Self-Similar Blow-up Structures in Unsteady Marginally Separated Flows
In the present study we consider unsteady three-dimensional marginally separated boundary layer flows in the limit of high Reynolds numbers. Special emphasis is placed on solutions which blow up within finite time. In analogy to strictly two-dimensional f
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$6 1 that contains a circular core Ωε of radius ε centered at x0 = (−1, 0). For this case, the exact mean flow velocity w ¯E for this eccentric annulus geometry can be written as a complicated infinite series as in Ward-Smith (1980). In contrast, we need only calculate three specific quantities for our hybrid formulation in (16). Firstly, the unperturbed solution is again given by W0H (r) = β(r02 − r2 )/4. Next, since the inner core cross-section is a circle of radius ε, then the logarithmic capacitance is d = 1, so that ν = −1/ log ε. Finally, using the method of images, we solve (13) analytically to obtain the Green’s function 1 |x − x0 |r0 Gd (x; x0 ) = − log . (32) 2π |x − x0 ||x0 | Here the image point x0 of x0 in the circle of radius r0 lies along the ray containing x0 and satisfies |x0 ||x0 | = r02 . Comparing (32) with (13b), we can then calculate the self-interaction term as 1 r0 Rd00 ≡ Rd (x0 ; x0 ) = − log . (33) 2π |x0 − x0 ||x0 | Substituting (32), (33), ν = −1/ log ε, and W0H (r), into (16a) we obtain the outer solution for the hybrid method. This solution is then used in (2) with AΩ ∼ πr02 to compute the mean flow velocity for the hybrid method. The integral in (2) is obtained from a numerical quadrature. For an eccentric annulus with pipe radius r0 = 2, and with β = 1, in Fig. 1(b) we plot the mean flow velocity w ¯ versus the circular core radius ε as obtained from the exact solution and from the hybrid solution. This plot shows that the hybrid method results compare rather well with the exact results. We remark that for an inner pipe core of an arbitrary shape centered at x0 = (−1, 0), the hybrid method solution as obtained above for the eccentric annulus still applies, provided that we replace ν = −1/ log ε in (16a) with
Asymptotic Methods for PDE Problems
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ν = −1/ log(εd), where d is to be computed from (7). In particular, if there is an ellipse with semi-axes ε and 2ε centered at x0 = (−1, 0) instead of the circle of radius ε, then from Table 1 we get d = 3/2. Hence, the plot in Fig. 1(b) for the hybrid solution still applies provided that we replace the horizontal axis in this figure by 3ε/2.
3 Some Related Steady-State Problems in Bounded Singularly Perturbed Domains In this section we extend the analysis of §2 to treat some related steadystate problems. The problem in 3.1, which concerns the distribution of oxygen partial press
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