698
V. TheoďŹ lis et al. / Computers & Fluids 32 (2003) 691–726
u M 2 iacĂ°1 bĂžKn Ă°^ gĂž p~ h1 1 i1=2 2 Ă°1 bĂž2 1 M1
as g ! 1;
Ă°24Ăž
2 1=2
2 where g^ Âź a½1 M1 Ă°1 bĂž Ă°1=f Ăž kf Ăž gĂž, the sign of which is chosen such that the real part of the argument is positive in order for boundedness as g ! 1 to be ensured, u1 is a constant and Kn Ă°z1 Ăž denotes the modiďŹ ed Bessel function of order n and argument z1 . Alternatively, one may combine the system (20) and (21), into a single equation ( #) " f 2 n2 2 2 Ă°W0 bĂž M1 T0 1 Ăž p~ 2 a2 1 Ăž kf2 Ăž fg ( ) T0 p~g p~g T0 fĂ°W0 bĂž d ; Ă°25Ăž Âź W0g Ă°W0 bĂž dg a2 Ă°W0 bĂž 1 Ăž kf2 Ăž fg a2 Ă°W0 bĂž
subject to the boundary conditions p~g Âź 0
for g Âź 0;
Ă°26Ăž
h i1=2 2 2 iacĂ°1 bĂžKn Ă°^ gĂž 1 M1 Ă°1 bĂž2 p~ u1 M1
as g ! 1:
Ă°27Ăž
2.3. Planar limiting cases To achieve the planar limits, we ďŹ rstly set k Âź 0 in (20) and (21), i.e., consider the cylinder form. Subsequently, the limit f ! 0 is applied, corresponding to ow in planar geometries. This yields h i W0g u i~ p 2 2 Ă°W bĂž ug Âź T M ; Ă°28Ăž 0 0 1 2 Ă°W bĂž W0 b cM1 0 ia2 Ă°W0 bĂž
p~g u Âź ; 2 T0 cM1
Ă°29Ăž
which describe the inviscid linear instability of compressible boundary-layer ow in planar geometries. Note, on applying the limit f ! 0, one collapses onto plane polar coordinates and not plane cartesian coordinates. The corresponding boundary conditions are u Ÿ p~g Ÿ 0 and
at g Âź 0;
Ă°30Ăž
qďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒ 2 g ; u u1 exp a 1 Ă°1 bĂž2 M1 2 iau1 cM1 Ă°1
p~
qďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒ 2 g bĂž exp a 1 Ă°1 bĂž2 M1 qďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒďŹƒ 2 1 Ă°1 bĂž2 M1
Ă°31Ăž
as g ! 1:
Ă°32Ăž