A HIGH ORDER POLYNOMIAL UPWIND SCHEME FOR NUMERICAL SOLUTION OF CONSERVATION LAWS AND FLUID DYNAMICS PROBLEMS

SIMMEC / EMMCOMP 2014 XI Simpósio de Mecânica Computacional II Encontro Mineiro de Modelagem Computacional A HIGH ORDER POLYNOMIAL UPWIND SCHEME FOR NUMERICAL SOLUTION OF CONSERVATION LAWS AND FLUID DYNAMICS PROBLEMS Rafael Alves Bonfim de Queiroz, Bernardo Martins Rocha rafael.bonfim@ice.ufjf.br, bernardomartinsrocha@ice.ufjf.br Universidade Federal de Juiz de Fora, Instituto de Ciências Eatas, Departamento de Ciência da Computação Rua José Lourenço Kelmer, São Pedro, 36036330, Juiz de Fora, MG, Brasil Abstract. A high order polynomial upwind scheme, namely TOPUS, for numerical solution of conservation laws and fluid dynamics problems is revisited in this paper. The scheme is based on TVD and CBC stability criteria. The performance of the scheme is investigated for solving the scalar convection equation with discontinuous initial data and 1D Riemann problem for Burger equation. The scheme is then applied in the simulation of 2D incompressible flow over a backward facing step. The numerical results show good agreement with other numerical and eperimental data. Keywords: TOPUS scheme, High-resolution schemes, Convection modeling

1 INTRODUCTION The development of numerical schemes for approimation of convection terms of conservation laws and transport equations in computational fluid dynamics (CFD) has presented a continuing challenge. The major problem is to prevent the unbounded growth of the unphysical spatial oscillations in the numerical solution near of discontinuities. To obtain stable, bounded and physically plausible solutions, the classical first order upwind (FOU) (Courant et al., 1952) is often adopted. However, this scheme is unsuitable for applications involving long time evolution of comple flows, mainly because etrema are clipped and numerical dissipation becomes dominant (Brandt and Yavneh, 1991). The cure for this has been to use conventional schemes, such as second-order upwind (SOU) (Warming and Beam, 1976), quadratic-upstream interpolation for convective kinematics (QUICK) (Leonard, 1979) and QUICK with estimated stream terms (QUICKEST) (Leonard, 1988) schemes, due to their intrinsic lack of numerical dissipation. However, they also inevitably can generate unphysical spatial oscillations and instabilities in regions where the convected variables eperience discontinuities. In order to overcome the defect produced by the aforementioned schemes, a number of monotonic upwind schemes has appeared in the published literature as, for eample, the sharp and monotonic algorithm for realistic transport (SMART) (Gaskell and Lau, 1988), the simple high accuracy resolution program (SHARP) (Leonard, 1987), the variable-order non-oscillatory scheme (VONOS) (Varonos and Bergeles, 1998), and convergent and universally bounded interpolation scheme for the treatment of advection (CUBISTA) (Alves et al., 2003) schemes. The main objective of these schemes is to recover smooth solutions from those that are contaminated by oscillations and, meanwhile, to improve the rate of convergence. It should be also noted, however, that these schemes, though performing well in some problems, cannot be bounded in situations such as shock phenomena in compressible flows (Kuan and Lin, 2000) and/or incompressible viscoelastic flow calculations with hyperbolic constitutive models (Xue et al., 2002; Alves et al., 2003). Hence, the need for accurate, efficient and robust upwind differencing schemes for approimating non-linear convective terms of conservation laws and related fluid dynamics equations continues to stimulate a great deal of research in CFD. This lead the motivation for the development of the high resolution upwind scheme, called TOPUS (Third-Order Polynomial Upwind Scheme) (Queiroz and Ferreira, 2010; Ferreira et al., 2012a,b), which is a generalization of the SMARTER (Waterson and Deconinck, 2007) and follows the basic idea of constructing the numerical flu function using a combination of low and high order schemes through some limiter function. This scheme approimates the advective flues at the cell boundaries with 1st, 2nd or 3rd order accuracy, and its numerical dissipation is highly scale selective, providing damping only at the highest wave numbers that are resolved, and thus automatically decreases numerical dissipation at higher grid resolutions. The epectation is that the use of this upwind scheme will enable not merely to capture the shock, but also to resolve the delicate features and structures of free surface flows. In the derivation of the TOPUS scheme, it is used the TVD (Total Variation Diminishing) (Harten, 1983) and CBC (Convection Boundedness Criterion) (Gaskell and Lau, 1988) criteria that offer great fleibility in the construction of the higher-order bounded schemes. In this paper, the TOPUS scheme is revisited for solving the following problems: 1D scalar

