Three-Dimensional Shock Wave Detection Basedon the Theory of Characteristics

Masashi Kanamori∗ and Kojiro Suzuki†

University of Tokyo, Kashiwa, Japan

DOI: 10.2514/1.J051922

This paper describes a method to detect three-dimensional, steady inviscid shock waves from computational fluid

dynamics data. Themethod is based on the principle that the collision of the characteristics of the same family causes

the generation of a shock wave in a two-dimensional flow. In a three-dimensional flowfield, however, there is an

infinite number of characteristics. Therefore, the plane on which the streamline can be treated locally as a planar

curve is introduced, and the characteristic as an intersection between the Mach cone and this plane is defined. As a

result, the shock waves in a three-dimensional, steady flowfield can be treated by a similar methodology as that in a

two-dimensional flowfield.

Nomenclature

A, P = coefficient matrices of the linearized equationsa = speed of soundb, q = vectors of the linearized equationsC = characteristic vector which induces the generation

of the shock waveC�∕C− = characteristics in two-dimensional flowfieldM = local Mach numberri = ith eigenvectorx, y, z = Cartesian coordinatesθ = argument of the flow velocityλi = ith eigenvalueμ = local Mach angleν = Prandtl-Mayer functionτ = pseudo time parameter in streamline equationsϕ = meridional angle of a sphere-coneξ, η, ζ = coordinates along the corresponding characteristics/

coordinates in computational space

Subscripts

i = the number of grid point/variable∞ = freestream values

I. Introduction

D ETECTION and visualization of shock waves are one of themost challenging problems in computational fluid dynamics

(CFD). A contour plot is usually used as a shock wave visualizationtool because of its simplicity and convenience, namely, a shock waveis interpreted as a zone, in which the contour lines are highlyconcentrated. This technique, however, faces some fatal deficiencies:there are no quantitative rules as to when packed contours can be

called a shockwave. In addition, the locationwhere the shockwave isformed or terminated cannot be exactly determined. Furthermore,contour plots have some difficulties in visualizing three-dimensionalshock waves. A two-dimensional method can be applied to a three-dimensional flowfield; consider a sliced plane and make a contourdiagram on the plane. However, such visualization results are uselessif one wants to know the information out of the plane, and thecombination with other technique will be necessary. As a result, onewill encounter a flood of visualization results, get confused, overlooksomething important, or even misunderstand the phenomena.Because a contour plot approach is, therefore, not adequate toaccurately investigate the properties of a shock wave, a new methodof the shock wave detection should be developed. Once such amethod is established, a position and a shape of a shock wave can beaccurately determined without having to rely on a contour plot toobscurely judge whether shock waves are present or not. Such amethod can also be applied to CFD techniques that requireinformation about the position of a shock wave, such as solutionadaptive techniques [1,2] or shock fitting techniques [3,4]. Thus, ashock detection method is useful, not only as a visualizationtechnique, but also as a flow analysis technique itself.Several methods of shock detection have been investigated to date

[5–12]. These techniques can be classified into three types: the firstone is to use a spatial derivative of the primitive variables [5–9], thesecond one is to fit the numerical result into the analytical solution ofthe local Riemann problem [10,11], and the third one is to find thelocationwhere the characteristics collidewith each other [12,13]. Thefirst approach is based on the assumption that a gradient of theprimitive variable, such as a pressure or a density, is perpendicular tothe shock front. Buning and Steger [5] mentioned for the first timethat shockwaveswere obtained as a locationwhere theMach numberalong the pressure gradient became unity. This method, however,detects not only shockwaves but also other types ofwaves. Liou et al.[6] introduced three filtering techniques for eliminating suchspurious detections. The extension to unsteady flows wassuccessfully performed by Lovely and Haimes [7]. Ma et al. [8]proposed another approach for detecting shock waves, in whichshock waves were extracted as isosurfaces corresponding to a zerosecond spatial derivative of the density along the flowdirection. In thesecondmethodology, they considered the local Riemann problem forone-dimensional, unsteady flows in each cell and detected shockwaves by fitting the numerical result with the analytical solution ofthe problem [10,11]. Themerit of this approach is that the other typesof waves, such as a contact discontinuity or an expansion wave, canalso be detected, as well as shock waves. Two-dimensional shockwaves can also be detected by determining a proper direction, inwhich the flowfield can be treated as one-dimensional flow [11]. Thethird approach is based on the most rigorous definition of shockwaves in inviscid flow from a theoretical point of view, that is, a shockwave is identical to the collision of the characteristics of the same

