Copyright © A. Filippone (1996-2001). All Rights Reserved.

Computational Aerodynamics

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.)

Reynolds Averaged Navier-Stokes Eqs.

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

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.

The Role of Turbulence

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.

Navier-Stokes Codes

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 Navier-Stokes

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

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.

Related Material

Selected References

The reference list below reports some fundamental works (additional references available therein). The technical literature on this topic is HUGE.

  • Ferziger JH, Peric M. Computational Methods for Fluid Dynamics, Springer, 1997.

  • Wilcox DC. Turbulence Modeling for CFD, DWC Industries Inc., La Canada, USA, 1993.

  • Anderson DA, Tannehill JC, Pletcher RH. Computational Fluid Mechanics and Heat Transfer, Taylor and Francis, Bristol, PA 19007, 1984.

  • Peyret R. (editor), Handbook of Computational Fluid Mechanics, Academic Press, 1996
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Copyright © A. Filippone (1996-2001). All Rights Reserved.