Navier–Stokes equations
The Navier–Stokes equations mathematically express momentum balance for Newtonian fluids and make use of conservation of mass.
Despite their wide range of practical uses, it has not yet been proven whether smooth solutions always exist in three dimensions—i.e., whether they are infinitely differentiable (or even just bounded) at all points in the domain.
Once the velocity field is calculated, other quantities of interest such as pressure or temperature may be found using dynamical equations and relations.
This is different from what one normally sees in classical mechanics, where solutions are typically trajectories of position of a particle or deflection of a continuum.
Studying velocity instead of position makes more sense for a fluid, although for visualization purposes one can compute various trajectories.
In particular, the streamlines of a vector field, interpreted as flow velocity, are the paths along which a massless fluid particle would travel.
The left side of the equation describes acceleration, and may be composed of time-dependent and convective components (also the effects of non-inertial coordinates if present).
need not be constant – in general, they depend on two thermodynamics variables if the fluid contains a single chemical species, say for example, pressure and temperature.
However, this difference is usually neglected most of the time (that is whenever we are not dealing with processes such as sound absorption and attenuation of shock waves,[11] where second viscosity coefficient becomes important) by explicitly assuming
one can finally condense the whole source in one term, arriving to the incompressible Navier–Stokes equation with conservative external field:
The discrete form of this is eminently suited to finite element computation of divergence-free flow, as we shall see in the next section.
It is desirable to choose basis functions that reflect the essential feature of incompressible flow – the elements must be divergence-free.
Similar considerations apply to three-dimensions, but extension from 2D is not immediate because of the vector nature of the potential, and there exists no simple relation between the gradient and the curl as was the case in 2D.
The rotating frame of reference introduces some interesting pseudo-forces into the equations through the material derivative term.
This single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematic viscosity as a parameter.
The nonlinearity makes most problems difficult or impossible to solve and is the main contributor to the turbulence that the equations model.
[26] The numerical solution of the Navier–Stokes equations for turbulent flow is extremely difficult, and due to the significantly different mixing-length scales that are involved in turbulent flow, the stable solution of this requires such a fine mesh resolution that the computational time becomes significantly infeasible for calculation or direct numerical simulation.
Attempts to solve turbulent flow using a laminar solver typically result in a time-unsteady solution, which fails to converge appropriately.
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.
This is partly because there is an enormous variety of problems that may be modeled, ranging from as simple as the distribution of static pressure to as complicated as multiphase flow driven by surface tension.
From this point onward, more quantities of interest can be easily obtained, such as viscous drag force or net flow rate.
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).
[32][33][34] Time-dependent self-similar solutions of the three-dimensional non-compressible Navier–Stokes equations in Cartesian coordinate can be given with the help of the Kummer's functions with quadratic arguments.
[36] Note that the existence of these exact solutions does not imply they are stable: turbulence may develop at higher Reynolds numbers.
[37] For example, in the case of an unbounded planar domain with two-dimensional — incompressible and stationary — flow in polar coordinates (r,φ), the velocity components (ur,uφ) and pressure p are:[38]
For example, in the case of an unbounded Euclidean domain with three-dimensional — incompressible, stationary and with zero viscosity (ν = 0) — radial flow in Cartesian coordinates (x,y,z), the velocity vector v and pressure p are:[citation needed]
Since both the solutions belong to the class of Beltrami flow, the vorticity field is parallel to the velocity and, for the case with positive helicity, is given by
Wyld diagrams are bookkeeping graphs that correspond to the Navier–Stokes equations via a perturbation expansion of the fundamental continuum mechanics.
Though comparatively more compact than other representations, this is still a nonlinear system of partial differential equations for which solutions are difficult to obtain.
[47][48] Many improvements have been proposed to Stam's original work, which suffers inherently from high numerical dissipation in both velocity and mass.
