# Buoyancy-Induced Heat Transfer inside Compressor Rotors: Overview of Theoretical Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Buoyancy-Induced Rotating Flow

## 3. Heat Transfer from Shrouds

#### 3.1. Calculation of Nusselt Numbers

#### 3.2. Maximum Nusselt Number

#### 3.3. Comparison with Experimental Measurements

## 4. Heat Transfer from Discs

#### 4.1. Assumptions for Buoyancy Model

#### 4.2. Modelled Nusselt Numbers

#### 4.3. Modelled Disc Temperatures

#### 4.4. Experimentally Derived Nusselt Numbers and Disc Temperatures

#### 4.5. Comparison with Experimental Measurements

## 5. Buoyancy-Induced Heat Transfer inside Compressor Rotors

#### 5.1. Model of Heat Transfer from Shroud to Core

#### 5.2. Model of Heat Transfer from Cob to Axial Throughflow

#### 5.3. Model of Temperature Rise of Axial Throughflow

#### 5.4. Comparison between Theoretical and Experimental Values

#### 5.4.1. Disc Nusselt Numbers and Temperatures

#### 5.4.2. Temperature Rise of Throughflow

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$a$ | inner radius |

${a}^{\prime}$ | inner radius of outer edge of cob |

$b$ | outer radius |

$c$ | empirical constant |

$C$ | empirical constant |

${C}_{p}$ | specific heat capacity at constant pressure |

$\mathrm{Co}$ | Coriolis parameter (see Equation (30)) |

${d}_{h}$ | hydraulic diameter ($=2\left(a-{r}_{s}\right)$) |

$\mathrm{Gr}$ | Grashof number for closed cavity ($=\frac{{\rho}^{2}{\mathsf{\Omega}}^{2}\left(a+b\right)/2{L}^{3}}{{\mu}^{2}}\frac{\mathsf{\Delta}T}{\left({T}_{a}+{T}_{b}\right)/2}$) |

${\mathrm{Gr}}_{\mathrm{a}}$ | Grashof number of inner surface of closed cavity (see Equation (12)) |

${\mathrm{Gr}}_{\mathrm{b}}$ | Grashof number of outer surface (shroud) of closed cavity (see Equation (13)) |

${\mathrm{Gr}}_{\mathrm{c}}$ | Grashof number in theory (see Equation (29)) |

${\mathrm{Gr}}_{\mathrm{f}}$ | Grashof number in experiment ($={\left(1-a/b\right)}^{3}{\mathrm{Re}}_{\mathsf{\varphi}}^{2}\beta \left({T}_{o,b}-{T}_{f}\right)$) |

${\mathrm{Gr}}_{\mathrm{sh}}$ | shroud Grashof number (see Equation (36)) |

$h$ | heat transfer coefficient |

${h}_{c}$ | heat transfer coefficient based on ${T}_{c}$(see Equation (26)) |

${h}_{f}$ | heat transfer coefficient based on ${T}_{f}$($={q}_{o}/\left({T}_{o}-{T}_{f}\right)$) |

${h}_{sh}$ | shroud heat transfer coefficient ($={q}_{sh}/\left({T}_{sh}-{T}_{c,b}\right)$) |

$i$ | disc number |

$I$ | integral (see Equation (31)) |

$k$ | thermal conductivity of air |

${k}_{s}$ | thermal conductivity of disc |

$l$ | axial length of cob |

$L$ | characteristic length |

${L}_{a}$ | characteristic length of inner surface of closed cavity |

${L}_{b}$ | characteristic length of outer surface of closed cavity |

${\dot{m}}_{f}$ | mass flow rate of axial throughflow (kg/s) |

$M{a}_{c}$ | Mach number in core (see Equation (A18)) |

${N}_{disc}$ | rotational speed of disc |

${N}_{shaft}$ | rotational speed of inner shaft |

$\mathrm{Nu}$ | Nusselt number of closed cavity (see Equation (8)) |

${\mathrm{Nu}}_{\mathrm{a}}$ | Nusselt number of inner surface of closed cavity (see Equation (10)) |

