The identification
of successive stages in the transition of unsteady viscous transonic
flow around an aerofoil was carried out by solving the time-dependent
Navier-Stokes equations for a compressible fluid in two-dimensional
approach. The numerical simulation was carried out at the Mach number
range (0.2-0.98). At a fixed Reynolds number (Re=10,000), this flow
undergoes the following four transition steps: It remains steady up
to the Mach number values (0.2-0.35) and develops afterwards without
any imposed artificial perturbation, an inherent unsteadiness corresponding
to a near wake von Kàrmàn instability, in the Mach number
range (0.35-0.9). In the Mach number range (0.9-0.95), there exists
a critical Mach number for which the flow returns to a steady state.
Furthermore, the flow is found to be governed by two instability processes
in the Mach number range (0.75-0.8), where, apart from the von Kàrmàn
mode (mode~I), a lower frequency mode~II appears, due to the formation
of weakly supersonic alternating zones in the region upstream of the
aerofoil, related to the buffeting phenomenon. The triple role played
by increasing compressibility effects to trigger the instability processes,
to maintain and to inhibit them in the transonic flow regime was analysed
in detail.
The first three figures show the Mach number distribution around the
airfoil at M=0.2, 0.85 and 0.9 respectively. A steady state flow
is clearly visible at M=0.2. At Mach number 0.85,
shock waves and vortex shedding are present. When the Mach number is
increased, the shock wave is pushed further downstream, interact with
the wake vortices and the upstream flow becomes steady.
The bottom figures show the time dependent evolution of the lift coefficient
and the
corresponding Strouhal numbers at specific Mach numbers (a) 0.5; (b)
0.7; (c) 0.75; (d) 0.8;(e) 0.85; (f) 0.9; (g) 0.95 and (h) 0.98; The
different flow stages mentioned above, are clearly demonstrated.
This study was carried out at the Fluid Mechanics Institute of Toulouse
(France) and was part of the Ph.D. thesis of Dr Latif Bouhadji.
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