Presented as Paper 2011-3681 at the 20th AIAA Computational FluidDynamics Conference, Honolulu, Hawaii, 27–30 June 2011; received 26February 2012; revision received 27March 2013; accepted for publication 28March 2013; published online 23 July 2013. Copyright © by the AmericanInstitute of Aeronautics and Astronautics, Inc. All rights reserved. Copies ofthis paper may be made for personal or internal use, on condition that thecopier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc.,222 Rosewood Drive, Danvers, MA 01923; include the code 1533-385X/13and $10.00 in correspondence with the CCC.

*Graduate Student, Department of Aeronautics and Astronautics, 5-1-5Kashiwanoha; kanamori@daedalus.k.u-tokyo.ac.jp. StudentMember AIAA;currently Researcher, Numerical Simulation Research Group, AerospaceResearch and Development Directorate, Japan Aerospace ExplorationAgency, 7-44-1, Jindaiji-Higashi, Chofu, Tokyo; kanamori.masashi@jaxa.jp.Member AIAA.

†Professor, Department of Advanced Energy, 5-1-5 Kashiwanoha;kjsuzuki@k.u-tokyo.ac.jp. Senior Member AIAA.

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family [14]. In the shock fitting code implemented by Moretti [12],shock waves were obtained by finding the intersection of thecharacteristics for one-dimensional, unsteady flow, that is,considering the characteristic velocity, which is equal to u� a inthe x direction, a shock wave emerges at the inflection point of thecharacteristic velocity. Kanamori and Suzuki [13] proposed themethod using the characteristics for two-dimensional, steady flow.They extracted the collision by considering the local linearization ofthe vector field of the characteristics. The advantage of the method isthat shock waves can be detected 1) based on the rigorous and simpledefinition and 2) using no empirical values or arbitrariness, such as athreshold. On the other hand, the extension to three-dimensional flowhas not been developed yet, which is the main topic of this paper. Theobjective of this study is to develop a three-dimensional shock wavedetection method based on the characteristics.An outline of the paper is as follows. Section II shows a review of

the shock wave detection method based on the characteristics in atwo-dimensional flowfield. The extension of the method to a three-dimensional flowfield will be explained in Sec. III. Section IV givesthe conclusion of this paper.

II. Two-Dimensional Shock Detection

Before proceeding with the three-dimensional shock wavedetection, an explanation about the concept of shock wave detec-tion based on the characteristics is necessary. Therefore, wereview two-dimensional shock wave detection based on thecharacteristics [13].

A. Characteristics in Two-Dimensional Flowfield

There are two families of characteristics for a two-dimensional,inviscid, steady supersonic, and homentropic flow: C� and C− asshown in Fig. 1. Each characteristic transports different information,which is called the Riemann invariant: the invariants θ� ν and θ − νare conserved along C� and C−, respectively. The transportequations for θ� ν and θ − ν are defined as follows [15]:

���������������M2 − 1p

cos θ� sin θ

M

∂∂x�θ� ν�

����������������M2 − 1p

sin θ − cos θ

M

∂∂y�θ� ν� � 0

���������������M2 − 1p

cos θ − sin θ

M

∂∂x�θ − ν�

����������������M2 − 1p

sin θ� cos θ

M

∂∂y�θ − ν� � 0 (1)

According to the theory of partial differential equations [16], thesecharacteristics can be drawn by solving the following equations,which are called characteristic equations:

d

dτx � f�x� �

�����������������M2 − 1p

cos θ� sin θ�∕M����������������M2 − 1p

sin θ∓ cos θ�∕M

�(2)

To find the convergence of the characteristics, or the solution curvesof Eq. (2), local linearization is considered, that is, the right-hand sideof Eq. (2) is replacedwith the linear term using three neighboring gridpoints as follows:

d

dτx � Ax� b (3)

