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

Presentation of the problem … a stable stratification of density with Brunt-Väisälä frequency z  A cylinder of diameter D tilted of an angle  in a flow U  3 parameters: - Reynolds number Re=UD/  = Angle of tilt  0-90  - Froude number F=U/ND=0.1-3 lengths dimensionalised by D time dimensionalised by D/U 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

D = 10 m U = 10 m/s N = /s Re = 10 8 F = 200 D = 10 m U = 1 m/s N = /s Re = 10 7 F = 20 Oceanic wakes D = 10 km U = 1 m/s N = /s Re = F =U/ND=0.02 Offshore platform Island wake Submarine wake

D = 1000 km U = 10 m/s N = /s Re = F = 0.1 D = 10 km U = 10 m/s N = /s Re = F = 10 Atmospheric wakes Island wake Mountain range

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,  ) function of x,y w is treated as an active scalar (wd/dz=0)

Axial velocity by PIV (  =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 /D=2  F - strong viscous decay at small wavelength F=1.7 F=0.57 F=0.28

Lighthill theory (at large F or small  ) In Fourier space: D(k,  )w=v 2D sin  avec - Theory x Num.

Lighthill theory The forcing term diverges for free waves :   non viscous viscous Residue theorem  In Fourier space: D(k,  )w=v 2D sin  avec

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

Comparison exp.-theory-num. (  =30°, Re=40) Amplitude of the axial velocityWavelength ● experiment + numerics - theory ● experiment + numerics - theory FF

Nearly horizontal cylinders (F=0.5,  =80°, Re=40) NumericsTheory Axial velocity: w/cos(  )

Presentation of the problem 2 3 parameters: - Reynolds number Re=U   /  = Angle of tilt  0-90  - Froude number F=U  /N  = Height of hills h=h*/  - Wave number k=k*  lengths dimensionalised by  time dimensionalised by  /U 

Divergence of lee waves h=0.06, F=1.046,  =45°, Re=1186, k= Strong transverse velocity above hills - Strong density above hills zczc __ w v... 

Critical altitude for various Re, h, , k, F - Critical altitude independent of Re, h - Critical altitude defined by zczc O varying Re  varying h

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

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

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

Conclusions Internal waves generated by a tilted cylinder wake: - Tilt induces axial velocity - Lighthill theory for large F, small tilt - Axial velocity ~ sin  cos  Internal waves generated by a tilted sinusoidal topography : - Tilt induces transverse velocity - Divergence at z c where kU(z c )=Nsin  - Maximum velocity scales as Re 1/3 - Thickness scales as Re -1/3 - 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) Perspectives:

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

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

Nearly horizontal cylinders (F=0.5,  =89°, Re=40) NumericsTheory Normal velocity