# Regimes of Axisymmetric Flow and Scaling Laws in a Rotating Annulus with Local Convective Forcing

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. The Numerical Model

#### 2.1. Dimensionless Parameters

#### 2.2. The Numerical Model

## 3. Results

#### 3.1. Non-Rotating Flow

#### 3.2. Rotating Flow Regimes

- No rotation ($P=0$): A density current flows beneath the heat sink, down the side of the inner cylinder and along the base towards the heat source. A corresponding density current is not seen above the heat source, presumably because of the larger area of the hot plate. At high z and for $r\gtrsim $ 10 cm the fluid interior is approximately isothermal at a temperature close to that of the hot plate. Since all fields are time averaged the plumes observed in Figure 3 are not seen here. The azimuthal velocity is zero everywhere.
- Weak rotation ($0.05\lesssim P\lesssim 0.5$): The density current is reduced and replaced by a more uniform thermal gradient. Close to the inner and outer cylinders the isotherms are approximately vertical and confined to Stewartson layers which are a few centimetres thick and on the vertical boundaries. The azimuthal velocity is now non-zero and takes a maximum value near the top of the tank, but outside the boundary layers, close to the inner cylinder. As the rotation increases this region of maximal velocity becomes more confined towards the top of the tank. Close to the bottom of the tank at small radii retrograde motion begins to develop. Using streaklines to visualise the flow, Scolan and Read [18] showed that for weak rotation the flow in the experiment remains approximately axisymmetric.
- Moderate rotation ($1\lesssim P\lesssim 10$): For moderate rotation P is of order unity and thus the thicknesses of the thermal boundary layer and the Ekman layer are comparable. Free convection results in well mixed, approximately isothermal regions above and below the heat source and sink respectively. Sandwiched between these two convective zones is a baroclinic region with approximately uniformly sloping isotherms. This thermal structure has also been observed in experiment as shown by Scolan and Read [18] (Figure 6 of that paper). As before, prograde azimuthal velocity is seen close to the top of the tank and retrograde motion near the bottom with the most intense movement occuring at small radii. The azimuthal velocity begins to transition towards geostrophic balance as the rotation rate increases.
- Strong Rotation ($P\gtrsim 20$): The Ekman layer thickness is less than that of the thermal boundary layer and thus the radial transport becomes inhibited. The suppression of vertical convection results in the replacement of the well mixed regions by statically unstable temperature gradients. The isotherms in the central baroclinic region steepen and, for the highest value of P, are seen to surpass the vertical and slope in the opposite direction. The flow fields approach those expected for a purely conducting sample as the rotation organises the flow via the Taylor Proudman effect and so mixing is suppressed [51]. The azimuthal velocity is now approximately zero in the regions directly over/under the heat source/sink but in the central zone an azimuthal flow which follows the applied rotation at the top of the tank and goes against it at the bottom is observed. This induces a thermal wind shear.

#### 3.3. Heat Transfer

#### 3.4. The Azimuthal Velocity Scale

## 4. Conclusion

## Supplementary Materials

^{−1}(same as Figure 2) and S2 has $\mathsf{\Omega}=1\mathrm{rad}$· s

^{−1}.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**(

**a**) Schematic of a whole hemispheric atmosphere. (

**b**) Vertical cross-section through the new rotating annulus laboratory experiment with thermal forcing at top and bottom. $\mathsf{\Omega}$ is the rotation vector and g the gravity vector. The dark blue regions represent the heat sink and the red regions represent the heat source.

**Figure 2.**The time series of total power (W) for $\Delta T=10\phantom{\rule{3.33333pt}{0ex}}$°C, $\mathsf{\Omega}=1\phantom{\rule{3.33333pt}{0ex}}$ rad$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$s${}^{-1}$ and $d=24.4$ cm. The red line shows the power dissipated by the heat sink and the black line the power input by the heat source, these converge after a stabilisation time of about 45,000 s.

**Figure 3.**The instantaneous temperature field after 100 s of simulation time for a run with $\Delta T=10\phantom{\rule{3.33333pt}{0ex}}$°C, $d=12.2$ cm and $\mathsf{\Omega}=0.001$ rad$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$s${}^{-1}$. Plumes above the heat source and below the heat sink are resolved. Supplementary Material movie S1 animates the first 250 s of this simulation.

**Figure 4.**The dependence of the Nusselt number for the non-rotating system, $N{u}_{0}$, on the Rayleigh number, $Ra$ with (

**a**) compensation by a $R{a}^{2/7}$ scaling law, (

**b**) compensation by a $R{a}^{1/3}$ scaling law. Darker colours represent higher applied temperature differences, ∘ for $\Delta T=0.5\phantom{\rule{3.33333pt}{0ex}}$°C, • for $\Delta T=1\phantom{\rule{3.33333pt}{0ex}}$°C, • for $\Delta T=2\phantom{\rule{3.33333pt}{0ex}}$°C, • for $\Delta T=4$ ° C and • for $\Delta T=10\phantom{\rule{3.33333pt}{0ex}}$°C. Shapes represent different aspect ratios, circles for $\Gamma =3.8$ whilst $\Gamma =1.9$ and $\Gamma =1.3$ are represented by squares and triangles respectively.

