Development and application of a multiphase Lagrangian structure function model in anisotropic turbulence

Technical Summary: Development and Application of a Multiphase Lagrangian Structure Function Model in Anisotropic Turbulence

Problem Statement
The energetic response of inertial particles to turbulent flow is critical for understanding multi-phase dynamics and developing reduced-order models for particle dispersion. While numerous studies have examined particle energetics in homogeneous isotropic turbulence (HIT), practical flows often exhibit spatial inhomogeneities and anisotropies, particularly near solid boundaries. In wall-bounded turbulent flows, such as turbulent boundary layers (TBL), large-scale anisotropy is generated by mean shear and boundary conditions. A key unresolved question is how this large-scale anisotropy is transferred to the particle phase and whether it persists into the formally isotropic small-scale ranges. Furthermore, the complex coupling between particle inertia, gravitational settling, and flow anisotropy in these environments requires further investigation to accurately model inertial settling particles.

Methodology
The authors employ a combined theoretical and numerical approach to investigate scale-wise particle energetics in a turbulent boundary layer under net deposition conditions.

1. Theoretical Framework:

   • The study utilizes a kinetic theory approach to derive exact, though unclosed, conservation laws for the second-order structure functions (SoSF) of the particle phase at arbitrary scales.
   • The particle dynamics are modeled using the Maxey-Riley equations for small, spherical, non-interacting particles in the limit of a large particle-to-fluid density ratio.
   • The authors define a metric for particle phase anisotropy, $\eta^{(p)}$, based on the difference between longitudinal and vertical second-order structure functions.
   • An asymptotic analysis is performed on the derived continuum equations to elucidate the balance of terms governing the energy fluxes and anisotropy, assuming power-law behaviors for relative velocity moments and the radial distribution function.

2. Numerical Simulations:

   • Direct Numerical Simulations (DNS) of one-way coupled particle-laden wall-bounded turbulent flow are conducted at a moderate friction Reynolds number ($Re_\tau = 315$).
   • The simulations cover the logarithmic and buffer layers of the TBL.
   • The study varies the Stokes number ($St^+$) to assess particle inertia and the settling parameter ($Sv^+$) to assess the effects of gravitational drift.
   • Both Eulerian (fluid) and Lagrangian (particle) perspectives are utilized, with a specific focus on "slab-wise" statistics where vertical separation is zero to isolate local vertical dynamics.

Key Contributions

• Derivation of Conservation Laws: The paper derives an exact conservation equation for particle phase energetics at arbitrary scales, explicitly identifying terms for transport, coupling to first-order moments, and energy fluxes. This provides a continuum framework for analyzing scale-wise energy transfer in anisotropic multiphase flows.
• Quantification of Anisotropy Transfer: The work systematically quantifies how large-scale anisotropy from the carrier fluid is "donated" to the particle phase and how particle inertia and settling modify this transfer.
• Asymptotic Analysis: The authors provide an asymptotic argument to interpret the DNS data, linking the observed non-monotonic behavior of anisotropy metrics to the competition between large-scale flow anisotropy and small-scale inter-scale energy transfer.

Results

• Inertia and Anisotropy: At small Stokes numbers, the particle phase exhibits near-isotropy at small scales, mirroring the fluid. However, as inertia increases ($St^+ = 10$), the particle phase retains significant anisotropy even at small scales, indicating that particle inertia preserves the memory of large-scale flow anisotropy.
• Settling Effects: The influence of the settling parameter ($Sv^+$) is highly dependent on the flow region. In the buffer layer ($z^+ = 30$), where the underlying flow is inherently anisotropic, increasing $Sv^+$ leads to a complex modification of structure functions where the vertical component increases while the longitudinal component decreases. In the logarithmic layer ($z^+ = 90$), where small-scale flow is isotropic, high settling rates ($Sv^+$) tend to drive the particle phase toward an isotropic state at small scales.
• Scale Non-Locality: The particle structure functions exhibit a shallower scaling (approaching $r^{2/3}$) in the dissipative range compared to the expected $r^2$ scaling of the fluid. This reflects the scale non-locality of inertial particles, driven primarily by inertia and modulated by settling.
• Anisotropy Metric Behavior: The particle anisotropy metric $\eta^{(p)}$ displays non-monotonic behavior as the separation scale $r$ decreases. It reaches a local minimum before increasing again as $r/z \to 0$. This suggests a competition between the anisotropy imposed by the large-scale flow and the anisotropy generated by the inter-scale energy transfer within the particle phase itself.

Significance and Claims
The paper claims to provide crucial insight into the scale-wise energy fluxes of inertial particles in anisotropic turbulence. By demonstrating that particle inertia can preserve large-scale anisotropy into the formally isotropic small scales, the authors highlight a limitation in models that assume local isotropy for the particle phase. The derived conservation laws and the identified mechanisms of anisotropy transfer have direct implications for the development of continuum models for inertial settling particles, particularly regarding how two-way coupled dynamics might be affected by these non-local anisotropic effects. The work emphasizes that for accurate modeling, the simultaneous consideration of particle inertia and gravitational drift is necessary, as their effects are not simply additive but are implicitly coupled within the particle dynamics.