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Three dimensional investigation of the shock train structure in a convergentdivergent nozzle Seyed Mahmood Mousavi, Ehsan Roohi n High Performance Computing (HPC) Laboratory, Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, P.O. Box 91775-1111, Mashhad, Iran article info Article history: Received 23 May 2014 Received in revised form 1 August 2014 Accepted 3 September 2014 Available online 16 September 2014 Keywords: Shock train Convergentdivergent nozzle Reynolds stress model Heat source Inlet flow total temperature abstract Three-dimensional computational fluid dynamics analyses have been employed to study the compressible and turbulent flow of the shock train in a convergentdivergent nozzle. The primary goal is to determine the behavior, location, and number of shocks. In this context, full multi-grid initialization, Reynolds stress turbulence model (RSM), and the grid adaption techniques in the Fluent software are utilized under the 3D investigation. The results showed that RSM solution matches with the experimental data suitably. The effects of applying heat generation sources and changing inlet flow total temperature have been investigated. Our simulations showed that changes in the heat generation rate and total temperature of the intake flow influence on the starting point of shock, shock strength, minimum pressure, as well as the maximum flow Mach number. & 2014 IAA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Shock waveboundary layer interaction in combination with the flow separation is an important phenomenon in modern aerodynamics. This topic is related to the air flow around supersonic vehicles as well as to nozzle or diffuser flows. The latter plays a major role in the design of supersonic ramjet or scramjet inlets, internal diffusers and supersonic ejectors, supersonic air-breathing engine inlets, internal diffusers, compressor cascades or super- sonic ejectors, to name some of them. Most of those applications have convergentdivergent walls or a rectan- gular cross-section, therefore; the focus of the current investigation lies at the shock train in the convergentdivergent duct. Under certain conditions, even one or more shocks can appear downstream of the first shock (for the Mach number over about 1.5). This series of shocks is also named shock train. Other names, e.g., X-shaped shocksor λ-shaped shocks, can be found in the literature, as well. A typical Schlieren photograph of the shock train is shown in Fig. 1 [1]. In contrast to other shock systems, the supersonic flow is decelerated at first by a shock system that is followed by a mixing area as shown in Fig. 2. At the centerline, the shocks are strong enough to decelerate the flow below Ma ¼ 1, whereas the flow remains supersonic between the core flow and the boundary layer. Therefore, the flow undergoes successive changes from supersonic regime to subsonic regime [2]. In this region, the transition from supersonic to subsonic conditions is very gradual. Further- more, the experiences had shown that the static pressure continues to rise after the shock train over a certain distance along the duct if the duct is long enough. In this case, the static pressure recovery is performed through both the shock train region and the subsequent static pressure recovery region after the shocks. According to the above discussions, many researchers have investigated the shock train phenomenon experi- mentally and numerically. In this regards, Katanoda et al. [3] experimentally examined shock train structures in a constant-area passage of the cold spray nozzle. Balu et al. [4] studied the performance of an isolator scramjet engine Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/actaastro Acta Astronautica http://dx.doi.org/10.1016/j.actaastro.2014.09.002 0094-5765/& 2014 IAA. Published by Elsevier Ltd. All rights reserved. n Corresponding author. Tel.: þ98 511 8805136; fax: þ98 511 8763304. E-mail address: [email protected] (E. Roohi). Acta Astronautica 105 (2014) 117127
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Page 1: Three dimensional investigation of the shock train …profdoc.um.ac.ir/articles/a/1042912.pdfThree dimensional investigation of the shock train structure in a convergent–divergent

Contents lists available at ScienceDirect

Acta Astronautica

Acta Astronautica 105 (2014) 117–127

http://d0094-57

n CorrE-m

journal homepage: www.elsevier.com/locate/actaastro

Three dimensional investigation of the shock trainstructure in a convergent–divergent nozzle

Seyed Mahmood Mousavi, Ehsan Roohi n

High Performance Computing (HPC) Laboratory, Department of Mechanical Engineering, Faculty of Engineering,Ferdowsi University of Mashhad, P.O. Box 91775-1111, Mashhad, Iran

a r t i c l e i n f o

Article history:Received 23 May 2014Received in revised form1 August 2014Accepted 3 September 2014Available online 16 September 2014

