Numerical and Theoretical Analysis of Shock Waves Interaction and Reflection

Fluid Structure Interaction and Moving Boundary Problems IV 299

Numerical and theoretical analysis of shock waves interaction and reflection

K. Alhussan Space Research Institute, King Abdulaziz City for Science and Technology, Saudi Arabia

Abstract

This paper will show numerical and theoretical analysis of shock waves interaction and reflection. In this paper some characteristics of non-steady, compressible flow are explored, including compression and expansion waves, interaction and reflection of waves. The work to be presented herein is a Computational Fluid Dynamics investigation of the complex fluid phenomena that occur inside a three-dimensional region, specifically with regard to the structure of the oblique shock waves, the reflected shock waves and the interactions of the shock waves. The flow is so complex that there exist oblique shock waves, expansion fans, slip surfaces, and shock wave interactions and reflections. The flow is non-steady, viscous, compressible, and high-speed supersonic. This paper will show a relationship between the Mach numbers and the angles of the reflected shock waves, over a double step and opposed wedges. The aim of this paper is to develop an understanding of the shock waves’ interaction and reflection in a confined space and to develop a relationship between the Mach number, the geometry of the channel and the strength of the reflected shock waves. In this paper a global comparison is made between the numerical method and the classical method. Keywords: compressible flow, oblique shock wave, shock wave interaction, shock wave reflection, numerical analysis, CFD.

Nomenclature

M Mach number p Static pressure po Total pressure

T Static temperature To Total temperature

Subscript i Region number

1 Introduction

This paper will explain the numerical analysis and the structure of the flow over a double step and opposed wedges. In this paper some characteristics of compressible, flow are explored, including compression and expansion waves. The work to be presented herein is a Computational Fluid Dynamics investigation of the complex fluid phenomena that occur inside a three- dimensional region, specifically with regard to the structure of oblique shock waves over double wedge and opposed wedges. The flow is so complex that there exist oblique shock waves, expansion fans, and slip surfaces. Solving these problems one can compare solutions from the CFD with analytical solutions. The problems to be solved involve formation of shock waves and expansion fans, so that the general characteristics of supersonic flow are explored through this paper. Shock waves and slip surfaces are discontinuities in fluid dynamics problems. It is essential to evaluate the ability of numerical technique that can solve problems in which shocks and contact surfaces occur. In particular it is necessary to understand the details of developing a mesh that will allow resolution of these discontinuities. Continuous compression waves always converge and the waves may coalesce and form a shock front. As more and more of the compression waves coalesce, the wave steepens and becomes more shock fronted. Discontinuities exist in the properties of the fluid as it flows through the shock wave, which may be treated as boundary for the continuous flow regions located on each side of it. Shock waves are also formed when the velocity of the fluid at the solid boundary of the flow field is discontinuous, as in the instantaneous acceleration of a piston. A moving shock wave may be transformed into a stationary shock wave by a relative coordinate transformation wherein the observer moves at the same velocity as the shock wave. The resulting stationary shock wave may, therefore, be analyzed as a steady state case [1–13]. In addition to the shock wave, there is another type of discontinuity termed a contact surface. The contact surface is an interface that separates two flow regions, but moves with those regions. The velocity and the pressure of the gas on each side of the contact surface are the same, but the other thermodynamic properties may be different. Unlike the shock wave, there is no flow of gas across a contact surface. It is clear that nothing is learned about the possibility of the formation of a contact surface from the velocity and pressure, because velocity and pressure are equal across the contact surface [1–13]. The results show a promising achievement, first, to understand the flow structure inside a supersonic confined region, second, to use this knowledge to interpolate the numerical results in order to achieve a design methodology that

WIT Transactions on The Built Environment, Vol 92, © 2007 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) Fluid Structure Interaction and Moving Boundary Problems IV 301 will benefit the industrial applications. Results including contour plots of static pressure, static temperature, and Mach number will show the structure of shock waves and the reflected oblique shock waves in a complex three-dimensional region. The results will show a relationship between the Mach numbers and the reflected shock waves configuration, over a double step and opposed wedges. A CFD analysis enables one to understand the complex flow structure inside this confined region. Through this computational analysis, a better interpretation of the physical phenomenon of the three-dimensional shock waves interaction and reflection can be achieved.