convection equation with discontinuous initial data, 1D Riemann problem for Burgers equation and 2D laminar flow over a backward facing step. The objective is to eamine the utility and effectiveness of the TOPUS scheme when incorpored into the CLAWPACK (Conservation LAWs PACKage) software (LeVeque, 2002; Clawpack, 2014) and Freeflow simulation system (Castelo et al., 2000). The organization of the paper is as follows. In Section 2, the mathematical formulation of high-resolution schemes is revised. A short outline of the governing equations of the problems considered in this paper is presented in Section 3. The versatility and robustness of the TOPUS scheme is considered in Section 4. Section 5 contains a few concluding remarks and guidelines for future work. 2 HIGH-RESOLUTION SCHEMES Before proceeding to provide the formulation of the ADBQUICKEST and TOPUS schemes, it is essential to present the normalized variables (NV) of Leonard (1988). Let u be the variation of a convected property through, for eample, the boundary face f between two control volumes can be represented by a functional relationship linking values u D, u U and u R, which represent, respectively, the Downstream, the U pstream and the Remote-upstream locations with respect to the mass flu V f at this face (see Figure 1). The neighbors of the g face can be similarly classified. If this functional relationship involving these three neighboring positions is prescribed, then the numerical flu at the interface can be determined. To this end, the original variable u is transformed into NV by û(, t) = u(, t) un R. (1) u n D u n R The advantage of this normalization is that the interface value û f depends on û n U and θ only, since û n D = 1 and û n R = 0. V f R U D g f Figure 1: The f face, together with its mass flu V f > 0 and neighboring nodes D, U and R. 2.1 ADBQUICKEST SCHEME The adaptative QUICKEST (Ferreira et al., 2009a) or ADBQUICKEST (Ferreira et al., 2009b) scheme was developed in the contet of the NV while enforcing the TVD property (Harten, 1983) and CBC conditions (Gaskell and Lau, 1988). According to Ferreira et al. (2009b), this scheme combines accuracy and monotonicity, while providing fleibility. In the simulations of this TVD

scheme, 2nd or 3rd order accurate in the smooth parts of the solution domain were obtained in the tests (Ferreira et al., 2009b), but only first-order near regions with large gradients (discontinuities). The ADBQUICKEST scheme in NV is given by (2 θ)û C, û C (0,a), û f = where θ = uδt δ a = û C + 1 2 (1 θ )(1 û C) 1 6 (1 θ2 )(1 2û C ), û C [a, b], 1 θ + θû C, û C (b,1), û C, û C / (0, 1), is the Courant number; and the a and b parameters are given by 2 3 θ + θ 2 7 6θ 3 θ + 2θ 2 and b = 4 + 6θ 3 θ + θ2 5 + 6θ 3 θ + 2θ 2. The flu limiter for the ADBQUICKEST scheme can be written as { [ ψ(r f ) = ma 0, min 2r f, 2 + θ2 3θ + (1 θ 2 ] } )r f, 2, (3) 3 3θ where r f for uniform meshes is defined by (2) r f = ûc 1 û C. (4) 2.2 CLASSICAL FLUX LIMITERS van Leer (van Leer, 1974): ψ(r f ) = r f + r f 1 + r f. Minmod (Roe, 1986): ψ(r f ) = ma [0, min (r f, 1)]. (6) 2.3 TOPUS SCHEME The detailed derivation of the TOPUS scheme has been previously described in (Queiroz and Ferreira, 2010; Ferreira et al., 2012a,b). In the following, we recapitulate the main features of this scheme for ease of reference. The TOPUS scheme in NV is given by αû 4 U + ( 2α + 1) û 3 U + ( ) 5α 10 û2 4 U + ( ) α+10 ûu, û 4 U [0, 1], û f = û U, û U / [0, 1], where α is an adjustable constant, α [ 2, 2], to ensure that TOPUS falls into the CBC region. If α = 0, then TOPUS corresponds to the SMARTER scheme Waterson and Deconinck (2007). (5) (7)

