Harish Dixit and Rama Govindarajan With Anubhab Roy and Ganesh Subramanian Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore September 2008 Instabilities in variable-property flows and the continuous spectrum An aggressive ‘passive’ scalar

Re=3000, unstratified Building block for inverse cascade

`Perpendicular’ density stratification: baroclinic torque (+ centrifugal + other non-Boussinesq effects) Heavy Light 1 2 ρ(y) ‏ y ρ Brandt and Nomura, JFM (2007): stratification upto Fr=2, Boussinesq Stratification aids merger at Re > 2000 At lower diffusivities, larger stratifications?

Re=3000, Pe=30000, Fr (pair) = 1

Large scale overturning: a separate story Why does the breakdown happen? Consider one vortex in a (sharp) density gradient In 2D, no gravity

Heavy Light point vortex Initial condition: Point vortex at a density jump

Homogenised within the yellow patch, if Pe finite A single vortex and a density interface Inviscid The locus seen is not a streamline!

Scaling Density is homogenised for e.g. Rhines and Young (1983) Flohr and Vassilicos (1997) (different from Moore & Saffman 1975) When Pe >>> 1, many density jumps between r h and r s Consider one such jump, assume circular

Linearly unstable when heavy inside light, Rayleigh-Taylor Vortex sheet of strength Rotates at m times angular velocity of mean flow Point vortex, circular density jump Radial gravity Non-Boussinesq, centrifugal Non-Boussinesq: e.g. Turner, 1957, Sipp et al., Joly et al., JFM 2005

m = 2 Vortex sheet at r j In unstratified case: a continuous spectrum of `non-Kelvin’ modes

Rankine vortex with density jumps at r j s spaced at r 3 r Kelvin (1880): neutral modes at r=a for a Rankine vortex

Vorticity and density: Heaviside functions

For j jumps: 2j+2 boundary conditions u r and pressure continuous at jumps and r c Green’s function, integrating across jumps For non-Boussinesq case: For one density jump

m = 5 Multiple (7) jumps

r j = 2 r c,  =0.1 Single jump Step vs smooth density change

Single jump: radial gravity (blue), non-Boussinesq (red) m = 2,  = 0.01

(circular jump: pressure balances, but) Lituus spiral Dominant effect, small   non-dimensional) KH instability at positive and negative jump growing faster than exponentially  In the basic flow

Simulations: spectral, interfaces thin tanh, up to periodic b.c. Heavy Light Non-Boussinesq equations

t= t=1.59 Boussinesq, g=0, density is a passive scalar

9.5 t=12.7

time=0 time=12.7 Vorticity

Non-Boussinesq A=0.2

t=4.5

t=5.1

5.73

t=3.2 Notice vorticity contours

t=4.5

5.1

5.73

A=0.12, t = 7.5Г/r c 2 λ ~ 2.5l d (λ stab ~ 4l d ) Viscous simulations: same instability

Re = 8000, Pe = 80000, rho1 = 0.9, rho2 = 1.1 (tanh interface), Circulation=0.8, thickness of the interface = 0.02, rc = 0.1, time = 2.5, N=1024 points Initial condition: Gaussian vortex at a tanh interface

Conclusions: Co-existing instabilities: `forward cascade’ unstable wins Beware of Boussinesq, even at small A What does this do to 2D turbulence?

Single jump: Boussinesq (blue), non-Boussinesq (red) m = 20,  = 0.1

Variation of u r eigenfunction with the jump location: r c = 0.1, m = 2

Effect of large density differences m = 2,  = 1

Reynolds number: Inertial / Viscous forces For inviscid flow, no diffusion of density, Re, Pe infinite 2D simulations of Harish: Boussinesq approximation Peclet number: Inertial / Diffusive Froude number: Inertial / Buoyancy (1/Fr = T I N)‏

Is the flow unstable? Consider radially outward gravity

m

m

Comparison: Boussinesq (blue), non-Boussinesq (red) m = 2,  = 0.1

Governing stability PDE’s:

Component equations Continuity equations Density evolution equations

Background literature:  Studying discontinuities of vorticity / densities or any passive scalar was initiated by Saffman who studies a random distribution of vortices as a model for 2D turbulence and predicted a k -4 spectrum  Bassom and Gilbert (JFM, 1988) studied spiral structures of vorticity and predicted that the spectrum lies between k -3 and k -4  Pullin, Buntine and Saffman (Phys. Fluid, 1994) verify the Lundgren’s model of turbulence based on vorticity spiral

 Batchelor (JFM, 1956) argued that at very large Reynolds number, the vorticity field inside closed streamlines evolves towards a constant value.  Rhines and Young (JFM, 1983) showed that any sharp gradients of a passive scalar will be homogenized at Pe 1/3  Bajer et al. (JFM, 2001) showed that the same holds true for the vorticity field, viz. t homo ~ Re 1/3  Flohr and Vassilicos (JFM, 1997) showed that a spiral structure unique among the range of vorticity distribution. Closed spaced spiral lead to an accelerated diffusion where Dk is the Kolmogorov capacity of the spiral

Density evolution Continuity Navier-Stokes: Boussinesq approximation, radial gravity

Navier-Stokes: Non-Boussinesq equations. For Boussinesq approxmiation,  =  0 Density evolution equation Continuity equation: valid for very high D

Linear stability: mean + small perturbation, e.g.

First: planar approximation, Rayleigh-Taylor instability When U 1 = U 2, always unstable if ρ heavy > ρ light If D=0, growth rate Using kinematic conditions and continuity of pressure at the interface