American Institute of Aeronautics and Astronautics

1

Shock Wave Detection based on the Theory of

Characteristics for CFD Results

Masashi Kanamori1 and Kojiro Suzuki.

2

The University of Tokyo, Kashiwa, Chiba, 277-8561, Japan

A method to detect the discontinuity of a shock wave from computational fluid dynamics

(CFD) data was developed based on the characteristics. A shock wave is mathematically

defined as a convergence of characteristics. Such convergences are interpreted as critical

lines of the streamlines, which are easily identified by calculating the eigenvectors of the

vector field for the propagation velocity of the Riemann invariants. Shock waves can be

successfully extracted using our method. Three-dimensional shock waves can also be

detected successfully by extending the idea for two-dimensional flows and defining the

characteristics which contribute the generation of shock waves.

Nomenclature

a = speed of sound

C+/C

- = characteristics in two-dimensional flow field

C = characteristic vector which induces the generation of the shock wave

M = local Mach number

x, y = Cartesian coordinates

iλ = ith eigenvalue

ir = ith eigenvector

θ = argument of the flow velocity

υ = Prandtl-Mayer function

µ = local Mach angle

τ = pseudo time parameter in streamline equations

ξ ,η ,ζ = coordinates along the corresponding characteristics/coordinates in computational space

I. Introduction

ISUALIZATION of shock waves is one of the most challenging problems in computational fluid dynamics

(CFD). Contour plots are usually used because of simplicity and convenience, namely, a shock wave is

interpreted as a zone where the contour lines are highly concentrated. This technique, however, faces some fatal

deficiencies: there are no quantitative rules as to when packed contours can be called a shock wave. In addition, the

point where the shock wave is formed or terminated cannot be exactly determined. Furthermore, contour plots are no

longer useful for visualizing three-dimensional shock waves. Because a contour plot approach is therefore not

adequate to accurately investigate the properties of a shock wave, a new method of the shock detection should be

developed. Once such a method is established, shock wave positions can be accurately determined, without having

to rely on a contour plot to obscurely judge whether shock waves are present or not. Furthermore, such a method can

be applied to CFD techniques that require information about the shock position, such as a solution adaptive

technique1. Thus, a shock detection method is useful not only as a visualization technique but also as a flow analysis

itself.

1 Graduate Student, Department of Aeronautics and Astronautics, The University of Tokyo, 5-1-5 Kashiwanoha,

kanamori@daedalus.k.u-tokyo.ac.jp, Student member AIAA. 2 Professor, Department of Advanced Energy, The University of Tokyo, 5-1-5 Kashiwanoha, kjsuzuki@k.u-

tokyo.ac.jp, Member AIAA.

V

20th AIAA Computational Fluid Dynamics Conference27 - 30 June 2011, Honolulu, Hawaii

AIAA 2011-3681

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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Several methods of the shock detection have been investigated to date2-8

. These techniques can be classified into

two types: one is to consider the Mach number perpendicular to the shock front2-6

and the other to fit the numerical

result into the analytical solution of the local Riemann problem7-8

. The former approach is based on the assumption

that a gradient of the primitive variable, such as a pressure, is perpendicular to the shock front. As a result, shock

waves can be detected as the point where the Mach number component perpendicular to the shock front is equal to

unity. As this method is easy to implement, they detect not only shock waves but also other types of waves. Some

complicated filters and thresholds therefore must be combined to the method to eliminate such waves2-3,5

.

Furthermore, the threshold values strongly affect the detected results and we therefore must adjust the value properly

for every problem we visualize. In the latter situation, they consider the local Riemann problem for one-dimensional,

unsteady flows in each cell and detect shock waves by fitting the numerical result with the analytical solution of the

problem. The merit of this approach is that other waves, such as a contact discontinuity or an expansion wave, can

also be detected as well as shock waves. Two-dimensional shock waves can be detected correctly by determining a

proper direction in which the flow field can be treated as one-dimensional flow8. Determining the direction consists

of two steps: estimate initial direction of a shock wave and correct more accurate direction based on the initial guess.