Keywords:Shock trainConvergent–divergent nozzleReynolds stress modelHeat sourceInlet flow total temperature

x.doi.org/10.1016/j.actaastro.2014.09.00265/& 2014 IAA. Published by Elsevier Ltd. A

esponding author. Tel.: þ98 511 8805136; faail address: [email protected] (E. Ro

a b s t r a c t

Three-dimensional computational fluid dynamics analyses have been employed to studythe compressible and turbulent flow of the shock train in a convergent–divergent nozzle.The primary goal is to determine the behavior, location, and number of shocks. In thiscontext, full multi-grid initialization, Reynolds stress turbulence model (RSM), and thegrid adaption techniques in the Fluent software are utilized under the 3D investigation.The results showed that RSM solution matches with the experimental data suitably. Theeffects of applying heat generation sources and changing inlet flow total temperature havebeen investigated. Our simulations showed that changes in the heat generation rate andtotal temperature of the intake flow influence on the starting point of shock, shockstrength, minimum pressure, as well as the maximum flow Mach number.

& 2014 IAA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Shock wave–boundary layer interaction in combinationwith the flow separation is an important phenomenon inmodern aerodynamics. This topic is related to the air flowaround supersonic vehicles as well as to nozzle or diffuserflows. The latter plays a major role in the design ofsupersonic ramjet or scramjet inlets, internal diffusersand supersonic ejectors, supersonic air-breathing engineinlets, internal diffusers, compressor cascades or super-sonic ejectors, to name some of them. Most of thoseapplications have convergent–divergent walls or a rectan-gular cross-section, therefore; the focus of the currentinvestigation lies at the shock train in the convergent–divergent duct. Under certain conditions, even one ormore shocks can appear downstream of the first shock(for the Mach number over about 1.5). This series of shocksis also named “shock train”. Other names, e.g., ‘X-shapedshocks’ or ‘λ-shaped shocks’, can be found in the literature,

ll rights reserved.

x: þ98 511 8763304.ohi).

as well. A typical Schlieren photograph of the shock train isshown in Fig. 1 [1].

In contrast to other shock systems, the supersonic flowis decelerated at first by a shock system that is followed bya mixing area as shown in Fig. 2. At the centerline, theshocks are strong enough to decelerate the flow belowMa¼1, whereas the flow remains supersonic between thecore flow and the boundary layer. Therefore, the flowundergoes successive changes from supersonic regime tosubsonic regime [2]. In this region, the transition fromsupersonic to subsonic conditions is very gradual. Further-more, the experiences had shown that the static pressurecontinues to rise after the shock train over a certaindistance along the duct if the duct is long enough. In thiscase, the static pressure recovery is performed throughboth the shock train region and the subsequent staticpressure recovery region after the shocks.

According to the above discussions, many researchershave investigated the shock train phenomenon experi-mentally and numerically. In this regards, Katanoda et al.[3] experimentally examined shock train structures in aconstant-area passage of the cold spray nozzle. Balu et al.[4] studied the performance of an isolator scramjet engine

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S.M. Mousavi, E. Roohi / Acta Astronautica 105 (2014) 117–127118

