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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
     The fluid fiow past a circular cylinder in crossfiow has attracted the attention of
several researchers, including Theodore 'von Karman. According to the ideal fiuid
theory, the maximum velocity is equal to twice the freestream velocity and occurs
at the maximum thickness point as shown in Fig. 1.3.
   The fiow pattern over a circular cylinder in a real fiuid flow depends on the
Reynolds number. At a low Reynolds number, i.e., when the Reynolds number is
below the critical value (subcritical), the fiow within the boundary layer is laminar
and the separation pointis located ahead of the maximum thickness point as shown
schematically in Fig.  1.8a. rfhis separation is of laminar type and is of permanent
nature. The separated flow rolls into a pair of alternating vortices, which are known
as Karman vortices. The pattern of alternating vortices is also known as Karman
vortex street. The wake flow is oscillatory and the drag coefficient is a function of
time. The frequency of the Karman vortex shedding is usually given in terms of a
8                 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Karman Vortex
a) Flow at subcritical Reynolds numbers
b) Flow at criticaIReynolds numbers
c) Flow at supercritical Reynolds numbers
Fig.1.8   Flow over circular cylinder.
-,
)
nondimensional number known as Strouhal number St, which is defined as
                                                 St = (/d                                      (1.11)
where  f  is the frequency of the vortex shedding, d is the diameter of the circular
cylinder, and Voo is the freestream velocity. For a circular cylinder, the Strouhal
number is approximately 0.21  for a Reynolds number range of l03 to  l04 (Ref.  1).
     As the freestream Reynolds number increases towards the critical value, there
is an increasing tenden/:y for the transition to occur in the separated boundary
layer. For a smooth circular cylinder, the critical Reynolds number, based on the
diameter, is in the range of4 x l05 to 8 x l05. As a result of flow transition, the
REVIEW OF BASIC AERODYNAMIC PRiNCIPLES                  9
Fig. 1.9    Schematic  variation of drag coefficient of circular cylinder with Reynolds
　
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