Symmetry-based Turbulence Modelling for Engineering Applications

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Symmetries are at the heart of all physical theories, such as classical and quantum mechanics and relativity, as they reflect the axiomatic properties of the underlying physics. For Navier-Stokes turbulence, this property was extended by the applicant to include statistical symmetries, which are a unitary measure of non-Gaussian statistics and intermittency - key properties of turbulence. He rigorously developed these from the infinite sequence of the multi-point correlation equation and the Lundgren-Novikov-Monin probability density function hierarchy. In a number of publications, the applicant was able to show that these symmetries are the axiomatic basis of all turbulent scaling laws and recently he extended this to scaling laws for arbitrary moments, e.g. for the log-region. In turbulence model development, however, symmetries have been used largely unknowingly, but at least it has been achieved that since the 1940s, with each new class of models, further symmetries have been included in the models. From the 1970s onwards, the most important models included all symmetries of classical mechanics, i.e. the Galilean group. However, this explicitly did not apply to the statistical symmetries, which have not been included in any turbulence model so far. It is the central working hypothesis that the explicit and rigorous implementation of all symmetries, i.e. the classical as well as the statistical symmetries, will lead to a significant improvement of the accuracy of the model prediction. This is to be implemented in a multi-equation turbulence model and for a Reynolds stress transport model. The rationale is that, because the new turbulence models include all central symmetries on which all scaling laws for the first and higher moments are based, these are also accurately represented by the model. Scaling laws usually describe only subregions in a turbulent flow, but their very accurate modelling implies that even more complex flows are modelled much more precisely. The machinery of symmetries in infinitesimal form allows an elegant and rigorous derivation of the model equations, whereby a first published preliminary work has shown that, as expected, certain model freedoms are retained, but in particular model parameters cannot be determined from the symmetries. For the final parameter determination of the new turbulence models a machine-learning (ML) approach is chosen, which allows the parallel use of data of different turbulent model flows. The models are implemented in the hp-accurate in-house code BoSSS, which, based on the discontinuous Galerkin method, allows a very precise separation of numerical and model errors. The ML concept is also implemented within BoSSS, which allows an efficient implementation.