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Investigation of injectant molecular weight effect on the transverse jet characteristics in supersonic crossflow
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Investigation of injectant molecular weight effect on the transverse jet characteristics in supersonic crossflow Yujie Zhang, Weidong Liu, Bo Wang, Yanhui Zhao, Duo Zhang
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S0094-5765(15)00189-7 http://dx.doi.org/10.1016/j.actaastro.2015.05.008 AA5435
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Received date: 5 March 2015 Revised date: 22 April 2015 Accepted date: 3 May 2015 Cite this article as: Yujie Zhang, Weidong Liu, Bo Wang, Yanhui Zhao, Duo Zhang, Investigation of injectant molecular weight effect on the transverse jet characteristics in supersonic crossflow, Acta Astronautica, http://dx.doi.org/ 10.1016/j.actaastro.2015.05.008 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.
Investigation of injectant molecular weight effect on the transverse jet characteristics in supersonic crossflow Yujie Zhang 1, 2∗, Weidong Liu 1, 2, Bo Wang 1, 2, Yanhui Zhao 1, 2, Duo Zhang 1, 2 1. College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, Hunan 410073, People’s Republic of China 2. Science and Technology on Scramjet Laboratory, National University of Defense Technology, Changsha, Hunan 410073, People’s Republic of China Abstract: The effect of injectant molecular weight on the transverse jet was investigated by the hybrid RANS/LES simulation, particle image velocimetry (PIV) and nanoparticle-based planar laser-scattering (NPLS) measurements with the nitrogen and helium jets at a constant jet-to-freestream momentum flux ratio involved. The recycling-rescaling procedure was applied to reproduce the turbulent boundary layer. Statistics obtained from the hybrid RANS/LES simulation with fine mesh shown good agreement with the experimental results. Two kinds of large-scale vortex structures are observed in the transverse jet of supersonic crossflow, namely leading edge vortices and hanging vortices, and they were previously noticed in the low speed transverse jet. The velocity gradient between jet and main flow is found to be the dominative factor determining the development of leading edge vortices in both nitrogen and helium jets. A larger velocity gradient in helium jet induces a quick breakup of the large-scale structures, and this produces small scale structures with small interval in jet plume. By contrast, a smaller velocity gradient in nitrogen jet induces a later breakup of the large-scale structures, and this produces large scale structures with large interval in jet plume. The development of large-scale vortices structures has also influenced the mixing and penetration height, with a better mixing achieved in helium jet and a higher penetration height in nitrogen jet.
∗
corresponding author, E-mail: [email protected], Phone: +86 731 84574756
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Keywords: transverse jet, supersonic flow, molecular weight, large-scale turbulent structures
1. Introduction As one of the most potential options for the hypersonic air-breathing propulsion system, the scramjet has attracted an increasing attention worldwide [1]. However, because of the supersonic crossflow, the residence time of fuel is too short to achieve the complete mixing between injectant and crossflow [2]. To improve the mixing efficiency between the fuel and air, injection strategies in scramjet engines have been investigated extensively , such as transverse jet [3-6], ramp jet [7, 8], aerodynamic ramp jet [9, 10], pylon jet [11, 12] and strut jet [13]. Compared with other injection strategies, transverse fuel injection through a wall orifice has been proved to be one of the simplest and most conventional methods for the scramjet engine. To satisfy the requirements in engineering, researchers mainly focus on time-averaged penetration height, mass fraction distribution, total pressure loss in the following four aspects, namely jet-to-freestream momentum flux ratio [4, 14-17], injection angle [4, 18-20], configuration of injection orifice [4, 21], molecular weight [21-23]. However, limited by experimental measurements and numerical simulations, the turbulent characteristics of sonic jet in supersonic crossflow are not the major contents in the early studies. Due to the lower speed and weaker instability in low-speed jet with low-speed crossflow, the development of large-scale turbulent structures are visualized [24-29] earlier than in sonic jet with supersonic crossflow. The experiment of New et al. [29] shown that leading edge vortices roll up periodically in the windward shear layer during the the three-dimensional interaction between the jet and crossflow. Kelso et al. [25] pointed out that the Kelvin-Helmholtz-like instability of shear layer induces the large scale, periodic structures roll up. Using the large eddy simulation (LES), Yuan et al. [28] observed that the counter rotating vortex pairs (CVPs) in the jet plume originate from hanging vortices which form in the skewed mixing layer on the lateral edges of the jet.
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Recently, with the development of the experimental measurements, large-scale turbulent structures of transverse jet in supersonic crossflow are visualized by the Rayleigh/Mie scattering imaging technique [30], planar laser induced fluorescence (PLIF) [31, 32] and ultrahigh-speed schlieren [33]. The shape of large-scale turbulent structures were obtained by the single-time two-point spatial correlations, and the results of spatial correlation revealed that the size of the large-scale structures tends to increase with the decrease of the compressibility of the shear layer [32]. At the same time, with the advancement of computational resources, LES and hybrid RANS/LES provide further insight into the mixing progress and development of eddies near the jet orifice. Won et al. [34, 35] found that a pair of interacting counter-rotating eddies is generated and detached from the upstream recirculation. Kawai et al. [36] reported that the jet surfaces are elongated and break down to finer structures along the large-scale vortices, accompanying with the deformation of barrel shock. In addition, Watanabe et al. [37, 38] pointed out that the large-scale vortex structures are the key factor to improve the mixing efficiency, and they observed that these structures lead to large protrusion of injectant toward the crossflow. Meanwhile, the significant influence of injectant’s molecular weight on the turbulent characteristics and mixing efficiency has been observed at a constant jet-to-freestream momentum flux ratio. The experiment performed by Gruber [30] indicated that the compressibility of the mixing layer is the main parameter affecting the turbulent behavior and mixing efficiency, and the lower compressibility mixing layer owns larger mixing potential. Later, Ben-Yakar et al. [33] figured out that the velocity difference is the key parameter influencing the mixing process by the stretching-tilting-tearing mechanism. Otherwise, they found that molecular weight can influence the jet penetration height, and this differs from Gruber’s viewpoint [30]. Also, the numerical results [39] revealed that the molecular weight has a great impact on the development of jet shear layer. The present study is to seek whether the large-scale turbulent structures in the supersonic crossflow have the same characteristics as previously noticed in subsonic crossflow and the effect of molecular weight on the
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development of large-scale structures, mixing, penetration height is evaluated as well. Based on the issues mentioned above, the characteristics of helium and nitrogen jets are investigated by hybrid RANS/LES approach, particle image velocimetry (PIV) and nanoparticle-based planar laser-scattering (NPLS) experimental measurements at a constant jet-to-freestream momentum flux ratio. First, the comparisons of statistics between simulation and experiment are made, and this is employed to confirm the accuracy of numerical approaches. Then, characteristics of the large-scale turbulent structures in the windward shear layer of helium and nitrogen jets are investigated in detail. Finally, the mixing and penetration height of the two jets are compared respectively.
