Numerical fluid mechanics

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Visualization of a CFD simulation of the Boeing X-43 at Mach 7

The numerical fluid mechanics ( English Computational Fluid Dynamics, CFD ) is an established method of fluid mechanics . Its aim is to solve fluid mechanical problems approximately with numerical methods. The model equations used are mostly the Navier-Stokes equations , Euler equations , Stokes equations or the potential equations .

The motivation for this is that important problems such as the calculation of the drag coefficient very quickly lead to non-linear problems that can only be solved precisely in special cases. The numerical fluid mechanics then offers an inexpensive alternative to experiments in the wind tunnel or water channel .


The most comprehensive model is the Navier-Stokes equations . It is a system of non-linear partial differential equations of the 2nd order. In particular, turbulence and the hydrodynamic boundary layer are also included, which, however, leads to the highest demands on computer performance, memory and numerical processes.

A simpler model are the Euler equations, which do not map the boundary layer due to the neglected friction and also do not contain any turbulence, which means that, for example, stall cannot be simulated using this model. Much coarser grids are suitable for solving the equations in a meaningful way. For those parts of the flow in which the boundary layer does not play an essential role, the Euler equations are very suitable.

Finally, the potential equations are particularly useful when rough predictions are to be made quickly. With them, the entropy is assumed to be constant, which means that no strong shock waves can occur, since the entropy is even discontinuous on these. Further simplification via constant density then leads to the Laplace equation .

In the case of multiphase flows , interaction forces between the phases play a role, and suitable simplifications can be made.

CFD methods also form the basis for numerical aeroacoustics , which deals with the calculation of flow noise.


The most common solution methods in numerical fluid mechanics are

The FEM is suitable for many problems, especially for elliptical and parabolic in the incompressible area, less so for hyperbolic. It is characterized by robustness and solid mathematical underpinning. FVM is suitable for conservation equations , especially for compressible flows. FDM is very simple and therefore primarily of theoretical interest.

Other common methods are

All methods are numerical approximation methods which have to be compared with quantitative experiments for validation . With the exception of the particle-based methods, the starting point of the methods mentioned above is the discretization of the problem with a computational grid .

Time-dependent equations

In the case of time-dependent equations, the sequence of place and time discretization leads to two different approaches:

The first method is mainly used for hyperbolic equations and compressible flows, the latter for incompressible flows. In addition, the Rothe method is more flexible in terms of implementing an adaptive grid refinement in place during the time evolution of the flow equations.

Turbulent currents

With turbulent flows there are still many open questions for numerical flow simulation : Either one uses very fine computational grids as in direct numerical simulation or one uses more or less empirical turbulence models in which, in addition to numerical errors, additional modeling errors occur. Simple problems can be solved in minutes on high-end PCs, while complex 3D problems can sometimes hardly be solved even on mainframes.


In the commercial sector, the market is dominated by the products of the company ANSYS (Fluent, CFX) and Siemens PLM Software (Simcenter ™ STAR-CCM +), both based on the finite volume method (FVM). In the open source area, OpenFOAM is the most popular software package, which is also based on the FVM.

In the field of grid-free solvers, which directly solve the Navier-Stokes equations analogously to the FEM or FVM, there is the commercial software LS-DYNA , MPMSim and Nogrid points . For solvers that solve the Boltzmann equation (so-called particle methods , Lattice-Boltzmann method ) there are other commercial and freely available solvers, such as B. Powerflow, OpenLB or Advanced Simulation Library. Software is also freely available for the smoothed particle hydrodynamics method (SPH), such as pysph or sphysics.

In addition, there is a large number of solvers that are geared to specific flow problems and are used there. Solvers are being developed at many universities and are particularly popular in academic circles.

See also

Web links

Details on the algorithms used can be found in the articles linked above under "Procedure". Comprehensive overviews of available applications and program codes can be accessed via the following links:

Individual evidence

  1. Hermann Schlichting , Klaus Gersten: boundary layer theory . Springer-Verlag, 1997, ISBN 978-3-662-07554-8 , pp. 73 .
  2. F. Durst: Fundamentals of fluid mechanics . Springer, 2006, ISBN 3-540-31323-0 , pp. 10-16 .
  3. LD Landau, EM Lifshitz: Fluid Mechanics-Course of Theoretical Physics , Volume 6, Institute of Physical Problems, Pergamon Press, 1966, pp. 47-53
  4. A. Chorin, J.-E. Marsden: A Mathematical Introduction to Fluid Mechanics . Springer Verlag, 2000
  5. B. Noll: Numerical Fluid Mechanics . Springer Verlag, 1993, ISBN 3-540-56712-7