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Large eddy simulation based studies of jet–cavity interactions in a supersonic flow
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Large eddy simulation based studies of jetcavity interactions in a supersonic flow Hongbo Wang, Zhenguo Wang, Mingbo Sun, Ning Qin
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S0094-5765(13)00219-1 http://dx.doi.org/10.1016/j.actaastro.2013.06.029 AA4786
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Acta Astronautica
Received date: 31 January 2013 Revised date: 11 April 2013 Accepted date: 21 June 2013 Cite this article as: Hongbo Wang, Zhenguo Wang, Mingbo Sun, Ning Qin, Large eddy simulation based studies of jet-cavity interactions in a supersonic flow, Acta Astronautica, http://dx.doi.org/10.1016/j.actaastro.2013.06.029 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Large eddy simulation based studies of jet-cavity interactions in a supersonic flow Hongbo Wang1, Zhenguo Wang1,*, Mingbo Sun1 and Ning Qin2 1
Science and Technology on Scramjet Laboratory
National University of Defense Technology, Changsha, 410073, China 2
Department of Mechanical Engineering
University of Sheffield, Sheffield S1 3JD, England, UK *Corresponding author: [email protected] Abstract: Interactions of a cavity flameholder with an upstream injected jet in a Ma 2.52 supersonic flow are investigated numerically. A hybrid RANS/LES (Reynolds-Averaged Navier-Stokes/Large Eddy Simulation) method acting as wall-modeled LES is adopted, for which the recycling/rescaling method is introduced to treat the unsteady turbulent inflow. Patterns of the fluid entrainment into the cavity and escape from the cavity are identified using a scalar-tracing method. It is found that the jet-cavity interactions remarkably enhanced the mass exchange between the fluids in and out of the cavity, resulting in reduced residence time of the cavity fluids. Increasing the distance between the fuel injection and the cavity leading edge tends to attenuate the jet-cavity interactions, leading to weaker mass exchange. Raising the injection pressure appears to enhance the jet-cavity interactions, resulting in a shorter residence time of the cavity fluids. Moreover, the mass decay processes for the fuel and air within the cavity are basically the same while the entrainment processes for the fuel and air into the cavity seem quite different. Keywords: cavity; jet; supersonic; mass exchange
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1. Introduction Supersonic combustion ramjet (scramjet) engine lets the air stream enter into the combustor supersonically and organizes combustion within supersonic flow, where robust flameholding schemes are necessary due to the short combustor residence time. One promising candidate for such a flameholder is the wall cavity which has been shown to be effective in stabilizing the flame without excessively decreasing total pressure[1]. When used as an integrated fuel injection/flameholding approach[2], cavity flameholders have become even more attractive in supersonic combustors and received more and more attention. In particular, flush-wall injection coupled with a downstream cavity flameholder is found to be a simple but efficient approach in maintaining combustion in supersonic flows. Ben-Yakar et al.[3] used high-speed framing schlieren and OH-PLIF (Planar Laser-Induced Fluorescence) to investigate hydrogen normal jet injected upstream of a cavity in air cross-flow simulating flight Mach 10 conditions, where autoignition was achieved and OH fluorescence appeared first in the recirculation upstream of the jet and extended along outer edge of the jet plume. Micka et al.[4-7] investigated the combustion characteristics of a dual-mode scramjet combustor with normal fuel injection upstream of cavity flameholder. It was found that the combustion was anchored at the leading edge of the cavity at low stagnation temperature and stabilized a short distance downstream of the fuel injection jet in the jet-wake at high stagnation temperature. Sun et al.[8, 9] studied the combustion in a supersonic combustor with normal hydrogen injection upstream of cavity flameholders using OH-PLIF and hybrid RANS/LES (Reynolds-Averaged Navier-Stokes/Large-Eddy Simulation). It was shown that an approximately steady flame
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existed in the cavity shear layer and hot combustion products were transported into the injection jet by the vortex interaction of the jet-with-cavity shear layer, where the counter-rotating vortex induced by the jet and the cavity shear layer played an important role. Kim et al.[10] carried out RANS simulations to investigate supersonic combustion with cavity-based fuel injection, where the cavity effect is discussed from a viewpoint of total pressure loss and combustion efficiency. Jeong et al.[11] studied the combustion characteristics of a scramjet engine using hydrogen injection upstream of the cavity and found the cavity acted as a flameholder. The analyses also indicated that the heat release is mostly initiated by the shock wave from the cavity’s trailing face and the ignition above the cavity does not have a strong influence on the downstream combustion. Wang et al.[12] studied the combustion characteristics in a supersonic combustor with hydrogen injection upstream of cavity flameholder both experimentally and numerically. It was observed that the flame or combustion zone spreading from the cavity to the main stream seemed to be dominated not only by the traditional diffusion process but also by the convection process associated with the extended recirculation flows resulting from the heat release and the interaction between the jet and the cavity shear layer. It is evident that the jet-cavity interactions play an important role in the flame holding and spreading processes, as has been pointed out by the previous studies. This is because the jet-cavity interactions to a large extent determine both the fuel transport from the jet into the cavity and the hot products transport from the cavity recirculation region to the fuel jet or to the main stream. Therefore, both the design of an efficient supersonic combustor and the development of a reliable theoretical model require a deep understanding of these interaction processes. However, there is a lack of detailed studies
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on this issue in the existing literatures. The present work numerically investigates the jet-cavity interactions in a supersonic flow, attempting to provide a useful understanding of the involved processes, such as cavity mass escape, fuel entrainment and jet-cavity shear layer interaction. Firstly, the theoretical models and numerical methods are briefly presented and validated. Next, the results are analyzed in detail and conclusions are drawn. 2. Physical models and numerical methods 2.1 Turbulence models LES has been increasingly used to study turbulent flow problems because it is undoubtedly more accurate than RANS in many complex flows, such as non-equilibrium, three-dimensional massively separated flows. However, it is still difficult to use LES in the simulations of wall-bounded flows at high Reynolds numbers due to the high mesh resolution required to resolve the small vortices in the near wall region at high Reynolds numbers. On the other hand, RANS is more suitable for the near wall flows because highly anisotropic meshes can be used to resolve the time-averaged viscous layer with high mesh density only in the wall normal direction. Accordingly, the total grid points required in RANS are much less than that required in LES. In order to combine the advantages of RANS and LES, many hybrid methods were proposed recently. In the present study, a hybrid RANS/LES method[13] blending the S-A RANS model[14] and Yoshizawa sub-grid scale(SGS) model[15] is adopted. The modeling equations are briefly described below. In the one-equation S-A RANS model[14], the eddy viscosity is directly calculated from the transport equation.
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D ρν 1 ∂ ∂ν ∂ν 2 ν2 (1) = Cb1 ρ Sν + [ ( ρ (ν + ν ) ) + Cb 2 ρ ( ) ] − ρ Cw1 f w 2 Dt σ ∂x j ∂x j ∂x j d where ρ is density, ν is molecular viscosity, d is the distance to the nearest solid wall,
f v1 =
χ3 χ +C 3
3 v1
, S=S+
ν κ d 2
2
fv 2 , fv 2 = 1 −
Sij = (∂ui / ∂x j + ∂u j / ∂xi ) / 2 , f w = g (
χ , χ = ν /ν , S = 2 Sij Sij , 1 + χ f v1
1 + Cw6 3 1/ 6 ν ) , g = r + Cw 2 ( r 6 − r ) , r = , 6 6 g + Cw3 Sκ 2 d 2
Cb1 = 0.1355 , Cb 2 = 0.622 , σ = 2 / 3 , Cv1 = 7.1 , Cw1 = Cb1 / κ 2 + (1 + Cb 2 ) / σ , Cw 2 = 0.3 , Cw3 = 2.0 , κ = 0.41 . The turbulent viscosity is obtained as ν t = ν f v1 . The one-equation Yoshizawa SGS model[15] for the LES region is
Dρ k ∂ ∂k = Pk + [ ρ (ν + σ kν t ) ] − Dk Dt ∂x j ∂x j where ν t = Cμ k 1/ 2 Δ , Pk = 2 ρν t Sij Sij , Dk = Cd ρ
(2)
k 3/ 2 , σ k = 1/ Prt . k is the sub-grid Δ
turbulent kinetic energy, Δ is the spatial filtering width, Prt = 0.9 is the turbulent Prandtl number, Pk and Dk are the production and dissipation of the sub-grid turbulent kinetic energy, respectively. Here, the values of Cμ and Cd need to be determined. According to the previous discussion[13], Cμ = 0.02075 and Cd = 1.0 are used in the present work. In order to blend the SGS model with the S-A RANS model, the turbulent kinetic energy transport SGS model is transformed to an eddy viscosity transport model based on the eddy viscosity hypothesis. According to the definition of the eddy viscosity, the k equation can be transformed to that of ν t as below. Replacing k withν t2 /(Cμ2 Δ 2 ) in the k equation, one obtains
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D ρν t 1 ∂ν t ∂ν t 2 Cd ρ ν t2 ∂ 2 2 [ ρ (ν + σ kν t ) ] + ρ (ν / ν t + σ k )( ) − = ρ Cμ Δ Sij Sij + + ρ PΔ 2 2Cμ Δ 2 Dt ∂x j ∂x j ∂x j
(3)
where PΔ represents the additional terms generated by the grid stretching or the non-uniformity of spatial filtering width.
PΔ =
3ν t 4 ∂Δ ∂Δ ∂ν t ν t ∂ ∂Δ (ν + σ kν t )( )2 − (ν + σ kν t ) [(ν + σ kν t ) ] − 2 Δ ∂x j Δ ∂x j ∂x j Δ ∂x j ∂x j
(4)
Keeping f v1 = 1.0 in the LES region, the ν t equation can be written in the form ofν equation usingν t = ν f v1 ∂ ∂ν ∂ν 2 Cd ρ ν 2 D ρν 1 2 2 [ ρ (ν + σ kν ) ] + ρ (ν / ν + σ k )( ) − = Cμ Δ ρ Sij Sij + + ρ PΔ (5) 2 2Cμ Δ 2 ∂x j ∂x j ∂x j Dt Based on the similarity of (1) and (5), these two equations can be blended by using a
blending function F, which is equal to one in the near-wall region and approaches zero for the region far away from the wall. The blended equation can be written as below.
