A computational framework for interface-resolved ...

Journal of Computational Physics 371 (2018) 751–778

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Journal of Computational Physics www.elsevier.com/locate/jcp

A computational framework for interface-resolved DNS of simultaneous atomization, evaporation and combustion Changxiao Shao, Kun Luo ∗ , Min Chai, Haiou Wang, Jianren Fan State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou 310027, PR China

a r t i c l e

i n f o

Article history: Received 8 March 2018 Received in revised form 22 May 2018 Accepted 2 June 2018 Available online 6 June 2018 Keywords: Level set method Integrated method Multiphase flows Atomization Evaporation Combustion

a b s t r a c t A computational framework for interface-resolved direct numerical simulations (DNS) of simultaneous atomization, evaporation and combustion process is proposed. The present work utilizes a level set method to implicitly capture the gas–liquid interface and the ghost fluid method (GFM) to accurately address jump conditions across the interface. Specific care has been devoted to the discretization of the convective term and diffusive term for the species and energy equations. The level set method with a sub-cell resolution of the interface is employed and a semi-Lagrangian scheme for the discretization of the level set equation with an evaporation source term is proposed. The one-step global reaction model of n-heptane is used for the vapor combustion. To the best of our knowledge, this is the first computational framework that has the capability to simulate spray combustion with an interface-resolved method. The present framework has been validated in several cases, including the one-dimensional evaporation and the static droplet evaporation for low ambient temperature, the Stefan problem, the Sucking problem, the evaporation of static and moving droplet for high ambient temperature, and the combustion of static and moving droplet. Finally, the integrated simulation of droplet collision and combustion is performed to evaluate the accuracy and robustness of the present method. © 2018 Elsevier Inc. All rights reserved.

1. Introduction Spray combustion exists in a variety of propulsion devices, such as aero engine and gas turbine. It features a multiphase, multi-scale, multi-physics process that involves liquid fuel atomization, fuel evaporation, air/fuel-vapor mixing, and fuel-vapor combustion. This complex process presents a great challenge for measuring techniques due to the extreme operating conditions within practical combustion devices. With the development of computational algorithm and computer capacity, numerical simulation becomes a promising tool for investigating spray combustion. In previous simulations of spray combustion, atomized droplets or atomization models were commonly used in the implementations. For example, Luo et al. [1] employed fully atomized droplets as the boundary condition of spray combustion in a model combustor. Irannejad et al. [2] and Esclapez et al. [3] utilized a stochastic approach to describe the atomization process in their spray combustion simulations. However, it is well established that spray combustion characteristics are influenced by the atomization condition. Som and Aggarwal [4] investigated the effects of different atomization models on combustion and found that the combustion characteristics exhibit significant discrepancies, while Greenberg [5] found the droplet size distribution has a great impact on the behavior of edge flames. In order to eliminate the possible errors from

*

Corresponding author. E-mail address: [email protected] (K. Luo).

https://doi.org/10.1016/j.jcp.2018.06.011 0021-9991/© 2018 Elsevier Inc. All rights reserved.

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C. Shao et al. / Journal of Computational Physics 371 (2018) 751–778

the atomization models to accurately simulate spray combustion, it is necessary to develop a computational framework to directly simulate the atomization/evaporation/combustion process without any atomization and evaporation models. The challenges for developing such a computational framework include an accurate interface tracking method for the atomization process, the exact discretization of various jump conditions across the interface for the mass and heat transfer, and the coupling of evaporation with combustion. There exist a variety of interface tracking methods in the literature, such as the fronting tracking method [6], the volume of fluid (VOF) method [7], the level set method [8], the phase field method [9] and the lattice Boltzmann method [10]. The level set method with a sharp interface representation seems to be a promising method due to its simplicity and efficiency. As demonstrated by Tanguy et al. [11], this method not only avoids the introduction of a fictitious interface thickness, but also improves the resolution of the jump conditions by providing an accurate discretization of discontinuous terms across the interface. As for the exact discretization of jump conditions instead of smearing out the discontinuous terms near the interface, the ghost fluid method (GFM) [12] is a powerful technique to address the jump conditions across the interface, which has been applied in incompressible flows with phase change [11,13,14] and in compressible reacting flows [15]. Recently, numerical methods for direct numerical simulation (DNS) of gas–liquid flows with phase change have been developed. In the framework of the level set method, Tanguy et al. [11,13] proposed an incompressible flow solver for simulations of droplet evaporation and boiling flows by using level set/GFM. They set up a projection method to impose a divergence-free condition for the extended velocity field across the interface and a variety of validations were conducted. Gibou et al. [14] developed a similar algorithm and applied it to the simulation of film boiling. Son [16] investigated the micro-droplet evaporation on a heated surface based on the level set method for incompressible flows and an iterative calculation procedure was proposed with the interface conditions of the temperature and species mass fraction. Son and Dhir [17] applied the GFM to simulate the film boiling on a horizontal surface and attempted to understand the mechanism of the transition in bubble release pattern. In the framework of the VOF method, Welch and Wilson [18] simulated the phase change process with only the mass transfer being considered. Strotos et al. [19] investigated the evaporation of two-component droplets by avoiding the discretization of the jump conditions across the interface meanwhile obtaining the distorted interface that is due to numerical artifacts. Sato and Niˇceno [20] proposed a sharp-interface phase change model coupled with the VOF method where the mass transfer rate is calculated directly from the heat flux at the interface with a large velocity jump across the interface captured. Schlottke and Weigand [21] performed DNS of evaporating droplets based on the VOF method and the emphasis was put on the correct calculation of the velocity both in the gas and liquid phases accounting for a divergence constraint. Duret et al. [22] investigated the evaporating turbulent mixing by using a coupled level set/VOF method, in which a passive scalar is used to represent the evaporation and mixing process instead of the energy equation. The film boiling flow was investigated by Tomar et al. [23] and Hens et al. [24] with only the heat transfer being considered. In the framework of the front tracking method, Juric and Tryggvason [25] simulated the film boiling by using an iterative procedure to address the temperature boundary condition at the interface. Irfan and Muradoglu [26] proposed a framework including both the temperature and species driven phase change processes. Koynov et al. [27] performed the simulation of bubble swarms rising due to buoyancy at different operating conditions by considering mass transfer and chemical reactions. Recently, the simulation of phase change has also been coupled with the lattice Boltzmann method and the interested readers could refer to [28–31]. Though significant progress in DNS of phase change has been made in recent years, there still exist limitations. Particularly, the species and energy equations should both be solved for simulations of spray combustion, which has not been implemented in most previous works. To the best of our knowledge, there is no computational framework to simultaneously consider the interface capturing, heat and mass transfer, and chemical reactions for spray combustion. In addition, no sufficient comparisons between the theoretical and experimental results were presented before. In the above context, the objective of this paper is to propose a variable density, low Mach number computational framework for interface-resolved DNS of integrated spray combustion process. The gas–liquid interface is captured by the level set method recently developed by Luo et al. [32] and the discretization of various jump conditions across the interface is utilized by the GFM. The one-step global reaction mechanism of n-heptane is used for the vapor combustion. The remainder of the work is organized as follows. In section 2, governing equations and the numerical details are introduced. In section 3, the verification and validation are conducted. 2. Numerical methods 2.1. Governing equations In the present computational framework, the Navier–Stokes equations with variable density using the low Mach number approximation are written as:

