added mass calculation
TRANSCRIPT

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2.016 HydrodynamicsProfessor A.H. Techet
by TA B. P. Epps
0.1 Derivation of Added Mass around a SphereWhen a body moves in a fluid, some amount of fluidmust move around it. When the body accelerates, sotoo must the fluid. Thus, more force is required to accelerate the body in the fluid than in a vacuum. Sinceforce equals mass times acceleration, we can think ofthe additional force in terms of an imaginary addedmass of the object in the fluid.
One can derive the added mass of an object byconsidering the hydrodynamic force acting on it as itaccelerates. Consider a sphere of radius, R, acceler
ating at rate U/t = U. We find the hydrodynamicforce in the xdirection by integrating the pressureover the area projected in the xdirection:
Fx =
p d Ax
where
d Ax = cos dA, dA = 2rds, r = R sin , ds = Rd
p = t +
1
22
, by unsteady Bernoullis equation
= Ucos R3
2r2 , for axisymmetric flow around a sphere
t r=R = Ucos
R3
2r2= Ucos R
2
1
22r=R =
1
2(Ucos R
3
r3 ,Usin R3
2r3)2 = 1
2[U2 cos2 + 1
4U2 sin2 ]
so
Fx =
0
t+
1
22
cos 2R2 sin d
=
0
Ucos
R
2+
1
2(U2 cos2 +
1
4U2 sin2 )
cos 2R2 sin d
= 2R2 UR
2
0
sin cos2 d
=2/3
2R2 1
2U2
0
[sin cos3 +1
4sin3 cos ]d
=0
= 2
3R3U
where U is the acceleration of the body, and the negative sign indicates that the force is in the negativexdirection, opposing the acceleration. Thus, the body must exert this extra force, and the apparent addedmass is
ma =2
3R3
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0.2 Derivation of Added Mass around a Cylinder
Similarly, for a cylinder of radius, R,and length, L, accelerating at rate U.We find the hydrodynamic force inthe xdirection by integrating the pressure over the area projected in the xdirection:
Fx =
Pd Ax
where
d Ax = cos dA
dA = Lds
ds = Rd
P = t + 12 2, by unsteady Bernoullis equation = UR
2
r cos , for flow around a cylinder
t r=R = U
R2
r cos = UR cos
1
22r=R =
1
2(UR
2
r2 cos ,UR2
r2 sin )2 = 1
2U2
so
Fx =
2
0
t+
1
22
cos RLd
= 2
0 UR cos +
1
2U2
) cos RLd= RL UR
2
0
cos2 d =
RL 1
2U2
2
0
cos d =0
= R2LU
where a = U is the acceleration of the body, and the negative sign indicates that the force is in the negativexdirection, opposing the acceleration. Thus, the body must exert this extra force, and the apparent addedmass is
ma = R2L
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