Physics - 142 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 Numerical Simulation to Study the Effects of Riemann Problems on the Physical Properties of the Astrophysics Gas Dynamics L. Y. S. Al-Mashhadani Department of physics, College of Education Ibn Al-Haithem, University of Baghdad Received in: 26 September 2011 Accepted in: 26 February 2012 Abstract In this work we run simulation of gas dynamic problems to study the effects of Riemann problems on the physical properties for this gas. We studied a normal shock wave travels at a high speed through a medium (shock tube). This would cause discontinuous change in the characteristics of the medium, such as rapid rise in velocity, pressure, and density of the flow. When a shock wave passes through the medium, the total energy is preserved but the energy which can be extracted as work decreases and entropy increases. The shock tube is initially divided into a driver and a driven section by a diaphragm. The shock wave is created by increasing the pressure in the driver section until the diaphragm bursts, sending normal shock waves down the shock tube into the low pressure driven section and at the same time sending an expansion wave into the high pressure driver section. Keywords: Riemann problems, numerical simulation, real gas flow Introduction From the solution of Riemann problems one finds directly how much velocity, density, and pressure flows into a cell from the interface under consideration. The initial value problem for any discontinuity is known as the Riemann problem. Where W represents the values of density, velocity and pressure. They can be thought of two constant states separated by a diaphragm at x0 which is removed at t = 0 and x0 is the interface between two meshes as it clear in fig. 1. The initial condition for the Riemann problems is set by specifying the left and right values of the density, velocity and pressure. Riemann problem for the Euler equations leads to two types of waves spreading. They are shock and expansion waves. The possible Riemann configurations are thus: 1. One shock and one expansion wave. 2. Two shock waves. 3. Two expansion waves. In each of the above cases there will be a contact discontinuity in between the two waves (due to the fact that we started with an initial discontinuity). [1] The collision between two flows, leading to the formation of three discontinuity as shown in fig.2. http://en.wikipedia.org/wiki/Pressure http://en.wikipedia.org/wiki/Density Physics - 143 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 The first Riemann problem we will consider is the so-called shock tube. The shock tube is a device in which a normal shock wave is produced by the sudden bursting of a diaphragm separating a gas at high pressure from one at lower pressure. When the diaphragm bursts a shock wave forms almost instantaneously and propagates into the driven section, while simultaneously an expansion wave propagates, in the opposite direction, into the driver section. The propagation of the shock front and expansion fan changes the gas pressure, and density, and sets the gas in motion relative to the shock-tube walls. The strength of the shock wave and expansion wave thus produced depends on the initial pressure ratio across the diaphragm and on the physical properties of the gases in the driver and the driven sections. [2] Jump Conditions Across a Standing Normal Shock Wave; Relationship Between Laboratory Fixed and Shock Fixed Coordinates. We first consider the case of a standing normal shock wave in a tube, which is clear in fig. 1. The conservation of mass, momentum, and energy for the flow through this wave are given by: uu 2211 ρρ = (1) uPuP 2 222 2 111 ρρ +=+ (2) ρερε 2 2 2 2 2 1 1 1 2 1 2 1 2 1 PuPu ++=++ (3) Where: pu 111 ,,ρ are the pre-shock density, velocity, and pressure and pu 222 ,,ρ are the post shock density, velocity, and pressure repectively. For an ideal gas pressure and internal energy are related by: )1( − = γ ρε P (4) is the specific internal energy. is the specific heat capacity. The entropy related quantity is ρ γ ξ − = P (5) The ratio of the shock parameters, are known as the Shock jump or Rankine –Hugoniot conditions (in the frame of Shock) and express by, [3, 4]: 2)1( )1( 2 1 2 1 1 2 +− + = M M γ γ ρ ρ (6) )1)(1( 2 1 2 11 2 −+ += MP P γ γ (7) Where M is the Mach number of the flow and it define by: (8) CS is the sound speed and it is equal to: Physics - 144 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 ρ γ P C S = (9) Numerical Results and Discussions In this work we applied numerical software to analyses Riemann problems of the Shock – tube problems. We run the simulation with the initial conditions in table 1. A stability condition is required and, assuming a constant grid size ( x∆ ), the time step ( t∆ ) is calculated by using: )max(λi CFL x t N ∆=∆ (10) Where )(maxλi is the maximum wave speed and NCFL the required Courant (CFL) number. [5] The numerical domain size ( x∆ ) of 1 is divided into 1000 computational cells of length 0.001. The computation has been performed using CFL number of 0.5. Shock Tube Problems Results The Sod shock tube is a Riemann problem used as a standard test problem, but in this case we did not exactly do the classical Sod problem because we do not consider any combustion reactions here, but are instead interested in the pattern of discontinuities. We started with the initial conditions shown in table 1, and the specific heat capacity is chosen to be (γ = 5/2). Applying the initial conditions and running the program leads to solution consisting of a shock wave propagates into the region of lower pressure, across which the density and pressure jump to higher values and all of the state variables are discontinuous. This is followed by a contact discontinuity, across