Special Edition PGCON-MECH-7 Development of high resolution methods for solving D Euler equation Ms.Dipti A. Bendale, Dr.Prof. Jayant H. Bhangale and Dr.Prof. Milind P. Ray ϯ Mechanical Department, SavitribaiPhule Pune University, Matoshri college of Engineering, Nashik, India. Mechanical Department, SavitribaiPhule Pune University, Matoshri college of Engineering, Nashik, India. ϯ Mechanical Department, SavitribaiPhule Pune University, Sandip Institute of Engineering and Management, Nashik, India. Abstract In this work high resolution schemes are developed to solve gas dynamics problems. Two dimensional Euler equations are used as governing equations. One of the method to solve the Euler equations that are hyperbolic in nature, is to treat them as initial value problem. The Riemann solvers are adopted for solving the initial value problems. Riemann solvers are mainly of two types, exact and approximate solvers. The Godunov solver provides an exact solution whereas HLL and HLLC an approximate solution. These methods are developed to solve gas dynamic problems. In this work Godunov solver, HLLC solvers are considered for obtaining solutions to several D Sod tests. First, these methods are benchmarked with the solutions to exact Riemann problems. Later, a qualitative comparison is presented for understanding the performance of these methods. Keywords:High Resolution Methods, Riemann Solvers, Exact Riemann Solvers, Approximate Riemann Solvers, HLLC.. Introduction In this work high resolution schemes are developed to solve gas dynamics problems. One of the method to solve the Euler equations that are hyperbolic in nature, is to treat them as initial value problem. Riemann problem offers a solution strategy to solve a system of hyperbolic conservation equations. Following are the equations that are considered for discretization, t ρ ρu ρv ρe + x ρu ρu + p ρuv ρu(e + p/ρ + y ρv ρuv ρv + p ρv(e + p/ρ = () In case of Euler equation the Riemann contain so called shock tube problem, the basic problem of gas dynamics. The Riemann problem solution of two types Exact and Approximate. The Godunov's method provides exact solution whereas, HLL (Harten, Lax and Van Leer) approach is an approximation for inter-cell numerical flux. It offers a very efficient and robust approximation to Godunov type methods. The shortcomings of HLL approach are overcome in a modified approach called the HLLC (C stands for contact) Riemann solver. First, all the three Riemann solvers the Godunov, HLL and HLLC are benchmarked with the exact solutions of the five Riemann problems, the Sod tests. The conditions in these tests are too hypothetical to exist in a real gas dynamic problem. An agreement to the results of these problems ensures robustness of the solver. Later these approximate Riemann solvers are used for solving another five -D tests In 988 S. F. Davis have done a number of algorithms for obtaining the bounds on smallest and the largest signal velocities in the solution to Riemann problem where bounds are used to construct the approximate Riemann solution of Harten, Lax, and van Leer which contains only one intermediate state. This approximate Riemann solution is used to construct first- and second-order Godunov type methods. The methods described in this paper require only the characteristic velocities of a hyperbolic system. Therefore, these methods are easily applied to any system of conservations laws. He has applied the method of this paper to the steady supersonic Euler equations, the unsteady Euler equations with a nonconvex equation of state on the nonlinear shallow water equations. In 988 B. Enfield.has describe a new approximate Riemann solver for compressible gas flow. In contrast to previous Riemann solvers, where a numerical approximation for the pressure and the velocity at the contact discontinuity is computed, he derived a numerical approximation for the largest and smallest signal velocity in the Riemann problem. Having obtained the numerical signal velocities, he used theoretical results by Harten, Lax and van Leer to obtain the full approximation. A stability condition for the numerical signal velocities is derived. He has also demonstrate a relation between the signal velocities and the dissipation contained in the corresponding Godunov-type method. Also given numerical results for the one- and two-dimensional