In the case of α = 2, TOPUS is entirely contained into the TVD region of Harten (1983), and its corresponding flu limiter ψ(r f ), with r f a sensor, satisfies Sweby s monotonicity preservation condition when r f tends to 0. The flu limiter ψ(r f ) for the TOPUS scheme is represented as follows ψ(r f, α) = ( r f + r f ) [ (α + 1)r 2 f + (α + 4)r f + (α + 3) ] 3 1D/2D Problems (1 + r f ) 3. (8) A short outline of the governing equations of the problems considered in this paper is presented below. 3.1 1D unsteady linear advection equation In this problem, the advection of a quantity u(, t) by a constant convecting velocity a is investigated. The equation for the unsteady linear advection problem is given by u t + a u = 0. (9) The eact solution is given by u(, t) = u 0 ( at), which states that the initial data u 0 () is simply translated with velocity a. 3.2 1D inviscid Burgers equation The inviscid Burgers equation is given by u t + ( ) u 2 = 0. t 2 As in the book (LeVeque, 2002), the equation is solved considering oscillatory initial data to demonstrate decay to an N wave. Also, it is adopted periodic boundary condition for solving (10). 3.3 2D instantaneous Navier-Stokes equations When the fluid is considered incompressible, the density ρ(, t) = ρ 0 of the particles does not change during its movement and the transport properties are constant. Considering numerical simulation of a laminar 2D flow over a backward facing step, the mathematical modeling of the conservation laws are the instantaneous Navier-Stokes and continuity equations, respectively, given by v i t + (v iv j ) = p + 1 j i Re j ( ) vi j (10) + 1 (F r) 2 g i, i = 1, 2, (11) v i i = 0, (12)

where t is the time, v i is the ith component of velocity, p is the kinematic pressure (pressure divided by density) and g i is the ith component of gravitational acceleration. The non-dimensional parameters Re = LV 0 /ν and F r = V 0 / L g denote the associated Reynolds and Froude numbers, respectively. The constant parameter ν is the kinematic viscosity coefficient of the fluid given by ν = µ/ρ (µ is the dynamic viscosity coefficient of the fluid), and V 0 is characteristic velocity and L is length scale. The system of equations formed by (11) and (12) cannot be solved without specifying appropriate initial and boundary conditions (Ferreira et al., 2009a). For initial conditions, a Dirichlet condition is used for all variables. There are four types of boundaries to be considered, namely: inlet, outlet, solid walls and free surfaces. At the inlet section - the velocity is known: v n = V 0 and v τ = 0, (13) where u n and u τ are the normal and tangential velocities to the boundary, respectively; At the outlet section - homogeneous Neumann (fully developed flow) conditions are specified for all variables: v n n = 0 and v τ = 0, (14) n where n and τ denote normal and tangential directions at the input and output of fluid, respectively; Free-slip walls - it is assumed that the fluid slips at the solid walls: v n = 0 and v τ n = 0; (15) 4 Numerical eperiments In this section, results obtained with the TOPUS scheme in numerical eperiments related to the three problems, namely, 1D advection of scalars, 1D Riemann problem for Burgers equation and 2D incompressible flow over a backward facing step are shown. The 1D problems were solved using the CLAWPACK software, which is equipped with high-resolution limiters (Minmod, TO- PUS, van Leer and other). The 2D incompressible flow was carried out using the Freeflow code, which was appropriately modified to incorporate the ADBQUICKEST and TOPUS schemes. 4.1 1D scalar advection problem Equation (9) is considered with a = 1, [0, 1] and initial condition given by (LeVeque, 2002) ep(100 ( 0.3) 2 ), [0, 0.6), u 0 () = 1, [0.6, 0.8], (16) 0, (0.8, 1]