∂ ρ ui u j ∂ σi j ∂p ∂ ρ ui + =− + , ∂t ∂xj ∂ xi ∂xj

(1)

∂ ρ ∂ ρ ui + = 0, ∂t ∂ xi

(2)

C. Shao et al. / Journal of Computational Physics 371 (2018) 751–778



σi j = μ

∂u j ∂ ui + ∂xj ∂ xi



2

− μ 3

753

∂ uk δi j . ∂ xk

(3)

In Eqs. (1)–(3), ρ is the density, u i is the velocity, p is the pressure, σi j is the stress tensor, μ is the dynamic viscosity and δi j is the Kronecker delta. Three additional scalar transport equations in the gas phase corresponding to the mass fractions of the fuel (F ), oxidizer (O ) and product ( P ) are solved coupled with the one-step global reaction mechanism that will be described later:

  ∂ρY F u j ∂ ∂ρY F ∂YF + ω˙ F , + = ρDF ∂t ∂xj ∂xj ∂xj   ∂ρY O u j ∂ ∂ρY O ∂Y O + ω˙ O , + = ρDO ∂t ∂xj ∂xj ∂xj   ∂ρY P u j ∂ ∂ρY P ∂Y P + ω˙ P , + = ρDP ∂t ∂xj ∂xj ∂xj

(4) (5) (6)

˙ i , i ∈ { F , O , P } are the species diffusion coefficient and chemical source term, respectively. where D i , i ∈ { F , O , P } and ω In addition, a transport equation for the temperature for both the gas and liquid phases is solved:

  ∂ ρc P T u j ∂T ∂ ∂ ρc P T λ + ω˙ T , + = ∂t ∂xj ∂xj ∂xj

(7)

˙ T is the chemical source term for where c p is the specific heat at constant pressure, λ is the thermal conductivity and ω temperature. 2.2. Jump conditions Since the level set method is a sharp interface method, the discretization of the governing equations should be carefully treated across the interface. In particular, the various jump conditions across the interface must be taken into consideration. The jump conditions across the interface can be classified into three categories: material property jump condition, zero-order jump condition and first-order jump condition. To declare the jump from the liquid phase (subscript l) to the gas phase (subscript g), the jump operator across the interface Γ , [·], is defined as:

[ A ]Γ = Al − A g .

(8)

The material jump conditions include the jumps of the density ρ , the dynamic viscosity μ, the thermal conductivity λ and the specific heat C p , i.e., [ρ ]Γ = ρl − ρ g , [μ]Γ = μl − μ g , [ D ]Γ = D l − D g , [λ]Γ = λl − λ g , and [c p ]Γ = c pl − c pg . The zero-order jump conditions include the jumps of the pressure and the velocity:

    1 [ p ]Γ = σ κ + 2 μnt · ∇ u · n Γ − ω˙ 2 , ρ Γ   1 , ˙ [ u] = ω n

ρ

(9) (10)

Γ

 is the interface normal and ω˙ is the evaporation where σ is the surface tension coefficient, κ is the interface curvature, n rate. The velocity jump condition is derived by accounting for the mass conservation across the interface, which can be referred to [33]. The first-order jump conditions refers to the Rankine–Hugoniot jumps [34,35] across the interface to satisfy the energy conservation and the mass conservation:

˙ − [λ∇ T · n]Γ = 0, hlg ω

(11)

ω˙ Y F l + [ρ D F ∇ Y F · n]Γ = ω˙ Y F g , Γ

Γ

(12)

where hlg is the latent heat of evaporation, and Y Fl and Y Fg are the fuel-vapor species mass fraction in the liquid and gas side, respectively. 2.3. Evaporation rate Evaporation rate is a crucial parameter to accurately predict the spray combustion. In general, two mechanisms are responsible for fuel evaporation, i.e., the fuel-vapor concentration gradient driven and temperature gradient driven. In the fuel-vapor concentration gradient driven condition, the evaporation rate is computed from the gradient of fuel-vapor mass fraction across the interface, which is valid when the ambient temperature is lower than the saturation temperature. As

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in the temperature gradient driven condition, the evaporation rate is computed from the gradient of temperature across the interface, which is valid when the ambient temperature is higher than saturation temperature. To summarize, the evaporation rate in different conditions can be obtained from Eqs. (11)–(12) as:

ω˙ = ω˙ =

ρ g D F ∇ Y F · n|Γg Γ 1 − Y Fvap

T g < T sat ,

,

λl ∇ T l · n − λ g ∇ T g · n hlg

(13)

T g ≥ T sat ,

,

(14)

Γ where Y Fvap and T sat are the fuel-vapor concentration and the saturation temperature, respectively. The fuel-vapor concenΓ tration Y Fvap at the interface is evaluated by the Clausius–Clapeyron relation:

   hlg mFvap 1 1 , p Fvap = patm exp − − B Γ Γ

R

Γ Y Fvap =

T

(15)

T

pΓ Fvap mFvap

( patm − p ΓFvap )m g + p ΓFvapmFvap

(16)

,

where p Γ Fvap is the fuel-vapor pressure at the interface, p atm is the ambient pressure, mFvap is the molar mass of the fuelvapor, R is the perfect gas constant, T Γ is the interface temperature, T B is the liquid boiling temperature for the ambient pressure and m g is the molar mass of the gas phase. In the present work, we only validate the conditions below and above the saturation temperature respectively, and the value of the evaporation rate calculated by Eqs. (13) and (14) is continuous when approaching from below and above the saturation temperature. It should be noted that Villegas [46] proposed a novel numerical method that accounts for both the concentration gradient driven and temperature gradient driven conditions, which works well in the situation that boiling and evaporation occur simultaneously in different regions of the liquid interface or successively at different times of simulating an evaporating droplet. The interested readers are referred to Villegas [46] for more details of this method. 2.4. Level set method The level set method is used to capture the interface, which is represented by an iso-surface of the smooth level set function φ(x, t ). Generally, the level set function φ denotes the distance to the interface:

  φ(x, t ) = min |x − xΓ |,

(17)

where xΓ is the location at the interface that is closest to x. The level set function is defined to be positive for the liquid phase and negative for the gas phase. In the simulation of atomization, it is fundamental to accurately capture the smallest scales. We employ the spectrally refined interface approach [36] to obtain the sub-cell resolution for the level set function. In this approach, by introducing a set of quadrature points corresponding to the locations where the nodal values of the level set function are specified, a polynomial reconstruction of the level set function is applied in each cell in a narrow band around the interface. The Gauss–Lobatto quadrature is employed to ensure some quadrature points are located on the cell faces. l,m,n l,m,n The level set value is denoted as φi , j ,k with its position vector xi , j ,k of the quadrature point (l, m, n) of the flow solver cell (i , j , k). The level set function reconstruction could be expressed as:

φi , j ,k (x) =

p p p l =1 m =1 n =1

l,m,n

Ll (x) L m ( y ) L n ( z)φi , j ,k ,

(18)

where p is the number of quadrature points per direction. The base polynomials are employed to simplify the expression and can be written as:

p L α (r ) =

β=1,β=α (r

p

β=1,β=α (r α

− rβ ) − rβ )

(19)

.

