1. Heat transfer made visible
Michel Speetjens
Energy Technology Laboratory
Mechanical Engineering Department
Outline
• Physical setting
• Fluid motion made visible
• Some illustrative examples
• Heat transfer made visible
• Some illustrative examples
• Conclusions and outlook
May 2008 PAGE 1

2. • Physical setting
• Fluid motion made visible
• Some illustrative examples
• Heat transfer made visible
• Some illustrative examples
• Conclusions and outlook
May 2008 PAGE 2
Physical setting: laminar transport
Laminar flow:
Viscous flows: high ν
• polymer/food processing
• heat-transfer fluids
• geophysical flows (magma, petroleum)
Small-scale flows: low U,L
• micro-fluidics
• compact heat exchangers
• physiological flows (lungs, blood)
May 2008 PAGE 3

3. • Physical setting
• Fluid motion made visible
• Some illustrative examples
• Heat transfer made visible
• Some illustrative examples
• Conclusions
May 2008 PAGE 4
Fluid motion made visible
Lagrangian approach: transport described by geometry of fluid paths
Governing equations:
• kinematic equation:
• mass conservation:
organises fluid paths into coherent structures
May 2008 PAGE 5

4. • Physical setting
• Fluid motion made visible
• Some illustrative examples
• Heat transfer made visible
• Some illustrative examples
• Conclusions
May 2008 PAGE 6
Example: visualising 2D steady flows
Steady flow: fluid paths streamlines:
flow past object flow inside lid-driven cavity
Organisation of streamlines into islands and/or open channels:
• islands => confine and circulate fluid
• channels => set up net throughflow
basic geometrical building blocks of 2D steady flows
May 2008 PAGE 7

5. Example: visualising 2D mixing
Lid-driven cavity flow: simplification of industrial mixer/heat exchanger
• flow forcing: time-periodic translation of sidewalls
• parameter: period time T
laminar flow (Re=1)
May 2008 PAGE 8
Visualisation time-periodic fluid paths: Poincaré-sections:
• release passive tracers in flow => “label” fluid parcels
• “illuminate” tracers at t = 0,T,2T,… => “stroboscopic map”
continuous flow Poincaré-section
May 2008 PAGE 9

6. Time-periodic forcing: basics
Visualising 2D mixing in lid-driven cavity by Poincaré-sections:
steady
transition to chaos with increasing T
Organisation Poincaré-sections into islands and/or chaotic seas:
• islands => poor mixing
• chaotic regions => good mixing May 2008 PAGE 10
Time-periodic forcing: basics
Organisation inside chaotic sea: manifolds:
Manifolds: principal transport directions:
• unstable: transport forward in time => asymptotic mixing pattern
• stable: transport backward in time => origin of material
May 2008 PAGE 11

7. • Physical setting
• Fluid motion made visible
• Some illustrative examples
• Heat transfer made visible
• Some illustrative examples
• Conclusions
May 2008 PAGE 12
Heat transfer made visible
Heat transfer as the “motion” of a “fluid”:
• fluid transport: Lagrangian representation:
• thermal transport:
- Eulerian representation:
- Lagrangian representation:
May 2008 PAGE 13

8. Fluid-motion representation of heat transfer: heat is transported along
trajectories xT delineated by total heat flux Q in same way as fluid is
transported along trajectories x delineated by fluid velocity u
This admits:
• heat-transfer visualisation by flow-visualisation methods
• heat-transfer analysis with geometrical methods of laminar mixing
based on organisation of trajectories into coherent structures
• unified approach to fluid transport and heat transfer
May 2008 PAGE 14
• Physical setting
• Fluid motion made visible
• Some illustrative examples
• Heat transfer made visible
• Some illustrative examples
• Conclusions
May 2008 PAGE 15

9. Example: 2D steady flows
Cooling of hot object by cold uniform flow:
fluid streamlines thermal streamlines
Thermal streamlines: also organised into islands and/or open channels:
• thermal island => confines and circulates heat
• thermal channels => heat exchange object flow => “thermal path”
basic geometrical building blocks of 2D steady heat transfer
May 2008 PAGE 16
Example: 3D steady flows
Cooling of 3D hot object by cold uniform flow:
Example: 2D hot object … steady
3D thermal streamlines emanating from object => 3D thermal path
May 2008 PAGE 17

10. Example: 2D transient behaviour
Heat transfer in lid-driven cavity: transient to steady state
Steady state: “high” Pe
“low” Pe
“moderate” Pe
COLD
HOT
Thermal streamlines: same organisation as before:
• thermal islands => confine and circulate heat
• thermal path => fluid-wall heat exchange
convection (higher Pe) promotes growth of islands
May 2008 PAGE 18
lid-driven cavity revisited
Transient (high Pe): evolution of T and instantaneous thermal streamlines:
steady state
Instantaneous thermal streamlines:
• attaching to top wall => formation of steady-state thermal path
• converging on instantanous stagnation points => formation of islands
visualise formation of steady state
May 2008 PAGE 19

11. Example: heat transfer versus mixing
fluid Poincare-section thermal Poincaré-section
Thermal Poincaré-section:
• thermal path (gray curves) => only marginal difference with steady case
• chaotic heat transfer (manifolds) => disintegration of thermal islands
non-trivial connection mixing and (chaotic) heat transfer!
May 2008 PAGE 20
• Physical setting
• Fluid motion made visible
• Some illustrative examples
• Heat transfer made visible
• Some illustrative examples
• Conclusions
May 2008 PAGE 21

12. Conclusions and outlook
Heat transfer can be described as “motion” of “fluid”; this admits:
• heat-transfer visualisation by flow-visualisation methods
• heat-transfer analysis by geometrical methods from laminar mixing
• unified approach to fluid transport and heat transfer
Affords new insight:
• isolation of heat transfer zones (thermal paths, …)
• fundamental connection mixing and heat transfer
• …
Challenges:
• further development of (unified) theoretical framework
• application to realistic (industrial) configurations
research in progress!
May 2008 PAGE 22