numerical simulation of shock tube generated vortex: effect of numerics

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Numerical simulation of shock tube generated vortex:effect of numericsSudipta De a & Murugan Thangadurai ba Simulation and Modelling Department, CSIR-Central Mechanical Engineering ResearchInstitute, Durgapur, 713209, Indiab Thermal Engineering Department, CSIR-Central Mechanical Engineering Research Institute,Durgapur, 713209, IndiaPublished online: 23 Aug 2011.

To cite this article: Sudipta De & Murugan Thangadurai (2011): Numerical simulation of shock tube generated vortex: effectof numerics, International Journal of Computational Fluid Dynamics, 25:6, 345-354

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Numerical simulation of shock tube generated vortex: effect of numerics

Sudipta Dea* and Murugan Thangaduraib

aSimulation and Modelling Department, CSIR-Central Mechanical Engineering Research Institute, Durgapur 713209, India;bThermal Engineering Department, CSIR-Central Mechanical Engineering Research Institute, Durgapur 713209, India

(Received 11 November 2010; final version received 7 June 2011)

Vortices generated at the open end of a planar shock tube are numerically simulated using the AUSMþ scheme. Thisscheme is known to have low numerical dissipation and therefore is suitable for capturing unsteady vortex motion.However, this low numerical dissipation can also cause oscillations in the vorticity field. Numerical experimentspresented here highlight the effect of numerical dissipation on the simulated vortex, as well as the role played byturbulence models. Two turbulence models – the shear-stress-transport (SST) and its modified version for unsteadyflows (SST-SAS) – are employed to observe the effect of including turbulence models in such complex flows wherethe vortex has an embedded shock.

Keywords: shock tube; vortex; AUSMþ; numerical dissipation; turbulence model

1. Introduction

The present study concerns the flow of air out of a two-dimensional shock tube when a membrane is ruptured,which initially separates air at two different pressures.The high-speed jet emerging from the shock tubegenerates a pair of vortices. A shock tube of circularcross-section produces a vortex ring. The two-dimen-sional version is chosen here to keep the numerical set-up simple. Air adjacent to the open end is at ambientconditions, whereas the gas on the other side of themembrane is at an elevated pressure. The section of thetube containing air at elevated pressure is known asthe driven section. The length of this section is smallerthan the one containing air at ambient conditions,known as the driven section. We keep these twolengths and the pressure of the driver section constantand focus on the effects of numerics by modifying theartificial dissipation term of the discretisation algo-rithm and also by adding turbulence models.

Vortices generated at the open end of a shocktube have been mostly subject to experimentalinvestigations. The shock tube experiment of Baird(1987) produced a shock Mach number (Ms) of 1.5,and is one of the early evidences of an embeddedshock in a shock tube generated vortex loop.Brouillette and Hebert (1997) have examined thevortex regimes created by emerging shock waves withMs varying between 1 and 2 using spark shadowgraphand schlieren photography. Within this Mach numberrange, three different vortex regimes were observed.

At the lower end, a simple vortex ring was produced.In the range of Ms¼ 1.43–1.60, an embedded shockappeared within the vortex ring. At still higher Machnumbers, a counter-rotating secondary vortex loopformed ahead of the primary loop. Importantparameters characterising the evolution of the vortex,such as the variation of the ring and core diameter,circulation and vortex position with time, transversevelocity profile within the vortex core, etc. arediscussed in Arakeri et al. (2004). Formation ofcounter-rotating vortex ring at Ms¼ 1.7 is studiedusing high-speed laser sheet-based flow visualizationtechnique in Murugan (2008) and Murugan and Das(2010). Zare-Behtash et al. (2008) discuss shock tubeswith elliptical cross-sections with various driverpressures. The authors noted strong disturbancesaccompanying the vortex loop development at adriver pressure of 12 bars.