In numerical simulations, a computational mesh with N = 100 cells (d = 1) was adopted. Two final times of simulation are considered: t f = and t f = 5.0. Figure 2 depicts the eact solution and numerical results obtained with Minmod, TOPUS and van Leer limiters. It can be seen from this figure that the TOPUS limiter have a good performance when compared to other limiters. Note that all limiters presented the problem of peak clipping near to = 0.3 at time t f = 5. Mainly, the Minmod which is symmetric flu limiter. 4.2 1D inviscid Burgers problem It is adopted the equation (10) with [ 8, 8] and initial condition given by (LeVeque, 2002) [cos() + 1] {2 sin(3) + [cos(2) + 0.2]}, [ π, π], u 0 () = (17) 0, otherwise. During the numerical simulations, meshes with N = 100 and N = 2000 computational cells and final time simulation t f = 1 were adopted. Figure 3 shows the numerical results obtained with Minmod, TOPUS and van Leer limiters. From numerical results, which are very similar, one can conclude that the TOPUS scheme is as robust strategy to capture shock as high-resolution limiters which eist in the literature. 4.3 2D incompressible flow over a backward facing step A flow, comprehensively studied over the years, is the laminar flow over a backward facing step. The geometry of this problem is illustrated in Figure 4. Flow separation and recirculation caused by a sudden epansion in a channel appear in many engineering applications. Heat echangers, reactors, combustion machines and industrial ducts are good eamples. This flow problem was chosen as a representative test bed because there are a great deal of data in the literature, both numerical and eperimental. With a fully developed Poiseuille parabolic velocity profile prescribed at the inlet section, we simulate numerically this fluid flow problem for a wide range of Reynolds numbers. These are based on the maimum velocity U ma = 1 m/s at the entrance section and the height of the step h (h = 0.1 m). The dimension of domain computational is 4.0 m 0.2 m and total time simulation 100 s. Table 1 shows values of the reattachment length 1 obtained by eperimental/numerical data of Armaly et al. (1983) and the present calculation using ADBQUICKEST and TOPUS schemes. From this table, one can see that the numerical method provided satisfactory results. In particular, Figure 5 graphically displays a plot of reattachment lengths 1 against Reynolds numbers using the data of Armaly et al. (1983) and the present results obtained using ADBQUICK- EST and TOPUS schemes. And, once more, this comparison shows a good agreement between the data for 0 < Re < 400. Besides, one can see from this same figure that for Re 400 the numerical results give poor agreement; this may be eplained by the three dimensional effects and, possibly, the turbulence transition in this high Reynolds number problem.

Minmod at t = 1 Minmod at t = 5 u(,t) u(,t) 0.2 0.4 0.6 0.8 (a) TOPUS at t = 1 0.2 0.4 0.6 0.8 TOPUS at t = 5 u(,t) u(,t) 0.2 0.4 0.6 0.8 (b) van Leer at t = 1 0.2 0.4 0.6 0.8 van Leer at t = 5 u(,t) u(,t) 0.2 0.4 0.6 0.8 (c) 0.2 0.4 0.6 0.8 Figure 2: Numerical (blue symbol) and eact (black line) of the unsteady linear advection equation.

6 5 4 3 Minmod at t = 0.1 N = 2000 N = 100 2.0 Minmod at t = 1 N = 2000 N = 100 u(,t) 2 1 u(,t) 0 1 2 (a) u(,t) (b) u(,t) 3 8 6 4 2 0 2 4 6 8 6 5 4 3 2 1 0 1 2 TOPUS at t = 0.1 N = 2000 N = 100 3 8 6 4 2 0 2 4 6 8 6 5 4 3 2 1 0 1 2 van Leer at t = 0.1 N = 2000 N = 100 u(,t) 8 6 4 2 0 2 4 6 8 2.0 TOPUS at t = 1 N = 2000 N = 100 8 6 4 2 0 2 4 6 8 u(,t) 6 5 4 3 2 1 0 1 2 van Leer at t = 0.1 N = 2000 N = 100 (c) 3 8 6 4 2 0 2 4 6 8 3 8 6 4 2 0 2 4 6 8 Figure 3: Solution of the Riemann problem for the inviscid Burgers equation.