D ρν ∂ ∂ν ∂ν [ ρ (ν + σ v1ν ) ] + ρσ v 2 ( )2 − Dv + (1 − F ) ρ PΔ = ρ Pv + Dt ∂x j ∂x j ∂x j 1 1 where Pv = F (Cb1Sν ) + (1 − F )( Cμ2 Δ 2 Sij Sij ) , σ v1 = F + (1 − F )σ k , σ 2
σ v2 = F
Cb 2
σ
+ (1 − F )(ν /ν + σ k ) , Dv = F (Cw1 f w
ν2 d
) + (1 − F )( 2
(6)
Cd ν 2 ). 2Cμ Δ 2
In order to recover the original SGS model in LES region, one can redefine fv1 as f v1 =
χ3 χ 3 + Γ( F )Cv31
(7)
Γ( F ) is supposed to be 1 in RANS region and 0 in the LES region. In order to make a
fast transition, a steep function is used as below. Γ( F ) = F 20 A blending function F similar to that used by Sánchez-Rocha and Menon[16] is adopted.
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(8)
y ≤ y start ⎧1.0 ⎪ ⎪ ⎪ y ≥ y end (9) F = ⎨0.0 ⎪1 C f 1 (d / d 0 − C f 2 ) ⎪ (1 − tanh( ) / tanh(C f 1 )) else (1 − 2C f 2 )d / d 0 + C f 2 ⎪⎩ 2 where C f 1 = 2.0, C f 2 = 0.2, d = y − ystart , d 0 = yend − ystart , y is the distance to the nearest
wall. ystart and yend are used to adjust the RANS-LES
transition. In the present study,
ystart = 0.1δ inf and yend = δ inf are adopted so that the hybrid model acts as a wall-modeled LES, where δ inf is the boundary-layer thickness of the turbulent inflow. These parameters are chosen to make the RANS-to-LES transition occur in the inner log-law region of the turbulent boundary layer so that large structures in the outer region of the boundary layer and in the free shear layer are resolved by LES. 2.2 Numerical schemes In the present study, the fifth-order WENO scheme developed by Jiang and Shu[17] is used for inviscid fluxes and the second-order centered scheme for viscous fluxes. In view of improving the computing efficiency, the time integration is performed by means of a second-order, implicit dual time step approach, the inner iteration of which is achieved by a lower-upper symmetric Gauss-Seidel (LU-SGS) method. 2.3 Turbulent inflow conditions A recycling/rescaling method[13] is used to treat the turbulent inflow condition, which is considered to be a promising way to prescribe time-dependent turbulent inflow conditions for LES or hybrid RANS/LES of spatially developing turbulent flows[18-21]. In the present study, a method similar to that of Xiao et al.[20] is used, which can be described as follows.
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1) Fix the mean profiles of velocity and temperature at inflow, and extract their fluctuations at the recycle station. Rescale and superimpose the fluctuations onto the mean inflow profiles at every time step. The pressure is fixed during the simulation and the density is given by the state equation. 2) Recycle and rescale both the mean and fluctuating values of k or ν at every time step. In the inner layer, ν is scaled according to the wall law. In the outer layer, where the simulation is basically carried out by LES, ν is scaled according to the law of k , based on the definitionν t = Cμ k 1/ 2 Δ . Then the scaling laws can be described as below. Inner layer: ' rec + ui'inf ( yinf , z, t ) ,in = β ui ' inf
+ Tin = β 2T ' rec ( yinf , z, t ) inf
+ kin = β 2 k rec ( yinf , z, t )
(10)
inf
+ ν in = ν rec ( yinf , z, t )
Outer layer: ' rec ui'inf (ηinf , z , t ) ,out = β ui ' inf
Tout = β 2T ' rec (ηinf , z , t )
(11)
inf
kout = β 2 k rec (ηinf , z , t ) inf
ν out = ( βΔ inf (ηinf , z, t ) / Δ rec (ηinf , z , t ))ν rec (ηinf , z , t ) where
β=
δ rec δ inf
uτ ,inf
=(
δ rec 1/10 ) δ inf
uτ ,rec x −x = [1 + ( rec inf )0.27 6/5 Reδ−inf1/5 ]5/6 δ inf
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(12) (13)
and y + = yuτ /ν w ,η = y / δ , and Δ is the filtering width. uτ ,inf and uτ ,rec are the friction velocities at the inflow and recycling station, respectively. δ inf and δ rec are the corresponding boundary-layer thickness. The complete field can be obtained from weighted averaging of inner and outer values
ϕ = ϕin (1 − w(ηinf )) + ϕout w(ηinf )
(14)
The weighting function is defined as w(η ) = where α = 4 and b = 0.2 .
⎫ 1⎧ α (η − b) ] / tanh(α ) ⎬ ⎨1 + tanh[ 2⎩ (1 − 2b)η + b ⎭
(15)
2.4 Mass-tracing procedure The mass exchange process between the main stream or fuel jet and the recirculation flow within the cavity to a large extent determines the eventual holding and spreading characteristics of the flame[22, 23]. In order to quantify the mass exchange process, the following procedure is adopted. Each simulation is initiated and stopped once the flow reaches a periodic state. At the point, the fluid within the cavity is tagged differently than the fluid in the main stream. In the present study, the fluids beneath the imaginary line that connects the top of the cavity fore wall to the top of the cavity aft wall are tagged as “inner” fluids, while the fluids above this line are tagged as “outer” fluids. The simulations are then restarted and the mass exchange process is monitored as a function of time. It is notable that the tagging process in no way changes the fluid state, it simply serves as a marker for monitoring how the fluids within the cavity interact with the main stream flow. Thus, there are two nominal species for the no-injection flow, the mass fractions of which are denoted by, Yair ,in and Yair ,out . Hence, one additional equation for
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scalar transport of Yair ,in needs to be solved. Similarly, there are four nominal species for the hydrogen-jet-cavity flows studied here, the mass fractions of which are denoted by YH 2 ,in , Yair ,in , YH 2 ,out , Yair ,out . Accordingly, three additional equations for scalar transport
of YH 2 ,in , Yair ,in and YH 2 ,out need to be solved simultaneously. 2.5 Code validation The general theoretical framework has been discussed and testified in the early works[8, 12, 13, 24-27]
. Additionally, a simulation of transverse jet in a supersonic turbulent crossflow,
experimentally studied by Santiago and Dutton[28], Everett et al.[29] and Vanlerberghe et al.[30], is carried out to validate the present numerical models. In order to get a reasonable LES resolution with acceptable computational costs, the present calculation uses a set of flow parameters leading to a lower Reynolds number ( Re D = 2.4 × 104 ) compared with the experiment, similar to that adopted by Kawai et al.[31, 32]. The flow Mach number is M ∞ = 1.6 , density and pressure ratio between the jet and crossflow are ρ 0 j / ρ ∞ = 5.55 and p0 j / p∞ = 8.4 , resulting in jet-to-crossflow momentum flux ratio of J = 1.7 , which is important to determine the jet penetration. The upstream boundary layer thickness, δ = 0.775 D ( D =4 mm), is matched to the experiment at x / D = -5. The schematic of the computational domain is shown in Fig. 1, where the width of the domain in the spanwise direction is 4.4 D and the length of the inflow region L is 9.5 D . The number of grid points is 481×151×151 , which leads to a grid resolution of Δx + ≈ 30, Δz + ≈ 20 and Δy + ≈ 1 ∼ 20 in the focused region.
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Fig.1 Schematic of computational domain for transverse jet.
(a) Experimental PLIF images[30]
(b) Calculation Fig.2 Representative snapshots of jet fluid in centerplane. Figure 2 shows the instantaneous flowfields of the numerical and experimental results. The vortical structures in the windward and leeward jet boundaries, observed in the experiments, are well captured by the present calculation. Figure 3 shows the comparisons of time-averaged velocity distributions between the calculation and experiment at jet downstream locations. The calculated results agree reasonably well with the experimental data except that significant discrepancy is observed close to the wall in
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the region immediately downstream of the jet in the streamwise velocity profiles. The same discrepancy is also observed in the LES of Kawai et al.[31, 32]. The possible reason may be the uncertainty involved in the experimental data and the unmatched Reynolds number used in the calculation.
Fig.3 Comparisons of streamwise and wall-normal velocities between calculation and experiment[28, 31, 32] at jet downstream locations x / D = 2, 3, 4 and 5. 3 Results and discussion 3.1 Geometry and boundary conditions The calculations simulate a model scramjet combustor recently designed at National University of Defense Technology. The combustor is connected to a Ma2.52 nozzle of the air heater by a 180 mm long isolator, and flow conditions at the nozzle exit and fuel jet exit are listed in Table 1. Since the computational region starts from the combustor entrance and the isolator is not simulated, a beforehand two dimensional RANS simulation is used to estimate the boundary-layer thickness at the combustor entrance, based on which a turbulent boundary layer with inflow thickness of δ inf = 2.5mm is adopted, resulting in a Reynolds number of Reδinf = 37539 . Schematic of the computational combustor is shown in Fig. 4. The computational domain has a constant width of 20 mm and height of 40 mm. A cavity with depth D = 8 mm, length-to-depth
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ratio L / D = 7 and aft angle of 45 degree is mounted on the bottom wall. The fuel is injected sonically from a 2-mm-diameter normal injector located s=10 or 30 mm upstream of the cavity leading edge. Three injection conditions Pjet=0.6, 1.2 and 1.8 MPa, resulting in equivalence ratios of 0.095, 0.19 and 0.285 are considered. A no-slip, no-penetration adiabatic condition is imposed at all lower walls, but a slip condition is used at the upper wall to reduce the computational costs. Table 1 : Inflow conditions parameter air stream M∞ T0 , K P0 , MPa δ inf , mm Injection distance s, m m
fuel jet
2.52 1486 1.6 2.5
1.0 300 0.6/1.2/1.8 0.0
-
10/30
Fig. 4 Schematic of combustor model For s=10 mm, the grid points are 513 ×121×121 and 129 × 61×121 for the regions above and inside the cavity, respectively. For s=30 mm, the computation domain is extended 24 mm upstream, and the grid points for the region above the cavity become
609 ×121×121 . These meshes lead to a grid resolution of Δx + ≈ 75, Δz + ≈ 50 and Δy + ≈ 1 ∼ 50 in the turbulent boundary layer and the cavity shear layer, which may be a little coarse for wall-resolved LES but is appropriate for the hybrid method used as wall-modeled LES.
