### Abstract

We present an in-depth analysis of De Laval nozzles, which are ideal for gas jet generation in a wide variety of experiments. Scaling behavior of parameters especially relevant to laser-plasma experiments as jet collimation, sharpness of the jet edges and Mach number of the resulting jet is studied and several scaling laws are given. Special attention is paid to the problem of the generation of microscopic supersonic jets with diameters as small as 150 μm. In this regime, boundary layers dominate the flow formation and have to be included in the analysis.

This work is supported by DFG-Project Transregio TR18 and by the Association EURATOM – Max-Planck-Institut fuer Plasmaphysik.

I. INTRODUCTION

II. SUPERSONIC FLOW THEORY

A. 1D isentropic flows

B. Effects of boundary layers

III. METHODS

A. Simulation

B. Measurement

IV. RESULTS

A. Comparison measurement and simulation

B. Scaling rules for De Laval nozzle geometry

C. Effects of nozzle-size and pressure

V. DISCUSSION AND CONCLUSION

### Key Topics

- Mach numbers
- 20.0
- High pressure
- 13.0
- Nozzle jets
- 13.0
- Supersonic flows
- 12.0
- Fluid jets
- 10.0

## Figures

Sketch of the De Laval Nozzle used in simulations and experiment. In the parameter study, the throat diameter *d* ^{*}, the exit diameter *d* _{ E }, and the length *L* of the conical section between throat and exit are varied. The throat contour has a radius of 4 times *d* ^{*}. The medium are helium or argon at a temperature of 300 K at varying pressures, the vacuum chamber has a background pressure of 10^{−2} mbar. The plot shows an idealized flat-top density profile that is generated by a De Laval nozzle at small Knudsen numbers (see Eq. (6)). r is the radial coordinate, ρ is the gas density.

Sketch of the De Laval Nozzle used in simulations and experiment. In the parameter study, the throat diameter *d* ^{*}, the exit diameter *d* _{ E }, and the length *L* of the conical section between throat and exit are varied. The throat contour has a radius of 4 times *d* ^{*}. The medium are helium or argon at a temperature of 300 K at varying pressures, the vacuum chamber has a background pressure of 10^{−2} mbar. The plot shows an idealized flat-top density profile that is generated by a De Laval nozzle at small Knudsen numbers (see Eq. (6)). r is the radial coordinate, ρ is the gas density.

Density, pressure, temperature, velocity, and Mach number obtained at the center of a helium gas flow through a De Laval nozzle as obtained from a computational fluid dynamics simulation. The nozzle shape is as displayed in Figure 1. The nozzle in this case has the geometric dimensions = 1 mm, *L* = 6 mm, and = 3 mm, the backing pressure is 50 bars, initial temperature is 300 K.

Density, pressure, temperature, velocity, and Mach number obtained at the center of a helium gas flow through a De Laval nozzle as obtained from a computational fluid dynamics simulation. The nozzle shape is as displayed in Figure 1. The nozzle in this case has the geometric dimensions = 1 mm, *L* = 6 mm, and = 3 mm, the backing pressure is 50 bars, initial temperature is 300 K.

Mach-Zehnder interferometer setup used for gas jet characterization. The laser beam (green line) consists of laser pulses of 20 ps duration at 10 Hz repetition rate and 523.5 nm wavelength, each pulse containing ∽300 μJ energy. The beam is split in two arms by beamsplitter BS1. Both interferometer arms pass through an evacuated chamber, the thin grey lines symbolize glass windows. The gas jet is placed into one arm inside the chamber. The beams are recombined by a second beamsplitter BS2 and the lens L1 images the gas jet onto the CCD camera where interference fringes form. The inset shows the round gas jet and the laser together with the coordinates used for Abel inversion.

Mach-Zehnder interferometer setup used for gas jet characterization. The laser beam (green line) consists of laser pulses of 20 ps duration at 10 Hz repetition rate and 523.5 nm wavelength, each pulse containing ∽300 μJ energy. The beam is split in two arms by beamsplitter BS1. Both interferometer arms pass through an evacuated chamber, the thin grey lines symbolize glass windows. The gas jet is placed into one arm inside the chamber. The beams are recombined by a second beamsplitter BS2 and the lens L1 images the gas jet onto the CCD camera where interference fringes form. The inset shows the round gas jet and the laser together with the coordinates used for Abel inversion.

Measured phase shift and density profiles of an argon gas jet produced by a De Laval nozzle with 86 μm throat and 176 μm exit diameter measured at a distance of 50 μm from the nozzle exit. (a) Shows the measured (solid black line) phase shift and the phase shift calculated from simulation data (dashed orange line) for a backing pressure of 130 bars. (b) Shows Abel-inversions of the phase-shift profiles shown in (a) together with measurements using backing pressures of 70 (dashed) and 100 bars (solid) and the simulated radial density distribution (dashed orange line).

Measured phase shift and density profiles of an argon gas jet produced by a De Laval nozzle with 86 μm throat and 176 μm exit diameter measured at a distance of 50 μm from the nozzle exit. (a) Shows the measured (solid black line) phase shift and the phase shift calculated from simulation data (dashed orange line) for a backing pressure of 130 bars. (b) Shows Abel-inversions of the phase-shift profiles shown in (a) together with measurements using backing pressures of 70 (dashed) and 100 bars (solid) and the simulated radial density distribution (dashed orange line).

