question: The Navier-Stokes equations - why does it not work with fluid motions that are highly turbulent?
comment: It is based on the assumption of that the fluid can no be squeezed and that the forces on the liquid is proportional to the viscous forces (honey is very viscous and liquid helium is very very low). So in very turbulent flows, these assumptions no long hold so you can't estimate away the effects of compressibility.
passages: The simulation of turbulent flows by numerically solving the Navier–Stokes equations requires resolving a very wide range of time and length scales, all of which affect the flow field. Such a resolution can be achieved with direct numerical simulation (DNS), but DNS is computationally expensive, and its cost prohibits simulation of practical engineering systems with complex geometry or flow configurations, such as turbulent jets, pumps, vehicles, and landing gear.
   Some exact solutions to the Navier–Stokes equations exist. Examples of degenerate cases — with the non-linear terms in the Navier–Stokes equations equal to zero — are Poiseuille flow, Couette flow and the oscillatory Stokes boundary layer. But also more interesting examples, solutions to the full non-linear equations, exist such as Jeffery–Hamel flow, Von Kármán swirling flow, Stagnation point flow, Landau–Squire jet, Taylor–Green vortex.
  
  Note that the existence of these exact solutions does not imply they are stable: turbulence may develop at higher Reynolds numbers.
  
  Under additional assumptions, the component parts can be separated.
  
  \end{align}/math
   Because applying mathematical equations to turbulent flow and diffusion is so difficult, research in this area has been lacking until recently. In the past, laboratory efforts have used data from steady flow in streams or from fluids, that have a high Reynolds number, flowing through pipes, but it is difficult to obtain accurate data from these methods. This is because these methods involve ideal flow, which cannot simulate the conditions of turbulent flow necessary for developing turbulent diffusion models. With the advancement in computer-aided modeling and programming, scientists have been able to simulate turbulent flow in order to better understand turbulent diffusion in the atmosphere and in fluids.
   Solutions of the Navier–Stokes equations for a given physical problem must be sought with the help of calculus. In practical terms only the simplest cases can be solved exactly in this way. These cases generally involve non-turbulent, steady flow in which the Reynolds number is small. For more complex cases, especially those involving turbulence, such as global weather systems, aerodynamics, hydrodynamics and many more, solutions of the Navier–Stokes equations can currently only be found with the help of computers. This branch of science is called computational fluid dynamics.
  
  Section::::Inviscid and viscous fluids.
   Failing that, one may have to resort to molecular dynamics or various hybrid methods.
  
  Another limitation is simply the complicated nature of the equations. Time-tested formulations exist for common fluid families, but the application of the Navier–Stokes equations to less common families tends to result in very complicated formulations and often to open research problems. For this reason, these equations are usually rewritten for Newtonian fluids where the viscosity model is linear; truly general models for the flow of other kinds of fluids (such as blood) do not exist.
  
  Section::::Application to specific problems.
   The nonlinearity is due to convective acceleration, which is an acceleration associated with the change in velocity over position. Hence, any convective flow, whether turbulent or not, will involve nonlinearity. An example of convective but laminar (nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a small converging nozzle. Such flows, whether exactly solvable or not, can often be thoroughly studied and understood.
  
  Section::::Properties.:Turbulence.
   It is believed that turbulent flows can be described well through the use of the Navier–Stokes equations. Direct numerical simulation (DNS), based on the Navier–Stokes equations, makes it possible to simulate turbulent flows at moderate Reynolds numbers. Restrictions depend on the power of the computer used and the efficiency of the solution algorithm. The results of DNS have been found to agree well with experimental data for some flows.
   Using Einstein notation, the Navier–Stokes equations for an incompressible fluid in Cartesian coordinates are
  
  Filtering the momentum equation results in
  
  If we assume that filtering and differentiation commute, then
  
  This equation models the changes in time of the filtered variables formula_42. Since the unfiltered variables formula_43 are not known, it is impossible to directly calculate formula_44. However, the quantity formula_45 is known. A substitution is made:
  
