poiseuille flow-hydrodynamic stability analysis
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Poiseuille Flow
Gohar KhokharLauriane Vilmin
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Navier-Stokes equations
Continuity
Boundary conditions
Base flow
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◦ Introduction of small perturbations
◦ Substraction of the base flow
◦ Linearization
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Orr-Sommerfeld equation
Squire equation
Boundary conditions
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Spectrum for Re = 2000 Spectrum for Re = 7000
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Growth versus timeResolvant norm versus frequency
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Spectrum for Re = 2000
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Growth versus timeResolvant norm versus frequency
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Maximum growth versus Reynolds’ number
Time for maximum growthversus Reynolds’ number
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Maximum Growth vs span wise wave number
Time for Maximum Growthvs Spanwise wave number
0 1 2 3 4 5 60
20
40
60
80
100
Beta
Tm
ax
Alpha = 0
Alpha =1
Time for maximum Growth vs span wise wavenumber (Re=1000)
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Maximum growth for α = 0 and β = 2 α = 1 and β = 0
◦ A, P and S-branches are visible◦ Transient growth; the eigenvectors are non orthogonal
α = 0 and β = 2◦ Spectrum: only the S-branch is present
Investigation of the Reynolds’ number◦ Maximum growth increases like the square of Reynolds’
number◦ Time of maximum growth grows linearly with Reynolds’
number
Investigation of the spanwise wave number◦ Maximum growth around β = 2, Gmax around 200◦ Time for that maximum growth is about 90s