Simulated jet divergence for different nozzle geometries. r_{95} is the radius that includes 95% of the total mass flow of the gas jet, r_{ E } is the nozzle exit radius. Plot (a) shows r_{95} versus distance from the nozzle exit for different opening halve-angles of the De Laval nozzle and different ratios of the nozzle throat to the exit diameter *d* ^{*}/*d* _{ E }. Plot (b) shows r_{95} normalized to r_{ E }. The angles of 22°, 27°, and 33° given in plot (b) correspond to the approximate jet divergence halve-angle for nozzle halve-angles between 5° and 10°.

Simulated jet divergence for different nozzle geometries. r_{95} is the radius that includes 95% of the total mass flow of the gas jet, r_{ E } is the nozzle exit radius. Plot (a) shows r_{95} versus distance from the nozzle exit for different opening halve-angles of the De Laval nozzle and different ratios of the nozzle throat to the exit diameter *d* ^{*}/*d* _{ E }. Plot (b) shows r_{95} normalized to r_{ E }. The angles of 22°, 27°, and 33° given in plot (b) correspond to the approximate jet divergence halve-angle for nozzle halve-angles between 5° and 10°.

Simulated radial density profiles at the nozzle exit for different nozzle geometries. The plots show the radial density profiles at the nozzle exit for different nozzle halve-angles of 5°, 7°, 10°, 14° and a constant ratio of nozzle throat to nozzle exit diameter of 1:2 (a), 1:3 (b), and 1:4 (c).

Simulated radial density profiles at the nozzle exit for different nozzle geometries. The plots show the radial density profiles at the nozzle exit for different nozzle halve-angles of 5°, 7°, 10°, 14° and a constant ratio of nozzle throat to nozzle exit diameter of 1:2 (a), 1:3 (b), and 1:4 (c).

Simulated scaling behavior of the exit gradient width for different nozzle geometries. The width of the density gradient (10%–90% of on-axis density) is plotted versus the product of nozzle length *L* times nozzle exit diameter *d* _{ E }. A linear correlation between these two parameters is discernible, the red line shows a linear fit. The parameters that are varied within this simulation series are the nozzle exit diameter and the length of the diverging nozzle section. All nozzle have a throat diameter of 1 mm.

Simulated scaling behavior of the exit gradient width for different nozzle geometries. The width of the density gradient (10%–90% of on-axis density) is plotted versus the product of nozzle length *L* times nozzle exit diameter *d* _{ E }. A linear correlation between these two parameters is discernible, the red line shows a linear fit. The parameters that are varied within this simulation series are the nozzle exit diameter and the length of the diverging nozzle section. All nozzle have a throat diameter of 1 mm.

Simulated scaling behavior of the radial density variations inside the expanding section of the nozzle for four different nozzle halve-angles of 5° (green), 7° (turquoise), 10° (blue), and 14° (dark blue). Radial density variations are given relative to the average core-flow value (excluding the boundary layers). In order to show all curves within the same x axis range, the parameter *d*/*d* ^{*} is used as a measure of longitudinal position within the expanding nozzle section.

Simulated scaling behavior of the radial density variations inside the expanding section of the nozzle for four different nozzle halve-angles of 5° (green), 7° (turquoise), 10° (blue), and 14° (dark blue). Radial density variations are given relative to the average core-flow value (excluding the boundary layers). In order to show all curves within the same x axis range, the parameter *d*/*d* ^{*} is used as a measure of longitudinal position within the expanding nozzle section.

Boundary layer as defined by the the displacement thickness (Eq. (5)) (a) and density profiles at the nozzle exit for different Knudsen numbers or equivalently different combinations of backing pressure and nozzle size (b). For both plots, the results of all nozzles are scaled to the same size and superimposed.

Boundary layer as defined by the the displacement thickness (Eq. (5)) (a) and density profiles at the nozzle exit for different Knudsen numbers or equivalently different combinations of backing pressure and nozzle size (b). For both plots, the results of all nozzles are scaled to the same size and superimposed.

Scaling behavior of important gas jet parameters with respect to the scaling parameter *d**p_{0}, the product of diameter of the nozzle throat (mm) and backing pressure (bar). Displayed are Mach number *M* at the jet center at the nozzle exit (a), divergence halve-angle of the free gas jet α (b), width (10%–90%) of the density gradient δ_{ρ} at the gas jet edge at the nozzle exit normalized to the nozzle exit radius r_{ E } (c) and gas density at the jet center at the exit of the nozzle ρ_{E} normalized to the reservoir density ρ_{0}. Black squares are CFD simulation results, solid red lines are fits to the simulation data (see text for details) and grey circles show values calculated by Eq. (4) using the Mach numbers in (a). Dashed blue lines in (a) and (b) show results of the 1D isentropic theory for this nozzle: *M* = 4.84, ρ_{E}/ρ_{0} = 3.83%.

Scaling behavior of important gas jet parameters with respect to the scaling parameter *d**p_{0}, the product of diameter of the nozzle throat (mm) and backing pressure (bar). Displayed are Mach number *M* at the jet center at the nozzle exit (a), divergence halve-angle of the free gas jet α (b), width (10%–90%) of the density gradient δ_{ρ} at the gas jet edge at the nozzle exit normalized to the nozzle exit radius r_{ E } (c) and gas density at the jet center at the exit of the nozzle ρ_{E} normalized to the reservoir density ρ_{0}. Black squares are CFD simulation results, solid red lines are fits to the simulation data (see text for details) and grey circles show values calculated by Eq. (4) using the Mach numbers in (a). Dashed blue lines in (a) and (b) show results of the 1D isentropic theory for this nozzle: *M* = 4.84, ρ_{E}/ρ_{0} = 3.83%.

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