  Let formula_47. The resulting set of equations are the LES equations:
  
  Section::::Filtered governing equations.:Compressible governing equations.
   Difficulties may arise when the problem becomes slightly more complicated. A seemingly modest twist on the parallel flow above would be the "radial" flow between parallel plates; this involves convection and thus non-linearity. The velocity field may be represented by a function that must satisfy:
   The principal idea behind LES is to reduce the computational cost by ignoring the smallest length scales, which are the most computationally expensive to resolve, via low-pass filtering of the Navier–Stokes equations. Such a low-pass filtering, which can be viewed as a time- and spatial-averaging, effectively removes small-scale information from the numerical solution. This information is not irrelevant, however, and its effect on the flow field must be modeled, a task which is an active area of research for problems in which small-scales can play an important role, such as near-wall flows 
  
  , reacting flows, and multiphase flows.
   The Navier–Stokes equations assume that the fluid being studied is a continuum (it is infinitely divisible and not composed of particles such as atoms or molecules), and is not moving at relativistic velocities. At very small scales or under extreme conditions, real fluids made out of discrete molecules will produce results different from the continuous fluids modeled by the Navier–Stokes equations. For example, capillarity of internal layers in fluids appears for flow with high gradients. For large Knudsen number of the problem, the Boltzmann equation may be a suitable replacement. 
   Unlike the laminar boundary layer equations, the presence of two regimes governed by different sets of flow scales (i.e. the inner and outer scaling) has made finding a universal similarity solution for the turbulent boundary layer difficult and controversial. To find a similarity solution that spans both regions of the flow, it is necessary to asymptotically match the solutions from both regions of the flow. Such analysis will yield either the so-called log-law or power-law.
   This ordinary differential equation is what is obtained when the Navier–Stokes equations are written and the flow assumptions applied (additionally, the pressure gradient is solved for). The nonlinear term makes this a very difficult problem to solve analytically (a lengthy implicit solution may be found which involves elliptic integrals and roots of cubic polynomials). Issues with the actual existence of solutions arise for (approximately; this is not ), the parameter R being the Reynolds number with appropriately chosen scales. This is an example of flow assumptions losing their applicability, and an example of the difficulty in "high" Reynolds number flows.
   Turbulent processes can be roughly combined into three categories: heat flux, moisture flux, and momentum flux. When calculating these fluxes, one ends up with more unknowns than equations, which means those equations cannot be solved directly. In order to calculate turbulent fluxes and close the equations, one must make parameterizations for the higher order terms. The sections below describe methods for parameterizing turbulent fluxes and turbulence closure.
  
  Section::::Turbulence closure.:Local and non-local closure.
   Although in this case we have assumed an inviscid fluid this assumption does not hold for flows where there is a boundary. The presence of a boundary causes some viscosity at the boundary layer which cannot be neglected and one arrives back at the Navier–Stokes equation. Finding the solutions to these governing equations under different circumstances and determining their stability is the fundamental principle in determining the stability of the fluid flow itself.
  
  Section::::Determining flow stability.:Linear stability analysis.
   Though Taylor's solution is still true for turbulent jet, for laminar jet or laminar plume, the effective Reynolds number for outer fluid is found to be of order unity since the entertainment by the sink in these cases is such that the flow is not inviscid. In this case, full Navier-Stokes equations has to be solved for the outer fluid motion and at the same time, since the fluid is bounded from the bottom by a solid wall, the solution has to satisfy the non-slip condition. Schneider obtained a self-similar solution for this outer fluid motion, which naturally reduced to Taylor's potential flow solution as the entrainment rate by the line sink is increased.
   For turbulent jet, formula_24, the linear terms in the equation can be neglected everywhere except near a small boundary layer along the wall. Then neglecting the non-slip conditions at the wall, the solution is given by
  
  Section::::Other considerations.
   In explicit filtering, an LES filter is applied to the discretized Navier–Stokes equations, providing a well-defined filter shape and reducing the truncation error. However, explicit filtering requires a finer grid than implicit filtering, and the computational cost increases with formula_92. Chapter 8 of Sagaut (2006) covers LES numerics in greater detail.
  
