104
P. Liu
Fig. 2.34 Early potential flow stage (accelerated)
process is an incompressible two-dimensional unsteady laminar boundary
layer differential equation
∂u
∂ x
+
∂v
∂ y
= 0
∂u
∂t
+ u
∂u
∂ x
+ v
∂u
∂ y
=
∂ V e
∂t
+ V e
∂ V e
∂ x
+ v
∂ 2 u
∂ y 2
where V e is the outflow velocity of the boundary layer. In the initial
stage of the airfoil starting from static condition, as shown in Fig. 2.34,
the boundary layer has not been formed, the viscous shear force is large,
the migration inertia force is small, and the unsteady inertia force of the
outflow field is the main one. The above equation can be simplified as
∂u
∂ x
+
∂v
∂ y
= 0
∂u
∂t
− v
∂ 2 u
∂ y 2 =
∂ V e
∂t
At the later stage of the start-up process, the boundary layer is basically
formed and nearly stable. At this time, the unsteady inertial force is in the
secondary position, and the boundary layer equation can be simplified as
∂u
∂ x
+
∂v
∂ y
= 0
∂u
∂t
− v
∂ 2 u
∂ y 2 = V e
∂ V e
∂ x
− u
∂u
∂ x
− v
∂u
∂ y
Now, according to the development of boundary layer, the evolution of
separation and vortex shedding, combined with the physical mechanism
of viscous flow, the starting process of unsteady airfoil flow can be divided
into the following stages.