Special Edition PGCON-MECH-7 compressible gas dynamics equations. In 3 Liska, R. and Wendroff, B. presented results of computations with eight explicit finite difference schemes on a suite of one-dimensional and two-dimensional test problems for the Euler equations. Both dimensionally split and two-dimensional schemes are represented, as are central and upwind-biased methods, and all are at least second-order. For D tests they had chosen five D (in x) Riemann problems. In 99 E. F. Toro investigated Riemann solvers and numerical methods for fluid dynamics. He considered more difficult set of D problems has been who describes in detail several popular methods and shows their behavior on his tests, all of which have easily computed exact solutions. In 994 E. F. Toro, M. Spruce and W. Spears presented new ways of obtaining wave speed estimates also implemented improved Riemann solver in the second order WAF method and tested for D problem.. Methodology. Understand shock tube problem and Initial value problems.. Use of Euler equations to solve IVPs and discretize these equations. 3.Obtain Exact solutions of IVPs (Five Sod tests), solving Riemann problems. 4. Use of these Riemann problems to obtain exact solutions, develop the Godunov's solver.. Benchmarking with Sod test D. 6. Develop approximate solution strategy, using HLLC approach for D. 7. Benchmarking with Sod test D. 8. Development of high resolution method for solving D U= [ρ, ρu, ρv, ρw,e] T, F(U)=[ ρu, ρu +p, ρuv, ρuw,u(e+p)] T, G(U)=[ ρv, ρuv, ρv +p. ρvw,v(e+p)] T, H(U)=[ ρw, ρuw, ρvw, ρw +p,w(e+p)] T. Here U = is a vector of conserved variables, F(U), G(U), and H(U) are flux vectors. ρ is density of fluid, u, v, w are velocity components in x, y, z directions. p is pressure and E is total energy per unit volume of the fluid. For one-dimensional case, Eqn.() reduces to U t+f(u) x= (3) Here, U = [ρ, ρu, E] T, F(U)=[ ρu, ρu +p,u(e+p)] T. Actually U, F, G, H are vectors but they are not be highlighted henceforth. 3. Basic Riemann Problem For one-dimensional non stationary flows, the state U of a perfect gas is specified completely by three independent variables ρ, u, p and two constants γ, R for a particular gas. From these variables and constants, all of the other state properties may be obtained, and the state variable U may completely be defined over a domain. The Riemann problem for one-dimensional time dependent Euler equations is represented as below: If the vectors of conserved variables U and flux vector F(U) have their usual form as in (3), solve U t+f(u) x= (3) Subjected to U x, = UL if x < ; UR if x <. Here, subscripts L and R represent the two discrete initial left and right states respectively.. Governing Equations The fluid, through which the wave propagating, is compressible and inviscid,. The fluid obeys the ideal gas equation,. No external forces act on the fluid, Thus reducing the Navier-Stokes equation to Euler equations which are written in the Cartesian vector form as, U +F(U) +G(U) Y+H(U) Z= () Where, Fig.Schematic representation of General Riemann problem To simplify the solution procedure, a vector of primitive variables W is often used instead of vector U: W = [ρ,u,p] T (4)
Special Edition PGCON-MECH-7 As shown in Fig.., the domain of interest is the (x-t) plane with x ranging from - to.as time progresses the discontinuities between the two initial states are break into leftward and rightward moving waves which are separated by the contact discontinuity. Each wave may either be a shock wave or a rarefaction wave (expansion wave) depending on the initial data, giving rise to four possible wave patterns as shown in Fig.. The region in between the left and right waves is called as the star region and the flow properties in this region are known as star properties.. On both side of contact discontinuity same gas exists.. The gas in the problem under consideration is a perfect gas. Properties of fluid flow across the shock are related by Rankine-Hugoniot relations, which are noting but applications of conservation equations across the shock wave, i.e., conservation of mass, momentum and energy. These are listed below. ρ u =ρ u () p +ρ u =p + ρ u (6) u (E +p )= u (E +p ) (7) 4.. Shock Wave As stated earlier, Rankine-Hugoniot conditions are applied across theshock. After certain mathematical manipulations one gets the relation for star properties as below. QL= ρ *Lu * (8) P *L= ρ L γ γ + +p pl γ p γ +pl + (9) u * = u L ƒ L (p *, W L) () ƒ L(p*,W L)=(p*p L) A L () p + B L A L= () γ+ ρ L Fig. Possible wave structures of Riemann problem The problem of determining the types of waves, their strengths and the flow properties in each region for a given set of initial conditions is called a Riemann problem. 