2 Numerical analysis

The governing equations are a set of coupled nonlinear, partial differential equations. In order to formulate or approximate a valid solution for these equations they must be solved using computational fluid dynamics techniques. To solve the equations numerically they must be discretized. That is, the continuous control volume equations must be applied to each discrete control volume that is formed by the computational grid. The integral equations are replaced with a set of linear algebraic equations solved at a discrete set of points. CFD-FASTRAN is used in the current research to model the flow over double step and opposed wedges. The CFD code is an integrated software system capable of solving diverse and complex multidimensional fluid flow problems. The fluid flow solver provides solutions for compressible, steady-state or transient, laminar or turbulent flow in complex geometries. The code uses block- structured, non-orthogonal grids to discretize the domain. CFD-FASTRAN is used in this current research to model the flow characteristics inside the three dimensional region. CFD-FASTRAN solves the three-dimensional Navier–Stokes equations by utilizing a finite element based finite volume method over structured hexahedral grids. The CFD code is an integrated software system capable of solving diverse and complex multidimensional fluid flow problems. The fluid flow solver provides solutions for compressible, steady-state or transient, laminar or turbulent single-phase fluid flow in complex geometries. The code uses block-structured, non-orthogonal grids. It should be possible to model the characteristic of the flow, the interaction of the shock waves, and expansion fans over double step and opposed wedges using the CFD analysis [14–24]. A numerical analysis must start with breaking the computational domain into discrete sub-domains, which is the grid generation process. A grid must be provided in terms of the spatial coordinates of grid nodes distributed throughout the computational domain. At each node in the domain, the numerical analysis will determine values for all dependent variables such as pressure and velocity components. Creating the grid is the first step in calculating the flow analysis. Three- dimensional Navier–Stokes equations are solved using fully implicit scheme with K-epsilon turbulence model. The grid is refined near the surfaces and in front of the body in order to model the large gradient.

Figure 1: Schematic view of a double step showing a 2-D structured hexahedral mesh.

Figure 2: Schematic view of a 3-D structured hexahedral mesh of opposed wedges.

Figure 3: Schematic view of opposed unequal wedges showing a 2-D structured hexahedral mesh.

A computational model that illustrates the physics of flow over double step and opposed wedges was developed. Through this computational analysis, a better interpretation of this physical phenomenon can be achieved. The results from the numerical analysis will be used to develop a design methodology so as to predict optimal performance.

3 Results and discussion

Flow over a double step with turning angles of θ=7o and β=13o was analyzed. Figure 1 shows a two-dimensional structured mesh for a flow over double step.

Note that the non-uniform structured mesh is used as shown in figure 1. The analysis was carried out based on the flow over double step for Mach number 2.0 and 1.8 and air, γ=1.4, is the working fluid. Figure 2 shows a schematic view of a 3-D structured hexahedral mesh of opposed equal wedges, β=θ=5o. Flow over opposed equal wedges was conducted using Mach number 2.0 and air, γ=1.4, is the working fluid. Figure 3 shows a schematic view of opposed unequal wedges, θ=3o and β=8o. Figure 3, also, shows a 2-D structured hexahedral mesh. The analysis was carried out based on the flow over opposed unequal wedges for Mach number 2.0 and air, γ=1.4, is the working fluid.

2 1

β=13o o θ=7

Figure 4: Flow over a double step working fluid air (γ=1.4).

Figure 4 shows the flow over a double step. The first and the second turning angles are θ=7o and β=13o, respectively. From analytical and classical method of compressible flow theory one can describe the flow structure of this kind as shown in figure 4. Table 1 shows a global comparison between the classical and numerical methods [1–13].

Figure 5: Contour plot of Mach number, flow over double step working fluid

air (γ=1.4, M free stream=2).

