Euler Equations Lab AA Computer Project 2

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1 Euler Equations Lab AA Computer Project 2 Mishaal Aleem February 26, 2015 Contents 1 Introduction Algorithms Lax-Wendroff MacCormack Roe (1st-order TVD scheme) Harten-Yee (2nd-order TVD scheme) Shock Tube Theoretical Numerical Supersonic Diverging Nozzle Theoretical Numerical Conclusion References Appendix A-1 1 Introduction The Euler Equations Lab is a MATLAB computational fluid dynamics (CFD) program that allows the user to study the behavior of several algorithms and compare the results to those that are physically expected for the pseudo-one-dimensional Euler equations as applied to a shock tube and a nozzle. 2 Algorithms The pseudo-one-dimensional Euler equations are written as where Q t + F x = S ρa ρua Q = ρua F = (ρu 2 + p)a S = ea u(e + p)a Each of the algorithms described briefly in the following sections were used to solve these equations for the case of a Sod shock tube and a nozzle. 2.1 Lax-Wendroff The Lax-Wendroff algorithm is a centered differencing scheme with artificial viscosity for stability that is applicable to the one-dimensional nonlinear Euler equations. However, this algorithm does not have enough dissipation to solve the psuedo-one-dimensional equations with even moderate expansions. 0 p da dx 0 1

2 2.2 MacCormack The MacCormack algorithm is a predictor-correct method. It is easily applicable to the pseudo-onedimensional Euler equations because it simply modifies the standard one-dimensional Euler MacCormack solver with a source term for S. 2.3 Roe (1st-order TVD scheme) Unlike the Lax-Wendroff and MacCormack algorithms which are centered-differencing methods with diffusion applied for stability, the Roe method is a first order total variation diminishing (TVD) scheme that applies the principle of flux-splitting into positive and negative components. 2.4 Harten-Yee (2nd-order TVD scheme) The Harten-Yee algorithm is also a TVD flux splitting method and is of second order. 3 Shock Tube For the Sod shock tube, the area is set to A = 1 throughout the nozzle making da dx = 0 and reducing the psuedo-one-dimensional Euler equations to the standard unsteady one-dimensional Euler equations. In the shock tube problem, a tube is filled with a gas and has a diaphragm in the middle. The gas is at different pressures and densities on either side of the diaphragm at t < 0. At t = 0 the diaphragm is removed. A diagram for the resulting motion in the tube is presented in Fig. 1. Figure 1: Motion in Sod shock tube [2] As can be seen from Fig 1, there is a left-moving shock wave within the tube resulting from the pressure difference between regions 1 and 4. For this shock tube, the diaphragm pressure ratio is given as p 4 = 4. The gas is air with a heat capacity ratio of γ = 1.4. The initial pressures are set to p 4 = 4/γ and p 4 = 1/γ and the initial densities are set to ρ 4 = 4 and ρ 1 = Theoretical The speed of the shock wave was found analytically using the shock strength and shock tube relationships. First, the speed of sound in regions 1 and 4 was calculated using Eq. 1. γp a = (1) ρ Note that because of the initial conditions, for region 1, a 1 = 1. Next, the pressure ratio between regions 1 and 2, p 2, was calculated using Eq. 2 and the known diaphragm pressure ratio. p 4 = p 2 [ ] 2γ 1 (γ 1)(a γ 1 1/a 4 )(p2/p1 1) 2γ 2γ + (γ + 1)(p2/p1 1) (2) 2

3 Next, Eq. 3 was used to find the Mach number of the shock wave, M s. γ 1 M s = 2γ + γ + 1 p 2 (3) 2γ Finally, the shock Mach was converted to speed using Eq. 4. u s = M s a 1 (4) Per this analysis, the shock speed was found to be u s = Note that because a 1 = 1, the numerical value of M s is equivalent to the numerical value of u s, and thus in the discussion, the physical units of the problem may be ignored. 3.2 Numerical The Euler Equations Lab was used to study each algorithms ability to simulate the conditions in the shock tube over time. To find the numerical shock speed, the program was run for 100 iterations and 200 iterations for each algorithm, and the time was recorded at the end of each run. The shock location was measured for each run using the pressure plots (for consistency). Finally, the shock speed was calculated using the definition of speed, Eq. 5. u s = x 2 x 1 (5) t 2 t 1 As an example, the plots used to calculate the numerical shock speed for the MacCormack algorithm is presented in Fig. 2. Figure 2: Shock speed calculation for MacCormack algorithm Presented in Tab. 1 are the numerical shock speed calculations from each algorithm and their errors. Note that all are within 3.13% of the theoretical value, supporting the ability of these algorithms to simulate the Sod shock tube problem well. Table 1: Shock Speed Calculations Algorithm MacCormack Lax-Wendroff Roe Harten-Yee Shock Speed Error 0.24% 1.42% 0.34% 3.13% Each algorithm was able to simulate the physics inside the Sod shock tube, though each did exhibit certain non-physical numerical computation phenomena. Figure 3 shows the plot of the pressure in the shock tube after 180 iterations for CFL=0.5 for all the algorithms. 3

