Numerical simulations of transverse liquid jet to a supersonic crossflow using a pure two-fluid model

Advances in Mechanical Engineering 2016, Vol. 8(1) 1–13

Ó_{The Author(s) 2016} DOI: 10.1177/1687814016629341 aime.sagepub.com

Numerical simulations of transverse

liquid jet to a supersonic crossflow

using a pure two-fluid model

Haixu Liu, Yincheng Guo and Wenyi Lin

Abstract

A pure two-fluid model was used for investigating transverse liquid jet to a supersonic crossflow. The well-posedness problem of the droplet phase governing equations was solved by applying an equation of state in the kinetic theory. A k-e-kpturbulence model was used to simulate the turbulent compressible multiphase flow. Separation of boundary layer in front of the liquid jet was predicted with a separation shock induced. A bow shock was found to interact with the separation shock in the simulation result, and the adjustment of shock structure caused by the interaction described the whipping phenomena. The predicted penetration height showed good agreement with the empirical correlations. In addition, the turbulent kinetic energies of both the gas and droplet phases were presented for comparison, and effects of the jet-to-air momentum flux ratio and droplet diameter on the penetration height were also examined in this work.

Keywords

Liquid injection, supersonic crossflow, kinetic theory, penetration height

Date received: 4 February 2015; accepted: 2 January 2016

Academic Editor: Moran Wang

Introduction

The description of liquid jet injection into a supersonic crossflow has become an important research area as the development of high-speed flight vehicles. Such a process is usually encountered in thrust vector control, external burning on projectiles, and particularly in the design of scramjet engines. For liquid fuel propulsion system, the combustion efficiency depends strongly on the breakup, atomization, and evaporation of liquid fuel jet.1–5 However, the complexity of mechanism in these physical processes makes it difficult for detailed analysis from a theoretical point, especially in compres-sible regimes.

Extensive experimental investigations have been car-ried out previously to study the spray structures.6–8 Kush and Schetz9used water as the liquid jet normally injected into a supersonic gas crossflow and found the liquid surface layers appear to be different in size,

shape, and depth with varying jet-to-air momentum flux ratio. Schetz et al.10employed photographic obser-vations to study the jet column breakup mechanism and found that the formation and behavior of the waves surrounding the jet column are insensitive to large variations in surface tension and viscosity. Lin et al.11investigated the structures of liquid jets injected vertically into a Mach 1.94 crossflow experimentally and used Phase Doppler Particle Analyzer (PDPA) measurement to establish distributions of droplet and spray plume properties. They found the normalized dis-tribution profiles for droplet size and droplet velocity

Department of Engineering Mechanics, Tsinghua University, Beijing, China

Corresponding author:

Yincheng Guo, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China.

Email: guoyc@tsinghua.edu.cn

exhibit S and mirrored-S shapes, respectively, which are caused by the presence of the bottom floor. Perurena et al.12 conducted an experimental investigation of a water jet into a crossing Mach 6 air flow, penetration heights for different shaped injectors were compared, and the liquid fracture was shown to be influenced by the bow shock–separation shock interaction. Based on the previous works, a jet plume model has been devised, and the schematic of the liquid jet into a high-speed crossflow is shown in Figure 1.

However, limited numerical investigations have been seen in the literatures, which is due to the complexity of the problem both in physical and theoretical aspects. There are two common approaches to investigate the two-phase flow problem. One is the Lagrangian approach, which takes the droplets as mass points, and traces the droplets to obtain their locations and veloci-ties individually. The other one is the Eulerian approach, which is a two-fluid model, and the droplet phase plays a role as continuum fluid. Comparing with the former approach, the Eulerian approach solves the governing equations in the Eulerian frame, which could avoid massive cost of computation for the droplet tra-jectories. The Eulerian approach has been widely used in simulations of incompressible regimes.13–17 Nevertheless, for modeling compressible multiphase flow, unlike the Navier–Stokes equations of the gas phase, the droplet phase governing equation system is found to be non-hyperbolic, which will lead to an ill-posed problem and produce non-physical results.18–21 In order to obtain a well-posed equation system, the kinetic theory of granular flow22–25is introduced in the current study.