2. Experimental installation All the experiments in this study have been performed in a low-noise wind-tunnel, with a test section of 120 × 380 × 56 mm3(width× length× height). The Mach number verified by wedge is 2.7 ± 0.02 . For the purpose of optical measurements, the tunnel is equipped with large optics windows of size 294 mm × 100 mm (height×length) on three sides of the test section. The stagnation pressure P0 of inflow is 77.6 kpa, and stagnation temperature T0 is 300 K, with the corresponding unit Reynolds number Rel being 6.95 ×106 . The transverse sonic injectant is from a circular orifice with a 2 mm diameter on a wall of the test section. The stagnation pressure Pjet of helium jet is 123 kpa, and the Pjet of nitrogen jet is 136 kpa. These parameters are set to ensure that the jet-to-freestream momentum flux ratio J of the helium and nitrogen jets is 2.9, and this is equal to the value of J in Gruber’s experiment [30]. The origin of the Cartesian coordinate system is at the center of the orifice, and the X, Y, Z denote streamwise, transverse and spanwise directions respectively, as shown in Fig. 1. In the experiments, NPLS is used to capture the fine turbulent structures by the adoption of nanoparticles [40, 41], and PIV is applied to obtain the velocity distribution. The NPLS is a new flow visualization based on Rayleigh-scattering, and the uncertainty about the fidelity of tracer particles in PIV and NPLS has been estimated
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in [42]. The laser sheet in NPLS and PIV is the XOY plane.
Fig. 1 Computational region sketch map (the characteristic length d is 4 mm, which is the boundary layer thickness).
3. Numerical Methods 3.1 LES equations In the present study, the filtered dimensionless compressible Navier-Stokes equations for an ideal nonreactive
：
gas are as follows
∂ρ ∂ ( ρ ui ) + =0 ∂t ∂ xi ∂ ( ρ ui ) ∂ + ∂t ∂ xj ∂
( ρ E ) + ∂t
1 1 sgs ρ ui u j + pδ ij − Re τ ij − Re τ ij = 0
~ ( ρ E + p ) ui − κ 0T0 (κ + κ t ) ∂ T − 1 u j τ ji − 1 ( µ + µt )∑ hm ∂Ym = 0 Re Sc Prt m µ0u0 2 Re ∂ xi Re ∂ xi ∂x
∂
(2)
(3)
i
∂
( ρY ) +
1 µ µ t ∂ Ym ∂ ρ Ym ui − ( + ) =0 ∂ xi Re S c S ct ∂ xi
(4)
∂ui ∂u j 2 ∂uk 2 sgs + − δ ij − ρ k δ ij ∂x 3 j ∂xi 3 ∂xk
(5)
m
∂t
Herein,
(1)
τ ijsgs is defined as τ ijsgs = µ t
5
1 2
In these equations, the filtered total energy is defined as E = e + uk uk + k sgs , and the turbulent Prandtl number Prt is 0.72. The blending function Γ [43] connects the one-equation Yoshizawa SGS [44, 45] and Menter’s SST turbulent model [46], as shown in eq. (6). ) ∂( ρ k ) ∂( ρ ku ∂ j + = ∂t ∂x j ∂x j
3 k 2 ∂k ( µl + µt ) + Pk − ρ ΓCd1 kω + (1 − Γ ) Cd2 ∂x j ∆
，
1
µt = ρ ΓµtRANS + (1 − Γ ) µtsgs vtsgs = Cv ( k ) 2 ∆ ∆ = ( ∆x∆y ∆z )
1 3
，v
RANS t
=
(6)
，
(7)
a1 k max ( a1ω , ΩF2 )
3.2 Numerical schemes and grids distributions Inviscid flux is solved with the fifth-order WENO scheme proposed by Jiang and Shu [47], while viscous flux is discretized by the second-order-accurate centered scheme. The time advancement is performed with the third-order Runge–Kutta scheme. To generate and sustain an unsteady incoming turbulent boundary layer, recycling-rescaling procedure of Xiao et al. [48, 49] is applied. The length of the recycling-rescaling region is 10d (the boundary layer thickness d is 4 mm, and this is the characteristic length in the study). The recycle plane, computational region and boundary condition are shown in Fig. 1. The number of fine grid, medium grid and coarse grid is N x × N y × N z = 589 × 193 × 138 , 407 × 172 × 122 , 210 × 130 × 100 , respectively. The grids distributions are shown in Table. 1. Table. 1 Grids distribution. Recycle region
Core region
X direction
Y direction
Z direction
X direction
Z direction
Fine
X+=50
Y/d=0~2.5: Y+=1~10
Z+=14.5
X+=5
Z+=5~12.5
Medium
X+=80
Y/d=0~2.5: Y+=1~20
Z+=20
X+=5~10
Z+=5~16
Coarse
X+=129
Y/d=0~2.5: Y+=1~30
Z+=25
X+=5~20
Z+=5~20
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4. Results and discussions 4.1 Comparisons with experiments First, the time-averaged streamwise velocity profile of inflow obtained by PIV, RANS and RANS/LES with recycling-rescaling procedure are compared to confirm the reliability of recycling-rescaling method, as shown in Fig. 2. The mean velocity profile on the fine and medium meshes shows good grid convergence and reasonable agreement with the RANS profile and PIV data, but the coarse mesh underpredicts the streamwise velocity in the logarithmic region. So the hybrid RANS/LES approach with fine and medium meshes can be used to achieve the average statistical data of turbulent boundary layer.
Fig. 2 Time-averaged streamwise velocity profile comparison of PIV, RANS and RANS/LES with recycling-rescaling. For helium jet, the time-averaged streamwise velocity distribution of PIV in median plane (Z/d=0) is shown in Fig. 3. The streamwise velocity u is normalized by incoming velocity magnitude U ∞ . As the particles can not enter the core region of jet near the orifice, a blue part in the average result of PIV emerges, and the velocity distribution can not be obtained at this location. The time-averaged streamwise velocity profiles of PIV extracted from Fig. 3 at X/d = 4 and X/d = 12 are compared with the hybrid RANS/LES velocity profiles. Since there is no particles in the blue region near the bottom wall at X/d=4, the velocity distribution can not be obtained under
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Y/d=0.8 as shown in Fig. 4 (a). Except the no particle region, it is clear that the fine and medium meshes show good agreement with the experimental statistics. However, the coarse mesh overpredicts the velocity at the upper part of the profile and underpredicts the velocity at the lower part in the two figures. Thus, the hybrid RANS/LES code with the fine and medium meshes can be applied to investigate the transverse jet in supersonic crossflow.
Fig. 3 Time-averaged streamwise velocity distribution of PIV in median plane (Z/d=0).
