NUMERICAL INVESTIGATION OF LIQUID JET INJECTION INTO A SUPERSONIC CROSSFLOW

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1 NUMERICAL INVESTIGATION OF LIQUID JET INJECTION INTO A SUPERSONIC CROSSFLOW Haixu Liu, Yincheng Guo, Wenyi Lin Deartment of Engineering Mechanics, Tsinghua University, Beiing , China Keywords: two-fluid model, liquid et, suersonic crossflow, enetration height Abstract Numerical study of liquid et inection into a suersonic crossflow was carried out using twofluid model. A k--k turbulence model is emloyed for simulating two-hase comressible turbulent flow. As the authors know, it is the first time to extend the two-hase turbulence model into the comressible multihase flow, desite its commonly alications in the incomressible regimes. Searation of the boundary layer in front of the liquid et was redicted with the induced searation shock wave. A bow shock wave caused by the inection was formed and interacts with the searation shock wave. The redicted enetration height is in good agreement with the exerimental data. In addition, the turbulent kinetic energy of both the gas and drolet hase were resented for comarisons, and the effects of the et-to-air momentum flux ratio and drolet diameters on the enetration height were also examined in this work. 1 Introduction The descrition of liquid et inection into a suersonic rimary flow has become an imortant research area as the develoment of high-seed flight vehicles. Such a rocess is encountered in thrust vector control, external burning on roectiles, and articularly in the design of scramet engines. For higherformance of roulsion, the combustion efficiency deends strongly on the breaku, atomization and evaoration of liquid fuel et [1-4]. The enetration of fuel et inection into the combustor has a great effect on the diffusive mixing of fuel and free stream air, which is needed for the reconditions of chemical reaction. However, the interaction between the two fluid streams is accomanied by boundarylayer searation and recirculation on the wall near the et inection. The comlexity of the mechanism makes it difficult for detailed analysis from a theoretical oint, esecially for the comressible two-hase flow. The liquid inection into a crossflow of air has been studied in numerous investigations reviously [5-7]. Based on these works, a et lume model has been devised, and the schematic of the liquid et into high-seed crossflow is shown in Fig. 1. In front of the inected liquid et, a bow shock is formed due to the resistance of the et, while the adverse ressure gradient causes the boundary layer to detach, resulting in the searation shock. The searation and bow shocks interact and modify the bow shock angle and et flow direction, which is called whiing henomena. The aearance of the searation shock is of imortance in cases involving fuel et mixing with the air and the combustion considering its flame-holding caability. As the et enetrates into the crossflow, the et is deflected and subected to wavelike disturbances, yielding the rimary breaku in the form of clum detachment. After that, the secondary breaku gives rise to smaller drolets under action of the aerodynamic forces of the crossflow. There are several arameters, such as drolet size, drolet velocity, and liquid volume flux distribution to be considered in the investigations, and one of the global arameters that indicates the mixing condition of the inected liquid with the freestream air is the enetration height of the liquid et, which is also 1

2 HAIXU LIU, YINCHENG GUO, WENYI LIN needed for the reconditions of chemical reaction. Bow shock Leading edge shock Shock interaction Searation shock Boundary searation Jet inection Breaku and atomization Boundary layer Fig. 1 Schematic of liquid et inection into suersonic flow. Many exerimental studies have been devoted to the liquid et s enetration height into suersonic flow, considering the imortance related to the mixing erformance and combustion efficiency. Several emirical correlations have been formulated exerimentally to redict the enetration height of the liquid et into subsonic or suersonic crossflows [8-12]. However, limited numerical investigations are seen in the literatures, which is an efficient way for the study of comressible two-hase flow. There are mainly two aroaches to investigate the two-hase flow roblem. One is the Euler-Lagrange method, which focuses on a single article traectory, and obtains the flow field after the comutation of interaction between the fluid and all of the articles, resulting in a lot cost of simulation time. The other one is the Euler-Euler method, which views all the articles as a continuum fluid hase, and averages out motion on the scale of individual articles, and it enormously reduces the simulation cost. The Euler-Euler method has been widely used in simulations of incomressible two-hase flow [13, 14]. However, for comressible twohase flow, most of the researches ignored the viscosity and fluctuations of the article hase, but the interaction between the two hases is so strong that it is necessary to adat aroriate model to take the turbulence of the article hase into account. In this work, the Navier-Stokes equations in a conservation form for comressible gas flow were solved. The mass, momentum and energy equations of the drolet hase in the Eulerian-Eulerian framework were conveniently exressed corresonding to the gas hase. A k- -k turbulence model was alied in the current turbulence simulation of liquid et inection into a suersonic crossflow. 2 Mathematical Modeling 2.1 Governing equations for gas hase The comressible conservative form governing equations for the gas hase are exressed as follows: Gas hase continuity equation ( v 0, (1 t x Gas hase momentum equation vi ( vv i t x, (2 i ( vi vi x x i r Gas hase energy equation ( E v E t x, (3 vi i T nq x x x where and n are the density of drolet hase and number of the drolets er unit volume, resectively, and the subscrit reresents the drolet hase. The stress i is given by u u i 2 uk i e i e. (4 u u i 3 uk In the resent simulations, the k- model is emloyed for turbulence modeling, and the two equations are given as: Turbulent kinetic energy equation ( k ( u k t x. (5 t k vi i x k x x Turbulent kinetic energy dissiation rate equation 2

