Flow Above the Lifting System and Its Wake
In the upper flow field Du, where (x, г/, г) = 0(1) for h0 —> 0 and є = O(/i0), both the wing and the wake approach the ground. In the limit, one comes in the upper half space to the problem for the flow, generated by the tangency conditions on the upper surfaces of the wing and the wake. Because є tends to zero (which practically means that, e. g., the relative ground clearance, angle of attack, curvature and thickness of the wing are small) the upper flow can be linearized, so that tangency conditions are satisfied upon the projections of the wing and wake sheet onto the plane у = 0.
In the region Du, we seek the upper flow perturbed velocity potential in the form
<Pu = h0<Pu1(x, y,z, t) + O(h%), <pUl= 0(1). (2.25)
Substitution of this expansion in the flow problem (2.22.6) leads to the following equation and boundary conditions for the upper flow problem:
d2<pUl d2<pUl dx2 dy2 
d2<pUl dz2 
= 0, 
(x, y, z) Є S; 
(2.26) 

% – U(t) dt v ’ 
dyu _ dx 
: , 
(x, z) Є 
S, у = 0 + 0; 
(2.27) 
~ 05 «2 c II 
"^Wi j 
(X, 2 
‘) Є W, 
У = 0 + 0; 
(2.28) 
9(Pui _ dy 
= 0, 
z) і 
І s + w, 
У = 0; 
(2.29) 
0, 
x2 
+ У2 + z2 
! —> со. 
(2.30) 
(auw) i?2(M) 2 W ~ 1zrvlnv + I V + у ; + 0(u2), 
where au = au//i0,aWl = aWl/h0. The channel flow and edge flow descriptions are lost in Du. Their influence will be recovered by asymptotic matching of the upper flow potential with that of the channel flow through edge regions.
Note that the boundary condition on the upper surface of the wake vortex sheet may be formulated as the flow tangency condition if the downwash olwx = ho&wx in the wake is known.
The upper flow potential (pUl is constructed in the form
The first term of (2.31) represents the induced velocity potential of the sourcesink distribution along the contours of the leading and side edges of the wing li and the edges of the wake /3. This contour distribution characterizes the influence of the channel flow upon the upper flow due to leakage of air from under the wing. The strength of the contour sources (sinks) Q(l, t) is determined as a result of matching the upper and channel flow potential through edges l and /3. The second and third terms of (2.31) correspond to the potential of the surface distributions of sources and the sinks of strength —2au upon S and —2aWl upon W. The latter result is based on a thin body theory.
Near the edges l and I2, the function ipUl has the following asymptotic representations:
where l is the arc coordinate measured along the planform contour, v is the external normal to the planform, and ( auw) = au(l, t) — aw(Z,£) is the jump of downwash on the upper surface of the wing across the trailing edge I2, (aUw) = (otUw)/h0. Parameters Ai, A2 and J5i, J52 characterize the influence of distant sources.
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