The kinetic theory of granular flow is on the basis of kinetic theory of non-uniform dense gas as described by Chapman and Cowling.26The main idea of the the-ory is to treat the particles in a dense state as conti-nuum fluid and apply the analogy between particles in granular flow and molecules in dense gas flow. The velocity distribution of a single particle is assumed to follow the Maxwellian distribution, and a pseudo tem-perature for the particle phase is derived to describe the

random motion of the particles, which is called granu-lar temperature27with respect to the thermal tempera-ture in gas phase. In particular, an equation of state is derived from the kinetic theory and modifies the char-acteristics of the governing equations of the particle phase. For the liquid injection in this article, the jet is supposed to be atomized into dense droplets plume. By assuming the breakup and atomization processes are completed quickly after the jet enters the supersonic flow, the kinetic theory is implemented consequently in the simulation of droplet phase.

The scope of this article is to construct a well-posed equation system for the droplet governing equations based on the kinetic theory and carry out the numerical simulation of transverse liquid jet to a supersonic cross-flow. For turbulence modeling of the two phases, the

k-emodel was applied to the gas phase with

compressi-ble correction,28 while the one-equationkpmodel was

used to describe the turbulence in droplet phase.24 Characteristics analysis was employed to investigate the well-posedness of the governing equation system, and the hyperbolic nature based on the kinetic theory was then demonstrated. As the authors know, it is the initial work to extent the kinetic theory withkp

turbu-lence model into the simulation of compressible multi-phase flow. The penetration height of the liquid jet from the numerical results was compared with the empirical correlations derived from the experimental data, and good agreement was obtained. The predicted distribution of the turbulent kinetic energy of both phases indicated the shear layer is the main mixing area for the liquid jet and the supersonic crossflow air. In addition, the jet-to-air momentum flux ratio showed to be important in predicting the penetration height, while the influence of the droplet diameters on the numerical results was slight.

Methodology

Governing equations for gas phase

In this work, the Navier–Stokes equations in conserva-tive form for the compressible gas phase were solved, and the expressions are presented as follows:

Gas phase continuity equation

∂

∂_t agrg

+ ∂

∂_xj(agrgvgj) =

0 _ð1Þ

Gas phase momentum equation

∂

∂_t agrgvgi

+ ∂

∂_xj(agrgvgivgj) = ag ∂pg

∂_xi + ∂tij

∂_xj +fgpi

ð2_Þ

Gas phase total energy equation

∂

∂_t(agrgEg) + ∂

∂_xj agrgvgj Eg+ pg agrg

!

" #

=∂vgitij

∂_xj ∂

∂_xj lg ∂_Tg

∂_xj

Qgp

ð3Þ

where the subscriptgrepresents the gas phase, and ag,rg, andvgare the volume fraction, density, and

velo-city of the gas, respectively. The total energy is given by

Eg=eg+ 1

2v

2

gi+k ð4Þ

andkis the turbulent kinetic energy. Besides,pgandTg

are the gas phase pressure and thermal temperature, which follow the equation of state

pg= (g1) Eg 1

2v

2 gik

ð5Þ

and g is the ratio of specific heats for the gas phase. The stresstijis given by

tij=m_g ∂vgi

∂_xj + ∂_vgj

∂_xi

dij2 3mg

∂_vgk

∂_xk ð6Þ

where mg is the gas viscosity coefficient. In equations

(2) and (3),fgpiandQgpare the interaction source terms

coupling the two phases, which will be discussed in sec-tion ‘‘Interacsec-tion modeling.’’

Governing equations for droplet phase

The kinetic theory introduces the concept of granular temperature Y _{used to describe the droplet random} motion as molecules in the gas phase.26,27Based on the kinetic theory,29the governing equations of the droplet phase can be expressed in conserved form in the Eulerian frame as follows:

Droplet phase continuity equation

∂

∂_t aprp

+ ∂

∂_xj aprpvpj

=0 _ð7Þ

Droplet phase momentum equation

∂

∂_t aprpvpi

+ ∂

∂_xj aprpvpivpj

= ap

∂_pg ∂_xi +

∂_Pi,j

∂_xj fgpi

ð8_Þ

Droplet phase granular temperature equation

∂

∂_t aprpY

+ ∂

∂_xj aprpvpjY

=2

3Pi,j

∂_vpj ∂_xi +

2

3

∂

∂_xj GY ∂Y

∂_xj

2₃v

ð9_Þ

Droplet phase energy equation

∂

∂_t(aprpC

p pTp) +

∂

∂_xj(aprpvpjC

p pTp)

