Numerical Analysis of Shock Tube Problem by using TVD and Schemes Dr. Mukkarum Husain, Dr. M. Nauman Qureshi, Syed Zaid Hasany IST Karachi, Email: mrmukkarum@yahoo.com Abstract Computational Fluid Dynamics (CFD) has grown from a mathematical curiosity to become an essential tool in almost every branch of fluid dynamics. Now, CFD as a computational technology is eminently suited to develop the concept of numerical test rig. Yee, Sandham and Djomehri developed a low dissipative high order artificial compression method () using characteristic based filters. The aim of these schemes is to minimize numerical dissipation for high speed compressible viscous flows containing shocks, shears and turbulence. In present studies, different shock tube test cases, namely, SOD, D LAX, and D inverse shock are simulated by using TVD and low dissipative high resolution artificial compression methods (). Comparison is made between numerical and analytical results. Computation shows that low dissipative high order is reasonably good for predicting shock waves, contact wave, and expansion fan. Key Words TVD, Low Dissipative High Order Artificial Compression Method, Shock Tube, SOD, D LAX, D Inverse Shock. Introduction TVD schemes are stable for nonlinear hyperbolic conservation laws. TVD schemes being able to capture discontinuities with ease and high resolution [-5]. While TVD formulations are very reliable, versatile, and quite accurate, they do have certain inherent accuracy limitations. At extrema points of the solution, TVD schemes automatically reduce to being first-order accurate discretization locally, while away from extrema they can be constructed to be of higher-order accuracy. Yee, Sandham and Djomehri [6-8] developed low dissipative high order shock capturing method using characteristic based filters. The aim of these schemes is to minimize numerical dissipation for high speed compressible viscous flows containing shocks, shears and turbulence. An artificial compression method [9] () of Harten is employed to identify non-smooth behavior and control the quantity of numerical dissipation to be added. Yee et al. utilized in an entirely different context than Harten originally intended. The fundamental idea of schemes consists of two steps, the first step is the high order temporal and spatial base scheme and the second step is the appropriate filtering for stability, shocks and for fine scale capturing. The shock tube problem [] is a general test case for the accuracy of computational fluid dynamics codes. It is often used as a test case for validation of numerical codes, because analytical solutions are available. In present work, numerical and analytical results of shock tube problem are computed and compared to analyze the capability of TVD scheme and low dissipative high order. Three different test cases are solved. Shock tube problem is sketched in figure and descriptions of test cases are given in Table. Table : Description of D Test Cases SOD LAX Inverse Shock p..57. ρ.5.5. v.. 5.968 p 4. 3.58 9. ρ 4..445 5. v 4..698.836
Driver Section High Region (4) Driven Section Low Region () P 4, ρ 4, V 4 P, ρ, V Diaphragm P 4 P Distance Fig : Initial Conditions in a Shock Tube Governing Equations The Navier-Stokes equations are used in present study. It can be written in conservation form under Cartesian coordinate system as Q F F + = v t x x () T Here, Q = ( ρ, ρu, E), T F = ρu ρu + p,( E + p) u () ( ) T Fv =, τ xx, uτ xx + k x the total energy E = p + ρ u (3) γ and kinematic viscosity μ is calculated using Sutherland formula as.5 μ T T + C = (4) μ T T + C where 5 μ =.7894 kg ( m s), T = 88.5K, C =.4K The thermal conductivity k is calculated by μc p k = Pr (5) Viscous stress tensor is calculated by following relations: 4 u τ xx = μ (6) 3 x Perfect gas assumption is used, which gives (7) Numerical Analysis Numerical analysis has been carried out using the following two different methods:. Second order [-3] scheme for convective part and second order central difference approximation for diffusive part of the governing equations T
. Forth order [6] for convective part and forth order central difference approximation for diffusive part of the governing equations Roe s approximate average state is used to calculate eigenvalues and eigenvector matrix [4]. Scheme The convective flux for scheme can be cast into the form F = Fi+ Fi x Δx (7) Here, _ F i + = [ F i + F i + + R Φ i + i + ] (8) where m m m m m m m Φ = σ a ( g j+ + g j) ψ a + γ α j+ j+ j+ j+ j+ (9) m =,, 3 z z δ ψ ( z ) = ( z + δ ) δ z < δ () ( U + U ) α () = R j j j+ j+ ( j+ j) m m m m g g α For α j+ j+ m () γ = j+ m For α = j+ σ ( z ) = ψ ( z ) + λβ ( θ ) z (3) For steady state calculation and/or implicit method, we have σ ( z ) = ψ ( z ) (4) limiter used for Eq. () in the present study is given as, m m m g j = min mod α, α (5) j j+ Low Dissipative High Order Method The fundamental idea of these shock-capturing schemes consists of two steps. The first step is a high-order spatial and temporal base scheme. Various standard high-order non dissipative or low dissipative base schemes fit in the current frame work. The second step is the appropriate filter for stability, shocks, contact discontinuities, and fine scale flow structure capturing. Various TVD, positive, WENO, and ENO dissipations, after a slight modification, are appropriate candidates as filters.. The Base Scheme Fourth order central differencing used for both convective and diffusion parts of Navier-Stokes equations is given below: F = Fi+, j, k 8Fi+, j, k+ 8Fi, j, k Fi, j, k (6) x Δx F 9 v = Fv Fv Fv 3 Fv 3 (7) x 8Δx i+,, j k i,, j k 4 x i+,, j k i,, j k Δ Other types of base schemes can also be formulated in the same manner.