${\mathrm{Nu}}_{\mathrm{b}}$ | Nusselt number of outer surface of closed cavity (see Equation (11)) |

${\mathrm{Nu}}_{\mathrm{c}}$ | Nusselt number based on ${h}_{c}$ (see Equation (25)) |

${\mathrm{Nu}}_{\mathrm{cob}}$ | cob Nusselt number (see Equation (40)) |

${\mathrm{Nu}}_{\mathrm{f}}$ | Nusselt number based on ${h}_{f}$ ($={h}_{f}r/{k}_{f}$) |

${\mathrm{Nu}}_{\mathrm{sh}}$ | shroud Nusselt number (see Equation (35)) |

$p$ | static pressure |

$P$ | reduced pressure |

$\mathrm{Pr}$ | Prandtl number |

${q}_{a}$ | heat flux from inner surface of closed cavity to air |

${q}_{b}$ | heat flux from outer surface of closed cavity to air |

${q}_{cob}$ | heat flux from cob to air |

${q}_{o}$ | heat flux from disc to air |

${q}_{sh}$ | heat flux from shroud to air |

$\dot{Q}$ | heat flow rate |

${\dot{Q}}_{a}$ | heat flow rate from inner surface of closed cavity to air |

${\dot{Q}}_{b}$ | heat flow rate from outer surface of closed cavity to air |

${\dot{Q}}_{cob}$ | heat flow rate from cob to air |

${\dot{Q}}_{cond}$ | heat flow rate due to conduction |

${\dot{Q}}_{d}$ | heat flow rate from downstream disc surface to air |

${\dot{Q}}_{u}$ | heat flow rate from upstream disc surface to air |

${\dot{Q}}_{sh}$ | heat flow rate from shroud to air |

$r$ | radius |

${r}_{s}$ | radius of inner shaft |

$R$ | gas constant |

$\mathrm{Ra}$ | Rayleigh number for close cavity ($=\mathrm{Pr}\text{}\mathrm{Gr}$) |

${\mathrm{Ra}}_{\mathrm{crit}}$ | critical Ra where $\mathrm{Nu}$ reaches maximum |

${\mathrm{Re}}_{\mathrm{crit}}$ | critical Re where $\mathrm{Nu}$ reaches maximum |

${\mathrm{Re}}_{\mathrm{T}}$ | Reynolds number for cob (see Equation (41)) |

${\mathrm{Re}}_{\mathrm{z}}$ | axial Reynolds number ($={\rho}_{f}W{d}_{h}/{\mu}_{f}$) |

${\mathrm{Re}}_{\mathsf{\varphi}}$ | rotational Reynolds number based on ${\mathsf{\rho}}_{\mathrm{f}}$($={\rho}_{f}\mathsf{\Omega}{b}^{2}/{\mu}_{f}$) |

$\mathrm{Ro}$ | Rossby number ($=W/\mathsf{\Omega}a$) |

$s$ | axial space between discs in cavity |

$S$ | area of convection surface |

${S}_{a}$ | area of inner surface of closed cavity |

${S}_{b}$ | area of outer surface of closed cavity |

$t$ | disc thickness |

$T$ | static temperature |

${T}_{c},{T}_{f},{T}_{o},{T}_{sh}$ | temperature of core, throughflow, disc, shroud |

$u,v,w$ | circumferential, radial, axial component of velocity in rotating frame |

$U$ | resultant velocity ($=\sqrt{{W}^{2}+{\left(\mathsf{\Omega}a\right)}^{2}}$) |

$V$ | speed of sound in core ($=\sqrt{\gamma R{T}_{c,a}}$) |

$W$ | axial component of velocity of throughflow |

$x$ | nondimensional radius ($=r/b$) |

${x}_{a}$ | radius ratio ($=a/b$) |

$\alpha $ | angle of gradient of disc surface |

$\beta $ | volume expansion coefficient $(=1/{T}_{ref}$) |

$\mathsf{\Delta}T$ | temperature difference between outer surface and inner surface of closed cavity |