4. Exact Solution of Riemann Solver A number of different exact Riemann solvers have been developed by different investigators. Godunov [] [], Hart Lax Leer pioneered the work in this field. He implicitly related the star pressure on both sides of the contact discontinuity to the mass fluxes passing through left and right waves. Thus a single equation is terms of star pressure are obtained. He used a fixed point iteration scheme for solving the pressure equation. In Godunov s second method, a higher order Newton s iterative method is used. Though this method is computationally accurate than the first one, it is time consuming due to the use of higher order derivatives. 4. Exact Riemann Solver Toro [] presented an exact Riemann solver for ideal gases. This method uses first order Newton- Raphson iterative method for solving the implicit pressure equation. For this analysis, the following assumptions are made: B L= γ γ+ p L(3) Thus the expression of f Lcontains all the known terms, i.e., the conditions to the left of the left nonlinear wave, and also the unknown star pressure. Above equations represents a case when the left wave is shock wave. Similar relations writtenfor the right moving wave is the short wave. 4... Expansion Wave The following equations are written assuming that right moving wave structure is an expansion wave. Similar equations exist for a case when the left moving wave is an expansion wave. ρ *=ρ R p p R C = γp ρ γ C *R= a R p p R γ γ () (6) (4) u *- c R = ur- c R (7) γ γ u *= u R+ ƒ R (p *,W R) (8) f R p, W R = C R γ p p R γ γ (9) Thus two expressions for star velocity () and (8). Eliminating the star velocity following expression is obtained: 3
Special Edition PGCON-MECH-7 f R p, W R f L p, W L + f R p, W R + u = () u = u R -u L () This () is the required expression in terms of star pressure. HLLC is the improved version of Hart-Lax-Leer Riemann solver by incorporating the contact surface in wave pattern. The wave estimates SL, S*, SR are assumed. The procedure to be adopted for obtaining the HLLC fluxes is as follows, Compute the wave speeds SL, S*, SR. 4. Godunov s Method Godunov's finite volume method for the specific case of time dependent one-dimensional Euler equations are be discussed []. In the discretized form Godunov's Scheme is written as, U n+ i = U n i + F i + F i+ () with inter cell numerical flux is given as F i+ = F U i+ (3) During the time marching one needs to obey the Courant Number (CFL) criterion. t = C cfl x S max (4) Ccfl is known as Courant number which must be in the range of to. Transmissive or transparent boundary arises from need to define finite or very small computational domain. 4..Solution algorithm for one dimensional Godunov's method is as follows:. Physical one dimensional space is discretized into the number of cells.. For the given unsteady problem, the initial flow variable values are assigned as the cell center values. 3. Outside the physical boundary, two fictitious cells, one outside left boundary and the other outside the right boundary are taken. The boundary conditions are taken to be trans- missive as per the requirement. 4. For each of the cell faces, an exact Riemann problem is created with the left cell values as left data set and the right cell values as right data set.. The solution of these Riemann problems gives the corresponding cell face values. 6. The solutions of the Riemann problems also give a maximum wave velocity fir that time step. 7. This maximum velocity is used in Courant number condition (6.4) of stability for deciding the time step size. 8. To calculate variable values at next time level, (6.) is used. 9. The code is be made to run for the desired number of time steps. 4.3 HLLC Riemann Solver Fig.3Solution in the star region for HLLC Riemann solver. Compute the approximate States. Compute the HLLC fluxes F L if S L Fi+ = F L = F L + S L U L U L ifs L S F R if S R F R = F R + S R U R U R ifs S R () Where, F L = F(U L) and F R = F(U R) 3. Compute new values of the states using the formula. U i n+ = U i n + Δt Δx (F i - F i + )(6). Results and Disscusions In order to illustrate the performance of solvers numerical results were presented for one dimensional test problems.. Sod s shock tube problem The test consists of solving one dimensional Euler s equations (), here the value of gamma is taken as.4, the left and right states are separated by diaphragm positioned in the middle of the tube. The domain is discretized with computing cells and the Courant number of computations is.8... One-dimensional Tests The performance of each high-resolution solver is assessed with ten one-dimensional (-D) tests. These include five standard Sod tests. The initial conditions for these nine tests are summarized in Table. A short description of the five sod tests respectively is given below. Test-: The solution of the mild test consists of shock, a contact and rarefaction, all moving towards right. Test-: The solution of this test consist of two symmetric rarefaction and a contact moving with zero velocity. The region between the rarefaction is near vacuum. 4
