The sodium-cooled fast nuclear reactor (SFR) is one of the most promising designs of the fourth generation (Gen IV) nuclear power reactors. Sodium-gas heat exchangers (SGHE) using nitrogen is being investigated as an alternative to improve operational safety associated with the use of steam Rankine cycles. This alternative eliminates the potential risk of chemical reactions. It is known that cracks inside an SGHE can cause the accidental leakage of nitrogen into the sodium-side. Due to the pressure difference between the secondary and tertiary loops, this nitrogen jet is therefore under-expanded. When the nitrogen leak is strong enough to flush the liquid sodium outside the SGHE channel, the nitrogen jet can be considered as single-phase. In this context, this work focuses on the influence of geometrical parameters of cracks (size, cross-section shape, transverse localization and inclination angle) on the impinging under-expanded nitrogen jet and its shock-wave system. A numerical study of impinging under-expanded nitrogen jet has been carried out using large eddy simulation (LES) technique. We applied a stagnation pressure upstream of the crack of 180 bar while the nozzle pressure ratio (NPR) ranged from 6.0 to 9.2. We were able to identify the link between the nozzle geometry and the Mach disk diameter and its localization. The vorticity distribution at the nozzle can be used to explain the structure of the jets and the entrainment. The central cross-section of the gas jet tends to turn 45° and 90° for square and rectangular cross-section nozzles respectively. The Taylor-Görtler instability is enhanced with a reduction in the nozzle diameter. These instabilities are also increased with square, rectangular and inclined nozzles.