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b) Concept of displacement tluckness
Fig.1.4 Schematic velocity distribuhonin boundarylayer.
REVIEW OF BASIC AERODYNAMIC PRINCIPLES 5
a) Given body
b) Gi-ven body with boundarylayer
c) Equwalent body
Fig.1.5 Concept equrvalent body.
He postulated that the thickness 8 of the boundary layer is small in relation to
the characteristic dimension L of the body (8]L << 1) so that the effects of fluid
viscosity are assumed to be essentially confined within this thin boundary layer.
Outside the boundary layer, the fiuid fiow practically oeha'ves as though it is in-
viscid. With this hypothesis, one can consider the effect of fiuid viscosit)r or the
boundary-layer effect as equivalent to shifting the inviscid (potential) flow by a
small amount equal to boundary-layer displacement thickness 8*, whic~ is def:nyed
as
1 .00
8*-- V1 j [Vi- V(y)ldy (i.io)
The concept of 8* is illustrated in Fig. 1.4b.
The actual body shape now consists of the given shape plus a displacement
thick:ness 8* as shown schematically in Fig. 1.5. The concept of boundary layer
leads to a simple and practical method of finding the pressure distribution over a
given body surface as follows: 1) using the ideal fluid theory, obtain the inviscid
velocity and pressure distribution and 2) with this velocity distribution, obtain the
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',
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