13.42
04/01/04:
Morrison’s Equation
SPRING 2004
©A. H. TECHET
1.
General form of Morrison’s Equation
Flow past a circular cylinder is a canonical problem in ocean engineering. For a purely inviscid,
steady flow we know that the force on any body is zero (D’Allembert’s paradox). For unsteady
inviscid flow this is no longer the case and added mass effects must be considered.
Of course in
the “real” world, viscosity plays a large role and we must consider, in addition to added mass
forces, viscous drag forces resulting from separation and boundary layer friction.
Following on Tuesday’s lecture, the resulting force on a body in an unsteady viscous
flow can be determined using Morrison’s Equation, which is a combination of an inertial term
and a drag term.
The force in the xdirection on a body in unsteady flow with velocity U(t) is
1
( )
=
D
t
F
t
( )
=
ρ
C
∀
U
±
+
ρ
C
AU U
(1.1)
x
m
2
d
In order to obtain rough estimates of the magnitude of the force of a body, it is advantageous to
use Morrison’s equation with constant coefficients. Supposing we want to find the estimates of
the wave forces on a fixed structure, then the procedure would be as follows:
1.) Select an appropriate wave theory (linear waves, or other higher order if necessary).
2.) Select the appropriate C
M
and C
D
based on Reynolds number, and other factors (see
table below).
3.) Apply Morrison’s Equation
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Wave Theory
C
d
C
m
Comments
Reference
Linear Theory
1.0
0.95
Mean values for ocean wave data
on 1324in cylinders
Wiegel, et al
(1957)
1.0
2.0
Recommended design values based
Agerschou and
1.4
on statistical analysis of published
Edens (1965)
data
Stokes 3
rd
order
1.34
1.46
Mean Values for oscillatory flow
Keulegan and
for 23in cylinders
Carpenter (1958)
Stokes 5
th
order
0.8
2.0
Recommended values based on
Agerschou and
1.0
statistical analysis of published data
Edens (1965)
We can see from the above table that for linear waves the recommended values for drag
and mass coefficients are 1.01.4 and 2.0, respectively. The range of drag coefficients allows us
to account for roughness and Reynolds number effects. These values are for rough estimates. In
reality these coefficients vary widely with the various flow parameters and with time.
Bretschneider showed that the values of C
D
and C
M
can even vary over one wave cycle. Even if
we ignore the time dependence of these coefficients we must account for the influence of other
parameters.
Reynolds number and roughness effects:
For smooth cylinders at Reynolds numbers
around 10
5
, laminar flow transitions to turbulent flow, and there is a dip in C
D
as a function of
Re. For larger Reynolds numbers the separation point remains essentially constant and thus so
does the drag coefficient. In this range C
D
is Reynolds number independent. Roughness causes
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 Spring '05
 AlexandraTechet
 Fluid Dynamics, Force, inertial force, sinh kH

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