a fixed path
in space
– A streamline is this fixed path

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Classification of Fluid Mechanics
Continuum Fluid Mechanics
• Inviscid
• Viscous
– Laminar
– Turbulent
• Compressible/Incompressible

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Classification of Flows
Compressible/Incompressible
Inviscid
Viscous
• Laminar
– Flow where the streamlines
are smooth and regular and a
fluid element moves smoothly
along a streamline
• Turbulent                          Turbulent Flow
– Flow where the streamlines
break up and a fluid element    Laminar Flow
moves in a random, irregular,
and tortuous fashion

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Exercise – Flow Past a Circular Cylinder

Model low speed
flow around a
cylinder

• Assumptions/Simplifications
– Incompressible: all density derivatives are zero
– Inviscid: neglect viscous forces
• Possible Objectives
– Practical application of the fundamental flow equations
– Examine assumptions and simplifications that can reduce complex
partial differential equations to simple analytic equations

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Exercise – Flow Past a Circular Cylinder
• Possible Objectives (cont)
– Understand when higher
fidelity tools are needed
to examine complex flow
phenomena
• Top image: - ideal flow
solution
• Bottom image:
Landsat 7 image
of
Juan Fernandez
islands off of the
coast of Chili

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Exercise – Flow Past a Circular Cylinder

Model low speed
flow around a
cylinder

• To study the ideal flow model, open Cylinder_ideal_flow.doc to see the
assignment
• Open Cylinder_CFD_portal.doc to see the assignment for the viscous,
CFD solution
• For the CFD assignment, follow steps through page 6 and stop after
SUBMIT button has been pressed and job is submitted
• Simulation takes about 45 minutes and results should be available
towards the end of the workshop
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Computational Fluid Dynamics
Partial Differential Equations
•Physical Classification
•Numerical Classification
Governing Equations
•Continuity
•Momentum
•Turbulence
Discretization
•Finite Difference
•Laplace’s Equation
•Finite Volume
Numerical Solutions

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Partial Differential Equations
Partial Differential Equations (PDEs)
– Many important physical processes are governed by
PDEs
– We will look at some PDEs commonly encountered in
fluid dynamics
Physical Classification
– Equilibrium Problems
– Marching Problems
Numerical Classification
– Elliptic
– Parabolic
– Hyperbolic

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PDEs – Physical Classification
Equilibrium Problems
– Solution desired in a closed domain
– Boundary value problem
– Governed by elliptic PDEs

Examples
– Incompressible, inviscid flow

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PDEs – Physical Classification
Marching Problems
– Transient
– Prescribed set of initial conditions in addition to
boundary conditions
– Solution computed by marching from the initial
conditions, constrained by boundary conditions
– Governed by parabolic or hyperbolic PDEs
Examples
– Boundary layer flow

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PDEs – Numerical Classification
Hyperbolic
– Fundamental property
is the limited domain of
dependence
– Solution at point P
depends
only on information in the
domain of dependence
– Any disturbance that
occurs
outside this interval can
never influence the
solution at point P
2 y     2 y
Example: Wave Equation            a2 2
t 2     x
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PDEs – Numerical Classification
Parabolic
– Unlike hyperbolic
equations,
solution at some time tn
depends upon the entire
physical domain at earlier
times, including side
boundary conditions
– Start at some initial data
plane and march forward
– Diffusion processes
u    2u
Example: 1D heat transfer equation       2
t   x

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PDEs – Numerical Classification
Elliptic
– Boundary value problem
– Subject to a prescribed set
of boundary conditions on
a closed domain
– Solution at any point
depends upon the specified
conditions at all points on
the boundary
Example: Laplace’s Equation

 2u  2u
 2 0
x 2
y

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Conservation Laws
Mass conservation
– Matter may neither be created
or destroyed
Conservation of momentum
– Newton’s 2nd law of motion
Conservation of energy
– First law of thermodynamics

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Continuity Equation
For a given control volumn

dm
 in m  out m
        
dt
The rate increase of mass within the control
volume is equal to the net rate at which mass
enters the control volume

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Continuity Equation
The partial differential form of the continuity
equation in cartesian coordinates is

D     u v w 
   
 x y z   0

Dt              
where

D               
    u    v    w
Dt   t    x    y    z

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Conservation of Momentum
Newton’s 2nd law
F   x
 m  ax
(in 2 dimensions)
F   y
 m  ay

