Development of new solution methods in fluid dynamics based on first integrals of Navier-Stokes equations
Mathematics
Final Report Abstract
In contrast to the previous project and its aim to develop the mathematical foundations for the first integrals of the Navier-Stokes equation, which were validated by benchmark tests against existing results available in open literature, the focus of the current project was the development of efficient analytical and numerical solution methods from the previously elaborated fundamentals in order to apply them to a broad spectrum of current scientific and technological problems. By utilising long-wave approximation to the 2D field equations of the potential-based first integral approach, analytical solutions has been obtained for gravity-driven film flows over step topographies, delivering a useful parameter map predicting the occurrence of capillary ridges/troughs depending on inclination angle and Capillary number. Such problems are relevant in particular for coating technologies. Numerical solutions for comparison have been obtained from a recently developed novel least square FE approach utilising a highly efficient optimally pre-conditioned algebraic multigrid solver, again based on the first integral field equations. The latter approach has also been applied to monolayer Couette flow along textured walls in order to investigate the possibility of friction reduction in combustion engines. Among others, a significant benefit of the potential-based first integral approach is the possibility to rewrite the dynamic boundary condition as a pure Dirichlet- Neumann condition for the potential. This facilitates the extension of the methodology to multi-layer systems, such as the two-layer Couette flows over profiled plates discussed in this project. The latter are useful w.r.t. biophysical questions, especially biological or artificial joints. Starting from the variation formulation of the first integral of the NS equations, 3D flows with axial symmetry could be addressed in a similar way as 2D flows. Also the long wave approximation could be applied in an analogous way to consider coating flows over cones and hemispheres with constant volume flux. After simple modifications, the approach also applied to unsteady flows with constant volume and partial wetting. In addition to the engineering applications mentioned above, the necessary further development of the mathematical fundamentals resulted in surprising perspectives for turbulent, compressible and relativistic flows. These findings are valuable for possible future work to obtain innovative, highly efficient computational methods. In particular, the relativistic four-dimensional tensor formulation suggests the use of Clifford algebras (quaternions, Dirac matrices) which are not applicable to the original field equations.
Publications
- Couette flow with geometrically induced unsteady effects. PAMM, 18(1):e201800239, Dec 2018
Markus Scholle and Florian Marner
(See online at https://doi.org/10.1002/pamm.201800239) - Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications. Journal of Mathematical Physics, 59(4):043101, Apr 2018
M. Scholle, P. H. Gaskell, and F. Marner
(See online at https://doi.org/10.1063/1.5031119) - A potential field description for gravity-driven film flow over piece-wise planar topography. Fluids, 4(2), 2019
Markus Scholle, Philip H. Gaskell, and Florian Marner
(See online at https://doi.org/10.3390/fluids4020082) - Potential-based formulations of the Navier-Stokes equations and their application. PhD thesis, Durham University, Durham, UK, 2019
F. Marner
- Thin liquid film formation on hemispherical and conical substrate. PAMM, 19(1):e201900111, 2019
Markus Scholle, Florian Marner, and Philip H. Gaskell
(See online at https://doi.org/10.1002/pamm.201900111) - A first integral form of the energy– momentum equations for viscous flow, with comparisons drawn to classical fluid flow theory. European Journal of Mechanics - B/Fluids, 84:262 – 271, 2020
M. Scholle, F. Marner, and P.H. Gaskell
(See online at https://doi.org/10.1016/j.euromechflu.2020.06.010) - Potential fields in fluid mechanics: A review of two classical approaches and related recent advances. Water, 12(5), 2020
Markus Scholle, Florian Marner, and Philip H. Gaskell
(See online at https://doi.org/10.3390/w12051241) - Multilayer modelling of lubricated contacts – a new approach based on a potential field description. In G.-P. Ostermeyer, V.L. Popov, E.V. Shilko, and O. Vasiljeva, editors, Multiscale Biomechanics and Tribology of Inorganic and Organic Systems — In memory of Professor Sergey Psakhie, Springer Tracts in Mechanical Engineering. Springer International Publishing, 2021
M. Scholle, M. Mellmann, P. H. Gaskell, L. Westerkamp, and F. Marner
(See online at https://doi.org/10.1007/978-3-030-60124-9_16)