**Figure 5.**(

**a**) r,z cross-sections of the equilibrated, time averaged temperature field for $\Delta T=0.5\phantom{\rule{3.33333pt}{0ex}}$°C, $d=12.2$ cm and the full range of rotation rates. The value of P for each run is shown on the left hand side. These correspond to rotation rates of 0, 0.001, 0.01, 0.05, 0.1, 0.5, 1 and 2 rad$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$s${}^{-1}$; (

**b**) r,z cross-sections of the equilibrated, time averaged azimuthal velocity fields for the same runs.

**Figure 6.**The isotherm gradient, normalised by the gradient of the conductive isotherms, as a function of P. Lower P data is not included as the interior of the annulus is approximately isothermal for low rotation and so the isotherm slope is not well defined. The dashed line shows the predicted dependence of the isotherm gradient on P and has a gradient of 3/2. Symbols have the same meaning as in Figure 4.

**Figure 7.**The reduced Nusselt number $(Nu-1)N{u}_{0}^{-1}$ plotted against P. The dashed line has a slope of −3/2 which can be compared with the slope of the datapoints. The Nusselt number is expected to decrease at this rate for high P. The dashed-dotted line has a gradient of −1/3, this rate of decrease in the Nusselt number may be expected if Stewartson layers were throttling the flow. The vertical dotted line marks the point where the Ekman layer and thermal boundary layer have approximately equal thickness at $P=1$. The symbols are the same as in Figure 4.

**Figure 8.**The azimuthal velocity scale, scaled following Read [16], as a function of P. The dashed-dotted line has a gradient of 1 and thus it is clear that the azimuthal velocity scale increases approximately linearly with P at low P. The dashed line has a gradient of −1 showing that for $P>1$ the velocity scale tends to the thermal wind scale. Symbols have the same meaning as in Figure 4.

Parameter | Symbol | Present range | Units |
---|---|---|---|

Rotation rate | $\mathsf{\Omega}$ | 0–2 | rad$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$s${}^{-1}$ |

Temperature difference | $\Delta T$ | 0.5–10 | K |

Fluid properties are listed at 20 ${}^{\circ}\mathrm{C}$: | |||

Density | $\rho $ | 1044 | kg$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$m${}^{-3}$ |

Thermal expansion coefficient | ${\alpha}_{v}$ | $2.76\times {10}^{-4}$ | K${}^{-1}$ |

Kinematic viscosity | $\nu $ | 1.71 $\times {10}^{-6}$ | m${}^{2}$$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$s${}^{-1}$ |

Thermal diffusivity | $\kappa $ | 1.28 $\times {10}^{-7}$ | m${}^{2}$$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$s${}^{-1}$ |

Channel geometry: | |||

Inner radius | a | 0.025 | m |

Outer radius | b | 0.488 | m |

Mean fluid depth | d | 0.122, 0.244, 0.366 | m |

Non-dimensional: | |||

Ekman number (Equation (2)) | $Ek$ | $>6\times {10}^{-6}$ | |

Prandtl number | $Pr$ | 12.6 | |

Rayleigh number (Equation (3)) | $Ra$ | $3.3\times {10}^{9}$ – $6.7\times {10}^{10}$ | |

Aspect ratio | $\Gamma $ | 3.8, 1.89, 1.27 |

**Table 2.**Least squares fit parameters for the non-rotating Nusselt number vs. the Rayleigh number. $\alpha $ is the exponent in the power law $N{u}_{0}\sim R{a}^{\alpha}$.

Aspect Ratio, $\mathsf{\Gamma}$ | $\alpha $ |
---|---|

1.3 | $0.301\pm 0.003$ |

1.9 | $0.297\pm 0.002$ |

3.8 | $0.34\pm 0.01$ |

All | $0.329\pm 0.018$ |

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**MDPI and ACS Style**

Wright, S.; Su, S.; Scolan, H.; Young, R.M.B.; Read, P.L. Regimes of Axisymmetric Flow and Scaling Laws in a Rotating Annulus with Local Convective Forcing. *Fluids* **2017**, *2*, 41.
https://doi.org/10.3390/fluids2030041

**AMA Style**

Wright S, Su S, Scolan H, Young RMB, Read PL. Regimes of Axisymmetric Flow and Scaling Laws in a Rotating Annulus with Local Convective Forcing. *Fluids*. 2017; 2(3):41.
https://doi.org/10.3390/fluids2030041

**Chicago/Turabian Style**

Wright, Susie, Sylvie Su, Hélène Scolan, Roland M. B. Young, and Peter L. Read. 2017. "Regimes of Axisymmetric Flow and Scaling Laws in a Rotating Annulus with Local Convective Forcing" *Fluids* 2, no. 3: 41.
https://doi.org/10.3390/fluids2030041