Nomenclature

A Area

a Sonic speed

h Height of separation bubble

L1Length of separation bubble

L2Distance between the top of separation bubble and reattachment point

（1）Asymmetry of confined SBLI is closely relevant to the strength of SBLI，i.e.，the shock intensity and the confinement level.

MacConvective Mach number

Mass flow rate

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NPR Nozzle pressure ratio，p0/pa

（2）Asymmetry of SBLI has a flipping phenomenon，i.e.，asymmetric shock system can flip between two sides of a nozzle.Flipping does not happen during agiven test run in nozzle experiments，but could takes place between runs.

r Velocity ratio across mixing layer

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s Density ratio across mixing layer

T Time

where L1is the length of separation bubble in the imaginary symmetric flow，pexthe pressure at nozzle exit，hexthe mainstream height at nozzle exit，αis the nozzle divergence angle，andβis the deflection angle of flow from horizontal direction on the nozzle symmetry plane （Fig.3）.Then，from Eq.（13）we obtain

V Volume

α Nozzle divergence angle

γ Specific heat ratio

where pmis the average pressure in the bubble.Then according to the first assumption，the pressure difference between two sides in the nozzle is obtained.

δ2 Vertical coordinate of the edge of mixing layer on low－velocity side

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Spreading rate of mixing layer

η Similarity variable

ρ Density of fluid

Φ Normalized spreading rate of mixing layer

Subscripts

0 Stagnation state at nozzle entrance

1 High－velocity side of mixing layer

2 Low－velocity side of mixing layer

a Ambient condition

b Separation bubble

e Entrainment

δwLocal thickness of mixing layer

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ex Nozzle exit

m Averaged

0 Introduction

Asymmetry phenomenon of shock/boundary layer interaction（SBLI）in a completely symmetric duct with symmetric flow conditions，e.g.，an asymmetric shock（see Fig.1），has been observed frequently by many researchers.The reason for the flow asymmetry is still an open question，and is clarified neither by experiment nor computational fluid dynamics（CFD）［1］.

Fig.1 Experimental schlieren of asymmetric lambda shock in a planar nozzle［2］

When a supersonic nozzle is operated at pressure ratio well below its design point，a shock system forms inside the nozzle and SBLI comes into being，which probably separates flow downstream of the shock from the nozzle walls and brings an asymmetric flowfield.The asymmetry in the nozzle flow can yield dangerous lateral forces，the so－called side－loads，which may damage the nozzle［1］.Lawrence［3］studied the nature of flow asymmetry in planar and axisymmetric nozzles.He observed different symmetric and asymmetric flow structures.Papamoschou et al.［2］ experimentally investigated the supersonic nozzle flow separation inside planar convergentdivergent nozzles.Their study shows that for the area ratio of nozzle exit to its throat Ae/At≥1.4 and NPR＞1.4，the flow pattern is asymmetric.This asymmetry does not flip during agiven test run，but it can change from one wall to the other and from one run to the next.This phenomenon was also observed by Shimshi et al.［4］.Bourgoing et al.［5］found that shock configuration in a Mach 2planar nozzle is transformed from a symmetric pattern to an asymmetric one and asymmetric one again with the variation of NPR.A series of largeeddy simulations（LES）were conducted by Olson et al.［6］to model the asymmetry and unsteadiness of the SBLI in the planar nozzles tested by Ref.［2］.

Based on the aforementioned research on the asymmetry of SBLI in nozzles，some basic conclusions can be drawn：

Ma Mach number，mass

p Pressure

The reason for the asymmetry of confined SBLI is not clear yet，but many researcher，e.g.，Lawrence［3］，Papamoschou et al.［2］， Myshenkov［7］， Wang［8］，attributed it to Coanda effect which is used for the tendency of a fluid jet issuing tangentially on to a curved or angled solid surface to adhere to it［9］.The entrainment of jet on ambient fluid is regarded as the cause for Coanda effect，but so far it has not been understood completely yet［10］，even though it has been applied widely in industry.Coanda effect was confirmed experimentally by Allery et al.［11］to work in symmetrical configurations as it does in the single wall case：the jet reattaches randomly to either of two walls.