3 NUMERICAL INVESTIGATION OF LIQUID JET INJECTION INTO A SUPERSONIC CROSSFLOW ( ( u t x,(6 t v i ( C1 i C2 x x k x where the turbulent viscosity is t, and the effective viscosity is defined as 2 k t C and e T. (7 2.2 Governing equations for drolet hase According to the two-fluid model, the governing equations for the drolet hase are exressed in Eulerian frame of reference, with resect to the gas hase Drolet hase continuity equation ( v, (8 t x x x Drolet hase momentum equation vi i, ( vvi t x x r vi v, (9 x x x i ( vi vi Drolet hase energy equation ( C T ( vc T t x T C x T x, (10 2 T C vcn3 rk x x x C T nq x x Drolet hase turbulent kinetic energy equation ( k ( vk t x k ( Gk G x x gk, (11 where the drolet hase stress i, is defined as: vi v 2 vk i, i, (12 x x i 3 xk and the roduction term of the drolet hase kinetic energy is given by vi v vi Gk (. (13 x x x i As described by Zhou [15], the last source term on the right side of Eq. (11 is the source term due to interaction between the gas hase and the drolet hase through momentum transfer rocess, which is modeled as: 2 Ggk ( C kk k rk. (14 1 ( vi vi x r i The drolet hase turbulent viscosity 0.5 and C k k 1.5 /. ( Interaction modeling between the two hases In this study, two-way couling between the gas hase and drolet hase, including the momentum and heat transfer. The viscous drag force is exressed in terms of the mechanical drolet resonse time r, which is also called drolet motion relaxation time, given as: 2 d ρm 1 24 τ r = 18μ C Re, (16 where C D is the drag coefficient, d is the drolet diameter, m is the material density of the drolets, and Re is the drolet Reynolds number based on the sli velocity between the drolets and the ambient gas flow, v v d Re, (17 The drag coefficient consists of three searate correlations based on the range of Re k, D 3

4 180 mm HAIXU LIU, YINCHENG GUO, WENYI LIN 24 / Re Re 1 1 2/3 CD 24 / Re (1+ Re 1 Re ( Re 1000 In Eq. (3, Q reresents the heat exchange between the gas hase and a single drolet, Q d Nu ( T T. (19 The Nusselt number is defined as: Nu Re Pr, (20 g where Pr is the Prandtl number Pr = C /, and the C and are the secific heat caacity g and the thermal conductivity of the surrounding gas hase, resectively. left side, and the transverse liquid et was inected from the wall. The inector orifice was located 80 mm downstream of the leading edge, with a diameter 2.0 mm. Comutations made here are run on a Cartesian grid, as shown in Fig. 3. Due to the strong interaction of the two hases around the inection orifice, a time ste t was chosen to be 2.e-9s in the current simulation. inlet 200 mm outlet 3 Numerical methods The governing equations of the gas and drolet hase are solved by finite differential method. The inviscid flux for the gas hase is comuted using Roe scheme, and for the drolet hase, because of the non-hyerbolic nature of the governing equation system, the Flux Vector Slitting (FVS method is alied to solve the inviscid flux. The face-states are reconstructed by Monotone Ustream-centred Schemes for Conservation Law (MUSCL method. To avoid un-realistic solutions, a simle modified Harten formulation is alied to satisfy the entroy condition for the Roe scheme [16]. For the viscid flux, second-order accurate central difference scheme is used. A two-ste Runge- Kutta method is used to integrate the equations temorally. 4 Results and discussions 4.1 system descrition The test domain is based on the exerimental facility in the Virginia Tech suersonic blowdown wind tunnel [17]. The exeriments were conducted under the inflow Mach number 3.0 configuration. The geometry of the test section mm is shown in Fig. 2. The suersonic air flow entered into the domain in the direction arallel to the flat late from the 80 mm inection Wall Fig. 2 Schematic of the inection into the suersonic crossflow. The inflow stagnation ressure in the exeriment maintained at 4.5 atm 2% and the stagnation temerature was close to the ambient air about 25. The rimary arameter used in inection investigations is the et-to-air 2 2 momentum flux ratio q v / v, where and are the density of the inected water and the inflow air. In the resent study, the comutation was conducted under the exerimental condition q=6, and the corresonding mass flow rate of the inected water was maintained at 90g/s 1.1% in the exeriment. The inlet boundary condition was treated as a suersonic inflow condition, and the outlet variables were obtained through extraolation. The uside boundary was considered as transarent, and the wall was suosed to be a no-sli boundary. 4