Figure 5 shows contour plot of Mach number for flow over double step. For free stream Mach number of 2.0 over the double step body the flow will generate multiple shock waves and shock wave interactions, as seen in figure 5. One can see from figure 5 that at the leading edge of the first step with an angle of θ=7o an oblique shock wave is created. The flow in this region – region 1 – is parallel to the solid surface of the first wedge. The flow in region 1 is still supersonic and the flow properties are given by the oblique shock wave relations. The second step also generates an oblique shock wave and the region down stream of the shock wave is noted as region 2. The conditions of the flow are given by the oblique shock wave relations [1–13]. The flow in region 2 is parallel to the solid surface of the second step. See figure 4 for the numbering system.

Table 1: Global comparison between classical and numerical solutions for double step.

Analytical values (region) Numerical values (region) i=1 i=2 i=3 i=1 I=2 i=3

M free stream 2 2 2 2 2 2 M i 1.74 1.28 1.27 1.7 1.35 1.2 Wave Angle 36.3 49.8 51.55 35 48 54

Pi /Po 1.47 2.8 2.83 1.5 2.9 2.9 Ti /To 1.12 1.36 1.37 1.2 1.4 1.5

Figure 6: Contour plot of static pressure, flow over double step working fluid

air (γ=1.4, M free stream=2).

Figure 6 shows a contour plot of static pressure for flow over double step with free stream Mach number 2.0. In this figure one can see the generation of multiple shock waves. The two shocks generated in regions 1 and 2 coalesce into a third shock in region 4 with downstream conditions. The flow in region 4 is parallel to the second step and the upstream conditions are the free stream conditions. Here the flow has passed through one oblique shock. But in region 2 the flow has gone through two oblique shock waves. Note that the static pressure in regions 2 and 4 is not the same, therefore an oblique shock wave is generated in the flow, and a new region is created, region 3. But the static pressure and the flow direction is the same in regions 3 and 4.Note that some flow variables in

WIT Transactions on The Built Environment, Vol 92, © 2007 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) Fluid Structure Interaction and Moving Boundary Problems IV 305 regions 3 and 4 are not the same since the flow in region 3 has encountered different number and strength of shock waves. A slip surface is generated between regions 3 and 4 and flow parameters such as density and temperature are not the same. See figure 4 for the numbering system. Figure 7 shows contour plot of Mach number for flow over double step. For free stream Mach number of 1.8 over the double step body the flow will generate multiple shock waves and shock wave interactions, as seen in figure 7. One can see from figure 7 that at the leading edge of the first step with an angle of θ=7o an oblique shock wave is created. One can notice from figure 7 that the shape of the shock that is generated from the interaction of the two oblique shock waves at the solid boundary is different than for Mach =2.0.

Figure 7: Contour plot of Mach number, flow over double step working fluid

air (γ=1.4, M free stream=1.8).

Figure 8 shows the flow over opposed wedges. From analytical and classical method of compressible flow theory one can describe the flow structure of this kind as shown in figure 8. Tables 2 and 3 show a global comparison between the classical and numerical methods [1–13]. Figure 9 shows a contour plot of Mach number for flow over opposed equal wedge. The free stream Mach number is 2.0 and air, γ=1.4, is working fluid. One can see the generation of multiple shocks and shock reflections. Figure 10 shows a contour plot of static pressure for a flow over opposed equal wedges. Figure 11 shows a contour plot of static pressure for flow over opposed unequal wedges. Figure 12 shows a contour plot of static temperature for flow over opposed unequal wedges. One can see the creation of the slip surface from the contour of static temperature, as shown in figure 12. For a free-stream Mach number of 2.0 over the opposed wedges body the flow will generate a multiple shock waves and shock waves interactions. At the leading edge of the bottom wedge with an angle of θ=3o an oblique shock wave is created. The flow in this region – region 1– is parallel to the solid surface of the bottom wedge. The flow in region 1 is still supersonic and the flow properties are given by the oblique shock relations. The top wedge with an angle of β=8o also generates an oblique shock wave and the region down stream of the shock

WIT Transactions on The Built Environment, Vol 92, © 2007 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) 306 Fluid Structure Interaction and Moving Boundary Problems IV wave is noted as region 2. The conditions of the flow are given by the oblique shock wave relation [1–13]. The flow in region 2 is parallel to the solid surface of the top wedge [4–9, 12–14]. See figure 8 for the numbering system.