4 Figure 3: Pressure in shock tube The MacCormack algorithm was clearly able to capture the shock, however it was visibly dispersive, as seen Fig 3. As the CFL value was decreased, the dispersion got worse, represented in Fig. 4. Figure 4: Density in shock tube, MacCormack algorithm, 300 grid points The Lax-Wendroff was very similar in behavior to MacCormack; the algorithm was able to capture the shock, however it was at the expense of dispersion which got worse as the CFL number was decreased from CFL=1 to CFL=0. For both the MacCormack and the Lax-Wendroff algorithm, because of the dispersion there are magnitude oscillations in the solutions for pressure, density, and energy. Dispersion is useful for shock capturing as it keeps the shock jump discontinuity relatively vertical and the expansion fan relatively linear. However, this is at the expense of diminishing true local information with the oscillations. The Roe algorithm, however, was more visibly diffusive. Shown in Fig 5 is the density solution using the Roe algorithm at t = 0.8 for various CFL numbers. Clearly, the diffusion did increase as the CFL number was varied, though not significantly. Diffusion is detrimental to shock capturing because shocks are locations of sharp discontinuity, but diffusion causes the discontinuity to spread over grid points rather than occurring at the single true location. As seen in Fig. 5, this causes the shock and contact discontinuity to be significantly non-vertical and the expansion fan to be non-linear (particularly near the top and bottom of the line). However, this diffusion keeps the algorithm very stable. 4

5 Figure 5: Density in shock tube, Roe algorithm, 300 grid points The Harten-Yee algorithm also had slight diffusion for the shock tube problem. There was also visible dispersion for higher CFL numbers. Interestingly, the dispersion became more pronounced as the grid points were increased ( 500+) for larger CFL numbers (.8+). This is represented in Fig. 6. Figure 6: Density in shock tube, Harten-Yee algorithm While all the algorithms were able to capture the shock, each had a trade-off with either diffusion or dispersion to varying degrees in the solution. 4 Supersonic Diverging Nozzle For the supersonic diverging nozzle problem, the area change of a nozzle is given by Eq. 6 for x=[0,10]. A(x) = tanh(0.8x4) (6) The inflow Mach number is 1.26 and the exit pressure is fixed such that the ratio gas is air with a heat capacity ratio of γ = Theoretical p exit p entrance = The The shock location for this nozzle was be found analytically by employing isentropic relations, shock jump relations, and principles of supersonic diverging nozzle. 5

6 First, Eq. 6 was used to calculate the area at the inlet, x = 0. Then, Eq. 7 was used to calculate A. ( ) A 2 A = 1 [ 2 M 2 γ + 1 (1 + γ 1 M 2 ) 2 ] γ+1 γ 1 (7) Next, a shock location, x s, was assumed. From this assumed location the area at the shock, A s, was calculated using Eq. 6 and from Eq. 7 an upstream Mach number, M 1, was calculated. Then, using the shock jump relations given by Eq. 8, the downstream Mach number, M 2, was calculated, with the subsonic solution chosen. M2 2 = 1 + γ 1 2 M 1 2 γm1 2 γ 1 (8) 2 Across the shock, the pressure jump was calculated using the Eq. 9 normal shock pressure jump relation. p 2 = 2γM12 (γ 1) (9) γ + 1 Using the downstream M 2 at x s and the nozzle area A s, A was once again calculated. Then, using the nozzle area at the exit calculated by Eq. 6 at x = 10, and the second calculated A, the exit Mach number was calculated. The total pressure to static pressure was calculated at the entrance, upstream of the shock, downstream of the shock, and at the exit using Eq. 10 and the Mach number at each of these locations. ( p t p = 1 + γ 1 ) γ M 2 2 p Finally, the pressure ratio exit p entrance was calculated using Eq. 11 and knowing that p t,exit p p t,2 = t,1 p t,entrance = 1. If the calculated value was higher than 1.931, the process was repeated using a shock location assumption closer to the entrance, and vice-versa. γ 1 (10) p exit p entrance = p exit p t,exit p t,exit p t,2 p t,2 p 2 p 2 p t,1 p t,1 p t,entrance p t,entrance p entrance (11) Per this analytical method, the shock location was found to be x s = The nozzle and shock location are represented in Fig 7. Figure 7: Supersonic Diverging Nozzle 4.2 Numerical The Euler Equations Lab was used to study the ability of the various algorithms to capture the supersonic diverging nozzle behavior and compare the numerical results to the theoretical shock location. 6