The experimental and numerical work cited aboveand references therein give us a fair idea of the complexflow configuration we are investigating – we have agrowing unsteady vortex with an embedded shock withKelvin-Helmholtz (KH) instability patterns trailingit and on its periphery. The numerically capturedvorticity field need not be clean with the possibility ofthe presence of oscillations and sharp gradients. It is inthis context that we venture into some numericalexperiments as presented here before using the devel-oped solver with some degree of confidence for highershock Mach numbers. The presented results are mostly

*Corresponding author. Email: [email protected]

International Journal of Computational Fluid Dynamics

Vol. 25, No. 6, July 2011, 345–354

ISSN 1061-8562 print/ISSN 1029-0257 online

� 2011 Taylor & Francis

DOI: 10.1080/10618562.2011.600694

http://www.informaworld.com

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vorticity contour plots. The original AUSMþ pro-duces oscillatory vorticity that can be linked to its lownumerical dissipation. A modified version of theAUSMþ scheme produces less vorticity disturbances,but also affects the KH-structures. Finally, we note theeffects turbulence models have on the evolving vortex.

2. Governing equations

We solve the Navier–Stokes equations of continuity,momentum and energy as listed below.

Continuity:

@r@tþ @

@xruð Þ þ @

@yrvð Þ ¼ 0 ð1Þ

x- and y-momenta:

@

@truð Þ þ @

@xru2 þ p� �

þ @

@yruvð Þ ¼ @txx

@xþ @txy

@yð2Þ

@

@trvð Þ þ @

@xruvð Þ þ @

@yrv2 þ p� �

¼ @txy@xþ @tyy

@yð3Þ

Energy:

@

@trEð Þ þ @

@xrEþ pð Þu½ � þ @

@yrEþ pð Þv½ �

¼ @

@xðutxx þ vtxyÞ þ

@

@yðutxy þ vtyyÞ �

@qx@x� @qy@y

ð4Þ

In the above, the stress terms are given by

txx ¼2m3

2@u

@x� @v@y

� �; tyy ¼

2m3

2@v

@y� @u@x

� �;

txy ¼ m@u

@yþ @v@x

� �:

The heat conduction terms are qx ¼ �k @T@x and qy ¼

�k @T@y .The energy term expands to E ¼ cvTþ 1

2 u2 þ v2� �

.Ideal gas behaviour is assumed with the equation ofstate p¼ rRT. Equations (1)–(4) are solved in a non-dimensionalised form with the ambient values ofdensity and temperature and the width of the shocktube as the reference values. Velocity is non-dimensio-nalised with respect to the ambient speed of sound.The non-dimensional form of the equations appearsslightly different. For example, the x-component of theheat conduction term becomes qx ¼ � ~m

ðg�1ÞM2 Re Pr@T@x,

Figure 1. Dissipation coefficient: (a) van Leer scheme, (b)AUSMþ with d¼ 0.5, (c) AUSMþ with d¼ 0.1 and (d)Original AUSMþ.

Figure 2. Computational domain and numerical shadowgraph.

346 S. De and M. Thangadurai

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where Ma is the reference velocity, a being thespeed of sound. em ¼ 1 if constant viscosity is used.In our simulations, the value of em is determined from

the Sutherland’s law. A detailed listing of equationsin various forms can be found in Anderson et al.(1984).

Figure 3. Mach number contours using (a) AUSMþ and (b) AUSMþ with increased numerical dissipation (d¼ 0.5). Thirteencontour levels are plotted starting from 0.3 with an increment of 0.1. (a) t¼ 39.0, (b) t¼ 39.0.

Figure 4. Contours of vorticity using AUSMþ at four different time instants. Same contour values (0.3, 0.5, 1, 2, 3) have beenplotted in all the frames. d¼ 0. (a) t¼ 31, (b) t¼ 39, (c) t¼ 47 and (d) t¼ 55.

International Journal of Computational Fluid Dynamics 347

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When a turbulence model is used, we also solve theequations for the turbulence kinetic energy (k) anddissipation rate (o).

@

@trkð Þ þ @

@xrukð Þ þ @

@yrvkð Þ

¼ tij@ui@xj� b�rokþ @

@xjðmþ skmtÞ

@k

@xj

� �ð5Þ

@

@troð Þ þ @

@xruoð Þ þ @

@yrvoð Þ

¼ guttij@ui@xj� bro2 þ @

@xjðmþ somtÞ

@o@xj

� �þ 2ð1� F1Þrso2

1

o@k

@xj

@o@xj

ð6Þ

The various associated parameters are available inMenter (1993) and will not be discussed here. Readersinterested in implementing turbulence models may find

useful information in Wilcox (2004). The the shear-stress-transport-scale-adaptive simulation (SST-SAS)model has an additional term in the o-equation. Thisterm (FSST-SAS, given in Menter and Egorov (2005))involves the von Karman length scale, which adjusts tolocal eddy-size and is responsible for flow-adaptivedissipation that is smaller than the dissipation pro-vided by standard turbulence models.

