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Numerical simulation of spray coalescence in an eulerian framework : direct quadrature method of moments and multi-fluid method

Rodney O. Fox, Frédérique Laurent, Marc Massot

arXiv:0706.3781v1math.NAphysics.class-ph

TL;DR

The paper addresses Eulerian simulation of polydisperse sprays undergoing evaporation and coalescence, where strongly coupled interactions make direct phase-space computation difficult. It applies DQMOM to the Williams spray equation and compares it with a multi-fluid model against a stochastic Lagrangian reference. Both Eulerian approaches describe spray coalescence adequately, while DQMOM is more efficient for coalescing distributions but requires further study of evaporative flux.

  • Problem

    Accurately modeling polydisperse sprays requires an Eulerian treatment of transport, evaporation, drag, and strongly coupled coalescence without directly discretizing the full volume-velocity phase space.

  • Method

    The study applies DQMOM to the Williams spray equation and compares it with an Eulerian multi-fluid model using nozzle test cases and a stochastic Lagrangian reference solution.

  • Results

    Both Eulerian models describe spray coalescence adequately; DQMOM is more efficient than the multi-fluid model for coalescing distributions because it has limited numerical diffusion in size space.

  • Takeaways & Limitations

    DQMOM is numerically robust and straightforward to implement for the Williams spray equation and is a candidate for more complex two-phase combustion applications.

  • Takeaways & Limitations

    DQMOM still requires further study to optimally treat evaporative flux from droplet disappearance, and moment constraints alone cannot determine that flux.

Abstract

from arXiv · show

The scope of the present study is Eulerian modeling and simulation of polydisperse liquid sprays undergoing droplet coalescence and evaporation. The fundamental mathematical description is the Williams spray equation governing the joint number density function f(v, u; x, t) of droplet volume and velocity. Eulerian multi-fluid models have already been rigorously derived from this equation in Laurent et al. (2004). The first key feature of the paper is the application of direct quadrature method of moments (DQMOM) introduced by Marchisio and Fox (2005) to the Williams spray equation. Both the multi-fluid method and DQMOM yield systems of Eulerian conservation equations with complicated interaction terms representing coalescence. In order to validate and compare these approaches, the chosen configuration is a self-similar 2D axisymmetrical decelerating nozzle with sprays having various size distributions, ranging from smooth ones up to Dirac delta functions. The second key feature of the paper is a thorough comparison of the two approaches for various test-cases to a reference solution obtained through a classical stochastic Lagrangian solver. Both Eulerian models prove to describe adequately spray coalescence and yield a very interesting alternative to the Lagrangian solver.

1 Introduction

The paper develops Eulerian approaches for polydisperse sprays with transport, evaporation, drag, and coalescence, applying DQMOM to the Williams equation and comparing it with a multi-fluid model. Tests against a stochastic Lagrangian reference show DQMOM is especially promising for strongly coalescing sprays.

  • Motivation: Eulerian spray models must jointly represent transport, evaporation, drag, and coalescence, whose strong coupling makes numerical simulation difficult.A direct solver in volume-velocity phase space is computationally intractable, motivating multi-fluid and moment-based closures.
  • Method: DQMOM is applied to the Williams spray equation as an Eulerian closure for nonlinear processes such as droplet coalescence.It reconstructs the number density from moments using weighted quadrature points, whose weights and abscissas evolve on an Eulerian mesh.
  • Model comparison: DQMOM and multi-fluid models use the same pressureless-gas-dynamics transport structure but differ in how they discretize and close the coalescence interactions.The multi-fluid model discretizes volume space with size-conditioned average velocity, whereas DQMOM uses a finite set of moments.
  • Validation: The study compares DQMOM and a multi-fluid model across smooth and Dirac-delta size distributions in a self-similar axisymmetric decelerating nozzle.The test configuration is designed to challenge numerical methods through significant coalescence rates and varied inlet distributions.
  • Results: DQMOM offers useful performance in several test cases, including strongly coalescing droplets, and is presented as a candidate for more complex configurations.The paper evaluates both Eulerian approaches against a classical Lagrangian solver.

2 DQMOM for Williams equation

DQMOM represents the Williams spray equation with Eulerian weights and abscissas, deriving conservation equations whose source terms account for evaporation, drag, and coalescence. Its moment choice controls solvability, while conservation is retained and numerical conditioning requires care.

