Navier-Stokes Equations
In this Chapter
The Navier-Stokes Equations (NSE) are regarded as the ultimate answer to fluid
dynamic problems. These equations may as well be the most widely studied equations in
applied physics. The range of validity of the Navier-Stokes is only limited by the
model used for the viscous stresses (Newtonian, non-Newtonian, turbulent eddies,
etc.)
The form most commonly used in CFD is called Reynolds averaged Navier-Stokes
equations (RANS). These momentum equations come short of turbulence modeling and
turbulent transition, which are fields of research on their own right. The momentum
equations are coupled with appropriate conservation equations for the mass and the
energy.
Sometimes the energy equation is not needed. This occurs when the flow is
incompressible, in thermal and chemical equilibrium (which is the case of low speed
aerodynamic flows). In the most general case the unknown variables to evaluate at
each grid vertex (or grid cell center) are: velocity (3 components); pressure,
density, temperature.
There are alternative formulations: the NSE in primitive variable (velocity-pressure)
are the ones considered in this section. Alternative formulations are obtained by
taking the curl the the original equations (which yields a formulation in terms of
vorticity). Neglecting second-order derivatives of the vorticity NSE in parabolic
form are obtained (parabolized NSE).
Numerical methods for the NSE are described in several textbooks of computational
fluid dynamics (for example ). Numerical methods for the NSE are a subset of
numerical methods for partial differential equations (PDE), a field of research that
has witnessed enormous progress in the last thirty years.
The methods can be roughly divided into finite difference, finite volume and finite
elements, spectral methods. The purpose of the present review is to avoid mathematical
details and provide phenomenological aspects of aerodynamic flows, along with the
latest achievements.
A comprehensive review for turbulence modeling in book form is found in Wilcox, 1993.
Models currently in use or being validated include: Balwin-Lomax,
Baldwin-Barth, Spalart-Almaras, K-epsilon, K-omega (various formu- lations),
K-l.
The role of turbulence is strongly related to the speed. Transitional flows are very
difficult to compute, because of the need to evaluate the conditions of the boundary
layer in that region extending from laminar to fully turbulent.
Transition modeling
is usually approached with semi-empirical methods. These include Michel’s formula,
the eN method, etc. These methods are applied for both airfoil and wing
flows. Computation of the correct transition point (whenever a transition point can
be defined) is absolutely important for low Reynolds number airfoil and wing flows.
Fully turbulent flows require modeling of the Reynolds stresses, whose intensity
controls boundary layer separation, velocity field in the boundary layer, and the
global force coefficients.
High Reynolds number flows are much less affected by the turbulence model, unless
they feature extensive areas of separated flow.
The most up-to-date NSE solvers are multi-grid and multi-block solvers. The methods
have reached a mature stage, and the computer codes are already so many that it is
not possible to make a full list (inquire directly about Navier-Stokes codes).
There
is a number of research codes widely documented in the technical literature,
along with commercial NSE codes, and in-house NSE codes developed at several research
institutes, government institutes, industry, etc.
A broad classification can be made between structured and unstructured codes (see the
grid generation chapter for details). There is still no agreement as to what method
is best, although structured codes are more developed, more documented and mode
reliable for the time being.
There is agreement on the fact that grid generation for unstructured codes is far
easier. Commercial codes (Star-CD, CFX) have sophisticated user interfaces and are
integrated with post-processing facilites. They are designed to perform computations
on very general configurations.
Research codes, on the other hand, generally lack the utilities of the codes
available in the market, but perform well on specific problems (wings, or rotors, or
internal flows, etc.)
Codes using hybrid grids and chimera schemes are at the stage of research codes, and
have not yet found commercial exploitation.
Direct numerical simulation (DNS) is a a direct solution of the unsteady
Navier-Stokes equations (with finite-difference, finite-element, etc.) that is
capable of resolving the smallest turbulent scales (Kolmogoroff) without requiring
additional closure equations. The method requires very fine grids and many time
steps; therefore the method is presently confined to simple problems and small
Reynolds numbers. DNS has been applied recently to simple box geometries to study
particle dispersion, even with two-way coupling (at a far higher computation
expense).
Large-eddy simulation (LES) is a method that can accurately predict the large scale
turbulent structures, that are the most important in the transport quantities. The
method has been applied successfully to homogeneous flows at relatively high
Reynolds numbers, on geometries more realistic than those feasible with DNS.
The reference list below reports some fundamental works (additional references
available therein). The technical literature on this topic is HUGE.
Reynolds Averaged Navier-Stokes Eqs.
Formulations
Numerical Methods
The Role of Turbulence
Navier-Stokes Codes
Classification
Direct Navier-Stokes
Large-Eddy Simulation
Related Material
Selected References