Hence, the reconstruction of the level set function reads as:

φi , j ,k (x, y , z) =

      p p p y − yj x − xi z − zk φil,,mj ,,kn . Ll Lm Ln x i +1 − x i y j +1 − y j zk+1 − zk l =1

m =1

n =1

(20)

C. Shao et al. / Journal of Computational Physics 371 (2018) 751–778

755

Fig. 1. Layout of the discretization across the interface in one dimension (red line represents the interface). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

The transport equation for the level set function can be written as:

∂φ + u s · ∇φ = 0, ∂t

(21)

 s is the interface velocity, which is sum of the liquid velocity u l and the surface regression velocity due to evapowhere u  l ) · n = ρ g (  g ) · n , we can ˙ = ρl ( ration and combustion. Considering the mass conservation across the interface ω us − u us − u rewrite the level set transport equation as:

∂φ ω˙ + u l · ∇φ = . ∂t ρl

(22)

In the present simulation, the re-initialization of the level set function is to interpolate the level set value at the quadrature points by using the tri-linear interpolation. The re-initialization process is not always necessary and is typically performed every 100 time steps with the present sub-cell resolution. The interested readers are referred to the work by Desjardins et al. [36] and Luo et al. [32] for the detailed information. 2.5. Discretization As mentioned above, the exact discretization of various jump conditions across the interface is significant in the present computational framework. The GFM [12] is utilized here to treat the jump conditions across the interface. The present method employs the uniform staggered grid in the whole computational domain, where the scalars are defined in the cell center whereas the velocity components are stored on the cell surfaces. Considering a velocity field which is discontinuous across the interface, we have to evaluate the ghost velocity in the ghost cell that is the extension of the gas/liquid velocity in the liquid/gas region, respectively. These ghost velocities can be expressed as: ghost

Ul

ghost

Ug

  1

= U g − ω˙

ρ

  = U l + ω˙

1

ρ

, n

(23)

. n

(24)

Γ

Γ

2.5.1. Discretization of the species mass fraction equations Since the species mass fractions of fuel-vapor, oxygen and product are zero in the mono-component liquid phase, the mass fraction equations are only required to solve in the gas phase. Eqs. (4)–(6) are solved with a Dirichlet boundary condition as Eq. (16) at the interface. We use the 2nd-order central differential scheme for spatial discretization and the 2nd-order semi-implicit Crank–Nicolson scheme for time integration. Let us consider the discretization of the convective term across the interface in one dimension as shown in Fig. 1. The ghost mass fraction in the liquid phase is evaluated as: ghost

Yi

=

Y I + (θ − 1)Y i −1

θ |x −x

(25)

, |

ghost

where θ is defined as θ = I xi−1 . Particularly, Y i is estimated to be equal to Y i if θ  1. Hence, the mass flux across the cell surface between i − 1 cell and i cell is computed as: ghost

ghost

F Xi = U g

·

Y i −1 + Y i 2

.

The discretization of the convective term can be expressed as:

(26)

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∇ uY =

F X i − F X i −1

x

(27)

.

For the discretization of the diffusive term, it can be evaluated as: ghost

Yi

∇ · (∇ Y ) =

− Y i −1 x

− x

Y i −1 − Y i −2 x

(28)

.

The above forms of discretization can be easily extended to two and three dimensions. 2.5.2. Discretization of the energy equation The discretization of the energy equation is performed in the whole computational domain. Similar with the discretization of the species mass fraction equations, we use the 2nd-order central differential scheme for spatial discretization and the 2nd-order semi-implicit Crank–Nicolson scheme for time integration. However, if we populate the ghost value of temperature on each side of the interface as the way of the mass fractions, the artificial heating of the gas/liquid phase would occur, which is not physically meaningful and could contaminate the numerical results. Here, we utilize a method proposed by Aslam [37] to address this issue. In this method, the ghost temperature is extrapolated to each side of the interface to satisfy the normal derivative continuity of the temperature by solving the following two sequent equations:

∂ Tn = ± n · ∇ Tn, ∂t ∂T = ±( n · ∇ T − T n ), ∂t

(29) (30)

 · ∇ T . The sign “+” means the extrapolation from the liquid where T n is the normal derivative of the temperature, i.e., T n = n phase to the gas phase and sign “−” means the extrapolation from the gas phase to the liquid phase. The discretization of the convective term is similar to Eq. (27). It is noteworthy that the temperature gradient, i.e., thermal flux, is discontinuous at the interface due to the latent heat from the phase change. Hence, when discretizing the diffusive term, the jump condition of Eq. (11) has to be taken into consideration and the diffusive term is expressed as: ghost

λT i

∇ · (λ∇ T ) =

−λ T i −1 x

−

λ T i −1 −λ T i −2 x

x

+ hlg ω˙

.

(31)

2.5.3. Discretization of the level set equation Before discretizing the level set transport equation, we should populate the evaporation rate that is only exist at the interface in the whole computational domain. An extrapolation procedure is then applied in each side of the interface in order to populate the normal continuity evaporation rate in the computational domain. As demonstrated by Aslam [37], this can be obtained by solving successively the following partial differential equations (PDE):

∂ ω˙ ˙ = 0, + H (φ) n · ∇ω ∂t ∂ ω˙ ˙ = 0, − H (−φ) n · ∇ω ∂t

(32) (33)

where H is the Heaviside function. Eq. (32) is used for the extrapolation from the liquid phase to the gas phase while Eq. (33) is used for the extrapolation from the gas phase to the liquid phase. We employ a semi-Lagrangian approach to transport the level set function. In the present simulation, a lower case letter is used for an Eulerian variable and an upper case letter for a Lagrangian variable. The Lagrangian form of the transport equation with Eq. (22) can be written as

dφ dt

= fω,

(34)

˙ · (1/ρ g − 1/ρl ) is the source term due to surface regression. The Crank–Nicolson semi-Lagrangian scheme of where f ω = ω the above transport equation reads