= ∂

∂_xj C

p p

m_p sT

∂_Tp ∂_xj

+Qgp

ð10_Þ

where the subscriptprepresents the droplet phase, ap,

rp, vp, andTpare the volume fraction, density, velocity

and thermal temperature of the droplet, respectively, andCp_pis the droplet specific heat capacity. In addition, Pi,jis the shear stress of droplet phase

Pi,j= ppdi,j+zpdi,j

∂_vpj ∂_xj

+m_p ∂vpi

∂_xj + ∂_vpj

∂_xi

2₃di,j

∂_vpj ∂_xj

ð11Þ

wheremp and zp are droplet phase viscosity and bulk

viscosity, respectively. It should be noted that accord-ing to the kinetic theory,zpcan be expressed as

zp= 4

3a

2

prpdpg0ð1+bÞ

ffiffiffiffi

Y

p

r

ð12Þ

andmpis defined in dilute and dense phase conditions,

respectively

Droplet phase dilute viscosity

m_p,dilute=5

ffiffiffiffi

p p

96 rpdpY

1=2

ð13Þ

Droplet phase dense viscosity

m_p,dense= 10_r

pdp

ffiffiffiffiffiffiffiffi

pY p

96 1_{ð +bÞg}0

1₊4 5

1_+b

ð Þg0ap

2

+4

5a

2

prpdpg0ð1+bÞ

ffiffiffiffi

Y

p

r ð14Þ

pp=aprp 1+2 1ð +bÞapg0

Y _ð15_Þ

with g0andbrepresenting radial distribution function

and the coefficient of restitution

g0=

3

5 1

ap ap,max

13

" #1

ð16Þ

Note that g0should reach infinite when the volume

fractionapequals the package limitap,max, which is 0.6

in the present study. When the droplet phase is in dilute condition and g0then approaches 0, the ‘‘kinetic

pres-sure’’ is obtained as described by Gidaspow29

pp=aprpY ð17Þ

The coefficient of restitution b ranges from 0 to 1 depending on the collision regime, and set to be 0.9 in the present study. In equation (9), GYis the transport

coefficient of the granular temperature, which is simi-larly defined for the dilute and dense conditions respec-tively as

Droplet phase dilute granular conductivity

GY_,dilute=

75 ffiffiffiffi

p p

384 rpdpY

1=2

ð18Þ

Droplet phase dense granular conductivity

GY_,dense=

150_r pdp

ffiffiffiffiffiffiffiffi

pY p

384 1_{ð +bÞg}0

1₊ 6

5ð1+bÞg0ap

2

+2_a2

prpdpg0ð1+bÞ

ffiffiffiffi

Y

p

r ð19Þ

and v is the dissipation term of the granular temperature

v=3 1 _b2_a2 prpg0Y

4 dp ffiffiffiffi Y p r

∂_∂vpk

xk

!

ð20_Þ

wheredpis the droplet diameter.

Interaction modeling

In this study, two-way coupling was established through the source terms in the governing equations between the gas phase and droplet phase, including the momentum exchange and heat transfer. The viscous drag force is caused by the velocity slip of the two phases and expressed in terms of the mechanical dro-plet response time trp, which is also called droplet

motion relaxation time

f_gp=aprp

trp (vgivpi) andtrp=

d2 prpm 18_m

g 1

CD 24

Rep ð 21_Þ

where rpm is the material density of the droplets, and

CD is the drag coefficient consisting of three separate

correlations based on the range of the droplet Reynolds numberRep

CD=

24₌Rep Rep\1 24_=Rep 1₊1

6Re 2=3

p

1_Rep1000_andRep=rgjvgvpjdp

mg

0_:44 _Rep.1000

8

> <

> :

ð22Þ

In equation (3), Qgp represents the heat transfer

between the two phases per unit volume, and the expression forQgpis30

Qgp=aprppdpNuplg(TgTp) ð23Þ

where the correlation of droplet Nusselt number is given by

Nup=2:0+0:6Re 0:5 p Pr

0:33

ð24_Þ

andPris the Prandtl numberPr=m_gCg p=lg.