Validity of these steps, however, was not explained in terms of the flow physics. In other words, the direction

obtained by the steps might not be the true direction of the shock wave.

The problem among these approaches is that they seek the location of a shock wave by scanning them one-

dimensionally; finding the direction of a shock wave in the former detection type or fitting the solution of a one-

dimensional Riemann problem in the latter type. Therefore, we adopt the definition of a shock wave as a

convergence of the characteristics of the same family9 in order to treat shock waves as multi-dimensional

phenomena rather than one-dimensional ones. The objective of this study is to develop a shock wave detection

method based on the characteristics for two-dimensional, steady flows and to assess the effectiveness of the method.

Extension of the method to three-dimensional flow field is also considered in this paper.

II. Two-Dimensional Shock Detection

A. Characteristics in Two-Dimensional Flow Field

There are two types of characteristics for two-dimensional, steady flow: C+ and C

- as shown in Fig.1. Each

characteristic transports different information, which is called the Riemann invariant, namely, the invariants υθ +

and υθ − are conserved along C+ and C

-, respectively. Transport

equations for υθ + and υθ − are defined as follows10

:

(1)

From Fig.1 and M/1sin =µ , differentiation with respect to ξ and η

can be rewritten, respectively, as follows:

(2)

According to the theory of partial differential equations11

, these characteristics can be drawn by solving the

following equations, which are called characteristic equations:

(3)

Comparing Eq.(3) with standard streamline equations, the right hand side of Eq.(3) can be interpreted as the velocity

of the characteristics. Thus we define the term as the propagation velocity for the Riemann invariants. Here, what we

want to know is the convergent sections of the characteristics. In the next section, we will propose a method of

detecting such sections automatically.

B. Behavior of Linear Ordinary Differential Equation System

The linearized equations of Eq.(3) are expressed in a vector form as:

0)(,0)( =−∂∂

=+∂∂

υθη

υθξ

0)(cossin1

)(sincos1

,0)(cossin1

)(sincos1

22

22

=−∂∂+−

+−∂∂−−

=+∂

∂−−++

∂

∂+−

υθθθ

υθθθ

υθθθ

υθθθ

yM

M

xM

M

yM

M

xM

M

−

±−=

=

MM

MM

y

x

d

dorxfx

d

d

/)cossin1(

/)sincos1(),(

2

2

θθ

θθττ m

Figure 1. Schematic of the

characteristics for two-dimensional

Steady flow

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(4)

The solution of Eq.(4) can be obtained as follows12

:

(5)

where 0x is called a fixed point of Eq.(4), which can be obtained by solving 00 =+ bxA . Figure 2 shows typical

solution curves for the case 21 0 λλ >> . They seem like hyperbolic curves with references to two straight lines.

These two lines are called critical lines and

the intersection of these lines is equal to the

fixed point. In this paper, the solution curves

are equivalent to the characteristics, which are

exactly the same as the critical lines. The

equation for critical lines crx can be

expressed as

(6)

where t denotes a parameter and the subscript

i is 1 or 2 corresponding to critical lines 1 or 2,

respectively. Therefore, all we have to do is to

calculate the fixed point and the eigenvector

in order to detect shock waves. That is the

reason why we replace Eq.(3) with a linear

ordinary differential equation system.

C. Shock Wave Detection Algorithm for Two-Dimensional Flow Fields

Based on the above arguments, we propose an algorithm for shock wave detection. The procedure for the

algorithm is summarized as follows:

1) Calculate the propagation velocity for the Riemann invariants as described in Eq.(3) at each grid point.

2) Construct triangular cells with three neighboring grid points and calculate the right hand side of Eq.(4)

from the vector )(xf at the three grid points.

