In this paper, orbital angular momentum (OAM) and distribution of intensity of vortex Sinh-Gaussian beam (ShGB) travelling through an atmosphere with turbulence are detailly studied by applying Fresnel-Kirchhoff diffraction integral equations. When the medium of propagation of an optical beam in the atmosphere, it encounters so many issues. One of the main issues is atmospheric turbulence. A turbulent atmosphere is composed of various types of aberrations. Here, we have undergone a detailed investigation of the effect of various aberrations present in the atmosphere, such as z-tilt aberration, defocus aberration, astigmatism aberration, coma aberration, and spherical aberration under zero, weak, medium, and strong turbulent conditions. Also, it noted its OAM and the distribution of intensity of the vortex ShGB with topological charge p = 1 at different propagation distances (z = 2000m & z = 5000m) under a turbulent atmosphere. Numerical analysis results of each aberration show that all aberrations have some sort of impact on vortex ShGB. But on comparing the results of all aberrations, it is observed that the consequence of tilt aberration is greater than the other four aberrations on vortex ShGB. From the simulation results, it is found that the beam gets converted into a Gaussian nature when the value of turbulence factor is increased from zero turbulent condition to strong turbulent condition. Therefore, the resultant OAM and intensity distribution of vortex ShGB due to the effect of z-tilt aberration contribute a crucial part when compared with the other four aberrations. Also, it is noticed that the beam is least affected by coma aberration. The results also show that as we increase the distance of propagation from 2000m to 5000m, the quality of the beam gets reduced.