Symmetries and Invariant Solutions of Turbulent Flows and their Implications for Turbulence Modelling

  • Martin Oberlack
Part of the International Centre for Mechanical Sciences book series (CISM, volume 442)


First a short introduction to the notion of symmetries of differential equations is given including infinitesimal transformations, invariant functions and invariant solutions. Then it is shown that the symmetry properties i.e. invariant transformations of the Navier-Stokes equations are pivotal to understand the physics of fluid flow. We demonstrate that all common symmetries “transfer” to the statistical equations such as the Reynolds stress transport equations or the multi-point correlation equations. From the knowledge of the symmetries we derive from the latter equations a broad variety of invariant solutions (scaling laws) using only first principles. These solutions comprise classical results such as the logarithmic-law-of-the-wall and other wall bounded shear flows. Also homogeneous and inhomogeneous time-dependent flows are analyzed and solutions are discussed. Since the symmetries of fluid motion are admitted by all statistical quantities of turbulent flows we give necessary conditions on turbulence models such that they “capture” the proper physics i.e. the symmetries and their corresponding invariant solutions. Particularly we will investigate two-equation models such as the κ-ε model as well as Reynolds stress transport models with respect to their symmetry properties. Finally we give conditions for the sub-grid scale model in large-eddy simulation of turbulence to obey the proper symmetries. For all of the latter turbulence models it is demonstrated that symmetry violation gives rather disadvantageous prediction capabilities of the model under investigation.


Symmetry Breaking Reynolds Stress Isotropic Turbulence Partial Differential Equation Invariant Solution 


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Copyright information

© Springer-Verlag Wien 2002

Authors and Affiliations

  • Martin Oberlack
    • 1
  1. 1.Hydromechanics and Hydraulics GroupDarmstadt University of TechnologyDarmstadtGermany

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