The study is motivated by the research of Piponniau et al.［12］who proposed a model based on the properties of fluid entrainment in the mixing layer to explain low frequency unsteadiness on shock induced separation.Due to the close relation between his model and Coanda effect，Piponniau′s model is developed further here to explain the reason for the asymmetry of SBLI in an overexpanded nozzle whose flowfield data were obtained by numerical simulations.

1 Theory

1.1 Aerodynamic scheme

According to Piponniau et al.［12］，when a separation bubble is produced by SBLI（Fig.2），in the first part of the bubble，that is from the separation line，eddies are formed in the mixing layer zone（the reversed flow in the separation bubble is regarded as the other stream）and grow as it moves downstream.Fluid from the separated zone is entrained by the mixing layer.Near the middle of bubble（where the mixing layer has the maximum thickness），these eddies are shed into the downstream flow，bringing with them their mass，momentum and vortices outside the separated region.This generates，in the recirculating region，a default of mass that increases over time.Therefore， when the flow reattaches downstream，the mass amount inside the bubble decreases.

Fig.2 Sketch of the entrainment of mixing layer on separation bubble［12］

Now we consider the situation of asymmetric SBLI in a 2Dnozzle，whose flowfield pattern is sketched in Fig.3.Coanda effect works in this flow，entrainment of mixing layer on separation bubble on the upper wall decreases the mass and pressure inside the bubble，and the opposite phenomena happen on the lower wall.The pressure difference between two sides deflects the flow behind the shock to the upper wall.To understand how the result of asymmetry takes place，an imaginary symmetric flowfield，which is supposed to be the situation before the asymmetric flow pattern appearing，is sketched in Fig.4.Prior to create the asymmetry model，three assumptions are made in this situation：

（1）Based on the feature of Conada effect introduced in Introduction，it is assumed that the entrainment of mixing layer only activates on one side，while the separation bubble on the other side keeps constant.

Fig.3 Asymmetric SBLI pattern in a 2Dnozzle

（2）After the shedding of eddies，the mass and pressure decrease inside the bubble but its size keeps constant.

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（3）If the final flow is still symmetric，a new amount of reversed flow from downstream enters the bubble at reattachment point Rto insure the balance［13］.

In the plane reflected shock case，large vortices are shed downstream at the position where the mixing layer reaches its maximum thickness，i.e.，the position with maximum separation bubble thickness，which is approximately at the middle of separation bubble［14］.But for SBLI in a nozzle，the situation is a little different，the divergent wall leads to slender rear of separation bubble，and the maximum thickness of the bubble lies in the front of it instead of the middle as the plane case.Therefore，the shedding position for large vortices is still regarded as the position with the maximum bubble thickness in the nozzle case，but not at the middle of bubble.The distance from large vortices shedding position to the reattachment point is denoted as L2（Fig.4）.

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Fig.4 Imaginary symmetric SBLI flowfield before asymmetric flow taking place in a 2Dnozzle

Based on the above assumptions and deduction，now we can discuss the total mass entrained from separation bubble.Total entrained mass is the product of entrainment mass rate m·eand entrainment time Te.The rate of mass entrainment can be obtained from Piponniau′s model，which will be deduced later.According to the third assumption，the distance between mass escaping from the bubble and mass returning into the bubble is L2，and then the time used for entrainment is given by

where a2is the sonic speed on the low－velocity side of the mixing layer.

With the total entrained mass，the pressure inside the bubble after entrainment can be obtained based on the second assumption.Finally，the deflection angle of the flow will be obtained from the longitudinal momentum conservation equation，which will be compared with the actual deflection angle of the flow.