φ n+1 − φdn 1  n +1 = f + f ωn d . t 2 ω

(35)

In the semi-Lagrangian method, a discrete set of particles arriving at the grid points are tracked backward over a time step along its characteristic line to its departure points, which can be shown in Fig. 2. It is noted that the departure point does not necessarily coincide with the grid point. In the backwards tracking process, the trajectory of a particle can be obtained by solving the kinematic equation

C. Shao et al. / Journal of Computational Physics 371 (2018) 751–778

757

Fig. 2. Semi-Lagrangian transport of the level set function.

dX dt

= U,

(36)

where X = X ( X 0 , t 0 |t ) is the location at time tof the particle that originates from the location X 0 at time t 0 . U = U ( X 0 , t 0 |t ) is the velocity of the particle at time t. Each quadrature point is advected backwards in time using a fourth order Runge– Kutta scheme. 2.6. Solving the Navier–Stokes equations The discretization of the Navier–Stokes equations is based on the staggered uniform grids mentioned above. The spatial discretization of the Navier–Stokes equations is performed using the second-order finite central-difference schemes. The second-order semi-implicit iterative procedure [38] for time integration is utilized, which is economical, stable and accurate. The iteration can be expressed as +1 ukn+ = un + t f 1



1 2

un + ukn+1





1

+ t 2



 ∂ f  n +1 uk+1 − ukn+1 , ∂u

(37)

where f is the right hand side of the Navier–Stokes equations, and ∂ f /∂ u is its Jacobian. The lower and upper index notations k and n denote the kth New–Raphson sub-iterative step and the current time step, respectively. The computations of the velocity and pressure fields are decoupled using the projection method. Note that the velocity is discontinuous across the interface, Eqs. (23)–(24) are employed to discretize the convective term of the momentum equation. The jump conditions for the pressure gradient in the Poisson equation as well as for the viscous terms and the evaporation effect are taken into account by the GFM. Consider an interface Γ located at xΓ between the two grid locations xi and xi +1 , where xi +1 is within the liquid phase. The pressure jump in the pressure Poisson equation is then written as

  1 1 [ p ]Γ ∂ 1 ∂ p  ρ ∗ ( p l , i +1 − p g , i ) − ρ g ( p g , i − p g , i −1 ) = − ∗ 2, ∂ x ρ ∂ x  g ,i x2 ρ x

(38)

where ρ ∗ = ρ g ζ + ρl (1 − ζ ) and ζ = (xΓ − xi )/x. A density-based flux correction scheme proposed by Desjardins et al. [40] is employed to treat the high density ratio and to ensure the robustness of the simulations. In addition, the divergence-free velocity should be ensured at the interface and the extrapolation method proposed by Villegas et al. [46] is employed here. The viscous term in the Navier–Stokes equation should be carefully treated due to its discontinuity across the interface. The Continuum Surface Force model proposed by Brackbill [39] is often used for the viscous term which smears out the viscous jump across the interface and it is employed in the present work for the sake of simplicity. However, this treatment can result to tricky interface thickening and spurious currents. It should be noted that Kang et al. [47] employed GFM to treat the viscous term in a sharp fashion. However, the implementation of this method is very complex and it is limited to configurations without any jump conditions of the velocity field. This implies that it cannot be used in the present simulations with phase change directly. Alternatively, Sussman et al. [48], Lalanne et al. [49] and Lepilliez et al. [50] proposed a much simpler implementation maintaining the sharp manner of the viscosity without loss of accuracy. These schemes can be used in simulations with phase change directly and the viscous term in the pressure jump of Eq. (9) needs no further consideration.

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2.7. Chemical mechanism In the present study, we focus on the global characteristics of spray combustion and the one-step global reaction mech˙ F , ω˙ O , ω˙ P and anism of n-heptane is chosen to represent the combustion process. The source terms in Eqs. (4)–(7), i.e. ω ω˙ T , are obtained from the following one-step irreversible reaction of n-heptane:

υF F + υO O → υP P + Q T ,

(39)

where the stoichiometric coefficients υi are obtained from the global reaction C7 H16 + 11O2 = 7CO2 + 8H2 O, and Q T is the heat release. The source terms could be written as:

⎛

⎞

⎛

⎞

ω˙ F υF W F ⎜ ω˙ O ⎟ ⎜ ⎟ ⎜ ⎟ = ω˙ che ⎜ υ O W O ⎟ , ⎝ ω˙ P ⎠ ⎝ υP W P ⎠ ω˙ T QT where



ω˙ che = A

ρg Y F



WF

ρg Y O WO



(40)

 exp −

TA T

 .

(41)

In the present simulation, Q T and the activation temperature T A are adaptive parameters. In order to retain as much as possible of the physical process, we employ the approach proposed by Fernandez-Tarrazo et al. [41], where the coefficients of the Arrhenius law are fitted to accurately predict the burning velocity. The pre-exponential factor A is set to be 9.7 × 108 m3 /(mol s). The mixture fraction Z is an important parameter for characterizing combustion, and it is defined as:

Z=

υˆ (Y F − Y F ,0 ) − (Y O − Y O ,0 ) υˆ (Y F ,1 − Y F ,0 ) − (Y O ,1 − Y O ,0 )

(42)

where υˆ = υ O W O /(υ F W F ). The subscript 0 and 1 denote the oxidizer and fuel streams, respectively. 3. Verification and validation In our previous work [32], we have developed a mass conserving level set method for atomization without evaporation and combustion, and a variety of validation have been conducted. In the present work, we only focus on the validation of evaporation and combustion. 3.1. Evaporation rate Evaporation rate is a crucial parameter that determines the rate of fuel evaporation and consequently the interface location. The first case is conducted to validate the accuracy of evaporation rate computation in the low ambient temperature condition. This case consists of a container filled with water up to a certain level and air the rest, and it is essentially a onedimensional evaporation problem. A schematic illustration of the configuration is found in Fig. 3. The vapor mass fraction Γ , is computed using the Clausius–Clapeyron relation. The air far from the interface is dry, Y L at the interface, Y Fvap Fvap = 0. Stefan flows occur due to the water vapor at the interface diffusing into the air. The interface location and the interface temperature are assumed to stay constant during the simulation. The mass conservation at the interface yields:

g  , ω˙ Fvap = ω˙ Fvap Y Fvap − ρ g D dy Γ dY Fvap 

(43)

with the following boundary conditions for Y Fvap : Γ Y Fvap | y = H = Y Fvap ,

L Y vap | y = H + L = Y Fvap .

(44)

˙ Fvap could be obtained as: The analytical solution for ω

ω˙ Fvap =

ρg D L

ln(1 + B ),

(45)

where B is the mass transfer number given as:

B=

Γ −YL Y Fvap Fvap Γ 1 − Y Fvap

(46)

C. Shao et al. / Journal of Computational Physics 371 (2018) 751–778

759

Fig. 3. A schematic illustration for the evaporation rate.

