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

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

- Abstract
- Where does buoyancy-induced flow occur?
- How does the buoyancy-induced flow occur?
- Why is the heat transfer from the solid surface to air difficult to measure or compute?
- What do designers want to measure during engine accelerations and decelerations?
- What kind of geometries can be considered?
- What kind of flows can be considered?

- Introduction
- What do we wish to calculate?
- What is needed in order to calculate this?
- Why is this difficult?
- What are the two regimes, between which transition must be predicted?
- What types of problem are present?
- What are the important non-dimensional parameters?
- What are the ranges of these non-dimensional parameters?
- Rayleigh Number
- Rossby Number
- Axial Reynolds Number

- Buoyancy-Induced Flow in Closed Cavities
- Coriolis Effects in Rotating Cavities
- What is radial convection in a rotating annulus
**analogous**to? - What does the heat transfer depend on?
- What are the Coriolis accelerations?
- Why can an inviscid linear set of equations be considered?
- How can radial flow occur in such a case?
- What are Ekman layers?
- What kind of radial flows are there?
- How can the flow become non-axisymmetric and unsteady?
- What is the surprising phenonmenon at large rotation speeds?
- Why is the critical Rayleigh number higher in a rotating cavity than a stationary one?
- How can Rayleigh-Benard convection occur?

- What is radial convection in a rotating annulus
- Heat Transfer in Closed Rotating Cavities
- Glossary

## 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
- Unsteady
- 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
- Radial inflow

## 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:
- Unsteady
- 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
- Axial flow is
**cooler**than shroud - For
**steady state**conditions **Buoyancy induced convection**can occur

- For

- Axial flow is

### 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?¶

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?¶

- \(W = \text{characteristic axial velocity}\)
- \(\Omega = \text{angular speed of rotor}\)
- \(L = \text{characteristic length}\)

#### 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?¶

- \(W = \text{characteristic axial velocity}\)
- \(L = \text{characteristic length}\)

#### 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:

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:

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

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

## 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

- For radial flow either:
- 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).