2.6 Forces Imposed by Fluid Flow
65
As added mass is the mass of fluid 'carried along' by the object which remains stationary in accelerating fluid, an extra force PwCaV(duo/dt) is required
to accelerate this added mass. The term PwV(duo/dt) in Eq. (2.87) is a manifestation of the presence of the object in the fluid. The effect of this presence
is the same as accelerating the volume of fluid equal to the object's volume, in
the direction opposite to the fluid movement, in order to reflect the fact that
the object is stationary.
It is instructive now to consider a case of a circular cylinder accelerating
through a quiescent fluid. Is the force exerted on the cylinder by the fluid the
same as in a case when the fluid accelerates past the cylinder? The cylinder
induces a fluid motion, and this motion tends to zero when the distance from
the cylinder tends to infinity. In each case, the exact rate of attenuation of the
induced motion depends upon the shape of the body involved. Using a similar
analysis to that above, we can find that the force induced by the fluid on the
accelerating cylinder is:
(2.88)
in which Ca = 1 for the cylinder, and V = 7ra 2 is the cylinder volume per unit
length. Because the cylinder is moving in a fluid, it is obvious that a force is
necessary to support its movement. The total force required to keep cylinder
in motion through the fluid is:
.
duo
duo
2 duo
F = Fa + PcV- = (Pc + CaPw)V- = (Pc + Ca Pw)7ra - ,
dt
dt
dt
(2.89)
in which Pc is the density of the cylinder, and Ca = Cm - 1.
Depending on whether the mass of the cylinder is greater or less than that
of the displaced water, the total force required to accelerate cylinder in quiescent, ideal fluid could be greater or less than for the case of water accelerated
past the cylinder. When dealing with various marine organisms either moving
through the water column or residing on the sea bottom, we encounter both
situations. There is quite extensive data of theoretically derived or empirically
measured values of the inertia coefficient, Cm, for a variety of shapes. Some of
them, mostly taken from Sarpkaya and Isaacson (1981), have been collected in
Fig. 2.29.
In the general case of a real fluid, the inertia force is accompanied by a drag
force induced by the fluid viscosity and its separation from the body. The
following empirical formula is used to represent the combination of both forces
acting on piles (Morison et ai., 1950):
1
2
duo
F = 2PwCdSUO + PwCmV ill'
(2.90)
It should be noted that in a real fluid both, Cd and Cm coefficients are not
constants but complex functions of the Reynolds number and frequency of flow
variation.
65
As added mass is the mass of fluid 'carried along' by the object which remains stationary in accelerating fluid, an extra force PwCaV(duo/dt) is required
to accelerate this added mass. The term PwV(duo/dt) in Eq. (2.87) is a manifestation of the presence of the object in the fluid. The effect of this presence
is the same as accelerating the volume of fluid equal to the object's volume, in
the direction opposite to the fluid movement, in order to reflect the fact that
the object is stationary.
It is instructive now to consider a case of a circular cylinder accelerating
through a quiescent fluid. Is the force exerted on the cylinder by the fluid the
same as in a case when the fluid accelerates past the cylinder? The cylinder
induces a fluid motion, and this motion tends to zero when the distance from
the cylinder tends to infinity. In each case, the exact rate of attenuation of the
induced motion depends upon the shape of the body involved. Using a similar
analysis to that above, we can find that the force induced by the fluid on the
accelerating cylinder is:
(2.88)
in which Ca = 1 for the cylinder, and V = 7ra 2 is the cylinder volume per unit
length. Because the cylinder is moving in a fluid, it is obvious that a force is
necessary to support its movement. The total force required to keep cylinder
in motion through the fluid is:
.
duo
duo
2 duo
F = Fa + PcV- = (Pc + CaPw)V- = (Pc + Ca Pw)7ra - ,
dt
dt
dt
(2.89)
in which Pc is the density of the cylinder, and Ca = Cm - 1.
Depending on whether the mass of the cylinder is greater or less than that
of the displaced water, the total force required to accelerate cylinder in quiescent, ideal fluid could be greater or less than for the case of water accelerated
past the cylinder. When dealing with various marine organisms either moving
through the water column or residing on the sea bottom, we encounter both
situations. There is quite extensive data of theoretically derived or empirically
measured values of the inertia coefficient, Cm, for a variety of shapes. Some of
them, mostly taken from Sarpkaya and Isaacson (1981), have been collected in
Fig. 2.29.
In the general case of a real fluid, the inertia force is accompanied by a drag
force induced by the fluid viscosity and its separation from the body. The
following empirical formula is used to represent the combination of both forces
acting on piles (Morison et ai., 1950):
1
2
duo
F = 2PwCdSUO + PwCmV ill'
(2.90)
It should be noted that in a real fluid both, Cd and Cm coefficients are not
constants but complex functions of the Reynolds number and frequency of flow
variation.