Flow Instability Due to Presence of Distributing Wall Heating

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Flow Instability Due to Presence of Distributing
Wall Heating
by Mohammad Zakir Hossain
Supervisor Dr. J. M. Floryan
Department of Mechanical Materials
Engineering University of Western Ontario
London, Ontario, Canada
Graduate Seminar 10 November 2008
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Distributed Heating
Figure Plane Poiseuille flow
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Distributed Heating
T gtTref
Figure Plane Poiseuille flow with uniform wall
heating
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Distributed Heating
Figure Plane Poiseuille flow with distributed
wall heating.
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Objectives

• To develop heat transfer augmentation strategies
• ? minimum streamwise pressure loss
• To determine optimal spatial distribution of heat
  load
• ? form streamwise vortices

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Streamwise Vortex

• Streamwise vortices create transverse convection
  resulting in a large increase of heat transfer

x
Figure Typical Streamwise (longitudinal)
vortices
Jacobbi and Shah (1995)
Benard and Avsec (1938)
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Presentation Sequence

• Overview of the analysis
• Stability diagrams for distributed wall heating
• With the presence of shear
• Without shear
• Conclusion Future work

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Overview
Reference flow Poiseuille Flow (2D)
Modified by distributed wall heating
Meanflow Analysis
Determine the flow modifications
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Meanflow Analysis

• Governing Equation
• Continuity, Navier-Stokes and Energy equations

• Assumption
• Steady 2D flow
• Flow is periodic in streamwise direction
• Newtonian Fluid Boussinesq approximation
• Neglected viscous dissipation

• Spatial discretization
• streamwise (x) direction ? Fourier expansions
• vertical (y) direction ? collocation points

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Overview (contd.)
Reference flow Poiseuille Flow (2D)
Modified by distributed wall heating
Meanflow Analysis
Determine the flow modifications
Modified flow
Add small 3D disturbances
Linearize the problem
Linear Stability Analysis
Determine the eigenvalue
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Linear Stability Analysis
Disturbances 3D
? streamwise wave number of disturbances, ?
streamwise wave number of heating, ?
spanwise wave number of disturbances, ?r
frequency of disturbances, ?i growth rate of
disturbances.
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Overview (contd.)
Reference flow Poiseuille Flow (2D)
Modified by distributed wall heating
Meanflow Analysis
Determine the flow modifications
Modified flow
Add small 3D disturbances
Linearize the problem
Linear stability Analysis
Determine the eigenvalue
Flow modifications Disturbance field
DNS
3D unsteady governing equations
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DNS Result
Instability induced by Periodic Heating
Figure Evolution of energy of disturbances for
Re1000, Ra105, ?3, ?1.5.
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Dimensionless numbers
Reynolds Number
Prandlt Number
Rayleigh Number

Uniform heating
Distributed heating
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Periodic Heating

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Linear Stability

• 2 types of instability
• - Vortex instability ? 0 and ?r 0
• Traveling Wave instability
• 2D waves
• 3D waves (oblique waves)

wave
Oblique angle

• 0 ? 2D wave
• ? 900 ? vortex

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Stability Result
No Heating case
Figure Neutral curve for plane Poiseuille flow
Schmid Henningson Stability Transition in
Shear Flow, Springer 2001
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Stability Result
Periodic Heating case
Unstable
Figure Stability diagram for a1
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Stability Result
Periodic Heating case
Figure Effect of heating wave number(?) for
vortices
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Stability Result
Periodic Heating case
Figure Effect of Reynolds number for vortices
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Meanflow streamlines
Periodic Heating case
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Meanflow
Re0
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Meanflow
Re0
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Meanflow
Re0
Figure Variation of dT/dy across the channel
at different heating wave numbers.
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Meanflow
Re0
conduction
convection
Figure Thickness of convection layer at
different Ra.
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Stability
Re0
Branch 3
Branch 4
Figure Critical stability diagram for Re0
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Stability
Re0
Figure Variation of d at critical stability.
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Stability
Re0
Figure Variation of b at critical stability.
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Conclusion

• Instability of channel flow modified by periodic
  heating applied at the lower wall has been
  analyzed.
• With the presence of shear, disturbances in the
  form of streamwise vortices are the most
  unstable.
• Without shear, stability characteristics changes
  and we identify four different branches of
  instability.

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Future work

• Capture more flow physics to make a conclusion
  about the effect of distributed heating.
• Modify the linear stability code to handle 3D
  mean flow and 3D disturbances.

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Thank You