It is well known that there exists only a few patterns of the solution forEq. (3) [17]. In this study,wewill focus on the pattern shown in Fig. 2.In this pattern, solution curves are distributed like hyperbolic curveswith references to two straight lines. These two lines and theintersection of these lines are called critical lines and a fixed point,respectively. It should be noted that the solution curves are identicalto the characteristics, andwhatwewant now is the convergent lines ofthese curves, that is, the critical line 1 in Fig. 2. The merit ofconsidering local linearization is that the critical line can be easilyobtained from linear equation: these lines are always straight for alinear system, and the direction of the line is equal to that of theeigenvector of the coefficient matrix A in Eq. (3). In addition, thefixed point can be obtained by solving the following equation:

Ax� b � 0 (4)

This means that shock waves are detected as critical lines, which canbe easily obtained by considering a local linearization.

B. Shock Wave Detection Algorithm for Two-Dimensional Flowfield

Based on the argument above, the algorithm for shock wavedetection is summarized as follows:1) Calculate the right-hand side of Eq. (2) at each grid point.2) Construct triangular cells with three neighboring grid points and

calculate the right-hand side of Eq. (3) from the vector f�x� at thethree grid points. Three grid points are necessary for determining thelinear interpolation uniquely.3) Obtain the critical lines for Eq. (3), and if it passes through the

cell, define the critical line as a shock wave.Note that the advantage of this method is that no threshold values

are used, and, thus, no adjustments are needed to eliminate theproblems associated with the use of thresholds.

C. Detection Results

Here is an example of application, which is a supersonic inviscidflow around a double wedge as illustrated in Fig. 3. This flowfieldwas obtained by solving two-dimensional, compressible Eulerequations with a finite volume approach. A simple high-resolutionupwind scheme [18] with a third-order MUSCL interpolation [19]and the Lower Upper Symmetric Gauss Seidel implicit scheme [20]

Fig. 1 Schematic of the characteristics for two-dimensional steady flow. Fig. 2 Typical solution curves of Eq. (3).

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were used for a numerical flux calculation and time integration,respectively. The number of grid points are 249 × 200. Two attachedshock waves emanate at each edge (i1 and i2 in Fig. 3) and intersecteach other, resulting in the generation of a reflected shock wave (rs inFig. 3) and a slip line (sl in Fig. 3) from the intersection. The rightfigure in Fig. 3 is the shock detection result. It should be noted thattwo oblique shock waves can be detected by the method, and sliplines are excluded correctly from the result.

III. Three-Dimensional Shock Detection

A. Characteristics in a Three-Dimensional Flowfield

There are an infinite number of the characteristics in three-dimensional inviscid supersonic flowfields. In fact, each character-istic is equivalent to the generating line of the local Mach cone. Froma mathematical point of view, such generating lines are calledbicharacteristics [21]. Thus, we should consider the collision of thesegenerating lines in three-dimensional space. The relation is, however,little understood between each generating line. In a two-dimensionalflowfield, there are only two invariants, θ� ν and θ − ν, cor-responding to C� and C−, respectively, and only the collision ofcharacteristics of the same family causes the generation of shockwaves. In a three-dimensional flowfield, on the other hand, thereexist an infinite number of characteristics, and the correspondencebetween each generating line is unknown. We cannot judge whetheror not it is a shock wave when a generating line of a Mach conecollides with that of another Mach cone. To overcome this difficulty,we introduce the idea of choosing the representative of the generatingline, which contributes to the generation of shock waves.Let us consider a streamline andMach cones on two points i and j

on the streamline as shown in Fig. 4. In general, if a shock wave islocated between two points i and j, a streamline turns its directionsuddenly, and two Mach cones should collide with each other. Here,we assume that the streamline can be treated as a planar curve on acertain plane in the neighborhood of the point of intersection betweenthe streamline and the shock wave. This becomes exact in the limit ofi and j to the point of intersection.We call such a plane the local planeof motion.Ci is one of the intersections between theMach cone i andthe local plane of motion, which are the characteristics by definition.There are two intersections between the Mach cone and the localplane of motion, Ci, and the other side from the velocity vector (redlines in Fig. 4.) Considering the collision of the two Mach cones,however, it is obvious that Ci is the first section of the Mach cone ithat collides with theMach cone j. Thus, we define the vectorCi as acharacteristic which induces the generation of a shock wave andconsiders the shock detection based on the vector field. Ci can beexpressed as the following relation:

