El lenguaje y sus perspectivas

Supersonic gas jets generated by de Laval nozzles are ideal targets for laser-plasma experi- ments due to their flat-top density profiles and steep density gradients at the jet edges, both of which are impossible to produce with a (subsonic) cylindrical nozzle. The variation of the most important flow parameters along the nozzle axis of a de Laval nozzle with a throat diameter of 1 mm, an exit diameter of 3 mm and a length of the diverging nozzle section of 6 mm are shown in figure 2.4. The medium is Helium with a backing pressure of 50 bar. For comparison with a cylindrical nozzle, approximate values for all flow parameters shown in 2.4 for the case of a cylindrical nozzle can be obtained by simply taking the val- ues at the nozzle throat. This is a quite good approximation because approximately there the flow becomes supersonic and becomes, therefore, independent of upstream conditions. Since for the de Laval nozzle the expansion of the gas mainly happens inside the guid- ing nozzle wall, the acceleration caused by the expansion is strongly guided into forward direction. The lower temperature at the nozzle exit of the de Laval nozzle, in the given example in figure 2.4 40 K as compared to 200 K, leads to the fact that at the same density

42 2. Numeric Flow Simulation

the pressure is much lower at the exit of the de Laval nozzle. Since the pressure induces the transversal spread of the gas jet it is, therefore, clear, that the supersonic jet emanating from a de Laval nozzle will diverge much less than the one from a cylindrical nozzle at the same density. This also leads to the fact that the flat top density profile at the exit of the conical de Laval nozzles studied here is preserved over a significant propagation distance. Figure 2.6 shows a comparison between a cylindrical subsonic nozzle with a diameter of

0 1 2 0 0.5 1 1.5 2 2.5 3 3.5 4 axial position (mm)

jet radius containing

95 % of mass flow (mm)

Figure 2.5: Divergence of jets emanating from a supersonic de Laval nozzle (solid line) and from a subsonic nozzle

(dashed line). The gas jet

from the subsonic nozzle is much less collimated, having a full opening angle of the 95% mass flow contour of

122◦. This compares to 56◦

for the de Laval nozzle.

0.75 mm and a de Laval nozzle with a throat diameter of 0.25 mm and an exit diameter of 0.75 mm. For both nozzles, the gas is Helium with 50 bar backing pressure. Only right at the nozzle exit is the subsonic nozzle able to maintain a steep gradient but even there the profile is not really flat top. Only 0.1mm away from the nozzle exit, for the subsonic nozzle, the gas already expanded into all directions producing broad gradients and a convex central shape. And 0.2 mm from the nozzle exit - this is 27 % of the diameter - the density line- out produced by the subsonic nozzle is already a very good approximation to a gaussian. By contrast, the supersonic jet produced by the de Laval nozzle maintains its flat top profile over a distance of more than its exit diameter of 0.75 mm. Figure 2.6(e) shows that this comes at a price, however.

The expansion of the gas that takes place inside the de Laval nozzle lowers the density at the nozzle exit signifi- cantly - in this case, it is approximately an order of magnitude lower than the density produced by the subsonic nozzle. It has been stated that for the subsonic nozzle the gas expands transversally much stronger than for the supersonic nozzle. This can be seen in figure 2.5 where the free jet divergence of the two nozzles is compared. The reason for the much larger divergence of the jet from the subsonic nozzle is the high pressure and comparatively low velocity at the nozzle exit. The gas leaves the nozzle and accelerates transversally, rapidly approaching an almost isotropic velocity distribution corresponding to a gaussian density profile. Only during the short period that the gas needs to accelerate transversally to speeds similar to the longitudinal one, there is a significant departure from this behavior.

For the de Laval nozzle, things are quite different. By the time the gas leaves the noz- zle, it has already converted a substantial part of its enthalpy into kinetic energy and the confining nozzle walls guided this expansion into the forward direction. Therefore, the

2.2 Simulation Results 43 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 0.8 1 radius (mm)

normalized density (a.u.)

Standard Shape Subsonic Nozzle (a) 0.0 mm 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 0.8 1 radius (mm)

normalized density (a.u.)

Standard Shape Subsonic Nozzle (b) 0.1 mm 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 0.8 1 radius (mm)

normalized density (a.u.)

Standard Shape Subsonic Nozzle (c) 0.2 mm 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 0.8 1 radius (mm)

normalized density (a.u.)

Standard Shape Subsonic Nozzle (d) 0.3 mm 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 10 20 30 40 50 60 70 80 radius (mm) density (10 19cm −3) Standard Shape Subsonic Nozzle

(e) not normalized 01 mm

Figure 2.6: Comparison of density line-outs at distances of 0, 0.1, 0.2, 0.3 mm from the nozzle exit. Black line-outs correspond to a de Laval nozzle with 0.25 mm throat and 0.75 mm exit diameter, red line-outs to a cylindrical subsonic nozzle with a diameter of 0.75 mm. Plots (a)-(d) are normalized

to one to demonstrate the qualitative differences between the profiles, plot (e) is not normalized

and shows that the cylindrical nozzle produces far higher densities than the de Laval nozzle. The backing pressure in both cases was 50 bar (Helium).

44 2. Numeric Flow Simulation

exit velocity is much larger (approximately a factor of two in this case) and the pressure is much smaller (approximately a factor of 20) so that the gas has no chance any more to accelerate transversally to velocities similar to its forward velocity. An approximate value for the divergence of the supersonic jet can be obtained by calculating the transversal spread velocity by equation (1.35) using the sound speed at the nozzle exit. For Helium and a de Laval nozzle that has an exit diameter of three times its throat diameter and a reservoir temperature of 300 K this gives a full divergence angle of 63◦which is actually quite close to the values obtained by simulation. Also far away from the nozzle exit, this divergence is approximately preserved because due to the low pressure at the exit of the nozzle within a very short distance from the nozzle the additional expansion that occurs is sufficient to render the jet essentially collisionless and the particles follow ballistic trajec- tories, therefore, preserving the collimated velocity distribution generated by the de Laval nozzle.

Here, the nomenclature should be clarified: The isentropic expansion that takes place after the gas has left the cylindrical "subsonic" nozzle rapidly accelerates the flow to su- personic speed. Therefore, by speaking of a subsonic nozzle or a subsonic gas jet or a supersonic nozzle, one always refers to the maximum mach number that the gas acquires

insidethe nozzle itself. Outside of it - provided that the gas emanates into vacuum, or at

least a sufficiently low pressure surrounding medium - supersonic conditions are always obtained.

The simplicity of the geometry of the cylindrical nozzle makes it quite easily possible - in contrast to the jet from a de Laval nozzle - to obtain fitting formulas that allow to calculate all important flow parameters outside the nozzle, Miller in [114], [150]. It should be noted, however, that the calculations presented there neglect the existence of boundary layers and, therefore, can be applied only to cases with high backing pressure and/or large nozzle diameters.