# Turbulence modelling in environmental flows: Improving the numerical accuracy of the k-epsilon model by a mathematical transformation

Turbulence modelling in environmental flows: Improving the numerical accuracy of the k-epsilon model by a mathematical transformation

Author ContributorPietrzak, J.D. (mentor)

Schuttelaars, H.M. (mentor)

Uittenbogaard, R.E. (mentor)

Van Kester, J.A.T.M. (mentor)

2014-11-06

AbstractNumerical modelling for environmental flow applications, such as for rivers, lakes, estuaries and coastal flows, faces a trade-off between the numerical accuracy and the required computation time. This trade-off results in grids which typically contain 10 to 100 layers in the vertical direction. Such a grid resolution poses severe limitations to the numerical accuracy of the model. The turbulence model determines a significant part of this accuracy. This research therefore investigates an unexplored method of using transformations to improve the numerical accuracy of two-equation turbulence models at a low resolution. The k-epsilon model is used as starting point for this method. The equation for epsilon is transformed to equations for omega and tau. This results in three turbulence models, the k-epsilon, k-omega and k-tau models, which are physically equivalent, but possess different numerical properties. This research identifies these different numerical properties in order to explain when and why a certain transformation is beneficial to the numerical accuracy. The three turbulence models are tested in six cases of homogeneous and stratified flows in a one-dimensional vertical (1DV) numerical model, which is representative for the implementation in the 3D simulation system Delft 3D-FLOW. It is shown that the k-tau model yields more accurate results than the k-epsilon and k-omega models in boundary friction dominated flows, such as those found in rivers, partially stratified estuaries and along the coast. This improved performance is explained from the profile of tau, which is linear near the frictional boundary and therefore accurately approximated on a low resolution grid. The profiles of epsilon and omega are hyperbolic near the frictional boundary and therefore not accurately represented on such a grid. The boundary condition for tau is well-posed, while no natural boundary conditions for epsilon and omega exist. Dirichlet boundary conditions for epsilon and omega are therefore inaccurate. The Neumann boundary condition is found to be the most accurate alternative boundary condition for epsilon and omega. An adjusted Dirichlet conditions used in Delft 3D-FLOW improves on the result of the ordinary Dirchlet condition, but shows bad convergence behaviour, with results being significantly worse at 100 vertical layers than at 10 vertical layers. A new adjusted Dirichlet condition is developed, which has better convergence behaviour, but is still somewhat worse than the Neumann condition. The k-tau and k-omega models contain a number of diffusive terms, the implementation of which may introduce numerical diffusion in the model. Some of these diffusive terms are essential to the stability of the model. Others are optional. It is argued that the choice whether or not to include such optional diffusive terms should be based on both physical and numerical arguments, because the numerical diffusion associated with the implementation of the terms may have a significant desired or undesired effects on the model results. It is found in the cases in this research that convergence of the turbulence models with increasing grid resolution is typically found between 100 and 1000 grid cells in the vertical direction. One case of temperature modelling of a lake has been tested in which convergence did not occur up to 2000 grid cells. So converged results are generally beyond the range of generally used vertical resolution in 3D models. Within the feasible range of 10 to 100 layers it is found that the results of the turbulence models do not necessarily become more accurate if higher resolution grids are used. So monotonous convergence of the turbulence models is not guaranteed.

Subjectturbulence modelling

numerical accuracy

stratification

k-epsilon model

k-tau model

convergence

http://resolver.tudelft.nl/uuid:251b4a5a-2823-4a2b-aa4a-1a2f52bc9272

Embargo date2015-11-06

Part of collectionStudent theses

Document typemaster thesis

Rights(c) 2014 Dijkstra, Y.M.