Emirates Journal for Engineering Research, 12 (2), 53-60 (2007)

(Regular Paper)

SHOCK- OVER SINGLE AND DOUBLE WEDGES

A. Bagabir1, A. Abdel-Rahman1 and A. Balabel2 1Faculty of Engineering, Sana’a University, Yemen, [email protected] 2Institute of Static, Technical University, Braunschweig, Germany

(Received January 2006 and accepted December 2006)

فى ھذه الدراسة تم عمل و إختبار نموذج التحليل العددي لمحاكات إنتشار و تصادم موجات إنفجارية (تصادميه) مع أسطح صلبة مائلة. النموذج المستخدم يعتمد على طريقة العناصر الحجمية المحددة والممثلة فى نظام أحداثي مطابق للجسم ثنائي األبعاد. حيث تم حل معادالت أويلر للغازات المضغوطة بواسطة الحلول الضمنية للزمن. وقد أخذ بعين األعتبار تحويل المعادالت التفاضلية الى منظومة من المعادالت الجبرية التى يسھل حلھا. تم التأكد من دقة نموذج التحليل العددي بالمقارنة مع نتائج سابقة لموجات إنفجارية مختلفة الشدة. النتائج وضحت أن الموجة اإلنفجارية تودي الى نظام معقد من الموجات نسبةً لتصادمھا مع األسطح الصلبة. كما تم التحقق من توزيع الضغوط على األسطح الصلبة و تحديد مواضع الضغط األعلى.

The performance of high-resolution Riemann solver is assessed in various reflection processes of shock-wedge interaction. The scheme has been implemented in conjunction with an implicit- unfactored method which is based on Newton-type sub-iterations and Gauss–Seidel relaxation. A modified MUSCL scheme has been employed. The simulations are conducted with the two- dimensional compressible Euler equations. The present study investigates unsteady diffraction over stationary single and concave-double wedges. The present results are compared with the previous numerical and experimental results for different types of reflection. It is found that the computed results show a good agreement with the existing data. An understanding of the system has been achieved using numerical schlieren-type images. The results include the pressure-load distribution and maximum overpressure.

Keywords: Compressible flow, Euler equations, shock wave, wedge

1. INTRODUCTION been carried out mostly by experiments using shock tubes. In these cases there occur unit with The interaction of shock waves with obstacles is one constant distributions of physical values behind the of the most important and unsolved problems of gas shock wave down to the contact surface. The unit dynamics. It attracted the interest of many theoretical, waves generate pseudo-stationary (self-similar) numerical and experimental studies. This interest is reflections, where incident shock wave propagate at a largely motivated by the need to understand the constant speed and the reflection patterns of shock physics of gasdynamic phenomena as well as from the waves over a wedge propagate almost with similarity. fact that shock reflection and diffraction appear in However, in the reality, shock waves propagate at a many applications. decreasing velocity due to expansion waves behind the The reflection of a planar shock wave over straight incident shock. Therefore, unsteady reflections of wedge depends upon the incident shock wave Mach shock wave occur[3,4]. number, MS, and the reflecting wedge angle, θ. Four It is necessary to establish the transition criteria types of shock wave have been between different reflection configurations. The observed[1,2] which are shown in Fig. 1: RR↔MR transition was first studied analytically by 1. Regular Reflection (RR) Von Neumann[5] for perfect gas. Subsequently, 2. Single-Mach Reflection (SMR). analytical and experimental researches were done to 3. Transition-Mach Reflection (TMR). establish transition criterion for different range of 4. Double-Mach Reflection (DMR). incident shock-wave Mach number[6-10]. Li and Ben- The last three types of diffraction are henceforth Dor[11] developed an analytical model for solving the labelled Mach Reflection (MR). The study of wave configuration resulting when two triple points reflection of a planar shock wave over a wedge has join together over a concave-double wedge.

53 A. Bagabir, A. Abdel-Rahman and A. Balabel

J stands for the Jacobian of the transformation from I I R Cartesian to curvilinear co-ordinates. The pressure, p, T is calculated by the perfect gas equation of state: R M S p = ρ()γ −1 i (3) where γ is the ratio of specific heats (γ = 1.4) and i is SMR internal energy. RR One of the challenges in the numerical simulations of such flows is to eliminate the numerical dissipation which may be responsible for suppressing the flow- field details. Therefore, the investigation has been performed using high resolution hybrid solver which I I R’ used in the past for the study of various compressible R R flows[14-17]. The hybrid solver is a combination of M [18] M S Riemann method and the modified Steger and S S’ Warming Flux Vector Splitting (FVS) method[19]. The flux, F, at the cell faces is given by[20]:

