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HAS OPTIMAL TRANSIENT GROWTH BEEN OBSERVED IN EXPERIMENTS?

• Transient growth is a physical phenomenon associated with the lift-up mechanism in shear flows. The first reference to transient growth in shear flows was by Ellingsen and Palm in 1975, and the phrase “optimal transient growth” has apearred in titles since 1992. The list of references includes the following publications

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• Google Scholar reports about 300 publications on optimal transient growth

(optimal disturbances). Do we have any evidence that optimal disturbances could be observed in experiments?

• M. V. Morkovin (‘‘Bypass transition research: issues and philosophy,’’ in Instabilities and Turbulence in Engineering Flows, edited by D. E. Ashpis, T. B. Gatski, and R. Hirsh Kluwer, Dordrecht, 1993, p. 3.) pointed out: “Models and experiments for … bypass disturbances e.g. Landahl (1980), Breuer and Landahl (1990), Henningson et al. (1990) are likely to contain essential elements of some bypass mechanisms. However, their ultimate validation must link them causally to external disturbances present in any given environment, i.e. to environmental realizability. That will be a major task.”

• Do we have an answer today regarding the realizability of “optimal disturbances?” Should we keep working on “optimal disturbances”?

• Today, it has become “a proven fact” that streaky structures observed in boundary layers at high free stream turbulence obey (and represent evidence for) the theory of optimal disturbances, e.g. their realizability in real experiments. For example, in a review paper by Hiroshi Maekawa (“Study of transition to turbulence in a supersonic boundary layer using DNS and transition prediction,” J. Fluid Science and Technology, V. 2., N. 3, 2007) we find: “The stream-wise velocity fluctuation increases as x1/2 of the downstream distance x and the fluctuation profiles normalized by the corresponding local displacement thickness are self-similar and very close to the transient growth analysis by Luchini.”

• Let’s find the source of the “evidence.” M. Matsubara and P. H. Alfredsson, (“Disturbance growth in boundary layers subjected to free-stream turbulence,” 2001, JFM, V. 430, pp. 149 – 168.) reported: “The measurements of stream-wise velocity disturbance … show how the initial growth is proportional to x1/2. This is in accordance with Luchini (2000). Also, the disturbance distribution is in accordance with this theory…”

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• Matsubara and Alfredsson conducted their experiment in the MTL wind tunnel at KTH, Stockholm, where Westin et al. carried out similar experiments in 1994 (K. J. A. Westin, A. V. Boiko, B. G. B. Klingmann, V. V. Kozlov, and P. H. Alfredsson, “Experiments in a boundary layer subjected to free stream turbulence. Part 1. Boundary layer structure and receptivity,” JFM, V. 281, pp. 193 – 218, 1994). Westin et al. reported the energy growth in the streaky structures depending on the frequency parameter

F = 2π fν

U∞2 ×106 .

Eδ F( ) = 1δ ′uF

2dy0

δ

∫

We want to point out that the optimal disturbances correspond to steady stream-wise vortices, whereas the experimental data by Matsubara and Alfredsson dealt with r.m.s. data. If we extrapolate the data of Westin et al. to very low frequencies F ≤1 , the energy ratio at x = 1000 mm and x = 500 mm is ≥4 . In other words, the stream-wise velocity perturbation grows approximately linearly with x!

• The interest in streaky structures generated by free-stream turbulence has a long history. The stream-wise velocity perturbations observed in boundary layers were referred to as Klebanoff’s mode. It was found that the velocity profile can be approximated by y(dU /dy) . The following figure shows a comparison of the optimal disturbance stream-wise velocity profile and y(dU /dy) .