. The Numerical Flux Filtering Scheme Non linear dissipation term R i + Φ i + of scheme in combination with Harten s switch applied to all characteristic waves as the filter numerical fluxes, * F% = kθ i + Ri i i + Φ + (8) + The parameter k is a problem dependent. The function θ i + is the Harten switch. For a general m+ points base scheme, Harten recommended θ = max ( θ j m j m ) i + +,....., θ (9) + p α α j+ j θ () = α + α j+ j Instead of varying k for the particular physics, one can vary p. The higher the p, the less the amount of numerical dissipation is added. Note that by varying the p one can essentially increase the order of accuracy of the dissipation term. The order of the dissipation depends on the value of p. For all the numerical examples, we use, θ = max ( θ, ) + j θ () i j + The new time level is defined as * * * Lf ( F% ) = F% F% () Δx i+, j, k i, j, k ) n+ n+ * * * Q = Q + Δt L F%, G%, H% (3) f ( ) Results and Discussion Figures -4 depict, respectively, the numerical results computed for SOD, D lax and D inverse shock problems and their comparisons with the respective analytical solutions. It is seen that the resolution for the shock wave is captured better than contact discontinuity for different schemes. The shock wave is a compressive wave, the characteristic lines of the conservation laws are convergent and so the dissipation near to the shock wave is controlled in a small level in the time marching steps regardless of the numerical dissipation. Near the contact wave the discrepancy of the curves by different schemes are noticeable. The contact discontinuity is a kind of linear wave in the theory of the hyperbolic conservation laws and the characteristic lines are parallel to each other so the dissipation can not be restrained. During the time marching the dissipation is accumulated, and the contact wave may span more and more grid points. Especially for the schemes that owned larger numerical dissipation, the jump is weakened more seriously. It is very difficult to resolve contact wave accurately by any scheme and it can also be seen from the results. The method is better than the TVD scheme especially in the vicinity of the contact wave. At the head/tail of the expansion fan, the method also gives better results than the TVD scheme..8.4 Density.8.4 a) Comparison of profile b) Comparison of Density Profile
.8 Mach.4 8 4 4 8 a) Comparison of Mach profile Fig : Numerical Results of SOD Problem. 3 Density.8.4 a) Comparison of profile b) Comparison of Density profile.8 Mach.4 c) Comparison of Mach profile Fig 3: Numerical Results for D LAX Problem 3 Density 4 a) Comparison of profile b) Comparison of Density profile
Mach 6 4 c) Comparison of Mach profile Fig 4: Numerical Results for D Inverse Shock Problem Conclusion In present work, shock tube problem have been solved to study the applicability and accuracy of two different numerical methods, namely, TVD and low dissipative high order. Three test cases composed of SOD, D LAX, and D inverse shock are simulated for comparative purpose. Comparison is made between numerical and analytical results. Computation shows that low dissipative high order is reasonably good for predicting shock waves, contact wave, and expansion fan. References [] John D. Anderson, Jr., Computational Fluid Dynamics, the Basics with Application. McGraw-Hill Series in Mechanical Engineering, 995 [].J. C. Tannehill, D. A. Anderson, R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer. Second Edition, January 997. [3] Toro EF, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction [M]. Springer, 997. [4] Yee, H.C., Upwind and Symmetric Shock-Capturing Schemes. NASA TM-89464, May 987. [5] Yee, H.C., A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods. VKI Lecture Series 989-4 Mar.6-, 989, also NASA TM-88, Feb.989. [6] Yee, H.C., N. D. Sandham, and M. J. Djomehri, Low Dissipative High Order Shock Capturing Methods Using Characteristic-Based Filters. Journal of Computational Physics, 5, pages99-38, 999. [7] Mukkarum Husain, Lee Chun Xuan, Comparative Study of Modern Shock Capturing Schemes. ICFP9, Hangzhu, China, 9 [8] Kamran R. Q., Numerical Simulation of High Speed Vehicle using High Resolution, High-Order, and Low Dissipative Scheme. PhD thesis, Beihang University, Beijing, China, 8. [9] Harten A., The Artificial Compression Method for Computation of Shocks and Contact Discontinuities, III Self-Adjusting Hybrid Schemes. Math Comp, 3:363-389, 978. [] Anderson, J. D., Jr., Modern Compressible Flow: With Historical Perspective. McGraw-Hill Book Co., New York, 98. [] Yee, H.C., G. H. Klopfer, and J. -L. Montagne, High-Resolution Shock-Capturing Schemes for Inviscid and Viscous Hypersonic Flows. Journal of Computational Physics, 88, pages 3-6, 99. [] A Harten, High Resolution Schemes for Hyperbolic Conservation Laws. J. Computational Phys.49, pages 357-393, 983. [3] Mukkarum Husain, Lee Chun Xuan, Comparative Study of Limiters for Scheme. IBCAST, Islamabad, Pakistan, 9. [4] P. L. Roe, Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes. Journal of Computational Physics, 43, 357(98)..