(4) TMR DMR F = αF()FVS + ()1−α F()Riemann [19] Figure 1. Schematic illustration of various types of shock-wave where F(FVS) and F(Riemann) are calculated by the FVS diffraction over a wedge and Riemann methods[18], respectively. The limiter α is defined as a function of the flow Mach number, M, Numerically, Hisley[12] studied the reflection of and the van Albada limiter, f [21]: planar shocks from wedge surface using BLAST2D 2 (5) code which is based on Roe’s approximate Riemann α = min{ 1,a()1− min()fi , fi+1 Mi+1 − Mi } solver with a total variation diminishing property and has second-order accuracy. Itoh et al.[13] simulated The coefficient a takes the value of 1 or 2 in various reflection processes of a shock wave over hypersonic flows. double wedge using TVD scheme. Kobayashi et al.[3] The FVS method provides additional numerical compared the numerical results of unsteady and dissipation which requires in the case of hypersonic pseudo-stationary blast wave diffraction over wedge. Mach numbers and in those regions where the The present study aims to validate the hybrid Riemann method does not provide sufficient numerical Riemann solver and to understand the reflection of dissipation to capture strong shock waves. A modified [22] shock wave over single and double wedges by a MUSCL scheme has been employed for calculating numerical simulation and quantitative comparison the conservative variables at the cell faces of the with the previous numerical and experimental results. control volumes. The time integration of the unsteady Moreover, the present study brings in an updated Euler equations has been obtained by the implicit- [23] description for the dynamics of the reflection of shock unfactored method which is second-order accuracy waves over wedge surfaces which go beyond that in time. provided by the previous studies. 3. RESULTS 2. NUMERICAL MODELLING 3.1 Code Validation The computational code used in the present study is The present numerical code is assessed against a test based on the finite volume method for the solution of case of a cylindrical shock wave which has exact the two-dimensional Euler compressible equations. solution[24]. A cylindrical shock wave of radius 0.4 The Euler equations can be written in matrix form and initiated at a centre of square structure of dimensions curvilinear co-ordinates ( , ) as: ξ ζ 2.0 × 2.0. The shock wave moves into a gas at rest. ∂Q ∂F ∂F The flow variables take constant values in the ambient + ξ + ζ = 0 (1) gas and inside the shock region as, respectively: ∂t ∂ξ ∂ζ ρ a = 0.125 ρ S = 1.0 F is the inviscid flux. Q contains the conservative ua = va = 0.0 uS = vS = 0.0 (6) variables: pa = 0.1 pS = 1.0 T T Q = J ()ρ,ρu,ρw,e = JU (2) The initial data was modified by assigning modified area-weighted values to the appropriate cells where ρ is the density, u and w are the velocity to avoid the staircase configuration of the data. The components, and e is the total energy per unit volume.

54 Emirates Journal for Engineering Research, Vol. 12, No.2, 2007 Shock-Wave Diffraction Over Single and Double Wedges mesh is of 100 × 100 computing cells. The density and pressure iso-surfaces at time 0.25 are shown in Figs. 2(a) and 2(b), respectively. The solution exhibits a circular shock wave and contact surface travelling away from the centre and a circular rarefaction travelling towards the centre. As shown in the pressure (a) (b) iso-surface (Fig. 2b) the pressure is continuous across the contact surface. Figure 2(c) shows density variation along the radial line compared with the exact 1 [24] solution . It is found that the present numerical Present result solver predicts the discontinuities that travel in all 0.8 Exact solution directions. The shock wave and contact surface is resolved with two cells. 0.6

3.2 Single Wedge centre Density 0.4 The problem under investigation is shown in Fig. 3. A planar, moving, incident shock wave of Mach number 0.2 MS encounters a sharp compressive wedge of fixed angle, θ, and is reflected by the wedge surface. The 0 Rankine-Hugoniot relations are used to obtain flow 0 0.2 0.4 0.6 0.8 1 Position properties behind the shock (pS, ρS, uS) in term of the shock Mach number, M [2]. (c) S 2 Figure 2. Cylindrical explosion at time t = 0.25; (a) density iso- pS 2γ M S − ()γ −1 = surface, (b) pressure iso-surface, (c) comparison between p γ +1 the present result and exact solution [24]. ρ γ +1 M 2 S ()S (7) = 2 Earfield ρ ()γ −1 M S + 2 u ρ S = u ρS The lower boundary is solid wall and the upper boundary is far field. The left boundary corresponds to the region behind the shock wave and the right boundary is considered to permit smooth outflow of any rightward-moving waves by maintaining a normal zero-gradient condition for all fluid variables. Wall Different shock reflection configurations are tested. The initial conditions used are tabulated in Table 1 (see also Fig. 1). Figure 3. Schematic of shock-wave diffraction over a wedge

