The sliding Couette flow problem

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The sliding Couette flow problem
The Nature of High Reynolds Number
Turbulence Issac Newton Institute for
Mathematical Sciences September 8-12, 2008
Cambridge

• T. Ichikawa and M. Nagata
• Department of Aeronautics and Astronautics
• Graduate School of Engineering
• Kyoto University

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Background
Sliding Couette flow

• Linearly stable
• for gt 0.14.15Gittler(1993)
• Nonlinear solution
• not yet known

( radius ratio)
UNDERGROUND

• Applications
• catheter through blood vessel
• train through a tunnel

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Background
Sliding Couette flow

• plane Couette flow
• 1

grad p

• pipe flow
• add axial pressure gradient
• adjust
• 0

• moderate
• pCf with curvature pf with solid core 1

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Energy (stability) surface
Joseph (1976)
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Background
Sliding Couette flow
Differential rotation
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Neutral surface(1)
U
S
7
Neutral surface (2)
U
S
U
8
Neutral surface (3)
U
S
U
3.61E06
( 0 )
Deguchi MN in preparation
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Spiral Couette flow

• Narrow gap limit
• Rigid-body rotation

Plane Couette fow with streamwise rotation or
Rotating plane Couette flow by Joseph (1976)
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Model

• Incompressible fluid between two parallel plates
  with infinite extent
• Translational motion of the plates with opposite
  directions

• Constant streamwise system rotation

dimensional quantities
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Energy method ReE
The Reynolds-Orr energy equation
Nondimensionalise
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Energy method ReE
Euler-Lagrange equation
eliminating
boundary condition
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Governing equations
time scale
length scale
velocity scale

• equation of continuity

• momentum equation

• boundary condition

• basic solution

Reynolds number
Rotation rate
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Disturbance equatuins
separation of the velocity field into basic flow,
mean flows and residual
Further decomposition of the residual into
poloidal and toroidal parts
substitute into the momentum equation
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Mean flow equations
-averages of the -components of the
momentum equation
Boundary conditions
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Linear stability analysis
Perturbation equations
expansion of
Chebyshev polynomials
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Numerical method(Linear analysis)
Evaluation of at the collocation points
Eigenvalue problem
unstable
neutral
stable
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Results (O40 )
Unstable region dark
A few largest eigenvalues
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Results (neutral curves)
Unstable
Stable
20.66
0
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Results
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Long wave limit with constant ß
consider
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Long wave limit with constant ß

boundary condition
ß1.558
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Non-linear Analysis
Expand
Boundary condition
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Non-linear Analysis
Evaluate at
Solve by Newton-Raphson method
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Nonlinear solution (bifurcation
diagram)
momentum transport
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Mean flows
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Steady spiral solution
flow field
x-component of the vorticity
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Summary

• Stability analysis
• Basic state is stable for small rotation and
  becomes unstable at O 17 to 3D perturbations.
• As O is increased, decreases.
• in the limit of O?8 and a?0.
• Nonlinear analysis
• Steady spiral solution bifurcates supercritically
  as a secondary flow.
• The mean flow in the spanwise direction is
  created.

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Future work

• Connection with cylinderical geometry

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End
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Spiral Couette flow
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Spiral Poiseuille flow
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Spiral Poiseuille flow (Joseph)
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Spiral Poiseuille flow (Joseph)
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Spiral Poiseuille flow

• Narrow gap limit
• Rigid-body rotation

Plane Poiseuille fow with streamwise rotation
Masuda, Fukuda Nagata (2008)
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Governing equations

• equation of continuity

• momentum equation

• boundary condition

Coriolis force
density
kinematic viscosity
pressure
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