The two-dimensional mode of the variable-density Kelvin-Helmholtz billow

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an author's

https://oatao.univ-toulouse.fr/608

Joly, Laurent and Fontane, Jérôme and Reinaud, Jean The two-dimensional mode of the variable-density

Kelvin-Helmholtz billow. (2007) In: 60th Annual Meeting of the Divison of Fluid Dynamics, 18 November 2007 - 20

November 2007 (Salt Lake City, United States). (Unpublished) Item availablity restricted.

The two-dimensional mode of the variable-density

Kelvin-Helmholtz billow

Laurent Joly

1

, Jérôme Fontane

1

and Jean Noël Reinaud

2

(1) ENSICA, Département de Mécanique des Fluides, Toulouse, France.

(2) Mathematical Institute, University of St Andrews, St Andrews, UK.

Outline of the talk

Previous works and motivations

The variable-density KH-billow

Results of the stability analysis

Focus on the 2D instability

Secondary modes of the shear-layer

Unequally addressed situations

Homogeneous mixing-layer

Standard scenario of the transition

◮

Core-centered

elliptic modes by translative instability,

Pierrehumbert and Widnall (1982)

◮

Braid-centered

hyperbolic modes, stream-wise vortices,

Metcalfe et al ; (1987)

Stratified mixing-layer

Specific biases

on secondary modes,

Klaasen and Peltier (1991)

Variable-density mixing-layer

No such deductive analysis

The Homogeneous KH-billow

Constant-density Vorticity pattern in the KH roll-up

x

y

Vorticity production in variable-density flows

d

t

ω

=

(ω · ∇)u

−

1

ρ

2

∇

P

× ∇ρ

−

(∇· u)ω

+ ν∆ω

◮

Dilatation/compression,

_{(negligible at low Mach number)}

◮

Vortex stretching,

_{(collapses in 2D)}

◮

Baroclinic torque,

_{(the main bias considered here)}

b

₌

1

ρ

2

∇

P

× ∇ρ

C

ρ

=

∆ρ

ρ

0

b

_{= O(}

u

2

ℓ

C

ρ

λ

ρ

)

|C

ρ

| 6 1

Spatial organization of the baroclinic torque

vorticity deposition along the braid between adjacent primary roll-ups :

The Variable-Density KH-billow

Reynolds number Re = 1500, density contrast from 0 to 0.5

x y (a) x y (b) x y (c) x y (d) x y (e) x y (f)

Formulation of the stability problem

J. Fontane PhD Thesis

◮

_{Modal representation of 3D perturbations ˆ}

u

_{= (ˆ}

u,

ˆ

v ,

w

ˆ

), ˆ

p,

ρ

ˆ

:

[ˆ

u,

ˆ

v ,

w ,

ˆ

p,

ˆ

ρ] (x, y , z, t) = [˜

ˆ

u,

˜

v ,

w ,

˜

p,

˜

ρ] (x, y )e

˜

i

(µx+kz)+σt

◮

_{Linearized NS equations around the base state U = (U, V ), P, R :}

D

t

ρ

ˆ

+ ˆ

u

· ∇R

= 0,

RD

t

ˆ

u

+ R ˆ

u

· ∇U + ˆ

ρU

· ∇U

= −∇ˆ

p

+

1

Re

∆ˆ

u

.

◮

_{Quasi-static approach (KP91) and a posteriori validation by}

σ

r

≫ σ

_KH

◮

_{Discrete eigenvalues σ issued from a Fourier-Galerkin method carried}

Growth rates versus k for increasing C

ρ

Stability analysis performed at saturation of the primary KH-wave

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.05 0.1 0.15 0.2 0.25 k σr (a) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.05 0.1 0.15 0.2 0.25 k σr (b) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.05 0.1 0.15 0.2 0.25 k σr (c) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.05 0.1 0.15 0.2 0.25 k σr (d) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.05 0.1 0.15 0.2 0.25 k σr (e) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.05 0.1 0.15 0.2 0.25 k σr (f)

Summary on secondary eigen-modes

At saturation of the primary KH-wave

Elliptic modes

Core centered modes insensitive to base flow

modifications,

Hyperbolic modes

Braid centered modes are promoted for growing

density contrasts,

Hyperbolic modes

Lying preferentially on the vorticity-enhanced

part of the braid,

Two-dimensional modes

In fair competition with most amplified 3D

modes beyond Cρ

= 0.4

Structure of the 2D mode

Maximum density contrast C

= 0.5, amplification rate σ

= 0.237

k = 0 (b) x y B A k = 0 (d) x y k = 0 (f) x y

Energy

Vorticity perturbation

Density perturbation

Kelvin-Helmholtz instability of the stretched vorticity density-gradient layer

Summary on secondary two-dimensional modes

Fair competition with 3D braid modes

assessment of the 2D mode

relevancy complementary to the Lagrangian inviscid

simulations by Reinaud et al. (2000)

Increase in mixing rate

Non-linear simulations indicate a jump in

exponential increase of the length of the central

isopycnic line,

Generic mechanism

2D instability of variable-density roll-up

condensing onto thin unstable shear layers → 2D

cascade