Emirates Journal for Engineering Research, 12 (2), 53-60 (2007) (Regular Paper) SHOCK-WAVE DIFFRACTION 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 اﻷبعاد. حيث تم حل معادﻻت أويلر للغازات المضغوطة بواسطة الحلول الضمنية للزمن. وقد أخذ بعين اﻷعتبار تحويل المعادﻻت التفاضلية الى منظومة من المعادﻻت الجبرية التى يسھل حلھا. تم التأكد من دقة نموذج التحليل العددي بالمقارنة مع نتائج سابقة لموجات إنفجارية مختلفة الشدة. النتائج وضحت أن الموجة اﻹنفجارية تودي الى نظام معقد من الموجات نسبةً لتصادمھا مع اﻷسطح الصلبة. كما تم التحقق من توزيع الضغوط على اﻷسطح الصلبة و تحديد مواضع الضغط اﻷعلى. 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 shock wave 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 waves 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 diffractions 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]: TMR F = αF + 1−α F (4) DMR ()FVS ()()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 [2] (c) shock Mach number, MS . 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]. 2 ρS ()γ +1 M 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.