Reynolds number dependence of an upper bound for the long-time-averaged buoyancy flux in plane stratified Couette flow

C. P. Caulfield, Wenbo Tang, S. C. Plasting

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

We derive an improved rigorous upper bound for the long-time-averaged vertical buoyancy flux for stably stratified Couette flow; i.e. the flow of a Boussinesq fluid (with reference density ρo, kinematic viscosity v, and thermal diffusivity K) confined between two parallel horizontal plates separated by a distance d, which are driven at a constant relative velocity ΔU, and are maintained at a constant (statically stable) temperature difference leading to a constant density difference Δρ. We construct the bound by means of a numerical solution to the 'background method' variation problem as formulated by Constantin and Doering using a one-dimensional uni-directional background. The upper bound so constructed is the best possible bound with the imposed constraints for streamwise independent mean flows that are statistically steady, and is calculated up to asymptotically large Reynolds numbers. We find that the associated (dimensional) upper bound ℬ*max on the long-time-averaged and volume averaged buoyancy flux ℬ* := lim[∞(1/t) ∫o t <ρu3>g/ρo dτ (where u3 is the vertical velocity, g is the acceleration due to gravity, and angled brackets denote volume averaging) does not depend on either the bulk Richardson number J = gΔρd/(ρoΔU2) of the flow, or the Prandtl number σ = v/k of the fluid. We show that ℬ*max has the same inertial characteristic scaling as the (dimensional) mechanical energy dissipation rate ℰ*B, and ℬB*max = 0.001267 ΔU3/d as Re → ∞. The associated flow structure exhibits velocity boundary layers embedded within density boundary layers, with local gradient Richardson numbers Ri = O(σ/Re)≪ 1 in the vicinity of the horizontal plates. There is a correspondence between the predicted flow structure and the flow structure at a lower Reynolds number associated with the upper bound on the mechanical energy dissipation rate ℰ*max in an unstratified fluid. We establish that, for the flow that maximizes the buoyancy flux, the flux Richardson number Rif → 1/3 as Re → ∞, independently to leading order of both Re and J. There is a generic partition of the energy input by the shear into the fluid into three equal parts: viscous dissipation of kinetic energy by the mean flow; viscous dissipation of kinetic energy by perturbation velocities; and vertical buoyancy flux.

Original languageEnglish (US)
Pages (from-to)315-332
Number of pages18
JournalJournal of Fluid Mechanics
Issue number498
DOIs
StatePublished - Jan 10 2004
Externally publishedYes

Fingerprint

Couette flow
Buoyancy
buoyancy
Reynolds number
Fluxes
Flow structure
Richardson number
Fluids
Kinetic energy
Energy dissipation
Boundary layers
fluids
Thermal diffusivity
Prandtl number
boundary layers
dissipation
energy dissipation
kinetic energy
Gravitation
calculus of variations

ASJC Scopus subject areas

  • Mechanics of Materials
  • Computational Mechanics
  • Physics and Astronomy(all)
  • Condensed Matter Physics

Cite this

Reynolds number dependence of an upper bound for the long-time-averaged buoyancy flux in plane stratified Couette flow. / Caulfield, C. P.; Tang, Wenbo; Plasting, S. C.

In: Journal of Fluid Mechanics, No. 498, 10.01.2004, p. 315-332.

Research output: Contribution to journalArticle

@article{e34cd672c5654aeaa7b61f4ced05fec0,
title = "Reynolds number dependence of an upper bound for the long-time-averaged buoyancy flux in plane stratified Couette flow",
abstract = "We derive an improved rigorous upper bound for the long-time-averaged vertical buoyancy flux for stably stratified Couette flow; i.e. the flow of a Boussinesq fluid (with reference density ρo, kinematic viscosity v, and thermal diffusivity K) confined between two parallel horizontal plates separated by a distance d, which are driven at a constant relative velocity ΔU, and are maintained at a constant (statically stable) temperature difference leading to a constant density difference Δρ. We construct the bound by means of a numerical solution to the 'background method' variation problem as formulated by Constantin and Doering using a one-dimensional uni-directional background. The upper bound so constructed is the best possible bound with the imposed constraints for streamwise independent mean flows that are statistically steady, and is calculated up to asymptotically large Reynolds numbers. We find that the associated (dimensional) upper bound ℬ*max on the long-time-averaged and volume averaged buoyancy flux ℬ* := lim[∞(1/t) ∫o t <ρu3>g/ρo dτ (where u3 is the vertical velocity, g is the acceleration due to gravity, and angled brackets denote volume averaging) does not depend on either the bulk Richardson number J = gΔρd/(ρoΔU2) of the flow, or the Prandtl number σ = v/k of the fluid. We show that ℬ*max has the same inertial characteristic scaling as the (dimensional) mechanical energy dissipation rate ℰ*B, and ℬB*max = 0.001267 ΔU3/d as Re → ∞. The associated flow structure exhibits velocity boundary layers embedded within density boundary layers, with local gradient Richardson numbers Ri = O(σ/Re)≪ 1 in the vicinity of the horizontal plates. There is a correspondence between the predicted flow structure and the flow structure at a lower Reynolds number associated with the upper bound on the mechanical energy dissipation rate ℰ*max in an unstratified fluid. We establish that, for the flow that maximizes the buoyancy flux, the flux Richardson number Rif → 1/3 as Re → ∞, independently to leading order of both Re and J. There is a generic partition of the energy input by the shear into the fluid into three equal parts: viscous dissipation of kinetic energy by the mean flow; viscous dissipation of kinetic energy by perturbation velocities; and vertical buoyancy flux.",
author = "Caulfield, {C. P.} and Wenbo Tang and Plasting, {S. C.}",
year = "2004",
month = "1",
day = "10",
doi = "10.1017/S0022112003006797",
language = "English (US)",
pages = "315--332",
journal = "Journal of Fluid Mechanics",
issn = "0022-1120",
publisher = "Cambridge University Press",
number = "498",

}

TY - JOUR

T1 - Reynolds number dependence of an upper bound for the long-time-averaged buoyancy flux in plane stratified Couette flow

AU - Caulfield, C. P.