1.2 Theoretical model for flow deflection with entrainment

According to entrainment model of Piponniau′s，the separation bubble is approximated by a triangle of length L1and height h （Fig.2），with an average density ofρm，then the air mass in the bubble by unit span is given by

And rate of mass entrainment［11］

whereδ2（x）is the vertical coordinate of the edge of the mixing layer on the low－velocity side，y0（x）the vertical coordinate of the centerline of the mixing layer，and x0＝L3＝L1－L2the position where large eddies shed，ηis the similarity variable，and the constant C0.14.δwandδ′ware the local thickness and the spreading rate of the mixing layer，respectively，and the latter can be expressed by the following relation according to Ref.［15］

where0.16is the spreading rate for subsonic half jet［16］.r＝u2/u1and s＝ρ2/ρ1are the velocity ratio and density ratio across the mixing layer，respectively.The functionΦ（Mac）is the normalized spreading rate of mixing layer and is dependent on the convective Mach number Mac

Φ （Mac）must be determined via experiment［17］and Fig.5gives its empirical value depending on the convective Mach number.Piponniau et al.［12］introduced a function gas

Fig.5 Normalized spreading rate as a function of the convective Mach number［12］

Finally，the rate of mass entrainment is obtained.

Consequently，we discuss the pressure variation in the bubble.The total mass of entrainment

Based on the second assumption，the volume of bubble keeps constant，and the temperature in the bubble can be considered as invariant，then the density and pressure in the bubble after entrainment

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Γ Index for flow asymmetry

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With the pressure difference obtained from the above，we can calculate the deflection angle of flow，concerning the vertical momentum conservation for the control volume in Fig.4.

u Velocity

The above asymmetry model is deduced based the nozzle flow with closed separation bubble， namely， restricted shock separation（RSS）［1］.When the separation zone is open to the ambiance，namely，free shock separation（FSS）（Fig.6），although the situation is a little different from RSS，the theoretical model created above is still applicable in this case.Now separation length L1is the distance from the onset of separation to the nozzle exit，and L2the distance from the position with maximum separation thickness to the nozzle exit.

Fig.6 Imaginary symmetric nozzle SBLI flowfield with FSS

2 Model Application Based on Numerical Results of a Nozzle

2.1 Nozzle geometry and numerical methods

Simulations were conducted to obtain the detailed features of asymmetric SBLI in a planar nozzle.The nozzle model chosen for simulation is aplanar nozzle tested by Papamoschou et al.［2］and simulated by Xiao et al.［18］，which has a throat height of 22.9mm，a length of 117mm and an area ratio（the area ratio of nozzle exit to throat）of 1.5.The nozzle is″trumpet－shaped″with the wall angle increasing monotonically from throat to exit.The maximum nozzle divergence angle is 3.83°.

To apply the asymmetry model proposed above，forced symmetric（half）and full nozzle model were simulated，respectively.The Reynolds－averaged governing equations for compressible turbulent flow with a two equation SST turbulence model in CFX software was employed to simulate idea gas（γ＝1.4）steady flows.The grid used in the simulations has a higher density near the wall and the minimum first grid point from the wall gives y＋＜1.The total number of grids used is 89 800for the full model and 51 900for the half model.Adiabatic，no slip wall boundary condition was specified for all the walls in these simulations.

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Computations were made for NPR between 1.20and 2.40by changing the total pressure at the nozzle entrance.Other boundary conditions were imposed as follows：The ambient pressure surrounding nozzle exit pa＝101 325Pa，total temperature at nozzle entrance Tt0＝290K.

2.2 Numerical results of asymmetric SBLI in full nozzle

Fig.7shows the pressure distributions on the two walls of the nozzle from simulations and experiments at two NPRs （the abscissa is normalized by the height of nozzle throat htand the vertical axis is normalized by total pressure at nozzle entrance pt0）.Note that the nozzle throat is located at x＝0.It can be seen that the flow is symmetric at NPR ＝ 1.27and is asymmetric at NPR＝1.61based on both numerical and experimental data.At NPR＝1.61，though both numerical and experimental results give the asymmetric flowfield，the deflection direction of flow is opposite：The flow from simulation is deflected downward while that from experiment is deflected upward，which is caused by the randomicity of Coanda effect（entrainment）mentioned in Section 1.The shock positions from simulations are a little more upstream than experimental data，except this difference numerical results are satisfactory.

Fig.7 Wall pressure distributions of the nozzle from simulations and experiments

Three typical flow patterns at different NPRs are shown with numerical schlieren in Fig.8.At NPR＝1.27，a symmetric flowfield with RSS on two sides is presented as Fig.8（a）.At NPR＝1.61，an asymmetric flowfield with FSS on one side and RSS on the other side appears as Fig.8（b）.At NPR＝2.40，a symmetric flowfield with FSS on two sides is obtained as Fig.8（c）.