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3.2 Characteristics of jet-cavity interactions In the present study, three injection pressures, Pjet=0.6, 1.2 and 1.8 MPa, together with two injection locations, s=10 and 30 mm, are considered. Here s denotes the distance that measures from the upstream injector to the cavity leading edge. Moreover, the cavity flow without injection is also calculated to clarify the effects of the jet on the cavity flow. First, the case with Pjet=1.2 MPa and s=10 mm is taken as a representative to analyze the basic flowfields and to compare with the no-injection case. Then, the effects of the injection pressure and injection location are addressed. Figure 5 shows the instantaneous flowfields of jet-cavity interactions in supersonic turbulent flow, where typical vortices associated with the transversely injected jet and the cavity flow are well captures. Hairpin-like vortices are evident in the windward jet boundary. Thanks to the formation of these vortices, the main stream air is entrained into the jet boundary and mixed with the jet fluid rapidly, leading to a quick decrease of the hydrogen mass fraction. As observed by Kawai et al.[31, 32], there should be two groups of hairpin-like vortices generated from the windward and leeward portion of the jet, respectively. In the present study, however, the hairpin-like vortices in the leeward portion of the jet are not evident, which may result from the interaction of the cavity shear layer and the leeward portion of the jet. From an average point of view, furthermore, a pair of counter-rotating vortices is formed with the axis approximately locating in the streamwise direction. Besides, there exist vortices in the turbulent boundary layer, cavity shear and cavity recirculation region. It is observed that a portion of the jet may begin to be drowned in the shear-layer vortices from some distance downstream of the cavity leading edge, leading to intense mass and momentum exchange between the jet fluid and
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cavity fluid. Thus the vortex interaction of the jet-with-cavity shear layer may play a key role in the jet-cavity interactions. All these vortices may have significant effects on the jet-cavity interactions, as will be seen below.
(a) Vorticity magnitude and hydrogen mass fraction contours in centerplane and wall-parallel plane close to the wall ( y / δ inf = 0.2 ).
(b) Isosurfaces of the second invariant of velocity gradient tensor Q colored by hydrogen mass fraction. Fig. 5 Instantaneous flowfields of jet-cavity interactions in supersonic turbulent flow.
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(a) No injection
(b) Pjet=1.2 MPa, s=10 mm
Fig. 6 Mass decay process of “inner” fluids. Vorticity magnitude overlapped with inner mass fraction contours; from top to bottom, t ∗ u∞ / δ inf = 0 , 5, 20 and 50.
Figure 6 shows the mass decay process of the fluids within the cavity with and without upstream fuel injection, where the instantaneous vorticity magnitude together with the “inner” mass fraction contours are displayed. The dashed line indicates the position of cavity mouth at each location. For the injection case, the “inner” fluids denote the sum of hydrogen and air that are tagged. In general, the “inner” fluids are well confined in the cavity by the cavity shear layer. Thus, the mass decay characteristics are basically determined by the unsteady behavior of the cavity shear layer. For the cavity flow
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without injection, the confinement of the cavity shear layer can be broken by two processes: the rolling-up of shear-layer vortices due to the development of Kelvin-Helmholtz instabilities and the impingement of the shear layer on the cavity aft wall. The first process is usually slow and weak, and the second one is more violent and determinative. As a result, the mass decay mainly occurs in the aft region of the cavity volume while the mass in the front portion maintains unabated for quite a long time. For the cavity flow with upstream injection, the vortex interaction of the jet-with-cavity shear layer introduces an additional mechanism to break the shear-layer confinement by tearing a gap on the shear layer intermittently. It is evident in the third snapshot in Fig. 6(b) that the “inner” fluids are entrained into the fuel jet at the second streamwise station. The achievement of this entrainment is determined by the unsteady behaviors of both the fuel jet and the cavity shear layer. Due to the oscillations of the entire flowfield, the fuel jet and the cavity shear layer oscillate up and down in the transverse direction. Once they form a close enough coupling during the oscillations, the “inner” fluids beneath the cavity shear layer can be entrained into the fuel jet by the upwash resulting from the counter-rotating vortices of the jet as shown in Fig. 7(c). This process may be important to the flame holding and spreading of the cavity-stabilized combustion as has been analyzed by Wang et al.[12]. Also can be observed is that the introduction of the jet enhances the shear-layer oscillations especially in the aft region of the cavity shear layer, which can accelerate the mass decay of the “inner” fluids. Furthermore, the vortex interaction of the jet-with-cavity shear layer pulls the cavity shear layer further into the mainstream, which tends to open a bigger “slot” around the cavity aft wall as shown in Fig. 7, and thus enhance the mass exchange between the fluids in and out of the cavity.
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Notably, once the hydrogen and air are entrained into the cavity recirculation region, they mix with each other and form an approximately uniform mixture soon due to the intense turbulent stirring and relatively long residence time within the recirculation region. Thus, the mass decay processes of the inner air and inner hydrogen are similar to that of the total inner fluids, as can be seen from their mass decay curves below. Figure 8 shows the entrainment process of “outer” fluids for Pjet=1.2 MPa, s=10 mm, where the instantaneous vorticity magnitude together with the “outer” mass fraction contours are displayed. Since the horseshoe vortex shells some fuel off the jet and into the boundary layer when it is formed around the jet exit. As the boundary layer separates at the cavity leading edge, the partially premixed mixture enters the cavity shear layer. Also, the jet boundary is partially premixed with outer hydrogen and air. Accordingly, a portion of the outer hydrogen and air enters into the cavity as partially premixed mixture, so their entrainment mechanisms are correlated. Similar to the mass decay processes, the entrainment of the outer hydrogen and air into the cavity is basically achieved via three processes. The first one is the rolling-up of shear-layer vortices due to the development of Kelvin-Helmholtz instabilities. The second one is the impingement of the cavity shear layer on the cavity aft wall. The third one is the interaction of the jet-with-cavity shear layer, i.e. interactions of the hairpin-like vortices in the leeward portion of the jet boundary with the cavity shear-layer vortices and the downwash around the jet boundary associated with the counter-rotating vortices as shown in Fig. 7(c).
However, most of
the outer air enters the cavity via the unsteady behavior of the entire cavity shear layer, while most of the outer hydrogen enters the cavity via the interaction of the
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jet-with-cavity shear layer. Therefore, there exist some differences between the entrainment processes of the outer hydrogen and air as will be observed below.
(a) Density gradient magnitude, no injection
(b) Density gradient magnitude overlapped with hydrogen mass fraction contours, Pjet=1.2 MPa, s=10 mm
(c) Hydrogen mass fraction overlapped with streamlines, Pjet=1.2 MPa, s=10 mm. Fig. 7 Time-averaged flowfields.
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(a) Hydrogen
(b) Air
Fig. 8 Entrainment process of “outer” fluids for Pjet=1.2 MPa, s=10 mm. Vorticity magnitude overlapped with outer mass fraction contours; from top to bottom, t ∗ u∞ / δ inf = 0 , 5, 20 and 50. The effects of the injection pressure and the injection distance are addressed. The mass exchange results are analyzed and presented in Figs. 9 and 10. The mass exchange rate
m is the mass of main stream that is gulfed into the cavity per second and the residence time τ is the time that the gulfed main stream resides in the cavity. The relation between them can be denoted as τ = m / m or m = −dm / dt = m / τ , and m is the total mass in the cavity. When calculating the residence time of the cavity, it is found that there is not a single exponential decay curve that can fit the mass decay history very well. As a result, a
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three-segment exponential fit method used by Baurle et al.[22, 23] is adopted here to calculate the residence time. We can see from Fig. 9 that the three-segment exponential fit is very proper to fit the raw data.
Fig. 9 Cavity fluid mass decay history and exponential fit. It is observed that the introduction of upstream fuel jet enhances the mass exchange between the fluids in and out of the cavity and reduces the residence time of the cavity fluids. Moreover, the residence time decreases with increasing injection pressure, indicating stronger jet-cavity interactions. For the highest injection pressure studied here, it is observed that the residence time of the cavity fluids is even reduced by more than 50%, which should be considered for the design of cavity flameholders that must provide long enough residence time to stabilize the combustion. However, as the injection distance away from the cavity leading edge increases, the residence time increases as shown in Fig. 9 and less fuel can be entrained into the cavity as displayed by Fig. 10(a), indicating weakened jet-cavity interactions. The time-averaged vorticity distributions displayed in Fig. 11 clearly show that the introduction of upstream fuel jet distorts the cavity shear layer and this distortion grows with increasing injection pressure or
21
decreasing injection distance. For s=10 mm, the counter-rotating vortices are very strong as the jet passes above the cavity, inducing strong interactions with the cavity shear layer. It is observed that a great portion of the jet is embedded in the cavity shear layer. For s=30 mm, however, the counter-rotating vortices have become rather weak as the jet passes above the cavity, resulting in diminished jet-cavity interactions. Notably, once the hydrogen and air are entrained into the cavity recirculation region, they mix with each other and form an approximately uniform mixture soon due to the intense turbulent stirring and relatively long residence time within the recirculation region. Thus, the mass decay processes of the inner air and inner hydrogen are similar to that of the total inner fluids, as can be seen from the mass decay curves shown in Figs. 9 and 10. However, the entrainment characteristics of the outer hydrogen and air into the cavity are quite different as can be seen in Fig. 10. Once the jet-cavity interactions are enhanced by increasing the injection pressure or decreasing the injection distance, the unsteady behaviors of the cavity shear layer tends to be tuned to enhance the mass exchange process. As has been pointed out above, most of the outer air enters the cavity via the unsteady behavior of the entire cavity shear layer. Thus the entrainment process of the outer air into the cavity is enhanced associated with the enhanced jet-cavity interactions. Most of the outer hydrogen, however, enters the cavity via the interaction of the jet-with-cavity shear layer. Though increasing the injection pressure can enhance the jet-cavity interaction, the jet penetrates further into the main stream at the same time. Therefore, the entrainment process of the outer hydrogen into the cavity is determined by the competition between the jet-cavity interaction and the jet penetration. As a result, it is observed that the entrainment of the outer hydrogen into the cavity for Pjet=1.2 MPa and
22
s=10 mm seems stronger than that for Pjet=1.8 MPa and s=10 mm. Though the mass exchange characteristics may become different when combustion occurs, the present analyses indicate that moderate injection pressure may be beneficial to transport more fuel into the cavity to promote the ignition process under certain conditions.
(a) Hydrogen
(b) Air Fig. 10 Histories of cavity fluid mass decay and main stream entrainment.
23
(a) No injection
(b) Pjet=0.6 MPa, s=10 mm
(c) Pjet=1.2 MPa, s=10 mm.
(d) Pjet=1.8 MPa, s=10 mm
(e) Pjet=1.8 MPa, s=30 mm Fig. 11 Time-averaged vorticity magnitude.
