5.2 Assessment of the Scaling Laws in Non-reacting Flows
5.2.2 Modelling Results
A wake structure develops directly downstream of the droplets. Due to the low relative velocities, the droplets do not induce significant small scale turbulence. The superposed turbulent field cannot disturb the quasi-laminar wakes near the droplet.
Few droplet diameters downstream, however, the smallest turbulent eddies start to act on the mixing field and can penetrate the wake. This is shown in Fig. 5.14 for two different turbulence intensities as defined in Tab. 5.3 for cases 1 and 2. The
Figure 5.14: Instantaneous mixture fraction fields on the plane across one layer of droplets for case 1 (left) and case 2 (right).
scaling relationships for the wake and for the zone affected by turbulence are fun-damentally different. The wake is clearly anisotropic and a directional dependence is integral to the analytical solution (see fN DZ (Eq. (2.37)). The distribution of mixture fraction is a function of x1 and also varies along the transversal distance.
This is different when mixing is dominated by the turbulent eddies. Here, scaling relationships assume local isotropy and the mixture fraction distribution is primarily a function of the distance from the plane where the droplets are located (cf. Eq.
(2.40)). The turbulent wake structure is instationary and fluctuates perpendicular to the wake. It is assumed to yield a statistically very weak dependence of mixture fraction on transversal direction [67]. The length of the laminar wake is determined by the intensity of the turbulence. The turbulence of case 1 is stronger than that of case 2, affecting the wake earlier and leading to a reduced area where the near droplet assumption is valid. The scaling relationships now allow to locate the tran-sition between the quasi-laminar wake zone and the zone dominated by turbulent mixing.
Three different cases are tested as introduced in Tab. 5.3 above. Typically, the
5.2. ASSESSMENT OF THE SCALING LAWS IN NON-REACTING FLOWS58 droplet diameters are no larger than the Kolmogorov scale and these scenarios are investigated here. If the scaling relations hold, the length of the near droplet zone is characterized byUdand varyingtkand the transition between the two zones is likely to start near the plane with x1 =Udtk [67], i.e. at locations of x1 = 2.4d, x1 = 4.4d andx1 = 6dfor the three different cases (also see Tab. 5.3). The results presented in Figs. 5.15-5.20 are obtained by temporal averaging over 8 flow through times after stationary conditions have been reached. Figure 5.15 shows the mixture fraction along the transversal direction for different planes. The symbols represent time averaged realizations from case 2 and lines show results from the analytical model as given in Eq. (2.37). It can be observed that the model gives very good agreement
0.05 0.1 0.15 0.2 0.25 0.3
0.5 1 1.5 2 2.5
Mixture Fraction
r/Diameter
x1/d = 2.0, DNS x1/d = 2.0, Analytical model x1/d = 3.0, DNS x1/d = 3.0, Analytical model x1/d = 4.0, DNS x1/d = 4.0, Analytical model x1/d = 5.0, DNS x1/d = 5.0, Analytical model x1/d = 7.0, DNS x1/d = 7.0, Analytical model
Figure 5.15: Comparison of the mean value of mixture fraction along the transversal distance for different planes perpendicular to the mean flow with the near droplet zone analytical model for case 2.
with the DNS data close to the droplet but accuracy diminishes from x1/d= 4 and is poor further downstream. Very similar trends can be observed for the cases with stronger and weaker turbulence, cases 1 and 3 as shown in Figs. 5.16 and 5.17. For the stronger turbulence, the wake structure is disturbed earlier and modelling with Eq. (2.37) only holds for zones x1 < 3d (cf. Fig. 5.16) while the trend for the weaker turbulence, case 3 is opposite as shown in Fig. 5.17. Figure 5.18 compares the scaling relationship for the Kolmogorov zone as presented in Eq. (2.40) with the DNS data. The range of validity of this scaling law can clearly be seen for all three cases. For the higher turbulence levels, the Kolmogorov zone moves closer to the droplet, and the model and DNS data agree well from x1 = 3d onwards. For case 3, the Kolmogorov scaling applies for regions more than 5-6 diameters downstream
0.05 0.1 0.15 0.2 0.25
0.5 1 1.5 2 2.5
Mixture Fraction
r/Diameter
x1/d = 1.5, DNS x1/d = 1.5, Analytical model x1/d = 2.0, DNS x1/d = 2.0, Analytical model x1/d = 3.0, DNS x1/d = 3.0, Analytical model x1/d = 4.0, DNS x1/d = 4.0, Analytical model x1/d = 5.0, DNS x1/d = 5.0, Analytical model
Figure 5.16: Comparison of the mean value of mixture fraction along transversal distance on different planes perpendicular to the mean flow with the near droplet zone analytical model for case 1.