(a) X/d=4
(b) X/d=12
Fig. 4 Time-averaged streamwise velocity profiles of PIV and RANS/LES (helium jet). The instantaneous results of NPLS and hybrid RANS/LES simulation in helium jet are shown in Fig. 5. The turbulent structures of the shear layer are similar, inclining toward the upstream direction, and this is because of the drag from the lower-speed crossflow. By comparison, it can be observed that the small-scale structures in the windward shear layer can be captured with fine mesh, instead of the medium and coarse meshes. To investigate characteristics of the turbulent structures, the fine mesh is adopted in this study. 8
(a) NPLS
(b) RANS/LES(fine)
(c) RANS/LES(medium)
(d) RANS/LES(coarse)
Fig. 5 Instantaneous NPLS and RANS/LES results in Z/d=0 (helium jet).
4.2 Development of large-scale vortex structures The NPLS results of nitrogen and helium jets are shown in Fig. 6. By comparing NPLS images, three marked differences are observed. First, the major distinctness is that the size of the large-scale turbulent structures and the interval between the large-scale turbulent structures in nitrogen jet are both larger than that in the helium jet. The similar phenomenon is also observed in the experiments of Gruber et al. [30], Benyaka et al. [33] and in simulations of Takahashi et al. [32], Watanabe et al. [37]. This point is to be discussed later in this section. Then, the second difference exists in density of the jet plume: since the density is in proportion to the light intensity qualitative in NPLS images, the light intensity of helium jet plume is weaker than that of nitrogen jet. Finally, the last difference lies in the waves of the flowfield: a series of compression waves are observed in helium jet, inclining at an angle of about 45 degrees to the wall surface, which may be due to the extraordinary instability of the bow shock.
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(a) nitrogen jet
(b) helium jet
Fig. 6 Instantaneous jet plume of NPLS in Z/d=0. To investigate the first major distinctness mentioned above, the second invariant of the velocity-gradient tensor Q [50] is used to capture the large-scale vortex structures in hybrid RANS/LES simulation. The expression of Q is shown as eq. (8). The instantaneous iso-surface of Q=5 colored by mass fraction y1 is shown in Fig. 7 (a). The instantaneous iso-surface of mass fraction y1=0.4 is shown in Fig. 7 (b). Leading edge vortices [29] marked L are denoted by the arrows in Fig. 7 (a). The leading edge vortices form periodically in the windward shear layer, which develop between injectant of Z direction and crossflow fluid of X direction. In Fig. 7 (b), the location of rolled up injectant coincides with the location of leading edge vortices in Fig. 7 (a), so the iso-surface of mass fraction can reflect the location and shape of large-scale vortex structures. In Fig. 7 (c), the arrows denote hanging vortices [28] marked H. The hanging vortices are produced in the skewed mixing layer on the lateral edges of the jet. As the hanging vortices move downstream, the CVPs are formed [24]. The leading edge vortices and hanging vortices are also shown in Fig. 8.
Q=
1 2
(W W ij
1 ∂u i ∂u j W ij= 2 ∂ − ∂ xi xj
ij
−
SS
，S
ij
= ij
ij
)
1∂
u 2 ∂ x
10
∂ i
j
+
u ∂x
j
i
(8)
(a) Q=5
(b) y1=0.4
(c) Q=5 (d) y1=0.4 Fig. 7 Iso-surface of second invariant of the velocity-gradient tensor Q and mass fraction y1.
Fig. 8 Slice of vorticity in X/d=0.5 in Fig. 7. The development of large-scale turbulent structures in windward shear layer of nitrogen and helium jets is shown in Fig. 9 and Fig. 10. In these figures, the iso-surface of mass fraction y1=0.4 is colored by
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non-dimensional transverse velocity v / U ∞ . At T1 in nitrogen jet, the leading edge vortices marked L-1 start to form as the arrow denotes in Fig. 9(a). At T1 + 2 ⋅ ∆t ( ∆t ≈ 3.8 ×10−6 s ), the injectant tends to roll up towards Y direction under the effect of leading edge vortices. At T1 + 8 ⋅ ∆t , the rolled up part (marked U-1) of jet reaches the upside of shear layer. At the same time, the new leading edge vortices L-2 form at front part of jet plume. At T1 + 10 ⋅ ∆t , the U-1 moves downstream as a whole, while the windward shear layer produces the new leading edge
vortices L-3 and roll-up part U-2. In this process, the leading edge vortices are formed steadily and periodically, and this is the main reason to roll up the periodic large-scale structures in shear layer.
(a) T1
(b) T1 + 2 ⋅ ∆t
(c) T1 + 8 ⋅ ∆t
(d) T1 + 10 ⋅ ∆t
Fig. 9 The development of windward shear layer in nitrogen jet. However, the shear layer of the helium jet in Fig. 10 is different from that of nitrogen jet. At T2 + 3 ⋅ ∆t ( ∆t ≈ 4.0 ×10−6 s ) the leading edge vortices L-1 are formed as the arrow denotes in Fig. 10 (b). At T2 + 5 ⋅ ∆t , part of the helium jet marked U-1 tends to roll up towards Y direction under the effect of leading edge vortices. But, the whole part of U-1 breaks up into smaller vortex structures quickly at T2 + 7 ⋅ ∆t , and this is quite different from the behavior of stable U-1 in nitrogen jet in Fig. 9.
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(a) T2
(b) T2 + 3 ⋅ ∆t
(c) T2 + 5 ⋅ ∆t
(d) T2 + 7 ⋅ ∆t
Fig. 10 The development of windward shear layer in helium jet. In response to the difference raised above, we investigate how the phenomena occurs. It is known that the molecular weight of nitrogen is 7 times heavier than helium, and the specific heat ratio of helium is 1.2 times larger than the nitrogen. Therefore, the velocity magnitude of sonic jet for helium jet is almost 3 times larger than that for nitrogen jet, and this leads to different velocity distributions of windward shear layer as shown in Fig. 11. To mark the region of shear layer, the 0.15 and 0.85 mass fraction tracks are drawn with black solid lines. For nitrogen jet in Fig. 11 (a), the velocity of the region in the blue dashed line is similar with the velocity value of crossflow, and this indicates that the magnitude of velocity gradient dU dY in shear layer is nearly 0, as shown in Fig. 12 (a). By comparison, the magnitude of velocity gradient dU dY is larger in windward shear layer of helium, due to the larger velocity difference between shear layer and crossflow, as shown in Fig. 11 (b). This difference leads to the obvious distinctness in the instantaneous large-scale vortex structures and shear layer between helium and nitrogen jets in Fig. 9 and Fig. 10.
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(a) nitrogen jet
(b) helium jet
Fig. 11 Time-averaged velocity distribution in nitrogen and helium jets.
