Mean dynamics of transitional boundary-layer flow

Abstract

The dynamical mechanisms underlying the redistribution of mean momentum and vorticity are explored for transitional two-dimensional boundary-layer flow at nominally zero pressure gradient. The analyses primarily employ the direct numerical simulation database of Wu & Moin (J. Fluid Mech., vol. 630, 2009, p. 5), but are supplemented with verifications utilizing subsequent similar simulations. The transitional regime is taken to include both an instability stage, which effectively generates a finite Reynolds stress profile, −ρuv(y), and a nonlinear development stage, which progresses until the terms in the mean momentum equation attain the magnitude ordering of the four-layer structure revealed by Wei et al. (J. Fluid Mech., vol. 522, 2005, p. 303). Self-consistently applied criteria reveal that the third layer of this structure forms first, followed by layers IV and then II and I. For the present flows, the four-layer structure is estimated to be first realized at a momentum thickness Reynolds number R θ = U ∞ θ/ν 780. The first-principles-based theory of Fife et al. (J. Disc. Cont. Dyn. Syst. A, vol. 24, 2009, p. 781) is used to describe the mean dynamics in the laminar, transitional and four-layer regimes. As in channel flow, the transitional regime is marked by a non-negligible influence of all three terms in the mean momentum equation at essentially all positions in the boundary layer. During the transitional regime, the action of the Reynolds stress gradient rearranges the mean viscous force and mean advection profiles. This culminates with the segregation of forces characteristic of the four-layer regime. Empirical and theoretical evidence suggests that the formation of the four-layer structure also underlies the emergence of the mean dynamical properties characteristic of the high-Reynolds-number flow. These pertain to why and where the mean velocity profile increasingly exhibits logarithmic behaviour, and how and why the Reynolds stress distribution develops such that the inner normalized position of its peak value, y + m , exhibits a Reynolds number dependence according to y + m 1.9 √ δ + .

Self-similar mean dynamics in turbulent wall flows

“…The four-layer structure and its attendant scaling behaviours constitute foundational elements of a mathematically cogent and mechanistically descriptive mean flow theory (Wei et al 2005Fife et al 2005aFife et al ,b, 2009Metzger, Adams & Fife 2008;Klewicki et al 2009Klewicki et al , 2011Klewicki et al , 2012. This subsection outlines and clarifies the elements of this theory most relevant to the present aims, and briefly describes some of the salient differences between channel, pipe and boundary layer flows.…”

“…In contrast, neither this process of mean vorticity annihilation nor the mean flux of vorticity into the flow at the wall are present in the zero-pressure-gradient boundary layer. The structure of figure 1 is an outcome of the flow development through the transitional regime for each of the canonical flows considered (Elsnab et al 2011;Klewicki et al 2011Klewicki et al , 2012. This structure arises due to the emergence of turbulent inertia.…”

“…where = (x + ) and b = b(y + , ) (Metzger et al 2008;Klewicki et al 2011). The unknown O(1) function b is constrained such that δ + 0 b dy + = O(δ + ) for all .…”

“…The appearance of robust nonlinear interactions associated with turbulent inertia mark the onset of the transitional regime. These interactions lead to rapid rearrangements of the mean momentum distribution during the transitional regimes of channels, pipes and boundary layers (Elsnab et al 2011;Klewicki, Ebner & Wu 2011;Klewicki et al 2012). The transitional regimes culminate (albeit in each case by a different route) with the establishment of the four-layer regime first described by Wei et al (2005).…”

Numerical investigation on two-dimensional boundary layer flow with transition

“…So, transitional flow has received much attention (Wang and Fu, 2009a, 2009bCao, 2009; © Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2014 Wassermann and Kloker, 2005). Besides experimental research (Klewicki et al, 2011), some numerical simulations based on Reynolds averaged numerical simulation (RANS) are conducted on the transition in the last decade (Biau et al, 2007;Jacobs and Durbin, 2000;Xiao et al, 2006;Yang, 2012). However, due to the short length of transition zone with rapid variation of flow parameters, it raises higher request for the turbulent model in RANS method (Wang and Guo, 2012;Chen and Chen, 2010;Fan et al, 2011).…”