II Encontro Mineiro de Modelagem Computacional Juiz de Fora, MG, 28-30 de maio de 2014
A HIGH ORDER POLYNOMIAL UPWIND SCHEME FOR NUMERICAL
SOLUTION OF CONSERVATION LAWS AND FLUID DYNAMICS
PROBLEMS
Rafael Alves Bonfim de Queiroz, Bernardo Martins Rocha
[email protected], [email protected]
Universidade Federal de Juiz de Fora, Instituto de Ciˆencias Exatas, Departamento de Ciˆencia da Computac¸˜ao
Rua Jos´e Lourenc¸o Kelmer, S˜ao 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 experimental data.
1
INTRODUCTION
The development of numerical schemes for approximation of convection terms of conserva-tion laws and transport equaconserva-tions in computaconserva-tional fluid dynamics (CFD) has presented a contin-uing 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 applica-tions involving long time evolution of complex flows, mainly because extrema 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 experience discontinuities.
In order to overcome the defect produced by the aforementioned schemes, a number of mono-tonic upwind schemes has appeared in the published literature as, for example, 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 objec-tive 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 approxi-mating non-linear convective terms of conservation laws and related fluid dynamics equations con-tinues 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 flux function using a combination of low and high order schemes through some limiter function. This scheme approximates the advective fluxes at the cell boundaries with 1st, 2nd or 3rd order accu-racy, 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 expectation 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 flexibility in the construction of the higher-order bounded schemes.
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 examine 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). Letube the variation of a convected property through, for example, the boundary face f between two control volumes can be represented by a functional relationship linking values uD, uU and uR, which represent,
respectively, the Downstream, the Upstream and the Remote-upstream locations with respect to the mass fluxVf at this face (see Figure1). The neighbors of thegface can be similarly classified.
If this functional relationship involving these three neighboring positions is prescribed, then the numerical flux at the interface can be determined. To this end, the original variableuis transformed into NV by ˆ u(x, t) = u(x, t)−u n R un D−unR . (1)
The advantage of this normalization is that the interface valueuˆf depends onuˆnU andθ only, since
ˆ unD = 1anduˆnR= 0. V R U D f g f
Figure 1: Thefface, together with its mass fluxVf >0and neighboring nodesD,U andR.
2.1
ADBQUICKEST SCHEME
The adaptative QUICKEST (Ferreira et al.,2009a) or ADBQUICKEST (Ferreira et al.,2009b) scheme was developed in the context of the NV while enforcing the TVD property (Harten,1983) and CBC conditions (Gaskell and Lau, 1988). According toFerreira et al. (2009b), this scheme combines accuracy and monotonicity, while providing flexibility. 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
ˆ uf = (2−θ)ˆuC, uˆC ∈(0,a), ˆ uC+ 12(1− |θ|)(1−uˆC)− 16(1−θ2)(1−2ˆuC), uˆC ∈[a, b], 1−θ+θuˆC, uˆC ∈(b,1), ˆ uC, uˆC ∈/ (0,1), (2)
whereθ= uδtδx is the Courant number; and theaandbparameters are given by
a= 2−3|θ|+θ
2
7−6θ−3|θ|+ 2θ2 and b=
−4 + 6θ−3|θ|+θ2
−5 + 6θ−3|θ|+ 2θ2.
The flux limiter for the ADBQUICKEST scheme can be written as
ψ(rf) = max ( 0,min " 2rf, 2 +θ2−3θ+ (1−θ2)r f 3−3θ ,2 # ) , (3)
whererf for uniform meshes is defined by
rf =
ˆ
uC
1−uˆC
. (4)
2.2
CLASSICAL FLUX LIMITERS
• van Leer (van Leer,1974):
ψ(rf) = rf +|rf| 1 +|rf| . (5) • Minmod (Roe,1986): ψ(rf) = max [0,min (rf,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 ˆ uf = αuˆ4 U + (−2α+ 1) ˆu3U + 5α−10 4 ˆ u2 U + − α+10 4 ˆ uU, uˆU ∈[0,1], ˆ uU, uˆU ∈/ [0,1], (7) where αis an adjustable constant, α ∈ [−2,2], to ensure that TOPUS falls into the CBC region. If α = 0, then TOPUS corresponds to the SMARTER schemeWaterson and Deconinck (2007).