with inlet Mach number 2.0 and found that for length toheight ratio of 4 to 5, shock train can be established withmaximum static pressure in the isolator. Grzona et al. [5]measured shock train generated turbulence inside an over-expanding rectangular nozzle with a small opening angleof 1.61. Weiss et al. [2,6] studied the behavior of shocktrains in a diverging duct, and then investigated thebehavior of a shock train under the influence ofboundary-layer suction by a normal slot. Numerical simu-lation and experiments on the Mach 2 Pseudo-shock wavein a square duct are performed by Sun et al. [7] on thebasis of the two-dimensional Navier–Stokes equations,using a Baldwin–Lomax turbulence model, and the Mach2 supersonic wind tunnel. The numerical results agree wellwith the experimental results. Based on these investiga-tions, the shock train characteristics, structure, pressureand velocity distributions, and the effect of flow confine-ment on the interaction are analyzed in detail. Somecomputational efforts on simulations of the shock trainin ducts have been reported. For example, Papamoschouand Johnson [8] numerically and experimentally examinedthe symmetry and asymmetry of the pseudo-shock systemin a planar nozzle. They stated that the separation of theshear layer on the side of the lambda shock foot creates anintense instability that grows into very large eddies at thenozzle exit. Kawatsu et al. [9] numerically simulated thepseudo shock wave in straight and diverging ducts withrectangular cross section and observed the separation ofthe boundary layer by the first shock wave of pseudo shockwave can be detected only near the corners of the duct. Incontrast, in the diverging duct case the large separationregion appeared at one corner of the upper wall and didnot reattach the wall in the test section. Lin and Tam [10]studied the impact of temperature and heat transfer on thestructure of the shock train inside parallel wall isolatorsboth experimentally and numerically. Their study discov-ered that heat addition to the low Mach number flow canchoke the flow and potentially decrease the isolatorperformance. They showed that heat addition to thesupersonic flow increases boundary-layer thickness anddecreases the flow Mach number. Allen et al. [11] studied

Fig. 1. Schlieren photograph showing shock train in a rectangularduct [1].

Fig. 2. Sketch of a pseudo

the shock train leading edge of a scramjet isolator usingRANS and LES numerical models and showed that RANSmodel provide reasonably suitable agreement with theexperimental results. Fotia et al. [12] examined the beha-vior of a ram-scram transition applying a direct-connectmodel scramjet experiment along with pressure measure-ments and high-speed laser interferometry. Their workshowed the wall static pressure profile and flame positionoccurring at the downstream boundary condition sud-denly changes when the flow becomes unchoked. Fischerand Olivier [13] experimentally studied the wall and totaltemperature influence on a shock train. They found thatvariations of the wall temperature influence on the pres-sure distribution in the shock train.

Gawehn et al. [14] investigated pseudo-shock systemsin a Laval nozzle with parallel side walls, both numerically(steady and unsteady simulation) and experimentally. Inthe steady case, good agreement is found between thecalculated and measured shock structure and pressuredistribution along the primary nozzle wall, except for aremaining slight deviation in the shock position. Theeffects of the divergence angle on the shock train in thescramjet isolator are investigated by Huanga et al. [15].They discovered that with increasing the divergence angleof the scramjet isolator, the static pressure along thecentral symmetrical line of the isolator decreases sharply.Grilli et al. [16] analyzed the unsteady behavior in shockwave turbulent boundary layer interaction. Their resultssupported the assumption that the observed shock-waveturbulent boundary layer interaction phenomena are aconsequence of the inherent dynamics between flowseparation and shock. Sridhar et al. [17] numericallyinvestigated the effect of geometry on the oscillatorperformance. They found that the length of the pseudoshock of square configuration is shorter than that of thecircular model. Giglmaier et al. [18] numerically andexperimentally investigated the pseudo-shock system ina planar nozzle in purpose to examine the impact ofbypass mass flow due to narrow gaps. Zhu and Jiang [19]investigated the entrainment performance and the shockwave structures in a 3D ejector investigated by CFD andSchlieren flows visualization. Their results show that theexpansion waves in the shock train do not reach themixing chamber wall when the ejector is working atthe sub-critical mode. Mousavi and Roohi [20] numericallyinvestigated the effect of working parameters such asthe inlet total pressure, back pressure, nozzle inletangle, and wall temperature on a 2D shock train in a

shock system [2].

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S.M. Mousavi, E. Roohi / Acta Astronautica 105 (2014) 117–127 119

convergent–divergent nozzle using the LES turbulencemodel. Their simulation found the exact location of thefirst shock train.

