Some CFD Books
Classics and General Purpose
Anderson, D. A., J. C. Tannehill, and R. H. Pletcher, Computational Fluid Mechanics
and Heat Transfer, 1984, Hemisphere Publishing Co., New York, ISBN 0-89116-471-5.
Ferziger, J. H. and M. Peric, Computational M

Computational Fluid Dynamics
Lecture 18
Shallow water equations and open boundaries
Boundary Conditions:
Real physical boundaries are not a major problem.
u ( o, t ) = u w
An artificial boundary is the difficult thing.
Open or artificial boundaries occur

Computational Fluid Dynamics
Lecture 19
We will examine open boundary conditions of 5 types.
1. Constant advection with one way wave equation with c* = 2 and 3.
2. Zero gradient condition
3. Rayleigh damping region with:
a. 10 grid points
b. 20 grid point

Computational Fluid Dynamics
Lecture 20
Outflow Boundary Conditions
Constant advection implementation
For the Boundary point i = nx use the one way wave equation:
u
u
+C
=0
t
x
with a stable discretization, one sided, up wind spatial derivative.
n
n
n
n

Computational Fluid Dynamics
Lecture 21
In flow B.C.s and Forcing
Often in CFD it is desired to simulate a steady or ongoing flow. Examples include: Spatially
developing Jet.
Wave breaking on beach
C ph
These situations require some kind of continuous fo

Computational Fluid Dynamics
Lecture 22
Subgrid Scale Modeling
The two most significant difficulties in doing CFD are:
1. What to do with unresolved large scale motions.
2. How to treat unresolved small scale motions.
Typical grid resolutions possible on

Computational Fluid Dynamics
Lecture 23
Wrap up on LES
Notes say (Ferziger).
v = C 2 2 s
but I said
v = C 2 2
1
2
how do these compare?
ui
= v
x j
+
u j ui
xi x j
eqn. 1-1-8 Hinze, pg 71, term IV.
u u
What does i + j look like? For i = 1, 3; j = 1, 3

Computational Fluid Dynamics
Lecture 24
General Curvilinear Coordinates
Computing flow around complex geometry is better accomplished by solving the PDEs of the N.S.
equations in a generalized coordinate system.
Transfer to Cartesian grid.
This causes the

Computational Fluid Dynamics
Lecture 25
J= 1
cfw_x
y z y z x y z y z + x y z y z
The metrics can be readily determined if analytical expressions are available for the inverse of the
transformation.
x = x ( , , )
y = y ( , , )
z = z ( , , )
For cases wh

EOC 6850
Numerical Simulation Techniques
For Coastal and Ocean Engineers
Civil and Coastal Engineering
Instructor:
Office:
Spring 2004
University of Florida
Don Slinn
575 I Weil Hall, 352-392-9537 x 1431, [email protected]
Class Time:
Office Hours:
We

Numerical Methods for Coastal and Oceanographic Engineering
Home Work Problem # 1.
Slinn Spring 2004
Due January 27, 2004.
Solve the one way wave equation,
u
u
+C
= 0 , using second and fourth order explicit
t
x
methods.
A. Use second order Leap Frog time

Numerical Methods for Coastal and Oceanographic Engineering
Home Work Problem # 2
Solve the one-way wave equation,
Slinn Spring 2004
Due February 17, 2004
u
u
+C
= 0 , using implicit finite difference schemes.
t
x
A. Use backward Euler, with = 0.5, 2, and

Numerical Methods for Coastal and Oceanographic Engineering
Home Work Problem # 3.
Slinn Spring 2004
Due March 4, 2004.
This problem set will give you experience with, 1) periodic boundary conditions, 2) the
compact scheme, 3) higher order explict time st

Numerical Methods for Coastal and Oceanographic Engineering
Home Work Problem # 4.
Slinn Spring 2004
Due March 30, 2004
The task is to simulate unsteady alongshore currents in the surf zone driven by breaking waves. The
governing equations are (e.g., Ozka

Numerical Methods for Coastal and Oceanographic Engineering
Home Work Problem # 5.
Slinn Spring 2004
Due April 8, 2004.
2 p
The task is to solve the equation
= R( x ) , for p using
x 2
1. Direct solvers (a) tri-diagonal inversion, (b) F.F.T.s,
2. Jacobi m