Table 1 Physical properties of gas and liquid phases at normal temperature and pressure.

ρ (kg m−3 ) Gas Liquid Gas Liquid

1.226 1000 m vap (kg mol−1 ) 0.029 0.018

μ (kg m−1 s−1 ) 10−5

1.780 × 1.137 × 10−3 hlg (J kg−1 ) 2.3 × 106

λ (W m−1 K−1 )

C p (J kg−1 K−1 )

0.046 0.600 D F (s−1 ) 3.750 × 10−5 1.435 × 10−7

1000 4180 σ (N m−1 ) 0.0728

Fig. 4. The vapor mass fraction at steady state with grid resolution 1282 .

The computational domain is 1 mm × 1 mm. Three simulations with different resolutions are performed with 322 , 642 , and 1282 grids, respectively. The liquid height is set to be H = 0.25 mm. The left and right boundaries (x = 0 and x = W ) are set to be periodic. The bottom and up boundaries are set to be the wall and free boundary condition, respectively. The gas and liquid properties are listed in Table 1. Fig. 4 shows the vapor mass fraction in the gas phase at the steady state for the simulation with the grid resolution of 1282 . It is observed that the maximum vapor mass fraction occurs at the interface. The vapor mass fraction decreases in the positive y direction due to the diffusion. Furthermore, we compare the numerical results with the analytical solution for different mass transfer number B and the results are shown in Fig. 5. It is shown that the numerical results approach the analytical solution with increasing grid number, which implies that the present method is grid convergent and precisely computes the evaporation rate.

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Fig. 5. The comparison of numerical and analytical non-dimensional evaporation mass fluxes for different values of mass number.

Fig. 6. Convergence of L 1 and L 2 norm error for the evaporative mass flux. The black solid and dashed lines show first-order and second-order convergence for comparison.

Next, the numerical error is quantified. Two errors, namely the one-norm L 1 and two-norm L 2 errors, are computed. They are defined for the variable Θ as:

L1 = and

N 1 

N

 Θ numerical − Θ exact , k

k

(47)

k =1

  N 1  numerical 2 Θk

L2 =  − Θkexact , N

(48)

k =1

where N is the number of cases (N = 9) for each resolution. The upper subscripts ‘numerical’ and ‘exact’ represent the numerical value and the theoretical value, respectively. The one-norm and two-norm errors for the evaporation rate are shown in Fig. 6. The errors decrease almost a first power as a function of the grid resolution, which indicates that the present method for this case has a first-order accuracy. 3.2. Wet bulb temperature comparison The second case of static droplet evaporation is conducted to validate the accuracy of wet bulb temperature computation compared with the psychrometric chart value in low ambient temperature condition. This case was also used by Irfan and Muradoglu [26] for the fuel-vapor concentration gradient driven model. In this case, the computational domain size is 1.25 mm × 1.25 mm and the initial droplet diameter is d0 = 0.25 mm as shown in Fig. 7. Initially, the uniform temperature

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Fig. 7. Schematic of the static droplet evaporation.

Fig. 8. The comparison of numerical and actual wet bulb temperature for different values of relative humidity.

(dry bulb temperature, DBT) is set throughout the computational domain. Evaporation occurs due to the vapor concentration gradient at the interface and the evaporation process results in a lower temperature at the interface. As the evaporation proceeds towards a steady state, the liquid temperature approaches an equilibrium value. This equilibrium temperature is called the wet bulb temperature (WBT). The WBT is a function of DBT and the relative humidity (RH) in the air. In this case, the Dirichlet boundary conditions apply at the computational domain boundaries for the temperature T b and vapor mass fraction Y b . Y b is estimated as Y b = ωh /(1 + ωh ), where ωh is the humidity ratio that is a function of DBT and RH, and can be read from the psychrometric chart. The physical properties of air and water that are listed in Table 1 are used except that the liquid density is taken as 10 kg/m3 in this case as in the work by Irfan and Muradoglu [26]. The liquid thermal conductivity is modified according to the thermal diffusivity value for water. We perform simulations of static droplet evaporation with DBT = 313 K under four conditions of RH, i.e. 30%, 40%, 50% and 60%. For each RH condition, three grid resolutions are considered, i.e. 322 , 642 and 1282 . The comparison of the numerical and actual wet bulb temperatures for different values of RH is presented in Fig. 8. It shows that the numerical results are in good agreement with the values from the psychrometric chart. With increasing grid resolution, the numerical results approach to the actual WBT values. The numerical errors of the WBT are also quantified following Eqs. (47) and (48). It is shown in Fig. 9 the accuracy for this case is less than first-order. To further characterize the droplet evaporation behaviors, we present the temporal evolution of contour plots for the temperature and vapor mass fraction fields with a grid resolution of 1282 , DBT = 313 K and RH = 60% in Fig. 10. It is shown that the temperature at the interface firstly decreases at early time. The liquid temperature decreases gradually until an equilibrium state is observed. It can be found that the liquid equilibrium temperature of 305.80 K agrees well with the

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Fig. 9. Convergence of L 1 and L 2 norm error for the wet bulb temperature. The black solid and dashed lines show first-order and second-order convergence for comparison.

value of 305.64 K from the psychrometric chart. The temporal evolution of temperature along the horizontal center line of the domain at various timings is shown in Fig. 11. Note that the temperature profiles are similar to those of Irfan and Muradoglu [26] (not shown). 3.3. The Stefan problem The Stefan problem is a well-known test case to validate the phase change model [14,18,20,30,52] that is driven by the temperature gradient. Fig. 12 shows the schematic of the configuration. Initially, the liquid temperature, T l , is set to be at saturation condition (T sat ) and the temperature of the left wall, T w , is higher than the liquid temperature. The temperature of the vapor phase varies linearly in space. The evaporation occurs at the interface due to the temperature gradient and results in the interface moving towards right. In this case, the vapor is assumed to remain stationary and the heat transfer from the wall to the interface is due to diffusion. Therefore, the temperature equation in the vapor phase can be written as

∂T ∂2T = αg 2 ∂t ∂x

0 ≤ x ≤ xΓ (t )

(49)

where T is the temperature, α g is the thermal diffusivity of vapor and xΓ (t ) is the interface location at time t. The analytical solution for the interface position at time t is [26]:

xΓ (t ) = 2β



αg t,

(50)

where β is the solution of the transcendental equation:

 c p , g ( T w − T sat ) β exp β 2 erf(β) = . √ hlg

(51)

π

In Eq. (51), erf is the Gauss error function [26]. The analytical value of temperature in the vapor phase can be expressed as:

 T g (x, t ) = T w +

T sat − T w erf(β)



 erf

x

2



αg t

 .

