fluid_mechanics

492 Chapter 9 ■ Flow over Immersed Bodies
F I G U R E 9.18 Flow visualization photographs of flow past an airfoil (the boundary layer
velocity profiles for the points indicated are similar to those indicated in Fig. 9.17b): (a) zero angle of attack,
no separation, (b) 5 angle of attack, flow separation. Dye in water. (Photograph courtesy of ONERA, France.)
Streamlined bodies
generally have no
separated flow.
zero. This is due to boundary layer separation. Similar concepts hold for other shaped bodies as
well. The flow past an airfoil at zero angle of attack 1the angle between the upstream flow and the
axis of the object2 is shown in Fig. 9.18a; flow past the same airfoil at a 5° angle of attack is shown
in Fig. 9.18b. Over the front portion of the airfoil the pressure decreases in the direction of flow—
a favorable pressure gradient. Over the rear portion the pressure increases in the direction of flow—
an adverse pressure gradient. The boundary layer velocity profiles at representative locations are
similar to those indicated in Fig. 9.17b for flow past a circular cylinder. If the adverse pressure
gradient is not too great 1because the body is not too “thick” in some sense2, the boundary layer
fluid can flow into the slightly increasing pressure region 1i.e., from C to the trailing edge in Fig.
9.18a2 without separating from the surface. However, if the pressure gradient is too adverse 1because
the angle of attack is too large2, the boundary layer will separate from the surface as indicated in
Fig. 9.18b. Such situations can lead to the catastrophic loss of lift called stall, which is discussed
in Section 9.4.
Streamlined bodies are generally those designed to eliminate 1or at least to reduce2 the effects
of separation, whereas nonstreamlined bodies generally have relatively large drag due to the
low pressure in the separated regions 1the wake2. Although the boundary layer may be quite thin,
it can appreciably alter the entire flow field because of boundary layer separation. These ideas are
discussed in Section 9.3.
9.2.7 Momentum Integral Boundary Layer Equation with Nonzero
Pressure Gradient
The boundary layer results discussed in Sections 9.2.2 and 9.2.3 are valid only for boundary layers
with zero pressure gradients. They correspond to the velocity profile labeled C in Fig. 9.17b.
Boundary layer characteristics for flows with nonzero pressure gradients can be obtained from
nonlinear, partial differential boundary layer equations similar to Eqs. 9.8 and 9.9, provided the
pressure gradient is appropriately accounted for. Such an approach is beyond the scope of this
book 1Refs. 1, 22.