Entrainment in stratified environments — an interface-based approach

Entrainment in stratified environments — an interface-based approach Dominik Krug 1 , Maarten van Reeuwijk 2 , Markus Holzner 3 and Ivan Marusic 1 Department of Mechanical Engineering, The University of Melbourne, dominik.krug@unimelb.edu. au Department of Civil and Environmental Engineering, Imperial College London Institute of Environmental Engineering, ETH Z¨ urich Abstract In this study, we focus on understanding stratification effects on the entrainment process in gravity currents via a multi-scale approach based on the turbulent/non-turbulent interface (TNTI). We find that a large part of the reduction of the entrainment parameter in the presence of a stable stratification can be explained through a reduction of the surface area of the TNTI. An analysis of the fractal geometry of the TNTI reveals that the decrease of the surface area with increasing stable stratification is predominantly due to a reduced effective fractal dimension. 1 Introduction Turbulent entrainment across stably stratified interfaces is widespread in many geophys- ical and engineering applications. In this paper, we focus on the case of gravity currents which is relevant to a wide range of flow scenarios (Simpson, 1999). It has long been known that in this kind of flow entrainment decreases with increasing Richardson num- ber Ri, i.e. with increasing relative strength of the stable stratification compared to the destabilizing shear (Ellison and Turner, 1959). Instead of providing a parametrization of entrainment in terms of bulk quantities, our goal here is to investigate the multi-scale processes that give rise to turbulent entrainment. It is widely observed that sharp and strongly contorted interfaces separate turbulent and non-turbulent regions in partly turbulent flows such as jets, wakes, boundary layers and many more (Corrsin and Kistler, 1954; Da Silva et al., 2014). In the present investigation, we study entrainment in terms of the volume flux through this turbulent/non-turbulent interface (TNTI). Defining the TNTI using an appropriate threshold on enstrophy ω 2 , where ω denotes the norm of the vorticity vector, the local entrainment velocity can be defined as the motion of the TNTI relative to the fluid, which can be computed from v n ≡ −|∇ω 2 | −1 Dω 2 /Dt (Holzner and uthi, 2011). The entrained volume flux is then R L¨ given by the surface integral Q ω ≡ A η v n dA = hv n iA η , where A η is the surface area of the TNTI and h·i denotes an average over A η . In Krug et al. (2015), we showed that the bulk entrainment coefficient E ≡ u e /u T , where u e is the global entrainment velocity and u T the top-hat velocity, can equivalently be obtained via a TNTI-based approach from E I = − v n A η h u T A 0 h ω Here, h is the top-hat thickness, h ω the mean TNTI position off the wall and the factor h/h ω represents a self-similarity correction of the entrained fluxes. Note also that the subscript I only serves to identify the way E was computed and it has been verified that E = E I by Krug et al. (2015) and van Reeuwijk et al. (2016). Based on this result, it is VIII th Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016