High-order finite difference methods with subcell resolution for 2D detonation waves

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1 Center for Turbulence Research Annual Research Briefs 7 High-order finite difference methods with subcell resolution for D detonation waves B W. Wang, C.-W. Shu, H. C. Yee AND B. Sjögreen. Motivation and objective In simulating hperbolic conservation laws in conjunction with an inhomogeneous stiff source term, if the solution is discontinuous, spurious numerical results ma be produced due to the different time scales of the transport part and the source term. This numerical issue often arises in combustion and high-speed chemical reacting flows. The reactive Euler equations in two dimensions have the form U t + F(U) + G(U) = S(U), (.) where U, F(U), G(U) and S(U) are vectors. If the time scale of the ordinar differential equation (ODE) U t = S(U) for the source term is orders of magnitude smaller than the time scale of the homogeneous conservation law U t + F(U) + G(U) = then the problem is said to be stiff. In high-speed chemical reacting flows, the source term represents the chemical reactions which ma be much faster than the gas flow. This leads to problems of numerical stiffness. Insufficient spatial/temporal resolution ma cause an incorrect propagation speed of discontinuities and nonphsical states in standard dissipative numerical methods. This numerical phenomenon was first observed b Colella et al. (986). Then LeVeque & Yee (99) showed that a similar spurious propagation phenomenon can be observed even with scalar equations. Colella et al. (986) and Majda & Rotburd (99) have successfull applied the random choice method of Chorin (976, 977) for the solution of underresolved detonation waves. However, it is difficult to eliminate completel the numerical viscosit in a shock-capturing scheme. Fractional step methods are commonl used for allowing an underresolved mesh size with a shock-capturing method. Chang (989, 99) applied the subcell resolution method of Harten (989) to the finite volume ENO method in the convection step, which is able to produce a zero viscosit shock profile in nonreacting flow. The time evolution is advanced along the characteristic line. Correct discontinuit speed was obtained in the one-dimensional scalar case. However, it is difficult to etend this approach to multi-dimensions and to sstem of equations because of the reliance on the eact time evolution via characteristics. Engquist & Sjögreen (99) proposed a simple temperature etrapolation method based on finite difference ENO schemes with implicit Runge-Kutta time discretization, which uses a first-/second-order etrapolation of the temperatures from outside the shock profile. The method is eas to etend to multi-dimensions. Their method is not a fractional step method. It does not seem to work well when the spatial scales are underresolved. Other first-/second-order Department of Mathematics and Statistics, Florida International Universit, Miami, FL 99 Division of Applied Mathematics, Brown Universit, Providence, RI 9 NASA Ames Research Center, Moffett Field, CA 95 Lawrence Livermore National Laborator, Livermore, CA 955

2 7 W. Wang et al. methods that are based on the fractional step method have been proposed b Bao & Jin (, ) and Tosatto & Vigevano (8). In our previous work b Wang et al. (), we developed a high-order finite difference method which can capture the correct detonation speed in an underresolved mesh and will maintain high-order accurac in the smooth part of the flow. Numerical eamples were presented for one-dimensional scalar problems and one-dimensional detonation waves. Our objective in this stud is to etend this method to two-dimensional reactive Euler equations. The first step of the proposed fractional step method is the convection step which solves the homogeneous hperbolic conservation law in which an high-resolution shock-capturing method can be used. The aim in this step is to produce a sharp wave front, but some numerical dissipation is allowed. The second step is the reaction step where an ODE solver is applied with modified transition points. Here, b transition points, we refer to the smeared numerical solution in the shock region, which is due to the dissipativit of a shock-capturing scheme. Because the transition points in the convection step will result in large erroneous values of the source term if the source term is stiff, we first identif these points and then etrapolate them b a reconstructed polnomial using the idea of Harten s subcell resolution method. Unlike Chang s approach, we appl Harten s subcell resolution in the reaction step. Thus our approach is fleible in allowing an shock-capturing scheme as the convection operator. In the reaction step, since the etrapolation is based on the high-order reconstruction, high-order accurac can be achieved in space. The onl drawback in our current approach is that the temporal accurac will onl be, at most, second-order due to the time splitting, which is common for most of the previous methods for stiff sources. We also remark that, in order to resolve the sharp reaction zone, sufficientl man grid points in this zone are still needed. The proposed method can capture the correct location and jump size of the reaction front, but it does not resolve the narrow reaction zone, as tpicall there is one or a few points in that zone.. Two-dimensional reactive Euler equations The considered two-dimensional problem is modeling the reaction with two chemical states: burnt gas and unburnt gas. The unburnt gas is converted to burnt gas via a single irreversible reaction. Without heat conduction and viscosit, the sstem can be written as t + (u) + (v) = (.) (u) t + (u + p) + (uv) = (.) (v) t + (uv) + (v + p) = (.) E t + (u(e + p)) + (v(e + p)) = (.) (z) t + (uz) + (vz) = K(T)z, (.5) where (,, t) is the miture densit, u(,, t) and v(,, t) are the miture - and - velocities, E(,, t) is the miture total energ per unit volume, p(,, t) is the pressure, z(,, t) is the mass fraction of the unburnt gas, K(T) is the chemical reaction rate and T(,, t) is the temperature. The pressure is given b p = (γ )(E (u + v ) q z), (.6) where the temperature T = p and q is the heat released in the reaction.