Figure 4: Geometry of the backward facing step problem. Table 1: Reattachment lengths 1 against Reynolds numbers. Eperimental Numerical Re Armaly et al. (1983) Armaly et al. (1983) ADBQUICKEST TOPUS 100 3.06 2.95 3.13 3.12 200 5.16 4.82 5.14 5.14 400 8.72 8.04 8.14 8.22 600 11.28 8.18 9.73 9.82 800 14.34 7.50 10.71 10.77 In addition, a convergence test of the numerical solution was performed with Reynolds number 400 and on three uniform meshes consisting of 200 10, 400 20 and 800 40 cells. This is illustrated in Figure 7, which shows how the reattachment length was estimated (the change in the sign of the u velocity profile adjacent to the lower bounding wall) in the code, i.e, 1 = (g 1)/0.1. 5 CONCLUDING REMARKS In this paper, it was presented the TOPUS scheme for numerical solution of time dependent conservation laws and related fluid dynamics problems. It was employed on three problems, namely, 1D advection of scalars, 1D Riemann problem for Burgers equation and 2D incompressible flow over a backward facing step. From 1D numerical results, one can conclude that the TOPUS scheme is a robust strategy to capture shock and it provides results in good agreement with other numerical and analytical data. From 2D incompressible flow, the TOPUS provided result compatible with that obtained using the ADBQUICKEST scheme and eperimental data. For the future, the authors are planning to use the CLAWPACK software equipped with TO- PUS scheme for solution of shock tube problem. This problem is used in the eperimental inves-

16 14 Ep. of Armaly et al. Num. of Armaly et al. ADBQUICKEST TOPUS 12 10 8 6 4 2 100 200 300 400 500 600 700 800 Re Figure 5: Comparison between eperimental data and numerical results. (a) ADBQUICKEST (b) TOPUS Figure 6: Numerical solution of backward facing step flow at Reynolds number Re = 800 using ADBQUICK- EST and TOPUS schemes - velocity u in the ais direction

0.3 ADBQUICKEST 0.2 mesh 200 10 mesh 400 20 mesh 800 40 u = 0 g = 2.005 0.1 u 0-0.1-0.2-0.3 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0.2 TOPUS 0.15 0.1 5 mesh 200 10 mesh 400 20 mesh 800 40 u = 0 g = 2.008 0 u -5-0.1-0.15-0.2-0.25-0.3 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Figure 7: Convergence test of the numerical solution of backward facing step flow at Reynolds number Re = 400.

