Wake behind a three-dimensional dry transom stern. Part 2. Analysis and modelling of incompressible highly variable density turbulence

Abstract

We analyse the turbulence characteristics and consider the closure modelling of the air entraining flow in the wake of three-dimensional, rectangular dry transom sterns obtained using high-resolution implicit large eddy simulations (iLES) (Hendrickson et al., J. Fluid Mech., vol. 875, 2019, pp. 854–883). Our focus is the incompressible highly variable density turbulence (IHVDT) in the near surface mixed-phase region ${\mathcal{R}}$ behind the stern. We characterize the turbulence statistics in ${\mathcal{R}}$ and determine it to be highly anisotropic due to quasi-steady wave breaking. Using unconditioned Reynolds decomposition for our analysis, we show that the turbulent mass flux (TMF) is important in IHVDT for the production of turbulent kinetic energy and is as relevant to the mean momentum equations as the Reynolds stresses. We develop a simple, regional explicit algebraic closure model for the TMF based on a functional relationship between the fluxes and tensor flow quantities. A priori tests of the model show mean density gradients and buoyancy effects are the main driving parameters for predicting the turbulent mass flux and the model is capable of capturing the highly localized nature of the TMF in ${\mathcal{R}}$.

Numerical investigation of mixed-phase turbulence induced by a plunging jet

“…Because the size of the mixed-phase region varies with the Froude number, to facilitate comparison of the results for different cases, we followed Hendrickson & Yue (2019) to define the conditioned average in the mixed-phase region as Here, represents the variable inside the mixed-phase region. The integration denoted by is performed over a cross-stream section – plane), and represents the area of the mixed-phase region in the corresponding plane.…”

“…This indicates that can be well estimated by the proposed model. Although the accuracy of the streamwise component estimated by the model is not as good as that of the vertical component, the correlation coefficients of the streamwise components are all above 0.55, which is overall higher than the correlation coefficient of the model that directly fits TMF using TKE in Hendrickson & Yue (2019).…”

“…To finalize this paper, we compare the main findings of the present study on the mixed-phase turbulence generated by the plunging jet with the results of Hendrickson & Yue (2019) on the mixed-phase turbulence in a wake flow. The purpose of providing this comparison is to evaluate whether the main conclusions of the present study are potentially common for different types of mixed-phase turbulence, or they are special in plunging jets.…”

Scaling effects on the free surface backward facing step flow

“…Although the geometry is far from a real ship stern, it was an attractive choice due to simplicity and availability of experimental data at various Froude numbers. The wave formation in the FSBFS had been previously studied analytically [82][83][84], experimentally [81] and numerically [79,85,86] but the focus of most of the studies had been on the Froude number effect. The only study where the Reynolds number effect was considered is the paper by Starke et al [79] where a steady state RANS solver was applied for the FSBFS in model (Re = UL/ν = 5.1 × 10 6 ) and full scale (Re = 4.5 × 10 8 ).…”

Wake behind a three-dimensional dry transom stern. Part 1. Flow structure and large-scale air entrainment

“…The objective of this part is to provide a physical understanding of the detailed flow features in the mixed region and the characteristics of the large-scale air entrainment. Part 2 (Hendrickson & Yue 2019) focuses on understanding and modelling of the highly mixed, air-water turbulent near-surface region of the wake by combining the surface fluctuations, spray and entrainment into a single mixed-phase region through an Eulerian framework. The objective of this part is to provide a statistical understanding of the mixed-phase region within the context of incompressible highly variable density turbulence and address the necessary turbulent mass flux closure modelling.…”