# New correlation of subsonic, supersonic and cryo gas jets validated by highly accurate schlieren measurements

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## Abstract

High-speed schlieren visualization at 20,000 fps was performed to investigate the transient penetration depth of helium gas jets into air at 8 different pressure ratios. The injection pressure investigated was between 1.5 and 40 bar. The pressure in the mixing chamber was varied between 1.0 and 4.0 bar. The different pressure ratios were chosen to examine subsonic and underexpanded jets. To investigate the density dependence of the process, cryo injections were performed at injection temperatures of 224, 198 and 173 K at different pressure ratios. Using the image processing technique for detecting the jet tip penetration distance, presented in this article, it can be shown that for a given injector the jet penetration behavior is the same for the same pressure ratio. A new self-similar model was developed, to describe the influence of the injection and chamber gas density on the jet penetration behavior more accurately. The model was verified with the experimental data sets presented in this paper and with literature data.

## Keywords

Penetration Depth Pressure Ratio Injection Pressure Nozzle Exit Discharge Coefficient## List of symbols

*a*Unobstructed height of source image (m)

*B*Empirical constant (−)

*C*_{D}Discharge coefficient (−)

*C*_{f}Fraction of centerline velocity (−)

*C*_{t}Scaling constant (−)

*D*Diameter of vortex ball (m)

*d*Nozzle diameter (m)

*d*_{e}Effective diameter (m)

*d*_{PMD}Diameter of the PMD (m)

*E*_{BG}Background illuminance (lux)

- \(\Updelta E\)
Differential illuminance (lux)

*f*_{2}Focal length of second mirror (m)

*h*Image height (−)

*I*Jet image (−)

- \(\tilde I\)
Normalized jet image (−)

*I*_{BG}Background image (−)

- \(\tilde I_{\rm BG}\)
Normalized background image (−)

*k*Gladstone-Dale constant (−)

*K*Entrainment constant (−)

*l*_{K}Kolmogorov length scale (m)

*l*_{T}Taylor length scale (m)

*M*_{jet}Momentum of steady-state jet region (kg m/s)

*M*_{vortex}Momentum of jet vortex ball (kg m/s)

*M*_{PMD}Mach number downstream of the PMD (−)

*M*_{u,s}Mach number upstream of normal shock (−)

- \(\dot M_{\rm n}\)
Momentum flux at nozzle exit (kg m/s

^{2})- \(\dot{m}\)
Massflow (kg/s)

- \(\dot{m}_{\rm n}\)
Massflow at nozzle exit (kg/s)

- \(\dot m_{\rm m}\)
Measured massflow through nozzle (kg/s)

*n*Refractive index (−)

*p*_{ch}Chamber pressure (bar)

*p*_{i}Injection pressure (bar)

*p*_{n}Pressure at nozzle exit (bar)

*R*Empirical constant (−)

*R*_{ch}Specific gas constant of chamber gas (J/kg K)

*R*_{n}Specific gas constant of injected gas (J/kg K)

*Re*Reynolds number (−)

*Re*_{T}Taylor-scale Reynolds number (−)

*Re*_{0}Characteristic jet Reynolds number (−)

*r*Radial distance from jet axis (m)

*S*Schlieren sensitivity (−)

*T*_{ch}Chamber temperature (K)

*T*_{i}Injection temperature (K)

*T*_{n}Temperature at nozzle exit (K)

*t*Time (s)

*t*^{+}Characteristic time scale (s)

- \(\tilde t\)
Non-dimensional time scale (−)

- \(\tilde{t}_{\rm P}\)
Normalized time according to Petersen (−)

*u*′r.m.s. velocity (m/s)

*U*_{c}Centerline velocity (m/s)

*U*_{c,m}Centerline mean velocity (m/s)

*U*_{n}Velocity at nozzle exit (m/s)

*U*_{PMD}Velocity downstream of the PMD (m/s)

*w*Image width (−)

*x*Position (m)

*x*_{0}Virtual origin (m)

*y*_{0.5}Half-width of jet (m)

*z*Position along the optical axis (m)

*z*^{+}Characteristic length scale (m)

*Z*_{t}Penetration depth (m)

- \(\tilde{Z}_{\rm t,l}\)
Normalized

*Z*_{t}(\(s^{\frac{1}{2}}\))- \(\tilde{Z}_{\rm t}\)
Non-dimensional

*Z*_{t}(−)- \(\tilde{Z}_{\rm t,H}\)
Normalized

*Z*_{t}according to Hill \((s^{\frac{1}{2}})\)- \(\tilde{Z}_{\rm t,P}\)
Normalized

*Z*_{t}according to Petersen (−)- \(\epsilon\)
Dissipation rate (m

^{2}/s^{3})- \(\hat{\epsilon}\)
Dissipation rate constant (−)

- \(\epsilon_y\)
Light ray deflection in y direction (rad)

- \(\Upgamma\)
Scaling constant (−)

- \(\tilde \Upgamma\)
Scaling constant (−)

- γ
Specific heat ratio (−)

- γ
_{n} Specific heat ratio at nozzle exit (−)

*μ*Mean pixel intensity of a image (−)

- ν
Kinematic viscosity (m

^{2}/s)- ν
_{n} Kinematic viscosity at nozzle exit (m

^{2}/s)- π
_{i} Injection pressure ratio (−)

- ρ
Gas density (kg/m

^{3})- ρ
_{ch} Chamber gas density (kg/m

^{3})- ρ
_{i} Injection gas density (kg/m

^{3})- ρ
_{n} Gas density at nozzle exit (kg/m

^{3})- ρ
_{PMD} Density downstream of the PMD (kg/m

^{3})*DI*Direct injection

*LNG*Liquefied natural gas

*PFI*Port fuel injection

*PMD*Pseudo-Mach disk

## Notes

### Acknowledgments

The authors would like to thank F. Gerbig from BMW for providing the cryo system for the low temperature injections.

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