3. Numerical scheme

The AUSMþ scheme belongs to the AUSM family offlux vector splitting schemes and its details arediscussed in Liou (1996). Here we will only commentupon its numerical dissipation term and its modifiedform. The governing equations can be written incompact notation as:

@Q

@tþ @ðFÞ

@xþ @ðGÞ

@y¼ @ðFvÞ

@xþ @ðGvÞ

@yð7Þ

Figure 5. Contours of vorticity using modified AUSMþ at four different time instants. Same contour values (0.3, 0.5, 1, 2, 3)have been plotted in all the frames. d¼ 0.5. (a) t¼ 31, (b) t¼ 39, (c) t¼ 47 and (d) t¼ 55.


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If fiþ12; j

represents F at the cell face, then in AUSMþscheme, it is computed as:

fiþ12; j¼ aiþ1

2; j

"1

2miþ1

2; jðFL þ FRÞ

� 1

2Ciþ1

2; jðFR � FLÞ

#þ piþ1

2; jð8Þ

In the above, miþ12; j

is an interface Mach number,and the subscript R/L denotes the right/left states to becomputed by the monotone-upstream-centred schemefor conversation laws (MUSCL) approach. Thecoefficient of the dissipation term, Ciþ1

2; j, is jmiþ1

2; jj in

the original AUSMþ, and can be modified to addextra dissipation (Radespiel and Kroll 1995) when theinterface Mach number goes to zero:

Figure 6. Contours of vorticity using modified AUSMþ at four different time instants in a finer grid. Same contour values (0.3,0.5, 1, 2, 3) have been plotted in all the frames. d¼ 0.1. (a) t¼ 31, (b) t¼ 39, (c) t¼ 47 and (d) t¼ 55.

Ciþ12; j¼

m2iþ1

2; jþ d2

2 dð9Þ

In some of our computations, the above has beeninvoked when jmiþ1

2; jj < d. This implies that as

jmiþ12; jj ! 0; Ciþ1

2; j! d

2 .

The recommended range of d is 05 d � 0.5 (Radespieland Kroll 1995). At the maximum value of 0.5, it is stillhalf as dissipative as van Leer flux vector splitting asinterface Mach number approaches zero. This is shownin Figure 1. It can be seen that the low dissipativecharacter of AUSMþ is hardly disturbed at low valuesof d. Time integration has been performed by a four-stage Runge-Kutta scheme (Arnone et al. 1993). If thegoverning equation is written as

@u

@t¼ FðuÞ ð10Þ

then the time advancement takes place by the followingfour stages:

uð0Þ ¼ uðnÞ

uð1Þ ¼ uð0Þ þ a1 Dt F uð0Þ� �




unþ1 ¼ uð4Þ

ð11Þ


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In the above, a1 ¼ 14 ; a2 ¼

13 ; a3 ¼

12 ; a4 ¼ 1. This

scheme is fourth order accurate for linear problemsbut second order accurate for non-linear ones. Unlessotherwise specified, the minmod limiter is used forlimiting the primitive variables during interpolation tocell faces.

4. Computational domain, initial and boundary

conditions

We explore a single shock tube set-up and varynumerical parameters to observe the effects numericshave on the captured flow physics. The computationaldomain is shown in Figure 2. The lengths of the driverand driven sections are 165 mm and 1200 mm,respectively. The inner width of the tube is 64 mm(W). These values have been taken from Murugan andDas (2010). When non-dimensionalised with respect tothe tube width, the total length of the shock tubebecomes 21.3281. The length of the domain down-stream of the open end is 12 W and in the vertical

direction 8 W, starting from the bottom line ofsymmetry. The simulations are initialised everywhereto ambient conditions (pressure 1.01325 bar andtemperature 300 K) except at the driver section, wherethe pressure is six times the ambient pressure. The gridhas been kept dense close to the open end andstretched away from it. The coarse grid has8006 400 cells and the fine grid has 15006 800 cellsin the domain. The shock speed inside the tube isfound to be around Ms¼ 1.42. The flow inside the tubeis viscous and laminar. The turbulence models, whenthey are used, are activated only in the region outsidethe tube. The grid distribution in the y-direction isuniform and there are equally spaced 50 points (i.e. 49cells). In the x-direction, the cell spacing is stretchedaway from the exit. Immediately outside the tube exit,400 cells are uniformly distributed over a length of4 W. This spacing is matched inside the tube near theexit and the spacing is increased away from exit withinthe tube using a stretching function described inVinokur (1983). There are 200 cells in the tube length