(52)

In this case, the top and bottom boundaries ( y = 0 and y = W ) are set to be periodic. The left and right boundaries are set to be the wall and free boundary condition, respectively. The computational domain size is 1 mm × 1 mm and three cases with grid resolutions of 642 , 1282 and 2562 are considered. The initial width of the vapor region is set to be H = 0.1 mm. The initial temperature profile in the vapor phase is specified using the theoretical solution. The fluid properties of water and vapor [20] used are as follows: ρ g = 0.597 kg m−3 , ρl = 958.4 kg m−3 , μ g = 1.26 × 10−5 kg m−1 s−1 , μl = 2.8 × 10−4 kg m−1 s−1 , λ g = 0.025 W m−1 K−1 , λl = 0.679 W m−1 K−1 , c p, g = 2030 J kg−1 K−1 , c p,l = 4216 J kg−1 K−1 , and hlg = 2.26 × 106 J kg−1 . The saturation temperature and wall temperature are set to be T sat = 373.15 K and T w = 383.15 K, respectively.

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Fig. 10. The temporal evolution of contour plots for temperature (left column, K) and vapor concentration (right column) fields for an static evaporating droplet with grid resolution 1282 at times (from top to bottom) t = 0.005 s, 0.020 s, 0.120 s; DBT = 313 K, RH = 60% (black line denotes the gas–liquid interface).

First we perform the grid convergence study of the interface location and compare the numerical results with the theoretical solutions in Fig. 13. It is seen that the numerical results with grid resolutions of 128 × 128 and 256 × 256 are in good agreement with the theoretical solution. We then perform the grid convergence study of the temperature profile and compare the numerical results with the theoretical solution in Fig. 14. It is also shown that the numerical results with grid resolutions of 128 × 128 and 256 × 256 are in excellent agreement with the theoretical solution. The L 1 and L 2 norm errors of the interface location and temperature are shown in Fig. 15. It is observed that the present method has a second-order accuracy for this case.

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Fig. 11. The temporal evolution of temperature along the horizontal center line of the domain at different times with grid resolution 1282 , DBT = 313 K and RH = 60%.

Fig. 12. Schematic of the Stefan problem.

Fig. 13. The comparison of numerical and theoretical interface location with different mesh resolution for the Stefan problem.

3.4. The Sucking problem The Sucking problem is also a well-known test case to validate the phase change models [18]. It is also called the boiling interface problem. Fig. 16 shows the schematic of the configuration. The liquid temperature, T g , and the wall temperature, T w , are set to be the saturation temperature (T sat ) and the liquid temperature far away from the interface, T ∞ , is higher than the saturation temperature. The evaporation occurs at the interface due to the temperature gradient and results in the

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Fig. 14. The comparison of numerical and theoretical profile of temperature with different mesh resolutions for the Stefan problem.

Fig. 15. (a) Convergence of L 1 and L 2 norm error for interface location of the Stefan problem. (b) Convergence of L 1 and L 2 norm error for temperature of the Stefan problem. The black solid and dashed lines show first-order and second-order convergence for comparison.

Fig. 16. Schematic of the Sucking problem.

interface moving towards right. The evaporation leads to the formation of a thermal boundary layer in the liquid side. The following temperature equation is solved for the liquid phase:

∂T ∂2T + u · ∇ T = αl 2 xΓ (t ) ≤ x ≤ 1 (53) ∂t ∂x where αl is the thermal diffusivity of the liquid phase. The Sucking problem is a more challenging case compared to the Stefan problem since the convective term needs to be solved.

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Fig. 17. The comparison of numerical and theoretical interface location with different mesh resolution for the Sucking problem.

The analytical solution for the interface position at time t reads [26]:

xΓ (t ) = 2β



αg t,

(54)

where β is the solution of the transcendental equation: ρ2 αg √  ( T ∞ − T sat )c p , g λl α g exp(−β 2 g2 )  c ( T − T ) ρl αl p, g w sat √ exp β erf(β) β − . = √ √ ρg αg hlg π hlg λ g παl erfc(β ρ √α )



2

l

(55)

l

In Eq. (55), erfc is the complementary error function. The analytical value of temperature in the liquid phase is written as:



T l (x, t ) = T ∞ −

T∞ − T w

√ ρg αg erfc(β √ )

ρl αl



erfc



x

2

√

αl t

+

β(ρ g − ρl )

ρl





αg . αl

(56)

In this case, the top and bottom boundaries ( y = 0 and y = W ) are set to be periodic. The left and right boundaries are set to be the wall and free boundary condition, respectively. The computational domain size is 1 m × 1 m and three cases with grid resolutions of 642 , 1282 and 2562 are considered. The initial width of the vapor region is set to be H = 0.1 m. The initial temperature profile in the liquid phase is specified using the theoretical solution. The fluid properties used are the same as those for the Stefan problem. The saturation temperature and wall temperature are set to be T sat = 373.15 K and T w = 378.15 K, respectively. First we perform the grid convergence study of the interface location and compare the numerical results with the theoretical solution in Fig. 17. It shows the numerical results with grid resolutions of 128 × 128 and 256 × 256 are in good agreement with the theoretical solution. We then perform the grid convergence study of the temperature profile and compare the numerical results with the theoretical solution in Fig. 18. It also shows the numerical results are in good agreement with the theoretical solution. The L 1 and L 2 norm errors of the interface location and temperature are shown in Fig. 19. It is observed that the present method has almost a first-order accuracy for this case. 3.5. Evaporation of a static droplet In this case, we perform simulations of static droplet evaporation in high ambient temperature conditions and compare the variation of droplet diameter with the experimental result [42]. The schematic of this case is the same as that of the case in Section 3.2, which is shown in Fig. 7. Initially, the droplet temperature is T l0 and the ambient temperature is T g0 . The droplet evaporates due to the temperature gradient across the interface. The liquid is n-heptane, the same with the experiment [42], and the liquid and gas properties are listed in Table 2. The boiling temperature T B of liquid is 371.6 K. The liquid saturation temperature is calculated using the following empirical correlation [43]:

 T sat = 137

TB 373.15

0.68 log10 ( T G ) − 45

where T sat and T G are the saturation and free stream temperature, respectively.

(57)

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Fig. 18. The comparison of numerical and theoretical profile of temperature with different mesh resolution for the Sucking problem.

Fig. 19. (a) Convergence of L 1 and L 2 norm error for interface location of the Sucking problem. (b) Convergence of L 1 and L 2 norm error for temperature of the Sucking problem. The black solid and dashed lines show first-order and second-order convergence for comparison.

Table 2 Physical properties of the gas phase and liquid n-heptane.

