13.021 Marine Hydrodynamics
Lecture 23
Wave steepness
Diffraction parameter
Wave Forces on a Body
CF =
h
F
A l
= f , , R , , roughness ,.
2
gAl
U = A body velocity, particle velocity
A
Ul Al
=
UT AT
A
Kc =
=
= 2
l
l
l
R=
h
A
Types of Forces
(1) Viscous
13.021 Marine Hydrodynamics
Lecture 21
Water Waves
Exact (nonlinear) governing equations for surface gravity waves assuming potential theory
y = (x,z,t) or F(x,y,z,t) = 0
y
x
z
B(x,y,z,t) = 0
Unknown variables:
v
v(x , y, z, t ) = (x , y, z, t )
(x, z, t
13.021 - Marine Hydrodynamics
Lecture 19
Turbulent Boundary Layers: Roughness Effects
So far, a smooth surface has been assumed.
In practice, it is rarely so due to fouling, rust, rivets, etc.
Viscous sublayer
Uo
v
k = characteristic
roughness height
Equi
Summary of Boundary Layer over a Flat Plate
Laminar (Blasius)
1
~ Rx 2
x
Turbulent (1/7 power law)
1
~ Rx 5
x
1
4
= 0.047 xRx 5 ~ x 5
8
1
o 0.0227U o2 R 4
1.72 Rx 2 ~ x
1
o 0.332U o2 Rx
1
2
0.029U o2 Rx
D 0.664U o2 ( BL) RL
C f = 1.328RL
1
2
1
1
5
D
y
Marine Hydrodynamics
Lecture 17
U, V potential
u, v viscous flow
x
4.6 Laminar Boundary Layers
L
Assume steady 0
t
2D
Uo
0
w,
z
u (x, y ), v (x, y ), p (x, y ), U (x, y ), V ( x, y )
For <L, use local coordinates (x,y)
u v
+
= 0
x y
Governing Equ
13.021 - Marine Hydrodynamics
Lecture 16
Vortex Shedding and Vortex Induced Vibrations
Uo
Consider a steady flow Uo on a bluff body with diameter D
D
We would expect the average forces to be:
Fx
Fy
F
Fx
Fy
t
The measured oscillatory forces are:
F
Fx
Avera
13.021 - Marine Hydrodynamics
Lecture 15
4.0 - Real Fluid Effects ( 0)
Potential Flow (under DAlemberts Condition): Drag = 0
Observed experiment (real fluid < 1 but 0): Drag 0
D (Drag)
4.1 - Drag on a Bluff Body
U
Consider a sphere
of diameter d
D
where S
13.021 Marine Hydrodynamics
Lecture 14
Some Properties of Added-Mass Coefficients
(1) mij= [function of geometry only]
&
F, M = [linear function of mij] * [function of instantaneous U, U, ]
Not of motion history
(2) Relationship to momentum of fluid:
B
B
13.021 Marine Hydrodynamics
Lecture 13
3.11 Unsteady Motion Added Mass
Dalembert: ideal, irrotational, unbounded, steady
4
Violations:
unsteady
wall
Koutta-Joakowski
Example 1: Force on a sphere accelerating (U=U(t), unsteady) in an unbounded fluid at
res
13.021 Marine Hydrodynamics
Lecture 12
3.8 Method of Images
1 source: =
m
ln x 2 + y 2
2
m
m
d
=0
dy
Source near wall:
=
y
b
x
m
2
ln x 2 + ( y b )
2
2
+ ln x 2 + ( y + b )
b
Added source for
symmetry
Note: Be sure to verify that the boundary condition
13.021 Marine Hydrodynamics
Lecture 11
3.9 Forces on a body undergoing steady translation DAlemberts paradox
3.9.1 Fixed bodies & translating bodies Galilean transformation.
y
o
y
o
x
x
U
z
z
Fixed in space
x = x + Ut
v
O : v, , p
Fixed in translating spa
13.021 - Marine Hydrodynamics
Lecture 10
Boundary Conditions for
(v )
Two types of Boundary Conditions: 1) Kinematic Boundary Conditions V
v
2) Dynamic Boundary Conditions F or P
Kinematic Boundary Conditions on an impermeable boundary (no flux conditio
13.021 Marine Hydrodynamics
Lecture 9
1
13.021 Marine Hydrodynamics
Lecture 9
Vorticity Equation
Return to viscous incompressible flow
v
p
v v v
2v
N-S equation: t + v v = + gy + v
v
v
vv
( )
+ (v v ) = 2 since = 0 for any (conservative forces)
t
Now:
v
13.021 Marine Hydrodynamics
Lecture 8
Vortex Lines, Tubes, etc.
v
A vortex line is a line everywhere tangent to
A vortex tube (filament) is a bundle of vortex lines.
Some Properties:
Mathematical
v
v
dV = 0 = ndS
V
v
( n )out
v
n = 0
Divergence Theore
13.021 Marine Hydrodynamics
Lecture 7
1
13.021 Marine Hydrodynamics
Lecture 7
Chapter 3 Ideal Fluid Flow
Re =
inertia UL
=
viscous
For typical problems we are interested in: (L 1m, U 1m/s) water = 10 6 m 2 s
= Re1 < 1; ( 10 6 )
UL
i.e. viscous effect < in
13.021 Marine Hydrodynamics
Lecture 6
13.021 - Marine Hydrodynamics
Lecture 6
2.2 Similarity Parameters (from governing equations)
Non-dimensionalize and normalize basic equations by scaling:
Identify characteristic scales for the problem
v
v
velocity
U
v
13.021 Marine Hydrodynamics
Lecture 5
Chapter 2 - Similitude
Similitude is a method that allows you to get a conceptual picture of a complicated idea,
occurrence or mechanism.
Similitude: Similarity of behavior of different systems.
Real world model
(prot
13.021 Marine Hydrodynamics
Lecture 4
1
13.021 - Marine Hydrodynamics
Lecture 4
Introduction
Governing Equations so far:
Knowns
Fi
Number of Equations
Continuity(conservation of mass)
Euler (conservation of momentum)
1
3
Number of Unknowns
3
i
f 9 elimina
13.021 Marine Hydrodynamics
Lecture 3
1
13.021 Marine Hydrodynamics
Lecture 3
1.2 Stress Tensor
Stress Tensor ij: The stress (force per unit area) at a point in a fluid needs nine
components so that it is completely specified. This is due to the two direc
13.021 Marine Hydrodynamics
Lecture 2
1
13.021 Marine Hydrodynamics
Lecture 2
Chapter 1 - Basic Equations
1.1 Description of a Flow
Flows are often defined either by a Eulerian description or a Lagrangian description.
- Eulerian description: This is a fie
1 3.021 Marine Hydrodynamics
Lecture 1
1
13.021 Marine Hydrodynamics
Lecture 1
Introduction
Marine hydrodynamics is a large and diverse subject and only a few topics can be
covered in an introductory course. Some course objectives to keep in mind througho