3.3 LES with the Single-Phase Model
3.3.4 Effect of Numerical Diffusion
In this section, we discuss results of grid convergence studies for the cases E4-T118 and E4-T128 using our baseline high-order numerical scheme as introduced in Sec. 2.4 (ALDM (Hickel et al., 2014) for velocity, pressure and kinetic energy and second-order TVD flux function for mass and internal energy advection, hereafter
3.3 LES with the Single-Phase Model
(a)Snapshots of hydrogen partial density distribution for case E4-T118.
y/Di
(b) Snapshots of hydrogen partial density distribution for case E4-T128.
Figure 3.8: Grid convergence study for the cases E4-T118 and E4-T128 with single-phase thermodynamics and the baseline numerical method SALD: Snapshots of hydrogen partial density distribution.
referred to as SALD) and for case E4-T118 with a more dissipative scheme (second-order upwind biased flux function for all governing equations, hereafter referred to as UW).
Grid Convergence Study with SALD
We uniformly refined (G3) and coarsened (G1) the base grid (G2) by a factor of 2 in all spatial directions (but only in the region of interest). For the highest (G3) and lowest (G1) grid resolution we obtain a total number of∼3.8·106 and∼101.5·106 computational cells, respectively. A single computation for 20 ms on the finest grid required about 3.1×106 CPU-hours. Figure 3.8a and 3.8b depict a contour plot of the instantaneous hydrogen partial density distribution at the different grid refinement levels for the cases E4-T118 and E4-T128, respectively. The qualitative comparison shows that with respect to thermodynamic phenomena all grid levels yield very similar results, however, we see that more fine-scale structure is added to the flow visualizations when the grid is refined and that the dense core becomes longer. A quantitative measurement of grid convergence is obtained from first order statistical moments such as mean density profiles and the integral property dark core lengthLC =x{hρi= 0.5(ρin+ρ∞)}. Figure 3.9 compares time-averaged nitrogen and hydrogen density profiles at different grid refinement levels to the
ex-0
(a) Nitrogen density for cse E4-T118.
0
(b) Hydrogen density for case E4-T118.
0
(c) Nitrogen density for case E4-T128.
0
(d) Hydrogen density for case E4-T128.
Figure 3.9: Grid convergence study for the cases E4-T118 and E4-T128 with single-phase thermodynamics and the baseline numerical method SALD: Mean solution for nitrogen and hydrogen partial densities.
perimental data of Oschwald et al. (1999). We now see clearly with the centerline nitrogen partial density in Fig. 3.9a, case E4-T118, and 3.9c, case E4-T128, that the dense core becomes longer when the grid is refined. In addition, we cannot exclude the possibility that the location of jet break-up is pushed even further downstream on the next finer grid. We obtain LC ={4Di,4.7Di,5.9Di} for case E4-T118 andLC ={4.6Di,5.4Di,6.4Di} for case E4-T128. Note that the value of the nitrogen density in the dense core is of course not affected by grid resolution.
Figures 3.9b and 3.9d show the corresponding hydrogen partial density plot for the cases E4-T118 and E4-T128, respectively. Again, to allow for a comparison to the data provided by the experimentalists, we must plot maximum values of the mean radial hydrogen density distribution max{ρH2(r)} as a function of the distance from the injector. In this representation the hydrogen partial density is not as sensitive to the grid resolution as the nitrogen centerline density. Similar to the instantaneous snapshots we can conclude that with respect to
character-3.3 LES with the Single-Phase Model
(a)Snapshots of hydrogen partial density distribution.
0
Figure 3.10: Grid convergence study for case E4-T118 using a more dissipative numerical scheme (second-order upwind biased flux function for all governing equations): Snapshots of hydrogen partial density distri-bution and mean solution for nitrogen and hydrogen partial densi-ties.
istic thermodynamic phenomena, i.e., exceeding hydrogen density and matching nitrogen density in the dense core, all grid levels yield similar results. The overall quantitative agreement to experimental data does not get better or worse for any of the three grid resolutions. To obtain grid converged results, we tried run LES on the next finer grid G4, however, these LES would come at a computational expense that exceeds our current possibilities. A single computation of case E4-T118 on the next finer grid resolution with 662.9×106 cells would require about
∼38×106 CPUh computing time. Within the current scope of this work nothing essentially new can be learned from higher-resolution LES.
Grid Convergence Study with UW
Figure 3.10 shows results of a grid convergence study using a more dissipative scheme, i.e., second-order upwind biased flux function together with the van Al-bada limiter (van AlAl-bada et al., 1982) for all governing equations. As to be
ex-T[K]
(a) Snapshot of the temperature field. Color map and range are the same as in Fig. 3.4a
0.2
(b)Binary phase diagram together with frozen TF and equilibriumTEQ adiabatic mixing temperature. Scattered points that are located within the two-phase re-gion are colored from blue to red shades by the vapor volume fraction.
Figure 3.11: Instantaneous temperature contours (a) and scatter plot (b) for case E4-T118 with the two-phase model and SALD.
pected, the instantaneous flow fields shown in Fig. 3.10a appear more laminar-like compared the results shown in Fig. 3.8a. In Fig. 3.10b, we show time-averaged nitrogen density profiles at different grid refinement levels. Using a more dissipa-tive scheme leads to a very different jet evolution and convergence rate. While we observed a monotonically increasing position of break-up/dark core length with increasing grid resolution for our high-order scheme SALD, the position of jet break-up is pushed upstream for G1 → G2 and it is again moving downstream for G2 → G3. Moreover, only for the highest grid resolution G3, we observe a fully mixed state at a similar axial position as with our higher-order scheme SALD, compare the gray line in Fig. 3.10b. It is noteworthy that such a non-monotonic convergence behavior can lead to a false conclusion whether or not LES results are grid converged. For this reason, results with the more dissipative scheme that have been published by our group in Müller et al. (2015) must be deemed non grid-converged. However, note that non-ideal thermodynamic phenomena like endothermic mixing or exceeding partial densities can be reproduced with any numerical scheme (and turbulence model) on any reasonably fine computational grid. Qualitative aspects discussed in Müller et al. (2015) remain therefore valid.
Only when it comes to quantitative comparisons, e.g. jet break-up position, jet spreading angle or even higher order statistics such as velocity fluctuations, care needs to be taken.
3.3 LES with the Single-Phase Model
(a)Overall hydrogen partial density.
ρN2[kg/m3]
(b)Overall nitrogen partial density.
ρH2,l=YH2,l·ρl[kg/m3]
(c) Hydrogen partial density in the liquid phase.
(d)Nitrogen partial density in the liq-uid phase.
(e) Hydrogen partial density in the va-por phase.
(f ) Nitrogen partial density in the va-por phase.
(g) Scatter plots of the hydrogen (left) and nitrogen (right) partial density as function of the hydrogen mole fraction. Scattered data located within the two-phase region are colored from blue to red shades with the vapor volume fraction. The black line labeledρi(TEQ)corresponds to the partial overall density of componenticalculated alongTEQ, compare Fig. 3.11b.
Solid red and blue lines represent vaporρi,vand liquidρi,lpartial densities of componenti.