3 High-order methods for D detonation waves 7 The reaction rate K(T) is modeled b an Arrhenius law ( ) Tign K(T) = K ep, (.7) T where K is the reaction rate constant and T ign is the ignition temperature. The reaction rate ma be also modeled in the Heaviside form { /ε T Tign K(T) =, (.8) T < T ign where ε is the reaction time and /ε is roughl equal to K. For simplicit, we onl consider the Heaviside source term (.8).. Numerical method for two-dimensional reactive Euler equations The general fractional step approach based on Strang-splitting (Strang 968) for equation U t + F(U) + G(U) = S(U) (.) is as follows. The numerical solution at time level t n+ is approimated b ( ) ( ) t t U n+ = A R( t)a U n. (.) The convection operator A is defined to approimate the solution of the homogeneous part of the problem on the time interval, i.e., U t + F(U) + G(U) =, t n t t n+. (.) The reaction operator R is defined to approimate the solution on a time step of the reaction problem: du dt = S(U), t n t t n+. (.) The convection operator is over a time step t and the reaction operator is over t/. The two half-step reaction operations over adjacent time steps can be combined to save cost. Net, we introduce the proposed fractional step methods for the convection step and the reaction step separatel... Convection operator In the convection step, we use fifth-order WENO (WENO5) with La-Friedrichs flu and third-order Runge-Kutta for time discretization... Reaction operator If there is no smearing of discontinuities in the convection step, an ODE solver can be used as the reaction operator. However, all the standard shock-capturing schemes will produce a few transition points in the shock when solving the convection equation. These transition points are usuall responsible for causing incorrect numerical results in the stiff case. Thus we cannot directl appl a standard ODE solver at these transition points. Here we use Harten s subcell resolution technique in the reaction step. The general idea is as follows. If a point is considered a transition point of the shock, information from its neighboring points which are deemed non-transitional points will be used instead.

4 7 W. Wang et al. In the two-dimensional case, we appl the subcell resolution procedure dimension b dimension. () Use a shock indicator to identif cells in which discontinuities are believed to be situated. We consider the minmod-based shock indicator in Harten (989) and Shu & Osher (989). Identif troubled cell I ij in both - and -directions b appling the shock indicator to the mass fraction z. Define the cell I ij as troubled in the -direction if s ij s i,j and s ij s i+,j with at least one strict inequalit where s ij = minmod{z i+,j z ij, z ij z i,j }. (.5) Similarl, we can define the cell I ij as troubled in the -direction if s ij s i,j and s ij s i,j+ with at least one strict inequalit where s ij = minmod{z i,j+ z ij, z ij z i,j }. (.6) If I ij is troubled in onl one direction, we appl the subcell resolution along this direction. If I ij is troubled in both directions, we choose the direction which has a larger jump. Namel, if s ij s ij, subcell resolution is applied along the -direction, otherwise it is done along the -direction. In the following steps ()-(), without loss of generalit, we assume the subcell resolution is applied in the -direction. () In a troubled cell identified above, we continue to identif its neighboring cells. For eample, we can define I i+,j as troubled if s i+,j s i,j and s i+,j s i+,j and similarl define I i,j as troubled if s i,j s i,j and s i,j s i+,j. If the cell I i s,j and the cell I i+r,j (s, r > ) are the first good cells from the left and the right (i.e., I i s+,j and I i+r,j are still troubled cells), we compute the fifth-order ENO interpolation polnomial p i s,j () and p i+r,j () for the cells I i s,j and I i+r,j, respectivel. Modif the point values z ij, T ij and ij in the troubled cell I ij b the ENO interpolation polnomials { zij = p i s,j ( i ; z), Tij = p i s,j ( i ; T), ij = p i s,j ( i ; ), if θ i, (.7) z ij = p i+r,j ( i ; z), Tij = p i+r,j ( i ; T), ij = p i+r,j ( i ; ), if θ < i where the location θ is determined b the conservation of energ E θ i / p i s,j (; E)d + i+/ θ p i+r,j (; E)d = E ij. (.8) Under certain conditions, it can be shown that there is a unique θ satisfing Eq. (.8), which can be solved using, for eample, Newton s method. If there is no solution for θ or more than one solution, we choose z ij = z i+r,j, T ij = T i+r,j and ij = i+r,j. However, in the sstem case we would like to have the shock travel ahead of the reaction zone, so we take the values of z, T and ahead of the shock. For simplicit, in the considered stiff problem, the value of z ij can be taken as {, θ i z ij =. (.9), θ < i () Use Ũij instead of U ij in the ODE solver if the cell I ij is a troubled cell. For simplicit, eplicit Euler is used as the ODE solver. (z) n+ ij = (z) n ij + ts( T ij, ij, z ij ). (.)

