Numerical modelling of the transition from laminar to turbulent stages in a simple parallel shear fl

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Numerical modelling of the transition from
laminar to turbulent stages in a simple
parallel shear flow

• M. Nagata T. Itano
• Kyoto University
• Japan

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Physical configuration

• Differentially heated side walls
• Boussinesq approxmation
• Cubic velocity profile
• Prandtl number 0
• gt purely hydrodynamic
• Periodicity in the streamwise(x) and the
  spanwise(y) directions

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Neutral curve and the stability of the transverse
vortex flowby Nagata Busse (1985)
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Decomposition of the velocity fields
?
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DNS
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The growth rate and the wall shear rate of the
basic state, 2DTV and 3DSS
Basic
2DTV
3DSS
Basic
2DTV
3DSS
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Transient states and the final state by DNS
3DSS-I
3DPS-I
2DTV
3DPS-I
3DPS-II
2DTV
Harmonic case
Subharmonic case
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The wall shear rate by DNS
Basic
Basic
3DPS-I
3DPS-I
3DSS-I
3DTW
2DTV
2DTV
Burst
3DPS-II
3DSS-II
Subharmonic case
Harmonic case
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Sporadic burst flow
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The spectrum of the wall friction.at Gr678, 684
and 690.
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3DSSII
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The vorticity on the mid-plane 3DSS-I
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The vorticity on the mid-plane 3DSS-II
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Bifurcation diagram
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Concluding remarks

• The bifurcation of flows of a Boussinesq fluid
  between two vertical plates with different
  temperatures in the vanishing Prandtl number
  limit has been examined.
• We have confirmed the results of Nagata Busse
  (1983), in which the secondary and tertiary flows
  are the two-dimensional steady transverse vortex
  flow and the three-dimensional steady subharmonic
  flow, respectively.
• We have also presented the results obtained by
  the numerical experiment. In particular, several
  unstable periodic solutions representing a
  transient state are captured in both the harmonic
  and the subharmonic cases.
• At high Grashof numbers different types of motion
  coexist as a stable solution of the system.
• The motion becomes quasi-periodic and a chaotic
  motion ensues eventually.