# 3.3.3. Review of Buoyancy-Induced Flow in Rotating Cavities¶

1. Michael Owen and Christopher A. Long, J. Turbomach 137(11), Aug 12, 2015.

## 3.3.3.1. Abstract¶

### 3.3.3.1.1. Where does buoyancy-induced flow occur?¶

In the cavity between two co-rotating compressor disks.

### 3.3.3.1.2. How does the buoyancy-induced flow occur?¶

• When the temperature of the disks and shroud is higher than that of the air inside the cavity.

• Coriolis forces in the rotating fluid create cyclonic and anti-cyclonic circulations inside the cavity

### 3.3.3.1.3. Why is the heat transfer from the solid surface to air difficult to measure or compute?¶

• The flows are:
• 3D

• Unstable - one flow structure can change quasi-randomly to another

### 3.3.3.1.4. What do designers want to measure during engine accelerations and decelerations?¶

• Transient temperature changes

• Thermal stresses

• Radial growth of the disks

### 3.3.3.1.5. What kind of geometries can be considered?¶

• Closed rotating cavities

• Open cavities

### 3.3.3.1.6. What kind of flows can be considered?¶

• Axial throughflow

## 3.3.3.2. Introduction¶

### 3.3.3.2.1. What do we wish to calculate?¶

The transient and steady clearance between the blades and the casing of a high pressure compressor in an aeroengine.

### 3.3.3.2.2. What is needed in order to calculate this?¶

• Radial growth of the compressor disks

• Transient temperatures of the disk

• Flow and heat transfer in the cavity between the corotating disks

### 3.3.3.2.3. Why is this difficult?¶

• Buoyancy-induced flow is:

• 3D

• Unstable

### 3.3.3.2.4. What are the two regimes, between which transition must be predicted?¶

• Axial flow is hotter than shroud
• Flow can be stably stratified

• Can occur in acceleration and deceleration of engine

• No buoyancy induced convection

• Heat transfer from disks is small

• Axial flow is cooler than shroud

• Buoyancy induced convection can occur

### 3.3.3.2.5. What types of problem are present?¶

• Inverse problem:
• Determination of heat fluxes from temperature measurements

• Ill-posed - small uncertainties in temperature create large errors in fluxes

• Conjugate problem:
• Buoyancy induced convection

• Temperature distribution on disks affects the flow in the cavity and vice versa

### 3.3.3.2.6. What are the important non-dimensional parameters?¶

• Rayleigh, $$Ra$$

• Rossby, $$Ro$$

• Axial Reynolds, $$Re_z$$

### 3.3.3.2.7. What are the ranges of these non-dimensional parameters?¶

• $$Ra \sim 10^{12}$$

• $$Ro \sim 10^{0}$$

• $$Re_z \sim 10^{5}$$

### 3.3.3.2.8. Rayleigh Number¶

#### 3.3.3.2.8.1. How is the Rayleigh Number defined?¶

$Ra = Pr Gr$
$Pr = {{\mu c_p} \over k}$
$Gr = {{\tilde{g} L^3 \beta \Delta T} \over {\nu^2}}$

where:

• $$L = \text{characteristic length}$$

• $$\beta = \text{volume expansion coefficient}$$

• $$k = \text{thermal conductivity of air}$$

• $$\tilde{g} = \text{charateristic acceleration}$$

#### 3.3.3.2.8.2. What does the Rayleigh Number mean?¶

• When $$Ra < Critical \rightarrow \text{conduction}$$

• When $$Ra > Critical \rightarrow \text{convection}$$

#### 3.3.3.2.8.3. What does the Prandtl Number measure?¶

• Momentum to thermal diffusivity

#### 3.3.3.2.8.4. What does the Grashof Number measure?¶

• Buoyancy to viscosity

### 3.3.3.2.9. Rossby Number¶

#### 3.3.3.2.9.1. How is the Rossby Number defined?¶

$Ro = {W \over {\Omega L}}$
• $$W = \text{characteristic axial velocity}$$

• $$\Omega = \text{angular speed of rotor}$$

• $$L = \text{characteristic length}$$

#### 3.3.3.2.9.2. What does the Rossby Number measure?¶

• Convection to Coriolis forces

### 3.3.3.2.10. Axial Reynolds Number¶

#### 3.3.3.2.10.1. How is the Axial Reynolds Number defined?¶

$Re_z = {W L \over {\nu}}$
• $$W = \text{characteristic axial velocity}$$

• $$L = \text{characteristic length}$$

#### 3.3.3.2.10.2. What does the Reynolds Number measure?¶

• Inertial to viscous forces

## 3.3.3.3. Buoyancy-Induced Flow in Closed Cavities¶

### 3.3.3.3.1. Heat Transfer in Closed Stationary Cavities¶

The Rayleigh number can be defined as:

$Ra^{'} = {Pr \beta \Delta T} {{g d^3} \over \nu^3}$

where:

• $$d$$ is the vertical distance between the plates

• $$\Delta T = T_H - T_C$$ ($$H$$ = hot and $$C$$ = cold)

### 3.3.3.3.2. What is the mechanism for Rayleigh-Benard convection?¶

• When the lower surface is hotter than the upper surface, the flow becomes unstable

• At a critical Rayleigh number, it breaks down into a series of counter-rotating vortices

• (When the upper surface is hotter, the fluid is thermally stratified and heat transfer is by conduction)

### 3.3.3.3.3. What is the critical Rayleigh Number?¶

• $$Ra^{'}_{crit} = 1708$$

### 3.3.3.3.4. What is the Nusselt Number?¶

• $$Nu^{'} = {\text{average heat flow at the surface} \over \text{heat flow due to conduction through the fluid}}$$