7 Expectedly, the Lax-Wendroff algorithm was unacceptable for the nozzle problem. This algorithm cannot handle even moderate area expansions which, as shown in the Fig 7 diagram of the nozzle, do occur for this nozzle. After just 50 iterations, the instability became apparent as the solution simply grew unchecked and, of course, did not capture the shock location. The Lax-Wendroff numerical solution after 1000 iterations is presented in Fig. 8. Figure 8: Lax-Wendroff Algorithm for Nozzle The other algorithms, however, were appropriate for the nozzle. Each of them were able to successfully capture the behavior in the nozzle. Reaching the steady solution, however, did take thousands of iterations. For stability, CFL numbers of greater than 1 were not used, and 1 was avoided due to potential non-linearity-induced stability issues. The MacCormack algorithm successfully simulated a solution, however with visible dispersion. With this method, after about iterations for a 600 grid point solution with CFL number = 0.99, the shock location was found to be 4.908, presented in Fig. 9. As the CFL number was incrementally decreased from CFL=1 to CFL=0, the solution became more and more dispersive. While varying the CFL number did change the amount of dispersion, the variations were not as significant as for the CFL variations seen for some algorithms in the Linear Advection Lab. Figure 9: MacCormack Algorithm for Nozzle The Roe algorithm neatly captured the behavior of the nozzle problem, and with much less numerical computation noise than the MacCormack algorithm. With this method, after about iterations for 7

8 a 600 grid point solution with CFL number = 0.99, the shock location was again found to be 4.908, presented in Fig. 10. Similar to the MacCormack algorithm, as the CFL number was incrementally decreased from CFL=1 to CFL=0, the solution became more and more dispersive. Figure 10: Roe Algorithm for Nozzle The Harten-Yee method was more unstable than the previous two methods. For large CFL numbers, the solution had a lot of noise. For this method, after about iterations for a 600 grid point solution with CFL number = 0.5, the shock location was found to be Figure 11: Harten-Yee Algorithm for Nozzle The shock location as computed by each algorithm is presented along with error value in Tab. 2. Note again the low error values, all less that 0.13%, pointing to the success of these algorithms in capturing the physical behavior in the nozzle. Table 2: Shock Location Calculations Algorithm MacCormack Roe Harten-Yee Shock Location Error 0.09% 0.09% 0.13% 8

9 5 Conclusion The Euler Equations Lab was used to study behavior in a Sod shock tube and in a supersonic diverging nozzle using the pseudo-one-dimensional Euler equations. Theoretical solution values were compared to numerical results and the ability of various algorithms to capture the physical behavior was explored. For the Sod shock tube, the Lax-Wendroff, MacCormack, Roe, and Harten-Yee algorithms were found to be able to capture the shock. The Lax-Wendroff and MacCormack algorithms tended to have dispersive behavior in the solutions. The Roe algorithm had diffusion in the solution, and the Harten-Yee algorithm had significant dispersion for high grid points coupled with high CFL numbers. However, each algorithm was able to capture the shock speed within 3.13% of the theoretical value pointing to their overall success. For the supersonic diverging nozzle, Lax-Wendroff was found to be unstable because there was not enough damping to compensate for the increasing area of the nozzle. However, the other three algorithms were able to simulate the physical behavior in the nozzle. The MacCormack algorithm showed dispersion and the Harten-Yee was again very noisy for large CFL numbers. The Roe algorithm was arguably the best with minimal diffusion, even as the CFL number was decreased. Each of the three algorithms, however, were able to numerically simulate the shock location within 0.13% of the theoretical solution. Through working with the Euler Equations Lab,the fundamental principle of using CFD to model model fluid dynamics equations was reiterated: there is an inherent numerical trade-off when capturing a physical solution. Diffusion caused loss of information but ensured stability of the algorithms. Dispersion caused oscillations in the solution but allowed for simulation sharp discontinuities. Ultimately, the appropriate algorithm is dependent on the application, the desired information to be retained, and the desired accuracy of the solution globally and locally. References [1] Eberhardt, D.S. and Shumlak, U. AA543 Computational Fluid Dynamics I, University of Washington, Seattle, WA, [2] Liepmann, H.W. and Roshko, A. Elements of Gasdynamics, Dover Publications, Inc., Mineola, NY,