$\mathsf{\Delta}{T}_{a}$ | temperature difference between core ($r=a$) and inner surface of closed cavity |

$\mathsf{\Delta}{T}_{b}$ | temperature difference between core ($r=b$) and outer surface of closed cavity |

$\mathsf{\Delta}{T}_{c}$ | temperature difference between core ($r=b$) and core ($r=a$) |

$\mathsf{\Delta}{T}_{f}$ | temperature rise of axial throughflow |

$\mathsf{\Delta}{\Theta}_{f}$ | nondimensional temperature rise of axial throughflow |

$\gamma $ | ratio of specific heats |

$\theta $ | nondimensional temperature (see Equation (28)) |

$\Theta $ | nondimensional disc temperature ($=\left({T}_{o}-{T}_{f}\right)/\left({T}_{o,b}-{T}_{f}\right)$) |

$\mu $ | dynamic viscosity |

$\xi $ | geometric parameter for close cavity(see Equation (18)) |

$\rho $ | density |

$\mathsf{\varphi},r,z$ | circumferential, radial, axial coordinates |

$\chi $ | compressibility parameter for close cavity (see Equation (19)) |

$\psi $ | compressibility parameter for close cavity(see Equation (20)) |

$\mathsf{\Omega},{\mathsf{\Omega}}_{\mathrm{c}}$ | angular speed of disc, core |

Subscripts | |

$a$ | value at $r=a$ |

$av$ | radially-weighted average value |

$A$ | cavity A from [9] |

$b$ | value at $r=b$ |

$c$ | value in core |

$cob$ | value on cob |

$cond$ | conduction |

$crit$ | critical |

$d$ | value on downstream disc surface |

$exp$ | Experimentally derived value |

$f$ | value in axial throughflow |

$i$ | values for the ${i}^{\mathrm{th}}$ disc in a multi-cavity system |

$o$ | value on disc surface |

$ref$ | reference value |

$sh$ | value on shroud |

$th$ | theoretical or modelled value |

$u$ | value on upstream disc surface |

$\mathsf{\varphi},r,z$ | circumferential, radial, axial direction |

## Appendix

#### Linear Equations for Inviscid Rotating Fluids

#### Compressible Adiabatic Flow in Rotating Core

## References

- Owen, J.M.; Long, C.A. Review of buoyancy-induced flow in rotating cavities. J. Turbomach.
**2015**, 137, 111001. [Google Scholar] [CrossRef] - Tang, H.; Owen, J.M. Theoretical model of buoyancy-induced heat transfer in closed compressor rotors. J. Eng. Gas Turbines Power
**2018**, 140, 032605. [Google Scholar] [CrossRef] - Owen, J.M.; Tang, H. Theoretical model of buoyancy-induced flow in rotating cavities. J. Turbomach.
**2015**, 137, 111005. [Google Scholar] [CrossRef] - Tang, H.; Owen, J.M. Effect of buoyancy-induced rotating flow on temperature of compressor discs. J. Eng. Gas Turbines Power
**2017**, 139, 062506. [Google Scholar] [CrossRef] - Tang, H.; Shardlow, T.; Owen, J.M. Use of fin equation to calculate nusselt numbers for rotating discs. J. Turbomach.
**2015**, 137, 121003. [Google Scholar] [CrossRef] - Tang, H.; Puttock, M.; Owen, J.M. Buoyancy-induced flow and heat transfer in compressor rotors. J. Eng. Gas Turbines Power
**2017**, in press. [Google Scholar] [CrossRef] - Farthing, P.R.; Long, C.A.; Owen, J.M.; Pincombe, J.R. Rotating cavity with axial throughflow of cooling air: Flow structure. J. Turbomach.
**1992**, 114, 237–246. [Google Scholar] [CrossRef] - Lloyd, J.R.; Moran, W.R. Natural convection adjacent to horizontal surface of various planforms. J. Heat Transf.