Due to immense applications of shock train phenomenonin the modern aerodynamics, the shock train structure stillrequires further investigations to understand its behaviorsunder different working parameters. In the present study, thestructure of the shock train in a convergent–divergent nozzleis investigated using the RSM turbulence model. In this work,after ensuring from the accuracy of the simulations, the effectof applying the heat generation source and the inlet flow totaltemperature (IFTT) are examined using the RSM turbulencemodel, full multi-grid (FMG) flow initialization, as well as thegrid adaption techniques under 3D investigation. These topicswere not considered in previous works.

Table 1Employed discretization schemes for different terms.

Pressure Second orderMomentum Second orderTurbulent kinetic energy Second orderTurbulent dissipation rate Second orderReynolds stress Second order

2. Numerical features

2.1. Reynolds stress model

The Reynolds stress model is a mighty turbulencemodel to predict the behavior of complex flow. This modelassumes an isotropic eddy-viscosity. Furthermore, theexact Reynolds stress transport equation, ðu0

iu0j Þ (Eq. (1)),

is directly used to compute the Reynolds stresses. It shouldbe noted that, in this model, the eddy viscosity approach isnot employed.

∂∂t

ρu0iu

0j

� �þ ∂∂xk

ρuku0iu

0j

� �¼ � ∂

∂xkρu0

iu0ju

0k þp δkju0

iþδiku0j

� �� �

þ ∂∂xk

μ∂∂xk

u0iu

0j

� �� ��ρ u0

iu0k

∂uj

∂xkþu0

ju0k∂ui

∂xk

� ��ρβ giu0

jθþgiu0iθ

� �

þp∂u0

i

∂xjþ∂u0

j

∂xi

� ��2μ

∂u0i

∂xk

∂u0j

∂xk�2ρΩk u0

ju0mεikmþu0

iu0mεjkm

� �þSuser

ð1Þ

Fig. 3. The FMG initial

Fig. 4. Pressure (Pa) contour obtai

This equation could be explained as follows:

Local Time DerivateþCij ¼DT ;ijþDL;ijþPijþGijþϕij�εijþFij

þUser�Defined Source Term

where Cij, DT ;ij, DL;ij, Pij, Gij, ϕij, εij, and Fij are the convec-tion-term, turbulent diffusion, molecular diffusion, stressproduction, buoyancy production, pressure strain, dissipa-tion, and turbulent production due to system rotation,respectively. It should be noted that, Cij, DL;ij, Pij, and Fijterms do not require modeling while DT ;ij, Gij, ϕij, and εijhave to be modeled to close Eq. (1).

In the present study, the “quadratic pressure–strainmodel” is employed for modeling the pressure strainparameter ðϕijÞ. This model has been demonstrated to givesuperior performance in a wider class of complex engi-neering flows [21,22]. The quadratic pressure–strainmodel can be selected as an option in the Viscous Modelpanel. This model is written as follows:

ϕij ¼ � C1ρεþCn

1P�

bijþC2ρε bikbkj�13bmnbmnδij

� �þ C3�Cn

3

ffiffiffiffiffiffiffiffiffiffibijbij

q� �ρkSij

þC4ρk bikSjkþbjkSik�23bmnSmnδij

� �þC5ρk bikΩjkþbjkΩik

� ð2Þ

where the isotropy tensor (bij), mean strain rate (Sij), andthe mean rate-of-rotation tensor ðΩijÞ are respectively

ization method.

ned after FMG initialization.

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Fig. 5. Sketch of the nozzle geometry considered in the present work, X-axis starts at the throat.

Table 2Boundary condition applied to the nozzle.

P0 (kPa) PStatic (kPa) T0 (K) v (m s�1) A (cm2)

Inlet 490 460 298 89.27 0.49Outlet – 325 298 – 1.96

Table 3Boundary conditions for turbulent characteristics.

