Laurent Joly, Jérôme Fontane & Jean Noël Reinaud. The 2D mode of the variable-density Kelvin-Helmholtz billow â APS DFD â Salt Lake City, November 2007 ...
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
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.
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Outline of the talk Previous works and motivations The variable-density KH-billow Results of the stability analysis Focus on the 2D instability Non-linear continuation of a 2D mode
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
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 ◮ 3D simulations Knio and Ghoniem (1992), Joly et al. (2001) Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
The Homogeneous KH-billow Constant-density Vorticity pattern in the KH roll-up
(a)
y
x Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Vorticity production in variable-density flows dt ω = (ω · ∇)u −
1 ∇P × ∇ρ − (∇· u)ω + ν∆ω ρ2
◮
Dilatation/compression,
◮
Vortex stretching,
(collapses in 2D)
◮
Baroclinic torque,
(the main bias considered here)
b=
1 ∇P × ∇ρ ρ2 Cρ =
∆ρ ρ0
(negligible at low Mach number)
b = O(
u 2 Cρ ) ℓ λρ
|Cρ | 6 1
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Spatial organization of the baroclinic torque
vorticity deposition along the braid between adjacent primary roll-ups :
sink on the denser side, source on the less-dense one Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
The Variable-Density KH-billow Reynolds number Re = 1500, density contrast from 0 to 0.5
y
y
x
y
x
x (f)
(e)
(d)
y
(c)
(b)
(a)
y
x
y
x
x
Are the secondary modes affected by the vorticity distribution ? Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Formulation of the stability problem J. Fontane PhD Thesis ◮
Modal representation of 3D perturbations u ˆ = (ˆ u , vˆ , w ˆ ), pˆ, ρˆ : [ˆ u , vˆ , w ˆ , pˆ, ρˆ] (x, y , z, t) = [˜ u , v˜ , w ˜ , p˜, ρ˜] (x, y )e i(µx+kz)+σt
◮
Linearized NS equations around the base state U = (U, V ), P, R : Dt ρˆ + u ˆ · ∇R RDt u ˆ + Ru ˆ · ∇U + ρˆU · ∇U
◮
= 0, = −∇ˆ p+
1 ∆ˆ u. Re
Quasi-static approach (KP91) and a posteriori validation by σr ≫ σKH
◮
Discrete eigenvalues σ issued from a Fourier-Galerkin method carried out for varying spanwise wave number k > 0
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Growth rates versus k for increasing Cρ Stability analysis performed at saturation of the primary KH-wave 0.25
0.25 (a)
0.25 (b)
0.2
(c)
0.2
0.15
0.2
0.15
0.15
σ
σ
σr
r
r
0.1
0.1
0.1
0.05
0.05
0.05
0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0
5
0.5
1
1.5
2
k
2.5
3
3.5
4
4.5
0 0
5
0.5
1
1.5
2
k
0.25 (d)
2.5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
k
0.25
0.25 (e)
0.2
0.2
0.2
0.15
0.15
0.15
σ
σ
r
σ
r
r
0.1
0.1
0.1
0.05
0.05
0.05 (f)
0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0
0.5
k
1
1.5
2
2.5
k
3
3.5
4
4.5
5
0 0
0.5
1
1.5
2
2.5
k
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
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
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Structure of the 2D mode Maximum density contrast Cρ = 0.5, amplification rate σr = 0.237 (b)
k=0
y
(d)
k=0
y
(f)
k=0
y
B A
x
Energy
x
Vorticity perturbation
x
Density perturbation
Kelvin-Helmholtz instability of the stretched vorticity density-gradient layer To follow : non-linear continuation of the 2D mode seeded before saturation
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Evolution of the vorticity field
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Evolution of the vorticity field
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Evolution of the vorticity field
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Evolution of the vorticity field
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Evolution of the vorticity field
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Evolution of the vorticity field
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Evolution of the density field
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Evolution of the density field
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Evolution of the density field
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Evolution of the density field
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Evolution of the density field
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
Evolution of the density field
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
Outline
Why
VDKH billows
Stability results
The 2D mode
Beyond linearity
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 Fractal Kelvin-Helmholtz break-ups – Poster number 80
Laurent Joly, Jérôme Fontane & Jean Noël Reinaud The 2D mode of the variable-density Kelvin-Helmholtz billow – APS DFD – Salt Lake City, November 2007