Ci � Ui cos μi �Wi sin μi (5)

where Ui andWi denote the unit vectors tangential to and normal tothe streamline at the grid point i, respectively. Ui is obtained bynormalizing the flow velocity ui.W i is on the local plane of motionand normal to the vector Ui. Any velocity vectors on the streamlineare always on the local plane of motion by definition, that is, the rateof change of these vectors, or an acceleration vector denoted by ai, isalso included on the plane. This means that the local plane of

motion consists of two independent vectors: velocity vector ui andacceleration vector ai. Acceleration vector ai can be obtained bycalculating the differentiation of ui with respect to τ or a parameteralong the streamline. Considering the linear interpolation of ui,namely ui � Pxi � qi, the differentiation can be calculated asfollows:

ai �d

dτui �

d

dτ�Pxi � qi� � P

d

dτxi � P�Pxi � qi� (6)

The vectorWi is obtained using Ui and ai as follows:

Wi �wijwij

; wi � ai − �Ui · ai�Ui (7)

In a subsonic region, however, the Mach cone cannot be defined; theMach cone at a certain point is defined as a boundary of the zone ofinfluence at the point. Unlike a supersonic case, the zone of influencespreads unboundedly in a subsonic region. That is the reason whywecannot define the Mach cone in a subsonic region. For such cases, arepresentative of the characteristics is considered as follows:

�Ci�subsonic � Ui cos �μi�subsonic �W i sin �μi�subsonic;

�μi�subsonic � 2 sin−12����������������

M2 � 4p

(8)

Equation (8) means that a representative of the characteristics insubsonic region points toward the direction almost opposite to theflow velocity. This is because the information, which induces thegeneration of a shock wave, propagates along such a direction. Suchconcept is derived from two-dimensional shock wave detection [13],which showed successful detections of nearly normal shockwaves. Itshould be noted that the choice of characteristics in a subsonic regionis not unique. But the insignificance of the choice has already beenconfirmed, as long as the direction of the characteristic points towardthe almost opposite to the flow velocity [13]. Equation (7), however,cannot be applied if the acceleration vector ai becomes zero orparallel to the flow velocity ui. Shock detection is successfullyachieved even for such cases. The former case is trivial: the velocityvector does not change in the cell considered, resulting in the absenceof any shock waves. Therefore, we skip the cell without evaluating

Fig. 3 Supersonic flow around a double wedge (left) schematic of the flowfield and (right) shock detection result.

Fig. 4 Definition of the characteristics in a three-dimensional flowfield.

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Eq. (7). The latter is the case of normal shockwaves, and the subsonicregion must be, therefore, formed behind the shock. As discussedabove, the characteristic in a subsonic region is defined, so that it isalmost directed toward the opposite direction of the flow velocityand, therefore, always collides with the characteristic in a supersonicregion for any direction of the vector ai.After calculating the vectorCi using Eq. (5), shock wave detection

is conducted, that is, calculate A and b for the vector Ci using fourneighboring grid points and obtain the eigenvalues of the matrix A.For negative eigenvalues, there exists a convergent plane of thesolution curves, which is called a critical plane. Figure 5 showstypical solution curves for a three-dimensional linear equation withone negative and two positive eigenvalues. As shown in Fig. 5, allsolution curves converge to the plane, which consists of theeigenvectors r2 and r3. In other words, the critical plane thatcorresponds to the negative eigenvalue can be obtained from theremaining two eigenvectors. A fixed point can be obtained fromEq. (4), and a normal direction of the plane is identical to that of theouter product of the eigenvectors r2 and r3.

B. Shock Wave Detection Algorithm for Three-Dimensional Flowfield

An algorithm for three-dimensional shock detection can besummarized as follows:1) Construct tetrahedral cells with four neighboring grid points

and make a linear interpolation for the flow velocity u, that is,u � Px� q. In three-dimensional space, four grid points are neededto determine the linear interpolation uniquely. Divide a hexahedralcell into six tetrahedral cells. The way of dividing the cell is notunique, butwe confirmed that the result was almost independent fromthe choice of division.2) Calculate the characteristics, which contribute to the generation

of a shock wave, denoted byCi, which is expressed as Eq. (5) at eachgrid point.3) Construct linear interpolation of the characteristics Ci in the

tetrahedral cell and obtain the critical plane for each negativeeigenvalue. If the plane passes through the cell, define the surface as ashock wave.

