**DNS
CFD-Simulation, Navier-Stokes Equations, and Turbulence
**By M. Kostic , June 1, 2007. This
page posted at: www.kostic.niu.edu/DNS-CFD-NS-Turbulence.htm

See the related Web Links at the end

The Direct
Numerical Simulation (DNS) is a simulation in computational fluid
dynamics (CFD) in which the Navier-Stokes equations are numerically solved
without any turbulence
model.

The Navier-Stokes equations are the basic
governing equations for a continuum medium, a viscous fluid. It is a vector
equation (or three scalar components) obtained by applying *momentum equation*. The solution is obtained by
integrating the governing equations, including conservation of mass and energy equations, with boundary
and constitutive fluid-property equations (stress-strain fluid response, etc.)
over fluid continuum domain.

Turbulence is a very
complex stochastic phenomenon which is not (and may never be) well understood.
Turbulence develops when the “loose” fluid-structure (constitutive property)
responds to flow instabilities (rapid variation of pressure and velocity in
space and time with appearance of unsteady vortices and eddies on many scales
which interact with each other) initiated by diverse extraneous and internal “disturbances.”
The large scale flow instability may develop by formation of eddies of many
different length scales, which will be “resisted” by very complex
fluid-structure (visco-elastic molecular particulate structure), which in turn
will "cascade" and break large scale flow instability into fine
turbulent structure with additional viscous dissipation of energy. Turbulence
is “damping” (stabilizing) the flow instabilities by the small-scale energy
dissipation and thus make “orderly disorder (how ironic!),” in similar (but
much more complex) way as viscosity (molecular momentum diffusion) does order
laminar flow.

For example, a simple, well-ordered, fully-developed laminar flow in a
smooth pipe may become unstable due to many reasons, like pipe surface small
irregularities (roughness) or small vibration, or fluid impurities (small
foreign particles or bubbles). For a given pipe size and fluid property the
flow will always be stabilized by viscosity if velocity (thus fluid inertia and
Reynolds number, Re) are billow certain value (Re<2000 for pipe flow). For higher velocity
(inertia, i.e. Re) due to diverse flow disturbances (imperfections) always
present in reality, the instability will develop and turbulence will occur to
“order” the flow by fluid “disorderly-stochastic” reaction. Careful experiments
in laboratory were conducted by minimizing all disturbances (roughness,
vibration, impurities, initial and boundary conditions) and laminar flow in
pipe was maintained up to 20,000, 40,000 and even 100,000 Reynolds number.
There is no upper theoretical limit for laminar pipe flow transition to turbulence,
but only the lower practical limit (i.e., Re about 2000 for pipe flow) where all practical
disturbances will be “damped” by fluid viscous forces.

Therefore, there is nothing in the Navier-Stokes equations to model real
turbulence phenomena, since “external” disturbances and irregularities at the
boundaries, which initiate flow instabilities, along with very complex
particulate fluid structure with impurities, which may promote and/or damp flow
instabilities, are not modeled as such. For example, the usual direct numerical
simulation (DNS) will never predict pipe turbulent flow, but only laminar,
regardless of Reynolds value, the same way the laminar pipe flow could be
maintained in reality up to much higher Re number values (up to 100,000 or
more!) than the (minimum) critical number of about 2,000.

The results of direct numerical simulation (DNS) in more complex flow
configurations will result in much more fine flow details including flow
instabilities and “its own turbulence,” due to instability and imperfections of
continuum media simulation and numerical discretization methods used to achieve
the solution. Such fine and transient flow fluctuations, as outcome of a very
detailed DNS simulation cannot be the same as the real turbulence (although it
may look similar in some instances) since reality with all “extraneous”
disturbances (including imperfect boundary conditions) and discrete (sub- and
molecular fluid structure) is not even modeled by the DNS governing and other
equations.

Even under idealized simulation conditions the existence and uniqueness of
classical solutions of the 3-D Navier-Stokes equations is still an open mathematical
problem and is one of the Clay
Institute's Millennium Problems:

* http://www.claymath.org/millennium/Navier-Stokes_Equations/Official_Problem_Description.pdf

The above is only to highlight limits
and uncertainties of CFD simulation which is in many ways similar to limits and
uncertainties of experimental investigation. It is important to repeat here,
that computational simulation and experimentation engineering have their
exclusive strengths and weaknesses and can not replace each-other, but if
properly integrated, will strongly complement each-other, resulting in a *synergistic result* which is much greater
than the simple sum of the two constituents.

Some interesting
(and useful) Web links:

*CFD-online/Wiki*CFD
Resources Online*****CFD Online Discussion Forums*****DNS-Direct Numerical Simulation*Turbulence*****Turbulence modeling***** **Turbulent Times
for Fluids*****Tackling
Turbulence with Supercomputers*