**1974**, 96, 443–447. [Google Scholar] [CrossRef] - Bohn, D.; Deuker, E.; Emunds, R.; Gorzelitz, V. Experimental and theoretical investigations of heat transfer in closed gas-filled rotating annuli. J. Turbomach.
**1995**, 117, 175–183. [Google Scholar] [CrossRef] - Owen, J.M.; Pincombe, J.R.; Rogers, R.H. Source–sink flow inside a rotating cylindrical cavity. J. Fluid Mech.
**1985**, 155, 233–265. [Google Scholar] [CrossRef] - Owen, J.M.; Pincombe, J.R. Vortex breakdown in a rotating cylindrical cavity. J. Fluid Mech.
**1979**, 90, 109–127. [Google Scholar] [CrossRef] - Atkins, N.R.; Kanjirakkad, V. Flow in a rotating cavity with axial throughflow at engine representative conditions. In Proceedings of the ASME Turbo Expo 2014: Turbine Technical Conference and Exposition, Düsseldorf, Germany, 16–20 June 2014. No. GT2014-27174. [Google Scholar] [CrossRef]
- Long, C.A.; Childs, P.R.N. Shroud heat transfer measurements inside a heated multiple rotating cavity with axial throughflow. Int. J. Heat Fluid Flow
**2007**, 28, 1405–1417. [Google Scholar] [CrossRef] - Atkins, N.R. Investigation of a radial-inflow bleed as a potential for compressor clearance control. In Proceedings of the ASME Turbo Expo 2013: Turbine Technical Conference and Exposition, San Antonio, TX, USA, 3–7 June 2013. No. GT2013-95768, V03AT15A020. [Google Scholar] [CrossRef]
- Bohn, D.; Deutsch, G.; Simon, B.; Burkhardt, C. Flow visualisation in a rotating cavity with axial throughflow. In Proceedings of the ASME Turbo Expo 2000: Power for Land, Sea, and Air, Munich, Germany, 8–11 May 2000. GT2000-280. [Google Scholar] [CrossRef]
- Bohn, D.; Dibelius, G.H.; Deuker, E.; Emunds, R. Flow pattern and heat transfer in a closed rotating annulus. J. Turbomach.
**1994**, 116, 542–547. [Google Scholar] [CrossRef] - Bohn, D.; Edmunds, R.; Gorzelitz, V.; Kruger, U. Experimental and theoretical investigations of heat transfer in closed gas-filled rotating annuli II. J. Turbomach.
**1996**, 118, 11–19. [Google Scholar] [CrossRef] - Bohn, D.; Gier, J. The effect of turbulence on the heat transfer in closed gas-filled rotating annuli. J. Turbomach.
**1998**, 120, 824–830. [Google Scholar] [CrossRef] - Bohn, D.; Ren, J.; Tuemmers, C. Investigation of the unstable flow structure in a rotating cavity. In Proceedings of the ASME Turbo Expo 2006: Power for Land, Sea, and Air, Barcelona, Spain, 8–11 May 2006. GT2006-90494. [Google Scholar] [CrossRef]
- Childs, P.R.N. Rotating Flow; Elsevier: Oxford, UK, 2011; ISBN 978-0-123-82098-3. [Google Scholar]
- Dweik, Z.; Briley, R.; Swafford, T.; Hunt, B. Computational study of the heat transfer of the buoyancy-driven rotating cavity with axial throughflow of cooling air. In Proceedings of the ASME Turbo Expo 2009: Power for Land, Sea, and Air, Orlando, FL, USA, 8–12 June 2009. GT2009-59978. [Google Scholar] [CrossRef]
- Farthing, P.R.; Long, C.A.; Owen, J.M.; Pincombe, J.R. Rotating cavity with axial throughflow of cooling air: Heat transfer. J. Turbomach.
