Turbulence Parameterisation in Hydrobiological Models for Natural Waters

In this thesis different turbulence models are tested with respect to their applicability to oceanographical and limnological problems. Two-equation models for rotating stratified flows are emphasized. After a short introduction in Chapter 1, the transport equations for the turbulent fluxes of momentum, heat and the variance of the temperature fluctuations are derived in Chapter 2. Several closure models for the pressure-strain correlation and the pressure-temperature-gradient correlation are introduced. After their algebraization, the closed transport equations are presented in the so-called boundary layer approximation. With this approximation it is possible to describe the essential features of turbulence models in terms of so-called stability functions. The chapter closes with the presentation of some stability functions, new or already known in the literature. In Chapter 3 different two-equation models (in particular the k-epsilon model, the k-omega model and the model of Mellor and Yamada are formulated and compared in some standard situations. Considered are: The logarithmic law-of-the-wall, the decay of homogeneous turbulence, homogeneously stratified and sheared homogeneous turbulence in full equilibrium and in structural equilibrium, and the balance between turbulent transport of turbulent kinetic energy and its rate of dissipation. The following results are presented: 1. Stability functions for the structural equilibrium, depending only on the Richardson number, are introduced. Analogous expressions for the turbulent Prandtl number and for the ratios of different length-scales are derived. Even though the two-equation models investigated are isomorphic in structural equilibrium, they are sensible with respect to different values of the model parameters. The best results are achieved with the k-omega model. 2. Analytical solutions (in agreement with numerical computations) of two-equation models for the balance between turbulent transport of turbulent kinetic energy and its dissipation are derived. It is demonstrated that the k-epsilon model exhibits a singularity for physically reasonable parameters and that the model of Mellor and Yamada is in accordance with the measurements only without its compulsory wall function. Only the k-omega model reproduces the experimental decay satisfactorily in all situations. Chapter 4 is concerned with applications of two-equation models to problems in limnology and oceanography. The main results are as follows: 1. The mixed layer depth and hence the temperature of the mixed layer in shear-driven entrainment situations is determined by the steady-state Richardson number, an intrinsic property of the models. This quantity, which can be adjusted by parameter calibration, is thus crucial for biological sub-models generally being very sensible with respect to temperature differences. 2. The turbulent bottom boundary layer in Lake Alpnach (Switzerland), induced by internal oscillations, could be modelled in agreement with all significant measurements. However, the phase-lag between the rate of dissipation and the current shear was underestimated by all models. This part of the work was based on a cooperation with the EAWAG (Switzerland) and includes the first reported comparison of continuous turbulence measurements and models in such a boundary layer. 3. A coupled oxygen-turbulence model is suggested that reproduces the measured oxygen profiles in Lake Ammer (Germany) adequately. In Chapter 5 the numerical Finite-Volume method is introduced. The properties of a new discretization of the boundary volumes are discussed. This chapter closes with some tests of the numerical robustness of two-equation models. In contrast to traditional program codes for the computation of turbulent flows, the program architecture suggested in Chapter 6 is based on an object-oriented technique. It is illustrated how turbulence models can be expressed by the abstract vocabulary of an object-oriented language, superior in terms of clarity, reliability, and extendibility compared to structural languages