Table 1. The initial conditions for different types of reflections over 7 single wedge. 6 Case Reflection type MS o Grid θ 5 1 RR 2.05 63.4o 300×300 2 SMR 1.2 30o 420×240 4 3 SMR 2.12 30o 420 240 × 3 4 TMR 7.19 20o Density 420×240 Present result o 5 DMR 2.8 47 420×240 2 Experimental data

1 The first case studied is the RR reflection type. Figure 4 shows the computed density distributions 0 along the wedge surface of angle θ = 63.4o and -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Position incident shock wave of MS = 2.05 compared with the experimental data of Deschambault and Glass[10]. Note Figure 4. Density distributions along the wedge surface of θ = 63.4o that the zero position in the graph represents the compared with the experimental data of Deschambault [10] location of the interaction of the reflected shock, R, and Glass for MS = 2.05 (Numerical image inserted).

Emirates Journal for Engineering Research, Vol. 12, No.2, 2007 55 A. Bagabir, A. Abdel-Rahman and A. Balabel

With surface and one is incident shock position (see the inserted image of Fig. 4). This formation is applicable to all subsequent graphs presented in this paper. It is obvious that the good agreement is already obtained for the RR reflection type. The sharp edge is located at the position 0.23. The inserted image of the (a) MS = 1.2 numerical shadowgraph illustrates that the type of the reflection is RR. The shock-system pattern does not vary as the incident shock wave progresses downstream. That is the self-similar phenomenon. Figure 5 shows the present numerical images of two different shock waves of Mach number 1.2 and 2.12 diffraction over wedge of fixed angle of 30o. As seen, the shock configuration for both cases is SMR.

The RR↔SMR transition occurs when the Mach (b) MS = 2.12 number behind the reflected shock is sonic in frame of reference to reflection point[5]. The four discontinuities Figure 5. Numerical images of density (left) and pressure (right) for shock wave diffraction over wedge of = 30o at two meet at the triple point which moves along a straight θ different shock intensities. line originating at the wedge edge. The path of the triple point is shown by dashed line presented in the 5 pressure images. The stronger shock wave reflection (M = 2.12 case) moves the triple-point further S 4 upwards while strengthening and flattening the reflected shock wave. The present code predicts all the 3 discontinuities, incident and reflected shocks, Mach stem and contact surface. The triple point path angle Present result Density 2 o BLAST2D for MS = 2.12 case is found to be 8.5 which is very [25] close the experimental data of Ben-Dor and Glass . 1 The computed density distributions along the wedge surface for MS=2.12 case are shown in Fig. 6. The 0 present result shows very good agreement with the -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 [12] BLAST2D-code numerical result of Hisley . Position

The SMR↔TMR transition takes place when the Figure 6. Density distributions along the wedge surface of θ = 30o Mach number behind the reflected shock is sonic in compared with BLAST2D numerical result of Hisley[12] [4] frame of reference to triple point . However, for MS = 2.12. Shirouzu and Glass[26] proposed another condition for SMR↔TMR transition is that the angle between the incident and reflected shock waves is equal or greater than 90o. The image of the present numerical shadowgraph for TMR reflection type is shown in Fig. 7(a). The wedge angle is selected as 20o and incident shock wave is 7.19. The numerical image illustrates that the angle between the incident and reflected shock waves is greater than 90o, which is the condition for (a) (b) [26] the SMR↔TMR transition . On the other hand, the Figure 7. SMR↔TMR for shock wave of MS = 7.19 diffraction over image of Fig. 7(a) shows that the bottom of the contact wedge of θ = 20o; (a) numerical image, (b) enlarged surface is disturbed. It was found that the curling of density contours. the contact surface is caused by compression wave[7]. As illustrated in the density contours (Fig. 7b), the curled contact surface catches up with the Mach stem which is pushed forward changing its orientation. For weak shock wave (results are not shown here), the front of the curled contact surface does not reach the Mach stem which remains straight and perpendicular to the reflecting wedge surface. Figure 8 shows the present numerical images for a Early time → Late time progressive series of refection of shock wave of MS = Figure 8. Numerical images show the progressive development of o 2.8 over wedge of θ = 47 . The three enlarged frames TMR↔DMR for shock wave of MS = 2.8 diffraction over show the TMR↔DMR transition as time progress. At wedge of θ = 47o at sequence of time instants.

