Viscous instability of parallel flows
Theorem 5.1 With any unstable oblique mode (k,ω) of temporal growth rate ω i for Reynolds number Re it is possible to associate a two-dimensional mode
5.5 Boundary layer on a flat surface .1 Experimental demonstration
5.5.3 Eigenmodes, marginal stability, and nonparallel effects
−
x x0
ki(ξ)dξ
. (5.27)
Since the local growth rate−ki(x) is not spatially uniform, this amplitude does not vary exponentially with downstream distance, in contrast with a strictly parallel flow.
5.5.3 Eigenmodes, marginal stability, and nonparallel effects
Figure 5.16 shows the Blasius velocity profile of the base flow, both measured and calculated, as well as an eigenmode of the velocity perturbation u measured and calculated using the local analysis. This particular eigenmode was isolated by the vibration of a ribbon close to the wall in a flow with very low noise.
10 u⬘(y)/u⬘max
Distance from the wall (mm)
8
6
4
2
0
0 0.2 0.4 0.6 0.8 1.0
U(y)/U`
Figure 5.16 Base flow and eigenfunction u(y) = | ˆu(y)| of the Blasius boundary layer: (—) calculation, (◦, •) measurements. Used with permission of Annual Reviews, Inc., from Reed et al. (1996); permission conveyed through Copyright Clearance Center, Inc.
û
y/δ1 y/δ1
v
Figure 5.17 Eigenfunctions of the Blasius boundary layer: (- - -) the parallel the-ory, (—) direct numerical simulation (Fasel and Konzelmann, 1990). Reδ= 1000, F= 140×10−6.
1.0
0.5 In
Reδ1 0
450 650 850 1050
A Amin
Figure 5.18 Variation of the maximum of the amplitude | ˆu(y)| with Reynolds number Reδ∼ x1/2, calculated by three methods: (- - -) the parallel theory(5.27), (– –) the Gaster nonparallel theory (1974), (—) direct numerical simulation (Fasel and Konzelmann,1990). F= 140×10−6.
We see that the agreement is excellent. Figure 5.17 compares the eigenfunc-tion obtained from the stability calculaeigenfunc-tion with the result of a direct numerical simulation of the Navier–Stokes equations (Fasel and Konzelmann, 1990). The agreement is again remarkable. Since the direct simulation is free of the local par-allel flow assumption, we see that nonparpar-allel effects on the eigenfunctions are very small.
Figure 5.18 shows the spatial evolution of the amplitude of a perturbation imposed upstream, with the evolution calculated by three methods: by the local analysis (5.27), by a nonparallel theory (Gaster, 1974), and by direct numeri-cal simulation (Fasel and Konzelmann, 1990). We see that the amplitude first decreases, then increases, and finally decreases again. The boundary layer is
3
2
1
0 400 600
Reδ F × 104
800 αi = 0.000 28
Figure 5.19 Marginal stability of the boundary layer on a flat surface: (—) the parallel theory, (– –) the nonparallel theory of Gaster (1974), (•) direct numerical simulations of Fasel and Konzelmann (1990).
therefore first stable, then unstable, and then again stable. The band of unstable Reynolds numbers lies between the minimum and maximum amplitude. Mea-surements display a similar evolution (Ross et al., 1970), as long as nonlinear or three-dimensional effects remain negligible even when the perturbation reaches its maximum amplitude.
The marginal stability curve in the Reδ− F plane can be obtained from the spa-tial evolution of a perturbation for various frequencies. This is shown in Figure 5.19,for the three different approaches. The “half-banana” shape corresponds to the restabilization of a wave packet of a given frequency. Such a packet is repre-sented by a point which moves to the right on a line F= constant. The packet is at first stable until it reaches the left-hand branch (called I) of the marginal curve, then unstable until it reaches the right-hand branch (II), after which it is again stable.
Theoretical and numerical results are compared with the measurements of Ross et al. (1970) in Figure 5.20. We see from these figures that the parallel theory gives good results for Reδ> 600, but that significant nonparallel effects are manifested for Reδ< 600.
The first calculation of a marginal stability curve (not shown in Figure 5.19) was done by Tollmien (1929) well before the first measurements of Schubauer
3
2
1
0 400 600 800
Reδ1 F × 104
αi = 0.000 28
Figure 5.20 Marginal stability of the boundary layer on a flat surface: (—) the nonparallel theory, (×) measurements, (•) simulations of Fasel and Konzelmann (1990).
and Skramstad (1947). Tollmien’s solution, obtained well before the widespread use of numerical methods, approximated the base velocity profile by two straight lines connected by an arc of a parabola (Drazin and Reid, 2004, §31.5). The first numerical calculation with the Blasius velocity profile is due to Jordinson (1970), who found Reδ,c= 520 for the critical Reynolds number, in good agreement with the observations.
In the end, nonparallel effects appear to be of little importance for the situations considered above of two-dimensional waves and negligible pressure gradient. This conclusion is no longer valid when the pressure gradient is significant (as occurs, for example, for nonzero incidence or a curved surface), or when the leading edge of the aerofoil is not perpendicular to the flow (as for swept or triangular wings).
In the latter case, the oblique waves are strongly amplified, with the wave vector
making an angle of nearly 90◦ to the flow (Arnal, 1994; Reed et al., 1996; Schmid and Hennigson, 2001).
5.5.4 Transient growth
Certain perturbations of the boundary layer can grow in a transient fashion for values of the Reynolds number below the critical value and induce transition to turbulence. As for plane Poiseuille flow, this transient growth is maximal for perturbations with a longitudinal vortex structure, and gives rise to the same lon-gitudinal streaks near the wall (Butler and Farrel, 1992). Figure 5.21a shows the amplitudes of the velocities v and w corresponding to the optimal perturbation, and Figure5.21b gives the corresponding streamwise perturbation u. The frequency of this perturbation is zero (ω = 0) and its transverse wave number is kzδ = 0.45.
Here the calculation takes into account the spatial evolution of the flow in the x-direction by integrating the boundary layer equations, which are parabolic in x (Luchini, 2000).
The amplitude of the transverse modulation of the velocity corresponding to the streaks is higher by a factor of order√
Re than that of the initial vortices (we note that as we are referring to a linear problem, the perturbation amplitude is defined only up to a multiplicative constant). These streaks can therefore be strong enough to significantly modify the base flow and in the end inhibit the growth of Tollmien–Schlichting waves. Note that Tollmien–Schlichting waves can neverthe-less be observed if the upstream perturbations are small, especially in boundary layers (see Figure5.4) where they are more amplified than in plane Poiseuille flow (owing to the stabilizing effect of a negative pressure gradient).
0 2 4 6 8
Figure 5.21 (a) Normalized amplitude profiles of v and w corresponding to the optimal perturbation (v is maximal and w is zero at the level of the vortex axis y/δ ≈ 2); (b) amplitude profile of u. From Luchini (2000).
We conclude by noting that for the boundary layer on a flat wall below the criti-cal Reynolds number, the optimal perturbation is dissipated along with the streaks, as in plane Poiseuille flow. On a concave wall the situation is different because the longitudinal vortices are maintained by the centrifugal instability (Luchini and Bottaro, 1998, Bottaro and Luchini, 1999). This situation corresponds to the G¨ortler and Dean vortices mentioned in the preceding chapter in connection with the Couette–Taylor instability.
5.6 Exercises