ρ (kg m−3 ) Gas Liquid Gas Liquid

1.13 649.38 m vap (kg mol−1 ) 0.028 0.1

μ (kg m−1 s−1 ) 10−5

1.780 × 4.090 × 10−4 hlg (J kg−1 ) 3.5 × 105

λ (W m−1 K−1 )

C p (J kg−1 K−1 )

0.0371 0.1768 D T (s−1 ) 3.118 × 10−5 1.142 × 10−7

1053 2383.89 σ (N m−1 ) 0.0216

In this case, all the boundaries are set to be free boundaries. The diameter d0 of the n-heptane droplet is 0.7 mm. The computational domain size is 2 mm × 2 mm and the grid resolutions are 322 , 642 and 1282 . The droplet temperature is T l0 = 300 K and the ambient temperature is T g0 = 471 K. The Dirichlet boundary conditions are used for temperature, which equals 471 K. The saturation temperature calculated from Eq. (57) is T sat = 321 K. First, we conduct the grid convergence test and compare the numerical results with the experimental results as shown in Fig. 20. It is observed that the numerical results approach the experimental results with increasing grid resolution. The numerical result with a grid resolution of 64 × 64 is sufficient to resolve the diameter variation. It should be noted that the level set method has the drawback of mass loss, which may also contribute to the variation of droplet diameter. To identify the possible error of the level set method, we perform a testing case of the same configuration with the evaporation module turned off. A grid resolution of 64 × 64 is used for the testing case. It is observed in Fig. 20 that the influence of the level set method on the mass loss is negligible in this case.

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Fig. 20. The comparison of diameter variation of numerical and experimental results with different mesh resolution for the evaporation of n-heptane droplet.

Fig. 21. The comparison of diameter variation of numerical and experimental results for the evaporation of n-heptane droplet under different ambient temperature condition.

It should be noted that droplet evaporation is a three-dimensional problem. Nevertheless, three-dimensional simulations of droplet evaporation are computationally expensive. So that only two-dimensional simulations are performed in the present work. Still, good agreements of the droplet diameter variation are observed for the two-dimensional simulations and the experiments. Furthermore, we perform the simulations of n-heptane droplet evaporation under different ambient temperature conditions and the results are shown in Fig. 21. The numerical results with the Dirichlet boundary conditions for temperature (fixed temperature at the boundary) are represented by the dashed lines Fig. 21. It can be seen that the numerical results agree well with the experimental results at the early stage, for example, t /d20 = 0∼3 for the simulation with an ambient temperature of 471 K. However, the numerical results deviate at the later stage for all the simulations. This should be attributed to the Dirichlet temperature boundary conditions in the present simulations. For example, Fig. 22 shows that, for the temperature profile across the interface with an ambient temperature of 471 K, the temperature gradient is different at two different timings during the later stage of droplet evaporation where the temperature inside the droplet is uniform. This results in different values of evaporation rate calculated by Eq. (14), e.g., the evaporation rate at t = 1.475 s is 5.5 × 10−5 kg m−2 s−1 while that at t = 2.975 s is 4.4 × 10−5 kg m−2 s−1 . The evaporation rate decreases with time at the later stage of droplet evaporation, which is not physically meaningful. Alternatively, we employ the variable Dirichlet temperature boundary conditions thus keeping the evaporation rate as constant at the later stage. The resultant diameter variations are shown as solid lines in Fig. 21. It can be seen that the

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Fig. 22. Temperature profile at different time for the ambient temperature 471 K.

Fig. 23. Temporal evolution of temperature field for the moving droplet evaporation.

numerical results are in good agreement with the experimental results, which obey the d square law. In this case, it is indicated that the temperature boundary conditions are crucial to capture the diameter variations of droplet evaporation. 3.6. Evaporation of a moving droplet In this case, we perform simulations of moving droplet evaporation in high ambient temperature condition. The schematic of this case is similar to that of the case in Section 3.2 as shown in Fig. 7. Initially, the droplet temperature is T l0 and the ambient temperature is T g0 . The initial droplet velocity is U 0 . The droplet evaporates due to the temperature gradient across the interface. The liquid is n-heptane and the liquid and gas properties are listed in Table 2. In this case, all the boundaries are set to be free boundaries. The diameter d0 of the n-heptane droplet is 0.5 mm. The computational domain size is 5 mm × 10 mm and the grid resolution is 128 × 256. The droplet temperature is T l0 = 300 K and the ambient gas temperature is T g0 = 1000 K. The Dirichlet boundary conditions are used for temperature, which equals 1000 K. The saturation temperature calculated from Eq. (57) is T sat = 421 K. The initial Reynolds number is defined as Re0 = ρ g U 0 d0 /μ g and Weber number is defined as We0 = ρ g U 02 d0 /σ . In this case, Re0 = 31.7 and We0 = 0.026. Due to the lack of the experimental data, we only present the temperature and vapor mass fraction fields, which are qualitatively compared with the previous numerical results [11,26]. Fig. 23 and Fig. 24 show the temporal evolutions of the

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Fig. 24. Temporal evolution of fuel concentration field for the moving droplet evaporation.

Fig. 25. Evaporation rate (kg m−2 s−1 ) calculated around the droplet at t = 0.0045 s, where black line represents the gas–liquid interface.

temperature field and vapor mass fraction field by using the present method. The temperature and vapor mass fraction fields are qualitatively in agreement with the numerical results in [11,26]. In particular, due to the evaporation, the temperature behind the droplet is lower than the ambient gas temperature, forming a temperature plume. In this case, due to the lower temperature concentrated behind the droplet, the corresponding evaporation rate driven by the temperature gradient is lower than in the front of droplet, which is highlighted in Fig. 25. The vapor mass fraction is more concentrated behind the droplet and a vapor plume is observed. The internal temperature field of the droplet is shown in Fig. 26 at t = 0.002 s. It is observed that a vortex pair occurs on the top of the droplet and a low temperature zone is right below the vortex pair. This indicates that the vortices could contribute to the non-uniform distribution of temperature inside the droplet. The temperature is high near the droplet surface owning to the heat flux from the high ambient temperature and is high in the center of the droplet due to convection. This temperature distribution is very similar to the experimental result by Wong and Lin [51] (not shown). 3.7. Combustion of a static droplet In the following, we perform simulations of static n-heptane droplet combustion in high ambient temperature conditions and compare the variation of droplet diameter with the experimental result [44]. The schematic of this case is similar to that in Fig. 7. Initially, the droplet temperature is T l0 and the ambient gas temperature is T g0 . The droplet evaporates due to the temperature gradient across the interface. The fuel vapor then mixes with the surrounding air and combustion occurs, which typically features a non-premixed mode. The liquid is n-heptane and the liquid and gas properties are listed in Table 2. The chemical reaction we use is the one-step irreversible reaction of n-heptane as mentioned in Section 2.7. In this case, all the

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Fig. 26. Temperature distribution inside the droplet at t = 0.002 s, where thick black line and thin arrow line represent interface and streamline, respectively.

