Revisiting Taylor's hypothesis in homogeneous turbulent shear flow

Taylor's hypothesis of frozen flow is revisited in homogeneous turbulent shear flow by examining the cancellation properties of Eulerian and convective accelerations at different flow scales. Using results of direct numerical simulations, vector-valued flow quantities, including the Lagrangian, Eulerian, and convective accelerations, are decomposed into an orthogonal wavelet series and their alignment properties are quantified through the introduction of scale-dependent geometrical statistics. Joint-probability density functions of the Eulerian and convective accelerations show antialignment at small scales of the turbulent motion, but this observation does not hold at large scales. Similarly, the angles of the scale-wise contributions of the Eulerian and convective accelerations were found to prefer an antiparallel orientation at small scales. Such antialignment, however, is not observed at the largest scales of the turbulent motion. The results suggest that Taylor's hypothesis holds at small scales of homogeneous turbulent shear flow, but not for large-scale motion. The Corrsin scale is proposed as a measure for the applicability of Taylor's hypothesis in such flows.

Figure 1

Evolution of the normalized turbulent kinetic energy $K/K_0$ with normalized time $St$ (left) as well as the growth rate of the turbulent kinetic energy $\gamma$, the normalized production rate $P/\left(SK\right)$, and the normalized dissipation rate $\epsilon /\left(SK\right)$ (right).

Figure 2

Joint probability distribution functions (pdf) of the the Eulerian acceleration ${\mathit{a}}_E$ and the convective acceleration ${\mathit{a}}_C$ at nondimensional time $St=10$ for the total fields.

Figure 3

Joint probability distribution functions (pdfs) of the the Eulerian acceleration ${\mathit{a}}_E$ and the convective acceleration ${\mathit{a}}_C$ at nondimensional time $St=10$ for the decomposed fields at scale indices $j=2$ (top, left), $j=4$ (top, right), $j=6$ (bottom, left), and $j=8$ (bottom, right).

Figure 4

Probability distribution functions (pdfs) of the cosine of the angle between the Eulerian acceleration ${\mathit{a}}_E$ and the convective acceleration ${\mathit{a}}_C$ (top, left), the Lagrangian acceleration ${\mathit{a}}_L$ and the convective acceleration ${\mathit{a}}_C$ (top, right), Lagrangian and Eulerian accelerations (center, left), Lagrangian acceleration and the pressure-gradient term (center, right), Eulerian acceleration and the pressure-gradient term (bottom, left), as well as the advection term and the pressure gradient term (bottom, right) at time $St=10$. Note that the pdfs are shown for the total flow fields (dashed lines), and the flow fields at the scale indices $j=2$ (large scale), 4, 6, and 8 (small scale).