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Instability due to three-dimensional disturbances

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Abstract

In the linear theory of the stability of parallel flows of a viscous fluid, most attention is usually given to plane-wave disturbances. The reason is the validity in many cases of the Squire theorem, which states that the critical Reynolds number R is determined by two-dimensional disturbances [1]. It is shown in the present paper that for large R the region generating the turbulence in the initial stage of its development is formed by three-dimensional disturbances. This feature applies both to the generating range of wave numbers and the dimension of the wall layer, where the fluctuating energy is produced. The consequences of the Squire transformations for parallel flows are analyzed. The contribution of resonant nonlinear triad coupling to the rapid growth of fluctuating energy is studied for the case of an explosive instability in an extended laminar mode. It is shown that the rate of turbulent energy production is not governed by the small derivatives of linear theory, but by nonlinear triad coupling of neutral and growing disturbances, with their three-dimensional nature playing an important role.

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Literature cited

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Authors and Affiliations

  1. Institute of High Temperatures, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk

    V. N. Shtern

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  1. V. N. Shtern
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Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 29–34, September–October, 1976.

The author thanks M. A. Gol'dshtik for his interest in the work and for discussion of the results.

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Shtern, V.N. Instability due to three-dimensional disturbances. Fluid Dyn 11, 678–682 (1976). https://doi.org/10.1007/BF01012956

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