Development of similarity relationships for energy dissipation rate and temperature structure parameter in stably stratified flows: a direct numerical simulation approach

Abstract

In this study, a newly developed direct numerical simulation (DNS) solver is utilized for the simulations of numerous stably stratified open-channel flows with bulk Reynolds number (Re _b ) spanning 3400–16,900. Overall, the simulated bulk Richardson number ($Ri_b$) ranges from 0.08 (weakly stable) to 0.49 (very stable). Thus, both continuously turbulent and (globally) intermittently turbulent cases are represented in the DNS database. Using this comprehensive database, various flux-based and gradient-based similarity relationships for energy dissipation rate (ε) and temperature structure parameter ($C_T^2$) are developed. Interestingly, these relationships exhibit only minor dependency on Re _b . In order to further probe into this Re _b -effect, similarity relationships are also estimated from a large-eddy simulation (LES) run of an idealized atmospheric boundary layer (very high Re _b ) case study. Despite the fundamental differences in the estimation of ε and $C_T^2$ from the DNS- and the LES-generated data, the resulting similarity relationships, especially the gradient-based ones, from these numerical approaches are found to be remarkably similar. More importantly, these simulated relationships are also comparable, at least qualitatively, to the traditional observational data-based ones. Since these simulated similarity relationships do not require Taylor’s hypothesis and do not suffer from mesoscale disturbances and/or measurement noise, they have the potential to complement the existing similarity relationships.

Appendix: Solver verification

In this Appendix, we first compare the simulation results produced by the current DNS solver (OF) against the DNS results from Nagaosa and Saito [54] (referred to as NS97 henceforth). A digitizer is used to extract the simulated data from the plots reported by Nagaosa and Saito [54]. Comparisons of mean and root-mean-square (rms) velocity, and turbulent momentum-flux for neutrally and stably stratified cases are presented in Fig. 10. One can see that the results from the current solver (OF) agree well with those from NS97 for all cases. The maximal discrepancy is found for $u_{rms}$ at $z/h\approx 0.3$, and the averaged and maximum differences are ~1.5 and ~2.2 %, respectively.

Fig. 10

Comparisons of variables for neutral and stratified flows, $Ri_{*}=0$, 10, 20. The lines are the results by current solver (OF), and the symbols are results from Nagaosa and Saito [54] (NS97). a Mean velocity, b rms streamwise velocity, c rms vertical velocity, and d turbulent momentum-flux

Next, we compare our results with those obtained by a spectral method in order to ascertain that the grid resolution utilized in the present DNS study is sufficient. For this purpose, we use the well-cited DNS results (neutrally stratified case) from Moser et al. [51] (referred to as MKM99 henceforth) as benchmarks. Comparisons of mean and rms velocity, turbulent momentum-flux, and production of turbulence kinetic energy for the neutral case are presented in Fig. 11. One can see that the results produced by the current solver (OF) agree well with those from MKM99 for both $Re_*=180$ and 395. We do point out that the mean velocity ($\overline{u}$, see Fig. 11a) is overestimated in the central-channel region ($z/h\approx 1$) with the averaged and maximum differences of ~2.2 and ~2.5 %, respectively.

Fig. 11

Comparisons of variables for neutral flows, $Ri_*=0$. The lines are the results from the current solver (OF), and the symbols are results from Moser et al. [51] (MKM99). The production of TKE are normalized by ${\nu }/u_{*}^4$. a Mean velocity, b rms velocity, c turbulent momentum-flux, and d production of turbulence kinetic energy

Although there are some differences between the results from the current DNS solver and those from NS97 and MKM99, this level of discrepancy is not unexpected from different DNS solvers (for example, see the code verification in Kasagi et al. [40], Vreman and Kuerten [72] for reference). In a nutshell, the current solver accurately simulates the mean and turbulent components of the flows for neutral and stratified cases. There is no indication that the current grid resolution is insufficient to fully resolve the turbulence phenomena in various stably stratified flows. Without any doubt, these solver verification results builds confidence for the simulations and analyses reported in the rest of the paper.