Inlet Outlet

Backflow turbulent intensity (%) 3.4 3.3Backflow hydraulic diameter (mm) 47.26 62.087Backflow UU Reynolds Stress (m2/s2) 1 1Backflow VV Reynolds stress (m2/s2) 1 1Backflow WW Reynolds stress (m2/s2) 1 1Backflow UV Reynolds stress (m2/s2) 0 0Backflow VW Reynolds stress (m2/s2) 0 0Backflow UW Reynolds stress (m2/s2) 0 0

Ave

rage

Mac

h N

umbe

r

1E+06 1.5E+06 2E+06 2.5E+06 3E+06 3.5E+06Number of Cells

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Ave

rage

Re δ

Num

ber

1400

1600

1800

2000

2200

2400

2600

2800

3000

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

Ave

rage

Cen

terl

ine

Pres

sure

(bar

)

Fig. 6. Effect of the grid size on the accuracy of different flow parameters.

S.M. Mousavi, E. Roohi / Acta Astronautica 105 (2014) 117–127120

defined as follows:

bij ¼ ��ρu0

iu0j þ2

3ρkδij2ρk

!ð3Þ

Sij ¼12

∂uj

∂xiþ∂ui

∂xj

� �ð4Þ

Ωij ¼12

∂ui

∂xj�∂uj

∂xi

� �ð5Þ

The constants are as follows:

C1 ¼ 3:4; Cn

1 ¼ 1:8; C2 ¼ 4:2; C3 ¼ 0:8; Cn

3 ¼ 1:3; C4 ¼ 1:25; C5 ¼ 0:4

This model does not require a correction to account for thewall-reflection effect in order to obtain as at is factory solutionin the logarithmic region of a turbulent boundary layer.

2.2. Discretization scheme

To get accurate simulation results, the choice of numer-ical scheme is almost as important as the choice ofturbulent model. For this reason, in the present work,the second order upwind scheme has been used fordiscretized, as reported in Table 1. The correspondingscheme is based on the fact that two upwind nodes aretaken into account when estimating the eastern face value.It is assumed that the gradient between the present nodeand the eastern face is the same as between the westernnode and the present node.

ϕe�ϕP

xe�xP¼ϕP�ϕw

xP�xw) ϕe ¼

ϕe�ϕP

� xe�xPð Þ

xP�xwþϕP ð6Þ

When second-order accuracy is desired, quantities at cellfaces are computed using a multidimensional linear recon-struction approach. In this approach, higher-order accu-racy is achieved at cell faces through a Taylor seriesexpansion of the cell-centered solution about the cellcentroid. Thus, when second-order upwinding is selected,the face value ϕf is computed using the following expres-sion:

ϕf ¼ϕþ∇ϕUΔ s! ð7Þ

where ϕ and ∇ϕ are the cell-centered value and itsgradient in the upstream cell, and Δ s! is the displacementvector from the upstream cell centroid to the face centroid.This formulation requires the determination of the gradi-ent ∇ϕ in each cell. This gradient is computed using thedivergence theorem, which in discrete form is written as,

∇ϕ¼ 1V

∑Nfaces

f

~ϕf A! ð8Þ

Here the face values ~ϕf are computed by averaging ϕ fromthe two cells adjacent to the face. Finally, the gradient ∇ϕ islimited so that no new maxima or minima are introduced.

2.3. Grid adaption

In the numerical simulations, grid adaptation is a numer-ical technique for increasing the accuracy of numericalsolutions in certain regions where there are strong flowgradients. This method begins with the entire computationaldomain covered with a coarsely resolved base-level regularCartesian grid. As the calculation progresses, individual gridcells are tagged for refinement, using a criterion that caneither be user-supplied or based on Richardson extrapola-tion. All tagged cells are then refined such that a finer grid isoverlaid on the coarse one. After grid refinement, individualgrid patches on a single fixed level of refinement are passedoff to an integrator which advances those cells in time.

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Fig. 7. Configuration of 3D cells at different locations of the nozzle. (a) Nozzle intel, (b) part of nozzle throat and (c) nozzle outlet.

Fig. 8. Grid size before (top frame) and after (bottom frame) adaptation.

S.M. Mousavi, E. Roohi / Acta Astronautica 105 (2014) 117–127 121

Finally, a correction procedure is implemented to correct thetransfer process along coarse–fine grid interfaces to ensurethat the amount of any conserved quantity leaving one cellexactly balances the amount entering the bordering cell. If atsome point the level of refinement in a cell is greater thanthe required value, the high-resolution grid may be removedand replaced with a coarser one.