(a) nitrogen jet
(b) helium jet
Fig. 12 Distribution of dU / dY in nitrogen and helium jets. Overall, the development of shear layer in nitrogen jet is as follows: with the leading edge vortices periodically forming around the windward shear layer, part of injectant is rolled up and gets into the main flow, which indicates that the behavior of leading edge vortices mainly influences the penetration of injectant. Due to the weak shear, the roll-up part is stable and moves downstream as a whole. In contrast, the leading edge vortices are also periodically formed in helium jet, but the large velocity gradient in windward shear layer makes the rolled up part break up quickly. Therefore, the turbulent structures in helium jet are smaller than that of nitrogen jet, and no large roll-up part are observed intruding into the main flow.
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4.3 Spatial correlation The spatial correlation of the mass fraction fluctuations can express the spatial extent of the turbulent scalar field [11], so this method is employed to capture the large turbulent structures in slices of X/d=0 and Z/d=0. The expression is as follows: 1 N Rxy ( x, y ) =
∑
y1′i ( x, y ) = y1i ( x, y ) −
N i =1
y1′i ( x0 , y0 ) ⋅ y1′i ( x, y )
′ ( x0 , y0 ) ⋅ yrms ′ ( x, y ) yrms 1 N
，i = 1, 2,3 ⋅⋅⋅ N (9)
′ ( x, y ) = ∑ i =1 y1i ( x, y ), yrms N
1 N
∑ i =1 y1i′2 ( x, y ) N
In eq. (9), y1′i ( x, y ) is the instantaneous injectant mass fraction fluctuation, y1′rms ( x, y ) is the root mean square(RMS) of the mass fraction fluctuation, and
( x0 , y0 ) is the reference location with which the features are correlated
( x, y ) is an arbitrary location in computational region. In Fig. 13, the center of orifice is at Z/d=0, and the reference point is on the iso-line of y1=0.12 at Y/d=0.7.
The L and H marked in the figures represent the region dominated by leading edge vortices and hanging vortices mentioned in section 4.2 respectively. The U represents the region where U-1 or U-2 influences.
(a) nitrogen jet
(b) helium jet
Fig. 13 The correlation of mass fraction in nitrogen and helium jets at X/d=0. For nitrogen jet, the correlation distribution is almost symmetric on both sides of the center line Z/d=0, and
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this reflects that the behavior of large-scale vortex structures on both sides of the center line is nearly symmetric. The correlation between L and H is opposite, indicating there is a phase difference between L and H. Moreover, the region L is negatively correlated with the region U. The reason is as follows: according to the analysis in section 4.2, it is revealed that as the rolled part U moves downstream, the mass fraction of injectant in this region is decreasing, while the next leading edge vortices start to form, which leads to the increasing of mass fraction at L. So the correlation between L and U is contrary. For the helium jet, the correlations distribution are mostly the same with nitrogen jet, as shown in Fig. 13(b). The mass fraction correlation maps of Z/d=0 plane are shown in Fig. 14. The center of orifice is located at the origin. The reference point is on the iso-line of y1=0.45 at X/d=1. In nitrogen jet, it is obvious that four regions marked 1, 2, 3, 4 are in the shear layer between X/d=0 and X/d=9. The correlation of region 1 and region 3 is positive, and the correlation of region 2 and region 4 is negative. Since the large-scale structures of nitrogen and air appear alternately as shown in Fig. 6 (a), the positive and negative regions are formed by turns. In helium jet, there are 7 regions in the shear layer, the positive and negative regions also appear alternately. Compared with nitrogen jet, the area of these regions is smaller, and the interval is shorter, and this has a good agreement with the instantaneous image in Fig. 6 (b).
(a) nitrogen jet
(b) helium jet
Fig. 14 The correlation of mass fraction in nitrogen and helium jets at Z/d=0. 4.4 Probability distribution of mass fraction To compare the difference in mixing effect between nitrogen and helium jet, this section focuses on the 16
probability distribution of mass fraction at X/d=2 as shown in Fig. 15. For facilitating the statistics, the mass fraction is divided into four intervals: 0 ≤ y1 < 0.25 , 0.25 ≤ y1 < 0.5 , 0.5 ≤ y1 < 0.75 , 0.75 ≤ y1 < 1 . The nitrogen plume core is mainly in 0.75 ≤ y1 < 1 , and the probability of mass fraction in core region is close to 0.75. For helium, the value is nearly 0.5, and this indicates that the mixing effect for helium jet is better than that of nitrogen jet.
(a) nitrogen jet
(b) helium jet Fig. 15 The probability distribution in nitrogen and helium jets at X/d=2.
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4.5 Penetration height By comparing ethylene and hydrogen jets, Ben-Yakar [30] observed that the penetration height of ethylene jet is higher than the hydrogen jet at the constant jet-to-freestream momentum flux ratio. Thus, she pointed out that the penetration height is not only related to the jet-to-freestream momentum flux ratio, but also affected by the characteristic of jet shear layer. In this section, the method measuring the penetration height by mass fraction [39] is adopted, and the 10% iso-line of mass fraction is regarded as the penetration height. With the constant jet-to-freestream momentum flux ratio, the results obtained by hybrid RANS/LES simulation are observed that the penetration height of nitrogen jet is higher than the helium jet, as shown in Fig. 16, and this is mainly due to the different development in shear layer between nitrogen and helium jet mentioned in section 4.2.
Fig. 16 Penetration height of nitrogen and helium jets.
5. Conclusions Combined with PIV and NPLS measurements, hybrid RANS/LES simulation was performed to investigate the development of large-scale turbulent structures and the effect of molecular weight on transverse jet in supersonic crossflow. We came to the following conclusions:
The hybrid RANS/LES numerical results with fine and medium meshes show good agreement with experimental data, and the simulation with fine mesh can capture the small-scale structures in jet plume clearly instead of medium and coarse mesh. The coarse mesh shows obvious deficiency in predicting the 18
velocity of inflow and jet plume. So the hybrid RANS/LES approach with fine mesh in the present study can be applied to investigate the issues of sonic transverse jet in supersonic crossflow.
By the hybrid RANS/LES simulation, two kinds of large-scale vortex structures are observed in the transverse jet of supersonic crossflow, namely leading edge vortices and hanging vortices. The mixing effect in the near field is influenced by these two large-scale structures, and penetration height of jet is mainly related to the behave of leading edge vortices.
The helium jet achieves a better mixing because of the earlier break-up turbulent structures, and the nitrogen jet has a higher penetration height due to the stabler and later break-up turbulent structures intruding into the main flow.
Through the spatial correlation analysis, there is a phase difference between leading edge vortices and hanging vortices. Additionally, the region L is negatively correlated with the region U.
Acknowledgements The authors would like to express their thanks for the support from the National Natural Science Foundation of China(No. 91216120). Also, the authors thank the anonymous reviewers for some very critical and constructive recommendations on this article.