5 y(m NUMERICAL INVESTIGATION OF LIQUID JET INJECTION INTO A SUPERSONIC CROSSFLOW x(m Fig. 3 Comutational grid. 4.2 Penetration height In the resent study, the time evolution of the cross inection into the suersonic flow is obtained. It is interesting to note that the boundary layer detachment, together with the bow shock and the searation shock are well described using the two-fluid model. Fig. 4 shows the consecutive contours of the Mach number. When the inectant enetrates into the suersonic flow, strong drag force due to the high velocity sli between the two hases is acted on the high-seed air, and adverse ressure gradient is formed in front of the inection column, which results in the boundary layer searation and gives rise to the searation shock. The formation of a bow shock attributes to the inection reducing the inflow cross section, and comresses the suersonic flow. Note that, initially, the searation region is small, and the searation shock and bow shock are weak. As the water inected into the main flow, the interaction of the two hases becomes stronger, and alterations in ressure distribution near the inection is induced. The searation oint is gradually moving ustream, and makes the searation shock inclined, creating a further change in bow shock angle and et flow. The continuous vibration of the shock system is called whiing henomena. The enetration height was obtained based on the density of the drolet hase. Fig. 5 shows the time evolution of the inection into the suersonic flow. It can be seen that, the inected liquid column maintains its direction at first. However, as the enetration evolves, the et is deflected due to the drag force and turns to the inflow direction. Several emirical correlations have been formulated to redict the enetration height of the liquid ets into suersonic crossflow. The exressions for the et traectories are commonly described as functions of the momentum flux ratio, the satial distance form the orifice, and the orifice diameter. The emirical correlations obtained by the shadowgrah and PDPA imaging techniques are, resectively. [7,17] h / d 3.75 q x / d, ( h / d 4.73 q x / d, ( h / d 3.94 q x / d. ( (a 0.4 ms (b 0.8 ms 5

6 HAIXU LIU, YINCHENG GUO, WENYI LIN (c 4.0 ms Fig. 4 Time evolutions of the Mach number contours. (a 0.4 ms (c 4.0 ms Fig. 5 Time evolutions of the drolet hase density of the et. The comarison of the numerical results with the exerimental emirical correlations for the et traectories is shown in Fig. 6. The enetration height obtained from the simulation is acquired by the centerline of the drolet hase density. It is observed that the redicted traectory is well matched with the emirical correlations. After about 8d 0 from the orifice, the et starts to incline, and the traectories turn in the inflow direction, while the increasing of enetration into the main flow slows down. However, in the resent study, the breaku of the et is not considered, so discreancies exist in the downstream of the traectories, where the exeriments detected the height of the et increasing more ositive than the numerical results, attributing to the smaller drolets stried from the detaching liquid clums. (b 0.8 ms h/d Exeriment (Nekad and Schetz 1983 Exeriment (Lin et al shadowgrah Exeriment (Lin et al PDPA Numerical results x/d 0 Fig. 6 Traectories of the numerical results with the redictions from the exerimental correlations. 6

