Stratification perturbation of a turbulent mixing. Effect on small

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[Proceeding] Stratification perturbation of a turbulent mixing. Effect on smallscale anisotropy
Original Citation:
Gallana, Luca; Iovieno, Michele; De Santi, Francesca; Richiardone, Renzo; Tordella, Daniela
Stratification perturbation of a turbulent mixing. Effect on small-scale anisotropy. In: 24th
International Congress of Theoretical and Applied Mechanics, Montreal (Canada), 21-26 Agosto
2016.
(In Press)
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XXIV ICTAM, 21-26 August 2016, Montreal, Canada
STRATIFICATION PERTURBATION OF A TURBULENT MIXING. EFFECTS ON
SMALL-SCALE ANISOTROPY
Luca Gallana1 , Michele Iovieno1 , Francesca De Santi1 , Renzo Richiardone2 , and Daniela Tordella ∗1
1
Dipartimento di Ingegneria Meccanica ed Aerospaziale, Politecnico di Torino, Torino, Italy
2
Dipartimento di Fisica, Università di Torino, Torino, Italy
Summary Modification of the small scale anisotropy generation inside a sheafree turbulent mixing following an impulsive local perturbation
of the temperature is observed. In PRL 2011, it was shown that the anisotropy induced by the presence of a kinetic energy gradient has a
very different pattern from the one generated by an homogeneous shear. The longitudinal velocity derivative moments departed from the
values found in homogeneous and isotropic turbulence with skewness variation of opposite sign for the components across the mixing layer
and parallel to it. Here, we show, first, that the stably stratified perturbation higly enhance the anisotropy of the neutral case. Second, that
the unstable perturbation behaves somehow counter-intuitively. While a tendence toward isotropy could be expected, very high anisotropy
is found. Compression on fluid filaments parallel to the mixing layer stays higher than that of orthogonal filaments, the opposite of what
happens in neutral and stably stratified cases.
x3
stable unstable
0.065
0.8
Fr2 = 69.2
Fr2 = 4.2
Fr2 = −4.2
0.06
0.6
hu3 Ei/E03/2
0.055
E/E0
0.05
0.4
0.045
±
Fr2 = 69.2
Fr2 = 4.2
Fr2 = −4.2
0.2
0.04
0.035
0
0.03
-0.2
0.025
0.02
-0.4 -0.3 -0.2 -0.1
E
(a)
(b)
-0.4
0
x3 /L
0.1
0.2
0.3
0.4
-4
-3
-2
-1
0
x3/δ
1
2
3
4
(c)
Figure 1: (a) Initial contition configuration for the mean kinetic energy distribution (black line) and the tempoerature profile
in case of stable and unstable stratification. (b) Mean kinetic energy distribution across the mixing. (c) Flux of kinetic energy
across the mixing.
(a) Stable strat. – Fr2 = 4.2
(b) Unstable strat. – Fr2 = −4.2
Figure 2: Streamlines for the stable and unstable
stratification perturbation after 6 time scales. The
starting position of each streamline is placed at a
fixed distance (equal to 2 times the initial mixing thickness) above (yellow/red tubes) and below
(cyan/blue tubes) the center of the interface. Color
saturation is proportional to the distance from the
starting position. In the stable case streamlines
crossing of the mixing layer are very few; the contrary can be seen for the unstable perturbation.
We study the evolution of a stratified mixing interface considering an initial value problem, in which two homogeneous
isotropic turbulences, with different level of energy (E1 /E2 = 6.7), interact inside a mixing sublayer. The stratification is
taken into account by an initial perturbation on the temparature distribution. The relevant parameter is the square Froude
number Fr2 = u2 /(`2 N 2 ) (where u2 is the velocity variance, ` the integral scale and N the Brunt-Väisälä frequency).
Here we consider, aside the non-stratified case (Fr2 = 69), both a stable (Fr2 = 4.2) and an unstable (Fr2 = −4.2) local
perturbation. Our simulations were performed though Direct Numerical Simulation by applying the Boussinesq approximation
to the Navier-Stokes momentum and energy equations together with a passive scalar transport equation [7].
The evolution of the system can be split in two main stages, according to the evolution of the ratio between buoyancy force
and kinematic forces (that are advection and diffusion). As long as the ratio remains small, there are no significant differences
∗ Corresponding
author. Email: [email protected]
0
-0.2
S (∂ui /∂xi)
S (∂ui /∂xi)
0
-0.2
-0.4
-0.6
-0.8
-1
S (∂u3 /∂x3 )
S (∂u1,2 /∂x1,2 )
-1.2
-0.4
-0.6
-0.8
-1
S (∂u3 /∂x3 )
S (∂u1,2 /∂x1,2 )
-1.2
-1.4
-1.4
-4
-3
-2
-1
0
1
2
3
4
-4
-3
-2
-1
x3/δ
-20
-40
-60
-80
Fr2 = 69.2
Fr2 = 4.2
Fr2 = −4.2
-120
-4
-3
-2
-1
0
1
2
3
4
3
4
(b) Reλ = 150, [6]
1
2
3
4
x3/δ
(c) Normalized third moment of longitudinal derivatives normal to the interface
h ∂u1,2/∂x1,2 3iδ30/hu31,2i
h(∂u3 /∂x3)3iδ30/hu33i
(a) Reλ = 45, [6]
-100
0
x3/δ
-50
-55
-60
-65
-70
-75
-80
-85
-90
-95
Fr2 = 69.2
Fr2 = 4.2
Fr2 = −4.2
-4
-3
-2
-1
0
1
2
x3/δ
(d) Normalized third moment of longitudinal derivatives parallel to the interface
Figure 3: (Above: longitudinal derivative of the skewness along the direction normal to the mixing (green lines) and parallel to
the interface (blue line) at Reλ = 45 (a) and Reλ = 150 (b), data from [6]. Below: third order moments derivative normalized
with the kinetic energy for the present case (Reλ = 250): (c) longitudinal derivative normal to the interface, (d) longitudinal
derivative parallel to the interface. Statistics at about 6 initial eddy turnover times.
with respect to a non-stratified case. On the contrary, as the stratification perturbation level become higher, buoyancy effects
are no more negligible [5]. The time required to have the buoyancy terms comparable with other forces depends directly on
the stratification intensity. When this condition is reached, it can be observed the formation, in correspondence of the local
temperature perturbation, of a sublayer layer with a pit (in stable cases) or a peak (in unstable cases)of kinetic energy, with
an intensity that depends on the stratification level, and grows over time (fig. 1, b-c). As a consequence, a stable stratification
induces a physical separation between the two external regions, greatly decreasing the interaction between them. On contrary,
in unstable case it can be observed an increment of the kinetic energy inside the mixing region, enhancing the mixing process.
A kinematic visualization of the mixing process is shown in figure 2 through instantaneous stremline patterns (see caption for
details).
The small scale anysotrophy is related to the longitudinal velocity derivative statistics [6]. The behavior of the third
moments of the longitudinal derivatives in the direction normal and parallel to the mixing is shown in figure 3, see panels c
and d. As a reference, panels (a-b) show the longitudinal derivative skewness at Reλ 45 and 150 [6] in the neutral case. With
respect to the isotropic situation, the turbulent mixing process produces a compression of fluid filaments across the layer. Here,
it is interesting to observe, see fig. 3, that any kind of stratification highly enhances the anisotropy. While stable stratification
perturbation greatly enhances the neutral anisotropy structure, the unstable one does not show the expected tendence toward
isotropy. In fact, compression on fluid filaments parallel to the mixing layer remains much higher than that of orthogonal
filaments, the opposite of what happens in the neutral and stably stratified cases.
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