Turbulence modeling

Turbulence of the two phases was considered in the cur-rent simulation. The two equationk-eturbulence model

with Wilcox compressibility correction was used for the gas phase.28 In order to incorporate the turbulence of the droplet phase, the one-equation kp turbulence

model was employed, while kp denotes the turbulent

kinetic energy of the droplet phase, and the concept is consistent with the turbulent kinetic energy of the gas phase. Therefore, the combined k-e-k_p turbulence

model24was presented for the compressible multiphase flow in this article.

Droplet phase turbulent kinetic energy

∂

∂_t(aprpkp) + ∂

∂_xj(aprpvpjkp)

= ∂

∂_xj

m_p sp

∂_kp ∂_xj

+Gpkaprpep

ð25Þ

where the production and dissipation term in the turbu-lent kinetic energy equation are given by

Gpk=mp

∂_vpi ∂_xj +

∂_vpj ∂_xi

_∂

vpi

∂_xj and

e_p=

2 trp cp ffiffiffiffiffiffiffi kkp p kp

ð26_Þ

and the turbulent viscosity is defined as

m_p=aprpcmpk 0_:5 p k

1_:5

The empirical constants for the two-phase governing equations above are listed in Table 1 It should be men-tioned that the empirical constants were chosen in the range of common values as presented in the previous studies,15,24,25where the constants have been examined with the experimental data.

Characteristics analysis

As in the compressible gas phase, the equation of state in the kinetic theory connected the granular tempera-ture with pressure in the droplet phase. Therefore, the continuity equation, momentum equation, and granu-lar temperature equation of the droplet phase were combined and solved simultaneously in the numerical simulation. Since the gas phase governing equations have already been demonstrated as a hyperbolic, thus well-posed system,31 characteristics analysis was employed to investigate the well-posedness of the dro-plet phase governing equations.

To be concise, the hyperbolic characteristic in x

direction of the governing equations was chosen to be demonstrated for representative in the present study, and the same analysis could be easily extended into the other two dimensions. Assuming the conservative vari-ables of the droplet phase are included inW_{p, and the} convection terms of the governing equations are included inF_p, the expressions are then given as

W_p= aprp aprpup aprpvp aprpwp aprpY

T

ð28_Þ

and

F_p= aprpup aprpupup+pp aprpvpup aprpwpup aprpYup

T

ð29Þ

Note that the pressure in the droplet phase shear stress in equation (8) is displaced to the convection term, and in this way, the characteristics analysis for the droplet phase governing equations can be carried out subsequently. By applying the chain rule to the derivative of the convection termF_{pwith respect to the} conservation variableW_p

∂F_p

∂_x = ∂F_p

∂W_p

∂W_p

∂_x ð30Þ

the terms on the left side of the equations (7)–(9) can be expressed in form of the Burgers’ equation

∂W_p

∂_t +A Wp ∂Wp

∂_x =0 ð31Þ

and the coefficient matrixA₍W_{p) is the Jacobian matrix} of the equation system

A W _p

= ∂Fp

∂W_p =

0 1 0 0 0

u2

p 2up 0 0 Rp

upvp vp up 0 0

upwp wp 0 up 0

Y_up Y 0 0 _u

p

2

6 6 6 6 4

3

7 7 7 7 5

ð32_Þ

where Rp=1+2(1+b)apg0, and up, vp, wp are the

velocities in three coordinate directions. The major rea-son of characteristics analysis is that the choice of numerical method and the stability of the equation sys-tem are intimately connected with the nature of the partial differential equations, which has been described by the Lax Equivalence Theorem.32This theorem states that the stability is both a necessary and a sufficient condition for convergence if a finite difference approxi-mation to a well-posed linear initial-value problem is consistent. Von Neumann analysis also predicts that no stable numerical scheme can be found at least for the linear system, since the partial differential equations are not hyperbolic or ill-posed.29

Based on the concept of diagonalization, a hyper-bolic system is often defined as a system with real eigenvalues and diagonalizable coefficient matrix.31 Therefore, diagonalization of the Jacobian matrix was tried to be conducted, and five real eigenvalues were obtained then

l1=u_pa_p, l2=u_p+a_p, l3=l4=l5=u_p ð33Þ

where ap=

ffiffiffiffiffiffiffiffiffi RpY

p

is called as ‘‘sound speed’’ of the droplet phase, and it is a function of the granular tem-perature. It is interesting to note that the expression of the ‘‘sound speed’’ of the droplet phase is similar with the sound speed in the gas phase, which is also propor-tional to the square root of temperature. Through the above analysis, the equation system of droplet phase is demonstrated a hyperbolic system, and a well-posed problem for droplet phase based on kinetic theory can be constructed.

