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
    • 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 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

3.3.3.2.8.1. 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 \(Ra > Critical \rightarrow \text{convection}\)

3.3.3.2.9. Rossby Number

3.3.3.2.9.1. How is the Rossby Number defined?

\[Ro = {W \over {\Omega L}}\]
  • \(W = \text{characteristic axial velocity}\)
  • \(\Omega = \text{angular speed of rotor}\)
  • \(L = \text{characteristic length}\)

3.3.3.2.10. Axial Reynolds Number

3.3.3.2.10.1. How is the Axial Reynolds Number defined?

\[Re_z = {W L \over {\nu}}\]
  • \(W = \text{characteristic axial velocity}\)
  • \(L = \text{characteristic length}\)

3.3.3.3. 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)

3.3.3.3.4. 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

  • 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!

3.3.3.4.11. 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).