Closure model for homogeneous isotropic turbulence in the Lagrangian specification of the flow field

In fluid mechanics, the closure problem in turbulence theory has for a long time remained an open case despite the numerous empirical and theoretical closure models that have been developed. In addition, the most progressive landmark in investigations regarding this issue has remained the Eulerian direct-interaction approximation (DIA). However, this tremendous technique is incapacitated since it is incompatible with the Kolmogorov −5/3 power law because the response integral diverges at the lower limit of the wavenumber as the Reynolds number approaches infinity.

Two alternative techniques that can be used to overcome the problem of divergence in the Eulerian DIA have been developed, namely: the local energy-transfer (LET) model and the abridged Lagrangian DIA. Both of these closure models are compatible with the Kolmogorov −5/3 power law. However, the first is incompatible with the characteristic time scale of the Lagrangian velocity correlation, and the second is very complicated. Therefore, it is imperative that a simpler closure model based on the direct manipulation of the equations of motion that is compatible with the Kolmogorov −5/3 power law in the inertial range without the inclusion of any adjustable free parameters or the introduction of the concept of eddy viscosity, be developed.

Recently, Dr. Okamura from the Research Institute for Applied Mechanics at Kyushu University developed a new two-point closure model that would be compatible with the Kolmogorov −5/3 power law without the inclusion of any adjustable free parameters or the introduction of the concept of eddy viscosity. He also created a model that yielded reasonable results for the Kolmogorov constant, the skewness of the longitudinal velocity derivative and the bottleneck effect. His work is currently published in Journal of Fluid Mechanics.

Briefly, the research method employed commenced with a thorough review of the Lagrangian velocity, the Lagrangian position function and equations describing their evolution. Next, the researcher then derived a closed set of three equations from the Navier–Stokes equation with no adjustable free parameters. Later, he then computed the Kolmogorov constant and the skewness using the closed set of equations. Lastly, he compared the model he had developed with other closure models, after which he evaluated the importance of using a mapping function.

Dr. Okamura observed that the Kolmogorov constant and the skewness of the longitudinal velocity derivative evaluated were found to be 1.779 and −0.49, respectively, using the proposed model. He also noted that the bottleneck effect was also reproduced in the near-dissipation range. Moreover, by considering that both the LET model and the Lagrangian DIA used a mapping function but the Eulerian DIA did not, he deemed it plausible to postulate that a key point in the derivation of the Kolmogorov −5/3 power law was the use of a mapping function in the closure procedure.

In summary, the Okamura study presented the advantages of the local interaction in frequency space (LIF) model over the Lagrangian DIA. The main technique employed in the work involved the derivation of a closed set of three equations from the Navier–Stokes equation with no adjustable free parameters. Altogether, the derivation of the closure equations in the LIF model is similar to that in the quasi-normal model and simpler than that in the Lagrangian DIA because the LIF model includes neither the response function nor the renormalized expansion.