2.6 Forces Imposed by Fluid Flow
63
problem of minimizing drag per unit volume, as organisms require a volume in
which to reside. This criterion, as well as the question of how marine organisms
adapt to withstand frequently changing directions of flow will be examined in
Chap. 11.
Let us give an example of the practical use of the lift coefficient and the lift
force. The Boeing 727 commercial jet aircraft during takeoff has a gross weight
of about 9.2 x 10 5 N (about 94 tonnes); its wing area is 153 m 2 (Redding and
Venne, 1983). What is the minimum speed to provide a necessary lift force for
takeoff? The lift force during takeoff must balance the aircraft's weight, thus:
(2.80)
where W is the aircraft weight, and Pa is the air density, Pa = 1.24 kg/m 3 . In
the commercial aircraft, the lift performance of aerofoils is improved by adding
flaps and slats. A combination of aerofoil and double-slatted flap results in the
lift coefficient Cz ~ 3.4. Now we are in the position to calculate the minimum
takeoff velocity U from Eq. (2.80) as:
(
2W )1/2
u- - -
C1PaSp
(2.81 )
Substituting all quantities into Eq. (2.81) yields the velocity U of about 190
km/hr., which is a realistic value. The above calculations are approximate
only, as they are based on very few data for this aircraft. In many commercial
aircraft, more complicated systems of flaps and leading-edge slots are used to
increase the lift force.
2.6.4 Inertia Force
We have assumed that fluid moves past a body in a steady manner. Under
steady conditions, the total force acting on a fixed body is nil for the case of a
perfect fluid (D'Alembert's paradox) and for a real fluid, the force is non-zero,
being a complex function of the Reynolds number, Re. What kind of additional
force is created if the fluid is accelerating or a body accelerates in quiescent
fluid? Such supposition is motivated by Newton's second law which states that
an acceleration of mass is proportional to the applied force, i. e. F = m (du / dt).
However, to apply Newton's second law, the effective mass of the body must
be known. The visualization of the motion of a body in a fluid shows that the
individual particles of fluid are pushed aside by the moving body. In addition
to pushing the particle aside, the body also accelerates the neighboring fluid
in the direction of its motion. This mass of fluid is called added mass, and is
only evident if the body accelerates through the fluid.
In order to provide some insight into the added mass phenomenon, let us
consider a fixed circular cylinder subjected to an unsteady ideal fluid flow
63
problem of minimizing drag per unit volume, as organisms require a volume in
which to reside. This criterion, as well as the question of how marine organisms
adapt to withstand frequently changing directions of flow will be examined in
Chap. 11.
Let us give an example of the practical use of the lift coefficient and the lift
force. The Boeing 727 commercial jet aircraft during takeoff has a gross weight
of about 9.2 x 10 5 N (about 94 tonnes); its wing area is 153 m 2 (Redding and
Venne, 1983). What is the minimum speed to provide a necessary lift force for
takeoff? The lift force during takeoff must balance the aircraft's weight, thus:
(2.80)
where W is the aircraft weight, and Pa is the air density, Pa = 1.24 kg/m 3 . In
the commercial aircraft, the lift performance of aerofoils is improved by adding
flaps and slats. A combination of aerofoil and double-slatted flap results in the
lift coefficient Cz ~ 3.4. Now we are in the position to calculate the minimum
takeoff velocity U from Eq. (2.80) as:
(
2W )1/2
u- - -
C1PaSp
(2.81 )
Substituting all quantities into Eq. (2.81) yields the velocity U of about 190
km/hr., which is a realistic value. The above calculations are approximate
only, as they are based on very few data for this aircraft. In many commercial
aircraft, more complicated systems of flaps and leading-edge slots are used to
increase the lift force.
2.6.4 Inertia Force
We have assumed that fluid moves past a body in a steady manner. Under
steady conditions, the total force acting on a fixed body is nil for the case of a
perfect fluid (D'Alembert's paradox) and for a real fluid, the force is non-zero,
being a complex function of the Reynolds number, Re. What kind of additional
force is created if the fluid is accelerating or a body accelerates in quiescent
fluid? Such supposition is motivated by Newton's second law which states that
an acceleration of mass is proportional to the applied force, i. e. F = m (du / dt).
However, to apply Newton's second law, the effective mass of the body must
be known. The visualization of the motion of a body in a fluid shows that the
individual particles of fluid are pushed aside by the moving body. In addition
to pushing the particle aside, the body also accelerates the neighboring fluid
in the direction of its motion. This mass of fluid is called added mass, and is
only evident if the body accelerates through the fluid.
In order to provide some insight into the added mass phenomenon, let us
consider a fixed circular cylinder subjected to an unsteady ideal fluid flow