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• It was noticed in experiments that boundary layer thickness and wall shear stress

had large span-wise variations in wind tunnels designed for two-dimensional flows (Klebanoff and Tidstrom, 1959). Bradshow (1965) suggested that these variations are caused by tiny span-wise variations in the free stream. Crow (S. C. Crow, “The span-wise perturbation of two dimensional boundary layers.” 1966, JFM, V. 24, pp. 153 – 164) analyzed steady streaky structures analytically using asymptotic perturbation methods. In the boundary layer, the problem was reduced to linearized boundary layer equations with a periodic span-wise velocity boundary condition at the edge of the boundary layer. He found the stream-wise velocity profile including the perturbation:

𝑢 = ℱ! 𝑠 − !!𝑅𝜖

!ξsin(ζ)𝑠ℱ′′(𝑠)+ 𝜖!(ℑ'(s)/2ξ)+ 𝑂(𝜖!) In his notation 𝜖 is a small parameter associated with the ratio of the boundary layer length scale νL/U∞ and λL , where L is the distance from the leading edge, and λ is the span-wise length scale of the imposed span-wise variation. The dimensionless variable ξ stands for dimensionless distance from the leading edge, and s is the dimensionless boundary layer coordinate. The function ℱ(𝑠) is the stream function of the Blasius solution. One can see that the steady stream-wise perturbation in the streaky structure grows proportionally to x. Also, the velocity profile has the shape of the

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Klebanoff mode y(dU /dy) . The span-wise variation is represented by sin(ζ), where ζ is the dimensionless span-wise coordinate.

• To summarize: 1) Experimental results by Westin et al. at F→0 agree with Crow’s solution (not with the optimal transient growth theory); 2) The stream-wise velocity perturbation profiles of Matsubara and Alfredsson (2001) and Westin et al. (1994) are close to the optimal perturbation of the velocity profiles. This raises a question: Why are experimental r.m.s. velocity profiles close to the optimal perturbation profile having zero frequency?

• Bertolotti (1997) explored the effect of free-stream turbulence on boundary layer flow using the PSE technique. However, it was not clear why his solutions did not show Crow’s result when steady large scale stream-wise vortices were present in the free stream. To build a link with Bertolotti’s model and Crows asymptotic solution. Tumin (“A model of the Blasius boundary layer response to free-stream vorticity,” EUROMECH Colloquium 380, Gottingen, 1998) considered a model of boundary layer flow past a flat plate in the presence of large non-steady stream-wise vortices in free stream. Non-steady perturbations in the boundary layer were governed by the linearized non-steady boundary-layer flow equations. The boundary conditions were matched outside the boundary layer accordingly to the inner limit of the outer solution. Because in the limit x→0 the frequency scaled with the boundary layer thickness tends to zero, Crows solution was used as the inflow. The following two figures show the downstream growth of the maximum stream-wise velocity in the streaks at different frequency parameters F and the corresponding velocity profiles.

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One can see that growth of umax is a linear function of the dimensionless downstream coordinate ξ for F = 0 and the growth depends on frequency. Also, the shape of the stream-wise velocity perturbation is insensitive to the frequency parameter F. This observation explains why the r.m.s. experimental data from the stream-wise velocity profile are close to the optimal perturbation shape (and to the Klebanoff mode shape derived in Crow’s analysis). Experimental data of Westin et al. indicate that the amplification of energy in streaky structures is noticeable only for low frequencies, F <50 . The next figure shows the r.m.s. of umax within

the interval 0< F <50 assuming a flat distribution in the inflow. Indeed, umax−rmscan be closely approximated by ξ

1/2 as it was observed in experiments of Matsubara and Alfredsson.

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These observations explain why the experimental velocity profiles are close to the Klebanoff mode (and to the optimal perturbation profile) and why the experimental growth of umax is proportional to x1/2.

• Summary: Experiments by Westin et al. (1994) and experiments by Matsubara and Alfredsson (2001) do not indicate that optimal perturbations had a role. Moreover, experimental data by Westin et al. (1994) clearly show that the stream-wise velocity at low frequencies is consistent with Crow’s solution and is not representative of the optimal transient growth theory. We still do not have any evidence today that optimal perturbations are realizable.

• Question: Why is so much effort still devoted to the study of optimal transient growth when the realizability issue raised by Morkovin has not yet been adequately addressed?