**1992**, 114, 229–236. [Google Scholar] [CrossRef] - Gunther, A.; Uffrecht, W.; Odenbach, S. Local measurements of disk heat transfer in heated rotating cavities for several flow regimes. J. Turbomach.
**2012**, 134, 051016. [Google Scholar] [CrossRef] - Gunther, A.; Uffrecht, W.; Odenbach, S. The effects of rotation and mass flow on local heat transfer in rotating cavities with axial throughflow. In Proceedings of the ASME Turbo Expo 2014: Turbine Technical Conference and Exposition, Düsseldorf, Germany, 16–20 June 2014. GT2014-26228. [Google Scholar] [CrossRef]
- He, L. Efficient computational model for nonaxisymmetric flow and heat transfer in rotating cavity. J. Turbomach.
**2011**, 133, 021018. [Google Scholar] [CrossRef] - Hollands, K.G.T.; Raithby, G.D.; Konicek, L. Correlation Equations for Free Convection Heat Transfer in Horizontal Layers of Air and Water. Int. J. Heat Mass Transf.
**1975**, 18, 879–884. [Google Scholar] [CrossRef] - Iacovides, H.; Chew, J.W. The computation of convective heat transfer in rotating cavities. Int. J. Heat Fluid Flow
**1993**, 14, 146–154. [Google Scholar] [CrossRef] - Johnson, B.V.; Lin, J.D.; Daniels, W.A.; Paolillo, R. Flow characteristics and stability analysis of variable-density rotating flows in compressor-disk cavities. J. Eng. Gas Turbines Power
**2006**, 128, 118–127. [Google Scholar] [CrossRef] - King, M.P. Convective Heat Transfer in a Rotating Annulus. Ph.D. Thesis, University of Bath, Bath, UK, 2003. [Google Scholar]
- King, M.P.; Wilson, M. Free convective heat transfer within rotating annuli. In Proceedings of the 12th International Heat Transfer Conference, Grenoble, France, 18–23 August 2002; Volume 2, pp. 465–470. [Google Scholar]
- King, M.P.; Wilson, M. Numerical simulations of convective heat transfer in Rayleigh-Benard convection and a rotating annulus. Numer. Heat Transf. Part A
**2005**, 48, 529–545. [Google Scholar] [CrossRef] - King, M.P.; Wilson, M.; Owen, J.M. Rayleigh-Benard convection in open and closed rotating cavities. J. Eng. Gas Turbines Power
**2007**, 129, 305–311. [Google Scholar] [CrossRef] - Kumar, V.B.G.; Chew, J.W.; Hills, N.J. Rotating flow and heat transfer in cylindrical cavities with radial inflow. J. Eng. Gas Turbines Power
**2013**, 135, 032502. [Google Scholar] [CrossRef] - Lewis, T.W. Numerical Simulation of Buoyancy-Induced Flow in a Sealed Rotating Cavity. Ph.D. Thesis, University of Bath, Bath, UK, 1999. [Google Scholar]
- Long, C.A. Disk heat transfer in a rotating cavity with an axial throughflow of cooling air. Int. J. Heat Fluid Flow
**1994**, 15, 307–316. [Google Scholar] [CrossRef] - Long, C.A.; Alexiou, A.; Smout, P.D. Heat Transfer in H.P. Compressor Internal Air Systems: Measurements from the Peripheral Shroud of a Rotating Cavity with Axial Throughflow. In Proceedings of the 2nd International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics (HEFAT 2003), Victoria Falls, Zambia, 23–25 June 2003. Paper No. LC1. [Google Scholar]
- Long, C.A.; Miche, N.D.D.; Childs, P.R.N. Flow measurements inside a heated multiple rotating cavity with axial throughflow. Int. J. Heat Fluid Flow