The governing equations of the gas and droplet phases were solved by finite differential method. The inviscid flux for the gas phase was computed using Roe-type Riemann solver. For the droplet phase, because of the application of kinetic theory, hyperbolic nature of the governing equation system was obtained, and the same numerical solver can be applied to the inviscid flux as in the gas phase. The face-states on the computational grid were reconstructed by Monotone Upstream-Centred Schemes for Conservation Law (MUSCL) method. To avoid un-realistic solutions, a simple modified Harten formulation was applied to

Table 1. Empirical constants in the governing equations.

sT sp b cp cmp

satisfy the entropy condition for the Roe scheme.33For the viscid flux, second-order accurate central difference scheme was used. A two-step Runge–Kutta method with second-order precision was applied to integrate the equations temporally. To ensure the temporal con-vergence, the time step is set to be 1.0 31028

s. Considering the Courant-Friedrichs-Lewy (CFL) con-dition, it is a quite small integration time step for expli-cit time marching in transient simulations.

Results and discussion

System description

The test domain is based on the experimental facility in the Virginia Tech supersonic blow down wind tunnel.34 The experiment was conducted under the inflow Mach number 3.0 configuration. The simulations in the pres-ent study were carried out under two-dimensional

con-dition, and the computational domain was

0.2 m₃ 0.18 m as shown in Figure 2. The supersonic air flow entered into the test section in the direction parallel to the flat plate, and water was used in the experiment as the transverse liquid jet, which was injected from an orifice located 80 mm downstream of the leading edge with its diameter d0 of 2.03 1023m.

A grid system of 251₃121 was used with refinement near the injection orifice and the flat plate wall, and the size of the minimum grid cell is 5 ₃1024

m as shown in Figure 3.

In the Eulerian approach, the uniform droplet size is used since only one droplet size can be calculated by one set of droplet phase governing equations. According to the liquid spray experiment by Less and Schetz,35the diameter of the droplets after atomization process mostly varies from 7₃ 1026

m to

100₃1026

m. In this article, the breakup and atomiza-tion of the liquid jet were assumed to be completed quickly in the supersonic crossflow.11The breakup and atomization processes were ignored and the uniform droplet diameter according to the liquid spray experi-ment was used in the present simulations, which was assumed to be 30₃1026

m. The effect of the droplet diameter on the numerical results will be discussed later in section ‘‘Effect of the momentum flux ratio and dro-plet diameter on penetration height.’’ The inflow stag-nation pressure in the experiment maintained at 4.5₃ 1056 2% Pa and the stagnation temperature was close to the ambient air. The primary parameter used in injection investigations is the jet-to-air momen-tum flux ratio

q=r_jv2 j=r‘v

2

‘ ð34Þ

where r_j and r_‘ are the density of the injected water and the inflow air, respectively. In the present study,

the computation was conducted under the experimental conditionq= 6, and the corresponding mass flow rate of the injected water was maintained at 0.09 kg/s as in the experiment. The inlet boundary was treated as a supersonic inflow condition, where the variables in the computation are given directly. The variables on the outlet and upside boundary were obtained through extrapolation. The bottom wall was set to be a no-slip boundary condition.

Penetration height

Since the penetration height of fuel jet injection into the combustor has great effect on the diffusive mixing of fuel and freestream air, which is needed for the pre-conditions of chemical reaction, many studies have been devoted to formulate empirical correlations for the liquid jet’s penetration height. The expressions for the penetration height are commonly described as

Figure 2. Schematic of the liquid injection into the supersonic crossflow.

