HydroMechanicsofOffshoreStructures
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Transcript of HydroMechanicsofOffshoreStructures
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Hydro Mechanics of
Offshore Structures
Hydro Mechanics of
Offshore Structures
Lecture (4)
MM771 M.A. Kotb
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Hydro Mechanics of
Offshore Structures MM771 M.A. Kotb
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In 1755, Leonard Euler applied Newtons second law (force = mass x acceleration) to non-viscous and incompressible fluids.
Euler Equations
Leonhard Euler
(1707-83)
Isaac Newton
(1642-1 727)
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Hydro Mechanics of
Offshore Structures
2
x
x
pp
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C v
w
u
x
z
y
CV
y
z
x
Dt
DudmFx
rgxyz
2
x
x
pp
Fluid Element
in the x-direction for a fluid mass element dm
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x
y
2
y
y
pp
2
y
y
pp
2
x
x
pp
2
x
x
pp
p
zy zy
zx
zx
C (xc , yc , zc)
Consider the flow in the x-y plane
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Dt
DudmFx
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Hydro Mechanics of
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z
uw
y
uv
x
uu
t
uzyxzyx
x
pypzyp
z
u
t
z
y
u
t
y
x
u
t
x
t
uzyxzy
x
x
ppzy
x
x
pp
r
r22
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2
x
x
pp
2
x
x
pp
zy zyDt
DudmFx
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Hydro Mechanics of
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z
uw
y
uv
x
uu
t
u
x
p
z
uw
y
uv
x
uu
t
u
x
p
z
uw
y
uv
x
uu
t
uzyxzyx
x
p
r
r
r
1
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z
uw
y
uv
x
uu
t
u
x
p
r
1
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z
vw
y
vv
x
vu
t
v
y
p
r
1
direction-Similarly in y
and in z-direction
z
ww
y
wv
x
wu
t
wg
z
p
r
1
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Hydro Mechanics of
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Thus, the Euler equations for a non-viscous and incompressible flow are:
in x-direction
z
uw
y
uv
x
uu
t
u
x
p
r
1
z
vw
y
vv
x
vu
t
v
y
p
r
1in y-direction
z
ww
y
wv
x
wu
t
wg
z
p
r
1in z-direction
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Hydro Mechanics of
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Bernoulli Equation
Swiss mathematician, son of Johann Bernoulli, who showed that as the velocity of a fluid increases, the pressure decreases, a statement known as the Bernoulli principle. He won the annual prize of the French Academy ten times for work on vibrating strings, ocean tides, and the kinetic theory of gases. For one of these victories, he was ejected from his jealous father's house, as his father had also submitted an entry for the prize. Daniel Bernoulli
(1700-1782)
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fSubstitute in terms of in x-direction
zxzyxyxxxxtx
p
xzzxyyxxxxtx
p
z
uw
y
uv
x
uu
t
u
x
p
fffffff
r
fffffff
r
r
1
1
1
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tzyConstzyxt
p
zyxt
p
x
zyxtx
p
x
zxyxxxtx
p
x
,,2
1
2
1
2
1
2
1
2
1
2
10
2
1
2
1
2
1
2
1
2
1
2
1
222
222
222
222
ffff
r
ffff
r
ffff
r
ffff
r
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Similarly in y-direction
tzConstzyxt
p
zyxt
p
y
,2
1
2
1
2
1
2
1
2
1
2
10
222
222
ffff
r
ffff
r
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Similarly in z-direction
tConstgzzyxt
p
zyxt
p
zg
222
222
2
1
2
1
2
1
2
1
2
1
2
1
ffff
r
ffff
r
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tConstV
gzt
p
tConstVgzt
p
tConstwvugzt
p
tConstzyx
gzt
p
2
2
1
2
1
2
1
2
2
222
222
f
r
f
r
f
r
ffff
r
Hence
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Hydro Mechanics of
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Stationary or Steady Flow
tConstV
gzt
p
2
2f
r
In steady flow ,the derivative of the potential, df/dt, becomes zero and the function of time, C (t), becomes a constant, C; all time functions are now absent. In this case, the Bernoulli equation becomes:
Recall
ConstV
gzp
2
2
r
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Hydro Mechanics of
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Constg
Vz
g
p
2
2
r
Often, the Bernoulli equation for a stationary flow is written in other forms:
ConstVgzp 2
2
1rr
Or
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Isaac Newton (1 642-1 727)
English mathematician, physicist
and astronomer; studied at the
University of Cambridge. His three
big discoveries of the spectral
analysis of light, universal
gravitation and differential and
integral calculus are only too well
known. There are so many
scientific terms named after
Newton (Newton's rings and
Newton's law of
otion/viscosity/resistance) that he
can be regarded as the greatest
contributor to the establishment of
modern natural science.
Isaac Newton (1 642-1 727)
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Leonhard Euler (1707-83)
Mathematician born near Basle in
Switzerland. A pupil of Johann Bernoulli
and a close friend of Daniel Bernoulli.
Contributed enormously to the
mathematical development of Newtonian
mechanics, while formulating the equations
of motion of a perfect fluid and solid. Lost
his sight in one eye and then both eyes, as
a result of a disease, but still continued
his research.
Leonhard Euler (1707-83)
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Daniel Bernoulli (1700-82)
Mathematician born in Groningen in the
Netherlands. A good friend of Euler. Made
efforts to popularise the law of fluid motion,
while tackling various novel problems in
fluid statics and dynamics. Originated the
Latin word hydrodynamics, meaning fluid
dynamics.
Daniel Bernoulli (1700-82)
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Thank you .!!!!!