boundaries are set to be free boundaries. The diameter d0 of n-heptane droplet is 3 mm. In order to eliminate the effect of boundary condition on the combustion process, we set the computational domain as large as 50 mm × 50 mm. Two grid resolutions with 2562 and 5122 are considered. The droplet temperature is initially set to be the saturation temperature T l0 = 366 K and the ambient gas temperature is T g0 = 1000 K. The Dirichlet boundary conditions are used for temperature, which equals 1000 K. The Dirichlet boundary conditions are also used for oxygen mass fraction, which equals 0.2. Fig. 27 shows the contours of temperature, fuel mass fraction, oxygen mass fraction and product mass fraction for static n-heptane droplet combustion at t = 1 s with a grid resolution of 5122 . It can be seen that a high temperature zone appears around the droplet due to the chemical reaction. The fuel mass fraction reaches its maximum at the interface and diffuses to the ambient gas. Due to the chemical reaction, the oxygen is consumed around the droplet. The product mass fraction has a maximum in the reaction zone and decreases away from the reaction zone. The corresponding radial profiles of various quantities are also shown in Fig. 28. It is interesting to see that the scale of the reaction zone is comparative to the droplet scale, which is consistent with the typical flame structures for the droplet combustion [45]. The grid convergence test is conducted and the variation of droplet diameter is compared to the experimental result in Fig. 29. It is shown that the numerical results approach the experimental results with increasing grid resolution. The variation of the droplet diameter with a grid resolution of 512 × 512 agrees very well with the experimental result. Also, we perform the simulation of a static droplet without combustion and it is observed in Fig. 29 that the mass loss is negligible for the present combustion case. 3.8. Combustion of a moving droplet In this case, we perform simulations of moving droplet combustion in high ambient temperature conditions. The schematic of this case is similar to that in Fig. 7. Initially, the droplet temperature is T l0 and the ambient gas temperature is T g0 . The initial droplet velocity is U 0 . The droplet evaporates due to the temperature gradient across the interface. The fuel vapor then mixes with the surrounding air and combustion occurs. The liquid is n-heptane and the liquid and gas properties are listed in Table 2. In this case, all the boundaries are set to be free boundaries. The diameter d0 of n-heptane droplet is 0.25 mm. The computational domain size is 2.5 mm × 5 mm and the grid resolution is 256 × 256. The droplet temperature is T l0 = 293 K and the ambient gas temperature is T g0 = 1000 K. The temperature boundaries are set to be Dirichlet (1000 K). The saturation temperature calculated from Eq. (57) is T sat = 366 K. The initial Reynolds number is Re0 = 17.2 and Weber number is We0 = 0.004. Lacking the experimental data, we only qualitatively display the temporal evolution of the temperature, vapor mass fraction, oxygen mass fraction, production fraction and mixture fraction fields as shown in Figs. 30–34. In Fig. 30, it can be seen that a high temperature zone occurs behind the droplet at t = 0.0001 s, implying the droplet ignition. This is due to the high concentration of fuel vapor behind the droplet as seen in Fig. 31. Then the droplet flame develops and high temperature region wrap the droplet. Finally, a temperature plume is observed at t = 0.003 s as shown in Fig. 30. 3.9. Droplet collision and combustion In order to further evaluate the capability of simultaneously treating to interface change and combustion, we perform simulations of collision and combustion of two n-heptane droplets. The initial droplet diameter is D = 250 μm. The computational domain size is 20D × 10D with a grid resolution of 256 × 256. The initial droplet temperature is T l = 293 K and

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Fig. 27. The contours of temperature, fuel concentration, oxygen concentration, product concentration of static n-heptane droplet combustion at t = 1 s with the grid resolution 5122 .

Fig. 28. The radial profiles of temperature, fuel concentration, oxygen concentration, product concentration of static n-heptane droplet combustion at t = 1 s with the grid resolution 5122 (gray zone represents the chemical reaction zone and yellow zone represents the droplet).

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Fig. 29. The comparison of diameter variation of numerical and experimental results with different mesh resolution for the combustion of n-heptane droplet.

Fig. 30. Temporal evolution of temperature field for the moving droplet combustion.

the ambient temperature is T g = 1000 K. The free boundary condition is used for all the boundaries. The initial Reynolds number is Re0 = 17.2 and initial Weber number is We0 = 0.004. Fig. 35 shows the temporal evolution of the temperature field and mixture fraction field. After a short time, ignition takes place in the wake with the maximum temperature rising up to about 2500 K as shown in Fig. 35(a1). The flame then develops and a high temperature zone around the droplet is observed. When the droplets collide with each other, the flame structure evolves accordingly as shown in Figs. 35(a2–a4). The temporal evolution of the mixture fraction field is presented in Fig. 35(b). Fig. 36 shows the correlations of the temperature and mixture fraction at different timings. The time is normalized as t ∗ = U 0 t / D 0 . It is seen that before ignition at t* = 0.4, the mixing process dominates. As combustion proceeds, the temperature profile approaches to the Burke–Schuman solution. As time evolves, large scattering is observed especially in the high mixture fraction zone, which is possibly explained by the inhomogeneous combustion around the droplet and the effect of the irregular interface.

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Fig. 31. Temporal evolution of fuel concentration field for the moving droplet combustion.

Fig. 32. Temporal evolution of oxygen concentration field for the moving droplet combustion.

4. Conclusions A computational framework that can simultaneously simulate atomization, evaporation and combustion processes is proposed. The present work utilizes the level set method to implicitly capture the gas–liquid interface and the GFM to accurately address jump conditions across the interface. Specific care has been devoted to the discretization of convective term and diffusive term for the mass fraction and energy transport equations. The level set method with sub-cell resolution is employed and a semi-Lagrangian scheme of the level set transport equation with source term is proposed. The one-step global reaction mechanism of n-heptane is used for the vapor combustion. The present method has been validated by several cases, such as, evaporation rate, wet bulb temperature comparison, the Stefan problem, the Sucking problem, evaporation of a static and moving droplet, and combustion of a static and moving droplet. Finally, droplet collision and combustion

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Fig. 33. Temporal evolution of product concentration field for the moving droplet combustion.

Fig. 34. Temporal evolution of mixture fraction field for the moving droplet combustion.

is simulated to evaluate the capability of simultaneously treating interface change and combustion. It is implied that the inhomogeneous combustion around droplet and the irregular interface could affect the overall combustion characteristics. It is therefore essential to investigate the spray combustion by using the present interface-resolved DNS computational framework. Acknowledgements This work is financially supported by the National Natural Science Foundation of China (Nos. 91541202 and 91741203).

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Fig. 35. Temporal evolution of the droplet collision and combustion (a) temperature field and (b) mixture fraction field.

Fig. 36. Correlations of temperature and mixture fraction at different time.

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