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Direct-Interaction Approximation - Principles andApplications
en
後藤 晋
ゴトウ ススム
GOTO Susumu
総研大甲第394号
It has been phenomenologically shown and widely supported by experiments that statistical properties in small scales of turbulence of incompressible fluids exhibit some universality irrespective of the kinds of fluids, boundary condition and the Reynolds number. On the other hand, this system is believed to be governed by the Navier-Stokes equations which consist of the equations of motion and of continuity. However, relationships between these equations and phenomenologies on statistical properties of small-scale motions have not been clearly understood primarily because such a statistical theory is hard to construct due to the nonlinearity of the basic equations. Since the nonlinearity causes an infinite hierarchy of moments, we never obtain a closed set of equations for a finite number of statistical quantities without any assumptions. This is the so-called closure problem in the statistical theory of turbulence. We adopt the direct-interaction approximation (DIA), which was originally proposed by Kraichnan (1959), to attack and solve the closure problem.<br /> The DIA is an excellent approximation in the sense that the nonlinearity is never neglected and no adjustable parameter is introduced in the formulation. Unfortunately, however, it is known that a closed set of equations obtained by a naive application of DIA (Kraichnan 1959) to the Navier-Stokes system is inconsistent with experimental observation. Especially, it is E(k) α k-3/2 that the closure equations predict as the energy spectrum E(k) in the inertial range, where the k-5/3 power law is observed by many experiments. This inconsistency implies incompleteness of the application of DIA to the Navier-Stokes system. Although Kraichnan (1965) improved the application method of DIA and succeeded in deriving the k-5/3 power law, the formulations are too complicated to be justified. Moreover, in spite of its long history and important role in the field of the statistical theory of turbulence, the essence of DIA may have been misunderstood by many researchers. This is due to the fact that validity conditions and applicability of DIA were not clear.<br /> We introduce a model equation, consisting of quadratic nonlinear and linear dissipative terms, which is simpler than the Navier-Stokes equation but still possesses its important mathematical structures. Then, it is shown that DIA is valid for such a system that has weak nonlinear couplings and large numbers of degrees of freedom even if nonlinearity of the system is strong (i.e., the nonlinear terms are larger than the linear ones in magnitude). Furthermore, we clarify similarities and differences between DIA and a Reynolds-number expansion so-called RRE (Reynolds-number reversed expansion) . For some known systems, including the Navier-Stokes system and the present model, these two approximations yield an identical set of equations for the correlation and the response functions. Owing to this fact, these two approximations have sometimes been identified erroneously. It must be stressed, however, that DIA and RRE are based upon completely different ideas and working assumptions. Hence, we should distinguish these two theories. This is reasonable because the validity conditions of DIA depend on the strength of nonlinear couplings and the number of degrees of freedom, but not on the Reynolds number, while the validity of RRE depends crucially on magnitude of the Reynolds number.<br /> We further investigate the validity condition of DIA and the relationships between DIA and RRE from a viewpoint of the strength of nonlinear couplings by extending the model equation. It is then shown that DIA is valid for systems such that the average number of direct interactions between a pair of modes is much smaller than the square root of the number of degrees of freedom, and that RRE may be regarded as an approximation under which the nonlinear terms are replaced by a joint-Gaussian random variables. The last approximation, called normal nonlinear term approximation, has the same validity conditions as DIA.<br /> Small-scale motions of turbulence may be statistically homogeneous, and the number of degrees of freedom of this system increases in proportion to the 9/4 power of the Reynolds number. Hence, small-scale motions of turbulent fields at high Reynolds number satisfy the two validity conditions of DIA, i.e., weakness of nonlinear couplings and largeness of the degrees of freedom. This implies that DIA is applicable to this system. As mentioned above, however, when we apply DIA to the Eulerian velocity correlation function and the Eulerian velocity response function (Kraichnan 1959), we encounter the difficulty that the resultant closure equations are incompatible with experiments. Here, we instead apply DIA to the Lagrangian velocity correlation function and the Lagrangian response function with the help of the position function (Kaneda 1981), which is a map between the Eulerian and the Lagrangian fields. The resultant equations yield not only the well-known k-5/3 power law predicted phenomenologically by Kolmogorov (1941) of the energy spectrum, but also the functional form in the entire universal range, which excellently agrees with experimental data.<br /> We next apply DIA to passive scalar fields (temperature, particle concentration, smoke, and so on) advected by turbulence without affecting fluid motions. Then it is systematically shown that solutions to the resultant closure equations by DIA for the Lagrangian correlation and the response functions for the velocity and the passive scalar fields are completely consistent with the phenomenologies on the scalar spectrum by Obukhov (1949) and Corrsin (1951) in the inertial-advective range, Batchelor, Howells & Townsend (1959) in the inertial-diffusive range, and Batchelor (1959) in the viscous-advective range.
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