Direct Numerical simulations (DNS) belong to the class of simulations that strive to emulate the real physical flow by trying to simulate the complete range of scales involved in the flow. Though the computational power has steadily increased in the past few decades, resolving the complete range of scales from the largest scales up to the Kolmogorov scales can be very complicated and time consuming for moderately high Reynolds numbers as well. Hence, DNS is still expected to be a research tool to study relatively simple flows and geometries for the forseeable future unlike its counterparts such as Large Eddy Simulations (LES) and Reynolds Averaged Navier--Stokes (RANS) methods. The main objective of DNS studies is to study the flows so that turbulence models which are parametrized can be improved upon with the physical insight from the results of DNS. It also helps in studying the effect of numerical methods and techniques applicable to real flows and their stand-alone effects without any type of modeling as done for LES or RANS.

This thesis is concerned with the DNS study of a compressible turbulent jet. A turbulent jet belongs to the class of free turbulent flows, in the sense that, it is not bounded physically. In this thesis, the transport of a passive scalar through the jet is studied. At higher Reynolds numbers, for convection dominated equations such as the scalar transport equation, discontinuities can occur due to the hyperbolic nature of the equations. When central methods are used, oscillations are observed in the regions of the discontinuities. These oscillations can make the solution unphysical. Therefore, in this thesis the scalar transport equation has been modeled with an additional Weighted Essentially Non-Oscillatory (WENO) interpolation to accurately capture the discontinuities without oscillations. The WENO interpolation is combined with central compact finite difference methods to reduce the numerical dissipation while maintaining the order of accuracy in the smooth regions which is essential in high Reynolds number flows.

It was observed that a high dissipation setting for the WENO interpolation removed the oscillations but introduced artificial numerical viscosity. Therefore a relatively large domain was used with a low dissipation which while removing the oscillations and reducing the numerical dissipation. The WENO method for the compressible turbulent jet was first validated with experimental results to ascertain the accuracy of the grid resolution. The same grid resolution was used to study some properties of the jet at two different Schmidt numbers at a slightly lower Reynolds number. It was observed that the results obtained with the WENO interpolation matched well with the experimental results while being physical valid solutions because they had no oscillations. The results also showed different modes of instabilities for different Schmidt numbers and that the decay rate is a good characterization of the properties of the turbulent jet.

In conclusion, the WENO methods can be a very helpful method to accurately capture the discontinuities in an efficient manner and is also suitable for methods such as LES and RANS to model flows that have some hyperbolic nature of terms in the governing equations.