ABSTRACT
We now apply with our hybrid method to a problem involving a nonlinear partial differential equation. As a model of this type of problem, we consider the problem of determining the steady, two-dimensional flow of an inviscid, compressible, perfect gas past a circular cylinder of unit radius without circulation (see, e.g., Van Dyke [10]). If we introduce the usual polar coordinates ( r, 8) and express the fluid velocity q in terms of a potential function 4> as
where
with cf>r = 0 on r = 1, and 4> = 0( r-1 ) as r -+ oo. Here M = U / c is the free stream Mach number, c is the speed of sound in the gas,
In particular, we find
(20)
N Perturbation Hybrid Me 1 0.420943 0.407258 0.5389 2 0.409239 0.401704 0.4990 3 0.404577 0.399699 0.4740 4 0.402274 0.398948 0.4606 5 0.400979 0.398608 0.4513 6 0.400187 7 0.399671 8 0.399319 9 0.399071 10 0.398891 11 0.398757
We consider a sink (or source) of strength Q located a distance h above the undisturbed height of a free surface. We introduce a rectangular coordinate system (hx, hy), with gravity acting in the negative y-direction and with the origin a distance h below the sink (see Figure 4). We let the velocity potential be q, = Q{ 4~ log[x2 + (y-1)2] + rp(x, y)} and denote the elevation of the free surface by y = 17( x ). Then the problem to determine ¢ and 17 becomes
with
(29)
(30)
(31)
£=5
N-1 N </> = ·l::>j </>(j) + 0( EN)", TJ = :L::>j'l/j + O(EN+l) as E-+ 0. (32)
(33)
(34)
(35)
3 x2(1-6x2 +x4) 2a 1-8x2 +3x4 1l2 = 21!"4[ (1 + x2)5 ] + 11"2[ (1 + x2)4 ]. (36)
128 Geer and Andersen
In particular, we find
411"4(1 + x2 )1 24 a2(2 - 33 x2 + 40 x4 - 5 x6 )