• $$Nu^{'}=1$$ $$\longrightarrow$$ $$\text{heat transfer is entirely by conduction}$$

### 3.3.3.3.5. What empirical correlations are possible?¶

King:

$Nu^{'} = C_1 Ra^{'A} + C_2 Ra^{'B}$

Grossmann and Lohse (where $$1/4$$ exponent is laminar convection at low $$Ra^{'}$$ and the $$1/3$$ with turbulent at high $$Ra^{'}$$):

$Nu^{'} = 0.27 Ra^{'1/4} + 0.038 Ra^{'1/3}$

Hollands (where $$Nu^{'} = 1$$ for $$Ra^{'} < Ra^{'}_{crit}$$):

$Nu^{'} = 1 + 1.44 max[1-1708/Ra^{'}, 0] + max[ (Ra^{'}/5830)^{1/3} - 1, 0]$

## 3.3.3.4. Coriolis Effects in Rotating Cavities¶

### 3.3.3.4.1. What is radial convection in a rotating annulus analogous to?¶

• Rayleigh-Benard convection that occurs in the air gap between two stationary horizontal plates

• g is replaced by the centripetal acceleration

### 3.3.3.4.2. What does the heat transfer depend on?¶

Whether the outer surface is hotter or colder than the inner one:

• If the outer surface is hotter than the inner, the density gradient stablizies the flow and heat transfer is by conduction

• If the outer surface is colder than the inner, the heat transfer is by convection

### 3.3.3.4.3. What are the Coriolis accelerations?¶

• $$\text{Radial acceleration} = -2 \Omega v$$

• $$\text{Tangential acceleration} = 2 \Omega u$$

where:

• $$u = \text{radial velocity}$$

• $$v = \text{tangential velocity}$$

These accelerations are associated with respective forces

### 3.3.3.4.4. Why can an inviscid linear set of equations be considered?¶

• $$u / \Omega r << 1$$

• $$v / \Omega r << 1$$

• The non-linear terms are much smaller than the linear Coriolis terms

• The Navier-Stokes equations reduce to inviscid linear equations

### 3.3.3.4.5. How can radial flow occur in such a case?¶

• $$u=0$$ in an inviscid axisymmetric rotating fluid

• It is confined to the boundary layers (where the Coriolis forces are produced by shear stresses)

• Or the flow is non-axisymmetric

### 3.3.3.4.6. What are Ekman layers?¶

• Circumferential shear stresses in the boundary layers on the two disks which create Coriolis forces

### 3.3.3.4.7. What kind of radial flows are there?¶

For unidirectional flows, such as source-sink flows, with a superposed radial outflow or inflow:

• Isothermal radial outflow - radial flow is confined to Ekman layers, between which there is a core of inviscid fluid that rotates at an angular speed slower than the disks

• Isothermal radial inflow - radial flow is confined to Ekman layers, between which there is a core of inviscid fluid that rotates at an angular speed faster than the disks

### 3.3.3.4.8. How can the flow become non-axisymmetric and unsteady?¶

• Inner surface is hotter than outer surface

• Rayleigh-Bernard convection occurs

• Contra-rotating vortices are created

• Cyclonic vortices create low pressure regions

• Anti-cyclonic vortices create high pressure regions

• Circumferential pressure gradients produce Coriolis forces for inflow and outflow of hot and cold fluids

• These flows are nonaxisymmetric and unsteady

### 3.3.3.4.9. What is the surprising phenonmenon at large rotation speeds?¶

• At large rotation speeds $$\Omega^2 b \gg g$$

• $$Ra \propto \beta \Delta T Re_{\phi}^2$$

• A given fluidic Rayleigh number can be produced by an infinite combination of $$Re_{\phi}$$ and $$\beta \Delta T$$

• Coriolis acceleration is proportional to $$\Omega$$

• Higher values of $$Re_{\phi}$$ result in lower values of $$Nu$$

• A given Rayleigh number could be produced by a wide variety of Nusselt numbers

• The value of Nusselt number could decrease as the Rayleigh number increases!

### 3.3.3.4.10. Why is the critical Rayleigh number higher in a rotating cavity than a stationary one?¶

• Coriolis forces tend to attenuate velocity fluctuations

### 3.3.3.4.11. How can Rayleigh-Benard convection occur?¶

• Can only occur if initial axisymmetry is broken to allow radial flow

• For an initially isothermal closed rotating cavity, the fluid will be in solid-body rotation

• If shroud is heated, heat transfer must initially be by axisymmetric conduction

• Only after axisymmetry is broken can convection begin

• Critical Rayleigh number for Rayleigh-Benard convection could depend on whether the cavity is initially rotating ot stationary before the shroud is heated

## 3.3.3.5. Heat Transfer in Closed Rotating Cavities¶

### 3.3.3.5.1. What kind of heat transfer can happen in closed rotating cavities?¶

• Axial - from a hot disk to a cold one

• Radial - from hot outer cylinder to cold inner one

### 3.3.3.5.2. How does axial heat transfer occur?¶

• Radial inflow in boundary layer on a hot disk

• Radial outflow in boundary layer on cold disk

### 3.3.3.5.3. What are the Nusselt Numbers and Rayleigh Numbers in closed cavities?¶

• Nusselt numbers are small

• Convection is the same magnitude as radiation

• Measured Nusselt numbers are less than $$10$$ when Rayleigh numbers are up to $$10^{11}$$

## 3.3.3.6. Glossary¶

Shroud: the surface defining the outer diameter of a turbomachine flow annulus (ring-shaped object).