functions of the momentum flux ratio, spatial distance from the orifice, and orifice diameter, such as12,35

h d0

=1_:15_{ð Þq}0:5_lnð1₊6_x=d0Þ ð35Þ

h d0

=3_:75_{ð Þq}0:414 x

d0

0:239

ð36Þ

h d0

=4_:73_{ð Þq}0:3 x

d0

0:3

ð37Þ

h d0

=3_:94_{ð Þq}0:47 x

d0

0:21

ð38Þ

Different definitions of the penetration height in experiments can be found in the previous investiga-tions.8,11,12,34,35 In this article, the penetration height was obtained by the liquid jet plume boundary based on critical density of 0.1 kg/m3 of the droplet phase. Since the density of droplet phase can be obtained through the droplet phase continuity equation (7), the critical density of the liquid plume boundary is then eas-ily recognized in the numerical results. Instead of track-ing the droplets individually in Lagrangian approach, for the Eulerian approach, the number density of the droplets in each grid cell can be represented by the den-sity of the droplet phase. Therefore, the statistical counting process for droplets in each grid cell interacted with the gas phase is unnecessary, avoiding the compu-tation cost increase and the resulted statistical conver-gence problems. It is very important to demonstrate the grid independence for the numerical simulation.36 Simulations were carried out on three different sets of grid systems, namely 2313 101, 2513 121, and 2713141. The comparison of the numerical results with the empirical correlations for the penetration height is shown in Figure 4(a) obtained with the grid set 251₃121. It is noted that the numerical result by the current two-fluid model is well matched with the empirical correlations. The injected liquid column

maintains its direction at first. After about 8d0

trans-verse distance from the orifice, the jet starts to incline, and the curves of penetration height turn to the inflow direction, with the progressive increasing of jet penetra-tion into the main flow slowing down. In Figure 4(b), the penetration heights resulted from the three grid sets are compared for grid independency analysis. It can be observed that the penetration height predicted is a little higher with coarse grid. However, the overall results do not change appreciably with the grid refinement. The grid set of 2513121 is then employed in the following numerical simulations.

Boundary layer separation

In the present study, time evolutions of the transverse liquid jet to the supersonic flow were obtained. Figure 5 shows the time evolutions of the Mach number con-tours from 0.4₃1023

s to 6.4₃ 1023

s. It is interesting to note that the boundary layer detachment, together with the bow shock and separation shock, is well described using the two-fluid model. Due to the bound-ary layer on the flat plate, the supersonic freestream is compressed when it enters into the computational domain, resulting in the existence of the leading edge shock near the inlet boundary. When the injected water penetrates into the supersonic flow, strong drag force acts on the high-speed air due to the high velocity slip between the two phases, and adverse pressure gradient is formed in front of the injection column, which results in the boundary layer separation and gives rise to the separation shock. Note that, initially, the separation region is small, and the separation shock and bow shock are weak. As the water injected into the main flow, the interaction force of the two phases becomes stronger, and alterations in pressure distribution near the injection are induced. The separation point is gradu-ally moving upstream and makes the separation shock inclined, creating a further change in bow shock angle. The continuous adjustment of the shock structure is

called whipping phenomena12 and has been presented numerically by the current simulation. In addition, it can be observed that the leading shock behaves weak compared with the bow shock and separation shock, and its interaction with the bow shock has little effects on the flow structures. Actually, in the present study, when the boundary layer separation develops further toward the inlet as shown in Figure 5(a)–(e), the leading shock wave was covered up by the induced separation shock wave. With the development of time, as shown in Figure 5(f)–(i), the difference of the time evolutions of

the flow structures in the Mach number contour can be found very small, indicating the numerical solution is converged.

As discussed above, separation of the boundary layer can be seen in front of the jet column. For a close inspection, the vector contour of the gas phase velocity is shown in Figure 6. Due to the separation, a circula-tion zone is formed, and the velocity of the gas phase decreases to subsonic state. The height of the circula-tion zone is observed to be about 8d0, the same as the

inclination point of the jet penetration curves shown in

Figure 5. Time evolutions of the Mach number contours: (a) 0.431023_{s, (b) 0.8}

31023_{s, (c) 1.6}

31023_{s, (d) 2.4}

31023_s,

(e) 3.231023_{s, (f) 4.0}

31023_{s, (g) 4.8}

31023_{s, (h) 5.6}

31023_{s, and (i) 6.4}

Figure 6, indicating that the reduced velocity of the gas phase caused by the boundary layer separation could benefit the penetration height.