Velocity Pressure Special Edition PGCON-MECH-7 Test-3: The solution for this test consists of a strong shock followed by a moving contact and refraction. It is designed for testing the robustness of solver. Test-4: The solution for this test consists of three strong discontinuities moving to right this test is also designed for assessing the robustness of a solver. Test-: This test similar to test 3 but with the nonlinear velocities, and it assesses a solver s ability to resolve slowly-moving contact discontinuities. Table Initial condition for -D Riemann problems D L U L P L D R U R P R..7..... -..4...4......999 9.97 46.894.994-6.9633 46.9. -9.97.. -9.97. Five test cases are checked using this code whose solution were available in the literature. The results obtained were exactly in agreement with literature []. Table represents the input condition for four test cases tested with the code. Table. Shows the exact solution of the star properties for the four test cases. These results match with the actual calculations and those published in the literature. Table Exact solution Test Ρ* L Ρ* R P* u*.46.67336.3984.97447.8 3 E- 4.8 4 3.99.8E-.89387E-3-3.77E-8 3.4 69.64 8.6897.777 46.9444-6.9638 476 Fig. 4 to 8 present a selection of results for test to. The density, Pressure Velocity and internal energy distributions are plotted and compared with the corresponding exact solutions...8.6.4. -....8.6.4. -...6.4..8.6.4. -.4 -. -...4.6.8
velocity Pressure Special Edition PGCON-MECH-7 4 3. 3... -.. HLLC GODUNOV EACT Fig.4 Property distribution for SOD Test..8.6.4. - -..... - -. -.. - -. - -..4.4.3.3..... - -....8.6.4. - -.. Fig. Property distribution for SOD Test 7 6 4 3 - -.. HLLC GODUNOV EACT 6
Velocity Pressure Velocity Pressure Special Edition PGCON-MECH-7 - -.. - 3 3 -.. 8 6 4 - -.. - 3 - -. -. Fig.6 Property distribution for SOD Test 3 -. -. - -.. - 7
Special Edition PGCON-MECH-7 3 3 -.. Fig.7 Property distribution for SOD Test 4 3 3 -.. HLLC GODUNOV EACT -.. - 3 3 -.. Fig.8 Property distribution for SOD Test -. -. - 6. Conclusion In case of Euler equation the Riemann contain the shock tube problem which is studied. Riemann solvers are used for solving five -D tests and allover five -D tests. Results obtained are exactly in agreement with literature. These results match with the actual calculations and those published in the literature. The conditions in these tests are too hypothetical to exist in a real gas dynamic problem. An agreement to the results of these problems ensures robustness of the solver. The Godunov's method and HLLC (Harten, Lax and Van Leer contact) are used further for benchmarking with SOD -D tests and remaining -D tests. References 8
Special Edition PGCON-MECH-7 S. F. Davis,(988), Simplfied second order Godunov Type methods, SIAM J. Sci. Stat. Comput., 9:44-473. B.Enfield.,(988),On Godunov Type methods for gas dynamics, SIAM J. Number Anal., (): 94-38. Harten, A.,(983),High resolution schemes for hyperbolic conversion laws,j. compute. Physso:3-69. Harten, A and Hyman, J.M.,(983), Self-adjusting grid method for one-dimensional hyperbolic conversion laws, s. Compute. Phys., SO: 3-69. C. B. Laney,(), Computational Gas dynamics, Cambridge University Press, First edition, New York, U.S., 998. R. J. Leveque, finite volume method for hyperbolic problems, Cambridge University Press, First edition, Cambridge, U.K. Liska,R.,Wendroff, B.,(3), Comparison of several differences schemes on -D and -D test problems fortheeuler equations, SIAM. Lax, P.D. and liu,. D.,(998), Solution of two dimensional Riemann Problems of gas dynamics by positive schemes. SIAMJ. scicomput., 9:39-34. Takayama,K. and Inoue,O.,(99),Shock wave diffraction over go degree sharp comes.posters presented at 8th ISSW. Shock waves, :3-3. E. F. Toro,(99), Riemann solvers and Numerical Methods for Fluid Dynamics: A Practical Approach, Springer Verlag, Berlin. E. F. Toro, M. Spruce and W. Spears,(994),Restoration of the contact surface in HLL- Riemann solver, shock waves, 4:-34. E. F. Toro and A. Chakraborty.,(994),Development of an Approximate Riemann solver for steady supersonic Euler equations, Aeronoutical Journal, 98:3-339. Woodword, P., colella,p.,(984),the numerical simulation of two dimensional fluid flow with strong shocks. J. Compute.Phys., 4:-73 9