Figure 7. Contours of vorticity using AUSMþ and SST turbulence model at four different time instants in the finer grid. Samecontour values (0.3, 0.5, 1, 2, 3) have been plotted in all the frames. d¼ 0. (a) t¼ 31, (b) t¼ 39, (c) t¼ 47 and (d) t¼ 55.


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of 21.3281 W. The tube wall thickness at y¼ 0.5 is notinfinitesimally thin, but has a thickness of 0.0408 W,covered by four cells, also uniformly spaced. No-slipboundary condition is applied at the left closed end ofthe shock tube. The bottom horizontal line is a line ofsymmetry. The remaining of the boundary lines has theNSCBC boundary condition of Poinsot and Lele(1992). This boundary condition at the open bound-aries seems to be most appropriate, given the unsteadynature of the flow.

5. Results and discussion

Figure 3 shows the Mach contours using the originalAUSMþ as well as the modified AUSMþ withadditional dissipation [Equation (9), d¼ 0.5]. Theeffect of increasing numerical dissipation shows in thedisappearance of oscillations in the Mach contours.Figure 4 shows vorticity contours computed usingAUSMþ in its original form. Two vorticity patchesnear the vortex centre are distinctly visible in the lasttwo frames. The downstream patch corresponds to the

embedded shock seen in Figure 3 but the origin of theupstream vorticity patch is unclear. With addeddissipation, these vorticity patches are nearly removed(Figure 5), with only a thin vorticity patch remainingcorresponding to the embedded shock. However, wealso notice the absence of Kelvin–Helmholtz roll uppatterns that are present in Figure 4. These resultsindicate that although we require a scheme with lessnumerical dissipation to capture the KH structures,such a scheme may also lead to oscillatory shockstructure. Next, we show the same contour plot for afiner grid and a numerical dissipation corresponding tod¼ 0.1 in Figure 6. Here we do not see any vorticitydisturbance as in Figure 4, except for a thin stretch ofvorticity indicative of the presence of the embeddedshock. The captured KH patterns are also moreprominent. Although this low level of added dissipa-tion seems satisfactory, it still remains to be seen howturbulence affects these structures.

It is well known that moving shocks generally leadto turbulent flows. First, there is a vorticity jumpacross a shock wave. This can occur due to shock

Figure 8. Contours of vorticity using AUSMþ and SST-SAS turbulence model at four different time instants in the finer grid.Same contour values (0.3, 0.5, 1, 2, 3) have been plotted in all the frames. d¼ 0. (a) t¼ 31, (b) t¼ 39, (c) t¼ 47 and (d) t¼ 55.


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curvature, pressure and density gradient misalignmentand angular momentum conservation (Kevlahan1997). It is noted in the cited reference that thisvorticity can affect turbulence evolution behind theshock. Moreover, moving shocks generate a slipstreamwhen they move out of shock tubes or just pass over acorner. Such slipstreams lead to rolled up shear layervortices, which often initiate a transition to turbulence(Murugan and Das 2010). It is, therefore, likely thatturbulence would dissipate the finest of the eddiescaptured in well-resolved numerical simulations. InSun and Takayama (2003), appearance of small scalevortices were found to be suppressed by the use of aturbulence model for shock diffraction over a 908 bend.The k – e turbulence model produced better agreementwith experimental findings compared to the laminarsimulation. The present situation is also expected toproduce similar results, and therefore we compute (inthe finer grid) with the SST turbulence model. SST waschosen because of the existence of a scale-adaptiveversion, which is less dissipative, and this enables us to