Since the liquid fuel droplets can be captured by the recirculation zone, the pre-mixing of the fuel with air may occur in this area. Furthermore, the low-speed flow in the recirculation zone provides the mechanism for stabilizing the combustion flame, as in the scramjet combustor where liquid fuel is injected into the superso-nic crossflow. Figure 7 shows the droplet phase density distributions in different streamwise locations of the recirculation zone, namelyx1= 0.065 m,x2= 0.070 m,

andx3= 0.075 m, respectively. The present simulation

obtains the droplet phase entraining into the recircula-tion zone in front of the liquid jet. The predicted dro-plet phase density increases when the streamwise location approaches the jet, and the captured droplets show to concentrate near the bottom wall. Although the captured droplets are limited, the existence of the

pre-mixing process plays an important role in holding the flame in supersonic freestream condition.

It can be expected that the liquid injection in the supersonic flow should result in great changes in pres-sure distribution in the flow field. Figure 8 shows the pressure profiles in differentylocations along the direc-tion of the supersonic inflow. The adverse pressure gra-dient begins with the separation shock and maintains high pressure in the circulation zone as exhibited on the wall at y= 0 m, which reveals the reason for the boundary layer detachment. Moreover, the separation shock gives rise to the first increase of pressure at

y= 0.03 m. After that, the pressure increases more rap-idly in the shock interaction point, resulting in the highest value in the pressure field. For pressure distri-bution at y= 0.1 m, the original flow field developed by the inflow condition is not disturbed until the bow shock appears above the jet plume, leading to the peak value in the pressure profile. It can be noted that the shock waves in the flow field may cause strong resis-tance for the inflow air.

In numerical simulations, the resistance of the liquid jet to the supersonic inflow air is exerted by the drag force in the source terms of the momentum governing equations (2) and (8). As a result, the two-way coupling computation will cause a great amount of total pressure loss for the supersonic crossflow. To evaluate the influ-ence of the drag force, the total pressure recovery ratio, defined as the ratio between the total pressurep0in the

internal flow field and the initial inflow total pressure

p0, total, has been employed in this study. Figure 9

shows the contours of the total pressure recovery ratio. It can be found that there is a large total pressure drop when the supersonic inflow air encounters the liquid jet. In addition, the induced bow shock and separation shock also cause the total pressure loss. Figure 10 shows the distributions of the total pressure recovery ratio along y direction in three different streamwise locations, namely, x1= 0.01 m, x2= 0.10 m, and

x3= 0.18 m. In thex1location, there is nearly no total Figure 6. Vector contours of the gas phase velocity.

Figure 7. Droplet phase density distributions in the recirculation zone.

pressure loss. However, for locationsx2andx3behind

the liquid jet, the total pressure recovery ratio decreases to 0.05 in the lower part of the spray plume. The

contributions of the bow shock to the total pressure loss can be also recognized easily in the distributions with a sudden drop identified in the profiles.

Turbulent kinetic energies for two phases

Figure 11 shows the turbulent kinetic energies for the air flow and liquid jet. It can be seen that the distribu-tion of the turbulent kinetic energies of the two phases is different. The intensity of the gas turbulence seems to be strong in the circulation zone in front of the jet col-umn and also in the shear layer between the liquid and air stream. This phenomenon can be explained by the large velocity gradient of the gas phase, which appears in these regions and results in strong turbulence there. However, for the droplet phase, the initial part of the jet from the orifice is not disturbed by the gas phase evi-dently. As the jet penetrates further, in the region where the jet starts to incline to the supersonic inflow direc-tion, the turbulent kinetic energy of the droplet phase is increasing. Note that the shear layer is the main area for the momentum exchange between the two phases, and the velocity of the droplet phase experiences great changes, which may account for the intensified turbu-lence of the droplet phase there. Therefore, the shear layer is found to possess great significance in the mixing of the fuel jet with the supersonic air flow.

Effect of the momentum flux ratio and droplet

diameter on penetration height

In the transverse gas injection regimes, there are many crucial parameters that have important impacts on the transverse flow field, such as the jet-to-momentum flux ratio, the geometric configuration of the injection ori-fice, the number of injection oriori-fice, and the injection

Figure 9. Contours of total pressure recovery ratio.

