# Classical 1/3 scaling of convection holds up to Ra = 10 ^{15}

## Abstract

The global transport of heat and momentum in turbulent convection is constrained by thin thermal and viscous boundary layers at the heated and cooled boundaries of the system. This bottleneck is thought to be lifted once the boundary layers themselves become fully turbulent at very high values of the Rayleigh number $\mathrm{R}\mathrm{a}$—the dimensionless parameter that describes the vigor of convective turbulence. Laboratory experiments in cylindrical cells for $\mathrm{R}\mathrm{a}\gtrsim 1{0}^{12}$have reported different outcomes on the putative heat transport law. Here we show, by direct numerical simulations of three-dimensional turbulent Rayleigh–Bénard convection flows in a slender cylindrical cell of aspect ratio $1/10$, that the Nusselt number—the dimensionless measure of heat transport—follows the classical power law of $\mathrm{N}\mathrm{u}=\left(0.0525\pm 0.006\right)\times {\mathrm{R}\mathrm{a}}^{0.331\pm 0.002}$up to $\mathrm{R}\mathrm{a}=1{0}^{15}$. Intermittent fluctuations in the wall stress, a blueprint of turbulence in the vicinity of the boundaries, manifest at all $\mathrm{R}\mathrm{a}$considered here, increasing with increasing $\mathrm{R}\mathrm{a}$, and suggest that an abrupt transition of the boundary layer to turbulence does not take place.

- Authors:

- Publication Date:

- Research Org.:
- Argonne National Lab. (ANL), Argonne, IL (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1606457

- Alternate Identifier(s):
- OSTI ID: 1625066

- Grant/Contract Number:
- AC02_06CH11357; AC02-06CH11357

- Resource Type:
- Published Article

- Journal Name:
- Proceedings of the National Academy of Sciences of the United States of America

- Additional Journal Information:
- Journal Name: Proceedings of the National Academy of Sciences of the United States of America Journal Volume: 117 Journal Issue: 14; Journal ID: ISSN 0027-8424

- Publisher:
- Proceedings of the National Academy of Sciences

- Country of Publication:
- United States

- Language:
- English

- Subject:
- Science & Technology - Other Topics

### Citation Formats

```
Iyer, Kartik P., Scheel, Janet D., Schumacher, Jörg, and Sreenivasan, Katepalli R. Classical 1/3 scaling of convection holds up to Ra = 10 15. United States: N. p., 2020.
Web. doi:10.1073/pnas.1922794117.
```

```
Iyer, Kartik P., Scheel, Janet D., Schumacher, Jörg, & Sreenivasan, Katepalli R. Classical 1/3 scaling of convection holds up to Ra = 10 15. United States. doi:https://doi.org/10.1073/pnas.1922794117
```

```
Iyer, Kartik P., Scheel, Janet D., Schumacher, Jörg, and Sreenivasan, Katepalli R. Wed .
"Classical 1/3 scaling of convection holds up to Ra = 10 15". United States. doi:https://doi.org/10.1073/pnas.1922794117.
```

```
@article{osti_1606457,
```

title = {Classical 1/3 scaling of convection holds up to Ra = 10 15},

author = {Iyer, Kartik P. and Scheel, Janet D. and Schumacher, Jörg and Sreenivasan, Katepalli R.},

abstractNote = {The global transport of heat and momentum in turbulent convection is constrained by thin thermal and viscous boundary layers at the heated and cooled boundaries of the system. This bottleneck is thought to be lifted once the boundary layers themselves become fully turbulent at very high values of the Rayleigh numberRa—the dimensionless parameter that describes the vigor of convective turbulence. Laboratory experiments in cylindrical cells forRa≳1012have reported different outcomes on the putative heat transport law. Here we show, by direct numerical simulations of three-dimensional turbulent Rayleigh–Bénard convection flows in a slender cylindrical cell of aspect ratio1/10, that the Nusselt number—the dimensionless measure of heat transport—follows the classical power law ofNu=(0.0525±0.006)×Ra0.331±0.002up toRa=1015. Intermittent fluctuations in the wall stress, a blueprint of turbulence in the vicinity of the boundaries, manifest at allRaconsidered here, increasing with increasingRa, and suggest that an abrupt transition of the boundary layer to turbulence does not take place.},

doi = {10.1073/pnas.1922794117},

journal = {Proceedings of the National Academy of Sciences of the United States of America},

number = 14,

volume = 117,

place = {United States},

year = {2020},

month = {3}

}

DOI: https://doi.org/10.1073/pnas.1922794117

*Citation information provided by*

Web of Science

Web of Science

#### Figures / Tables:

*A*) Turbulent heat transport law Nu(Ra) for two datasets on log–log coordinates. The present data (open circles) are for Pr = 1 and Γ=0.1 for 10

^{8}≤ Ra ≤10

^{15}. Open squares representing DNS at Pr=0.7 and Γ=1 formore »

*
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(6 total)
*

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##
Efficiency of Heat Transfer in Turbulent Rayleigh-Bénard Convection

##
Convective Self-Aggregation in Numerical Simulations: A Review

##
Turbulent convection at very high Rayleigh numbers

##
Large granulation cells on the surface of the giant star π1 Gruis

##
Scaling of hard thermal turbulence in Rayleigh-Bénard convection

##
Zhu

##
Resolving the fine-scale structure in turbulent Rayleigh–Bénard convection

##
Irminger Sea deep convection injects oxygen and anthropogenic carbon to the ocean interior

##
New perspectives in turbulent Rayleigh-Bénard convection

##
Turbulent rotating convection confined in a slender cylinder: The sidewall circulation

##
Lithium–antimony–lead liquid metal battery for grid-level energy storage

##
An Overlapping Schwarz Method for Spectral Element Solution of the Incompressible Navier–Stokes Equations

##
Turbulent Thermal Convection at Arbitrary Prandtl Number

##
Transition to the Ultimate State of Turbulent Rayleigh-Bénard Convection

##
Wall-bounded turbulent flows at high Reynolds numbers: Recent advances and key issues

##
Influence of 2D and 3D convection–diffusion flow on tritium permeation in helium cooled solid breeder blanket units

##
High Rayleigh Number Convection

##
Confined turbulent convection

##
Absence of Evidence for the Ultimate Regime in Two-Dimensional Rayleigh-Bénard Convection

##
Turbulent Rayleigh–Bénard convection in gaseous and liquid He

##
Predicting transition ranges to fully turbulent viscous boundary layers in low Prandtl number convection flows

##
Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection

##
Convection in Stars I. Basic Boussinesq Convection

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Figures / Tables found in this record:

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*