56 Emirates Journal for Engineering Research, Vol. 12, No.2, 2007 Shock-Wave Diffraction Over Single and Double Wedges

early time, the compression wave (CW) is formed. As 40 M2 the time progress (see the frame at late time), the CW 35 is replaced by the Mach stem, M2. The frames also 30 show the formation of the second triple point and Pressure contact wave, S2. It is seen in the images the 25 expansion waves behind the incident shock wave. The 20 [4] 15 analytical work showed that the TMR↔DMR edge transition is attributed to the supersonic flow behind 10 Density the reflected shock with respect to the second triple 5 point. The pressure distribution over the wedge 0 surface at late time, shown in Fig. 9, reveals that the -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 maximum pressure occurs around the interaction of Position the second Mach stem, M2, with the surface. The density distribution presented in Fig. 9 exhibits density Figure 9. Density and pressure distributions along the wedge surface for shock wave of MS = 2.8 diffraction over wedge jump after Mach stem M2. of θ = 47o. 3.3 Double Wedge R I The present study also investigates the reflection of a R T planar shock wave over concave double wedge. Figure M 10 illustrates the progressive development of shock- I S1 S2 wave diffraction over concave double wedge. The details of the numerical experiments are shown in T

Table 2. R T I M Table 2: Details of numerical experiments of shock double-wedge R S diffraction. T1 θ2 M S

Case MS θ1 θ2 Grid θ1

Reflection type over surface 1 Reflection type over surface 2 Figure 10. Schematic illustration of progressive development of shock-wave diffraction over concave-double wedge. 6 2.16 15o SMR 35o SMR 420×240

7 2.16 20o SMR 55o RR 420 240 × ↔SMR

Figure 11 shows numerical images at sequence of time instants for shock wave of MS = 2.16 diffraction o over double wedge whose angles θ1 and θ2 are 15 and 35o, respectively (case 6). The time has been normalised using shock-wave velocity and horizontal distance of the inclined surface. The zero time is the instance when the incident shock wave hits the first edge. The dashed lines shown in the pressure images are the path of the triple points. For the consider Mach number and wedge angles, the reflection type is SMR in both surfaces. The first row shows the formation of the first triple-shock structure, T1, at the first wedge. The second triple point is formed due to the interaction of the first Mach stem with the second wedge surface (see the second frame). The first Mach stem at the first surface is straight. However, the second Mach stem has a curved shape (see the third frame). The third and forth frames show that as the time progress, the second Mach stem swallows the first one. It is found that the path of the triple point T1 is parallel to the second surface. The new forming triple point, T, (see the forth Figure 11. Numerical images showing progressive development of frame) is a junction of six discontinuities, incident the reflection patterns for shock of MS = 2.16 diffraction shock, Mach stem, two reflected shocks and two over double wedge of angles 15o and 35o.

Emirates Journal for Engineering Research, Vol. 12, No.2, 2007 57 A. Bagabir, A. Abdel-Rahman and A. Balabel

contact surfaces (see also Fig. 10). The present 15 numerical simulation produces clearly the first and 14 second contact surfaces. 13 12 For the above case, the variation of maximum 11 pressure load with time at the wedge surface is shown 10 in Fig. 12. It is obvious that the pressure load increase 9 suddenly when the moving shock interacts with the pressureMax. 8 wedge edges. At the first surface the maximum load is 7 almost constant. However, it is fluctuated at the 6 5 second surface. The maximum overpressure load of 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 about 14.6 times the ambient pressure occurs at time Time 6.6 when the incident shock moving at the second Figure 12. Maximum pressure load at the surface against time for surface. The pressure distribution along the wedge shock of MS = 2.16 diffraction over double wedge of surface, shown in Fig. 13, reveals that the maximum angles 15o and 35o. overpressure occurs at the second edge. This is also clear in the pressure image (Fig. 11) as black spot at 15 the second edge at t = 0.66. The maximum overpressure can be attributed to the interaction of the reflected sock with the second edge and enhanced by 10 the interaction of the contact surface, S1, (see the Pressure frames at t = 0.66). The density and pressure 5 distributions along the double-wedge surface, shown Density in Fig. 13, exhibit similar trend. edge 1 edge

