Drift and pseudomomentum in bounded turbulent shear flows

This paper is concerned with the evaluation of two Lagrangian measures which arise in oscillatory or fluctuating shear flows when the fluctuating field is rotational and the spectrum of wave numbers which comprise it is continuous. The measures are the drift and pseudomomentum. Phillips [J. Fluid Mech. 430, 209 (2001)] has shown that the measures are, in such instances, succinctly expressed in terms of Lagrangian integrals of Eulerian space-time correlations. But they are difficult to interpret, and the present work begins by expressing them in a more insightful form. This is achieved by assuming the space-time correlations are separable as magnitude, determined by one-point velocity correlations, and spatial diminution. The measures then parse into terms comprised of the mean Eulerian velocity, one-point velocity correlations, and a family of integrals of spatial diminution, which in turn define a series of Lagrangian time and velocity scales. The pseudomomentum is seen to be strictly negative and related to the turbulence kinetic energy, while the drift is mixed and strongly influenced by the Reynolds stress. Both are calculated for turbulent channel flow for a range of Reynolds numbers and appear, as the Reynolds number increases, to approach a terminal form. At all Reynolds numbers studied, the pseudomomentum has a sole peak located in wall units in the low teens, while at the highest Reynolds number studied, ${\mathrm{Re}}_{\tau }=5200$, the drift is negative in the vicinity of that peak, positive elsewhere, and largest near the rigid boundary. In contrast, the time and velocity scales grow almost logarithmically over much of the layer. Finally, the drift and pseudomomentum are discussed in the context of coherent wall layer structures with which they are intricately linked.

Figure 1

Profiles of drift $D_1$ and pseudomomentum $P_1$ as a function of distance from the boundary in (a) wall units and (b) outer units. The profiles are for turbulent channel flow for ${\mathrm{Re}}_{\tau }=180$, 550, 1000, 2000, 4200, and 5200; previous calculations of $D_1⧫$ and $P_1+$ at ${\mathrm{Re}}_{\tau }=180$ by [3].

Figure 2

Profiles of drift $D_1$ relative to the mean velocity $U$ with distance $z$ from the boundary in wall units for ${\mathrm{Re}}_{\tau }=180$, 550, 1000, 2000, 4200, and 5200.

Figure 3

Profiles of the component terms $\text{I}, \text{II,}$ and $\text{III}$ as defined in the text for (a) the drift $D_1$ Eq. (19) and terms $i, ii, iii, iv$ for the (b) pseudomomentum $P_1$ Eq. (20) as a function of distance $z$ from the boundary in wall units, for channel flow for ${\mathrm{Re}}_{\tau }=5200$.

Figure 4

Profiles of the first $\text{I}+\text{II}$ and second $\text{III}$ component terms of the drift $D_1$ defined in Eq. (19) as a function of distance $z$ from the boundary in wall units, for channel flow at ${\mathrm{Re}}_{\tau }=180$, 550, 1000, 2000, 4200, and 5200.

Figure 5

Profiles of the time scales $\mathcal{A}, \mathcal{B}$, and $\mathcal{C}$ in wall units as a function of distance $z$ from the boundary in outer units, for channel flow at (a) ${\mathrm{Re}}_{\tau }=180$ and (b) ${\mathrm{Re}}_{\tau }=5200$.

Figure 6

Profiles of the differential time scale $d\mathcal{A}/dz$ as a function of distance $z$ from the boundary in wall units, for channel flow at ${\mathrm{Re}}_{\tau }=180$, 550, 1000, 2000, 4200, and 5200.

Figure 7

Profiles of the velocity scales $\mathcal{U}, \mathcal{V}$, and $\mathcal{W}$ in wall units as a function of distance $z$ from the boundary in outer units for channel flow at (a) ${\mathrm{Re}}_{\tau }=180$ and (b) ${\mathrm{Re}}_{\tau }=5200$.

Figure 8

Profiles of the differential pseudomomentum $d{P}_1/dz$ with distance $z$ from the boundary in wall units for ${\mathrm{Re}}_{\tau }=180$, 550, 1000, 2000, 4200, and 5200.

Figure 9

Stability boundaries for the onset of streamwise vortex structure depicted by the linear steady states of CLg theory [17] describing periodic Poiseuille flow for ${\mathrm{Re}}_{\tau }=180$, 550, 1000, 2000, 4200, and 5200, expressed as vortex spacing $L$ (2 rolls) in wall units against ${\mathrm{Re}}_{\tau }$. The asymptotes for instability to two- and three-dimensional waves determined numerically are drawn at ${\mathrm{Re}}_{\tau }=45$ and ${\mathrm{Re}}_{\tau }=64.8$ [65]. Numerical results for $•$ roll spacing and $▵$ ensemble average of streak spacing at ${\mathrm{Re}}_{\tau }=80$ [41].