Reynolds-number dependence of the longitudinal dispersion in turbulent pipe flow

In Taylor's theory, the longitudinal dispersion in turbulent pipe flows approaches, on long time scales, a diffusive behavior with a constant diffusivity $K_L$, which depends empirically on the Reynolds number Re. We show that the dependence on Re can be determined from the turbulent energy spectrum. By using the intimate connection between the friction factor and the longitudinal dispersion in wall-bounded turbulence, we predict different asymptotic scaling laws of $K_L$(Re) depending on the different turbulent cascades in two-dimensional turbulence. We also explore numerically the $K_L$(Re) dependence in turbulent channel flows with smooth and rough walls using a lattice Boltzmann method.

Figure 1

Longitudinal concentration of tracers in the comoving frame. Gaussian fit of the core distribution.

Figure 2

Time-dependence of the rescaled longitudinal diffusivity measured from the mean-squared displacement of passive tracers at different Re numbers measured in a turbulent flow generated (a) from wall roughness, and (b) behind a grid.

Figure 3

Lagrangian frequency spectrum ${\mathrm{\Phi }}_L\left(\omega \right)$ and Eulerian frequency spectrum ${\mathrm{\Phi }}_E\left(\omega \right)$ of the transverse velocity fluctuations in a turbulent flow at $\mathrm{Re}=30000$ that develops (a) from the wall roughness and (b) behind a grid.

Figure 4

Longitudinal diffusivity $K_L$ with Re numbers for the grid- and rough-wall-induced turbulence.