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P. Meunier M. Bosco, P-Y Passaggia, S. Le Dizès

Lee waves of a tilted object. P. Meunier M. Bosco, P-Y Passaggia, S. Le Dizès Institut de Recherche sur les Phénomènes Hors-Equilibre, Marseille, France. Presentation of the problem. z. a. … a stable stratification of density with Br u nt-Väisälä frequency. A cylinder of diameter D

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P. Meunier M. Bosco, P-Y Passaggia, S. Le Dizès

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  1. Lee waves of a tilted object P. Meunier M. Bosco, P-Y Passaggia, S. Le Dizès Institut de Recherche sur les Phénomènes Hors-Equilibre, Marseille, France

  2. Presentation of the problem z a … a stable stratification of density withBrunt-Väisälä frequency A cylinder of diameter D tilted of an angle a in a flow U D U  3 parameters: - Reynolds number Re=UD/n = 30-200 - Angle of tilt a=0-90 ° - Froude number F=U/ND=0.1-3 lengths dimensionalised by Dtime dimensionalised by D/U • Large Froude correspond to small N, i.e. to weak stratification (homogeneous fluid) • Small Froude correspond to large N, i.e. to strong stratification

  3. Oceanic wakes D = 10 km U = 1 m/s N = 0.005 /s D = 10 m U = 10 m/s N = 0.005 /s D = 10 m U = 1 m/s N = 0.005 /s Re = 108 F = 200 Re = 107 F = 20 Re = 1010 F =U/ND=0.02 Island wake Offshore platform Submarine wake

  4. Atmospheric wakes D = 1000 km U = 10 m/s N = 10-4 /s D = 10 km U = 10 m/s N = 10-4 /s Re = 1012 F = 0.1 Re = 1010 F = 10 Island wake Mountain range

  5. Materials and methods • Cylinder on a translation bench: - D є [0.3 ; 1cm] - U є [0.4 ; 4cm/s] - transient regimes- Linear density profile (salted water: N=1.5-3 s-1) • - PIV measurements - 2D numerial simulations (Comsol, pseudo spectral): NS in the Boussinesq approx. (u,v,w,p,r) function of x,y w is treated as an active scalar (wd/dz=0)

  6. Axial velocity by PIV (a=30°,Re=40) • Axial velocity forced by the tilted flow around the cylinder • wavelength decreases when strat. increases • oscillations of fluid particles at frequency N • advection at U leads to wavelength l/D=2pF • strong viscous decay at small wavelength F=1.7 F=0.57 F=0.28

  7. Lighthill theory (at large F or small a) In Fourier space: D(k,w)w=v2Dsina avec - Theory x Num.

  8. Lighthill theory In Fourier space: D(k,w)w=v2Dsina avec  The forcing term diverges for free waves : w=0  non viscous viscous Residue theorem 

  9. Axial velocity by theory (a=30°,Re=40) F=1.7 F=0.57 F=0.28

  10. Comparison exp.-theory-num. (a=30°, Re=40) Wavelength Amplitude of the axial velocity ● experiment + numerics - theory ● experiment + numerics - theory F F

  11. Nearly horizontal cylinders (F=0.5, a=80°, Re=40) Numerics Theory Axial velocity: w/cos(a)

  12. Presentation of the problem 2 3 parameters: - Reynolds number Re=Ud/n = 30-200 - Angle of tilt a=0-90 ° - Froude number F=U/Nd=0.1-3 - Height of hills h=h*/d - Wave number k=k* d lengths dimensionalised by dtime dimensionalised by d/U

  13. Divergence of lee waves h=0.06, F=1.046, a=45°, Re=1186, k=1.04 zc __ w - - - v . . . r - Strong transverse velocity above hills - Strong density above hills

  14. Critical altitude for various Re, h, a, k, F O varying Re  varying h zc - Critical altitude independent of Re, h - Critical altitude defined by

  15. Profile of normal velocity - Theory o Numerics In Fourier space: - Third term diverges for kU=sin(a)/F - Logarithmic divergence of w'  jump of w' of ip - Divergence of v~w/(sin(a)-kFU)

  16. Profile of transverse velocity - Theory o Numerics Adding viscous terms Rescaling inside critical layer: Re1/3 Airy equation v''+zv = 1 Re-1/3 with Jet profile and shear profile at different x

  17. Profile of transverse velocity Thickness Amplitude Scalings as Re-1/3 and Re-1/3 for thickness and amplitude

  18. Conclusions • Internal waves generated by a tilted cylinder wake: - Tilt induces axial velocity - Lighthill theory for large F, small tilt - Axial velocity ~ sina cosa • Internal waves generated by a tilted sinusoidal topography : - Tilt induces transverse velocity - Divergence at zc where kU(zc)=Nsina - Maximum velocity scales as Re 1/3 - Thickness scales as Re-1/3 Perspectives: • 3D instabilities • Zig-zag instability of a cylinder wake • Internal waves generated by the wake • Experiment on critical layer • Experiment on radiative instability of boundary layer • Influence of the background rotation (Rossby number)

  19. How to make a stratification? Fresh water Salted water H floater

  20. Bluff body wakes • Bluff bodies: separated layer • Drag reduction, energy savings • Robustness of bridges, buildings • Vortex induced-vibration

  21. Nearly horizontal cylinders (F=0.5, a=89°, Re=40) Numerics Theory Normal velocity

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