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Research Papers: Fundamental Issues and Canonical Flows

Challenging Mix Models on Transients to Self-Similarity of Unstably Stratified Homogeneous Turbulence

[+] Author and Article Information
Benoît-Joseph Gréa

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

Alan Burlot, Jérôme Griffond, Antoine Llor

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

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received March 27, 2015; final manuscript received September 4, 2015; published online April 12, 2016. Assoc. Editor: Daniel Livescu.

J. Fluids Eng 138(7), 070904 (Apr 12, 2016) (12 pages) Paper No: FE-15-1209; doi: 10.1115/1.4032533 History: Received March 27, 2015; Revised September 04, 2015

The present work aims at expanding the set of buoyancy-driven unstable reference flows—a critical ingredient in the development of turbulence models—by considering the recently introduced “Unstably Stratified Homogeneous Turbulence” (USHT) in both its self-similar and transient regimes. The previously established accuracy of an anisotropic Eddy-Damped Quasi-Normal Markovian Model (EDQNM) on the USHT has allowed us to: (i) build a data set of well defined transient flows from Homogeneous Isotropic Turbulence (HIT) to late-time self-similar USHT and (ii) on this basis, calibrate, validate, and compare three common Reynolds-Averaged Navier–Stokes (RANS) mixing models (two-equation, Reynolds stress, and two-fluid). The model calibrations were performed on the self-similar flows constrained by predefined long range correlations (Saffman or Batchelor type). Then, with fixed constants, validations were carried out over the various transients defined by the initial Froude number and mixing intensity. Significant differences between the models are observed, but none of them can accurately capture all of the transient regimes at once. Closer inspection of the various model responses hints at possible routes for their improvement.

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Figures

Grahic Jump Location
Fig. 1

Visualization of the buoyancy parameter ϑ in a vertical plane of an USHT DNS. The acceleration points to the bottom in the figure. White zones, corresponding to lighter fluid, move upward while heavier darker parcels of fluid move downward.

Grahic Jump Location
Fig. 2

Evolution of nondimensional numbers as a function of time for the different EDQNM simulations corresponding to Table 1; (top) Reynolds number, (middle) Froude number, (bottom) mixing parameter. (Left) Saffman k2 spectra with symbols corresponding to initial Froude number; (circle) Fr = 3.1, (cross) Fr = 0.62, (square) Fr = 31, (diamond) Fr = 0.155, and (star) Fr = 0.078. Lines correspond to the initial mixing parameter; (plain) Λ=0.184, (dashed) Λ=0.92, (dashed–dotted) Λ=1.84, (dotted) Λ=9.2, and (none) Λ=18.4. (right) Batchelor k4 spectra with symbols corresponding to initial Froude number; (circle) Fr = 6.61, (cross) Fr = 1.28, (square) Fr = 0.67, (diamond) Fr = 0.337, and (star) Fr = 0.168. Lines correspond to initial mixing parameter; (plain) Λ=18.1, (dashed) Λ=1.81, (dashed–dotted) Λ=0.184, and (dotted) Λ=9.2.

Grahic Jump Location
Fig. 3

(Top) Time-evolution of the kinetic energy K (plain curve), the variance of buoyancy 〈ϑϑ〉 (dashed curve) and the vertical buoyancy flux 〈u3ϑ〉 (dotted curve) corresponding to run S15. (Bottom): Energy spectrum E(k) (plain curve), buoyancy spectrum B(k) (dashed curve), and vertical velocity-buoyancy spectrum F(k) (dotted curved) as a function of wavenumber k for run S15. Different times are shown corresponding to the initialization at Nt < 0, when stratification effects are applied Nt=0 and during self-similar regime Nt=5. F(k) is nonzero only for t > 0.

Grahic Jump Location
Fig. 4

Nondimensional transition delay NTI as a function of initial Froude and mixing parameter and obtained from Eq. (2) for the k2 Saffman spectra. The square symbols correspond to the simulations points of Table 1.

Grahic Jump Location
Fig. 5

Nondimensional time shift NTs as a function of initial Froude and mixing parameter and obtained from Eq. (2) for the k4 Batchelor spectra. The square symbols correspond to the simulations points of Table1.

Grahic Jump Location
Fig. 6

Transition delays TIKE and time shifts TsKE as a function of the initial mixing parameter Λ (abscissa) and the Froude number Fr (ordinate) obtained from Eqs. (2) and (3) for the K−E model. Also, the differences between EDQNM and the K−E model on transition delays ΔTIKE=TI−TIKE and on time shifts ΔTsKE=Ts−TsKE are represented. Cases for Saffman (top) and Batchelor (bottom) initial spectra are represented.

Grahic Jump Location
Fig. 7

Transition delays TIGSG+ and time shifts TsGSG+ as a function of the initial mixing parameter Λ (abscissa) and the Froude number Fr (ordinate) obtained from Eqs. (2) and (3) for the RSM GSG+ model. Also, the differences between EDQNM and the GSG+ model on transition delays ΔTIGSG+=TI−TIGSG+ and on time shifts ΔTsGSG+=Ts−TsGSG+ are represented. Cases for Saffman (top) and Batchelor (bottom) initial spectra are represented.

Grahic Jump Location
Fig. 8

Transition delays TI2SFK and time shifts Ts2SFK as a function of the initial mixing parameter Λ (abscissa) and the Froude number Fr (ordinate) obtained from Eqs. (2) and (3) for the 2SFK model. Also, the differences between EDQNM and the 2SFK model on transition delays ΔTI2SFK=TI−TI2SFK and on time shifts ΔTs2SFK=Ts−Ts2SFK are represented. Cases for Saffman (top) and Batchelor (bottom) initial spectra are represented.

Grahic Jump Location
Fig. 9

Evolution of the kinetic energy K as a function of time Nt for different cases S* with Saffman initial spectra. (Thick red plain line) EDQNM, (thin plain line) 2SFK, (dashed line) GSG+, and (dotted line) K−E.

Grahic Jump Location
Fig. 10

Evolution of the Froude number Fr=E/KN as a function of time Nt for different cases S* with Saffman initial spectra. (Thick red plain line) EDQNM, (thin plain line) 2SFK, (dashed line) GSG+, and (dotted line) K−E.

Grahic Jump Location
Fig. 11

Evolution of the kinetic energy K as a function of time Nt for different cases B* with Batchelor initial spectra. (Thick red plain line) EDQNM, (thin plain line) 2SFK, (dashed line) GSG+, and (dotted line) K−E.

Grahic Jump Location
Fig. 12

Evolution of the Froude number Fr=E/KN as a function of time Nt for different cases B* with Batchelor initial spectra. (Thick red plain line) EDQNM, (thin plain line) 2SFK, (dashed line) GSG+, and (dotted line) K−E.

Grahic Jump Location
Fig. 13

(Left) Evolution of the vertical buoyancy flux 〈u3ϑ〉, (right) evolution of the mixing parameter Λ as a function of time Nt for the S23 case. (Thick red plain line) EDQNM and (dashed line) GSG+.

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