24
Additionally, mixing efficiency and total pressure recovery are also compared in Fig. 12. Quantitative evaluation of the mixing degree can be made by using a mixing-efficiency parameter, which indicates the fraction of the reactant that would react if brought to chemical equilibrium with air. The fraction of the reactant refers to the least-available reactant, air or fuel, depending on whether the mixture is lean or rich. The mixing efficiency is defined as
ηm =
∫ Y ρudA ∫ Y ρudA react
(16)
and
Y Y ≤ Ystoic ⎧ ⎪ (17) Yreact = ⎨ 1−Y ⎪Ystoic (1 − Y ) Y > Ystoic stoic ⎩ where Y is the fuel mass fraction, Yreact is the fuel fraction mixed in a proportion that can react, and Ystoic is the fuel stoichiometric mass fraction. The total pressure recovery is used to evaluate the total pressure loss during the mixing process, which is defined as
Ptrec =
∫ Pt ρudA ∫ Pt ρudA
(18)
∞
Basically, the mixing efficiency decreases with increasing injection pressure. For quite a long distance, the fuel-air mixing can only take place around the interfaces formed between the fuel jet and air while the fuel in the jet core can not readily contact with the air. As the injection pressure increases, the total mass flow rate of the fuel increases. The area of the interfaces that responsible for the fuel-air mixing, however, does not increase as fast as the fuel mass flow rate does. As a result, the mixing efficiency decreases with
25
increasing injection pressure. For Pjet=1.8 MPa, the mixing distance with s=10 mm is actually 20 mm shorter than that with s=30 mm, but the mixing efficiency with s=10 mm tends to approach that with s=30 mm near the exit of the domain. It can be inferred that, if the jet exit for s=10 and 30 mm are located at the same position, the mixing efficiency with s=10 mm will exceed that with s=30 mm. That is, stronger jet-cavity interactions with shorter injection distance are beneficial to enhance the fuel-air mixing. The total pressure loss here is mainly induced by shock waves and injection-related separations. Thus, the total pressure loss tends to increase as the injection pressure is raised or the injection position is shifted upstream.
Fig. 12 Mixing efficiency and total pressure recovery. 4
Conclusions The present study concerns the characteristics of flow over a cavity flameholder
coupled with a fuel jet for scramjet applications. Interactions of a cavity flameholder with an upstream injected jet in a Ma 2.52 supersonic flow are investigated numerically. Patterns of the fluid entrainment into the cavity and escape from the cavity are identified using a scalar-tracing method. Significant conclusions that can be made are as follows.
26
1)
The jet-cavity interactions remarkably enhanced the mass exchange between the fluids in and out of the cavity, resulting in reduced residence time of the cavity fluids. This should be considered for the design of cavity flameholders that must provide long enough residence time to stabilize the combustion.
2)
Increasing the distance between the fuel injection and the cavity leading edge tends to attenuate the jet-cavity interactions, leading to weaker mass exchange. Raising the injection pressure appears to enhance the jet-cavity interactions, resulting in a shorter residence time of the cavity fluids.
3)
The mass decay processes of the inner air and inner hydrogen are similar to that of the total inner fluids due to the relatively rapid hydrogen-air mixing within the cavity recirculation region.
4)
Most of the outer air enters the cavity via the unsteady behavior of the entire cavity shear layer, while most of the outer hydrogen enters the cavity via the interaction of the jet-with-cavity shear layer. The entrainment process of the outer hydrogen into the cavity is determined by the competition between the jet-cavity interaction and the jet penetration. The results indicate that moderate injection pressure may be beneficial to transport more fuel into the cavity to promote the ignition process under certain conditions.
Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant Nos. 50906098 and 91016028, and Fok Ying Tung Education Foundation under Grant No.131055. References
27
[1] [2] [3] [4] [5] [6] [7] [8] [9]
[10] [11] [12] [13]
[14] [15]
Mathur T, Gruber M, Jackson K, Donbar J, Donaldson W, Jackson T, Billig F. Supersonic Combustion Experiments with a Cavity-Based Fuel Injector [J]. Journal of Propulsion and Power 2001 17 (6):1305-1312 Ben-Yakar A, Hanson R K. Cavity Flame-Holders for Ignition and Flame Stabilization in Scramjets: An Overview [J]. Journal of Propulsion and Power, 2001, 17 (4):869-878 Ben-Yakar A, Hanson R K. Supersonic Combustion of Cross-Flow Jets and the Influence of Cavity Flame-Holders [R]. AIAA Paper 99-0484, 1999 Micka D J, Driscoll J F. Reaction Zone Imaging in a Dual-Mode Scramjet Combustor Using CH-PLIF [R]. AIAA Paper 2008-5071, 2008 Micka D J, Driscoll J F. Combustion characteristics of a dual-mode scramjet combustor with cavity flameholder [J]. Proceedings of the Combustion Institute, 2009, 32:2397-2404 Micka D J, Torrez S M, Driscoll J F. Heat Release Distribution in a Dual-Mode Scramjet Combustor - Measurements and Modeling [R]. AIAA Paper 2009-7362, 2009 Micka D J. Combustion Stabilization, Structure, and Spreading in a Laboratory Dual-Mode Scramjet Combustor [D]. The University of Michigan, 2010 Sun M B, Wang Z G, Liang J H, Geng H. Flame Characteristics in a Supersonic Combustor with Hydrogen Injection Upstream of a Cavity flameholder [J]. Journal of Propulsion and Power, 2008, 24 (4):688-696 Sun M B, Wang H B, Bai X S, Wang Z G, Geng H, Liang J H, Liu W D. Experimental and Numerical Study on Flame Stabilization in a Supersonic Combustor with Hydrogen Injection Upstream of Cavity Flameholders [R]. AIAA Paper 2009-5187, 2009 Kim K M, Baek S W, Han C Y. Numerical study on supersonic combustion with cavity-based fuel injection [J]. International Journal of Heat and Mass Transfer, 2004, 47:16 Jeong E, O’Byrne S, Jeung I-S, Houwing A F P. Investigation of Supersonic Combustion with Angled Injection in a Cavity-Based Combustor [J]. Journal of Propulsion and Power, 2008, 24 (6):1258-1268 Wang H B, Wang Z G, Sun M B, Qin N. Combustion characteristics in a supersonic combustor with hydrogen injection upstream of cavity flameholder [J]. Proceedings of the Combustion Institute, 2013, 34:2073-2082 Wang H B, Qin N, Sun M B, Wang Z G. A dynamic pressure-sink method for improving large eddy simulation and hybrid Reynolds-averaged Navier-Stokes/large eddy simulation of wall-bounded flows [J]. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 2012, 226 (9):1107-1120 Spalart P R, Allmaras S R. A one-equation turbulence model for aerodynamic flows [R]. AIAA Paper 92-0439, 1992 Yoshizawa A, Horiuti K. A Statistically-Derived Subgrid Scale Kinetic Energy Model for Large-Eddy Simualtion of Turbulent Flows [J]. Journal of the Physical Society of Japan, 1985, 54 (8):2834-2839
28
[16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
[26] [27]
[28] [29] [30] [31] [32]
Sánchez-Rocha M, Menon S. The compressible hybrid RANS/LES formulation using an additive operator [J]. Journal of Computational Physics, 2009, 228:2037-2062 Jiang G, Shu C W. Efficient implementation of weighted ENO schemes [J]. Journal of Computational Physics, 1996, 126:917-923 Lund T S, Wu X, Squires K D. Generation of turbulent inflow data for spatially-developing boundary layer simulation [J]. J. Comput. Phys., 1998, 140: 233-258 Urbin G, Knight D. Large-eddy simulation of a supersonic boundary layer using an unstructured grid [J]. AIAA Journal, 2001, 39:1288-1295 Xiao X, Edwards J R, Hassan H A, Baurle R A. Inflow Boundary Conditions for Hybrid Large Eddy/Reynolds Averaged Navier-Stokes Simulations [J]. AIAA Journal, 2003, 41 ( 8):1481-1489 Liu K, Pletcher R H. Inflow conditions for the large eddy simulation of turbulent boundary layers: A dynamic recycling procedure [J]. J. Comput. Phys., 2006, 219:1-6 Winterfield G. On processes of turbulent exchange behind flameholders [J]. In Tenth International Symposium on Combustion, 1965:1265-1275 Baurle R A, Tarn C-J, Dasgupta S. Analysis of Unsteady Cavity Flows for Scramjet Applications [R]. AIAA 2000-3617, 2000 Sun M B, Geng H, Liang J H, Wang Z G. Mixing Characteristics in a Supersonic Combustor with Gaseous Fuel Injection Upstream of a Cavity Flameholder [J]. Flow, Turbulence and Combustion, 2009, 82:271-286 Wang H B, Qin N, Sun M B, Wu H Y, Wang Z G. A hybrid LES (Large Eddy Simulation)/assumed sub-grid PDF(Probability Density Function) model for supersonic turbulent combustion [J]. Sci China Tech Sci, 2011, 54 (10):2694-2707 Wang H B, Sun M B, Qin N, Wu H Y, Wang Z G. Characteristics of Oscillations in Supersonic Open Cavity Flows [J]. Flow Turbulence and Combustion, 2013, 90:121-142 Wang H B, Wang Z G, Sun M B, Qin N. Simulations of combustion with normal and angled hydrogen injection in a cavity-based supersonic combustor [J]. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 2013, in press, Santiago J G, Dutton J C. Velocity measurements of a jet injected into a supersonic crossflow [J]. Journal of Propulsion and Power, 1997, 13 (2):264-273 Everett D E, Woodmansee M A, Dutton J C, Morris M J. Wall Pressure Measurements for a Sonic Jet Injected Transversely Into a Supersonic Crossflow [J]. Journal of Propulsion and Power, 1998, 14 (6):861-868 VanLerberghe W M, Santiago J G, Dutton J C, Lucht R P. Mixing of a Sonic Transverse Jet Injected into a Supersonic Flow [J]. AIAA Journal, 2000, 38 (3):470-479 Kawai S, Lele S K. Mechanisms of jet mixing in a supersonic crossflow: a study using large-eddy simulation [R]. AIAA Paper 2008-4575, 2008 Kawai S, Lele S K. Large-Eddy Simulation of Jet Mixing in a Supersonic Turbulent Crossflow [R]. AIAA Paper 2009-3795, 2009
29
► Jet-cavity interactions in a supersonic flow are studied numerically. ► Patterns of fluid entrainment and escape are identified using scalar-tracing method. ► Jet-cavity interactions remarkably enhance the cavity mass exchange. ► Entrainment processes for the fuel and air into the cavity seem quite different.