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Investigation of injectant molecular weight effect on the transverse jet characteristics in supersonic crossflow Yujie Zhang, Weidong Liu, Bo Wang, Yanhui Zhao, Duo Zhang
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Received date: 5 March 2015 Revised date: 22 April 2015 Accepted date: 3 May 2015 Cite this article as: Yujie Zhang, Weidong Liu, Bo Wang, Yanhui Zhao, Duo Zhang, Investigation of injectant molecular weight effect on the transverse jet characteristics in supersonic crossflow, Acta Astronautica, http://dx.doi.org/ 10.1016/j.actaastro.2015.05.008 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.
Investigation of injectant molecular weight effect on the transverse jet characteristics in supersonic crossflow Yujie Zhang 1, 2∗, Weidong Liu 1, 2, Bo Wang 1, 2, Yanhui Zhao 1, 2, Duo Zhang 1, 2 1. College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, Hunan 410073, People’s Republic of China 2. Science and Technology on Scramjet Laboratory, National University of Defense Technology, Changsha, Hunan 410073, People’s Republic of China Abstract: The effect of injectant molecular weight on the transverse jet was investigated by the hybrid RANS/LES simulation, particle image velocimetry (PIV) and nanoparticle-based planar laser-scattering (NPLS) measurements with the nitrogen and helium jets at a constant jet-to-freestream momentum flux ratio involved. The recycling-rescaling procedure was applied to reproduce the turbulent boundary layer. Statistics obtained from the hybrid RANS/LES simulation with fine mesh shown good agreement with the experimental results. Two kinds of large-scale vortex structures are observed in the transverse jet of supersonic crossflow, namely leading edge vortices and hanging vortices, and they were previously noticed in the low speed transverse jet. The velocity gradient between jet and main flow is found to be the dominative factor determining the development of leading edge vortices in both nitrogen and helium jets. A larger velocity gradient in helium jet induces a quick breakup of the large-scale structures, and this produces small scale structures with small interval in jet plume. By contrast, a smaller velocity gradient in nitrogen jet induces a later breakup of the large-scale structures, and this produces large scale structures with large interval in jet plume. The development of large-scale vortices structures has also influenced the mixing and penetration height, with a better mixing achieved in helium jet and a higher penetration height in nitrogen jet.
∗
corresponding author, E-mail: [email protected], Phone: +86 731 84574756
1
Keywords: transverse jet, supersonic flow, molecular weight, large-scale turbulent structures
1. Introduction As one of the most potential options for the hypersonic air-breathing propulsion system, the scramjet has attracted an increasing attention worldwide [1]. However, because of the supersonic crossflow, the residence time of fuel is too short to achieve the complete mixing between injectant and crossflow [2]. To improve the mixing efficiency between the fuel and air, injection strategies in scramjet engines have been investigated extensively , such as transverse jet [3-6], ramp jet [7, 8], aerodynamic ramp jet [9, 10], pylon jet [11, 12] and strut jet [13]. Compared with other injection strategies, transverse fuel injection through a wall orifice has been proved to be one of the simplest and most conventional methods for the scramjet engine. To satisfy the requirements in engineering, researchers mainly focus on time-averaged penetration height, mass fraction distribution, total pressure loss in the following four aspects, namely jet-to-freestream momentum flux ratio [4, 14-17], injection angle [4, 18-20], configuration of injection orifice [4, 21], molecular weight [21-23]. However, limited by experimental measurements and numerical simulations, the turbulent characteristics of sonic jet in supersonic crossflow are not the major contents in the early studies. Due to the lower speed and weaker instability in low-speed jet with low-speed crossflow, the development of large-scale turbulent structures are visualized [24-29] earlier than in sonic jet with supersonic crossflow. The experiment of New et al. [29] shown that leading edge vortices roll up periodically in the windward shear layer during the the three-dimensional interaction between the jet and crossflow. Kelso et al. [25] pointed out that the Kelvin-Helmholtz-like instability of shear layer induces the large scale, periodic structures roll up. Using the large eddy simulation (LES), Yuan et al. [28] observed that the counter rotating vortex pairs (CVPs) in the jet plume originate from hanging vortices which form in the skewed mixing layer on the lateral edges of the jet.
2
Recently, with the development of the experimental measurements, large-scale turbulent structures of transverse jet in supersonic crossflow are visualized by the Rayleigh/Mie scattering imaging technique [30], planar laser induced fluorescence (PLIF) [31, 32] and ultrahigh-speed schlieren [33]. The shape of large-scale turbulent structures were obtained by the single-time two-point spatial correlations, and the results of spatial correlation revealed that the size of the large-scale structures tends to increase with the decrease of the compressibility of the shear layer [32]. At the same time, with the advancement of computational resources, LES and hybrid RANS/LES provide further insight into the mixing progress and development of eddies near the jet orifice. Won et al. [34, 35] found that a pair of interacting counter-rotating eddies is generated and detached from the upstream recirculation. Kawai et al. [36] reported that the jet surfaces are elongated and break down to finer structures along the large-scale vortices, accompanying with the deformation of barrel shock. In addition, Watanabe et al. [37, 38] pointed out that the large-scale vortex structures are the key factor to improve the mixing efficiency, and they observed that these structures lead to large protrusion of injectant toward the crossflow. Meanwhile, the significant influence of injectant’s molecular weight on the turbulent characteristics and mixing efficiency has been observed at a constant jet-to-freestream momentum flux ratio. The experiment performed by Gruber [30] indicated that the compressibility of the mixing layer is the main parameter affecting the turbulent behavior and mixing efficiency, and the lower compressibility mixing layer owns larger mixing potential. Later, Ben-Yakar et al. [33] figured out that the velocity difference is the key parameter influencing the mixing process by the stretching-tilting-tearing mechanism. Otherwise, they found that molecular weight can influence the jet penetration height, and this differs from Gruber’s viewpoint [30]. Also, the numerical results [39] revealed that the molecular weight has a great impact on the development of jet shear layer. The present study is to seek whether the large-scale turbulent structures in the supersonic crossflow have the same characteristics as previously noticed in subsonic crossflow and the effect of molecular weight on the
3
development of large-scale structures, mixing, penetration height is evaluated as well. Based on the issues mentioned above, the characteristics of helium and nitrogen jets are investigated by hybrid RANS/LES approach, particle image velocimetry (PIV) and nanoparticle-based planar laser-scattering (NPLS) experimental measurements at a constant jet-to-freestream momentum flux ratio. First, the comparisons of statistics between simulation and experiment are made, and this is employed to confirm the accuracy of numerical approaches. Then, characteristics of the large-scale turbulent structures in the windward shear layer of helium and nitrogen jets are investigated in detail. Finally, the mixing and penetration height of the two jets are compared respectively.