3) Obtain the critical lines for Eq.(4), and if it passes through the cell, consider the shock-crossing condition.

Namely, calculate the velocity component perpendicular to the critical line nV at each grid point and check

whether the Mach number Mn satisfies

the following relation or not:

(7)

The point L is determined as the

furthermost point from the critical line.

Point R is set so that points L and R

possess line symmetry with respect to

the critical line. (Mn)R is calculated by

interpolating Ru and aR from the

information at the grid points 1, 2 and 3. It should be noted that the shock-crossing condition is equivalent

to the entropy condition, or Lax’s shock condition13

.

4) If the shock-crossing condition is satisfied, define the critical line as a shock wave.

However, we encounter a problem: )(xf cannot be calculated in subsonic region (see Eq.(3) with 1<M ). The

characteristics are defined as envelope curves of the region where the information propagates. In subsonic region,

bxAxd

d+=

τ

2221110 )exp()exp( rCrCxx τλτλ ++=

icr rtxx += 0

Figure 2. Typical solution curves of Eq.(4)

Figure 3. Application of the shock-crossing condition

( ) ( )

( ) ( ) 11

11

><

<>

RnLn

RnLn

MandM

or

MandM

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information can propagate throughout the entire region regardless of its flow direction. This is why the

characteristics cannot be defined in subsonic region. In this region, however, information that induces a shock wave

propagates along the almost opposite direction to the flow velocity. Therefore, in the region, we calculate the

characteristics that have almost the opposite direction to the flow velocity, and let them represent the characteristics

in subsonic region.

The procedure for calculating the characteristics in subsonic region is as follows:

1) Calculate the new velocities 1'V and

2'V , as illustrated in Fig.4. Here, a is

defined as a vector that is orthogonal to

the vector V and the length of which is

equal to the speed of sound.

2) Calculate )(xf described as Eq.(3) for

the velocity 1'V and 2'V . This yields

four characteristics described as red

arrows in Fig.4. Choose the two that are

directed against the flow velocity,

which are described as solid arrows in

Fig.4.

Note that the advantage of this method is that no threshold values are used, and thus no adjustments are needed

to eliminate the problems associated with the use of thresholds, as discussed in section I.

D. Detection Results

Here, the shock detection method is applied to various numerical results. All flow fields in this section were

obtained by solving two-dimensional, compressible Euler equations. Simple High-resolution Upwind Scheme

(SHUS) 14

with 3rd

order MUSCL interpolation15

and the LU-SGS implicit scheme16

were used for numerical flux

calculation and time integration, respectively.

The first application is for a supersonic flow around a sphere-cone. In this flow field, a strong bow shock wave

exists in front of the body, thus forming subsonic region. We therefore can assess the effect of the characteristics in

subsonic region by applying our shock

detection method to this problem. The

number of the grid points is 115 × 80.

Freestream Mach number is set to 3. Figure

5 shows the pressure contour and the shock

detection result, clearly indicating that a

shock wave in this flow field can be

successfully detected.

The second application is for a

supersonic flow around a double wedge, as

illustrated in Fig.6. Two attached shock

wave emanate at each edge (i1 and i2 in

Fig.6), and intersect each other, resulting in

the generation of a reflected shock wave (rs

in Fig.6) and a slip line (sl in Fig.6) from the

intersection. Therefore, this flow field is

suitable for validating the capability to

discriminate slip lines from shock waves.

The number of grid points is 249 × 200.

Note that no slip line is evident in the results

from our shock detection method. This

means that our method can distinguish slip

lines from shock waves.

Figure 4. Calculation of the characteristics for subsonic

region

Figure 5. Supersonic flow around a sphere-cone

(left) Pressure contour, (right) shock detection result

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The third application is for a transonic flow of Mach number of 0.82 around an NACA0012 airfoil

17 with the

angle of attack of 2 degrees. A pressure contour and a shock wave detection result are shown in Fig.7. We can

obviously recognize the start and the terminal point of the shock wave from the detection result. The other method,

such as a contour plot, cannot show us this kind of information.