30
Large eddy simulation based studies of jetcavity interactions in a supersonic flow Hongbo Wang, Zhenguo Wang, Mingbo Sun, Ning Qin
www.elsevier.com/locate/actaastro
PII: DOI: Reference:
S0094-5765(13)00219-1 http://dx.doi.org/10.1016/j.actaastro.2013.06.029 AA4786
To appear in:
Acta Astronautica
Received date: 31 January 2013 Revised date: 11 April 2013 Accepted date: 21 June 2013 Cite this article as: Hongbo Wang, Zhenguo Wang, Mingbo Sun, Ning Qin, Large eddy simulation based studies of jet-cavity interactions in a supersonic flow, Acta Astronautica, http://dx.doi.org/10.1016/j.actaastro.2013.06.029 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Large eddy simulation based studies of jet-cavity interactions in a supersonic flow Hongbo Wang1, Zhenguo Wang1,*, Mingbo Sun1 and Ning Qin2 1
Science and Technology on Scramjet Laboratory
National University of Defense Technology, Changsha, 410073, China 2
Department of Mechanical Engineering
University of Sheffield, Sheffield S1 3JD, England, UK *Corresponding author: [email protected] Abstract: Interactions of a cavity flameholder with an upstream injected jet in a Ma 2.52 supersonic flow are investigated numerically. A hybrid RANS/LES (Reynolds-Averaged Navier-Stokes/Large Eddy Simulation) method acting as wall-modeled LES is adopted, for which the recycling/rescaling method is introduced to treat the unsteady turbulent inflow. Patterns of the fluid entrainment into the cavity and escape from the cavity are identified using a scalar-tracing method. It is found that the jet-cavity interactions remarkably enhanced the mass exchange between the fluids in and out of the cavity, resulting in reduced residence time of the cavity fluids. Increasing the distance between the fuel injection and the cavity leading edge tends to attenuate the jet-cavity interactions, leading to weaker mass exchange. Raising the injection pressure appears to enhance the jet-cavity interactions, resulting in a shorter residence time of the cavity fluids. Moreover, the mass decay processes for the fuel and air within the cavity are basically the same while the entrainment processes for the fuel and air into the cavity seem quite different. Keywords: cavity; jet; supersonic; mass exchange
1
1. Introduction Supersonic combustion ramjet (scramjet) engine lets the air stream enter into the combustor supersonically and organizes combustion within supersonic flow, where robust flameholding schemes are necessary due to the short combustor residence time. One promising candidate for such a flameholder is the wall cavity which has been shown to be effective in stabilizing the flame without excessively decreasing total pressure[1]. When used as an integrated fuel injection/flameholding approach[2], cavity flameholders have become even more attractive in supersonic combustors and received more and more attention. In particular, flush-wall injection coupled with a downstream cavity flameholder is found to be a simple but efficient approach in maintaining combustion in supersonic flows. Ben-Yakar et al.[3] used high-speed framing schlieren and OH-PLIF (Planar Laser-Induced Fluorescence) to investigate hydrogen normal jet injected upstream of a cavity in air cross-flow simulating flight Mach 10 conditions, where autoignition was achieved and OH fluorescence appeared first in the recirculation upstream of the jet and extended along outer edge of the jet plume. Micka et al.[4-7] investigated the combustion characteristics of a dual-mode scramjet combustor with normal fuel injection upstream of cavity flameholder. It was found that the combustion was anchored at the leading edge of the cavity at low stagnation temperature and stabilized a short distance downstream of the fuel injection jet in the jet-wake at high stagnation temperature. Sun et al.[8, 9] studied the combustion in a supersonic combustor with normal hydrogen injection upstream of cavity flameholders using OH-PLIF and hybrid RANS/LES (Reynolds-Averaged Navier-Stokes/Large-Eddy Simulation). It was shown that an approximately steady flame
2
existed in the cavity shear layer and hot combustion products were transported into the injection jet by the vortex interaction of the jet-with-cavity shear layer, where the counter-rotating vortex induced by the jet and the cavity shear layer played an important role. Kim et al.[10] carried out RANS simulations to investigate supersonic combustion with cavity-based fuel injection, where the cavity effect is discussed from a viewpoint of total pressure loss and combustion efficiency. Jeong et al.[11] studied the combustion characteristics of a scramjet engine using hydrogen injection upstream of the cavity and found the cavity acted as a flameholder. The analyses also indicated that the heat release is mostly initiated by the shock wave from the cavity’s trailing face and the ignition above the cavity does not have a strong influence on the downstream combustion. Wang et al.[12] studied the combustion characteristics in a supersonic combustor with hydrogen injection upstream of cavity flameholder both experimentally and numerically. It was observed that the flame or combustion zone spreading from the cavity to the main stream seemed to be dominated not only by the traditional diffusion process but also by the convection process associated with the extended recirculation flows resulting from the heat release and the interaction between the jet and the cavity shear layer. It is evident that the jet-cavity interactions play an important role in the flame holding and spreading processes, as has been pointed out by the previous studies. This is because the jet-cavity interactions to a large extent determine both the fuel transport from the jet into the cavity and the hot products transport from the cavity recirculation region to the fuel jet or to the main stream. Therefore, both the design of an efficient supersonic combustor and the development of a reliable theoretical model require a deep understanding of these interaction processes. However, there is a lack of detailed studies
3
on this issue in the existing literatures. The present work numerically investigates the jet-cavity interactions in a supersonic flow, attempting to provide a useful understanding of the involved processes, such as cavity mass escape, fuel entrainment and jet-cavity shear layer interaction. Firstly, the theoretical models and numerical methods are briefly presented and validated. Next, the results are analyzed in detail and conclusions are drawn. 2. Physical models and numerical methods 2.1 Turbulence models LES has been increasingly used to study turbulent flow problems because it is undoubtedly more accurate than RANS in many complex flows, such as non-equilibrium, three-dimensional massively separated flows. However, it is still difficult to use LES in the simulations of wall-bounded flows at high Reynolds numbers due to the high mesh resolution required to resolve the small vortices in the near wall region at high Reynolds numbers. On the other hand, RANS is more suitable for the near wall flows because highly anisotropic meshes can be used to resolve the time-averaged viscous layer with high mesh density only in the wall normal direction. Accordingly, the total grid points required in RANS are much less than that required in LES. In order to combine the advantages of RANS and LES, many hybrid methods were proposed recently. In the present study, a hybrid RANS/LES method[13] blending the S-A RANS model[14] and Yoshizawa sub-grid scale(SGS) model[15] is adopted. The modeling equations are briefly described below. In the one-equation S-A RANS model[14], the eddy viscosity is directly calculated from the transport equation.
4
D ρν 1 ∂ ∂ν ∂ν 2 ν2 (1) = Cb1 ρ Sν + [ ( ρ (ν + ν ) ) + Cb 2 ρ ( ) ] − ρ Cw1 f w 2 Dt σ ∂x j ∂x j ∂x j d where ρ is density, ν is molecular viscosity, d is the distance to the nearest solid wall,
f v1 =
χ3 χ +C 3
3 v1
, S=S+
ν κ d 2
2
fv 2 , fv 2 = 1 −
Sij = (∂ui / ∂x j + ∂u j / ∂xi ) / 2 , f w = g (
χ , χ = ν /ν , S = 2 Sij Sij , 1 + χ f v1
1 + Cw6 3 1/ 6 ν ) , g = r + Cw 2 ( r 6 − r ) , r = , 6 6 g + Cw3 Sκ 2 d 2
Cb1 = 0.1355 , Cb 2 = 0.622 , σ = 2 / 3 , Cv1 = 7.1 , Cw1 = Cb1 / κ 2 + (1 + Cb 2 ) / σ , Cw 2 = 0.3 , Cw3 = 2.0 , κ = 0.41 . The turbulent viscosity is obtained as ν t = ν f v1 . The one-equation Yoshizawa SGS model[15] for the LES region is
Dρ k ∂ ∂k = Pk + [ ρ (ν + σ kν t ) ] − Dk Dt ∂x j ∂x j where ν t = Cμ k 1/ 2 Δ , Pk = 2 ρν t Sij Sij , Dk = Cd ρ
(2)
k 3/ 2 , σ k = 1/ Prt . k is the sub-grid Δ
turbulent kinetic energy, Δ is the spatial filtering width, Prt = 0.9 is the turbulent Prandtl number, Pk and Dk are the production and dissipation of the sub-grid turbulent kinetic energy, respectively. Here, the values of Cμ and Cd need to be determined. According to the previous discussion[13], Cμ = 0.02075 and Cd = 1.0 are used in the present work. In order to blend the SGS model with the S-A RANS model, the turbulent kinetic energy transport SGS model is transformed to an eddy viscosity transport model based on the eddy viscosity hypothesis. According to the definition of the eddy viscosity, the k equation can be transformed to that of ν t as below. Replacing k withν t2 /(Cμ2 Δ 2 ) in the k equation, one obtains
5
D ρν t 1 ∂ν t ∂ν t 2 Cd ρ ν t2 ∂ 2 2 [ ρ (ν + σ kν t ) ] + ρ (ν / ν t + σ k )( ) − = ρ Cμ Δ Sij Sij + + ρ PΔ 2 2Cμ Δ 2 Dt ∂x j ∂x j ∂x j
(3)
where PΔ represents the additional terms generated by the grid stretching or the non-uniformity of spatial filtering width.
PΔ =
3ν t 4 ∂Δ ∂Δ ∂ν t ν t ∂ ∂Δ (ν + σ kν t )( )2 − (ν + σ kν t ) [(ν + σ kν t ) ] − 2 Δ ∂x j Δ ∂x j ∂x j Δ ∂x j ∂x j
(4)
Keeping f v1 = 1.0 in the LES region, the ν t equation can be written in the form ofν equation usingν t = ν f v1 ∂ ∂ν ∂ν 2 Cd ρ ν 2 D ρν 1 2 2 [ ρ (ν + σ kν ) ] + ρ (ν / ν + σ k )( ) − = Cμ Δ ρ Sij Sij + + ρ PΔ (5) 2 2Cμ Δ 2 ∂x j ∂x j ∂x j Dt Based on the similarity of (1) and (5), these two equations can be blended by using a
blending function F, which is equal to one in the near-wall region and approaches zero for the region far away from the wall. The blended equation can be written as below.