AU - Tang, Wenbo

AU - Plasting, S. C.

PY - 2004/1/10

Y1 - 2004/1/10

N2 - We derive an improved rigorous upper bound for the long-time-averaged vertical buoyancy flux for stably stratified Couette flow; i.e. the flow of a Boussinesq fluid (with reference density ρo, kinematic viscosity v, and thermal diffusivity K) confined between two parallel horizontal plates separated by a distance d, which are driven at a constant relative velocity ΔU, and are maintained at a constant (statically stable) temperature difference leading to a constant density difference Δρ. We construct the bound by means of a numerical solution to the 'background method' variation problem as formulated by Constantin and Doering using a one-dimensional uni-directional background. The upper bound so constructed is the best possible bound with the imposed constraints for streamwise independent mean flows that are statistically steady, and is calculated up to asymptotically large Reynolds numbers. We find that the associated (dimensional) upper bound ℬ*max on the long-time-averaged and volume averaged buoyancy flux ℬ* := lim[∞(1/t) ∫o t <ρu3>g/ρo dτ (where u3 is the vertical velocity, g is the acceleration due to gravity, and angled brackets denote volume averaging) does not depend on either the bulk Richardson number J = gΔρd/(ρoΔU2) of the flow, or the Prandtl number σ = v/k of the fluid. We show that ℬ*max has the same inertial characteristic scaling as the (dimensional) mechanical energy dissipation rate ℰ*B, and ℬB*max = 0.001267 ΔU3/d as Re → ∞. The associated flow structure exhibits velocity boundary layers embedded within density boundary layers, with local gradient Richardson numbers Ri = O(σ/Re)≪ 1 in the vicinity of the horizontal plates. There is a correspondence between the predicted flow structure and the flow structure at a lower Reynolds number associated with the upper bound on the mechanical energy dissipation rate ℰ*max in an unstratified fluid. We establish that, for the flow that maximizes the buoyancy flux, the flux Richardson number Rif → 1/3 as Re → ∞, independently to leading order of both Re and J. There is a generic partition of the energy input by the shear into the fluid into three equal parts: viscous dissipation of kinetic energy by the mean flow; viscous dissipation of kinetic energy by perturbation velocities; and vertical buoyancy flux.

AB - We derive an improved rigorous upper bound for the long-time-averaged vertical buoyancy flux for stably stratified Couette flow; i.e. the flow of a Boussinesq fluid (with reference density ρo, kinematic viscosity v, and thermal diffusivity K) confined between two parallel horizontal plates separated by a distance d, which are driven at a constant relative velocity ΔU, and are maintained at a constant (statically stable) temperature difference leading to a constant density difference Δρ. We construct the bound by means of a numerical solution to the 'background method' variation problem as formulated by Constantin and Doering using a one-dimensional uni-directional background. The upper bound so constructed is the best possible bound with the imposed constraints for streamwise independent mean flows that are statistically steady, and is calculated up to asymptotically large Reynolds numbers. We find that the associated (dimensional) upper bound ℬ*max on the long-time-averaged and volume averaged buoyancy flux ℬ* := lim[∞(1/t) ∫o t <ρu3>g/ρo dτ (where u3 is the vertical velocity, g is the acceleration due to gravity, and angled brackets denote volume averaging) does not depend on either the bulk Richardson number J = gΔρd/(ρoΔU2) of the flow, or the Prandtl number σ = v/k of the fluid. We show that ℬ*max has the same inertial characteristic scaling as the (dimensional) mechanical energy dissipation rate ℰ*B, and ℬB*max = 0.001267 ΔU3/d as Re → ∞. The associated flow structure exhibits velocity boundary layers embedded within density boundary layers, with local gradient Richardson numbers Ri = O(σ/Re)≪ 1 in the vicinity of the horizontal plates. There is a correspondence between the predicted flow structure and the flow structure at a lower Reynolds number associated with the upper bound on the mechanical energy dissipation rate ℰ*max in an unstratified fluid. We establish that, for the flow that maximizes the buoyancy flux, the flux Richardson number Rif → 1/3 as Re → ∞, independently to leading order of both Re and J. There is a generic partition of the energy input by the shear into the fluid into three equal parts: viscous dissipation of kinetic energy by the mean flow; viscous dissipation of kinetic energy by perturbation velocities; and vertical buoyancy flux.

UR - http://www.scopus.com/inward/record.url?scp=1242344853&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1242344853&partnerID=8YFLogxK

U2 - 10.1017/S0022112003006797

DO - 10.1017/S0022112003006797

M3 - Article

AN - SCOPUS:1242344853

SP - 315

EP - 332

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

IS - 498

ER -