Table 2: Global comparison between classical and numerical solutions for opposed wedges.

Analytical value (region) Numerical values (region) i=1 i=3 i=5 i=1 i=3 i=5

M free stream 2 2 2 2 2 2 M i 1.83 1.65 1.48 1.7 1.6 1.4 Wave Angle 34.28 37.93 42.36 35 39 45

Pi /Po 1.33 1.72 2.19 1.5 1.9 2.4 Ti /To 1.09 1.18 1.25 1.3 1.4 1.5

The oblique shock waves generated by the two wedges intersect and form two additional oblique shock waves. The flow in region 4 is parallel to bottom wedge, while the flow in region 3 is parallel to top wedge. The static pressure in regions 3 and 4 must be the same and slip surface is generated between region 3 and region 4. The oblique shock wave for region 3 then intersects and reflects from the top wedge, while the oblique shock wave from region 4 intersects and reflects from bottom wedge. Note the Mach number is decreased in all of the shock reflections and intersections.

β=8o

θ=3o Figure 8: Flow over opposed wedges working fluid air (γ=1.4).

Table 3: Global comparison between classical and numerical solutions for opposed unequal wedges.

Analytical values (region) Numerical values (region) i=1 i=2 i=3 I=1 I=2 i=3

M free stream 2 2 2 2 2 2 M i 1.89 1.72 1.62 1.8 1.65 1.55 Wave Angle 32.56 37.23 39.53 34 38.5 40

Pi /Po 1.19 1.55 1.82 1.2 1.6 1.7 Ti /To 1.06 1.14 1.19 1.1 1.2 1.3

Figure 9: Contour plot of Mach number, flow over opposed equal wedges

working fluid air (γ=1.4, M free stream=2).

Figure 10: Contour plot of static pressure, flow over opposed wedges working

fluid air (γ=1.4, M free stream=2).

Figure 11: Contour plot of static pressure, flow over opposed unequal wedges

working fluid air (γ=1.4, M free stream=2).

Figure 12: Contour plot of static temperature, flow over opposed unequal

wedges working fluid air (γ=1.4, M free stream=2).

4 Conclusion

A computational model that illustrates the physics of flow through shock waves, expansion fans and slip surfaces was developed. The flow is compressible viscous high speed. In this situation, one should expect oblique shock waves, expansion fans, shock wave interactions, and slip surface generation. The results of the numerical data from this paper, such as static pressure, static temperature, and Mach number were used to show the good agreement between the numerical and the analytical solutions. Through this computational analysis, a better interpretation of this physical phenomenon of the can be achieved. The results from the numerical analysis are used to study the flow structure and compared it to the analytical solution. From the results illustrated in tables 1 to 3 and also from study of the detailed results from figures 5 to 7 and 9 to 12 one can conclude that CFD is capable of predicting accurate results and is also able to capture the discontinuities in the flow, e.g. the oblique shock waves and slip surfaces.

Acknowledgements

The author gratefully acknowledges sponsorship of this research from the Space Research Institute of the King Abdulaziz City for Science and Technology.

References

[1] Gaydon, A.G. and Hurle, I. R. The Shock Tube in High-Temperature Chemical Physics, Reinhold, New York, 1963. [2] Henshall, B.D. The Use of Multiple Diaphragms in Shock Tubes, A.R.C. National Physical Laboratory, England, 1955. [3] Anderson, J. D. Jr. Modern Compressible Flow, McGraw-Hill, New York, 1990. Saad, M. A., Compressible Fluid Flow, 2nd ed., 1993.

