Hierarchical parcel-swapping representation of turbulent mixing. III. Origins of correlation patterns observed in turbulent boundary layers

Patterns observed in correlation measurements provided early indications of organized structures in turbulent boundary layers, later supplemented by imaging studies. It is recognized that correlation patterns could in some instances reflect transient flow processes rather than structures, but discrimination among the possible origins of various correlation patterns has not been addressed systematically. Here this is done from three complementary perspectives. First, a conceptual framework termed cascade analogy is shown to imply three processes, all devoid of organized-structure influences, that might induce such patterns. Second, hierarchical parcel swapping (HiPS), a stochastic model that is likewise devoid of such influences, is shown to produce the correlation patterns that are implied by these processes. The HiPS formulation used here is a highly reduced version of a previous formulation, highlighting the elementary nature of the pattern-inducing processes. Third, experimental indications of all these patterns are noted. One pattern, an obliquely oriented ridge of peak cross-correlation between streamwise velocity at various heights and wall stress, is known by other means to be associated with organized structure. The results indicate that the oblique ridge does not in itself uniquely imply this interpretation. This inference is shown to have substantive consequences. The other two patterns are marginally discernible in reported measurements, but there has been no prior assessment of their significance, nor are they obviously associated with organized structure in boundary layers. Here they are identified as possible analogs of the inertial-range inverse cascade (backscatter) and the effects of the fluctuating rate of forward cascading, respectively. Further investigations to delineate more clearly the origins and the circumstances of occurrence of such patterns are suggested.

Figure 1

The implementation of a representative swap illustrates the difference between the full and reduced HiPS channel-flow formulations. A five-level tree is shown. In full HiPS, there are 16 parcels at the base of the tree, but only the eight left-of-center parcels (squares) are shown. The equivalent reduced formulation is shown to the right of the vertical dashed-dotted line that separates the two halves of the flow domain. In full HiPS, a pair of adjacent parcels has the same state (schematically, $b$, $c$, or $d$) because adjacent parcels mix instantly if any operation produces adjacent parcels with different states. Whenever the leftmost pair is mixed, the wall parcel is then reset to $u=v=w=0$, denoted as state 0, to enforce the wall boundary condition, but its nearest neighbor remains at some different state $a$. The tree structure (triangles and line segments connecting them) identifies allowed swaps, adjacent parcel pairs, and statistically equivalent parcel groups, which are collections of parcels with the same proximity to the nearest wall parcel. The filled triangle is the apex of the illustrative swap and the apex of each swapped subtree is denoted by a striped triangle, where thick line segments denote the paths from the former to each of the latter. Associated parcel displacements, indicated by dashed arrows, yield the modified system state as illustrated. Below that, additional operations corresponding to reduced HiPS are shown. In reduced HiPS, all parcels in a statistically equivalent group are fully mixed, so $c$ and $d$ must be equal initially. This implies that the group of parcels whose states are $(c,c,b,b)$ mix to identical states $(m,m,m,m)$, as shown. This enforced uniformity of parcel groups allows each group to be represented by one parcel, as shown right of center in a mirror-image rendering of the left-of-center operations. Parcels thus omitted have dotted borders. As explained in the Appendix, an eddy event consists of a swap followed by an additional operation that is not illustrated in the sketch, requiring generalization of the indicated algebraic expression for $m$.

Figure 2

Absolute value of wall-normalized Reynolds shear stress for Table 1 cases: $+$, case R; $\times$, case F; ——, exact result in the absence of viscous transport. Distance $y$ from the wall is normalized by the channel half-height $h$.

Figure 3

Mean velocity profile in wall-normalized coordinates: $+$, case R; $\times$, case F; $◯$, measurements for $R{e}_{\tau }=2000$ [5].

Figure 4

Velocity variances ${\overline{u^{\prime 2}}}^{+}$ ($\square$) and ${\overline{v^{\prime 2}}}^{+}$ ($◯$) for cases R (filled symbols) and F (open symbols), connected by lines for both cases. The symbols not connected by lines are measurements for $R{e}_{\tau }=2000$ [5].

Figure 5

Positive $y^{+}$: TKE budget, normalized by $u_{\tau }^4/\nu$: $◯$, case R; $\square$, case F. Negative $y^{+}$: DNS for $R{e}_{\tau }=590$ [6]. Budget terms: ——, production; $---$, dissipation; $-·-$, sum of viscous and advective transport.

Figure 6

PDF of wall-normalized wall stress: $+$, case R; $\times$, case F; $◯$, DNS for $R{e}_{\tau }=1000$ [7]. The curve connecting the case R symbols shows the smoothness of the PDF for that case.

Figure 7

Sketch of the propagation of the influence of a fluctuation originating at a nominal time $t=0$ at height $Y$ above the wall (filled star). After a time $t^{*}$, it has arrived at the wall at a distance $X_w$ downstream of its origin and concurrently advanced a distance $U\left(Y\right)t^{*}$ directly downstream of its origin (open stars). At a representative point on the trajectory toward the wall (solid curve), the velocity vector tangent to the trajectory (dashed) is decomposed into its streamwise and downward components $U\left(y\right)$ and $V\left(y\right)$, respectively. As explained in Sec. 7, the resulting near-wall perturbation at time $t^{*}$ can initiate reverse (upward, downstream) propagation (dotted curve).

Figure 8

Correlation coefficient specified by Eq. (1). Time lag is normalized by $\delta /U_0$, where $\delta$ is the boundary-layer thickness and $U_0$ is the free-stream velocity. Thick lines, boundary-layer measurements by BT; thin lines, model channel-flow results with time lag scaled for consistency with the measurements using the same scaling factor for all model profiles. Height $Y$ in Eq. (1), normalized by the half-height $h$ for the channel flow and by $\delta$ for the boundary layer: blue dotted, $Y/h=0.04$, $Y/\delta =0.05$; red dashed, $Y/h=0.18$, $Y/\delta =0.25$; black solid, $Y/h=0.37$, $Y/\delta =0.50$.