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Research Papers: Fundamental Issues and Canonical Flows

Strain and Stratification Effects on the Rapid Acceleration of a Turbulent Mixing Zone

[+] Author and Article Information
Benoît-Joseph Gréa

CEA, DAM, DIF,
F-91297 Arpajon, France
e-mail: benoit-joseph.grea@cea.fr

Jérôme Griffond

CEA, DAM, DIF,
F-91297 Arpajon, France

Fabien Godeferd

LMFA, UMR5509,
Ecole Centrale de Lyon,
Université de Lyon, CNRS,
69130 Écully, France

At this stage, we have to add that an effective mixing model is here defined as one that rightly predicts the growth of a TMZ. This is less restrictive than giving the correct values for all turbulent quantities (which of course is often better).

An analogy can be made with sheared and stratified media. The free parameter $S/N$ corresponds to the inverse of the square root of a Richardson number.

We use only the vertical turbulent velocity $u3$ to characterize the ratio $Epot/Ekin$ to be consistent with Eq. (3). If the initial condition were not isotropic, it would be necessary to take into account the kinetic energy in the other directions of course.

Contrary to the sheared problem, in the strained problem a remeshing procedure of the domain is not easily possible as explained in Ref. [30].

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received January 29, 2013; final manuscript received January 24, 2014; published online July 9, 2014. Assoc. Editor: Oleg Schilling.

J. Fluids Eng 136(9), 091203 (Jul 09, 2014) (10 pages) Paper No: FE-13-1059; doi: 10.1115/1.4026856 History: Received January 29, 2013; Revised January 24, 2014

Abstract

We consider the problem of a turbulent mixing zone (TMZ), initially submitted to coupled effects of axisymmetric strain and stratification, then subsequently accelerated. The TMZ grows in the latter stage due to a rapid mixing induced by the Rayleigh-Taylor instability. It is shown that the short time dynamics is simply determined by only two parameters expressing the structure of the turbulent density field, one related to the mixing, the other to the dimensionality of the flow. These quantities are studied by rapid distortion theory and by several homogeneous direct numerical simulations performed in the moving frame of the mean flow. The implications for modeling are discussed, the influence of anisotropy is presented.

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Figures

Fig. 6

Evolution of the dimensionality parameter for two different initializations Ekin≪Epot (a) and Epot≪Ekin (b), from DNS with resolution 5123

Fig. 7

Evolution of the normalized variance of concentration for two different initializations Ekin≪Epot (a) and Epot≪Ekin (b), from DNS with resolution 5123

Fig. 1

Evolution of the buoyancy coefficient Cb = 4 sin2γ(1-Θ) represented in the plane (Nt,St) from the rapid theory. The two images correspond to different initial conditions with (a) Ekin≪Epot and (b) Ekin>>Epot. The straight lines starting from center are associated to the simulations described in Sec. 3.2.

Fig. 2

Evolution of the dimensionality parameter sin2γ represented in the plane (Nt,St) from the rapid theory. The two images correspond to different initial conditions with (a) Ekin≪Epot and (b) Ekin>>Epot. The straight lines starting from center are associated to the simulations described in Sec. 3.2.

Fig. 3

Evolution of the normalized variance of concentration (buoyancy) proportional to (1-Θ) (see Eq. (7)) and represented in the plane (Nt,St) from the rapid theory. The two images correspond to different initial conditions with (a) Ekin≪Epot and (b) Ekin>>Epot. The straight lines starting from center are associated to the simulations described in Sec. 3.2.

Fig. 4

Visualization of the turbulent density field in physical space on simulations: from (a) through (c) the initialization from HIT, a case of axisymmetric contraction (case N16S4mpot at St = -1) and a case of axisymmetric expansion (case N16mS4pot at St = 1)

Fig. 5

Evolution of the buoyancy coefficient for two different initializations Ekin≪Epot (a) and Epot≪Ekin (b), from DNS with resolution 5123

Fig. 8

Comparison to RDT of Cb derived from DNS. (a) corresponds to axisymmetric contraction unstable stratification with S/N = -0.5 (simulations N16S4mpot, N8S2mpot, N4S1mpot corresponding to initial Froude number, respectively, Fr = 0.08,0.16,0.32). (b) corresponds to axisymmetric expansion, stable stratification also with S/N = -0.5 (simulations N16mS4pot, N8mS2pot, N4mS1pot corresponding to initial Froude number, respectively, Fr = 0.08,0.16,0.32).

Fig. 10

Comparison to RDT of 〈cc〉 derived from DNS. (a) corresponds to axisymmetric contraction unstable stratification with S/N = -0.5 (simulations N16S4mpot, N8S2mpot, N4S1mpot corresponding to initial Froude number, respectively, Fr = 0.08,0.16,0.32). (b) corresponds to axisymmetric expansion, stable stratification also with S/N = -0.5 (simulations N16mS4pot, N8mS2pot, N4mS1pot corresponding to initial Froude number, respectively, Fr = 0.08,0.16,0.32). Inset: The evolutions of the Froude number function of Nt.

Fig. 9

Comparison to RDT of sin2γ derived from DNS. (a) corresponds to axisymmetric contraction unstable stratification with S/N = -0.5 (simulations N16S4mpot, N8S2mpot, N4S1mpot corresponding to initial Froude number, respectively, Fr = 0.08,0.16,0.32). (b) corresponds to axisymmetric expansion, stable stratification also with S/N = -0.5 (simulations N16mS4pot, N8mS2pot, N4mS1pot corresponding to initial Froude number, respectively, Fr = 0.08,0.16,0.32).

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