2. Experimental installation All the experiments in this study have been performed in a low-noise wind-tunnel, with a test section of 120 × 380 × 56 mm3(width× length× height). The Mach number verified by wedge is 2.7 ± 0.02 . For the purpose of optical measurements, the tunnel is equipped with large optics windows of size 294 mm × 100 mm (height×length) on three sides of the test section. The stagnation pressure P0 of inflow is 77.6 kpa, and stagnation temperature T0 is 300 K, with the corresponding unit Reynolds number Rel being 6.95 ×106 . The transverse sonic injectant is from a circular orifice with a 2 mm diameter on a wall of the test section. The stagnation pressure Pjet of helium jet is 123 kpa, and the Pjet of nitrogen jet is 136 kpa. These parameters are set to ensure that the jet-to-freestream momentum flux ratio J of the helium and nitrogen jets is 2.9, and this is equal to the value of J in Gruber’s experiment [30]. The origin of the Cartesian coordinate system is at the center of the orifice, and the X, Y, Z denote streamwise, transverse and spanwise directions respectively, as shown in Fig. 1. In the experiments, NPLS is used to capture the fine turbulent structures by the adoption of nanoparticles [40, 41], and PIV is applied to obtain the velocity distribution. The NPLS is a new flow visualization based on Rayleigh-scattering, and the uncertainty about the fidelity of tracer particles in PIV and NPLS has been estimated
4
in [42]. The laser sheet in NPLS and PIV is the XOY plane.
Fig. 1 Computational region sketch map (the characteristic length d is 4 mm, which is the boundary layer thickness).
3. Numerical Methods 3.1 LES equations In the present study, the filtered dimensionless compressible Navier-Stokes equations for an ideal nonreactive
：
gas are as follows
∂ρ ∂ ( ρ ui ) + =0 ∂t ∂ xi ∂ ( ρ ui ) ∂ + ∂t ∂ xj ∂
( ρ E ) + ∂t
1 1 sgs ρ ui u j + pδ ij − Re τ ij − Re τ ij = 0
~ ( ρ E + p ) ui − κ 0T0 (κ + κ t ) ∂ T − 1 u j τ ji − 1 ( µ + µt )∑ hm ∂Ym = 0 Re Sc Prt m µ0u0 2 Re ∂ xi Re ∂ xi ∂x
∂
(2)
(3)
i
∂
( ρY ) +
1 µ µ t ∂ Ym ∂ ρ Ym ui − ( + ) =0 ∂ xi Re S c S ct ∂ xi
(4)
∂ui ∂u j 2 ∂uk 2 sgs + − δ ij − ρ k δ ij ∂x 3 j ∂xi 3 ∂xk
(5)
m
∂t
Herein,
(1)
τ ijsgs is defined as τ ijsgs = µ t
5
1 2
In these equations, the filtered total energy is defined as E = e + uk uk + k sgs , and the turbulent Prandtl number Prt is 0.72. The blending function Γ [43] connects the one-equation Yoshizawa SGS [44, 45] and Menter’s SST turbulent model [46], as shown in eq. (6). ) ∂( ρ k ) ∂( ρ ku ∂ j + = ∂t ∂x j ∂x j
3 k 2 ∂k ( µl + µt ) + Pk − ρ ΓCd1 kω + (1 − Γ ) Cd2 ∂x j ∆
，
1
µt = ρ ΓµtRANS + (1 − Γ ) µtsgs vtsgs = Cv ( k ) 2 ∆ ∆ = ( ∆x∆y ∆z )
1 3
，v
RANS t
=
(6)
，
(7)
a1 k max ( a1ω , ΩF2 )
3.2 Numerical schemes and grids distributions Inviscid flux is solved with the fifth-order WENO scheme proposed by Jiang and Shu [47], while viscous flux is discretized by the second-order-accurate centered scheme. The time advancement is performed with the third-order Runge–Kutta scheme. To generate and sustain an unsteady incoming turbulent boundary layer, recycling-rescaling procedure of Xiao et al. [48, 49] is applied. The length of the recycling-rescaling region is 10d (the boundary layer thickness d is 4 mm, and this is the characteristic length in the study). The recycle plane, computational region and boundary condition are shown in Fig. 1. The number of fine grid, medium grid and coarse grid is N x × N y × N z = 589 × 193 × 138 , 407 × 172 × 122 , 210 × 130 × 100 , respectively. The grids distributions are shown in Table. 1. Table. 1 Grids distribution. Recycle region
Core region
X direction
Y direction
Z direction
X direction
Z direction
Fine
X+=50
Y/d=0~2.5: Y+=1~10
Z+=14.5
X+=5
Z+=5~12.5
Medium
X+=80
Y/d=0~2.5: Y+=1~20
Z+=20
X+=5~10
Z+=5~16
Coarse
X+=129
Y/d=0~2.5: Y+=1~30
Z+=25
X+=5~20
Z+=5~20
6
4. Results and discussions 4.1 Comparisons with experiments First, the time-averaged streamwise velocity profile of inflow obtained by PIV, RANS and RANS/LES with recycling-rescaling procedure are compared to confirm the reliability of recycling-rescaling method, as shown in Fig. 2. The mean velocity profile on the fine and medium meshes shows good grid convergence and reasonable agreement with the RANS profile and PIV data, but the coarse mesh underpredicts the streamwise velocity in the logarithmic region. So the hybrid RANS/LES approach with fine and medium meshes can be used to achieve the average statistical data of turbulent boundary layer.
Fig. 2 Time-averaged streamwise velocity profile comparison of PIV, RANS and RANS/LES with recycling-rescaling. For helium jet, the time-averaged streamwise velocity distribution of PIV in median plane (Z/d=0) is shown in Fig. 3. The streamwise velocity u is normalized by incoming velocity magnitude U ∞ . As the particles can not enter the core region of jet near the orifice, a blue part in the average result of PIV emerges, and the velocity distribution can not be obtained at this location. The time-averaged streamwise velocity profiles of PIV extracted from Fig. 3 at X/d = 4 and X/d = 12 are compared with the hybrid RANS/LES velocity profiles. Since there is no particles in the blue region near the bottom wall at X/d=4, the velocity distribution can not be obtained under
7
Y/d=0.8 as shown in Fig. 4 (a). Except the no particle region, it is clear that the fine and medium meshes show good agreement with the experimental statistics. However, the coarse mesh overpredicts the velocity at the upper part of the profile and underpredicts the velocity at the lower part in the two figures. Thus, the hybrid RANS/LES code with the fine and medium meshes can be applied to investigate the transverse jet in supersonic crossflow.
Fig. 3 Time-averaged streamwise velocity distribution of PIV in median plane (Z/d=0).