Body forces
•Gravity
•Centrifugal
•Corolis
•Electromagnetic
Surface forces
•Normal stress
•Tangential stress
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Conservation of Momentum
Combining forces yields

Du  xx  yx  zx
                     Fx
Dt   x   y    z

Surface forces   Body forces

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Navier-Stokes Equations
Replacing stress terms with stress-strain relationship
Du        1 p   2  u v w     u v     w u 
 gx               x y z   y   y  x   z   x  z 
   2                            
Dt         x x  3                                             
Dv         p    v u    2  v u w     v w 
                     x y  y 3  y x z   z   z  y 
 g y           2                             
Dt         y x                                        
Dw         p    w u     v w    2  w u v 
       g z                      z y   z  3   2 z  x  y 
                                         
Dt         z x   x z  y                                    

Assumptions/simplifications can reduce complexity

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Navier-Stokes for Incompressible Flow
For an incompressible, constant viscosity flow, the viscous
terms simplify significantly (more applicable to gasses
than fluids)
u    u    u    1 P    2u  2u
u    v           2  2
t    x    y     x   x   y
v   v  v    1 P    2v  2v
u v           2  2
t   x  y     y   x   y

How much detail was skipped?
1st year graduate course in continuum mechanics in 5 slides

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Turbulence Models
Turbulence depends on the ratio of the inertia force to viscous force
Laminar flows can be described by the continuity and momentum
equations
Rotational flow structure have a wide range of length and velocity scales,
called turbulent scales
Several popular techniques for accounting for turbulence are:
• Direct Numerical Simulation (DNS)
• Large Eddie Simulation (LES)
• K-epsilon

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Large Eddy Simulation (LES)
Large scale motions are generally much more energetic and
transport most of the conserved properties
• Large eddies are modeled exactly
• Small eddies are approximated
Smaller universal scales, called sub-grid scales, are modeled
using a sub-grid scale (SGS) model

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K-epsilon Model
Time averaged governing equations yield the Reynolds-
averaged Navier-Stokes equations (RANS)
– Point velocities are considered to be comprised of two
components
• Fluctuating component
Additional unknowns due to turbulent fluctuations can be
handled with transport equations
Two important turbulent quantities in these transport
equations:
k – turbulent kinetic energy
epsilon – dissipation of turbulent kinetic energy

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Discretization
Conversion of the governing equations into a
system of algebraic equations
Two popular discretization techniques in CFD are
– Finite difference method
– Finite volume method

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Finite Difference Method
First order derivatives                                                  x
• Fordward difference
u ui , j 1  ui , j
                   O(x)
x         x
ui,j+1
• Backward difference
y
u ui , j  ui , j 1                                   Ui-1,j          Ui+1,j
                   O(x)                                    ui,j
x        x
ui,j-1
• Central difference
u ui , j 1  ui , j 1
                      O(x 2 )
x         2x
y(j)

x(i)

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Finite Difference Approximations
Second order derivative
x
• Central difference
 2u ui 1, j  2ui , j  ui 1, j
                              O (( x) 2 )
x 2           ( x ) 2
ui,j+1

y                 Ui-1,j          Ui+1,j
For time derivatives
ui,j

ui,j-1
u u n 1i , j  u n i , j
                        O(t )
t           t                                          y(j)

x(i)

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Finite Volume Method
Unstructured mesh offers more flexibility
Control volumes are defined by the surfaces of the elements
Control volume integrals can be converted to discretized
equations base on face area and flow across boundaries

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Numerical Solution
Discretization results in a system of linear or non-
linear equations
Numerical methods are applied to solve these
equations
– Direct methods
– Iterative methods

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Convergence
With iterative methods, as progress proceeds
towards a solution, the equations are determined
to have converged to a solution when certain
values do not change between iterations by a
specified tolerance
– Numerical solution does not change with additional
iterations
– Mass, momentum and energy balances are obtained

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Residuals
Residuals are the errors of the discretized equations
Residuals are calculated for each equation
(Ux, Uy, P, …)
Residuals should
diminish as the
numerical process
progresses
They are often used
to monitor the
behavior of the
numerical process

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Exercise – Flow Past a Circular Cylinder
• Open Cylinder_CFD_portal.doc
• Continue where you left off (~page 6 after initial simulation submitted)

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Exercise – Turbulent Flow over a Backward Facing
Step

• Open Backward_step.doc for instructions on how to solve this problem
with the OpenFOAM CFD solver
• Examine the model assumptions and setup
• Run the model in its current form
• Includes turbulence modeling

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