Fig.8 Typical flow patterns in an over－expanded nozzle with numerical schlieren

Deflection angle of nozzle exit flow at different NPRs are shown in Fig.9 （squares）.It can be seen that the flow in the nozzle is symmetric as Fig.8（a）when NPR≤1.27，while it is asymmetric as Fig.8（b）after that and deflection angle becomes larger and larger with increasing NPR up to 2.10where the peak asymmetry reaches.Then，the flow returns to be symmetric as Fig.8（c）when NPR＞2.20.

Fig.9 Deflection angle of flow in the nozzle exit from full nozzle simulations and theoretical results

2.3 Model application based on numerical results of half nozzle with forced symmetry

The nozzle flowfields with forced symmetry were computed to apply the asymmetry model.The numerical results show that the flows with RSS （Fig.10（a））were obtained when NPR≤1.61，while the flowfields with FSS （Fig.10（b））were attained when NPR≥1.70.According to Eq.（14），the theoretical values of deflection angle of flow at nozzle exit at different NPRs have been calculated，which have been shown in Fig.9（triangles）.One can see that there is a large difference between the results from theoretical model and actual deflection from full nozzle simulations.Theoretical results give a peak of deflection angle at NPR＝1.47and then decrease gradually，which are close to the deflection of full nozzle only around NPR ＝1.70and NPR ≥2.30.The results show that there may be other factors to control the flow asymmetry besides Coanda effect.Table 1gives the aerodynamic parameters of the separation at typical NPRs.

Fig.10 Numerical schlieren and streamlines of nozzle flow with forced symmetry

Fig.11gives averaged pressure in the larger separation of full nozzle and in the separation of half nozzle with force symmetry，which shows averaged pressure in the separation on the side without entrainment is nearly constant before and after flow deflection at most NPRs except NPR＝1.34，and consequently proves the validity of the first assumption in Section 1.1.

Fig.11 Averaged pressure in the separation

Table 1 Aerodynamic parameters of separation at typical NPRs for half nozzle with forced symmetry

NPR T1/T2Ma1rsg（r，s）/10－2Mac Φ（Mac）h/mm L1/m L2/mm p2/pexMaexhex/mm 1.27 0.86 0.93 －0.033 0.84 2.09 0.46 0.67 0.15 11.4 6.8 0.69 0.53 34.35 1.47 0.84 1.01 －0.107 0.81 1.89 0.54 0.58 1.2 49.6 39 0.71 0.67 34.35 1.61 0.81 1.06 －0.128 0.79 1.80 0.57 0.55 1.7 58.0 46 0.74 0.76 30.95 2.2 0.80 1.16 －0.160 0.73 1.64 0.63 0.49 3.2 32.9 15 0.76 1.15 27.95 2.3 0.79 1.21 －0.157 0.80 1.70 0.66 0.47 3.4 29.7 11 0.77 1.15 27.55 2.4 0.76 1.26 －0.161 0.78 1.67 0.68 0.46 3.5 27.0 7 0.79 1.15 27.35

3 Conclusions

A model for the asymmetry of SBLI in nozzle flows has been proposed based on the properties of fluid entrainment in the mixing layer and momentum conservation.Deflection angles obtained from the theoretical model based on the simulation results of a half nozzle with forced symmetry show a large difference from those of the actual full nozzle，which shows there should be other factors to control the flow asymmetry besides Coanda effect（or entrainment），and the entrainment of shear layer on the separation induced by SBLI is just one of causes for the asymmetry.

式中:分别对应的是温度为T0、T时的k阶Stokes波中心频率。可以看出,k阶Stokes波相对于BP的中心频移量是1阶Stokes波的k倍,即前者的温度灵敏度系数是后者温度灵敏度系数CT的k倍。这也意味着,利用Brillouin多波长输出可以实现更高的温度灵敏度系数,如果Stokes波达到10阶或以上,则理论上可以使温度灵敏度系数提高一个数量级。

Acknowledgements

This work was supported by the National Natural Science Foundations of China（Nos.51476076，51776096）.

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