Turbulence modeling of stably stratified wall-bounded flows

Abstract

The subject of wall-bounded flows has been a matter of discussion and has received considerable attention in the past few decades. This is mainly attributed to the fact that the presence of the solid wall has profound effects on the turbulence and hence results in anomalous mixing and transport of momentum, scalar and heat in environmental flows. This is much more intense in the vicinity of the solid wall commonly known as the near-wall region compared to regions away from the wall. This effect will be more complicated in the presence of density stratification which has a strong influence on the development of turbulence. Therefore, numerous field, laboratory, numerical and theoretical studies are performed in a quest to gain a better understanding of wall-bounded flows especially in the presence of stratification. However, there is still a lack of a clear picture on the near-wall flow properties, the onset of turbulence and the resulting mixing in wall-bounded flows. The aim of this dissertation is to employ both theory and numerical simulations to revisit mixing in wall-bounded flows, especially in the near-wall region. The main objectives are: • To investigate the unstratified near-wall turbulence and revisit the turbulent (eddy) viscosity (νt) formulation in unstratified wall-bounded flows. This will be followed by derivation of a novel proposition for the appropriate velocity, length and time scales in unstratified wall-bounded flows. • To revisit the fundamentals of common Reynolds-averaged Navier-Stokes (RANS) closure schemes such as the standard k-ε model and investigate their capability to model near-wall turbulence. • To investigate the turbulent mixing in stably stratified wall-bounded flows. The mixing of momentum, scalar and the efficiency of the mixing are evaluated. • To study wall-bounded turbulent flows in the presence of stable stratification by performing one-dimensional RANS simulations. In particular, this includes introduction of a modified turbulent Prandtl number (Prt) for wall-bounded flows and calibration of the standard k-ε model. In this dissertation, a novel formulation for the turbulent (eddy) viscosity given by ν=ε/(S2) is derived by assuming equilibrium between the turbulent kinetic energy production rate P and the dissipation rate of the turbulent kinetic energy (ε), where S is the mean shear rate. Also, the relevant scales of length and velocity are derived. The propositions are tested with the direct numerical simulation (DNS) data of unstratified turbulent channel flow of Hoyas & Jiménez (2006) and unstratified turbulent boundary layer flow of Sillero et al. (2013). The comparisons of the propositions with the exact computations from the DNS data are excellent. Furthermore, the suitability of the equilibrium assumption (i.e. P ≈ ε) for modeling near-wall turbulence is revisited. This is important as most widely used turbulent viscosities such as the formulation of the standard k-ε model are developed by using the equilibrium assumption. It is analytically shown that such νt formulations are not suitable for modeling near-wall turbulence. Also, the turbulent mixing in stably stratified wall-bounded flows is studied by employing analytical arguments. 'A priori' tests are performed by using highly resolved stably stratified channel flow DNS data of García-Villalba & del Álamo (2011). It is shown that in such flows assuming P ≈ ε + εPE, where εPE is the dissipation rate of the turbulent potential energy, holds in a big fraction of the flow depth. Also, the results show that an irreversible flux Richardson number as R*f = εPE/(ε + εPE) can properly predict the flux Richardson number (Rf =-B/P), where B is the buoyancy flux. It is also shown that neglecting the transport rate of εPE and assuming equilibrium as -B ≈ εPE is not a suitable assumption. Furthermore, the ideas discussed are utilized to perform 'a posteriori' tests and to simulate stably stratified wall-bounded flows by using RANS numerical models. To do this, first a simple one-dimensional zero-equation as well as two-equation k-ε RANS models are developed. It is shown that turbulent Prandtl numbers based on the homogeneous assumption are not capable of providing a good estimation of the mixing and therefore an inhomogeneity correction must be introduced. It is analytically shown that commonly used homogeneous turbulent Prandtl numbers should be modified for a wall-bounded flow using a correction as (1-z/D), where D is the total flow depth. This work is extended by revisiting the buoyancy parameter (Cε3) in the standard k-ε closure scheme. Analytical arguments are used to show that Cε3 ≈ 0. RANS results show the suitability of the propositions for modeling of stably stratified turbulent channel flows. The ultimate goal of this research is to enhance understanding of the fundamental aspects of wall-bounded environmental flows and develop appropriate turbulence models that can capture the physics of stably stratified wall-bounded turbulent flows.