ILASS-Americas 29th Annual Conference on Liquid Atomization and Spray Systems, Atlanta, GA, May 2017

Numerical Study of Liquid Jet Atomization in Supersonic Crossflows

O. Desjardins∗, M. Natarajan, and M. KuhnSibley School of Mechanical and Aerospace Engineering

Cornell UniversityIthaca, NY 14853 USA

AbstractCompressible fuel injection is a critical process in the operation of scramjet engines, but a challengingarea for experimentation. Accurate numerical simulations based on first principles can help elucidate thelimiting physics in these flows and provide guidance for engineering design. However, most computationalstudies of liquid jet atomization so far have been conducted in an incompressible setting. While a variety ofaccurate and robust simulation techniques have been developed for atomization, their transfer to the realm ofcompressible flows presents major challenges. In this work, we combine a recently developed, unsplit volume-of-fluid technique (VOF) with second order accuracy (Owkes and Desjardins, J. Comp. Phys. 2014) with afully conservative finite volume compressible flow code that solves discontinuous equations for the densityand energy of each phase. To maximize computational performance with atomizing flows in low supersonicconditions, we employ an implicit equation for pressure, following Kwatra et al. (J. Comp. Phys. 2009),allowing us to avoid any acoustic CFL limitation. We then use this multiphase compressible flow solver tosimulate a liquid jet in supersonic crossflow studied experimentally at the Air Force Research Laboratory.Qualitative comparison with experimental results and mesh convergence properties are discussed.

∗Corresponding Author: olivier.desjardins@cornell.edu

Introduction

The design and development of scramjet enginespresent many outstanding challenges, one of whichbeing fuel delivery. In a scramjet, the fuel injec-tion strategy has a direct impact of the atomizationprocess, which controls the evaporation characteris-tics of the spray and ultimately the spatial distri-bution of the fuel vapor (see [1]). This, in turn,controls the propensity for the combustible mixtureto ignite. While it is simplest to inject fuel in liq-uid form directly in the supersonic crossflow, little isknown about the characteristics of liquid break-upin highly compressible flow conditions, and thereforemore work is needed to design scramjet fuel injec-tion schemes that can lead to predictable ignition.To that end, both experimental and computationalstudies are needed, yet the state of the art in compu-tational modeling of compressible liquid atomizationhas been insufficient to allow such simulations.

The problem of supersonic liquid jet atomiza-tion has received much less attention than the cor-responding subsonic atomization problem. Only afew studies have attempted to simulate this phe-nomenon, and with limited success. The simulationsof [2] and [3], for example, were able to capture afew of the relevant features, but were unable to rep-resent some of the shock waves observed in experi-ments. Recently, [4] used a ghost fluid method com-bined with both exact and linear Riemann solvers onan axisymmetric domain, and were able to capturemuch of the relevant physics responsible for early-stage jet instabilities. However, this study, while auseful and novel advancement of modern simulationcapabilities, considered only the initial instability ofthe jet and did not analyze processes that are par-ticularly relevant for supersonic combustion, such aslater-stage ligament formation or droplet breakup.

In the incompressible realm, we have recentlyintroduced a formally second-order, unconditionallystable, formally conservative and bounded volume-of-fluid (VOF) scheme [5, 6], as well as a second-order VOF-based interface curvature calculationtechnique [7]. Such methods, combined with afully conservative discretization of the Navier-Stokesequations [8] have made high-fidelity simulations ofincompressible liquid atomization possible [9]. Here,we combine these techniques with the implicit fullycompressible flow solver proposed by Kwatra etal. [10, 11, 12], and demonstrate that the resultingframework is capable of simulating turbulent liquidatomization in supersonic environments.