5 High-order methods for D detonation waves 75 Here implicit methods cannot be used in this step because the troubled values need to be modified eplicitl. However, there is no small time-step restriction in the eplicit method used here, because once the stiff points have been modified, the modified source term S( T ij, ij, z ij ) is no longer stiff. Therefore, a regular CFL number is allowed in the eplicit method. In general, a regular CFL=. can be used in the proposed scheme to produce a stable solution. But the solution is ver coarse in the reaction zone because of the underresolved mesh in time. In order to obtain more accurate results in the reaction zone, we evolve one reaction step via N r sub steps, i.e., u n+ = A in some numerical eamples. ( t ) R ( t N r ) R ( t N r ) A.. A numerical eample of D detonation waves ( ) t u n (.) This eample is taken from Bao & Jin (). The chemical reaction is modeled b the Heaviside form with the parameters γ =., q =.596, ε =.585, T ign =.55 in CGS units. Consider a two-dimensional channel of width.5, and the upper and lower boundaries are solid walls. The computational domain is [,.5] [,.5]. The initial conditions are { (l, u (, u, v, p, z) = l,, p l, ), if ξ(), (.) ( r, u r,, p r, ), if > ξ(), where ξ() = {..5., <., (.) and u l = 8.6, l =. and p l = Similar problems are also computed in Engquist & Sjögreen (99). One important feature of this solution is the appearance of triple points, which travel in the transverse direction and reflect from the upper and lower walls. The mechanisms driving this solution are discussed in Kailasanath et al. (985). Figures - show densit contours computed b WENO5/SR with 5 ( = = 5 5 ), CFL=. and N r = at eighteen evolutionar times from t = to t =.7 7. We can see the movement of the triple points. The same case b WENO5/SR with a much coarser grid ( = =.5 ) with CFL=. and N r = at three evolutionar times is shown in Figure. We can see WENO5/SR with the ver coarse mesh can still capture the correct shock location, although the shocks are smeared due to the lack of resolution. It is more apparent to compare the computed results with the reference solution in a D cross section. The reference solutions are computed b standard WENO5 with grid points and CFL=.. The results b WENO5/SR and the splitting WENO5 are compared with the same mesh and CFL=.5. Figures -7 show the D cross section at =.5 at evolutionar times t = 8, t = 6 8, t =. 7 and t =.7 7 separatel, where the left subplots are computed b WENO5/SR and the right subplots are b splitting WENO5. We can see WENO5/SR has ecellent agreement with the reference solutions

6 76 W. Wang et al Figure. Computed densit: WENO5/SR with 5, CFL=. and N r = at nine different evolutionar times t =, t = 8, t = 8, t = 8, t = 8, t = 5 8, t = 6 8, t = 7 8 and t = 8 8. ecept that it cannot capture the waves sharpl due to the underresolved mesh. However, the splitting WENO5 method produces spurious waves in front of the detonation shock starting at time t = 8 (right subplot of Figure ) and after that the solutions move at a wrong speed (right subplots of Figures 5-7).

