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 Newton's Law of Motion to a fluid element and is also called the 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:
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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*