2.4. Overview of the full multi-grid (FMG) initialization

After initializing, the solutions were further improvedusing the Text User Interface (TUI) command of the Full

Multi-Grid (FMG) initialization of the Fluent package[23,24]. FMG has two features that speed up the conver-gence of the simulations. First, the flow is temporarilyassumed to be inviscid, which reduces the number ofequations being solved. Thus, less time is spent periteration. Based on numerical experimentation, time periteration of an inviscid case seems to be about half of thatof a turbulent case. Second, FMG utilizes a strategy knownas multi-gridding. This temporarily merges adjacent cellsinto larger cells, and then combines those new cells intolarger ones yet. The solver then converges (or partiallyconverges) a solution for the courses grid, splits to the

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X- Coordinate (m)

Wal

lpre

ssur

e(b

ar)

0 0.1 0.2 0.3 0.4 0.5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 Experiment (Weiss)RSM- 1RSM- 2K-ω SSTK-ε RNG

X- Coordinate (m)

Mac

hnu

mbe

r

0 0.05 0.1 0.15

0

0.5

1

1.5

K-K-

Experiment (Weiss)Present Work- 1Present Work- 2

ω SSTε RNG

Fig. 9. Comparison between the current numerical solutions and experi-mental data of Ref. [6]. (a) Wall static pressure and (b) enterline Machnumber.

S.M. Mousavi, E. Roohi / Acta Astronautica 105 (2014) 117–127122

intermediate refinement and reconverges, and finallyrestores the original grid and reconverges. This processdecreases the cell count, making all iterations less timeconsuming. At FMG, five tetrahedral cells make up a largertetrahedron, so the iteration time for each multi-grid levelis on the order of 5 times faster than the level just fine. TheFMG initialization iteration is illustrated in Fig. 3. An FMGinitialized solution is far closer to the final convergedsolution than a simple initialized one, i.e., Fig. 4 showsthe contour of pressure obtained for the current test caseafter FMG initialization.

2.5. Geometric configurations and boundary conditions

The nozzle geometry considered in this work isschematically shown in Fig. 5. The design consists of

the first nozzle with parallel side walls, and a lateraldivergent section right downstream. The height of thethroat and the total length of the nozzle are 6 mm and0.65 m, respectively. The distance between the nozzlethroat and the compression region is 0.16 m. The size ofthe simulated geometry and the magnitude of specifiedboundary conditions shown in Table 2 are exactly thesame as the experimental geometry and test conditionsreported in Ref. [6]. Dry air, assuming the ideal gasbehavior, is used as the working fluid. The viscosity andthermal conductivity are evaluated using a mass-weighted mixing law [15]. No-slip and adiabatic bound-ary conditions are imposed along the walls of thegeometry. The boundary conditions for turbulent char-acteristics are presented in Table 3.

3. Result and discussion

3.1. Grid properties

In order to obtain a grid independent solution, themodel was solved using six different grid sizes. The resultsof different grids for average centerline pressure in theaxial direction, average centerline Mach number, andaverage Reynolds number based on the boundary layerthickness is shown in Fig. 6. According to this figure, thedata obtained from the numerical solutions with a gridsize finer than 272�104 are almost identical. Consideringthe suitable accuracy of this grid and also the computa-tional costs, the mesh with 272�104 cells was used toperform the simulations reported in this work, sample cellconfigurations are shown in Fig. 7. The value of wall Y-plusfor this grid is around 3.1 for all locations. Fig. 8 shows theadapted grid before and after adaptation in a portion of thedoamin where the shock waves occur. The initial grid isfine near the walls while the adapted grid becomes muchfiner in regions of high gradient which include shock waveregions.