7 y(m P (10 3 Pa NUMERICAL INVESTIGATION OF LIQUID JET INJECTION INTO A SUPERSONIC CROSSFLOW 4.3 Boundary layer searation In the rocess of the liquid et enetrating into the suersonic flow, searation of the boundary layer can be seen in front of the et column. The vector contour of the gas hase velocity is shown in Fig. 7. Due to the searation, a circulation zone is formed, and the cross section of the tunnel is reduced for the suersonic inflow. Thus, the resulting searation shock wave is induced. Moreover, the gas hase behind the et has transferred the momentum to the liquid, and the velocity of the gas hase is decreasing. A low seed area resents in the backside of the inection. In the boundary layer searation region, the flow velocity decreases to subsonic state which is imortant to the mixing of the fuel with the air ustream of the inection, and also contributes to the stability of the combustion. The ressure contour is shown in Fig. 8 to illustrate the formation of the adverse ressure gradient leading to the searation of the boundary layer. Note that, the liquid et give rise to the resistance to the inflow air, and the strong interaction between the gas hase and the liquid et increases the ressure in front of the et column. Due to the ressure driven in the boundary layer, the air flows ustream and results in the circulation zone as shown in Fig. 7, and further causes the boundary layer searation. In addition, Fig. 8 gives the interaction oint of the searation shock with the bow shock, where high ressure exists and contributes to the inclination of the et column v m/s Bow shock wave Searation shock wave x(m Fig. 7 Vector contours of the gas hase velocity. Fig. 9 shows the ressure rofiles in different y locations along the direction of the suersonic inflow. It can be seen that the highest ressure in the inection rocess aears in the shock interaction oint. The adverse ressure gradient in the circulation zone is also clearly exhibited on the wall at y=0 mm. Fig. 8 Pressure contours of the gas hase Interaction of the shock waves y=0mm y=30mm y=100mm Circulation zone Bow shock x (m Fig. 9 Pressure rofiles along the inflow direction in different vertical locations. 4.4 Turbulent kinetic energy for two hases Fig. 10 shows the turbulent kinetic energy for the air flow and liquid et. The intensity of the gas turbulence seems to be strong in the circulation zone in front of the et column, and also in the dividing layer between the liquid and the air. This henomenon can be exlained by the large velocity gradient of the gas hase, which aears in these regions, resulting in strong turbulence there. However, for the drolet hase, the initial art of the et from the 7

8 HAIXU LIU, YINCHENG GUO, WENYI LIN orifice is not disturbed by the gas hase evidently. As the et enetrates, in the region where the et starts to incline to the suersonic inflow direction, the turbulent kinetic energy is increasing. It is consistent with the exeriment where the instability of the liquid et is observed and breaku of the et column is suosed to be in this region. It is shown that the enetration heights are increasing with the momentum flux ratio. The initial art of the inection out of the orifice is similarly erendicular. However, the inclination oints of the ets vary, while the ets with larger momentum flux ratios seem to enetrate more deely. After the inclination, all of the et traectories are almost arallel to each other, and roagate downstream h/d q=3 q=4 q=5 q= (a Gas hase x/d 0 Fig. 11 Drolet traectories of different et-to-air momentum flux ratios. Fig. 12 shows the contours of the Mach number resulted from the momentum flux ratio q=2.0 and q=4.0. It is noted the momentum flux ratio also affects the gas hase flow field, in articular the angles of the shock waves. As the enetration height increasing with q, the strength of the bow shock is intensified, and the boundary layer searation becomes larger as the searation oint moves ustream, which causes a stronger searation shock. (b Drolet hase Fig. 10 Turbulent kinetic energy of the gas hase and drolet hase. 4.5 Effects of the momentum flux ratio on the enetration height The et-to-air momentum flux ratio is an imortant arameter that influences the liquid et enetration height. The emirical correlations also reveal its significance in the exeriments. In Fig.11, the redicted traectories of different momentum flux ratios are comared. (a q=2.0 8

9 NUMERICAL INVESTIGATION OF LIQUID JET INJECTION INTO A SUPERSONIC CROSSFLOW Fig. 14 shows the contours of the Mach number resulted from the drolets of d=10 m and d=40 m. The difference of the angles of the shock waves caused by the different enetration heights is shown visibly in the contours. In addition, the smaller drolets are suosed to follow the gas closely to get to the backside of the liquid et, and aear to lead to a lower gas hase velocity zone behind the et. (b q=4.0 Fig. 12 Mach number contours of different momentum flux ratios. 4.6 Effects of the drolet diameter on the enetration height There are different sizes of drolets in the liquid inection because of the atomization and breaku. In this section, different diameters of drolets are tested to examine the relationshi between the enetration heights with the drolet sizes. In Fig. 13, the traectories of the et of drolets varying from 10 m to 40 m are obtained from the simulation. It is shown that the larger drolets seem to enetrate higher than the smaller ones. It indicates that the atomization of the et initially out of the orifice is unfavorable to the et enetration, though it is considered to contribute to the mixing of the liquid with the crossflow. (a d=10 m 16 h/d d=10m d=20m d=30m d=40m x/d 0 Fig. 13 The traectories of different drolets. (b d=40 m Fig. 14 Mach number contours of different drolets. 5 Conclusion Numerical investigations of the liquid et into suersonic crossflow have been conducted using the two-fluid method. The boundary layer searation induced by the adverse ressure gradient in front of the inection is catured, and the bow shock wave and the searation shock 9

