The specification of boundary conditions for open boundaries of the numerical space is a subtlety in DNS. One must consider that given the turbulent flow, the only boundary conditions for a specific DNS realization is the solution itself, on that is of course unknown a-priori…

The major difficulty (considering incompressible flow) is posed in inflow and outflow boundary conditions. Statistically homogeneous flows (whether spanwise for 2-D boundary layers are a convenient task as we simply impose periodic conditions.

This is definitely not the case for far-field vortical flows, mixing layers and boundary layers. except channel or pipe flow which are homogeneous in the streamwise direction, most flows are simply not as convenient like that. For these cases special and careful care should be given to inflow and outflow boundary conditions.

In the early days of DNS Taylor’s Hypothesis (An assumption that advection contributed by turbulent circulations themselves is small and that therefore the advection of a field of turbulence past a fixed point can be taken to be entirely due to the mean flow; also known as the Taylor “frozen turbulence” hypothesis – easy explained at J. M. McDonough Turbulence models book) assisted in overcoming some of the subtleties since the turbulence was assumed homogeneous in the streamwise direction and as a direct consequence evolved statistically in time, meaning actually a temporal simulation. The temporal evolution was then related to the spatial one by a convection velocity such that the spatial evolution is seen in the realization.

The approach worked extremely well for homogeneous shear flows, turbulence passing through axisymetric contractions/expansions and decaying grid turbulence.

Non-paraallel streamlines that are present in experiments were not seen in the DNS realization since the mean flow was actually constant for the temporal simulation in the streamwise direction.

One giant leap in the evolution in DNS was Philippe spalart idea of coordinate transformation that overcame this limitation in DNS of a turbulent boundary layer.

By using periodic boundary conditions in the streamwise direction the simulation allowed generating stationary turbulent flow of which statistics correspond to a single stramwise station. The major drawback of Splart’s idea is a restriction to flows with small mean streamwise variation in comparison with the transverse one.

So we reach to a conclusive decision that DNS of complex flows must be accompanied with as much as possible supportive inlet and outlet boundary conditions.

An Amazing DNS by P. Moin and the Stanford group of collaborators

### The beginning of a new era in DNS

The first breakthrough in the field was made by Stanford group lead by Parviz Moin were 3-D, divergence free field of random fluctuations that were both homogeneous streamwise and had intentionally prescribed second order statistics. Than Taylor’s hypothesis came to the assist to convect the field of turbulence through the inflow plane. For the prevention of fluctuations being time-periodic the disturbances were also randomized in time. Such inflow conditions evolved into a realization of DNS, though it took quite a long distance to occur rendering an amount of about 50 displacement thickness in the inlet channel.

The gain of such a methodology is that it allowed calculations such as an isotropic turbulence/shock wave interaction and the known backward facing step.

high-quality movie of a turbulent boundary layer studied by direct numerical simulation (DNS) – KTH

P. Moin and his collaborators then kept improving the methodologies by convection of an instanious turbulent field extracted from a temporal simulation as oppsed to convected turbulent field extracted from random numbers. This shortened the evolution distance by a more than a half and allowed for a wider range of increasingly complex flows to be calculated.

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