3.3 LES with the Single-Phase Model
3.3.1 Baseline Case E4-T118
Figure 3.4a depicts an instantaneous snapshot of the temperature distribution for case E4-T118 (TN2 = 118 K). Contour levels are shown for 118 K< T <300 K, from dark to light shades, superimposed by a second group of contour levels with 110 K< T <118 K, from blue to red shades. Cryogenic nitrogen at118 K can be identified as ’dark core’, surrounded by a co-flow of warm hydrogen at270 K. It is interesting to see that the temperature within in the mixing layer drops below its inflow value of 118 K with a minimum value of approximately 110 K. A related effect can be observed also for the hydrogen partial densityρH2, which is depicted in Fig. 3.4b. Hydrogen is injected with a density ofρH2 = 3.55 kg/m3 and reaches a much higher partial density of almostρH2 = 5.35 kg/m3within the mixing layer.
In order to analyze changes in the thermodynamic state throughout the mixing process in greater depth, we show a scatter plot of temperature as function of hydrogen mole fractionzH2 in Fig. 3.4c for the same dataset as used for Fig. 3.4a.
On the left-hand side (zH2 = 0), pure nitrogen at either 118 K (injection temper-ature) or 298.15 K (reservoir temperature) can be identified. On the right-hand side (zH2 = 1), we observe pure hydrogen at 270 K. In between, either cryo-genic nitrogen from the main injection (follow the line labeled ’adiabatic mixing temperature’) or warm nitrogen from the reservoir (follow the straight line, top of the figure) is mixed with warm hydrogen. The line labeled ’adiabatic mixing temperature (TF)’ is calculated analytically as solution to the adiabatic mixing model, see Sec. 2.3.1 for details. This analytical model (which neglects the kinetic energy of the flow and the transport of heat and mass) shows, that the temper-ature decrease in the mixing layer can be attributed to real-gas mixing effects, i.e., non-linear interactions between molecules introduced via mixing rules, and not, as one could also assume, to heat transfer and diffusion mechanisms or to compressibility effects. The isobaric mixing of cryogenic nitrogen and warm hy-drogen is endothermic. The scattered points in Fig. 3.4c are colored in gray-scale with the streamwise distance from the injector. We observe a fast mixing of the warm hydrogen with the warm atmospheric nitrogen such that forx/Di >1.2 no pure hydrogen (zH2 <1) is present in the chamber. Therefore, scatter away from the adiabatic mixing line can be attributed to mixing with the warm atmospheric nitrogen and transport phenomena (e.g., heat and mass diffusion) not covered by the model assumptions. In the scatter plot of Fig. 3.4c it is indicated that the LES data include thermodynamic mixture states that lie within the mixture two-phase region, although the operating pressure is well above the critical pressure of the pure components. This aspect is discussed separately in Sec. 3.4.
A corresponding scatter plot for the hydrogen partial density is shown in Fig. 3.4d.
3.3 LES with the Single-Phase Model
(a) Snapshot of the temperature field.
ρH2[kg/m3]
(b)Snapshot of the hydrogen partial density (ρH2=YH2·ρ) field.
(c) Binary phase diagram together with the adiabatic mixing temperature. The scat-tered points, colored with the streamwise distance from the injector, depict the ther-modynamic states that are obtained in the LES (instantaneous data, taken from Fig. 3.4a).
(d)Scatter plot of the hydrogen partial den-sity ρH2, colored with temperatureT, as function of the hydrogen mole fraction zH2. The dashed line labeledρH2(TF) cor-responds to the hydrogen partial density calculated along the adiabatic mixing line TF.
Figure 3.4: Instantaneous contour and scatter plots for baseline case E4-T118 with the assumed single-phase model.
Again, there is pure nitrogen (meaning ρH2 = 0) on the left hand side and pure hydrogen at its corresponding density at injection ρH2 = 3.55 kg/m3 on the right hand side. The scattered data is colored by temperature in the same way as done for Fig. 3.4a. The dashed line labeled ρH2(TF) now corresponds to the hydro-gen partial density (ρH2 = YH2 ·ρ) calculated along the adiabatic mixing line of Fig. 3.4c. The scattered data follow closely the analytical solution, and again, this result shows that, in the LES, the hydrogen partial density exceeds its pure com-ponent density due to real-fluid mixing effects. We note that the overall hydrogen partial density calculated along the adiabatic mixing line ρH2(TF) as shown in
0
Figure 3.5: Axial (centerline) and radial nitrogen hρN2iand hydrogen hρH2i par-tial density profiles for test case E4-T118 with single-phase thermody-namics. ( ) LES results; ( ) experimental data of Oschwald et al. (1999). Radial profiles are extracted at 4 mm. Note that Fig. 3.5c displays the maximum values of the mean radial hydrogen density distributionmax{hρH2i(r)}as a function of the distance from the injector that have been recorded experimentally. ( ) LES data at the corresponding streamwise locations.
Fig. 3.4d is not sensitive to meaningful values of the binary interaction parameter k0ij.
In the following, we compare our numerical results with the experimental data of Oschwald et al. (1999). A total time interval of 20ms has been simulated for all simulations, which corresponds to5flow through times (FTT) with respect to the nitrogen bulk velocity uN2 = 5 m/s and Lx = 20 mm. A fully developed flow field from coarser grids served as initial solution. Statistical properties have been obtained by averaging in circumferential direction and in time after an initial tran-sient of 6 ms (1.5 FTT). Figures 3.5a and 3.5b depict the axial (centerline) and radial nitrogen density profiles for test case E4-T118. Radial data are extracted at x/Di = 2.1. We observe significant differences in the potential core region (x/Di <3) with experimental and numerical nitrogen densities of∼390.18 kg/m3 and ∼608.78 kg/m3, respectively. Recall Fig. 3.2: a specification of the inflow boundary condition in terms of temperature and pressure must yield a density within the potential core much higher than what is observed experimentally. With an error of about 4% when comparing the PR EOS to the NIST reference data, it becomes apparent that the observed differences in nitrogen density of approxi-mately200 kg/m3 can not be attributed to an inaccurate equation of state (or the
3.3 LES with the Single-Phase Model
simulation) but rather to measurement uncertainties. A more thorough discussion on this issue is given in Sec. 3.3.3.
Figures 3.5c and 3.5d show the corresponding hydrogen partial density profiles.
Figure 3.5c displays the observed maximal values of the mean radial hydrogen density distribution max{hρH2i(r)} at several stations. Oschwald et al. (1999) report an increase in hydrogen density downstream of the jet break-up that ex-ceeds its pure-component value at injection, which is qualitatively reproduced by the numerical simulation. Quantitatively, we observe a very good agreement for x/Di > 6. In the immediate vicinity of the injector, however, large deviations between measured and simulated hydrogen density are undeniably present.