observe effects of less turbulent eddy-viscosity with aminor change in the solver. In Figure 7, the presence ofvorticity oscillations shows that this phenomenon isstrongly dependent on the numerical dissipation in thelow Mach number range and not on the grid or thepresence or absence of a turbulence model. We also donot observe much of KH roll up vortices, which arepresent even in the coarser grid simulation without anyturbulence model (Figure 4). It appears that in realitythe KH-vortices could possibly be present, but thestrength and appearance may not be as displayed inFigure 6. Effect of turbulence is to diminish their size,and a turbulence model is likely to produce betterresults, as already noted in Sun and Takayama (2003).There are models, which are suited for predictingunsteady phenomena – the SST-SAS being oneexample. Figure 8 shows vorticity contours using thismodel. SST-SAS turbulent eddy viscosity is signifi-cantly less compared to that of SST, and this results inthe appearance of the roll up vortices, though not tothe extent present in Figure 6.

Figure 9. Contours of vorticity using AUSMþ coupled to third order MUSCL with a total variation diminishing (TVD) limiter(a, b) and a multi-dimensional limiter (c, d). Same contour values (0.3, 0.5, 1, 2, 3) have been plotted in all the frames. d¼ 0. (a)t¼ 39, (b) t¼ 55, (c) t¼ 39 and (d) t¼ 55.


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So far, we have shown results produced by secondorder accurate MUSCL scheme with the minmodlimiter. Results can vary depending on the kind oflimiter used (see, e.g. Juntasaro and Marquis (2004)for a comparison of limiters for some pure convectiontest cases). To check the effects limiters might have onthe solution, we show the outcome of applying a TVDlimiter and a multi-dimensional limiter (MLP3) (Kimand Kim 2005) with a third order accurate MUSCLscheme (Figure 9). These are laminar solutions on thecoarser grid. We notice that the vorticity disturbanceis somewhat reduced in both, but the TVD limiter andthe multi-dimensional limiter produce almost similarlevels of disturbance. The presence of the disturbancepatches in the finer grid results shows that the presentproblem does not occur due to lack of grid resolution.In Figure 10, this is further confirmed. This figureshows the results produced by the third order accurateMUSCL implementation with the TVD limiter (toprow) and the AUSMþ-up (bottom row) scheme(Liou 2006) with the minmod limiter, both on the

finer grid and without any turbulence model. If thevorticity disturbances were related to grid resolution,they would have diminished with grid refinement. Thisobviously is not the case, as the figure suggests. TheAUSMþ-up scheme has the ability to handle flow atall speeds. If the disturbances originated at low speedregions close to the solid wall of the tube, theAUSMþ-up would have produced improved results.The problem is likely to be associated with themodified wave number of the second and third orderschemes. It can be shown that the former has greaterdispersive errors at lower wave numbers. Furthernumerical experimentation is necessary to resolve thisissue.

6. Conclusions

The complex flow emerging at the open end of a two-dimensional shock tube has been simulated numeri-cally using the low dissipation AUSMþ scheme withand without a turbulence model. Low dissipation

Figure 10. Contours of vorticity using AUSMþ coupled to third order MUSCL with a TVD limiter (a, b) and second orderaccurate AUSMþ-up (c, d), both on the finer grid and without any turbulence model. Same contour values (0.3, 0.5, 1, 2, 3) havebeen plotted in all the frames. d¼ 0. (a) t¼ 39, (b) t¼ 55, (c) t¼ 39 and (d) t¼ 55.


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schemes are better suited for computing unsteadyphenomena such as these but when the numericaldissipation goes to zero linearly with the cell-faceMach number (as in AUSMþ), the presented resultsindicate that a small amount of additional numericaldissipation in the low Mach number range removesunphysical oscillations. However, this dissipation hasto be low enough so that it does not smooth out theKH roll up vortices, which might also be present athigher shock Mach numbers. Laminar solvers tend tocapture these vortices in a very spectacular manner,particularly in a fine grid. Turbulence model-generatededdy viscosity has been found to dissipate the vorticesto smaller sizes, which might be closer to reality. TheSST-SAS model in particular has a capacity to providelow eddy viscosity for unsteady phenomena, and hasbeen found to dissipate the roll up vortices lesscompared to the original SST model.

Based on these findings, we hope to computecircular shock tube-generated vortices using an ax-isymmetric version of our solver with a propercombination of numerical dissipation and turbulenteddy viscosity.

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numerical simulation of shock tube generated vortex: effect of numerics

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