3.2. Validation of the numerical results

Fig. 9(a–b) shows a comparison between the numer-ical results of the centerline Mach number and wallpressure with the experimental data reported by Weisset al. [6]. These figures compare numerical solutions fromthe k-ω SST, k-ɛ RNG, and RSM turbulence models as wellas experimental data [6]. The employed boundary condi-tions are reported in Table 2. As shown in the figures, theresults obtained from the k-ω SST and k-ɛ RNG turbulencemodels show deviation in predicting the flow behavior incomparison with the experimental data. However, theresults of RSM models match the experimental resultswith reasonable accuracy. Comparing the results of twoexperimental data and RSM models confirm the suitabil-ity of this model for simulation of the shock trainphenomena. Therefore, in this paper, the RSM turbulencemodel has been applied. The benefit of the RSM model isthat it provides quite accurate solutions using a reason-able number of grid cells for 3D simulations. The constantvalues for Cn

1 are set either as 1.8 or 1.6. As Fig. 8 shows,case-2 (with Cn

1¼1.6) has higher accuracy than case-1

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Fig. 10. Comparison between numerical method (Mach contour) and experimental picture [6] for demonstrating number and length of the shock train,LFS: Location of the first shock.

Fig. 11. Structure of the shock: comparison of experiments and current numerical results. (a) Experimental visualization [15] and (b) numerical solution.

S.M. Mousavi, E. Roohi / Acta Astronautica 105 (2014) 117–127 123

(with Cn

1¼1.8) for predicting the wall pressure and thelocation of the first shock. Then, we used Cn

1¼1.6 for all ofthe investigations. This figure shows that the RSM turbu-lence model with ¼1.6 can specify the shock trainstarting point accurately.

Fig. 10 reveals the behavior of the shock train in theconvergent–divergent nozzle from the numerical simula-tion and Weiss et al. experiment [6]. The figure shows thatthe employed numerical solver could predict the locationof the first shock and the number of shocks suitably. Inaddition, the employed numerical method could indicatethe structure of the shock train (lambda shape) once

compared with the corresponding experimental picture[25], see Fig. 11. As this figure shows, the bifurcation of theshock foot could be captured quite accurately using the3-D RSM simulations.

Following Ref. [26], Fig. 12 shows the behavior of theshock parameter upstream of the shock wave regionand inside the shock train. The shock parameter isdefined as multiplication of flow Mach number by thenormalized pressure gradient

Ua:∇p

jj∇pjj ð9Þ

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Table 4Impacts of heat generation on the shock train.

Heat generationrate (kW/m3)

Shock startingpoint (mm)

Pmin (kPa) Mmax

Case 1 0 172 80 1.72Case 2 25 154 120 1.52Case 3 75 121 140 1.43Case 4 170 41 240 1.39

Fig. 12. Contour of shock parameter upstream of the shock train andinside the shock train.

Fig. 13. Impacts of the heat generation on the shock train starting point: Cont

S.M. Mousavi, E. Roohi / Acta Astronautica 105 (2014) 117–127124

This figure also shows the position of the first shockand emitted shocks with a suitable agreement withexperimental data [6].

3.3. Effect of heat source

The effects of heat generation on the shock trainstarting point are investigated and shown in Fig. 12(a–d)and Table 4. The boundary conditions are the same asTable 2 but the nozzle's walls are not considered adiabaticnow. Shown in Fig. 13 are the contours of the pressurewith arrows depicting the shock train starting point. Asportrayed, by increasing the heat generation rate from 0 to170 kWm�3, the shock starting location approaches to thethroat up to 126 mm. In addition, the shock wave strengthdiminishes for higher heat generation rate cases. This isbecause of variations in flow properties due to heatgeneration effects. The mass flow rate decreases withincreasing the heat generation rate. One reason for thisbehavior is the decrease of the flow density with the heat

our of pressure (Pa). (a) Case- 1, (b) Case- 2, (c) Case- 3 and (d) Case- 4.

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Fig. 14. Impacts of heat generation on the flow separation: Velocity contour. a Case- 1, b Case- 2, c Case- 3, d Case- 4.

S.M. Mousavi, E. Roohi / Acta Astronautica 105 (2014) 117–127 125

generation. Additionally, Fig. 14(a–d) displays that flowseparation has occurred if the rate of heat genera-tion becomes higher than 170 kW m�3. This could beattributed to the interaction between the shock waveand the boundary layer. As shown in Fig. 14d, the startingand terminal points of separation zone are 35.23 and148.75 mm, respectively.