Figure 10. Distributions of total pressure recovery ratio in different streamwise locations.

angle. Among these parameters, the jet-to-crossflow momentum flux ratio implies the energy of the trans-verse jet in comparison with the crossflow. For the gas injection, the jet-to-crossflow momentum flux ratio shows to be a key parameter which influences the trans-verse flow field, including the shapes of the bow shock wave, the separation point of boundary layer, and the O_{-shaped vortices.}37 _{For the liquid injection regime,} the jet-to-crossflow momentum flux ratio shows its sig-nificance by the penetration height of the liquid jet as the empirical correlations (35)–(38) have revealed, which is the most concerned parameter in liquid jet experiments. In Figure 12, the predicted penetration heights of different momentum flux ratio q varying from 3 to 6 are compared. The inflow condition in dif-ferent cases remains the same, while the jet velocity is modified with respect toq. It is shown that the penetra-tion heights are increasing with the momentum flux ratio. The initial states of the injection out of the orifice are similarly perpendicular. However, the jet with larger injection velocity gains more inertia and is sup-posed to penetrate deeper into the main flow, leading to the changes of jet inclination points. After the jets inclination, all of the penetration height curves are almost parallel to each other propagating downstream.

In this article, the breakup and atomization of the liquid jet were assumed to be completed quickly, and uniform diameter of droplets was used in the current simulation. To examine the effect of droplet size on the penetration height, in this section, different diameters of droplet phases were tested. In Figure 13, the pre-dicted penetration heights of the liquid jet with droplet diameter varying from 10₃1026

m to 40₃ 1026

m were obtained under the condition of momentum flux ratioq= 6. It is shown that the larger droplets seem to penetrate a little higher than the smaller ones because of the inertia effect. However, the drag force acted on droplets also increases as the droplets become larger, and the difference in penetration heights appears to be

slight as the droplet diameter further increasing. Therefore, it can be the seen that the uniform droplet diameter assumption has small influence on the numer-ical results.

Conclusions

A liquid jet into supersonic crossflow has been investi-gated numerically using the Eulerian approach. The kinetic theory of granular flow was introduced to simu-late the droplet phase, and hyperbolic nature of the governing equation system for the droplet phase was demonstrated through characteristics analysis. Therefore, a well-posed problem was established for compressible multiphase flow based on the kinetic the-ory. The comparison between the penetration height predicted by the numerical work with the empirical cor-relations derived from experiments was conducted and showed good agreement. The boundary layer separa-tion induced by the adverse pressure gradient in front of the liquid injection was observed with droplet phase captured in the recirculation zone. The whipping phe-nomenon caused by the bow shock and separation shock interaction was also well described numerically. In addition, a great amount of total pressure loss was predicted when the supersonic freestream enters the spray plume. The present simulation shows that the k-e-k_pturbulence model is adaptable in compressible

mul-tiphase flow and gives a good description of the turbu-lent flow field of liquid jet injection. The shear layer between the liquid jet plume and the inflow air is found to be an important area for the momentum exchange and mixing process between the two phases. The effects of jet-to-air momentum flux ratio and droplet diameter on the penetration height were examined in the present study, and the liquid jet shows to penetrate deeper into the supersonic flow with larger momentum flux ratio, while the influence of the droplet diameter is slight.

Figure 12. Droplet penetration height of different jet-to-air momentum flux ratios.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial sup-port for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China under Grant Nos 51176099 and 51390493.

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Appendix 1

Notation

ap sound speed of the droplet phase

cp,cmp empirical coefficients

CD drag coefficient

Cp specific heat capacity

dp droplet diameter

d0 orifice diameter

e internal energy

E total energy

f _{drag force vector}

F convection term vector of the governing equations

g0 radial distribution function

Gpk production term of the turbulent kinetic energy

h penetration height

k turbulent kinetic energy

Nup droplet Nusselt number

p pressure

Pr Prandtl number

q momentum flux ratio

Q heat transfer

Rep droplet Reynolds number

T temperature

up,vp,

wp

droplet velocities in the coordinate directions

v _{velocity vector}

W conservative variable vector of the governing equations

x,y,z coordinate directions

Greeks

a volume fraction

b coefficient of restitution

g ratio of specific heats

GY transport coefficient of the granular

temperature

e dissipation rate of the turbulent kinetic

energy

zp bulk viscosity of the droplet phase

Y _{granular temperature}

l thermal conductivity

m viscosity coefficient

Pi,j shear stress of the droplet phase

r density

sp,sT empirical coefficients tij shear stress of the gas phase

trp droplet response time

v dissipation term of the granular temperature

Suffix

g,p gas and droplet phase