Investigation of droplet motion in turbulent flows by a VoF-DNS method | NHR-Verein: Wir sind Hochleistungsrechnen

For the simulation of turbulent dispersed liquid-liquid flows at large scales, coalescence and breakup of droplets is approximated with sub-grid scale closures. For these closures, the root mean square (RMS) droplet fluctuation velocity Urms,d is a decisive input quantity. Recently, Solsvik & Jakobsen [1] proposed an enhanced model to predict Urms,d, which has not been verified yet. Hence, Direct Numerical Simulations (DNS) together with a Volume-of-Fluid (VoF) approach were employed to study the motion of single droplets in a Forced Homogeneous Isotropic Turbulent (FHIT) flow. A parameter study was conducted to investigate the effect of the initial droplet diameter D on Urms,d, and the DNS results were used to assess the model from [1].

Body:

Method

The software OpenFOAM ESI-v2112 with a segregated pressure-based time-implicit flow solver is used. A second-order central difference scheme for spatial discretization is used. A low Courant-Friedrich-Lewy number below 0.05 and a second-order Crank-Nicolson scheme enabled the omission of non-linear outer loops of the predictor-corrector method. A cubic computational domain with edge length L=2π [mm], periodic boundaries and 300³=27 Mio. cubical cells with a constant cell width Δx in each direction is employed (Figure 1). A Taylor-scale Reynolds number of Re_λ≈58 is investigated. An accurate spatial resolution by a ratio of Δx and the Kolmogorov length η below 2.1 [2] is evaluated by on-the-fly monitoring [3]. A linear forcing term [4] is customized and applied to drive the FHIT.

Preliminary investigations on a static droplet test case showed that the algebraic VoF method [6] was outperformed by the geometric VoF method by Scheufler & Roenby [5], since the magnitude of spurious currents was reduced by more than a factor of 10. Thus, the geometric VoF method is applied for the two-phase FHIT simulations. The droplet is initialized in statistically converged single-phase flow after 50 eddy turn-over times (single-phase flow precursor simulation). Density and viscosity ratio between the continuous and droplet phase are kept at unity. The Weber number is kept at a small value of 0.5 to prevent droplet breakup. The initial droplet diameter D was varied between approximately 10η and 50η, to cover a broad range of the spectrum of turbulent length scales. Statistical convergence of U_rms,d was achieved after 30 eddy turn-over times.

Results and Discussion

A comparison of the spectral energy E of our single-phase flow DNS results with the model spectrum by Pope [2] and the DNS results by Komrakova [7] in Figure 2a illustrates the reasonability of our FHIT simulation method. Our two-phase DNS results in terms of the RMS droplet fluctuation velocity U_rms,d show alarge disparity to the results obtained by the model equation [1]. According to our DNS results, U_rms,d decreases monotonically with increasing droplet diameter, in contrast to the model results. Thus, the DNS results provide suggestions for enhancing the model equation from [1] for the flow conditions at hand. In our future research, we plan a variation of the viscosity ratio between the droplet and the continuous phase and an increase of Re_λ by employing an LES.

[1] Solsvik, J. & Jakobsen, H. A. (2016). A Review of the Statistical Turbulence Theory Required Extending the Population Balance Closure Models to the Entire Spectrum of Turbulence. AIChE J., 62(5), 1795-1820. DOI: 10.1002/aic.15128.

[2] Pope, S. B. (2000). Turbulent flows. Cambridge University Press.

DOI: 10.1017/CBO9780511840531.

[3] Cadieux, F.; Sun, G. & Domaradzki, J. A. (2017). Effects of numerical dissipation on the interpretation of simulation results in computational fluid dynamics. Comput. Fluids, 154, 256-272. DOI: 10.1016/j.compfluid.2017.06.009.

[4] Bassenne, M.; Urzay, J.; Park, G. I. & Moin, P. (2016). Constant-energetics physical-space forcing methods for improved convergence to homogeneous-isotropic turbulence with application to particle-laden flows. Phys. Fluids, 28(3), 035114. DOI: 10.1063/1.4944629.

[5] Scheufler, H. & Roenby, J. (2023). TwoPhaseFlow: A Framework for Developing Two Phase Flow Solvers in OpenFOAM. OpenFOAM J, 3, 200-224. DOI: 10.51560/ofj.v3.80.

[6] Deshpande, S. S.; Anumolu, L. & Trujillo, M. F. (2012). Evaluating the performance of the two-phase flow solver interFoam. Comput. Sci. Disc, 5(1), 014016. DOI: 10.1088/1749-4699/5/1/014016.

[7] Komrakova, A. E. (2019): Single drop breakup in turbulent flow. Can. J. Chem. Eng., 97(10), S. 2727-2739. DOI: 10.1002/cjce.23478.