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variation of the boundary-layer displacement thickness using the boundary-layer
theory. The new surface now consists of the given shape plus the displacement
thickness. For this new body shape, use the ideal fluid theory once again and
predict the velocity and pressure distribution. Repeat above steps until there is a
con'vergence within a certain specified tolerance.
This approach is generally applicable as long as the body is streamlined and the
fluid fiow is attached to the body along its surface. K the flow is separated from
the surface of the body, this approach cannot be used.'
6 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
~
Fig.1.6 Conceptoftotalenergy.
1.2.1 FlowSeparation
The ideal fiuid theory predicts that the fiuid flow closes behind any body no
matter what the body shape is. Wc have two stagnation points Si and S2 as shown
in Fig. 1.2. Positive pressures acting in f:ront and rear of the body balance out each
other so that the net drag force is zero. Similarly, negative pressures on the top
and bottom surfaces balance out, resulting in a zero net lift force. However, in real
fluid flow, the flow pattem will be differenL To understand this, let us consider a
mechanical analogy forwarded by Prandtl.
Consider a roller coaster starting from rest at an elevation A and rolling down
along a track as shown in Fig. 1.6. During this motion, the potential energy at A
is transferred to kinetic energy at B and back to potential energy while ascending
the hilj towards C. The roller coastcr would regain the same elevation at C as that
at point A if there is no loss of energy during its motion. Because there is friction
between the wheels of the roller coaster and the track, the roller coaster can only
make it to point C' and not to point C.
Now let us consider the flow in the boundary layer as schematically shown m
Fig. 1.7. The innermost fluid particles traveling within the boundary layer experi-
ence a retardation and come toPa halt before the rear stagnation point S2 iS reached.
Assume that at point A the velocity is maximum and falls gradually to zero at
the rear stagnation point S2. According to Bernoulli's theorem, when the velocity
decreases, the pressure must increase so that the total pressure, which is the sum
of static and dynamic pressures, remains constant Thus, moving from A towards
S2, the pressure increaCes in the streamwise direction. Such a pressure rise in the
streamwise direction is caUed adverse pressure gradient.lf the pressure decreases
in the streamwise direction, it is callecta favorable pressure gradient. At point A',
the boundary-layer velocity profile has a considerably differerit shape than that at
point A because of retardation.
To overcome the adverse pressure gradient, fluid particles need to have sufficient
energy in the reserve. However, because of friction, fluid particles will have lost
part of their energy and hence cannot reach the stagnation point S2, which woulcl
have been the case if the fluid flow were frictionless. As a result, fluid particles
momentarily come to rest at point Au. Downstream of A", the fiow is reversed in
direction. Thus, downstream of A", the flow is separated.
REVIEW OF BASIC AERODYNAMIC PRINCIPLES
Fig. 1.7 Concept of boundary-layer separation.
7
ed Flow
Once the flow separates from the body, the pressure distribution is altered, partic-
ularly in the region ofseparated flow. It is different from that predicted by Lhe ideal
fluid theory. The adverse pressure gradient is no longer there. In other words, the
picture presented in Fig. 1.7, based on the ideal fluid theory, existed only during the
initial moments of the flow over the body. To understand this concept,let us assume
that at t < O there is no :flow over the body and let the fiow start impulsively at t = 0.
For the "first batch" of fluid particles that arrive at the surface of the body, there is
no adverse pressure gradient to overcome. So the fiow is smooth over the body and
closes behind the body, forming front and rear stagnation points Si and S2 (Fig. 1.2)
as postulated by the ideal fluid theory. Once this happens, adverse pressure gradi-
ents are established. The immediate "next batch" of fiuid particles faces the adverse
pressure gradient and separates from the body surfacein a manner discussed earlier.
1.2.2 Flow Past Circular Cylinder
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