D ρν ∂ ∂ν ∂ν [ ρ (ν + σ v1ν ) ] + ρσ v 2 ( )2 − Dv + (1 − F ) ρ PΔ = ρ Pv + Dt ∂x j ∂x j ∂x j 1 1 where Pv = F (Cb1Sν ) + (1 − F )( Cμ2 Δ 2 Sij Sij ) , σ v1 = F + (1 − F )σ k , σ 2
σ v2 = F
Cb 2
σ
+ (1 − F )(ν /ν + σ k ) , Dv = F (Cw1 f w
ν2 d
) + (1 − F )( 2
(6)
Cd ν 2 ). 2Cμ Δ 2
In order to recover the original SGS model in LES region, one can redefine fv1 as f v1 =
χ3 χ 3 + Γ( F )Cv31
(7)
Γ( F ) is supposed to be 1 in RANS region and 0 in the LES region. In order to make a
fast transition, a steep function is used as below. Γ( F ) = F 20 A blending function F similar to that used by Sánchez-Rocha and Menon[16] is adopted.
6
(8)
y ≤ y start ⎧1.0 ⎪ ⎪ ⎪ y ≥ y end (9) F = ⎨0.0 ⎪1 C f 1 (d / d 0 − C f 2 ) ⎪ (1 − tanh( ) / tanh(C f 1 )) else (1 − 2C f 2 )d / d 0 + C f 2 ⎪⎩ 2 where C f 1 = 2.0, C f 2 = 0.2, d = y − ystart , d 0 = yend − ystart , y is the distance to the nearest
wall. ystart and yend are used to adjust the RANS-LES
transition. In the present study,
ystart = 0.1δ inf and yend = δ inf are adopted so that the hybrid model acts as a wall-modeled LES, where δ inf is the boundary-layer thickness of the turbulent inflow. These parameters are chosen to make the RANS-to-LES transition occur in the inner log-law region of the turbulent boundary layer so that large structures in the outer region of the boundary layer and in the free shear layer are resolved by LES. 2.2 Numerical schemes In the present study, the fifth-order WENO scheme developed by Jiang and Shu[17] is used for inviscid fluxes and the second-order centered scheme for viscous fluxes. In view of improving the computing efficiency, the time integration is performed by means of a second-order, implicit dual time step approach, the inner iteration of which is achieved by a lower-upper symmetric Gauss-Seidel (LU-SGS) method. 2.3 Turbulent inflow conditions A recycling/rescaling method[13] is used to treat the turbulent inflow condition, which is considered to be a promising way to prescribe time-dependent turbulent inflow conditions for LES or hybrid RANS/LES of spatially developing turbulent flows[18-21]. In the present study, a method similar to that of Xiao et al.[20] is used, which can be described as follows.
7
1) Fix the mean profiles of velocity and temperature at inflow, and extract their fluctuations at the recycle station. Rescale and superimpose the fluctuations onto the mean inflow profiles at every time step. The pressure is fixed during the simulation and the density is given by the state equation. 2) Recycle and rescale both the mean and fluctuating values of k or ν at every time step. In the inner layer, ν is scaled according to the wall law. In the outer layer, where the simulation is basically carried out by LES, ν is scaled according to the law of k , based on the definitionν t = Cμ k 1/ 2 Δ . Then the scaling laws can be described as below. Inner layer: ' rec + ui'inf ( yinf , z, t ) ,in = β ui ' inf
+ Tin = β 2T ' rec ( yinf , z, t ) inf
+ kin = β 2 k rec ( yinf , z, t )
(10)
inf
+ ν in = ν rec ( yinf , z, t )
Outer layer: ' rec ui'inf (ηinf , z , t ) ,out = β ui ' inf
Tout = β 2T ' rec (ηinf , z , t )
(11)
inf
kout = β 2 k rec (ηinf , z , t ) inf
ν out = ( βΔ inf (ηinf , z, t ) / Δ rec (ηinf , z , t ))ν rec (ηinf , z , t ) where
β=
δ rec δ inf
uτ ,inf
=(
δ rec 1/10 ) δ inf
uτ ,rec x −x = [1 + ( rec inf )0.27 6/5 Reδ−inf1/5 ]5/6 δ inf
8
(12) (13)
and y + = yuτ /ν w ,η = y / δ , and Δ is the filtering width. uτ ,inf and uτ ,rec are the friction velocities at the inflow and recycling station, respectively. δ inf and δ rec are the corresponding boundary-layer thickness. The complete field can be obtained from weighted averaging of inner and outer values
ϕ = ϕin (1 − w(ηinf )) + ϕout w(ηinf )
(14)
The weighting function is defined as w(η ) = where α = 4 and b = 0.2 .
⎫ 1⎧ α (η − b) ] / tanh(α ) ⎬ ⎨1 + tanh[ 2⎩ (1 − 2b)η + b ⎭
(15)
2.4 Mass-tracing procedure The mass exchange process between the main stream or fuel jet and the recirculation flow within the cavity to a large extent determines the eventual holding and spreading characteristics of the flame[22, 23]. In order to quantify the mass exchange process, the following procedure is adopted. Each simulation is initiated and stopped once the flow reaches a periodic state. At the point, the fluid within the cavity is tagged differently than the fluid in the main stream. In the present study, the fluids beneath the imaginary line that connects the top of the cavity fore wall to the top of the cavity aft wall are tagged as “inner” fluids, while the fluids above this line are tagged as “outer” fluids. The simulations are then restarted and the mass exchange process is monitored as a function of time. It is notable that the tagging process in no way changes the fluid state, it simply serves as a marker for monitoring how the fluids within the cavity interact with the main stream flow. Thus, there are two nominal species for the no-injection flow, the mass fractions of which are denoted by, Yair ,in and Yair ,out . Hence, one additional equation for
9
scalar transport of Yair ,in needs to be solved. Similarly, there are four nominal species for the hydrogen-jet-cavity flows studied here, the mass fractions of which are denoted by YH 2 ,in , Yair ,in , YH 2 ,out , Yair ,out . Accordingly, three additional equations for scalar transport
of YH 2 ,in , Yair ,in and YH 2 ,out need to be solved simultaneously. 2.5 Code validation The general theoretical framework has been discussed and testified in the early works[8, 12, 13, 24-27]
. Additionally, a simulation of transverse jet in a supersonic turbulent crossflow,
experimentally studied by Santiago and Dutton[28], Everett et al.[29] and Vanlerberghe et al.[30], is carried out to validate the present numerical models. In order to get a reasonable LES resolution with acceptable computational costs, the present calculation uses a set of flow parameters leading to a lower Reynolds number ( Re D = 2.4 × 104 ) compared with the experiment, similar to that adopted by Kawai et al.[31, 32]. The flow Mach number is M ∞ = 1.6 , density and pressure ratio between the jet and crossflow are ρ 0 j / ρ ∞ = 5.55 and p0 j / p∞ = 8.4 , resulting in jet-to-crossflow momentum flux ratio of J = 1.7 , which is important to determine the jet penetration. The upstream boundary layer thickness, δ = 0.775 D ( D =4 mm), is matched to the experiment at x / D = -5. The schematic of the computational domain is shown in Fig. 1, where the width of the domain in the spanwise direction is 4.4 D and the length of the inflow region L is 9.5 D . The number of grid points is 481×151×151 , which leads to a grid resolution of Δx + ≈ 30, Δz + ≈ 20 and Δy + ≈ 1 ∼ 20 in the focused region.
10
Fig.1 Schematic of computational domain for transverse jet.
(a) Experimental PLIF images[30]
(b) Calculation Fig.2 Representative snapshots of jet fluid in centerplane. Figure 2 shows the instantaneous flowfields of the numerical and experimental results. The vortical structures in the windward and leeward jet boundaries, observed in the experiments, are well captured by the present calculation. Figure 3 shows the comparisons of time-averaged velocity distributions between the calculation and experiment at jet downstream locations. The calculated results agree reasonably well with the experimental data except that significant discrepancy is observed close to the wall in
11
the region immediately downstream of the jet in the streamwise velocity profiles. The same discrepancy is also observed in the LES of Kawai et al.[31, 32]. The possible reason may be the uncertainty involved in the experimental data and the unmatched Reynolds number used in the calculation.
Fig.3 Comparisons of streamwise and wall-normal velocities between calculation and experiment[28, 31, 32] at jet downstream locations x / D = 2, 3, 4 and 5. 3 Results and discussion 3.1 Geometry and boundary conditions The calculations simulate a model scramjet combustor recently designed at National University of Defense Technology. The combustor is connected to a Ma2.52 nozzle of the air heater by a 180 mm long isolator, and flow conditions at the nozzle exit and fuel jet exit are listed in Table 1. Since the computational region starts from the combustor entrance and the isolator is not simulated, a beforehand two dimensional RANS simulation is used to estimate the boundary-layer thickness at the combustor entrance, based on which a turbulent boundary layer with inflow thickness of δ inf = 2.5mm is adopted, resulting in a Reynolds number of Reδinf = 37539 . Schematic of the computational combustor is shown in Fig. 4. The computational domain has a constant width of 20 mm and height of 40 mm. A cavity with depth D = 8 mm, length-to-depth
12
ratio L / D = 7 and aft angle of 45 degree is mounted on the bottom wall. The fuel is injected sonically from a 2-mm-diameter normal injector located s=10 or 30 mm upstream of the cavity leading edge. Three injection conditions Pjet=0.6, 1.2 and 1.8 MPa, resulting in equivalence ratios of 0.095, 0.19 and 0.285 are considered. A no-slip, no-penetration adiabatic condition is imposed at all lower walls, but a slip condition is used at the upper wall to reduce the computational costs. Table 1 : Inflow conditions parameter air stream M∞ T0 , K P0 , MPa δ inf , mm Injection distance s, m m
fuel jet
2.52 1486 1.6 2.5
1.0 300 0.6/1.2/1.8 0.0
-
10/30
Fig. 4 Schematic of combustor model For s=10 mm, the grid points are 513 ×121×121 and 129 × 61×121 for the regions above and inside the cavity, respectively. For s=30 mm, the computation domain is extended 24 mm upstream, and the grid points for the region above the cavity become
609 ×121×121 . These meshes lead to a grid resolution of Δx + ≈ 75, Δz + ≈ 50 and Δy + ≈ 1 ∼ 50 in the turbulent boundary layer and the cavity shear layer, which may be a little coarse for wall-resolved LES but is appropriate for the hybrid method used as wall-modeled LES.