The present results show that the pressure rises occurredjust due to the normal or oblique shock waves if heatgeneration rate is equal to 170 kWm�3, whereas, the con-tinuous pressure increase is due to the shock train for caseswith the heat generation rates lower than 170 kW m�3.

Furthermore, adverse pressure gradient (APG) exists just forthe cases with heat generation rates higher than 170 kWm�3.

3.4. Effect of inlet flow total temperature (IFTT)

In this section, the effect of the inlet flow total tempera-ture (IFTT) on the shock train in the convergent–divergentnozzle with a constant back pressure Pb¼325 kPa is dis-cussed. The values of flow inlet total temperature areconsidered as 170, 220, 300, 500, and 700 K, respectively.In this regard, Figs. 15(a–b) and 16 show the dependency offlow (pressure, Mach number, as well as separation flow) on

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-0.02 -0.01 0 0.01 0.020

100

200

300

400

500IFT= 800 K

-0.02 -0.01 0 0.01 0.02

50

100

150

200IFT= 170 K

-0.02 -0.01 0 0.01 0.02

50

100

150

200IFT= 220 K

Y- Direction (m)

X- V

eloc

ity (m

.s-1)

velocity at x= 0.5 mvelocity at x= 0.4 m

velocity at x= 0.3 m

-0.02 -0.01 0 0.01 0.02

50

100

150

200IFT= 300 K

-0.02 -0.01 0 0.01 0.02

50

100

150

200

250

300IFT= 500 K

Fig. 16. dependency of separation to the inlet total temperature.

X- Coordinate (m)

P wal

l (bar

)

0 0.1 0.2 0.3 0.4 0.5 0.60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

IFT= 170 KIFT= 220 KIFT= 300 KIFT= 500 KIFT= 700 K

X- Coordinate (m)

Mac

h N

umbe

r

-0.04 0 0.04 0.08 0.12 0.16 0.2 0.24 0.280

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

IFT= 170 KIFT= 220 KIFT= 300 KIFT= 500 KIFT= 700 K

Fig. 15. Dependency of pressure and Mach number to the inlet total temperature. (a) Pressure profile and (b) Mach number profile.

S.M. Mousavi, E. Roohi / Acta Astronautica 105 (2014) 117–127126

the IFTT. Our results show that the shock system movesdownstream by increasing the IFTT. Furthermore, as theIFTT becomes larger, the maximum Mach number andminimum pressure magnitude increases and decreases,respectively. Nevertheless, the wall pressure distribution

and Mach number upstream of the shock system areidentical in all cases. The viscosity of gas flow dependson the temperature through the Sutherland law and foll-ows the changes in inlet temperature. This subsequentlyaffects the flow Reynolds number. Variation in the Reynolds

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S.M. Mousavi, E. Roohi / Acta Astronautica 105 (2014) 117–127 127

number changes the boundary-layer thickness and shocktrain behavior. Fig. 16 shows the velocity profiles at variousIFTT. The figure show that the flow velocity increases byincreasing IFTT. Our results show that increasing theflow temperature does not result in the flow separationfor any cases.

4. Conclusion

In this paper, behavior of the shock train in the con-vergent–divergent nozzle has been numerically investigatedusing the RSM turbulence model. The results demonstratedthat the RSM turbulence model could predict the exactlocation of the shock train compared with the experimentaldata [6]. An additional benefit of this turbulence model isthat it can predict the location of the first shock wave quiteaccuracy. The effects of heat generation source and inletflow temperature on the behavior of the shock train werealso surveyed. The results showed that by increasing the rateof heat generation, the location of the first shock trainmoves closer to the nozzle throat. The results of variationof the IFTT showed that increasing the temperature resultsin a shock train closer to the nozzle outlet.

Acknowledgment

The authors would like to acknowledge the financialsupports of “Iranian Elite foundations” under Grant no.100666 for equipping the HPC Laboratory.

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