[Paper Review] Nonlinear waves in stratified Taylor--Couette flow. Part 2. Buoyancy flux

Colin Leclercq, Jamie Partridge+3 more|arXiv (Cornell University)|Sep 9, 2016

Fluid Dynamics and Turbulent Flows6 references3 citations

TL;DR

This study identifies nonlinear wave structures—specifically ribbons and mixed-ribbons—in stratified Taylor–Couette flow as the primary drivers of upward buoyancy flux, even at high Reynolds numbers. The mechanism arises from a positive correlation between density and vertical velocity in these coherent structures, driven by diffusion at low Schmidt numbers and nonlinear coupling at higher Sc, demonstrating that chaotic advection—not turbulence—dominates efficient mixing.

ABSTRACT

This paper is the second part of a two-fold study of mixing, i.e. the formation of layers and upwelling of buoyancy, in axially stratified Taylor--Couette flow, with fixed outer cylinder. In a first paper, we showed that the dynamics of the flow was dominated by coherent structures made of a superposition of nonlinear waves. (Mixed)-ribbons and (mixed)-cross-spirals are generated by interactions between a pair of linearly unstable helical modes of opposite `handedness', and appear to be responsible for the formation of well-mixed layers and sharp density interfaces. In this paper, we show that these structures are also fully accountable for the upwards buoyancy flux in the simulations. The mechanism by which this occurs is a positive coupling between the density and vertical velocity components of the most energetic waves. This coupling is primarily caused by diffusion of density at low Schmidt number Sc, but can also be a nonlinear effect at larger Sc. Turbulence was found to contribute negatively to the buoyancy flux at Sc=1,10,16, which lead to the conclusion that mass upwelling is a consequence of chaotic advection, even at large Reynolds number. Artificially isolating the coherent structure therefore leads to excellent estimates of the flux Richardson numbers Ri_f from the DNS. We also used the theoretical framework of Winters et al. (1995) to analyse the energetics of mixing in an open control volume, shedding light on the influence of end effects in the potential energy budget. The potential connection with the buoyancy flux measurements made in the recent experiment of Oglethorpe et al. (2013) is also discussed.

Motivation & Objective

• To identify the physical mechanism responsible for upward buoyancy flux in axially stratified Taylor–Couette flow.
• To determine whether turbulent or coherent wave structures dominate mixing efficiency in the weakly turbulent regime.
• To assess the role of end-plates and boundary conditions in distorting potential energy and buoyancy flux budgets.
• To evaluate the relevance of the findings to experimental observations, particularly Oglethorpe et al. (2013), despite Schmidt number mismatches.
• To investigate whether the observed flux mechanism could underlie the universal flux law reported in experiments.

Proposed method

• Direct numerical simulations (DNS) of axially periodic stratified Taylor–Couette flow with fixed outer cylinder.
• Spectral decomposition of the buoyancy flux term to isolate contributions from coherent structures.
• Application of the Winters et al. (1995) framework for open control volume energetics to separate available and background potential energy.
• Analysis of flux Richardson number (Rif) by isolating coherent wave structures and comparing with full DNS.
• Use of Newton-type solvers to converge nonlinear wave solutions along bifurcating branches.
• Comparison of numerical results with experimental data from Oglethorpe et al. (2013), focusing on buoyancy Reynolds number and flux scaling.

Experimental results

Research questions

• RQ1 What physical mechanism drives the upward buoyancy flux in weakly turbulent, axially stratified Taylor–Couette flow?
• RQ2 To what extent do coherent nonlinear waves—specifically ribbons and mixed-ribbons—account for the observed buoyancy flux?
• RQ3 How does the Schmidt number influence the efficiency of mass transport via nonlinear waves versus turbulence?
• RQ4 What role do end-plates and boundary conditions play in distorting potential energy and buoyancy flux diagnostics?
• RQ5 Can the coherent wave mechanism explain the universal flux law observed in Oglethorpe et al. (2013) despite differences in Schmidt number?

Key findings

• Nonlinear wave structures—ribbons and mixed-ribbons—dominate the upward buoyancy flux, accounting for nearly all observed flux in simulations.
• At low Schmidt numbers (Sc), the positive correlation between density and vertical velocity is primarily driven by molecular diffusion, enabling efficient buoyancy transport.
• At higher Schmidt numbers, a nonlinear coupling mechanism sustains the positive correlation, though with significantly reduced efficiency, leading to lower flux Richardson numbers.
• Turbulence contributes negatively to the buoyancy flux at Sc = 1, 10, and 16, indicating that mass upwelling is not due to turbulent mixing but to chaotic advection by coherent structures.
• The flux Richardson number (Rif) can be accurately estimated by isolating the coherent structure, confirming that it is the primary agent of mixing.
• End-plates cause spurious accumulation of potential energy and unstratified layers due to impermeability and no-flux density boundary conditions, distorting bulk energy budgets.