Numerical Methods

The compressible multiphase flow solver used inthis work is based on volume of fluid (VOF) con-cepts (i.e., a variable α is transported that tracks theliquid volume fraction of each computational cell),along with a fractional step time integrator that al-lows for implicit handling of the pressure term (i.e.,flow acoustics). Equations are written in each phasefor α, ρα, ραE (i.e., volume, mass, and energy ofeach phase — we will use the i subscript below torepresent the phase), and an additional equation formixture momentum ρu completes the system. It isfully given as

∂αi

∂t+∇ · (αiu) = αi∇ · u, (1)

∂ρiαi

∂t+∇ · (ρiαiu) = 0, (2)

∂ρiEiαi

∂t+∇ · (ρiEiαiu) +∇ · (pαiu) =

∇ · (u · τ + k∇T ) + u · Fαi, (3)

and

∂ρu

∂t+∇ · (ρu⊗ u) +∇ · (pI) = ∇ · (τ ) + F , (4)

where

τ = µ

(∇u+∇uT − 2

3∇ · uI

)(5)

is the viscous stress tensor, the mixture densityis ρ = α1ρ1 + α2ρ2, and the mixture energy isρE = α1ρ1E1 +α2ρ2E2. F contains all body forces,including gravity and surface tension.

The convective transport step for all vari-ables is performed using a recently-developed, for-mally second-order, unconditionally stable, formallyconservative and bounded volume-of-fluid (VOF)scheme [5]. This transport step assumes that the in-terface can be represented within each cell as a plane,which is known as the PLIC (piecewise linear inter-face construction) model. Using this assumption, itis straightforward to construct the geometry of theliquid fraction and of the gas fraction of each cellin the computational domain, with second order ac-curacy (since planes are used to build the cut-cells).Then, all transported variables are reconstructed lin-early [6]: liquid variables are reconstructed in allliquid cells and cut-cells, and gas variables are re-constructed in all gas cells and cut-cells. Linearreconstruction is achieved using the average valueof the variable in each cell at the phase-barycenter

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Figure 1. Semi-Lagrangian VOF scheme for convec-tive transport [5, 6]. For each cell where 0 < α < 1,PLIC is used to build a planar interface. Each trans-ported variable is reconstructed with second order accu-racy, then transport is performed using semi-Lagrangianstreaktubes emitted from each cell-face. The integralover the liquid fraction of the streaktube of each recon-structed liquid variable provides a second-order accurateconvective flux (same for the gas).

(standard finite volume concept) and the gradientof the variable within the cell, obtained with a min-mod limiter for stability. Finally, semi-Lagrangianstreaktubes are built to calculate convective fluxes,as illustrated in Fig. 1. Note that because convectionis performed in a semi-Lagrangian framework, thisconvection scheme is unconditionally stable, hencethe convective CFL restriction does not constrainthe time step size.

A fractional step approach is then used for tem-poral integration [10, 11, 12], wherein all flow vari-ables are first convected at the flow velocity, thenmodified to account for viscous transport along withany other source terms, and finally corrected to in-clude the effect of acoustic transport. The acousticstep involves the solution of an implicitly formulatedHelmholtz equation for pressure, which is free of anyCFL restriction, thereby allowing to take time stepsthat are larger than the acoustic time scale in bothphases. That equation can be written as

pn+1 − ρn+1 (c?)2

∆t2∇ ·(

1

ρn+1∇pn+1

)=

p? − ρn+1 (c?)2

∆t∇ · u?, (6)

where p? is a pressure prediction obtained from ap-plying the equation of state to the transported vari-ables (identified with ?). Note that a simple har-monic mixing rule is used to obtain a unique pres-sure in mixed cells [13]. Similarly, the speed of soundc? is obtained from the equation of state with thesame mixing rule for mixed cells. For both phases,a stiffened gas equation of state is employed. TheHelmholtz equation for pressure is solved using a

Krylov solver preconditioned by a Black-Box Multi-Grid solver [14]. Because acoustic phenomena arehandled implicitly, this fractional-step time integra-tor is ideally suited to handle the nearly incompress-ible flow conditions found inside the liquid phase.

