Poiseuille Flow

Gohar KhokharLauriane Vilmin

Navier-Stokes equations

Continuity

Boundary conditions

Base flow

◦ Introduction of small perturbations

◦ Substraction of the base flow

◦ Linearization

Orr-Sommerfeld equation

Squire equation

Boundary conditions

Spectrum for Re = 2000 Spectrum for Re = 7000

Growth versus timeResolvant norm versus frequency

Spectrum for Re = 2000

Growth versus timeResolvant norm versus frequency

Maximum growth versus Reynolds’ number

Time for maximum growthversus Reynolds’ number

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)

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