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side. It causes the transition to occur prior to separation and, therefore, moves the
separation point downstream (Fig. 8.50c). On the top side, the moving wall effect
remains the same as said above for the case of p < pcr However, the net result is
that the Magnus lift starts dropping and assumes negative values for some cases
as indicated by the data for group 1 in Fig. 8.49.
Turbulent flow. Consider the effect ofrotation rate when the flow is turbulent
corresponding to the data of group 3 in Fig. 8.49. For a stationary cylinder, when the
Reynolds number exceeds the critical Reynolds number, the flow over the circular
cylinder is of the supercritical type, which is characterized by the flow transition
ahead of the lateral meridian point and a turbulent separation downstream of the
lateral meridian point as shown in Fig. 8.51a. When p < Pcr, the moving wall
effect does not do much to alter the supercritical-type separation on the top side.
However, on the bottom side, the moving wall effect causes the flow separation
point to move upstream towards the subcritical locaOon as shown in Fig. 8.51b.
The net effect is to generate a positive Magnus lift similar to that observed for
laminar fiow, but the magnitude of the Magnus l/tft is much higher.
When p > Pcr, the downstream moving walleffect on the top side causes a delay
in the transition to such an extent that separation occurs before transition, thereby
moving the separation point towards the subcritical (laminar-type) location. On the
bottom side, the moving wall effect does not do much to alter the supercritical flow
separation. As a result, the Magnus lift starts building up in the opposite direction
as observed for the laminar fiow.
Transition Reyno/ds numbers. rfhis case corresponds to the data of group
2 in Fig. 8.49. Let us first consider the flow pat'tem for p = O. On a stationary
720 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Laminar
Separat
v
~
laminar
Separati
a) p=0
Tra n sition
b) p < Pcr
Laminar
c) p > pcr
Sepa ration
Fig. 8 50 Schematic itlustration of mOYing wall effect in laminar flOW.36
STABILITY AND CONTROL PROBLEMS AT HIGH ANGLES OF ATTACK 721
Transit
Flow
a) p=0
Separation
Separation
b) p < pcr
c) p > Pcr
Fig. 8.51 Schematic iHustration of moving wall effectin turbulent floW.36
722 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
a) p=0
Laminar Type
Separation
~ .agnus ~ift
V
-
Laminar
Reynolds Number
b)
Separation
c) p < pcr d) p > Pa
Fig 8.52 Schematic illustration of moving waU effect at transition Reynolds
numbers.36
circular cylinder, at transition Reynolds numbers, the flow is characterized by the
presence of a separation bubble as schematically shown in Fig. 8.52a. The flow
separates at Pi just ahead of the lateral meridian, and this separation is of the
subcritical or laminar type. A transition occurs in the lifted shear layer at P2,
causing the flow to become turbulent. This turbulent flow reattaches to the surface
of the cylinder at P3, forming a bubble. The reattached ftow eventually separates
at P4. An important feature of this type of fiow pattern is that the separation point
P4 iS further downstream of the supercritical (t;bulent-type) separation point. As
a consequence of this type of flow pattern, the drag coefficient assumes minimum
value, forming a drag bucket as shown schematically in Fig. 8.52b. When the
freestream Reynolds numberincreases further, the transition point gradually moves
ahead in the bubble so that, at onc point,it jumps ahead of the bubble and wipes it
out completely, establishing the supercnitical flow pattern of the type shown earlier
in Fig. 8.51a.
Now consider a rotating cylinder ( p 7/ 0) when the freestream Reynolds number
is in the transition range. The data in Fig. 8.49 for group 2 give the Magnus lift
for this range of Reynolds numbers. We observe that the Magnus lift is negative
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