13
3.2 Characteristics of jet-cavity interactions In the present study, three injection pressures, Pjet=0.6, 1.2 and 1.8 MPa, together with two injection locations, s=10 and 30 mm, are considered. Here s denotes the distance that measures from the upstream injector to the cavity leading edge. Moreover, the cavity flow without injection is also calculated to clarify the effects of the jet on the cavity flow. First, the case with Pjet=1.2 MPa and s=10 mm is taken as a representative to analyze the basic flowfields and to compare with the no-injection case. Then, the effects of the injection pressure and injection location are addressed. Figure 5 shows the instantaneous flowfields of jet-cavity interactions in supersonic turbulent flow, where typical vortices associated with the transversely injected jet and the cavity flow are well captures. Hairpin-like vortices are evident in the windward jet boundary. Thanks to the formation of these vortices, the main stream air is entrained into the jet boundary and mixed with the jet fluid rapidly, leading to a quick decrease of the hydrogen mass fraction. As observed by Kawai et al.[31, 32], there should be two groups of hairpin-like vortices generated from the windward and leeward portion of the jet, respectively. In the present study, however, the hairpin-like vortices in the leeward portion of the jet are not evident, which may result from the interaction of the cavity shear layer and the leeward portion of the jet. From an average point of view, furthermore, a pair of counter-rotating vortices is formed with the axis approximately locating in the streamwise direction. Besides, there exist vortices in the turbulent boundary layer, cavity shear and cavity recirculation region. It is observed that a portion of the jet may begin to be drowned in the shear-layer vortices from some distance downstream of the cavity leading edge, leading to intense mass and momentum exchange between the jet fluid and
14
cavity fluid. Thus the vortex interaction of the jet-with-cavity shear layer may play a key role in the jet-cavity interactions. All these vortices may have significant effects on the jet-cavity interactions, as will be seen below.
(a) Vorticity magnitude and hydrogen mass fraction contours in centerplane and wall-parallel plane close to the wall ( y / δ inf = 0.2 ).
(b) Isosurfaces of the second invariant of velocity gradient tensor Q colored by hydrogen mass fraction. Fig. 5 Instantaneous flowfields of jet-cavity interactions in supersonic turbulent flow.
15
(a) No injection
(b) Pjet=1.2 MPa, s=10 mm
Fig. 6 Mass decay process of “inner” fluids. Vorticity magnitude overlapped with inner mass fraction contours; from top to bottom, t ∗ u∞ / δ inf = 0 , 5, 20 and 50.
Figure 6 shows the mass decay process of the fluids within the cavity with and without upstream fuel injection, where the instantaneous vorticity magnitude together with the “inner” mass fraction contours are displayed. The dashed line indicates the position of cavity mouth at each location. For the injection case, the “inner” fluids denote the sum of hydrogen and air that are tagged. In general, the “inner” fluids are well confined in the cavity by the cavity shear layer. Thus, the mass decay characteristics are basically determined by the unsteady behavior of the cavity shear layer. For the cavity flow
16
without injection, the confinement of the cavity shear layer can be broken by two processes: the rolling-up of shear-layer vortices due to the development of Kelvin-Helmholtz instabilities and the impingement of the shear layer on the cavity aft wall. The first process is usually slow and weak, and the second one is more violent and determinative. As a result, the mass decay mainly occurs in the aft region of the cavity volume while the mass in the front portion maintains unabated for quite a long time. For the cavity flow with upstream injection, the vortex interaction of the jet-with-cavity shear layer introduces an additional mechanism to break the shear-layer confinement by tearing a gap on the shear layer intermittently. It is evident in the third snapshot in Fig. 6(b) that the “inner” fluids are entrained into the fuel jet at the second streamwise station. The achievement of this entrainment is determined by the unsteady behaviors of both the fuel jet and the cavity shear layer. Due to the oscillations of the entire flowfield, the fuel jet and the cavity shear layer oscillate up and down in the transverse direction. Once they form a close enough coupling during the oscillations, the “inner” fluids beneath the cavity shear layer can be entrained into the fuel jet by the upwash resulting from the counter-rotating vortices of the jet as shown in Fig. 7(c). This process may be important to the flame holding and spreading of the cavity-stabilized combustion as has been analyzed by Wang et al.[12]. Also can be observed is that the introduction of the jet enhances the shear-layer oscillations especially in the aft region of the cavity shear layer, which can accelerate the mass decay of the “inner” fluids. Furthermore, the vortex interaction of the jet-with-cavity shear layer pulls the cavity shear layer further into the mainstream, which tends to open a bigger “slot” around the cavity aft wall as shown in Fig. 7, and thus enhance the mass exchange between the fluids in and out of the cavity.
17
Notably, once the hydrogen and air are entrained into the cavity recirculation region, they mix with each other and form an approximately uniform mixture soon due to the intense turbulent stirring and relatively long residence time within the recirculation region. Thus, the mass decay processes of the inner air and inner hydrogen are similar to that of the total inner fluids, as can be seen from their mass decay curves below. Figure 8 shows the entrainment process of “outer” fluids for Pjet=1.2 MPa, s=10 mm, where the instantaneous vorticity magnitude together with the “outer” mass fraction contours are displayed. Since the horseshoe vortex shells some fuel off the jet and into the boundary layer when it is formed around the jet exit. As the boundary layer separates at the cavity leading edge, the partially premixed mixture enters the cavity shear layer. Also, the jet boundary is partially premixed with outer hydrogen and air. Accordingly, a portion of the outer hydrogen and air enters into the cavity as partially premixed mixture, so their entrainment mechanisms are correlated. Similar to the mass decay processes, the entrainment of the outer hydrogen and air into the cavity is basically achieved via three processes. The first one is the rolling-up of shear-layer vortices due to the development of Kelvin-Helmholtz instabilities. The second one is the impingement of the cavity shear layer on the cavity aft wall. The third one is the interaction of the jet-with-cavity shear layer, i.e. interactions of the hairpin-like vortices in the leeward portion of the jet boundary with the cavity shear-layer vortices and the downwash around the jet boundary associated with the counter-rotating vortices as shown in Fig. 7(c).
However, most of
the outer air enters the cavity via the unsteady behavior of the entire cavity shear layer, while most of the outer hydrogen enters the cavity via the interaction of the
18
jet-with-cavity shear layer. Therefore, there exist some differences between the entrainment processes of the outer hydrogen and air as will be observed below.
(a) Density gradient magnitude, no injection
(b) Density gradient magnitude overlapped with hydrogen mass fraction contours, Pjet=1.2 MPa, s=10 mm
(c) Hydrogen mass fraction overlapped with streamlines, Pjet=1.2 MPa, s=10 mm. Fig. 7 Time-averaged flowfields.
19
(a) Hydrogen
(b) Air
Fig. 8 Entrainment process of “outer” fluids for Pjet=1.2 MPa, s=10 mm. Vorticity magnitude overlapped with outer mass fraction contours; from top to bottom, t ∗ u∞ / δ inf = 0 , 5, 20 and 50. The effects of the injection pressure and the injection distance are addressed. The mass exchange results are analyzed and presented in Figs. 9 and 10. The mass exchange rate
m is the mass of main stream that is gulfed into the cavity per second and the residence time τ is the time that the gulfed main stream resides in the cavity. The relation between them can be denoted as τ = m / m or m = −dm / dt = m / τ , and m is the total mass in the cavity. When calculating the residence time of the cavity, it is found that there is not a single exponential decay curve that can fit the mass decay history very well. As a result, a
20
three-segment exponential fit method used by Baurle et al.[22, 23] is adopted here to calculate the residence time. We can see from Fig. 9 that the three-segment exponential fit is very proper to fit the raw data.
Fig. 9 Cavity fluid mass decay history and exponential fit. It is observed that the introduction of upstream fuel jet enhances the mass exchange between the fluids in and out of the cavity and reduces the residence time of the cavity fluids. Moreover, the residence time decreases with increasing injection pressure, indicating stronger jet-cavity interactions. For the highest injection pressure studied here, it is observed that the residence time of the cavity fluids is even reduced by more than 50%, which should be considered for the design of cavity flameholders that must provide long enough residence time to stabilize the combustion. However, as the injection distance away from the cavity leading edge increases, the residence time increases as shown in Fig. 9 and less fuel can be entrained into the cavity as displayed by Fig. 10(a), indicating weakened jet-cavity interactions. The time-averaged vorticity distributions displayed in Fig. 11 clearly show that the introduction of upstream fuel jet distorts the cavity shear layer and this distortion grows with increasing injection pressure or
21
decreasing injection distance. For s=10 mm, the counter-rotating vortices are very strong as the jet passes above the cavity, inducing strong interactions with the cavity shear layer. It is observed that a great portion of the jet is embedded in the cavity shear layer. For s=30 mm, however, the counter-rotating vortices have become rather weak as the jet passes above the cavity, resulting in diminished jet-cavity interactions. Notably, once the hydrogen and air are entrained into the cavity recirculation region, they mix with each other and form an approximately uniform mixture soon due to the intense turbulent stirring and relatively long residence time within the recirculation region. Thus, the mass decay processes of the inner air and inner hydrogen are similar to that of the total inner fluids, as can be seen from the mass decay curves shown in Figs. 9 and 10. However, the entrainment characteristics of the outer hydrogen and air into the cavity are quite different as can be seen in Fig. 10. Once the jet-cavity interactions are enhanced by increasing the injection pressure or decreasing the injection distance, the unsteady behaviors of the cavity shear layer tends to be tuned to enhance the mass exchange process. As has been pointed out above, most of the outer air enters the cavity via the unsteady behavior of the entire cavity shear layer. Thus the entrainment process of the outer air into the cavity is enhanced associated with the enhanced jet-cavity interactions. Most of the outer hydrogen, however, enters the cavity via the interaction of the jet-with-cavity shear layer. Though increasing the injection pressure can enhance the jet-cavity interaction, the jet penetrates further into the main stream at the same time. Therefore, the entrainment process of the outer hydrogen into the cavity is determined by the competition between the jet-cavity interaction and the jet penetration. As a result, it is observed that the entrainment of the outer hydrogen into the cavity for Pjet=1.2 MPa and
22
s=10 mm seems stronger than that for Pjet=1.8 MPa and s=10 mm. Though the mass exchange characteristics may become different when combustion occurs, the present analyses indicate that moderate injection pressure may be beneficial to transport more fuel into the cavity to promote the ignition process under certain conditions.
(a) Hydrogen
(b) Air Fig. 10 Histories of cavity fluid mass decay and main stream entrainment.
23
(a) No injection
(b) Pjet=0.6 MPa, s=10 mm
(c) Pjet=1.2 MPa, s=10 mm.
(d) Pjet=1.8 MPa, s=10 mm
(e) Pjet=1.8 MPa, s=30 mm Fig. 11 Time-averaged vorticity magnitude.