0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
0.5 1 1.5 2 2.5
Mixture Fraction
r/Diameter
x1/d = 3.0, DNS x1/d = 3.0, Analytical model x1/d = 4.0, DNS x1/d = 4.0, Analytical model x1/d = 5.0, DNS x1/d = 5.0, Analytical model x1/d = 6.0, DNS x1/d = 6.0, Analytical model x1/d = 7.0, DNS x1/d = 7.0, Analytical model
Figure 5.17: Comparison of the mean value of mixture fraction along transversal distance on different planes perpendicular to the mean flow with the near droplet zone analytical model for case 3.
of the droplet. Thus the results in Figs. 5.15-5.18 demonstrate that the transition starts within the interval 2 < x1/d < 3, 4 < x1/d < 5 and 5 < x1/d < 6 for the three cases, respectively, and these are consistent with the predictions by the scaling relationship as shown in Tab. 5.3. It can be stated that
1. the scaling laws approximate the DNS data well in their respective zones, 2. the transition regions between the regions are rather small and
5.2. ASSESSMENT OF THE SCALING LAWS IN NON-REACTING FLOWS60
0.02 0.04 0.06 0.08 0.1
2 3 4 5 6 7 8
Mixture Fraction
x1/Diameter
case 1, DNS case 1, Analytical model case 2, DNS case 2, Analytical model case 3, DNS case 3, Analytical model
Figure 5.18: Comparison of average mixture fraction variation along longitudinal distance with analytical model in the Kolmogorov scale zone for three cases.
3. the location where the transition is likely to start can be relatively well es-timated by the product of the Kolmogorov time scale and the relative mean velocity Ud.
Scalar dissipation modelling for the near droplet zone was investigated in [159]
and is not repeated here. We focus instead on the Kolmogorov zone and compare the conditionally averaged scalar dissipation, Nf = 2D(∇f)2, from the DNS with the scaling relationship given by Eq. (2.42). As can be seen in Fig. 5.19, case 1 reveals the highest scalar dissipation rates due to the highest turbulence levels. This is reflected in Eq. (2.42) by the proportionality of the scalar dissipation to energy dissipation. For mixture fraction smaller than 0.02, the scalar dissipation rates are similar for all three cases. Figure 5.19 also demonstrates that the conditionally averaged scalar dissipation is very well captured by the scaling relationship and could serve as a model for flamelet or CMC closures in turbulent spray flames.
Finally, Fig. 5.20 shows a comparison of the PDF obtained from the DNS with the scaling laws applicable to the Kolmogorov zone, Eq. (2.44). The PDFs are constructed from DNS data with x1 > Udτη. Again, the analytical expression cap-tures the dependence of the turbulence field quite well and provides a quantitively satisfactory approximation of the data. It is noted that cases 2 and 3 do not show much difference which may be attributed to the rather late disturbance of the quasi-laminar wake structure. The consistency of the results between scalar dissipation (Fig. 5.19) and the PDF modelling (Fig. 5.20) demonstrates the validity of their quasi-steady-state relationships. These assumptions are often applied to the
0 0.5 1 1.5 2 2.5
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Scalar dissipation (1/s)
Mixture fraction
case 1, DNS case 1, Analytical model case 2, DNS case 2, Analytical model case 3, DNS case 3, Analytical model
Figure 5.19: Comparison of scalar dissipation with the analytical model in the Kol-mogorov scale zone for three cases.
oration of droplet clouds and simulated here, but some care should be taken since quasi-steady state conditions may be inaccurate in real droplet clouds.
0 10 20 30 40 50 60 70 80
0.02 0.03 0.04 0.05 0.06
Mixture fraction
case 1, DNS case 1, Analytical model case 2, DNS case 2, Analytical model case 3, DNS case 3, Analytical model
Figure 5.20: Comparison of PDF with the analytical model in the Kolmogorov scale zone for three cases.