  • Governing equation: The Williams equation includes real-space transport, evaporation, drag, and a nonlocal nonlinear coalescence term in volume-velocity phase space.A direct Eulerian discretization of the full phase space is computationally intractable, motivating multi-fluid and moment closures.
  • DQMOM representation: DQMOM approximates the volume-velocity number density with weighted delta functions and evolves Eulerian weights and abscissas through closed transport equations.The resulting equations describe number, mass, and momentum densities, with source terms determined from moment constraints.
  • Moment closure: DQMOM source terms are obtained from a linear moment system, requiring 5N independent moments for N weights, N volume abscissas, and 3N velocity abscissas.The coefficient matrix depends on the selected moments and must be non-singular.
  • Conservation: Coalescence preserves mass and momentum in the DQMOM representation when evaporation and drag are absent.The corresponding moment constraints match the conservation properties of the original Williams equation.
  • Evaporative flux: Evaporative flux at zero droplet size cannot be determined from moment constraints alone and requires additional constraints to avoid poor estimates for smooth distributions.Neglecting this flux can produce singular behavior or poorly estimated moments, especially when evaporation and coalescence interact.
  • Numerical conditioning: With velocity moments limited to first order, the DQMOM matrices are non-singular when volume abscissas are distinct, but coalescence can make them increasingly ill-conditioned as abscissas grow.Scaling and up to three iterative solver improvements eliminate round-off error and can reduce overall computational cost.

3 Nozzle Test Problem

The test problem uses a self-similar 2D axisymmetrical conical decelerating nozzle designed to challenge Eulerian spray models under strongly coalescing conditions.

  • Test configuration: The test configuration is a 2D axisymmetrical conical decelerating nozzle selected to validate DQMOM against a Lagrangian solver and compare it with the multi-fluid model.The nozzle is designed to provide a sufficiently difficult problem for exposing method limitations.

3.1 Definition of configuration

The configuration is a stationary, self-similar nozzle problem with one-dimensional droplet-size variation and straight trajectories near the centerline. It compares monomodal and bimodal initial droplet distributions under specified heptane spray conditions.

  • The test case is stationary, two-dimensional and axisymmetric in space, with one-dimensional variation in droplet size.
  • The nozzle carries heptane droplets in a fixed-temperature heptane–nitrogen gas mixture through a conical diverging geometry.At the entrance, 99% of fuel mass is liquid and 1% is gaseous; the gas temperature is 400 K.
  • Straight droplet trajectories follow from co-linear injection and gas velocities, remaining valid near the centerline even when coalescence occurs.
  • The monomodal spray spans radii from 0 to 35 µm, with mean radius 12 µm, variance 5 µm, and Sauter mean radius 15.6 µm.
  • The bimodal spray contains two radius groups, 10 and 30 µm, with equal mass density.The distribution represents alumina-particle conditions in solid-propellant rocket boosters.
  • The initial liquid volume fraction is 0.57%, and nozzle deceleration produces size-dependent droplet deceleration that induces coalescence.

3.2 DQMOM model equations in nozzle configuration

In the nozzle configuration, the DQMOM formulation becomes ordinary differential equations for quadrature weights, abscissas, and velocity variables. Transport, coalescence, evaporation, and drag enter through distinct terms, with moment choices used to simplify the coalescence system.

  • The nozzle equations reduce to ordinary differential equations in z for quadrature weights, volumes, and axial and radial velocity variables.
  • The DQMOM equations can be reduced to three nonlinear ODEs when the velocity relation preserves straight droplet trajectories.
  • The left-hand sides describe transport changes, while right-hand-side terms represent coalescence, evaporation, and drag.
  • The coalescence source terms are computed by solving a linear system whose scaling includes the nozzle factor (z0/z)^2.

3.3 Test cases

The test suite varies evaporation laws and coalescence, including non-evaporating, linearly evaporating, and nonlinearly evaporating sprays. These cases examine agreement with Lagrangian dynamics and the effects of coalescence–evaporation competition.

  • The test matrix combines three evaporation cases—none, linear, and nonlinear—with and without coalescence.Linear evaporation scales as Rv ∝ v, whereas nonlinear evaporation scales as Rv ∝ v1/3.
  • Without coalescence, DQMOM is expected to agree closely with the Lagrangian solver because corresponding transport equations are identical.
  • With coalescence and no evaporation, droplets are expected to grow substantially, making differences between coalescence treatments more pronounced.
  • Linear evaporation couples droplet volume and velocity through evaporation and drag while leaving volume independent of quadrature weight without coalescence.
  • With linear evaporation and coalescence, competition between growth and evaporation produces smaller droplets than in the non-evaporating case.
  • Nonlinear evaporation generally produces a nonzero evaporative flux, and the study compares predictions with alternative flux treatment for the bimodal distribution.

3.4 Reference Lagrangian solution

A stochastic Lagrangian solver supplies reference solutions by alternating parcel transport and evaporation with Monte Carlo collision processing. Its accuracy depends on parcel, mesh, and time-step choices, with additional constraints when coalescence is present.

  • The reference method is a stochastic representation of the Williams kinetic equation using weighted parcels that approximate the droplet distribution.
  • Each particle-method time step first advances parcel position, velocity, and volume for transport and evaporation, then discretizes the collision operator.
  • The collision algorithm randomly pairs parcels within computational cells and samples coalescence counts from a Poisson distribution.
  • The collision algorithm costs O(NJ) per cell and time step, compared with O(NJ^2) for the O’Rourke method.
  • Accurate reference solutions require sufficient parcel numbers, suitable cell sizes, and small time steps; the test cases use Δt ≤ 10^-6 s.
  • Coalescence cases require finer cells near the nozzle entrance because steep gas-velocity gradients can rapidly alter the droplet distribution.