7 High-order methods for D detonation waves Figure. Computed densit: WENO5/SR with 5, CFL=. and N r = at nine different evolutionar times t = 9 8, t = 7, t =. 7, t =. 7, t =. 7, t =. 7, t =.5 7, t =.6 7 and t = Future plans In this report, we demonstrated that the proposed high-order finite difference schemes with subcell resolution are able to capture the correct detonation wave speed in two-

8 78 W. Wang et al Figure. Computed results: WENO5/SR with, CFL=. and N r = at t =.5 7, t =.6 7 and t = Figure. D cross-section at t = 8 b different WENO schemes with. Solid line: reference solution; dashed line: numerical solution. Left: WENO5/SR with CFL=., N r = ; right: splitting WENO5 with CFL= Figure 5. D cross-section at t = 6 8 b different WENO schemes with. Solid line: reference solution; dashed line: numerical solution. Left: WENO5/SR with CFL=., N r = ; right: splitting WENO5 with CFL=.5.

9 High-order methods for D detonation waves Figure 6. D cross-section at t =. 7 b different WENO schemes with. Solid line: reference solution; dashed line: numerical solution. Left: WENO5/SR with CFL=., N r = ; right: splitting WENO5 with CFL= Figure 7. D cross-section at t =.7 7 b different WENO schemes with. Solid line: reference solution; dashed line: numerical solution. Left: WENO5/SR with CFL=., N r = ; right: splitting WENO5 with CFL=.5. dimensional sstem cases. Future work will etend this approach to multiple reaction models. Acknowledgments The authors acknowledge the support of the DOE/SciDAC SAP grant DE-AI- 6ER5796. The work b Björn Sjögreen was performed under the auspices of the U.S. Department of Energ at Lawrence Livermore National Laborator under Contract DE- AC5-7NA7. The research of C.-W. Shu is also partiall supported b ARO grant W9NF The research of H. C. Yee is also partiall supported b the NASA Fundamental Aeronautics Hpersonic program. REFERENCES Bao, W. & Jin, S. The random projection method for hperbolic conservation laws with stiff reaction terms. J. Comp. Phs. 6, 6 8.

10 8 W. Wang et al. Bao, W. & Jin, S. The random projection method for stiff detonation capturing. J. Sci. Comput., 5. Chang, S.-H. 989 On the application of subcell resolution to conservation laws with stiff source terms. NASA Technical Memorandum 8, ICOMP Report Chang, S.-H. 99 On the application of subcell resolution to conservation laws with stiff source terms. NASA Lewis Research Center, Computational Fluid Dnamics Smposium on Aeropropulsion pp Chorin, A. 976 Random choice solution of hperbolic sstems. J. Comp. Phs., Chorin, A. 977 Random choice methods with applications for reacting gas flows. J. Comp. Phs. 5, 5 7. Colella, P., Majda, A. & Rotburd, V. 986 Theoretical and numerical structure for numerical reacting waves. SIAM J. Sci. STAT. Comp. 7, Engquist, B. & Sjögreen, B. 99 Robust difference approimations of stiff inviscid detonation waves. Technical Report CAM 9-, UCLA.. Harten, A. 989 ENO schemes with subcell resolution. J. Comp. Phs. 8, 8 8. Kailasanath, K., Oran, E. S., Boris, J. P. & Young, T. R. 985 Determination of detonation cell size and the role of transverse waves in two-dimensional detonations. Combust. Flame 6, LeVeque, R. J. & Yee, H. C. 99 A stud of numerical methods for hperbolic conservation laws with stiff source terms. J. Comp. Phs. 86, 87. Majda, A. & Rotburd, V. 99 Numerical stud of the mechanisms for initiation of reacting shock waves. SIAM J. Sci. STAT. Comp., Shu, C.-W. & Osher, S. 989 Efficient implementation of essentiall non-oscillator shock capturing schemes, ii. J. Comp. Phs. 8, 78. Strang, G. 968 On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, Tosatto, L. & Vigevano, L. 8 Numerical solution of under-resolved detonations. J. Comp. Phs. 7, 7. Wang, W., Shu, C.-W., Yee, H. C. & Sjögreen, B. High-order finite difference methods with subcell resolution for hperbolic conservation laws with stiff reaction terms: preliminar results. Annu. Res. Briefs, Center for Turbulence Research, Stanford Universit pp. 9 6.