Shock Wave Detection based on the Theory of Characteristics for CFD Results

Transcription

1 0th AIAA Computational Fluid Dynamics Conference 7-30 June 011, Honolulu, Hawaii AIAA Shock Wave Detection based on the Theory of Characteristics for CFD Results asashi Kanamori 1 and Kojiro Suzuki. The University of Tokyo, Kashiwa, Chiba, , 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. a C + /C - Nomenclature = speed of sound = characteristics in two-dimensional flow field C = characteristic vector which induces the generation of the shock wave = local ach number x, y = Cartesian coordinates λ = ith eigenvalue i r i = ith eigenvector θ = argument of the flow velocity υ = Prandtl-ayer function µ = local ach 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 V (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 technique 1. 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, Kashiwanoha, kanamori@daedalus.k.u-tokyo.ac.jp, Student member AIAA. Professor, Department of Advanced Energy, The University of Tokyo, Kashiwanoha, kjsuzuki@k.utokyo.ac.jp, ember AIAA. 1 Copyright 011 by the, Inc. All rights reserved.

2 Several methods of the shock detection have been investigated to date -8. These techniques can be classified into two types: one is to consider the ach number perpendicular to the shock front -6 and the other to fit the numerical result into the analytical solution of the local Riemann problem 7-8. 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 ach 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 waves -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 flow 8. 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 onedimensionally; finding the direction of a shock wave in the former detection type or fitting the solution of a onedimensional Riemann problem in the latter type. Therefore, we adopt the definition of a shock wave as a convergence of the characteristics of the same family 9 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 θ + υ and θ υ are conserved along C + and C -, respectively. Transport equations for θ+ υ and θ υ are defined as follows 10 : Figure 1. Schematic of the characteristics for two-dimensional Steady flow ( θ+ υ) = 0, ( θ υ) = 0 ξ η From Fig.1 and sin µ =1/, differentiation with respect to ξ and η can be rewritten, respectively, as follows: 1 cosθ + sinθ ( θ + υ) + x 1 cosθ sinθ ( θ υ) + x y () According to the theory of partial differential equations 11, these characteristics can be drawn by solving the following equations, which are called characteristic equations: d x= f ( x), 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: or d x ( = y ( 1cosθ± sinθ ) / 1sinθ m cosθ ) / 1sinθ + cosθ ( θ υ) = 0 (1) 1 sinθ cosθ ( θ + υ) = 0, y (3)

3 The solution of Eq.(4) can be obtained as follows 1 : d x = Ax+ b (4) x= x λ τ + λ τ r 0 + C1 exp( 1 ) r1 C exp( ) (5) where x 0 is called a fixed point of Eq.(4), which can be obtained by solving A x0 + b= 0. Figure shows typical solution curves for the case λ 1 > 0> λ. 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 expressed as xcr can be Figure. Typical solution curves of Eq.(4) x + cr = x 0 tr i (6) where t denotes a parameter and the subscript i is 1 or corresponding to critical lines 1 or, 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. ) Construct triangular cells with three neighboring grid points and calculate the right hand side of Eq.(4) from the vector f (x) 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 V n at each grid point and check whether the ach number n satisfies the following relation or not: ( ) > 1 and ( ) n < 1 ( ) < 1 and ( ) > 1 n L L or (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. ( n ) R is calculated by n n R R Figure 3. Application of the shock-crossing condition interpolating u R and a R from the information at the grid points 1, and 3. It should be noted that the shock-crossing condition is equivalent to the entropy condition, or Lax s shock condition 13. 4) If the shock-crossing condition is satisfied, define the critical line as a shock wave. However, we encounter a problem: f (x) cannot be calculated in subsonic region (see Eq.(3) with < 1). The characteristics are defined as envelope curves of the region where the information propagates. In subsonic region, 3

4 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 V ' 1 and 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. ) Calculate f (x) described as Eq.(3) for the velocity V ' 1 and 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 3 rd order USCL interpolation 15 and the LU-SGS implicit scheme 16 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 Freestream ach 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 i 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 Figure 5. Supersonic flow around a sphere-cone (left) Pressure contour, (right) shock detection result Figure 4. Calculation of the characteristics for subsonic region 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. 4

5 Figure 6. Supersonic flow around a double wedge (left) density contour, (right) shock detection result The third application is for a transonic flow of ach number of 0.8 around an NACA001 airfoil 17 with the angle of attack of 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. Figure 7. Transonic flow around an airfoil (left) Pressure contour, (right) shock detection result 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 ach 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 θ + υ and θ υ 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. 5