Pressure and density images showing progressive 2 edge development of the reflection patterns for M =2.16, 0 S θ1 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 o o = 20 and θ2 = 55 (case 7) are illustrated in Fig. 14. Position The first frame at t = 0.35 shows the formation of the Figure 13. Density and pressure distributions along the surface for first triple point, T1, of SMR reflection. The second shock of MS = 2.16 diffraction over double wedge of and third frames exhibit the RR ↔SMR transition on angles 15o and 35oat time 0.66. the second wedge surface. According to the shock diffraction over wedge configurations[4] and numerical experiments (not shown here), for the consider incident shock wave, the reflection over the 55o wedge should be RR. However, for the present case, the reflection is RR ↔SMR. This is not caused by the incident shock attenuation; because even for lower incident shock wave (MS < 2.16) the reflection type remains RR[4]. Nevertheless, this could be attributed to the interaction of the first Mach stem with second edge. At t = 0.62 (the third frame of Fig. 14), the two triple points coincide. Also at this time, the second triple point, T2, gets its highest Mach stem. The forth and fifth frames demonstrate the collision of T1 with the wedge surface. It is noticed that the contact surfaces, S1 and S2, are very close to the surface. Because this is unsteady case, it is observed the reflection and of expansion waves. Figure 15 shows comparison of maximum pressure load at different time for case 6 (15o and 35o) and case 7 (20o and 55o). It is obvious that case 7 exhibit higher maximum overpressure of about 20 due to higher wedge angle. It occurs at the second edge as shown in the pressure distribution along the surface (Fig. 16). However, both cases show almost similar trends except two observations. The first one is that for case 6 the maximum load decreases after the interaction of the incident shock with the second edge at about t Figure 14. Numerical images showing progressive development of =0.5. The second observation is that for case 7, the the reflection processes for shock of MS=2.16 diffraction maximum pressure increases again after the maximum over double wedge of angles 20o and 55o

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20 overpressure (at t = 0.62). This is attributed to the case 7 interaction of the first triple point, T1, with the wedge surface. This is illustrated in the pressure distribution 15 at the wedge surface at t = 0.75 shown in Fig. 17. It case 6 demonstrates that the location of the maximum pressure is at T1. Furthermore, the enlarged density

Max. pressure 10 contours shown in Fig. 18 depicts the interaction of T1 with the surface at t = 0.75. It is worth mentioning that the generation of high pressure due to the 5 interaction of triple point with structure was also 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 observed for confined explosion[17]. Figure 18 also Time verifies that the second Mach stem, M2, is not straight Figure 15. Comparison of maximum pressure load at different time due to compression wave. for shock of MS = 2.16 diffraction over double wedge of case 6 (15o and 35o) and case 7 (20o and 55o). 4. CONCLUSIONS 20 In this study the reflection patterns of diffraction of shock waves over single and double wedges are 15 numerically investigated. The research is performed using implicit-unfactored high-order Riemann solver in associated with a modified MUSCL scheme. A 10 Pressure cylindrical shock problem is first tested in order to verify the solver accuracy. It is found that the 5 Density computed result agrees well with the exact solution. The scheme is validated against different reflection edge 1 edge 2 0 configuration of diffraction of shock wave over single -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 wedge. The present results show very good agreement Position with the previous numerical and experimental Figure 16. Density and pressure distributions along the surface for investigations. Moreover, the present study shock of MS = 2.16 diffraction over double wedge of investigates the SMR↔TMR and TMR↔DMR angles 20o and 55oat time 0.62. transitions. The numerical experiments show that the Mach stem of TMR reflection type changes its 20 T1 orientation from straight to convex shape due to the disturbances of the curled contact surface. For 15 TMR↔DMR transition, the compression wave is replaced by the second Mach stem. 10 The present study also simulates diffraction of a Pressure planar shock wave over concave-double wedge. It is observed that the second Mach stem formed in the 5 Density second wedge has a curved shape while the first Mach edge 1 edge 2 stem produced by the first surface is straight. For case 0 o o -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 7 (MS=2.16, θ1 = 20 and θ2=55 ), the reflection over

Position the second wedge surface is RR ↔SMR transition. It is

Figure 17. Density and pressure distributions along the surface for found that the maximum overpressure occurs at the shock of MS = 2.16 diffraction over double wedge of second edge. However, high pressure zone is angles 20o and 55oat time 0.75. developed due to the interaction of the first triple point with the wedge surface.

ACKNOWLEDGEMENT The first author would like to thank the German Academic Exchange Service (DAAD) for the visiting- scholarship grant (A/05/06681) at Technical University, Braunschweig, Germany for the period July-September 2005.

Figure 18. Density contours for shock of MS = 2.16 diffraction over double wedge of angles 20o and 55o at time 0.75.

Emirates Journal for Engineering Research, Vol. 12, No.2, 2007 59 A. Bagabir, A. Abdel-Rahman and A. Balabel

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