10 Appendix MacCormack Algorithm The MATLAB code used to implement the MacCormack algorithm in the Euler Equations Lab is presented below. if strcmp(algorithm,'maccormack')==1 % Start MacCormack Algorithm here. [flux]=calfx(q,flux,1,imax,gamma1); for i=2:imax for nx=1:3 qbar(i,nx) = (q(i,nx)) - (dt/dx)*(flux(i,nx)-flux(i-1,nx)); end end [flux]=calfx(qbar,flux,1,imax-1,gamma1); for i=2:imax-1 for nx = 1:3 dq(i,nx) = -q(i,nx)/2 +qbar(i,nx)/2 -(dt/dx)/2 * (flux(i+1,nx) - flux(i,nx)); end end for i = 2:imax rhoa = q(i,1) + dq(i,1); rhoua = q(i,2) + dq(i,2); ea = q(i,3) + dq(i,3); u = rhoua / rhoa; p = gamma1 * (ea - (.5 * rhoua * u)) / area(i,1); dq(i,2) = dq(i,2) + (dt * p * s(i,2)); end Shock Tube Shock Speed The MATLAB code used to calculate the shock speed for the Sod shock tube problem is presented below. %gamma g=1.4; %initial conditions p1=1/g; p4=4/g; rho1=1; rho4=4; %calculate speed of sound a1=sqrt(g*p1/rho1); a4=sqrt(g*p4/rho4); %solve for p2/p1 p4 p1=4; syms p2 p1 alpha=((g-1)*(a1/a4)*(p2 p1-1))/(sqrt(2*g)*sqrt(2*g+(g+1)*(p2 p1-1))); p2 p1=solve(p4 p1==p2 p1*(1-alpha)ˆ(-2*g/(g-1)),p2 p1); p2 p1=double(p2 p1(2)); %solve for shock Mach Ms=((g-1)/(2*g)+(g+1)/(2*g)*p2 p1)ˆ(1/2); %solve for shock speed us=ms*a1 A-1

11 Supersonic Nozzle Shock Location The MATLAB code used to calculate the shock location for the supersonic diverging nozzle is presented below. % Calculate A* from the LHS M0=1.26; A0= *tanh(0.8*0-4); syms Astar1 Astar1=solve((A0/Astar1)ˆ2==1/M0ˆ2*(2/(g+1)*(1+(g-1)/2*M0ˆ2))ˆ((g+1)/(g-1)),Astar1); Astar1=double(Astar1(1)); % calculate M1 based on a guess x x guess=4.9036; A xs= *tanh(0.8.*x guess-4); syms M1 M1=solve((A xs/astar1)ˆ2==1/m1ˆ2*(2/(g+1)*(1+(g-1)/2*m1ˆ2))ˆ((g+1)/(g-1)),m1); M1=double(M1(2)); % calculate M2 M2=sqrt((1+(g-1)/2*M1ˆ2)/(g*M1ˆ2-(g-1)/2)); % calculate A* from RHS syms Astar2 Astar2=solve((A xs/astar2)ˆ2==1/m2ˆ2*(2/(g+1)*(1+(g-1)/2*m2ˆ2))ˆ((g+1)/(g-1)),astar2); Astar2=double(Astar2(1)); % calculate Mexit syms Mexit Aend= *tanh(0.8*10-4); Mexit=solve((Aend/Astar2)ˆ2==1/Mexitˆ2*(2/(g+1)*(1+(g-1)/2*Mexitˆ2))ˆ((g+1)/(g-1)),Mexit); Mexit=double(Mexit(1)); % pressure ratios pe pte=((1+(g-1)/2*mexitˆ2)ˆ(g/(g-1)))ˆ(-1); pte pt2=1; pt2 p2=((1+(g-1)/2*m2ˆ2)ˆ(g/(g-1))); p2 p1=(2*g*m1ˆ2-(g-1))/(g+1); p1 pt1=((1+(g-1)/2*m1ˆ2)ˆ(g/(g-1)))ˆ(-1); pt1 pint=1; pint pin=((1+(g-1)/2*m0ˆ2)ˆ(g/(g-1))); calc pratio=pe pte*pte pt2*pt2 p2*p2 p1*p1 pt1*pt1 pint*pint pin A-2