**2007**, 28, 1391–1404. [Google Scholar] [CrossRef] - Long, C.A.; Morse, A.P.; Tucker, P.G. Measurement and computation of heat transfer in high pressure compressor drum geometries with axial throughflow. J. Turbomach.
**1997**, 119, 51–60. [Google Scholar] [CrossRef] - Long, C.A.; Tucker, P.G. Numerical computation of laminar flow in a heated rotating cavity with an axial throughflow of air. Int. J. Numer. Methods Heat Fluid Flow
**1994**, 4, 347–365. [Google Scholar] [CrossRef] - Long, C.A.; Tucker, P.G. Shroud heat transfer measurements from a rotating cavity with an axial throughflow of air. J. Turbomach.
**1994**, 116, 525–534. [Google Scholar] [CrossRef] - Niemela, J.J.; Skrbek, L.; Sreenivasan, K.R.; Donnelly, R.J. Turbulent Convection at Very High Rayleigh Numbers. Nature
**2000**, 404, 837–840. [Google Scholar] [CrossRef] [PubMed] - Owen, J.M. Thermodynamic analysis of buoyancy-induced flow in rotating cavities. J. Turbomach.
**2010**, 132, 031006. [Google Scholar] [CrossRef] - Owen, J.M.; Abrahamsson, H.; Linblad, K. Buoyancy-induced flow in open rotating cavities. J. Eng. Gas Turbines Power
**2007**, 129, 893–900. [Google Scholar] [CrossRef] - Owen, J.M.; Powell, J. Buoyancy-induced flow in a heated rotating cavity. J. Eng. Gas Turbines Power
**2006**, 128, 128–134. [Google Scholar] [CrossRef] - Owen, J.M.; Rogers, R.H. Flow and Heat Transfer in Rotating Disc Systems, Volume 2—Rotating Cavities; Research Studies Press: Baldock, UK; John Wiley: New York, NY, USA, 1995; ISBN 086380179X. [Google Scholar]
- Pitz, D.B.; Marxen, O.; Chew, J.W. Onset of convection induced by centrifugal buoyancy in a rotating cavity. J. Fluid Mech.
**2017**, 826, 484–502. [Google Scholar] [CrossRef] - Shevchuk, I.V. Convective Heat and Mass Transfer in Rotating Disc Systems; Springer: Heidelberg, Germany, 2009; ISBN 978-3-642-00717-0. [Google Scholar]
- Sun, X.; Kilfoil, A.; Chew, J.W.; Hills, N.J. Numerical simulation of natural convection in stationary and rotating cavities. In Proceedings of the ASME Turbo Expo 2004: Power for Land, Sea, and Air, Vienna, Austria, 14–17 June 2004. GT2004-53528. [Google Scholar] [CrossRef]
- Sun, X.; Linblad, K.; Chew, J.W.; Young, C. LES and RANS investigations into buoyancy-affected convection in a rotating cavity with a central axial throughflow. J. Eng. Gas Turbines Power
**2007**, 129, 318–325. [Google Scholar] [CrossRef] - Tan, Q.; Ren, J.; Jiang, H. Prediction of flow features in rotating cavities with axial throughflow by RANS and LES. In Proceedings of the ASME Turbo Expo 2009: Power for Land, Sea, and Air, Orlando, FL, USA, 8–12 June 2009. GT2009-59428. [Google Scholar] [CrossRef]
- Tan, Q.; Ren, J.; Jiang, H. Prediction of 3D unsteady flow and heat transfer in rotating cavity by discontinuous Galerkin method and transition model. In Proceedings of the ASME Turbo Expo 2014: Turbine Technical Conference and Exposition, Düsseldorf, Germany, 16–20 June 2014. GT2014-26584. [Google Scholar] [CrossRef]
- Tian, S.; Tao, Z.; Ding, S.; Xu, G. Investigation of flow and heat transfer in a rotating cavity with axial throughflow of cooling air. In Proceedings of the ASME Turbo Expo 2004: Power for Land, Sea, and Air, Vienna, Austria, 14–17 June 2004. GT2004-53525. [Google Scholar] [CrossRef]
- Tritton, D.J. Physical Fluid Dynamics; OUP: New York, NY, USA, 1988; ISBN 0-19-854489-8. [Google Scholar]
- Tucker, P.G. Temporal behaviour of flow in rotating cavities. Numer. Heat Transf. Part A
**2002**, 41, 611–627. [Google Scholar] [CrossRef]

