Inertial Layer Asymptotics in Wall Bounded Turbulence


Generally, turbulent flow describes a flow in which the fluid undergoes irregular fluctuations i.e. the speed of the fluid at a point is continuously undergoing changes in both magnitude and direction. In terms of Reynolds number (Re), this is the flow with Re greater than 4000. Research has revealed that with the aid of Direct Numerical Simulations (DNS) and experiments, the streamwise component of the mean velocity distribution in the inertial sub-layer of a fully developed turbulent flow obeys a universal log-law. This region, henceforth termed: log-layer, includes the intermediate zone between the buffer layer along the wall and the outer layer in the core region. Recently, an asymptotic analysis implemented using the RANS based turbulence energy transport equation showed that the energy dissipation equals its production in the inertial layer of wall-induced turbulence (Brouwers et al). However, a study preceding this report had already established that the pressure and energy diffusion terms appearing in the energy transport equation were of the same order of magnitude especially in the inertial layer thus leading to a contradiction (Tennekes and Lumley). Consequently, it is imperative that the turbulence energy budgets be re-estimated.

In this view, Dr. Kannan Sundaravadivelu from the Institute of High-Performance Computing at Agency for Science, Technology and Research in Singapore re-estimated the turbulence energy budgets in a different way by invoking the Kolomogrov’s similarity hypotheses and (4/5)th law. The main aim of his work was to theoretically establish the local energy equilibrium in the inertial layer of wall bounded turbulent flows. His work is currently published in  Journal of Turbulence.

An overview of his approach revealed that turbulence equilibrium in the inertial layer was established theoretically using RANS based turbulent energy equation. In fact, he made use of the two-point correlations in order to quantify the one-point correlations, by also assuming a small separation distance so that it mimics the one-point statistics. Also, the solution of the Poison equation for the pressure was expressed in terms of Green’s function so as to determine the direction of the pressure velocity correlation.

The author reported that the explicit expression for both the diffusion terms were obtained and found to agree well; the best available direct numerical simulations. Additionally, he found out that the obtained diffusion terms had the same order of magnitudes but oriented in opposite directions thus resulting in negligible/cancelling effect when combined.

In summary, the study theoretically attempted to establish the local energy equilibrium in the inertial layer of wall bounded turbulent flows. The prominent feature of Dr. Kannan Sundaravadivelu’s analysis was that no prior assumption was enforced on the mean velocity unlike in the earlier study. In an interview with Advances in Engineering, Dr. Kannan Sundaravadivelu further emphasized that the two main advantages of his work were: first, reasonable estimates for both the diffusion terms that were obtained explicitly as they were unavailable before and second, the obtained estimate enabled him to tweak the production/dissipation terms to reflect the influence of the turbulent diffusion mechanisms without the necessity to model them as in the case of elliptic relaxation and Reynold stress RANS models.


K. Sundaravadivelu. Local equilibrium in the inertial layer of wall bounded turbulence. Journal of Turbulence 2019, Volume 20, No. 6, page 381–392.

Go To Journal of Turbulence 2019

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