III. Three-Dimensional Shock Detection

A. Characteristics in Three-Dimensional Flow Field

There are more than two characteristics in three-dimensional flow fields. In fact, each characteristic is equivalent

to the generating line of the local Mach cone. Thus, we should consider the collision of these generating lines. The

relation, however, is little understood between each generating line, namely, we do not know what is the invariant

conserved along each generating line. In two-dimensional flow fields, there are only two invariants υθ + and υθ −

corresponding to C+ and C

-, respectively. Three-dimensional flow fields have infinite number of characteristics and

thus the corresponding invariants should be defined, but we do not know what the invariants are. In order to

overcome this difficulty, we introduce the idea of determining the generating line which contributes a generation of

shock waves from a physical point of view.

Figure 7. Transonic flow around an airfoil

(left) Pressure contour, (right) shock detection result

Figure 6. Supersonic flow around a double wedge

(left) density contour, (right) shock detection result

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B. Relation between Mach Cone, Characteristics and Shock Waves

As shown in Fig.8, Mach cones emanate from a stream line with the flow velocity vector as its axis: Mach cones

turn their directions as the stream line turns. Highly bent stream line causes a collision of Mach cones and, as a

result, a generation of a shock wave.

Here, we consider the local region where the

collision of Mach cones occurs, as illustrated in Fig.9.

Under assumption of a small region, the stream line

should be contained in a certain plane. We define the

plane as a plane of motion. Namely, the stream line can

be considered as a planar curve on the plane of motion.

Two Mach cones and the corresponding stream line are

drawn in Fig.9. iC indicates the intersection between

the Mach cone i and the plane of motion, which is one

of the characteristics by definition. It is obvious that

1C is the first section that collides with 2C when the

two Mach cones collide with each other. Thus we

define the vector iC as a characteristic in three-

dimensional flows and consider the shock detection

based on the vector field. iC can be expressed as the

following relation:

(8)

where U and α denote the unit vectors tangential to

and normal to the stream line, respectively. U is

obtained by normalizing the flow velocity. α is

identical to the acceleration vector of the flow, namely,

we can obtain α by considering the differentiation of

u with respect to τ . Considering the linear

interpolation of u , namely bxAxddu +== )/( τ , the differentiation can be calculated as follows:

(9)

C. Shock Wave Detection Algorithm for Three-Dimensional Flow Fields

An algorithm for three-dimensional shock detection can be summarized as follows:

1) Construct triangular cells with three neighboring grid points and make an interpolation for the flow velocity

u , i.e., bxAu += .

2) Calculate the characteristics which contribute the generation of the shock wave, denoted by C , which is

expressed as Eq.(8) at each grid point.

3) Construct linear interpolation of the characteristics C in the triangular cell and obtain the critical surface.

4) Define the critical surface as a shock wave if the critical surface satisfies the shock-crossing condition,

which was introduced in Section II. C.

It should be noted that the critical line in two-dimensional space is replaced with the critical surface in three-

dimensional space. As a result, the shock-crossing condition should also be modified, namely we have to replace the

velocity component nV with a velocity normal to the critical surface. In order to calculate the normal vector of the

surface, the eigenvectors of the vector field for the characteristics are needed: the normal vector of the critical

surface which corresponds to the eigenvalue 1λ is identical to the outer product of the eigenvectors 2r and 3r ,

which correspond to the eigenvalue 2λ and 3λ , respectively.

µαµ sincos ±=UC i

)()(, bxAAxd

dAbxA

d

du

d

d+==+===

τττβ

β

βα

Figure 8. Relation between streamline and Mach

cones

Figure 9. Definition of the characteristics in three-

dimensional space

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D. Detection Results

Here, the shock detection method is applied to various numerical results. All calculating conditions are the same

as that for the two-dimensional cases. The first application is a supersonic, inviscid flow around a blunt-nose cone.