Computational Fluid Dynamics
Lecture 17
Stopping Criteria Continued
There is a solution that is as accurate as possible with a given difference equation and grid.
The number of iterations required to improve the solution one decimal place (for the Jacobi

Computational Fluid Dynamics
Lecture 16
Topics:
1. Iterative solvers stopping criteria.
2. Pressure Projection Method
3. Forms of Nonlinear terms
1. Stopping Criteria:
It is not enough to describe an iterative scheme; we also must discuss how to decide wh

Numerical Simulations Techniques applied to Coastal and
Oceanographic Engineering Computational Fluid Dynamics (CFD)
Lecture 1
1.
2.
3.
4.
See syllabus finalize time/ location
See text book Fletcher Vols. 1 & 2
See list of CFD books
Standard grading, Unde

1
time index
du
u n +1 u n
=
+ ( t )
dt
t
n
1st order accuracy in time
Leap - Frog difference (central difference)
du n u n +1 u n 1
=
+ ( t 2 )
dt
2t
2nd order accuracy in time
Many higher order schemes with different properties have been developed, we w

Computational Fluid Dynamics
Lecture 3
Discretization Continued.
A fourth order approximation to
f
can be found using Taylor Series.
x
a f ( x0 + 2x ) + b f ( x0 + x ) + c f ( x0 ) + d f ( x0 x ) + e f ( x0 2x ) = f ( x0 )
a + b + c + d + e = 0 : f ( x0 )

Computational Fluid Dynamics
Lecture 4
Tri-diagonal matrices are very efficient to solve computationally using the Thomas algorithm. Pg
183-184 using forward substitution and backward elimination.
b1c1
a b c
u1 R1
222
u R
a3b3c3
2 2
u3 R3
=
ai bi

Computational Fluid Dynamics
Lecture 5
Time differencing continued
A. Three level schemes.
B. Modified L-F schemes.
C. Higher order methods.
Three level schemes
n +1 = 1 n + 2 n 1 + 1tF ( n ) + 2 tF ( n 1 )
t
consistent schemes if 1 + 2 = 1 and 1 + 2 = 1

Computational Fluid Dynamics
Lecture 6
Space differencing errors.
+C
=0
t
x
Seek traveling wave solutions.
e(
i kx t )
k is wave number and is frequency.
=kC is dispersion relation.
where C is phase speed.
k
= C , true solution is non dipersive for const

Computational Fluid Dynamics
Lecture 7
Artificial Dissipation
Lack of dissipation in centered spatial differencing can be a disadvantage.
Dispersed, small scale waves propagate in arbitrary directions without loss of amplitude.
Adding scale selective diss

Computational Fluid Dynamics
Lecture 8
Combining Time and Space Differencing
u
u
+C
=0
t
x
1.
2.
3.
4.
Suppose forward time and centered space unstable.
Leap Frog and centered space neutral, conditionally stable
Forward time and one sided space could be s

Computational Fluid Dynamics
Lecture 9
Spectral Methods
The most accurate method for calculating spatial derivatives is to use Fast Fourier transforms.
2
6
n
Instead of being ( x ) or ( x ) or ( x ) , the spectral, semi-spectral or Galerkin methods
conver

Computational Fluid Dynamics
Lecture 10
In general, spectral methods belong to a class of methods called Weighted Residual Methods
(Chapter 5 in Fletcher), where a continuous function
N
( x, y, z, t ) = ak ( t ) k ( x, y, z )
k =1
is represented in a ser

Computational Fluid Dynamics
Lecture 11
Finite Elements:
j ( x)
1
0
j-2
x
x
downslope =
j
j+1
j+2
0
upslope =
j-1
x
0
2x
x
2x
on interval 0 < x < x.
x x
x
on interval 0 < x < x.
j
j +1
0 x
Region where both are non zero.
x
1
x x x
I j , j +1 = j +1 j

Computational Fluid Dynamics
Lecture 12
Finite Elements in 2-D
Straightforward extension of 1-D ideas
Simplest FE is rectangle (also consider triangles)
1. Bilinear interpolation function: C1 + C2 x + C3 y + C4 xy
Coefficients determined by function value