(a) X/d=4
(b) X/d=12
Fig. 4 Time-averaged streamwise velocity profiles of PIV and RANS/LES (helium jet). The instantaneous results of NPLS and hybrid RANS/LES simulation in helium jet are shown in Fig. 5. The turbulent structures of the shear layer are similar, inclining toward the upstream direction, and this is because of the drag from the lower-speed crossflow. By comparison, it can be observed that the small-scale structures in the windward shear layer can be captured with fine mesh, instead of the medium and coarse meshes. To investigate characteristics of the turbulent structures, the fine mesh is adopted in this study. 8
(a) NPLS
(b) RANS/LES(fine)
(c) RANS/LES(medium)
(d) RANS/LES(coarse)
Fig. 5 Instantaneous NPLS and RANS/LES results in Z/d=0 (helium jet).
4.2 Development of large-scale vortex structures The NPLS results of nitrogen and helium jets are shown in Fig. 6. By comparing NPLS images, three marked differences are observed. First, the major distinctness is that the size of the large-scale turbulent structures and the interval between the large-scale turbulent structures in nitrogen jet are both larger than that in the helium jet. The similar phenomenon is also observed in the experiments of Gruber et al. [30], Benyaka et al. [33] and in simulations of Takahashi et al. [32], Watanabe et al. [37]. This point is to be discussed later in this section. Then, the second difference exists in density of the jet plume: since the density is in proportion to the light intensity qualitative in NPLS images, the light intensity of helium jet plume is weaker than that of nitrogen jet. Finally, the last difference lies in the waves of the flowfield: a series of compression waves are observed in helium jet, inclining at an angle of about 45 degrees to the wall surface, which may be due to the extraordinary instability of the bow shock.
9
(a) nitrogen jet
(b) helium jet
Fig. 6 Instantaneous jet plume of NPLS in Z/d=0. To investigate the first major distinctness mentioned above, the second invariant of the velocity-gradient tensor Q [50] is used to capture the large-scale vortex structures in hybrid RANS/LES simulation. The expression of Q is shown as eq. (8). The instantaneous iso-surface of Q=5 colored by mass fraction y1 is shown in Fig. 7 (a). The instantaneous iso-surface of mass fraction y1=0.4 is shown in Fig. 7 (b). Leading edge vortices [29] marked L are denoted by the arrows in Fig. 7 (a). The leading edge vortices form periodically in the windward shear layer, which develop between injectant of Z direction and crossflow fluid of X direction. In Fig. 7 (b), the location of rolled up injectant coincides with the location of leading edge vortices in Fig. 7 (a), so the iso-surface of mass fraction can reflect the location and shape of large-scale vortex structures. In Fig. 7 (c), the arrows denote hanging vortices [28] marked H. The hanging vortices are produced in the skewed mixing layer on the lateral edges of the jet. As the hanging vortices move downstream, the CVPs are formed [24]. The leading edge vortices and hanging vortices are also shown in Fig. 8.
Q=
1 2
(W W ij
1 ∂u i ∂u j W ij= 2 ∂ − ∂ xi xj
ij
−
SS
，S
ij
= ij
ij
)
1∂
u 2 ∂ x
10
∂ i
j
+
u ∂x
j
i
(8)
(a) Q=5
(b) y1=0.4
(c) Q=5 (d) y1=0.4 Fig. 7 Iso-surface of second invariant of the velocity-gradient tensor Q and mass fraction y1.
Fig. 8 Slice of vorticity in X/d=0.5 in Fig. 7. The development of large-scale turbulent structures in windward shear layer of nitrogen and helium jets is shown in Fig. 9 and Fig. 10. In these figures, the iso-surface of mass fraction y1=0.4 is colored by
11
non-dimensional transverse velocity v / U ∞ . At T1 in nitrogen jet, the leading edge vortices marked L-1 start to form as the arrow denotes in Fig. 9(a). At T1 + 2 ⋅ ∆t ( ∆t ≈ 3.8 ×10−6 s ), the injectant tends to roll up towards Y direction under the effect of leading edge vortices. At T1 + 8 ⋅ ∆t , the rolled up part (marked U-1) of jet reaches the upside of shear layer. At the same time, the new leading edge vortices L-2 form at front part of jet plume. At T1 + 10 ⋅ ∆t , the U-1 moves downstream as a whole, while the windward shear layer produces the new leading edge
vortices L-3 and roll-up part U-2. In this process, the leading edge vortices are formed steadily and periodically, and this is the main reason to roll up the periodic large-scale structures in shear layer.
(a) T1
(b) T1 + 2 ⋅ ∆t
(c) T1 + 8 ⋅ ∆t
(d) T1 + 10 ⋅ ∆t
Fig. 9 The development of windward shear layer in nitrogen jet. However, the shear layer of the helium jet in Fig. 10 is different from that of nitrogen jet. At T2 + 3 ⋅ ∆t ( ∆t ≈ 4.0 ×10−6 s ) the leading edge vortices L-1 are formed as the arrow denotes in Fig. 10 (b). At T2 + 5 ⋅ ∆t , part of the helium jet marked U-1 tends to roll up towards Y direction under the effect of leading edge vortices. But, the whole part of U-1 breaks up into smaller vortex structures quickly at T2 + 7 ⋅ ∆t , and this is quite different from the behavior of stable U-1 in nitrogen jet in Fig. 9.
12
(a) T2
(b) T2 + 3 ⋅ ∆t
(c) T2 + 5 ⋅ ∆t
(d) T2 + 7 ⋅ ∆t
Fig. 10 The development of windward shear layer in helium jet. In response to the difference raised above, we investigate how the phenomena occurs. It is known that the molecular weight of nitrogen is 7 times heavier than helium, and the specific heat ratio of helium is 1.2 times larger than the nitrogen. Therefore, the velocity magnitude of sonic jet for helium jet is almost 3 times larger than that for nitrogen jet, and this leads to different velocity distributions of windward shear layer as shown in Fig. 11. To mark the region of shear layer, the 0.15 and 0.85 mass fraction tracks are drawn with black solid lines. For nitrogen jet in Fig. 11 (a), the velocity of the region in the blue dashed line is similar with the velocity value of crossflow, and this indicates that the magnitude of velocity gradient dU dY in shear layer is nearly 0, as shown in Fig. 12 (a). By comparison, the magnitude of velocity gradient dU dY is larger in windward shear layer of helium, due to the larger velocity difference between shear layer and crossflow, as shown in Fig. 11 (b). This difference leads to the obvious distinctness in the instantaneous large-scale vortex structures and shear layer between helium and nitrogen jets in Fig. 9 and Fig. 10.
13
(a) nitrogen jet
(b) helium jet
Fig. 11 Time-averaged velocity distribution in nitrogen and helium jets.