**Figure 2.**Schematic of flow structure in heated rotating cavity with axial throughflow of cooling air [7].

**Figure 3.**Simplified diagram of temperature distribution inside closed cavity [2].

**Figure 6.**Simplified diagram of instrumented disc of Sussex rig (Dimensions in mm) [4].

**Figure 7.**Distributions of temperature and Nusselt numbers for Ro ≈ 0.6. (Symbols denote measured temperatures; broken and solid lines represent experimental and theoretical results respectively; shading shows 95% confidence intervals on experimental Nusselt numbers) [4].

**Figure 8.**Distributions of temperature and Nusselt numbers for Ro ≈ 0.3. (Symbols denote measured temperatures; broken and solid lines represent experimental and theoretical results respectively; shading shows 95% confidence intervals on experimental Nusselt numbers) [4].

**Figure 9.**Control volume for temperature rise of axial throughflow [6].

**Figure 10.**Distributions of nondimensional temperature and Nusselt numbers for Case a ($\mathrm{Ro}\approx 0.6,{\text{}\mathrm{Gr}}_{\mathrm{f}}=2.5\times {10}^{11},{\text{}\mathrm{Re}}_{\mathsf{\varphi}}=1.6\times {10}^{6},{\mathrm{Re}}_{\mathrm{z}}=5.1\times {10}^{4}$) Solid lines are theoretical curves; broken lines are results from Bayesian model, and shading shows uncertainty in ${\mathrm{Nu}}_{\mathrm{f}}$; symbols denote temperature [6].

**Figure 11.**Distributions of nondimensional temperature and Nusselt numbers for Case c1 ($\mathrm{Ro}\approx 0.2,{\text{}\mathrm{Gr}}_{\mathrm{f}}=4.2\times {10}^{11},\text{}{\mathrm{Re}}_{\mathsf{\varphi}}=3.0\times {10}^{6},\text{}{\mathrm{Re}}_{\mathrm{z}}=2.5\times {10}^{4}$). Solid lines are theoretical curves; broken lines are results from Bayesian model, and shading shows uncertainty in Nu; symbols denote temperature measurements [6].

**Figure 12.**Comparison between modelled and experimental nondimensional temperature rise of axial throughflow [6].

**Table 1.**Flow parameters and average Nusselt numbers for experiments of Atkins and Kanjirakkad [4].

Cases | Ro ≈ 5 | Ro ≈ 1 | Ro ≈ 0.6 | Ro ≈ 0.3 | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1a | 1b | 1c | 1d | 1e | 1f | 2a | 2b | 2c | 2d | 2e | 2f | 2g | 3a | 3b | 4a | 4b | 4c | 4d | |

$\mathrm{Ro}$ | 4.7 | 4.7 | 4.9 | 4.9 | 4.5 | 4.5 | 0.8 | 0.8 | 0.9 | 0.9 | 1.0 | 1.0 | 1.0 | 0.6 | 0.6 | 0.3 | 0.3 | 0.3 | 0.3 |

${\mathrm{Gr}}_{\mathrm{f}}/{10}^{11}$ | 0.0017 | 0.0030 | 0.0085 | 0.015 | 0.062 | 0.14 | 0.065 | 0.10 | 0.44 | 0.84 | 1.7 | 2.5 | 3.9 | 5.7 | 9.1 | 0.4 | 1.0 | 3.7 | 7.8 |

${\mathrm{Re}}_{\mathsf{\varphi}}/{10}^{6}$ | 0.078 | 0.077 | 0.19 | 0.19 | 0.46 | 0.45 | 0.46 | 0.45 | 1.1 | 1.1 | 2.1 | 2.1 | 2.1 | 3.5 | 3.1 | 1.4 | 1.4 | 3.1 | 3.0 |

$\mathsf{\beta}\mathsf{\Delta}\mathrm{T}$ | 0.08 | 0.17 | 0.05 | 0.11 | 0.09 | 0.24 | 0.09 | 0.16 | 0.11 | 0.23 | 0.13 | 0.19 | 0.32 | 0.15 | 0.32 | 0.06 | 0.16 | 0.12 | 0.29 |