C. Detection Results

In this section, we shall show several results of application. Themethod proposed by Lovely and Haimes [7] is also considered as aconventional method in order to validate and compare with ourresults. Through the application, we shall show the competitivenessof our method. The governing equation is a three-dimensional,compressible Euler equation, and the calculation methods are thesame as that for two-dimensional flows. The first application is asupersonic, inviscid flow around a cone. The aim of this applicationis to validate Eq. (5) to be an appropriate representative of thecharacteristics in three-dimensional space.We consider the casewithangles of attack of 10 deg in order to assess the capability of themethod.A computational grid is shown in Fig. 6. A half-cone angle is

set to 15 deg. The number of grid points is 151 × 91 × 151 in ξ, η, andζ directions, respectively, which is about 2.1 million grid points. Theerrors of both the lift and drag coefficients between this grid and thefiner one arewithin 1%. A perspective of the shock detection result isdepicted in Fig. 7, which clearly shows the applicability of ourmethod to such a true three-dimensional shock wave. Figure 8illustrates the detection results by our shock detection (lower figures)and the conventional method (upper ones) with a pressure contour

Fig. 5 Typical solution curves of Eq. (3) for a three-dimensional case.

Fig. 6 Computational grid around a cone.

Fig. 7 Shock wave detection result (red surfaces) for a supersonic flowaround a cone (body surface is shown as a white, broken line).

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diagram (green lines) for several meridional angles ϕ. Comparedwith the conventional method, our method successfully shows shockwaves, which indicates that the characteristic for a three-dimensionalflowfield works properly as a characteristic which induces thegeneration of a shock wave. Figure 9 illustrates the closeup view ofthe case ϕ � 180 deg with grid lines, in which the shock wave is

successfully extracted as a continuous line. This is because thecomputational grid ismade so that the grid line is parallel to the shockwave. The effect of the relation between the shock wave and the gridline will be discussed in the next example.The second example is a supersonic, inviscid flow around a sphere

cone with angles of attack of 10 deg. In this flowfield, a strong bow

Fig. 8 Shockdetection results (red lines) andpressure contourdiagrams (green lines) for supersonic flowaround a cone (upper: conventionalmethod [7],lower: present method).

Fig. 9 Closeup view of the shock detection result (red lines) and pressure contours (green lines) for a supersonic flow around a cone with computationalgrid (dotted lines).

Fig. 10 Computational grid around a sphere cone.Fig. 11 Shock wave detection result (red surfaces) for a supersonic flowaround a sphere cone (body surface is shown as a white, broken line).

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shock wave emanates in front of the body, resulting in a formation ofsubsonic region. This example is, therefore, appropriate to confirmthe characteristics in a subsonic region, as in Eq. (8). In addition, acurved shock wave is suitable for the discussion on the effect of gridlines. A computational grid is shown in Fig. 10. The nose radius andthe half-cone angle are set to 1 and 10 deg, respectively. The number ofgrid points is 150 × 91 × 120 in ξ, η, and ζ directions, respectively,which is about 1.8 million grid points. The errors of both the lift anddrag coefficients between this grid and the finer one arewithin 1%. Thedetection result is illustrated in Fig. 11. The bow shock wave is also

detected successfully, as well as the attached shock wave in theprevious example. Figure 12 shows the shock detection results andpressure contour on several meridional planes. The upper results areobtained from the conventionalmethod and the lower fromourmethod.It is obvious that our method detects only shock waves, whereas theconventional method fails. It should be noted that the upper results canbe modified by introducing filters or thresholds to eliminate spuriousdetections into the conventional method. Our method, on the otherhand, detects only shock waves without using such techniques, whichis one of the most advantageous points of our method.

Fig. 12 Shock detection results (red lines) and pressure contour diagrams (green lines) for supersonic flow around a sphere cone (upper: conventionalmethod [7], lower: present method).

Fig. 13 Closeup view of the shock detection result (red lines) and pressure contours (green lines) for a supersonic flow around a sphere cone withcomputational grid (dotted lines).

