# Matching Conditions and High-Re Anomalies in Hydrodynamic Turbulence

## TL;DR

This paper extends the understanding of turbulence transition by analyzing matching conditions in hydrodynamic turbulence, explaining non-monotonic velocity derivative moments, and comparing different production mechanisms across Reynolds numbers.

## Contribution

It introduces a unified theory for turbulence transition in Rayleigh-Bénard convection, linking low and high Reynolds number behaviors and explaining observed non-monotonic moments.

## Key findings

- Transition at critical Reynolds number around 8.91 confirmed.
- Non-monotonic behavior of velocity derivative moments explained.
- High-Reynolds moments scale with Re with specific exponents.

## Abstract

{\bf Direct} transition from low Reynolds number "weak" Gaussian turbulence to fully developed "strong" turbulence at a critical Reynolds number $R^{tr}_{\lambda}\approx 8.91$ has recently been theoretically predicted and tested in high resolution numerical simulations of V. Yakhot \& D. A. Donzis, Phys. Rev. Lett. {\bf 119}, 044501 (2017) \& PhysicaD, {\bf 384-385}, 12 (2018) on an example of a flow excited by a Gaussian random force. The matching between the low-Reynolds number Gaussian asymptotic ($Re<<Re^{tr}$) and the multi-scaling one, dominating the high-Re limit ($Re>>Re^{tr}$), led to closed approximate equation for exponents of moments of derivatives in a good agreement with experimental data. In this paper we study transition to turbulence in Benard (RB) convection where, depending on the Rayleigh number, turbulence is produced by both weak instabilities of the bulk flow and, the plume-generating instabilities of the wall boundary layers. The developed theory explains non-monotonic behavior of the low-Reynolds - number moments of velocity derivatives $M_{2n (Re)=\frac{\overline{(\partial_{x}v_{x})^{2n}}}{[\overline{(\partial_{x}v_{x})^{2}}]^{n}}$ observed in direct numerical simulations of Schumacher et.al (Phys.Rev.E, {\bf 98},033120 (2018)). In the high-Reynolds number limit, the moments are given by $M_{2n}\propto Re^{\rho_{2n}}$ with the exponents $\rho_{2n}$ slightly different from those in a Gaussian-stirring case of Refs. [3]-[4]. This may be related to universality classes defined by production mechanisms.