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

3.3.3.2.1. 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 cooler than shroud

  • For steady state conditions

  • 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¶

3.3.3.2.9. Rossby Number¶

3.3.3.2.10. Axial Reynolds Number¶

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 \(Ra^{'} < Ra^{'}_{crit}\)):

3.3.3.4.1. 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

3.3.3.5.1. 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 \(10^{11}\)