Introduction

The complex behavior of turbulence is the consequence of a fairly simple set of equations - the Navier-Stokes equations. However, analytical solutions to even the simplest turbulent flows do not exist. A complete turbulent flow, where the flow variables like velocity and pressure are known as a function of space and time can therefore only be obtained by numerically solving the Navier-Stokes equations. These numerical solutions are termed as Direct Numerical Simulations (DNS) [10]. The main purpose of DNS is to solve (to best of our ability) for the turbulent velocity field without the use of ``turbulent modeling''. This condition means that the Navier-Stokes momentum equation for fluid must be solved exactly, which is not a simple task. Turbulence modeling involves the estimation of the increased fluid stresses due to the swirling motion of a turbulent flow field. Before the advent of powerful supercomputers, the effect of high-frequency velocity fluctuations within a flow field was estimated using modeling techniques, resulting in a macro-scale, or an integral-sense of the flow properties. Nowadays, the Navier-Stokes equation can be solved directly through the use of fast, large memory capacity computers using highly specialized numerical techniques, i.e. we can now examine fully developed turbulence flow fields at a micro-scale and perform extremely accurate calculations of flow properties. For the discretization, relatively fine grid sizes are necessary. Thus, any DNS code is very time consuming and has extensive storage requirements. Until now, only DNS computations at moderate Reynolds numbers are possible.

DNS using high-performance computers is an economical and mathematically appealing tool for study of fluid flows with simple boundaries which become turbulent. DNS is used to compute fully nonlinear solutions of the Navier-Stokes equations which capture important phenomena in the process of transition, as well as turbulence itself. DNS can be used to compute a specific fluid flow state. It can also be used to compute the transient evolution that occurs between one state and another. DNS is mathematical, and therefore, can be used to create simplified situations that are not possible in an experimental facility, and can be used to isolate specific phenomena in the transition process. As we are aiming at a nearly exact solution (and not ``the'' exact) to specific turbulent flows utilizing limited computational resources, DNS is stressed as a research tool and not as a brute-force solution. The objective of DNS is not necessarily to reproduce real-life flows (say the flow over an airplane), but to perform controlled studies that allow better insight, scaling laws, and turbulent models to develop. In aerodynamics, DNS is associated with a large-scale computationally intensive solution procedure which may consume hundreds to thousands of Cray Super-computing resources. The earliest use of DNS began in the 1970's and with the growth in the computational power today, it is getting more and more popular day by day [4]. Current computations typically use finite-difference schemes, or a combination of spectral and finite-difference schemes, although finite element approaches using unstructured grids are also being explored.

However, the main technical challenges of DNS remains the memory and computational speed requirements. A DNS of the flow past a complete airfoil would require a computer with exaflop (10¹⁸ flops) capacity to be practical, which is still not available now. The instantaneous range of scales in turbulent flows increases rapidly with the Reynolds number and hence most practical engineering problems (e.g. flow around a car) have too wide a range of scales to be directly computed using DNS. The difficulty in the DNS is that the turbulence contains wide spectrum of vortices with an equal physical importance. With increase of the Reynolds number, the size ratio of the largest to the smallest vortices increases. This makes it difficult to perform the DNS of turbulence with a higher Reynolds number. Thus for most high Reynolds number applications today, approximations like Large Eddy Simulation (LES) which computes only the large energy-containing scales, and Reynolds averaged Navier-Stokes solutions (RANS) are more prevalent than DNS. DNS can be thought of as the most desirable solution to a turbulent flow problem which is much more computationally intensive, followed by LES which is less complex and then RANS which is the least complex (and also the coarsest approximation).