24
Additionally, mixing efficiency and total pressure recovery are also compared in Fig. 12. Quantitative evaluation of the mixing degree can be made by using a mixing-efficiency parameter, which indicates the fraction of the reactant that would react if brought to chemical equilibrium with air. The fraction of the reactant refers to the least-available reactant, air or fuel, depending on whether the mixture is lean or rich. The mixing efficiency is defined as
ηm =
∫ Y ρudA ∫ Y ρudA react
(16)
and
Y Y ≤ Ystoic ⎧ ⎪ (17) Yreact = ⎨ 1−Y ⎪Ystoic (1 − Y ) Y > Ystoic stoic ⎩ where Y is the fuel mass fraction, Yreact is the fuel fraction mixed in a proportion that can react, and Ystoic is the fuel stoichiometric mass fraction. The total pressure recovery is used to evaluate the total pressure loss during the mixing process, which is defined as
Ptrec =
∫ Pt ρudA ∫ Pt ρudA
(18)
∞
Basically, the mixing efficiency decreases with increasing injection pressure. For quite a long distance, the fuel-air mixing can only take place around the interfaces formed between the fuel jet and air while the fuel in the jet core can not readily contact with the air. As the injection pressure increases, the total mass flow rate of the fuel increases. The area of the interfaces that responsible for the fuel-air mixing, however, does not increase as fast as the fuel mass flow rate does. As a result, the mixing efficiency decreases with
25
increasing injection pressure. For Pjet=1.8 MPa, the mixing distance with s=10 mm is actually 20 mm shorter than that with s=30 mm, but the mixing efficiency with s=10 mm tends to approach that with s=30 mm near the exit of the domain. It can be inferred that, if the jet exit for s=10 and 30 mm are located at the same position, the mixing efficiency with s=10 mm will exceed that with s=30 mm. That is, stronger jet-cavity interactions with shorter injection distance are beneficial to enhance the fuel-air mixing. The total pressure loss here is mainly induced by shock waves and injection-related separations. Thus, the total pressure loss tends to increase as the injection pressure is raised or the injection position is shifted upstream.
Fig. 12 Mixing efficiency and total pressure recovery. 4
Conclusions The present study concerns the characteristics of flow over a cavity flameholder
coupled with a fuel jet for scramjet applications. Interactions of a cavity flameholder with an upstream injected jet in a Ma 2.52 supersonic flow are investigated numerically. Patterns of the fluid entrainment into the cavity and escape from the cavity are identified using a scalar-tracing method. Significant conclusions that can be made are as follows.
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1)
The jet-cavity interactions remarkably enhanced the mass exchange between the fluids in and out of the cavity, resulting in reduced residence time of the cavity fluids. This should be considered for the design of cavity flameholders that must provide long enough residence time to stabilize the combustion.
2)
Increasing the distance between the fuel injection and the cavity leading edge tends to attenuate the jet-cavity interactions, leading to weaker mass exchange. Raising the injection pressure appears to enhance the jet-cavity interactions, resulting in a shorter residence time of the cavity fluids.
3)
The mass decay processes of the inner air and inner hydrogen are similar to that of the total inner fluids due to the relatively rapid hydrogen-air mixing within the cavity recirculation region.
4)
Most of the outer air enters the cavity via the unsteady behavior of the entire cavity shear layer, while most of the outer hydrogen enters the cavity via the interaction of the jet-with-cavity shear layer. The entrainment process of the outer hydrogen into the cavity is determined by the competition between the jet-cavity interaction and the jet penetration. The results indicate that moderate injection pressure may be beneficial to transport more fuel into the cavity to promote the ignition process under certain conditions.
Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant Nos. 50906098 and 91016028, and Fok Ying Tung Education Foundation under Grant No.131055. References
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[1] [2] [3] [4] [5] [6] [7] [8] [9]
[10] [11] [12] [13]
[14] [15]
Mathur T, Gruber M, Jackson K, Donbar J, Donaldson W, Jackson T, Billig F. Supersonic Combustion Experiments with a Cavity-Based Fuel Injector [J]. Journal of Propulsion and Power 2001 17 (6):1305-1312 Ben-Yakar A, Hanson R K. Cavity Flame-Holders for Ignition and Flame Stabilization in Scramjets: An Overview [J]. Journal of Propulsion and Power, 2001, 17 (4):869-878 Ben-Yakar A, Hanson R K. Supersonic Combustion of Cross-Flow Jets and the Influence of Cavity Flame-Holders [R]. AIAA Paper 99-0484, 1999 Micka D J, Driscoll J F. Reaction Zone Imaging in a Dual-Mode Scramjet Combustor Using CH-PLIF [R]. AIAA Paper 2008-5071, 2008 Micka D J, Driscoll J F. Combustion characteristics of a dual-mode scramjet combustor with cavity flameholder [J]. Proceedings of the Combustion Institute, 2009, 32:2397-2404 Micka D J, Torrez S M, Driscoll J F. Heat Release Distribution in a Dual-Mode Scramjet Combustor - Measurements and Modeling [R]. AIAA Paper 2009-7362, 2009 Micka D J. Combustion Stabilization, Structure, and Spreading in a Laboratory Dual-Mode Scramjet Combustor [D]. The University of Michigan, 2010 Sun M B, Wang Z G, Liang J H, Geng H. Flame Characteristics in a Supersonic Combustor with Hydrogen Injection Upstream of a Cavity flameholder [J]. Journal of Propulsion and Power, 2008, 24 (4):688-696 Sun M B, Wang H B, Bai X S, Wang Z G, Geng H, Liang J H, Liu W D. Experimental and Numerical Study on Flame Stabilization in a Supersonic Combustor with Hydrogen Injection Upstream of Cavity Flameholders [R]. AIAA Paper 2009-5187, 2009 Kim K M, Baek S W, Han C Y. Numerical study on supersonic combustion with cavity-based fuel injection [J]. International Journal of Heat and Mass Transfer, 2004, 47:16 Jeong E, O’Byrne S, Jeung I-S, Houwing A F P. Investigation of Supersonic Combustion with Angled Injection in a Cavity-Based Combustor [J]. Journal of Propulsion and Power, 2008, 24 (6):1258-1268 Wang H B, Wang Z G, Sun M B, Qin N. Combustion characteristics in a supersonic combustor with hydrogen injection upstream of cavity flameholder [J]. Proceedings of the Combustion Institute, 2013, 34:2073-2082 Wang H B, Qin N, Sun M B, Wang Z G. A dynamic pressure-sink method for improving large eddy simulation and hybrid Reynolds-averaged Navier-Stokes/large eddy simulation of wall-bounded flows [J]. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 2012, 226 (9):1107-1120 Spalart P R, Allmaras S R. A one-equation turbulence model for aerodynamic flows [R]. AIAA Paper 92-0439, 1992 Yoshizawa A, Horiuti K. A Statistically-Derived Subgrid Scale Kinetic Energy Model for Large-Eddy Simualtion of Turbulent Flows [J]. Journal of the Physical Society of Japan, 1985, 54 (8):2834-2839
28
[16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
[26] [27]
[28] [29] [30] [31] [32]
Sánchez-Rocha M, Menon S. The compressible hybrid RANS/LES formulation using an additive operator [J]. Journal of Computational Physics, 2009, 228:2037-2062 Jiang G, Shu C W. Efficient implementation of weighted ENO schemes [J]. Journal of Computational Physics, 1996, 126:917-923 Lund T S, Wu X, Squires K D. Generation of turbulent inflow data for spatially-developing boundary layer simulation [J]. J. Comput. Phys., 1998, 140: 233-258 Urbin G, Knight D. Large-eddy simulation of a supersonic boundary layer using an unstructured grid [J]. AIAA Journal, 2001, 39:1288-1295 Xiao X, Edwards J R, Hassan H A, Baurle R A. Inflow Boundary Conditions for Hybrid Large Eddy/Reynolds Averaged Navier-Stokes Simulations [J]. AIAA Journal, 2003, 41 ( 8):1481-1489 Liu K, Pletcher R H. Inflow conditions for the large eddy simulation of turbulent boundary layers: A dynamic recycling procedure [J]. J. Comput. Phys., 2006, 219:1-6 Winterfield G. On processes of turbulent exchange behind flameholders [J]. In Tenth International Symposium on Combustion, 1965:1265-1275 Baurle R A, Tarn C-J, Dasgupta S. Analysis of Unsteady Cavity Flows for Scramjet Applications [R]. AIAA 2000-3617, 2000 Sun M B, Geng H, Liang J H, Wang Z G. Mixing Characteristics in a Supersonic Combustor with Gaseous Fuel Injection Upstream of a Cavity Flameholder [J]. Flow, Turbulence and Combustion, 2009, 82:271-286 Wang H B, Qin N, Sun M B, Wu H Y, Wang Z G. A hybrid LES (Large Eddy Simulation)/assumed sub-grid PDF(Probability Density Function) model for supersonic turbulent combustion [J]. Sci China Tech Sci, 2011, 54 (10):2694-2707 Wang H B, Sun M B, Qin N, Wu H Y, Wang Z G. Characteristics of Oscillations in Supersonic Open Cavity Flows [J]. Flow Turbulence and Combustion, 2013, 90:121-142 Wang H B, Wang Z G, Sun M B, Qin N. Simulations of combustion with normal and angled hydrogen injection in a cavity-based supersonic combustor [J]. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 2013, in press, Santiago J G, Dutton J C. Velocity measurements of a jet injected into a supersonic crossflow [J]. Journal of Propulsion and Power, 1997, 13 (2):264-273 Everett D E, Woodmansee M A, Dutton J C, Morris M J. Wall Pressure Measurements for a Sonic Jet Injected Transversely Into a Supersonic Crossflow [J]. Journal of Propulsion and Power, 1998, 14 (6):861-868 VanLerberghe W M, Santiago J G, Dutton J C, Lucht R P. Mixing of a Sonic Transverse Jet Injected into a Supersonic Flow [J]. AIAA Journal, 2000, 38 (3):470-479 Kawai S, Lele S K. Mechanisms of jet mixing in a supersonic crossflow: a study using large-eddy simulation [R]. AIAA Paper 2008-4575, 2008 Kawai S, Lele S K. Large-Eddy Simulation of Jet Mixing in a Supersonic Turbulent Crossflow [R]. AIAA Paper 2009-3795, 2009
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► Jet-cavity interactions in a supersonic flow are studied numerically. ► Patterns of fluid entrainment and escape are identified using scalar-tracing method. ► Jet-cavity interactions remarkably enhance the cavity mass exchange. ► Entrainment processes for the fuel and air into the cavity seem quite different.
30