Wave Particle Turbulent Simulation of Spatially Developing Round Jets Using a Non Equilibrium Transport Model with a Mixing Length Characteristic Time Closure

TL;DR

This work tackles turbulence modeling for spatially developing round jets on coarse grids by introducing a wave–particle turbulent simulation (WPTS) framework that couples a grid-resolved wave component with stochastic particles representing unresolved motions. A turbulence characteristic-time closure, $\tau_t$, derived from Prandtl’s mixing-length concept, governs particle transport via $\tau_n = \tau + \tau_t$ and uses $\nu_t = (C_{ml} l)^2 |\mathbf{S}|$ with $l = \sqrt{D x} + b_{ml}$. The method reproduces jet self-similarity and key turbulence statistics at $Re=5{,}000$ and $Re=20{,}000$, requiring only modest parameter adjustments and grid refinement, demonstrating robustness and computational efficiency for practical jet simulations. The results highlight non-local transport and laminar–turbulent regime coupling within a unified framework, offering a promising tool for predicting jet flows on coarse grids with potential applicability to a broader class of turbulent flows.

Abstract

In this paper, the wave-particle turbulent simulation (WPTS), a recently developed multiscale, non-equilibrium turbulence modeling approach, is coupled with a turbulence characteristic-time closure derived from Prandtl mixing-length hypothesis and applied to spatially developing round jets. In WPTS, fluid elements in strongly turbulent regions are represented by Lagrangian particles that travel a finite distance before interacting with the background flow field represented in a wave-like (Eulerian) form. This mechanism bears conceptual similarity to the discrete fluid parcels invoked in the Prandtl mixing-length picture. WPTS differs from conventional mixing-length-based turbulence models in two key respects. First, particle evolution follows a non-equilibrium transport mechanism, rather than the equilibrium assumptions typically embedded in eddy-viscosity closures. Second, WPTS advances the wave and particle components in a coupled manner, with the particle fraction governed primarily by the modeled turbulence characteristic time, enabling laminar and turbulent regimes to be represented within a unified framework. Because spatially developing jets provide a canonical test case with well-established similarity behavior, they are used here for evaluation. Specifically, this work (1) develops a mixing-length-based characteristic-time model tailored to jet flows and (2) incorporates it into WPTS to assess predictive performance. The resulting WPTS framework accurately reproduces the jet similarity solution and other characteristic features at Reynolds numbers of 5,000 and 20,000, demonstrating the promise of WPTS as a practical tool for turbulence modeling and simulation.

Figures (7)

• Figure 1: The instantaneous snapshot of vorticity $\omega_z = V_x - U_y$ in WPTS at $t=1650$ for $xoy$ plane with $z=0$. The presented domain is around $[0,32D]\times[-11D,11D]$.
• Figure 2: (a) The modeled mixing length $C_{ml}l$ in Eq.\ref{['nut']} with $C_{ml}^2 = 6.1\times10^{-4}$ and $b_{ml} = 1.60$, normalized by the jet diameter $D$. (b) The instantaneous snapshot of $\tau_n$, the sum of $\tau_p$ and $\tau_t$, at $t=1650$ for $xoy$ plane with $z=0$. The presented domain is around $[0,32D]\times[-11D,11D]$.
• Figure 3: The instantaneous snapshot of $\rho_p$, indicating the particle component in WPTS, at $t=1650$ for $xoy$ plane with $z=0$. The presented domain is around $[0,32D]\times[-11D,11D]$.
• Figure 4: The inverse of the averaged centerline mean velocity, and the reference is from Tur-case-jet-DNS-sharan2021investigation.
• Figure 5: The profiles of (a) mean streamwise velocity, and Reynolds stress associated terms: (b) the r.m.s. of $U' U'$, (c) the r.m.s. of $U'_r U'_r$, (d) the cross-stress term of $U' U'_r$. $U$ and $U_r$ denote the velocity in the streamwise direction and radial direction, respectively.

Theorems & Definitions (2)

• Remark 1
• Remark 2