which the density is again discontinuous but the velocity and pressure are constant. The third wave moves into the opposite direction and have a very different structure. This wave is called an expansion wave, and across which the materials streams in with a high density, low velocity, and high pressure. Results of solution of Riemann problems for Euler equations is shown in figure 3, and figure 5. On the plot in figure 3 one can see the density jumps through the contact discontinuity while the speed of the flow and pressure are the same. In the same figure, the expansion wave moves to the left into the high pressure region, while the shock and contact discontinuity move to the right into the low pressure region. Using the shock jump conditions for the pre- and post shock states, one can calculate the Mach number characterizing the strength of the shock in the shock frame 1.48, and the shock velocity in the lab frame is 1.8 cm s -1. The time evolution of the right travelling shock and the left expansion wave shown in figure 4. In this figure the density profile is plotted for three instants in time. The contact discontinuity in figure 5, travelling slowly to the right with speed of 0.8 cm s- 1, Shock moves fast to the right with speed of 1.83 cm s-1, and the expansion wave goes to the left with speed of - 1.28 cm s-1. If we compare the value for the shock here, it is close to the previous value. So, the numerical solution is close to the exact solution in case of shock tube problem. Figure 5, illustrates results of Riemann problem of the shock tube for the Mach number μ in the lab frame, specific internal energy, total energy density, and entropy related quantity. It is Physics - 145 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 clear that there is a jump in the Mach number, specific internal energy density, the total energy density, and the entropy related quantity across the shock and at the contact discontinuity. Conclusions This work discusses the propagation of shock waves through a shock tube. It was found that the gas to the left and right of the diaphragm is initially at rest. The pressure and density are discontinuous across the diaphragm. At t = 0, the diaphragm is broken. Three types of discontinuity then propagate through the gas: • Contact discontinuity: The pressure p and velocity u are continuous, but the density ρ, Mach number, specific internal energy density, the total energy density, and the entropy related quantity are discontinuous. • Shock waves: All quantities p, u, ρ, Mach number, specific internal energy density, the total energy density, and the entropy related quantity are in general discontinuous across the shock front. • Expansion wave: which is basically the reverse of a shock wave. It was found also that the numerical calculations of the velocities of the three discontinuities give similar speed of the 3 discontinuities by looking at a range of outputs (figure 4). References 1. Clarke, C. and Carswell, R. ( 2007), Principle of Astrophysical Fluid Dynamics, Cambridge University Press, UK. 2. Sod, G. (1991), Numerical relativistic hydrodynamics: Local characteristic approach, J. Comp. Phys., 27,(1). 3. Kamali, R., et al, (2006), Numerical Solution of Compressible Euler Equations for Gas Mixture Applications, Scientia Iranica, 13( 3),University of Technology. 4. Browne, S.; Browne, J. and Shepherd, J. E. GALCIT Report FM 2006.006, July (2004)-Revised August 29, (2008), Numerical Solution Methods for Shock and Detonation Jump Conditions, California Institute of Technology. 5. Deconinck, H. and Dick E. (2009), Computational Fluid Dynamics, Springer - Verlag Berlin Heidelberg. Regions Density (g cm-1 s-1) Velocity (cm s-1) Pressure (g cm-1 s-2) Right 0.125 0 0.1 Left 1.0 0 1.0 Table(1): Initial conditions for shock tube problems Physics - 146 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 Fig. (1): The Riemann Problem Fig( 2): Collision of two flows Physics - 147 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 Fig.(3): Results of the shock tube problem. The figure is show the density profile in blue color, velocity profile in green color, and the pressure profile in red color. The velocity and pressure are continuous at the contact discontinuity. The size of the time step was controlled by the CFL condition with CFL number of 0.5 Fig.( 4): Density profile along the shock tube for different time steps. Initial discontinuity at x = 0 Physics - 148 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 Fig.( 5): The Mach number, specific internal energy, total energy density, and entropy related quantity, as a function of position Physics - 149 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 الخصائص الفيزيائية للغاز في تحليالت عددية لدراسة تأثيرات مشاكل ريمان الداينميكي الفيزيائي الفلكي ليالي يحيى صالح ألمشهداني جامعة بغدادابن ألهيثم، –كلية التربية قسم الفيزياء، 2012شباط 26قبل البحث في 2011ايلول 26استلم البحث في الخالصة الخصائص الفيزيائية فيحث نفذ برنامج تحليل عددي لمشاكل الغاز الداينميكي لدراسة تأثيرات مشاكل ريمان في هذا الب لهذا الغاز. معين مثلمنتشرة بسرعة عالية خالل وسط (Shock wave) موجة اصطدام اعتيادية التي تسمى تلقد درس (Shock tube) الكثافة، و الضغط. و عند و يحدث ارتفاع سريع لقيم السرعة، اذ،. ما يسبب تغيير في خصائص ذلك الوسط مرور هذه لموجة خالل الوسط فان اجمالي الطاقة اليتبدد و لكن كمية الطاقة التي تتحول الى شغل هي التي سوف تتبدد و .ايضا" يحدث زيادة في عشوائية النظام في تحطم الحاجر الذي يفصل بين جزئي االنبوب مولدا موجات ان زيادة الضغط داخل انبوب االصطدام الذي يتسببكما expansion)اصطدام اعتيادية منتشرة الى مناطق الضغط العالي و موجات اخرى تنتقل الى مناطق الضغط الواطئ وتسمى waves). مشاكل ريمان، تحليالت عددية، جريان الغاز الحقيقيمفتاحية: الكلمات ال 2. Sod, G. (1991), Numerical relativistic hydrodynamics: Local characteristic approach, J. Comp. Phys., 27,)1(.