Direct numerical simulation–based Reynolds-averaged closure for bubble-induced turbulence

Budgets of the turbulent kinetic energy from direct numerical simulations (DNSs) of disperse bubbly channel flows are used to develop a model for bubble-induced turbulence in the Euler-Euler Reynolds-averaged framework. First, an appropriate time scale is selected. Second, links between the unclosed terms in the transport equations of the turbulence quantities and the DNS data for small bubbles are established. Third, a suitably chosen iterative procedure employing the full Reynolds-averaged model provides suitable coefficients for the closure of the terms resulting from bubble-induced turbulence while largely removing the influence of others. At the same time these results validate the closure, exhibiting very good agreement with the DNS and better performance than the standard closures. The model is now ready for use and can be employed in practical Euler-Euler simulations.

Figure 1

Schematic representation of the DNS configuration (not to scale). The marked region indicates the location where the spatial energy spectra were computed.

Figure 2

Spectra of liquid velocity from DNS and experiment of bubbly flow, with vertical lines illustrating the reference scales indicated. (a) Spatial spectra from DNS SmMany. (b) Temporal spectra from DNS SmMany. (c) Temporal spectra from the experiment of [27] and DNS of [17], labeled Expt. and DNS, respectively. Here - - - - - - - denotes $u_r/d_p$ from experiment and $\text{–}·\text{–}·\text{–}$ denotes $u_r/d_p$ from DNS.

Figure 3

Comparison between the interfacial term $S_k$ according to (3) and the a priori evaluation of $S_k^{\text{RANS}}/C_I$ (5) for all cases, both normalized with $U_b^3/H$.

Figure 4

One-point statistics from the present EE RANS model and DNS data for SmMany. (a) Gas void fraction. (b) Liquid streamwise velocity and gas streamwise velocity. (c) Liquid TKE and comparison with standard models.

Figure 5

Comparison of the present EE RANS model with the DNS data for the case of SmMany for (a) the $k$ budget with all terms normalized by $U_b^3/H$ and (b) the $\omega$ budget with all terms normalized by $U_b^2/H^2$.

Figure 6

(a) Comparison of $k$ from simulations with the present model to DNS data for LaMany, SmFew, and BiDisp. (b) Values of $C_I$ as a function of ${\text{Re}}_p$ for all cases considered. The dashed line shows the fit. (c) Values of $C_{\varepsilon }/C_D$ as a function of $C_D$ for all cases considered. The dashed line shows the fit.

Figure 7

Comparison of the simulation RANS2 ($C_I=0.4$ and $C_{\varepsilon }=0.17$) with DNS data for the case of SmMany for (a) liquid TKE, (b) the $k$ budget normalized by $U_b^3/H$, and (c) the $\omega$ budget normalized by $U_b^2/H^2$.