6 B. Relation between ach Cone, Characteristics and Shock Waves As shown in Fig.8, ach cones emanate from a stream line with the flow velocity vector as its axis: ach cones turn their directions as the stream line turns. Highly bent stream line causes a collision of ach cones and, as a result, a generation of a shock wave. Here, we consider the local region where the collision of ach 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 ach cones and the corresponding stream line are drawn in Fig.9. C i indicates the intersection between the ach cone i and the plane of motion, which is one of the characteristics by definition. It is obvious that C 1 is the first section that collides with C when the two ach cones collide with each other. Thus we define the vector C i as a characteristic in threedimensional flows and consider the shock detection based on the vector field. following relation: C i can be expressed as the Figure 8. Relation between streamline and ach cones C i = U cos µ ± α sinµ (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 u = ( d / ) x= Ax+ b, the differentiation can be calculated as follows: α = Figure 9. Definition of the characteristics in threedimensional space β d d d, β = u = ( Ax+ b) = A x = A( Ax+ b) β (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., u = Ax+ b. ) 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 threedimensional space. As a result, the shock-crossing condition should also be modified, namely we have to replace the velocity component V n 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 λ 1 is identical to the outer product of the eigenvectors r and r 3, which correspond to the eigenvalue λ and λ 3, respectively. 6

7 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 Figure 10. Computational grid around a sphere-cone Figure 11. Shock detection result for supersonic flow around a sphere-cone 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 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 threedimensional 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 ach number, as was reported by many researchers both experimentally and numerically 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 direction 0. 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.1. Freestream ach number is set to.5 and the angle of attack is 30 degrees. According to the classification by iller 18, the flow field is classified as shock with vortex for this condition. The number of grid points is Figure 1. Computational grid around a delta wing in ξ,η, and ζ direction respectively, about 7.7 million grid points. Reynolds number is and laminar flow in the entire region is assumed. Viscous terms in the Navier-Stokes equations are evaluated with nd 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 0% 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. 7

8 Figure 13. Shock detection results for a supersonic viscous flow around a delta wing (left)three-view, (right)perspective view with a pressure contour 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 ach 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 of the Japan Society for the Promotion of Science. asashi Kanamori is supported by a Research Fellowship of Japan Society for the Promotion of Science for Young Scientists. References 1 Nakahashi, K., Deiwert, G., Selfadaptive-grid method with application to airfoil flow, AIAA Journal, Vol. 5, No. 4, 1987, pp. 513, 50. Liou, S. P., ehlig, S., Singh, A., Edwards, D., Davis, R., An image analysis based approach to shock identification in CFD, AIAA Paper, Lovely, D., Haimes, R., Shock detection from computational fluid dynamics results, AIAA Paper, Darmofal, D., Hierarchal visualization of three-dimensional vertical flow calculation, Ph.D. Dissertation, Dept. of Aeronautics and Astronautics,.I.T., Cambridge, A, a, K. L., Rosendale, J. V., Vermeer, W., 3D Shock Wave Visualization on Unstructured Grids, Proceedings of the 1996 symposium on Volume visualization, 1996, pp., 87, Glimm, J., Grove, J. W., Kang, Y., Lee, T., Li, X., Sharp, D., H., Yu, Y., Ye, K., Zhao,., Statistical Riemann problems and a composition law for errors in numerical solutions of shock physics problems, SISC 6, 004, pp., 666, Glimm, J., Grove, J. W., Kang, Y., Lee, T., Li, X., Sharp, D., H., Yu, Y., Ye, K., Zhao,., Errors in numerical solutions of spherically symmetric shock physics problems Contemporary athematics, Vol. 371, 005, pp., 163,

9 9 Zel dovich, Y. B., Raizer, Y. P., Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Dover Publications, Liepmann, H. W., Roshko, A., Elements of Gas Dynamics, Dover Publications, John, F., Partial Differential Equations, Springer Verlag, Hirsch,. W., Smale, S., Devaney, R. L., Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, Toro, E. F., Riemann Solvers and Numerical ethods for Fluid Dynamics, Springer Verlag, Shima, E., Jounouchi, T., Role of CFD in aeronautical engineering AUS type upwind scheme, Proceedings of the 14 th NAL Symposium on Aircraft Computational Aerodynamics, 1999, pp., 7, van Leer, B., Toward the ultimate conservative difference scheme. 4 A new approach to numerical convection, Journal of Computational Physics, Vol. 3, 1977, pp., 76, Yoon, S., Kwak, D., An implicit three-dimensional Navier-Stokes solver for compressible flow, AIAA Journal, Vol. 30, No. 11, 199, pp., 635, Abbott, I. H., von Doenhoff, A. E., Theory of wing section, Dover Publications, iller, D., Wood, S. R.., Leeside flows over delta wings at supersonic speeds, Journal of Aircraft, Vol. 1, No. 9, 1984, pp., 680, Stanbrook, A., Squire, L. C., Possible types of flow at swept leading edges, Aeronautical Quarterly, Vol. 15, No., 1964, pp., 7, 8. 0 Imai, G., Fujii, K., Oyama, A., Computational Analyses of Supersonic Flows over a Delta Wing at High Angles of Attack, 5 th International Congress of the Aeronautical Sciences, 006 9