  Section::::Boundary conditions of large eddy simulations.
  
  Inlet boundary conditions affect the accuracy of LES significantly, and the treatment of inlet conditions for LES is a complicated problem. Theoretically, a good boundary condition for LES should contain the following features: 
   Section::::Methodology.:Turbulence models.:Direct numerical simulation.
  
  Direct numerical simulation (DNS) resolves the entire range of turbulent length scales. This marginalizes the effect of models, but is extremely expensive. The computational cost is proportional to formula_24. DNS is intractable for flows with complex geometries or flow configurations.
  
  Section::::Methodology.:Turbulence models.:Coherent vortex simulation.
   It is important to note that the large eddy simulation filtering operation does not satisfy the properties of a Reynolds operator.
  
  Section::::Filtered governing equations.
  
  The governing equations of LES are obtained by filtering the partial differential equations governing the flow field formula_11. There are differences between the incompressible and compressible LES governing equations, which lead to the definition of a new filtering operation.
  
  Section::::Filtered governing equations.:Incompressible flow.
  
  For incompressible flow, the continuity equation and Navier–Stokes equations are filtered, yielding the filtered incompressible continuity equation,
  
  and the filtered Navier–Stokes equations,
   The incompressible Navier–Stokes equation is a differential algebraic equation, having the inconvenient feature that there is no explicit mechanism for advancing the pressure in time. Consequently, much effort has been expended to eliminate the pressure from all or part of the computational process. The stream function formulation eliminates the pressure but only in two dimensions and at the expense of introducing higher derivatives and elimination of the velocity, which is the primary variable of interest.
  
  Section::::Properties.
  
  Section::::Properties.:Nonlinearity.
   Fortunately there exists a closure approximation in introducing the concept of eddy diffusivity and its statistical approximations for the random concentration and velocity components from turbulent mixing.
  
  formula_13
  
  Where K is the eddy diffusivity.
  
  Substituting into the first continuity equation and ignoring reactions, sources, and molecular diffusion results in the following differential equation considering only the turbulent diffusion approximation in eddy diffusion:
  
  formula_14
  
  Unlike the molecular diffusion constant D, the eddy diffusivity is a matrix expression that may vary in space, and thus may not be taken outside the outer derivative.
  
  Section::::Modeling.:Lagrangian approach.
   The second method involves a separate and precursor calculation to generate a turbulent database which can be introduced into the main computation at the inlets. The database (sometimes named as ‘library’) can be generated in a number of ways, such as cyclic domains, pre-prepared library, and internal mapping. However, the method of generating turbulent inflow by precursor simulations requires large calculation capacity. 
  
  Researchers examining the application of various types of synthetic and precursor calculations have found that the more realistic the inlet turbulence, the more accurate LES predicts results.
  
  Section::::Modeling unresolved scales.
   Section::::Properties.:Applicability.
  
  Together with supplemental equations (for example, conservation of mass) and well formulated boundary conditions, the Navier–Stokes equations seem to model fluid motion accurately; even turbulent flows seem (on average) to agree with real world observations.
   The higher-order term, namely the shear stress divergence , has simply reduced to the vector laplacian term . This laplacian term can be interpreted as the difference between the velocity at a point and the mean velocity in a small surrounding volume. This implies that – for a Newtonian fluid – viscosity operates as a "diffusion of momentum", in much the same way as the heat conduction. In fact neglecting the convection term, incompressible Navier–Stokes equations lead to a vector diffusion equation (namely Stokes equations), but in general the convection term is present, so incompressible Navier–Stokes equations belong to the class of convection-diffusion equations.
  