Viscous fluxes are discretized using second-orderfinite differences, and are integrated implicitly intime using an approximate factorization technique[8], again avoiding all viscous CFL restrictions. Fi-nally, surface tension is embedded as a pressurejump inside the Helmholtz equation for pressure, fol-lowing [15]. Interfacial curvature is obtained withsecond order accuracy using a recently-developedheight function scheme [7].

Verification and Validation

Overall, this approach is expected to be uncon-ditionally stable (except for surface tension whichstill limits the time step size, although it is not foundto be a hindrance in a typical simulation of liquidjet in supersonic crossflow), formally second orderin space, and formally conservative. It was verifiedand validated against published data, both in single-phase and multiphase settings. In particular, its per-formance was assessed using the 1D shock-interfacetest of Shukla [13], then using the water column-shock interaction experiments by Igra et al. [16, 17]for 2D code validation, and finally using the aero-breakup data of Theofanous et al. [18] for millimet-ric droplets for 3D code validation. In the interestof conciseness, only the last two cases are discussedhere.

Figure 2. Validation of numerical method. Top imagesare simulations, bottom images are experiments. Thefour leftmost columns correspond to a water column-shock interaction problem described experimentally byIgra et al. [16, 17]. Both Schlieren and interface are visi-ble on these images. The rightmost column correspondsto a water droplet-shock interaction problem describedexperimentally by Theofanous et al. [18]. Only the in-terface is visible on these images.

Figure 2 shows the simulation data on top, and

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the corresponding experimental snapshot on the bot-tom. The water column problem [16, 17] is shownin the four leftmost columns, and both the interfaceand the Schlieren are provided. The water dropletproblem [18] is shown on the rightmost column, andonly the interface is presented. Overall, the agree-ment with the experiments is excellent in both thewater column and droplet case. The shock structureshown by the numerical Schlieren compares favor-ably to that of Igra et al. [16, 17]. Similarly, thedrop shape in the 2D case compares well to the col-umn shape reported by Igra. Similarly, significantbreak-up and surface deformation is visible in thesimulation of the 3-D droplet problem of Theofanouset al. [18], in agreement with the experiments. Over-all, the simulations capture interfacial dynamics ac-curately, without robustness issue.

Liquid Jet in Supersonic Crossflow

Finally, the computational framework describedabove is used to simulate a liquid jet in supersoniccrossflow (LJSC). A 9 million cells uniform carte-sian mesh is employed, using 288 cores for 24 hours.The incoming crossflow Mach number is 1.5, and theliquid-to-gas momentum flux ratio is 8. Top bound-ary is a slip wall, bottom boundary is a non-slipwall (except for the liquid injection at 50 m/s overa circular orifice), leftmost boundary is a Dirichletinflow condition, and rightmost boundary is a Neu-mann/sponge outflow. Finally, the last boundaryconditions are specified as periodic.

Figure 3 shows an instantaneous snapshot of thissimulation. A numerical Schlieren allows to visu-alize expected compressible flow features, includinga lambda shock and a bow shock before the liq-uid jet, a Mach reflection at the top of the channel,and turbulence in the spray. These features are fur-ther highlighted in Fig. 4. This simulation is foundto be robust and relatively inexpensive, and there-fore confirms that our newly developed high-fidelitymethodology for two-phase compressible flows is ca-pable of capturing some of the key physics of LJSC.The ILASS presentation will focus on validating ourLJSC simulation against AFRL experimental data,performing a mesh refinement analysis, and improv-ing the specification of flow boundary conditions inorder to quantitatively assess the predictive capabil-ities of the computational approach.

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Figure 3. Demonstration simulation of LJSC, showing the isosurface is the liquid-gas interface, while the colormaprepresents a numerical Schlieren, showing typical compressible flow features including a bow shock in front of theliquid jet and a Mach reflection at the top boundary.

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Figure 4. Simulation snapshot highlighting various LJSC features. In this image, the solid white surface shows theliquid-gas interface, while the transparent surface shows an isosurface of the numerical Schlieren, colored by the flowvelocity.

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