(a) nitrogen jet
(b) helium jet
Fig. 12 Distribution of dU / dY in nitrogen and helium jets. Overall, the development of shear layer in nitrogen jet is as follows: with the leading edge vortices periodically forming around the windward shear layer, part of injectant is rolled up and gets into the main flow, which indicates that the behavior of leading edge vortices mainly influences the penetration of injectant. Due to the weak shear, the roll-up part is stable and moves downstream as a whole. In contrast, the leading edge vortices are also periodically formed in helium jet, but the large velocity gradient in windward shear layer makes the rolled up part break up quickly. Therefore, the turbulent structures in helium jet are smaller than that of nitrogen jet, and no large roll-up part are observed intruding into the main flow.
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4.3 Spatial correlation The spatial correlation of the mass fraction fluctuations can express the spatial extent of the turbulent scalar field [11], so this method is employed to capture the large turbulent structures in slices of X/d=0 and Z/d=0. The expression is as follows: 1 N Rxy ( x, y ) =
∑
y1′i ( x, y ) = y1i ( x, y ) −
N i =1
y1′i ( x0 , y0 ) ⋅ y1′i ( x, y )
′ ( x0 , y0 ) ⋅ yrms ′ ( x, y ) yrms 1 N
，i = 1, 2,3 ⋅⋅⋅ N (9)
′ ( x, y ) = ∑ i =1 y1i ( x, y ), yrms N
1 N
∑ i =1 y1i′2 ( x, y ) N
In eq. (9), y1′i ( x, y ) is the instantaneous injectant mass fraction fluctuation, y1′rms ( x, y ) is the root mean square(RMS) of the mass fraction fluctuation, and
( x0 , y0 ) is the reference location with which the features are correlated
( x, y ) is an arbitrary location in computational region. In Fig. 13, the center of orifice is at Z/d=0, and the reference point is on the iso-line of y1=0.12 at Y/d=0.7.
The L and H marked in the figures represent the region dominated by leading edge vortices and hanging vortices mentioned in section 4.2 respectively. The U represents the region where U-1 or U-2 influences.
(a) nitrogen jet
(b) helium jet
Fig. 13 The correlation of mass fraction in nitrogen and helium jets at X/d=0. For nitrogen jet, the correlation distribution is almost symmetric on both sides of the center line Z/d=0, and
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this reflects that the behavior of large-scale vortex structures on both sides of the center line is nearly symmetric. The correlation between L and H is opposite, indicating there is a phase difference between L and H. Moreover, the region L is negatively correlated with the region U. The reason is as follows: according to the analysis in section 4.2, it is revealed that as the rolled part U moves downstream, the mass fraction of injectant in this region is decreasing, while the next leading edge vortices start to form, which leads to the increasing of mass fraction at L. So the correlation between L and U is contrary. For the helium jet, the correlations distribution are mostly the same with nitrogen jet, as shown in Fig. 13(b). The mass fraction correlation maps of Z/d=0 plane are shown in Fig. 14. The center of orifice is located at the origin. The reference point is on the iso-line of y1=0.45 at X/d=1. In nitrogen jet, it is obvious that four regions marked 1, 2, 3, 4 are in the shear layer between X/d=0 and X/d=9. The correlation of region 1 and region 3 is positive, and the correlation of region 2 and region 4 is negative. Since the large-scale structures of nitrogen and air appear alternately as shown in Fig. 6 (a), the positive and negative regions are formed by turns. In helium jet, there are 7 regions in the shear layer, the positive and negative regions also appear alternately. Compared with nitrogen jet, the area of these regions is smaller, and the interval is shorter, and this has a good agreement with the instantaneous image in Fig. 6 (b).
(a) nitrogen jet
(b) helium jet
Fig. 14 The correlation of mass fraction in nitrogen and helium jets at Z/d=0. 4.4 Probability distribution of mass fraction To compare the difference in mixing effect between nitrogen and helium jet, this section focuses on the 16
probability distribution of mass fraction at X/d=2 as shown in Fig. 15. For facilitating the statistics, the mass fraction is divided into four intervals: 0 ≤ y1 < 0.25 , 0.25 ≤ y1 < 0.5 , 0.5 ≤ y1 < 0.75 , 0.75 ≤ y1 < 1 . The nitrogen plume core is mainly in 0.75 ≤ y1 < 1 , and the probability of mass fraction in core region is close to 0.75. For helium, the value is nearly 0.5, and this indicates that the mixing effect for helium jet is better than that of nitrogen jet.
(a) nitrogen jet
(b) helium jet Fig. 15 The probability distribution in nitrogen and helium jets at X/d=2.
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4.5 Penetration height By comparing ethylene and hydrogen jets, Ben-Yakar [30] observed that the penetration height of ethylene jet is higher than the hydrogen jet at the constant jet-to-freestream momentum flux ratio. Thus, she pointed out that the penetration height is not only related to the jet-to-freestream momentum flux ratio, but also affected by the characteristic of jet shear layer. In this section, the method measuring the penetration height by mass fraction [39] is adopted, and the 10% iso-line of mass fraction is regarded as the penetration height. With the constant jet-to-freestream momentum flux ratio, the results obtained by hybrid RANS/LES simulation are observed that the penetration height of nitrogen jet is higher than the helium jet, as shown in Fig. 16, and this is mainly due to the different development in shear layer between nitrogen and helium jet mentioned in section 4.2.
Fig. 16 Penetration height of nitrogen and helium jets.
5. Conclusions Combined with PIV and NPLS measurements, hybrid RANS/LES simulation was performed to investigate the development of large-scale turbulent structures and the effect of molecular weight on transverse jet in supersonic crossflow. We came to the following conclusions:
The hybrid RANS/LES numerical results with fine and medium meshes show good agreement with experimental data, and the simulation with fine mesh can capture the small-scale structures in jet plume clearly instead of medium and coarse mesh. The coarse mesh shows obvious deficiency in predicting the 18
velocity of inflow and jet plume. So the hybrid RANS/LES approach with fine mesh in the present study can be applied to investigate the issues of sonic transverse jet in supersonic crossflow.
By the hybrid RANS/LES simulation, two kinds of large-scale vortex structures are observed in the transverse jet of supersonic crossflow, namely leading edge vortices and hanging vortices. The mixing effect in the near field is influenced by these two large-scale structures, and penetration height of jet is mainly related to the behave of leading edge vortices.
The helium jet achieves a better mixing because of the earlier break-up turbulent structures, and the nitrogen jet has a higher penetration height due to the stabler and later break-up turbulent structures intruding into the main flow.
Through the spatial correlation analysis, there is a phase difference between leading edge vortices and hanging vortices. Additionally, the region L is negatively correlated with the region U.
Acknowledgements The authors would like to express their thanks for the support from the National Natural Science Foundation of China(No. 91216120). Also, the authors thank the anonymous reviewers for some very critical and constructive recommendations on this article.
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