${\mathrm{Re}}_{\mathrm{z}}/{10}^{5}$ | 0.19 | 0.19 | 0.50 | 0.50 | 1.1 | 1.1 | 0.19 | 0.18 | 0.51 | 0.50 | 1.1 | 1.1 | 1.1 | 1.1 | 1.1 | 0.20 | 0.20 | 0.48 | 0.48 |

${\mathrm{Nu}}_{\mathrm{av},\mathrm{exp}}$ | 28.7 | 30.7 | 51.6 | 58.2 | 92.3 | 109 | 47.4 | 53.4 | 96.7 | 126 | 131 | 170 | 233 | 126 | 225 | 45.3 | 82.5 | 72.1 | 146 |

${\mathrm{Nu}}_{\mathrm{av},\mathrm{th}}$ | 32.5 | 37.2 | 56.2 | 66.5 | 103 | 128 | 59.4 | 64.7 | 109 | 135 | 149 | 175 | 216 | 133 | 214 | 50.5 | 83.5 | 78.0 | 142 |

**Table 2.**Flow parameters and experimental and theoretical average disc temperatures [6].

Case | Ro ≈ 0.6 | Ro ≈ 0.3 | Ro ≈ 0.2 | Ro ≈ 0.1 | ||||||
---|---|---|---|---|---|---|---|---|---|---|

a | b1 | b2 | c1 | c2 | c3 | c4 | c5 | c6 | d | |

Ro | 0.61 | 0.31 | 0.31 | 0.16 | 0.16 | 0.18 | 0.17 | 0.17 | 0.17 | 0.10 |

Gr_{f}/10^{11} | 2.5 | 10 | 10 | 4.2 | 4.5 | 6.8 | 7.2 | 7.2 | 7.8 | 4.1 |

$\beta \left({T}_{b}-{T}_{f}\right)$ | 0.32 | 0.35 | 0.35 | 0.15 | 0.16 | 0.33 | 0.32 | 0.32 | 0.34 | 0.29 |

Re_{ϕ}/10^{6} | 1.6 | 3.0 | 3.0 | 3.0 | 3.0 | 2.5 | 2.7 | 2.7 | 2.7 | 2.1 |

Re_{z}/10^{4} | 5.1 | 5.0 | 5.0 | 2.5 | 2.5 | 2.4 | 2.4 | 2.4 | 2.5 | 1.1 |

${N}_{shaft}/{N}_{disc}$ | 1.0 | 0.34 | 1.0 | 0.34 | −0.34 | 1.0 | 0.34 | −0.34 | 0 | 0 |

${\Theta}_{av,exp}$ | 0.347 | 0.349 | 0.330 | 0.499 | 0.494 | 0.357 | 0.377 | 0.385 | 0.368 | 0.413 |

${\Theta}_{av,th}$ | 0.334 | 0.332 | 0.334 | 0.540 | 0.518 | 0.364 | 0.370 | 0.367 | 0.357 | 0.427 |

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**MDPI and ACS Style**

Owen, J.M.; Tang, H.; Lock, G.D.
Buoyancy-Induced Heat Transfer inside Compressor Rotors: Overview of Theoretical Models. *Aerospace* **2018**, *5*, 32.
https://doi.org/10.3390/aerospace5010032

**AMA Style**

Owen JM, Tang H, Lock GD.
Buoyancy-Induced Heat Transfer inside Compressor Rotors: Overview of Theoretical Models. *Aerospace*. 2018; 5(1):32.
https://doi.org/10.3390/aerospace5010032

**Chicago/Turabian Style**

Owen, J. Michael, Hui Tang, and Gary D. Lock.
2018. "Buoyancy-Induced Heat Transfer inside Compressor Rotors: Overview of Theoretical Models" *Aerospace* 5, no. 1: 32.
https://doi.org/10.3390/aerospace5010032