In this flow field, a strong bow shock emanates in front of the body. We consider the case with

angle of attack in order to assess the capability for non-axisymmetric shock detection. Computational grid is shown

in Fig.10. The number of grid point is 200× 91× 100 in ξ ,η , and ζ direction respectively, about 1.8 million grid

points. The angle of attack is set to 10 degrees. Shock detection result for this flow field is shown in Fig.11. In

Fig.11, shock detection results are shown as semitransparent, red surfaces. As can be seen from Fig.11, the shock

waves can be detected correctly, which indicates the correctness of the choice of the characteristics for three-

dimensional flows.

The second example is a supersonic viscous flow around a delta wing. The existence of vortices plays an

important role on the position of the shock waves, especially in the leeward side of the wing. The condition of the

flow field is governed by the angle of attack and the

freestream Mach number, as was reported by many

researchers both experimentally and numerically18-20

.

Almost all researchers, however, visualized the flow

field with contour plots, resulting in the lack of the

knowledge for the overview of the shock waves. Some

researchers stated that the shock shape changed

dramatically not only in the cross-flow direction but

also in its station direction20

. This paper will therefore

reveal the whole shape of the shock wave in this flow

field, which has never been revealed before.

Computational grid is shown in Fig.12. Freestream

Mach number is set to 2.5 and the angle of attack is 30

degrees. According to the classification by Miller18

,

the flow field is classified as “shock with vortex” for

this condition. The number of grid points is

191× 201× 201 in ξ ,η , and ζ direction respectively,

about 7.7 million grid points. Reynolds number is 3.18× 105 and laminar flow in the entire region is assumed.

Viscous terms in the Navier-Stokes equations are evaluated with 2nd

order central differencing scheme. Figure 13

shows the shock detection result for this problem. Red, green and blue surfaces indicate the leeward shock, the

upwind shock and the shock wave from the trailing edge, respectively. Figure 13 clearly shows the overview of the

shock wave: leeward shock emanates from about 20% chord and gets away from the surface while expanding its

wave surface gradually. As a result, the shock wave interacts with the shock wave from the trailing edge (the blue

surface). We can recognize the above information at a glance.

Figure 10. Computational grid around

a sphere-cone

Figure 11. Shock detection result for supersonic flow around

a sphere-cone

Figure 12. Computational grid around a delta wing

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As seen above, our shock detection method is useful as a tool to understand the complex flow structure

accompanying shock waves.

IV. Concluding Remarks

In this paper, a method for shock wave detection based on the characteristics for two-dimensional, steady flow

was proposed. The method calculates the critical lines of the vector field of the characteristics without using any

threshold values, which was one of the problems in the past studies. As a result, shock waves were clearly and

accurately detected, and other types of discontinuities were properly excluded.

Extension of the method to three-dimensional flows was also considered. We selected the generating lines of the

local Mach cone as the characteristics which contributed the generation of the shock waves. As a result, we could

determine shock waves in three-dimensional flow fields even for viscous flows. This paper showed the possibility of

our shock detection method as a tool to understand the complex flow structure accompanying shock waves.

Acknowledgments

This work was supported by Grant-in-Aid for Scientific Research No. 21.7903 of the Japan Society for the

Promotion of Science. Masashi Kanamori is supported by a Research Fellowship of Japan Society for the Promotion

of Science for Young Scientists.

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Figure 13. Shock detection results for a supersonic viscous flow around a delta wing

(left)three-view, (right)perspective view with a pressure contour

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No. 11, 1992, pp., 2635, 2659. 17Abbott, I. H., von Doenhoff, A. E., Theory of wing section, Dover Publications, 1949. 18Miller, D., Wood, S. R. M., “Leeside flows over delta wings at supersonic speeds,” Journal of Aircraft, Vol. 21, No. 9,

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