[Paper Review] Entrainment, diffusion and effective compressibility in a self-similar turbulent jet

Thomas Basset, Bianca Viggiano+6 more|arXiv (Cornell University)|Jan 7, 2022

Aerodynamics and Acoustics in Jet Flows56 references14 citations

TL;DR

This study introduces a nozzle-seeding experimental approach in a self-similar turbulent round jet to isolate the Lagrangian dynamics of fluid parcels injected through the nozzle from those entrained from the quiescent environment. By leveraging mass conservation and self-similarity, the authors analytically derive the radial velocity profile of the nozzle-seeded flow, revealing an effective compressibility (∇·⟨Uϕ⟩≠0) despite global incompressibility. The key contribution is a direct link between turbulent diffusion (KT) and momentum transport (νT) and entrainment, enabling experimental determination of non-uniform KT, νT, and the turbulent Prandtl number σT from simple mean concentration and axial velocity profiles alone.

ABSTRACT

An experimental Lagrangian study based on particle tracking velocimetry has been completed in an incompressible turbulent round water jet freely spreading into water. The jet is seeded with tracers only through the nozzle: inhomogeneous seeding called nozzle seeding. The Lagrangian flow tagged by these tracers therefore does not contain any contribution from particles entrained into the jet from the quiescent surrounding fluid. The mean velocity field of the nozzle seeded flow, $\langle \boldsymbol{U_\varphi} angle$, is found to be essentially indistinguishable from the global mean velocity field of the jet, $\langle \boldsymbol{U} angle$, for the axial velocity while significant deviations are found for the radial velocity. This results in an effective compressibility of the nozzle seeded flow for which $\boldsymbol{ abla \cdot} \langle \boldsymbol{U_\varphi} angle eq 0$ even though the global background flow is fully incompressible. By using mass conservation and self-similarity, we quantitatively explain the modified radial velocity profile and analytically express the missing contribution associated to entrained fluid particles. By considering a classical advection-diffusion description, we explicitly connect turbulent diffusion of mass (through the turbulent diffusivity $K_T$) and momentum (through the turbulent viscosity $ u_T$) to entrainment. This results in new practical relations to experimentally determine the non-uniform spatial profiles of $K_T$ and $ u_T$ (and hence of the turbulent Prandtl number $\sigma_T = u_T/K_T$) from simple measurements of the mean tracer concentration and axial velocity profiles. Overall, the proposed approach based on nozzle seeded flow gives new experimental and theoretical elements for a better comprehension of turbulent diffusion and entrainment in turbulent jets.

Motivation & Objective

• To isolate the dynamics of fluid parcels injected through the nozzle from those entrained from the surrounding quiescent fluid in a turbulent jet.
• To investigate how entrainment modifies the Eulerian mean velocity and concentration fields of the jet, particularly in the context of self-similarity.
• To establish a quantitative link between entrainment, turbulent diffusion (KT), and momentum transport (νT) in self-similar jets.
• To develop experimentally accessible relations for determining spatially non-uniform turbulent transport coefficients (KT, νT) and the turbulent Prandtl number σT from simple mean field measurements.
• To demonstrate that the nozzle-seeded flow exhibits effective compressibility even though the global background flow is incompressible.

Proposed method

• Using particle tracking velocimetry (PTV) in a water jet to measure Lagrangian trajectories of tracers injected only at the nozzle (nozzle seeding), ensuring no contribution from ambient fluid.
• Applying mass conservation and self-similarity assumptions to derive the radial velocity profile of the nozzle-seeded flow, ⟨Uϕ⟩, from the axial velocity profile and tracer concentration.
• Deriving the continuity equation for the nozzle-seeded flow: ∇·(⟨ϕ⟩⟨Uϕ⟩) = 0, which leads to a modified radial velocity profile due to the non-zero divergence of ⟨Uϕ⟩.
• Using the boundary-layer approximation for turbulent jets to express the axial and radial mean velocity profiles in terms of self-similar functions f(η) and g(η), with η = r/z.
• Applying the gradient closure model ⟨uv⟩ = −νT ∂U/∂r and ⟨vϕ⟩ = −KT ∂ϕ/∂r to relate turbulent viscosity νT and diffusivity KT to the mean velocity and concentration gradients.
• Deriving analytical expressions for νT(η) and KT(η) from the self-similar velocity and concentration profiles, leading to σT(η) = νT(η)/KT(η) = [Φ′(η)/Φ(η)] · [f(η)/f′(η)].

Experimental results

Research questions

• RQ1 How does entrainment modify the Eulerian mean velocity field of a self-similar turbulent jet when only the nozzle-injected fluid is tracked?
• RQ2 To what extent does the nozzle-seeded flow exhibit effective compressibility despite the global incompressibility of the background flow?
• RQ3 Can turbulent diffusivity KT and viscosity νT be determined independently from the same experimental measurements of mean axial velocity and tracer concentration profiles?
• RQ4 What is the analytical relationship between entrainment, turbulent diffusion, and momentum transport in self-similar jets?
• RQ5 How can the spatial non-uniformity of the turbulent Prandtl number σT be experimentally accessed without requiring simultaneous measurements of velocity and scalar fluctuations?

Key findings

• The nozzle-seeded flow exhibits effective compressibility, with ∇·⟨Uϕ⟩ ≠ 0, due to the absence of radial inflow from the ambient, even though the global flow is incompressible.
• The radial velocity profile of the nozzle-seeded flow is analytically derived as gϕ(η) = ηfϕ(η), which differs from the global radial velocity profile g(η) = ηf(η) − (1/η)∫₀^η x f(x) dx.
• The turbulent diffusivity KT(η) and viscosity νT(η) are explicitly expressed in terms of the self-similar axial velocity profile f(η) and concentration profile Φ(η), via bKT(η) = −1/S · [Φ(η)/Φ′(η)] · (1/η)∫₀^η x f(x) dx and bνT(η) = −1/S · [f(η)/f′(η)] · (1/η)∫₀^η x f(x) dx.
• The turbulent Prandtl number is given by σT(η) = [Φ′(η)/Φ(η)] · [f(η)/f′(η)], enabling direct experimental determination from mean velocity and concentration profiles.
• The method allows experimental determination of non-uniform KT and νT without requiring simultaneous measurements of velocity and scalar fluctuations, unlike classical cross-correlation methods.
• The approach provides a new framework to disentangle the roles of nozzle-injected and entrained fluid in turbulent mixing, with direct implications for modeling passive scalar dispersion in jets.