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The effect of grid lines on the detection is shown in Fig. 13. Unlikethe previous example shown in Fig. 9, the detected shock waves aresomewhat “dispersive,” or like a group of line segments, in the regiontop of the body (region A in Fig. 13), whereas a perfectly continuousshockwave is obtained in regionB. The difference between these twois the angle between the shock wave and grid line; from a discussionin the previous example, our method can detect shock wavesperfectly if the grid line is parallel to the shock wave. Region B is justa desirable case from such a viewpoint, that is, a grid line is parallel tothe shockwave, resulting in a successful detection of a shockwave asa perfectly continuous line. In region A, on the other hand, the shockwave cannot be detected correctly, because the grid line is not parallelto the shock. This can be summarized that the more the grid linebecomes parallel to the shockwave, themore the detection results areimproved. In other words, shock waves cannot be detected correctlyif they are not treated correctly in the CFD analysis. This means thatour method does not give a perfect prediction of shock wavesregardless of the quality of shock capturing.

IV. Conclusions

Amethod for detecting a three-dimensional, steady inviscid shockwave from computational fluid dynamics (CFD) data is proposed inthis paper. Thismethod is based on the equivalence of a shockwave tothe collision of the characteristics of the same family. Difficultiesarise, however, when extending this idea to a three-dimensionalflowfield, that is, the appropriate relation between the characteristicson differentMach conesmust be determined correctly. Therefore, thelocal plane of motion is introduced, in which the streamline can betreated locally as a planar curve, and define a characteristic, whichcauses the generation of shock waves as an intersection between thelocal Mach cone and the local plane of motion.The application of the proposed method is illustrated using two

examples, namely, inviscid flow around a sharp cone and a sphere-one, respectively. Three-dimensionality is enforced via a nonzeroangle of attack. Compared with the results obtained with aconventional shock detection technique, the proposed method showsalmost the same or even better results, reflecting the validity of theassumption on the local plane of motion.The two applications reveal not just a remarkable ability of the

proposed method as a visualization tool but also some interestingfeatures. Specifically, the more the grid line becomes parallel to theshock wave, the better the shock detection becomes, namely, shockwaves can be obtained as perfectly continuous surfaces. The featurecan be reinterpreted from the view point of the CFD technique itself.Needless to say, the combination of the appropriate mesh and solveris necessary to perform a correct CFD calculation. The proposedmethod detects a shock wave as a set of surface segments and aperfectly continuous set of segments can be obtained under thecondition that the grid lines be parallel to the shock wave. When thegrid lines are not aligned with the shock wave, detected shock wavesbecome “dispersive,” resulting in a poor quality CFD calculation.This indicates that our method is also useful as an evaluation tool toassess the quality of the CFD solution containing shockwaves. Somereaders may think that the above feature is not desirable from a viewpoint of a visualization technique. Such a concern, however, seemsnot so critical, because one can easily grasp a whole shape of theshockwaves from thevisualization result, although the grid line is notaligned to the shock wave. On the other hand, some remedies, like agrouping or smoothing of the detected surfaces, must be necessarywhen the proposed method is used as a shock fitting method, whichwill be considered in a future work.

Acknowledgments

This work was supported by Grant-in-Aid for Scientific ResearchNo. 21.7903 of the Japan Society for the Promotion of Science.Masashi Kanamori is supported by a Research Fellowship of theJapan Society for the Promotion of Science for Young Scientists.

References

[1] Brackbill, J. U., and Saltzman, J. S., “Adaptive Zoning for SingularProblems in Two Dimensions,” Journal of Computational Physics,Vol. 46, No. 3, 1982, pp. 342–368.doi:10.1016/0021-9991(82)90020-1

[2] Nakahashi, K., and Deiwert, G., “Self Adaptive-Grid Method withApplication to Airfoil Flow,” AIAA Journal, Vol. 25, No. 4, 1987,pp. 513–520.doi:10.2514/3.9655

[3] Morton,K.W., and Paisley,M. F., “AFiniteVolume Schemewith ShockFitting for the Steady Euler Equations,” Journal of Computational

Physics, Vol. 80, No. 1, 1989, pp. 168–203.doi:10.1016/0021-9991(89)90094-6

[4] Moretti, G., “Computation of Flows with Shocks,” Annual Reviews ofFluid Mechanics, Vol. 19, 1987, pp. 313–337.doi:10.1146/annurev.fluid.19.1.313

[5] Buning, P. G., and Steger, J. L., “Graphics and Flow Visualization inComputational Fluid Dynamics,” AIAA Paper 85-1507, 1985.

