HIGH RESOLUTION 
SIMULATIONS OF TUR-
BULENT FLOWS

Turbulence, "the last great unsolved 
problem of classical physics" (Feyn-
man), has been under extensive investi-
gation for over 100 years since 
Reynolds studied the transition from 
laminar to turbulent flows in 1893. The 
major difficulties that arise in the statis-
tical theory of turbulence are caused by 
the strongly dissipative character of the 
dynamical system and the nonlinearity 
of the equation of motion. During the 
last two decades, direct numerical sim-
ulation has played an important role in 
our understanding of structures and 
statistical properties of turbulent fluid 
flows.

The main limitation encountered in 
previous simulations has been the 
restricted computational memory and 
computational speed, dictating limited 
spatial resolution. The large memory 
and high computational speed on the 
CM-200 allows, for the first time, the 
implementation of turbulent simula-
tions on a spatial grid of 5123. Taylor 
microscopic Reynolds numbers  
can be resolved. 

Using this new computational power, 
the built-in parallel computational 
method, and the accompanying 3D 
visualization capability, we have stud-
ied spatial structures and statistical 
properties of isotropic turbulence. The 
computational speed of the present 
code on the CM-200 is about three 
times faster than that of the CRAY-2 
simulations using four processors.

In other studies, we have seen that the 
probability distribution function of 
pressure is very skewed, meaning that 
most of the 3D space has below-aver-
age pressure. In contrast, the pressure 
head, defined as the summation of 
pressure and velocity magnitude, has a 
symmetric distribution function. More-
over, it appears that the Bernoulli law 
(pressure head = constant) for inviscid 
fluid flows is approximately correct for 
high Reynolds number turbulent flows. 
High vorticity regions are strongly cor-
related with high velocity regions 
because of spin-up. Therefore, the Ber-
noulli law leads to low pressure in the 
high vorticity regions.

In the movie sequences, we present iso-
pressure contours from a pseudo-spec-
tral simulation of the 3D Navier-
Stokes equations. The system size is 
5123 and . The red, blue, and 
green colors represent pressure con-
tours of 50% above average, 5% above 
average and 30% below average, 
respectively. The low-pressure region 
forms worm-like structures, while the 
high-pressure region is randomly ori-
ented. In other visualizations, we 
observe that the spatial structure of 
the pressure field is strongly corre-
lated with the vorticity field. The 
shape of the high-vorticity region is 
substantially the same as the low-
pressure field. In this connection, it 
can be noted that the pressure satisfies 
a Poisson equation, and the vorticity 
contributes to the source term in this 
equation.

Acknowledgement: Shi-yi Chen, LANL, T13