10 HAIXU LIU, YINCHENG GUO, WENYI LIN wave are obtained as a result. The enetration height redicted by the numerical work has been comared with the emirical correlations derived from exeriments, and good agreements are obtained. The resent simulation demonstrates that the k--k turbulence model is adatable in two-hase comressible flow and gives a reasonable descrition of the flow field of liquid et inection into suersonic crossflow. The effects of the et-to-air momentum flux ratio and the drolet diameter on the enetration height are also investigated in the resent work, and larger momentum flux ratio and drolet diameter seem to contribute to the et enetrate height in the suersonic crossflow. Acknowledgment This work was suorted by the National Natural Science Foundation of China under Grant No References [1] Rachner M, Becker J, Hassa C, and Doerr T. Modelling of the atomization of a lain liquid fuel et in crossflow at gas turbine conditions. Aeros. Sci. Technol, Vol. 6, No. 7, , [2] Mathur T, Gruber M, Jackson K, Donbar J, Donaldson W, Jackson T, et al. Suersonic combustion exeriments with a cavity-based fuel inector. J. Proul. Power, Vol. 17, No. 6, , [3] Vinogradov VA, Kobigski SA and Petrov MD. Exerimental investigation of kerosene fuel combustion in suersonic flow. J Proul Power, Vol. 11, No. 1, 130-4, [4] Li J, Ma F, and Yang V. A comrehensive study of combustion oscillations in a hydrocarbon-fueled scramet engine. 45th AIAA Aerosace Sciences Meeting and Exhibit, Reno, Nevada, , [5] Edward AK, Joseh AS. Liquid et inection into a suersonic flow. AIAA J., Vol. 11, No. 9, [6] Schetz JA, Kush EA and Joshi PB. Wave henomena in liquid et breaku in a suersonic crossflow. AIAA J., Vol. 18, No. 7, , [7] Perurena JB, Asma CO, Theunissen R, and Chazot O. Exerimental investigation of liquid et inection into Mach 6 hyersonic crossflow. Ex. Fluids., Vol. 46, No. 3, , [8] Kyoung SI, Lin KC, and Lai MC. Sray atomization of liquid et in suersonic crossflows. 43th AIAA Aerosace Sciences Meeting and Exhibit, Reno, Nevada, , 1-9. [9] Lin KC, Kennedy PJ, and Jackson TA. Sray structures of aerated-liquid ets in subsonic crossflows. 39th AIAA Aerosace Sciences Meeting and Exhibit, Reno, Nevada, , [10] Lin KC, Kennedy PJ, and Jackson TA. Structure of aerated-liquid ets in high-seed crossflows. 32th AIAA Fluid Dynamics Conference and Exhibit, St. Louis, Missouri, , [11] Sallam KA, Aalburg C, Faeth GM, Lin KC, Carter CD and Jackson TA. Breaku of aerated-liquid ets in suersonic crossflows. 42th AIAA Aerosace Sciences Meeting and Exhibit, Reno, Nevada, , [12] Lin KC, Kennedy PJ, and Jackson TA. Structure of water ets in a Mach 1.94 suersonic crossflow. 42th Aerosace Sciences Meeting and Exhibit, Reno, Nevada, , [13] Zheng Y, Wan, XT, Qian Z, Wei F and Jin Y. Numerical simulation of the gas-article turbulent flow in riser reactor based on k--k - - two-fluid model. Chem. Eng. Sci., vol. 56, , [14] Enwald H, Peirano E and Almstedt AE. Eulerican two-hase flow theroy alied to fluidization. Int. J. Multih. Flow, Vol. 22, 21-66, [15] Zhou LX. Theory and numerical modeling of turbulent gas-article flows and combustion. Science Press and CRC Press, [16] Kermani MJ and Plett EG. Modified entroy correction formula for the Roe scheme. AIAA, Inc [17] Thomas RH and Schetz JA. Distributions across the lume of transverse liquid and slurry et in suersonic airflow. AIAAJ., Vol 23, No. 12, , Contact Author Address to: guoyc@tsinghua.edu.cn Coyright Statement The authors confirm that they, and/or their comany or organization, hold coyright on all of the original material included in this aer. The authors also confirm that they have obtained ermission, from the coyright holder of any third arty material included in this aer, to ublish it as art of their aer. The authors confirm that they give ermission, or have obtained ermission from the coyright holder of this aer, for the ublication and distribution of this aer as art of the ICAS 2014 roceedings or as individual off-rints from the roceedings. 10