3.5 Eulerian multi-fluid solver

The multi-fluid solver derives an Eulerian conservation model by assuming size-conditioned mono-kinetic velocity, then discretizing droplet volume into finite sections. In the stationary self-similar nozzle, the resulting equations are solved section by section with evaporation, drag, and coalescence source terms.

  • 3.5 Eulerian multi-fluid solver: The multi-fluid formulation is equivalent to an Eulerian conservation model whose coalescence source terms are evaluated using the finite-volume size discretization.Its derivation uses the semi-kinetic model and a presumed distribution within each section.
  • 3.5 Eulerian multi-fluid solver: The model assumes that droplets with the same volume share one velocity, an assumption that can fail after coalescence creates velocity dispersion.The nozzle is mono-kinetic without coalescence, but coalescence can make the size-conditioned velocity variance nonzero.
  • 3.5 Eulerian multi-fluid solver: The multi-fluid model discretizes the droplet-size distribution into finite sections after assuming a single averaged velocity for each section.This produces conservation equations for section-wise mass and momentum densities.
  • 3.5 Eulerian multi-fluid solver: Evaporation, drag, and coalescence enter the section equations through source terms and interaction coefficients computed for the chosen size discretization.The coalescence terms include disappearance and appearance integrals over rectangular domains and symmetric strips in volume space.
  • 3.5 Eulerian multi-fluid solver: In the prescribed stationary one-way-coupled flow, each section reduces to a one-dimensional stiff initial-value problem integrated from injection until 99.9% of the mass evaporates.The simulations use LSODE for the stiff ordinary differential equations.

4 Results and Discussion

The results compare DQMOM and the multi-fluid model with a Lagrangian reference across evaporation and coalescence cases. Agreement is generally strong, while differences arise mainly when finite-section discretization or neglected velocity dispersion affects coalescence and droplet-size predictions.

  • 4 Results and Discussion: The Eulerian simulations require CPU seconds, whereas the time-dependent Lagrangian cases require several CPU hours, although the timings are not directly comparable.The Eulerian results are stationary in time, while the Lagrangian simulations depend on time and axial position.
  • 4.1 Monomodal case: linear evaporation without coalescence: For monomodal linear evaporation without coalescence, DQMOM, the multi-fluid model, and the Lagrangian solver produce essentially identical statistics.DQMOM uses N = 4 and the multi-fluid model uses N = 10 in this representative case.
  • 4.3 Bimodal case: nonlinear evaporation without coalescence: For bimodal nonlinear evaporation without coalescence, DQMOM with N = 2 accurately reproduces the two-peak behavior, while the multi-fluid model requires N = 30 sections.The multi-fluid model captures mass density well but has difficulty representing sharp peaks with finite-volume sections.
  • 4.4 Monomodal case: linear evaporation with coalescence: With monomodal coalescence and linear evaporation, all three methods predict similar velocity differences, but finite-section and mono-kinetic approximations shift relaxation relative to the Lagrangian result.The multi-fluid model predicts slightly slower relaxation and DQMOM slightly faster relaxation than the Lagrangian solver.
  • 4.5 Bimodal case: linear evaporation with coalescence: For bimodal coalescence, the multi-fluid model needs N = 500 sections to suppress spurious coalescence, whereas DQMOM avoids this numerical artifact but still faces a difficult two-peak distribution.The spurious coalescence arises when numerical diffusion creates droplets between the original 10 and 30 µm peaks.
  • 4.7 Monomodal case: coalescence with no evaporation: In pure coalescence, DQMOM and the multi-fluid model overpredict relaxation because both underpredict the mean droplet size relative to the Lagrangian solution.The Lagrangian distribution contains more droplets with radii above 80 µm.

5 Conclusions

The study implements DQMOM for the Williams spray equation and compares it with a multi-fluid model against a Lagrangian reference. DQMOM performs better for coalescence, while evaporation requires further study because evaporative flux affects moment dynamics.

  • 5 Conclusions: DQMOM was implemented for evaporation, acceleration, and coalescence in the Williams spray equation, producing a linear source-term system whose matrix depends on the selected moments.The system’s right-hand side is nonzero only for coalescence or nonlinear evaporation.
  • 5 Conclusions: DQMOM outperformed the multi-fluid model for coalescence, especially for bimodal distributions, because of its lower numerical diffusion in size space.For evaporation, DQMOM was comparable to the multi-fluid model, but evaporative flux from droplet disappearance remains insufficiently understood.
  • 5 Conclusions: Evaporative flux associated with droplet disappearance has an important effect on moment dynamics and must be addressed before broader applications.The study identifies DQMOM as a candidate for more complex two-phase combustion applications once this issue is resolved.
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