Lagrangian modeling of a nonhomogeneous turbulent shear flow: Molding homogeneous and isotropic trajectories into a jet

Turbulence is prevalent in nature and industry, from large-scale wave dynamics to small-scale combustion nozzle sprays. In addition to the multiscale nonlinear complexity and both randomness and coherent structures in its dynamics, practical turbulence is often nonhomogeneous and anisotropic, leading to great modeling challenges. In this paper, an efficient model is proposed to predict turbulent jet statistics with high accuracy. The model leverages detailed knowledge of readily available velocity signals from idealized homogeneous turbulence and transforms them into Lagrangian trajectories of a turbulent jet. The resulting spatiotemporal statistics are compared against experimental jet data showing remarkable agreement at all scales. In particular, the intermittency phenomenon is accurately mapped by the model to this inhomogeneous situation, as observed by higher-order moments and velocity increment probability density functions. Crucial to the advancement of turbulence modeling, the transformation is simple to implement for the jet configuration, with possible extensions to other inhomogeneous flows such as wind turbine wakes and canopy flows, to name a few.

Figure 1

(a) The proposed modeling technique: obtain HIST trajectories from an online database, such as JHTDB, to input into the Batchelor transformation, resulting in Lagrangian trajectories which are compared with experimental trajectories. Self-similar profiles of the mean (b) axial and (c) radial velocity as well as (d) the normal axial stress and (e) the shear stress.

Figure 2

Lagrangian two-time statistics of the model and experiment at four locations downstream: (a) axial and (b) radial velocity correlations, (c) axial and (d) radial second-order structure function and the flatness for the (e) axial and (f) radial components. The input is trajectories obtained from the DNS.

Figure 3

PDF of the velocity increment of the model and the experiment for various $\tau$ separations. PDFs of the axial velocity increments at (a) $z/D=15$, (b) $z/D=25$, (c) $z/D=35$, and (d) $z/D=45$. PDFs (e)–(h) correspond to the same downstream locations for the radial velocity increments.

Figure 4

Schematic of the conditioning for statistical analysis of the nonstationary trajectories.