Characteristics of the Entrainment Velocity in a Developing Wake

Characteristics of enstrophy generation and entrainment velocity at the turbulent non-turbulent interface of a temporally evolving wake are studied using Direct Numerical Simulations (DNS), starting from the time instance when the instabilities develop to a fully turbulent state. Results reveal the dominant role of viscous effects compared to the production of enstrophy at the interface at all time instances, although Kolmogorov scaling is found not to apply. The entrainment velocity vn is dominated by a balance of viscous diffusion and dissipation of enstrophy, and the pdf of vn shows large positive tails in fully turbulent regimes which are absent during the initial development of the wake. Many salient features of the entrainment velocity can also be explained by the analytically tractable solution of a laminar wake. Furthermore, the propagation velocity of the interface seems to be dictated by the local fluid velocity for all time instances. Finally, we introduce the dependence of entrainment velocity on the local geometry through the geometric parameters of ‘curvedness’ and ‘shape index’. It is found that TNTI dominated by concave regions with radii of curvature similar to Taylor microscale, with entrainment happening mostly through these concave lower curvedness or flatter regions. INTRODUCTION AND OBJECTIVES Over the past few years there has been an increased number of investigations on characteristics of entrainment at the turbulent non-turbulent interface (TNTI) of turbulent shear flows, such as wakes, jets and boundary layers (e.g, Bisset et al., 2002; Westerweel et al., 2005; Holzner & Lüthi, 2011; Philip et al., 2014; Chauhan et al., 2014; da Silva et al., 2014a). There are many interesting features that characterise the TNTI, and one important feature is the entrainment velocity the difference between the propagation of the TNTI and the fluid velocity. Entrainment velocity has been studied for example by Wolf et al. (2012) experimentally for the case of a turbulent jet. Here we intend to (i) examine the various characteristics of entrainment velocity in the ‘developing’ and the ‘fully turbulent’ regimes of a temporally evolving wake using DNS. After corroborating some of the recent findings, we show features of the viscous and inviscid components of entrainment velocity at the TNTI as the wake develops in time, as well as the strong correlation of the propagation velocity of TNTI with the local fluid velocity. (ii) Quantify the geometric features of the TNTI and their relation to the entrainment velocity. NUMERICAL SIMULATIONS The numerical simulations follow the pseudo-spectral DNS study of da Silva & Pereira (2008). The domain is periodic in all three directions with a size of Lx × Ly × Lz = 4× 6× 4 , with number of points Nx ×Ny ×Nz = 384× 576×384. The Navier-Stokes equations are solved by projecting onto Fourier modes where the nonlinear terms are treated explicitly and viscous terms implicitly. The initial condition involves a step velocity profile (c.f. equation 5) along with a random noise of about 8% to induce transition. The Reynolds number (Re) based on the initial centreline velocity (u0) and width of the velocity profile (L) is 3200. The streamwise, normal and spanwise directions are x, y and z. The temporally evolving wake starts undergoing transition by time of t ≈ 500 and simulations continues till t ≈ 2500 before the box effects come into play. Figure 1 shows contours of enstrophy production at four different time instances across a z=constant plane. CHARACTERISTICS OF ENTRAINMENT VELOCITY The turbulent non-turbulent interface (TNTI) is defined here using an isosurface of enstrophy, ω2, where the vorticity ω = ∇×u, and normal to TNTI, n = ∇ω2/|∇ω2|, see figure 2. The evolution of this isosurface, which is propagating at a velocity, say, vsur f is given by,