[6] Liou, S. P.,Mehlig, S., Singh,A., Edwards,D., andDavis, R., “An ImageAnalysis BasedApproach to Shock Identification in CFD,”AIAAPaper95-0117, 1995.

[7] Lovely, D., and Haimes, R., “Shock Detection from ComputationalFluid Dynamics Results,” AIAA Paper 99-3285, 1999.

[8] Ma, K. L., Rosendale, J. V., and Vermeer, W., “3D Shock WaveVisualization on Unstructured Grids,” Proceedings of the 1996

Symposium on Volume Visualization, IEEE Computer Society Press,Los Alamitos, CA, 1996, pp. 87–104.

[9] Darmofal, D., “Hierarchal Visualization of Three-Dimensional VerticalFlow Calculation,” Ph.D. Dissertation, Dept. of Aeronautics andAstronautics, Massachusetts Inst. of Technology, Cambridge, MA, 1991.

[10] Glimm, J., Grove, J. W., Kang, Y., Lee, T., Li, X., Sharp, D. H., Yu, Y.,Ye, K., and Zhao,M., “Statistical Riemann Problems and aCompositionLaw for Errors in Numerical Solutions of Shock Physics Problems,”SIAM Journal on Scientific Computing, Vol. 26, No. 2, 2004, pp. 666–697.doi:10.1137/S1064827503427534

[11] Glimm, J., Grove, J. W., Kang, Y., Lee, T., Li, X., Sharp, D. H., Yu, Y.,Ye, K., and Zhao, M., “Errors in Numerical Solutions of SphericallySymmetric Shock Physics Problems,” Contemporary Mathematics,Vol. 371, 2000, pp. 163–179.

[12] Moretti, G., “A General-Purpose Technique for Two-dimensionalTransonic Flows,” NASA CR-194186, 1987.

[13] Kanamori, M., and Suzuki, K., “Shock Wave Detection in Two-Dimensional Flow Based on the Theory of Characteristics From CFDData,” Journal of Computational Physics, Vol. 230, No. 8, 2011,pp. 3085–3092.doi:10.1016/j.jcp.2011.01.007

[14] Zel’dovich, Y. B., and Raizer, Y. P., Physics of Shock Wave and High-

Temperature Hydrodynamics Phenomena, Dover, New York, 2002,pp. 19–25.

[15] Liepmann, H. W., and Roshko, A., Elements of Gas Dynamics, Dover,New York, 2002, pp. 284–289.

[16] John, F., Partial Differential Equations, Springer–Verlag, Berlin, 1981,pp. 11–23.

[17] Hirsch, M. W., Smale, S., and Devaney, R. L., Differential Equations,Dynamical Systems, and an Introduction to Chaos, Academic Press,New York, 2003, pp. 73–107.

[18] Shima, E., and Jounouchi, T., “Role of CFD in AeronauticalEngineering –AUSM Type Upwind Scheme,” Proceedings of the 14thNAL Symposium on Aircraft Computational Aerodynamics, NationalAerospace Laboratory, Tokyo, 1999, pp. 7–24.

[19] van Leer, B., “Toward the Ultimate Conservative Difference Scheme. 4ANewApproach to Numerical Convection,” Journal of ComputationalPhysics, Vol. 23, No. 3, 1977, pp. 276–299.doi:10.1016/0021-9991(77)90095-X

[20] Yoon, S., and Kwak, D., “An Implicit Three-Dimensional Navier–Stokes Solver for Compressible Flow,” AIAA Journal, Vol. 30, No. 11,1992, pp. 2635–2659.doi:10.2514/3.48957

[21] Coburn, N., “Characteristic Directions in Three-DimensionalSupersonic Flows,” Proceedings of the American Mathematical

Society, Vol. 1, No. 2, 1950, pp. 241–245.doi:10.2307/2031930

M. ChoudhariAssociate Editor

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