REVISITING THE STABILITY OF PULSATILE PIPE FLOW

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REVISITING THE STABILITY OF PULSATILE PIPE FLOW

• F. Fedele Department of Civil Engineering
• D.L. Hitt R.D. Prabhu Department of Mechanical
  Engineering
• University of Vermont,
  Burlington, VT 05405 USA

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Womersley solution

• The flow is linearly stable for axisymmetric
  perturbations (Tozzi von Kerczek, 1986)
• Slightly more stable than Poiseiulle flow
• Presence of inflection rings occur during an
  oscillation cycle for sufficiently strong flow
  pulsation in relation to the mean flow

Womersley J.R., Method for the Calculation of
Velocity, Rate of Flow and Viscous Drag in
Arteries When the Pressure Gradient is Known. J.
Physiol., 127 (1955), pp. 553-563.
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Reynolds pipe flow experiment
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The Orr-Sommerfeld Equation
Linear stability analysis axisymmetric
perturbations
Stokes Stream function
Result Orr-Sommerfeld Equation
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The Long-Wave Orr-Sommerfeld basis
Longwave limit of the Orr-Sommerfeld equation
Analytical solution in longwave limit
Orthogonal with respect to the Scalar product
?
Non orthogonal with respect to
?
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Galerkin method
Boundary conditions satisfied a priori
Galerkin projection
steady flow
oscillation
Energy of the perturbation
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Convergence analysis
Steady flow case First eigenvalue
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Stability by Floquet theory
Fundamental matrix at tT
Eigenvalue Plot
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Transient Energy Growth
Initial conditions
Set of non orthogonal eigenmodes
Non-orthogonal modes energy growth possible
orthogonal modes no growth
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Optimal Disturbances
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Time evolution Optimal Disturbances
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FREE-STREAM TURBULENCE AND STREAK BREAKDOWN