[4] Shapiro, A. E. The Dynamics and Thermodynamics of Compressible Fluid Flow, 2nd vol., Ronald, New York, 1953. [5] Zucrow, M. J. and Hoffman, J. D. Gas Dynamics, 2nd vol., Wiley, New York, 1977. [6] Cheers, F. Elements of Compressible Flow, Wiley, London, 1963. [7] Liepmann, H. W. and Roshko, A. Element of Gas Dynamics, Wiley, New York 1957. [8] Thompson, P. Compressible-Fluid dynamics, McGraw-Hill, 1972. [9] Glarke, T. F. and McChesney, M. The Dynamics of Real Gases, Butterworth, London, 1964. [10] Oosthuizen, P. H. and Carscallen, W. E. Compressible Fluid Flow, McGraw-Hill, 1997. [11] Schreier, S. Compressible Flow, Wiley, New York, 1982. [12] Courant, R. and Friedrichs, K. O. Supersonic Flow and Shock Waves, Interscience Publishers, Inc. New York, 1948. [13] Haluk, A. Gas Dynamics, McGraw-Hill, New York, 1986. [14] Alhussan, K. and Garris, C. A.: “Non-Steady Three-Dimensional Flow Field Analysis in Supersonic Flow Induction”, Fluids Engineering Summer Conference Montreal, Canada, Paper No. FEDSM2002-13088, July 2002. [15] Alhussan, K. and Garris, C. “Computational Study of Three- Dimensional Non-Steady Steam Supersonic Pressure Exchange Ejectors” IASME Transactions Issue 3, Volume 1, pp 504-512, ISSN 1790-031X, July 2004. [16] Alhussan, K. and Garris, C. “ Comparison of Cylindrical and 3-D B- Spline Curve of Shroud-Diffuser for a Supersonic Pressure Exchange Ejector in 3-D, Non-Steady, Viscous Flow” IASME Transactions Issue 3, Volume 1, pp 474-481, ISSN 1790-031X, July 2004. [17] Alhussan, K. and Garris, C. “Study the Effect of Changing Area Inlet Ratio of a Supersonic Pressure-Exchange Ejector” 43rd AIAA Aerospace Science Meeting and Exhibit, Paper No. AIAA-2005-519, Reno, NV, USA, January 2005. [18] Alhussan, K. “Application of Computational Fluid Dynamics in Discontinuous Unsteady Flow with Large Amplitude Changes; The shock Tube Problem” IASME Transaction Issue 1 Volume 2, pp 98-104, January 2005. [19] Alhussan, K. and Garris, C. 2005, “Study the effect of changing throat diameter ratio of a supersonic pressure exchange ejector” 6th KSME- JSME Thermal and Fluids Engineering Conference, Jeju City, South Korea, Paper Number: tfec6-406, March 20-23, 2005 [20] Alhussan, K. and Garris, C. 2005, “Computational analysis of flow induction phenomena in three-dimensional, non-steady supersonic pressure exchange ejectors” 6th KSME-JSME Thermal and Fluids Engineering Conference, Jeju City, South Korea, Paper Number: tfec6- 404, March 20-23, 2005. [21] Alhussan, K. “Computational Analysis of High Speed Flow over a Double-Wedge for Air as Working Fluid”, Proceedings of FEDSM2005

ASME Fluids Engineering Division Summer Meeting and Exhibition FEDSM2005-77441 June 19-23, 2005, Houston, TX, USA. [22] Alhussan, K. “Study the Structure of Three Dimensional Oblique Shock Waves over conical rotor-Vane surfaces”, Proceedings of FEDSM2005 ASME Fluids Engineering Division Summer Meeting and Exhibition FEDSM2005-77440 June 19-23, 2005, Houston, TX, USA. [23] Alhussan, K. “Oblique Shock Waves Interaction in a Non-Steady Three Dimensional Rotating Flow”, Proceedings of FEDSM2005 ASME Fluids Engineering Division Summer Meeting and Exhibition FEDSM2005- 77442 June 19-23, 2005, Houston, TX, USA. [24] Alhussan, K. “Application of Computational Fluid Dynamics in Discontinuous Unsteady Flow with Large Amplitude Changes; The shock Tube Problem” IASME Transaction Issue 1 Volume 2, pp 98-104, January 2005.