tigation of several physical phenomena, such as chemical reaction kinetics, shock structure, and aero-thermodynamics of supersonic/hypersonic vehicles. Moreover, this problem is very useful for assessing the entropy satisfaction property of numerical method Toro (1999). REFERENCES Alves, M., Oliveira, P., & Pinho, F., 2003. A convergent and universally bounded interpolation scheme for the treatment of advection. International Journal for Numerical Methods in Fluids, vol. 41, pp. 47 75. Armaly, B., Durst, F., Periera, J., & Schonung, B., 1983. Eperimental and theoretical investigation of backward facing step flow. Journal of Fluid Mechanics, vol. 127, pp. 473 496. Brandt, A. & Yavneh, I., 1991. Inadequacy of first-order upwind difference schemes for some recirculating flows. Journal of Computational Physics, vol. 93, pp. 128 143. Castelo, A., Tom, M., McKee, S., Cuminato, J. A., & Cesar, C., 2000. Freeflow: an integrated simulation system for three-dimensional free surface flows. Computing and Visualization in Science, vol. 2, pp. 1 12. Clawpack, 2014. Clawpack 5.. Available from http://clawpack.github.io/inde.html. Courant, R., Isaacson, E., & Rees, M., 1952. On the solution of nonlinear hyperbolic differential equations by finite differences. Communications on Pure and Applied Mathematics, vol. 5, n. 243-255. Ferreira, V., Kurokawa, F., Oishi, C., Kaibara, M., Castelo, A., & Cuminato, J., 2009a. Evaluation of a bounded high order upwind scheme for 3D incompressible free surface flow computations. Mathematics and Computers in Simulation, vol. 79, pp. 1895 1914. Ferreira, V., Kurokawa, F., Queiroz, R., Kaibara, M., Oishi, C., Cuminato, J., Castelo, A., Tomé, M., & Mckee, S., 2009b. Assessment of a high-order finite difference upwind scheme for the simulation of convection-diffusion problems. International Journal for Numerical Methods in Fluids, vol. 60, pp. 1 26. Ferreira, V., Queiroz, R., Candenazo, M., Lima, G., Corrła, L., Oishi, C., & Santos, F., 2012a. Simulation results and applications of an advection bounded scheme to practical flows. Computational and Applied Mathematics, vol. 31, pp. 591 616. Ferreira, V., Queiroz, R., Lima, G., Cuenca, R., Oishi, C., Azevedo, J., & Mckee, S., 2012b. A bounded upwinding scheme for computing convection-dominated transport problems. Computers & Fluids, vol. 57, pp. 208 224. Gaskell, P. & Lau, A. K., 1988. Curvature-compensated convective transport: SMART, a new boundedness-preserving transport algorithm. International Journal for Numerical Methods in Fluids, vol. 8, pp. 617 641. Harten, A., 1983. High resolution schemes for conservation laws. Journal of Computational Physics, vol. 49, pp. 357 393.

Kuan, K. & Lin, C., 2000. Adaptive QUICK-based scheme to approimate convective transport. AIAA Journal, vol. 38, n. 12, pp. 2233 2237. Leonard, B., 1979. A stable and accurate convective modelling provedure based on quadratic upstream interpolation. Computer Methods in Applied Mechanics and Engineering, vol. 19, pp. 59 98. Leonard, B., 1987. Sharp simulation of discontinuities in highly convective steady flow. NASA Techn. Mem., vol.. Leonard, B., 1988. Simple high-accuracy resolution program for convective modeling of discontinuities. International Journal for Numerical Methods in Fluids, vol. 8, pp. 1291 1318. LeVeque, R., 2002. Finite Volume Methods for Hyperbolic Problems. Cambridge Tets in Applied Mathematics. Queiroz, R. & Ferreira, V., 2010. Development and testing of high-resolution upwind schemes: Upwind schemes for incompressible free surface flows. VDM Verlag Dr. Muller. Roe, P., 1986. Characteristic-based schemes for the Euler equations. In Annual Review of Fluid Mechanics, pp. 337 365. Toro, E., 1999. Riemann Solves and Numerical Methods for Fluid Dynamics. Springer-Verlag, 2nd edition edition. van Leer, B., 1974. Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. Journal of Computational Physics, vol. 14, pp. 361 370. Varonos, A. & Bergeles, G., 1998. Development and assessment of a variable-order non-oscillatory scheme for convection term discretization. International Journal for Numerical Methods in Fluids, vol. 26, pp. 1 16. Warming, R. & Beam, R., 1976. Upwind second-order difference schemes and applications in aerodynamic flows. AIAA Journal, vol. 14, pp. 1241 1249. Waterson, N. & Deconinck, H., 2007. Design principles for bounded higher-order convection schemes - a unified approach. Journal of Computational Physics, vol. 224, pp. 182 207. Xue, S.-C., Phan-Thien, N., & Tanner, R. I., 2002. Upwinding with deferred correction (UPDC): and effective implementation of higher-order convection schemes for implicit finite volume methods. Journal of Non-Newtonian Fluid Mechanics, vol. 108, pp. 1 24.