In the case of α = 2, TOPUS is entirely contained into the TVD region of Harten (1983), and its corresponding flux limiter ψ(rf), withrf a sensor, satisfies Sweby’s monotonicity preservation
condition when rf tends to 0. The flux limiter ψ(rf) for the TOPUS scheme is represented as
follows ψ(rf, α) = 0.5 (|rf|+rf) h (−0.5α+ 1)r2 f + (α+ 4)rf + (−0.5α+ 3) i (1 +|rf|)3 . (8)
3
1D/2D Problems
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(x, t) by a constant convecting velocity a is investigated. The equation for the unsteady linear advection problem is given by
∂u
∂t +a
∂u
∂x = 0. (9)
The exact solution is given byu(x, t) =u0(x−at), which states that the initial datau0(x)is simply
translated with velocitya.
3.2
1D inviscid Burgers’ equation
The inviscid Burgers’equation is given by∂u ∂t + ∂ ∂t u2 2 ! = 0. (10)
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(x, t) = ρ0 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 ∂vi ∂t + ∂(vivj) ∂xj = −∂p ∂xi + 1 Re ∂ ∂xj ∂v i ∂xj + 1 (F r)2gi, i= 1,2, (11) ∂vi ∂xi = 0, (12)
wheretis the time,viis the ith component of velocity,pis the kinematic pressure (pressure divided
by density) andgi is the ith component of gravitational acceleration. The non-dimensional
param-eters Re = LV0/ν and F r = V0/
q
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(µis the dynamic viscosity coefficient of the fluid), andV0 is characteristic velocity and
Lis length scale.
The system of equations formed by (11) and (12) cannot be solved without specifying appro-priate 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:
vn=V0 and vτ = 0, (13)
whereunanduτ 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:
∂vn
∂n = 0 and
∂vτ
∂n = 0, (14)
where n andτ denote normal and tangential directions at the input and output of fluid, re-spectively;
• Free-slip walls - it is assumed that the fluid slips at the solid walls:
vn= 0 and ∂vτ
∂n = 0; (15)
4
Numerical experiments
In this section, results obtained with the TOPUS scheme in numerical experiments 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, x ∈ [0,1] and initial condition given by (LeVeque,
2002) u0(x) = exp(100 (x−0.3)2), x∈[0,0.6), 1, x∈[0.6,0.8], 0, x∈(0.8,1] (16)
In numerical simulations, a computational mesh withN = 100cells (dx= 0.01) was adopted. Two final times of simulation are considered: tf = 1.0and tf = 5.0. Figure 2depicts the exact
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 tox= 0.3at timetf = 5. Mainly,
the Minmod which is symmetric flux limiter.
4.2
1D inviscid Burgers problem
It is adopted the equation (10) withx∈[−8,8]and initial condition given by (LeVeque,2002)
u0(x) =
[cos(x) + 1]× {2×sin(3x) + [cos(2x) + 0.2]}, x∈[−π, π],
0, otherwise.
(17) During the numerical simulations, meshes withN = 100andN = 2000computational cells and final time simulationtf = 1were adopted. Figure3shows 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 exist 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 expansion in a channel appear in many engineering applications. Heat ex-changers, reactors, combustion machines and industrial ducts are good examples. This flow prob-lem was chosen as a representative test bed because there are a great deal of data in the literature, both numerical and experimental.
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 maximum velocityUmax = 1m/s at the entrance section and the height of the steph
(h = 0.1m). The dimension of domain computational is 4.0 m×0.2 m and total time simulation 100 s.
Table 1shows values of the reattachment length x1 obtained by experimental/numerical data
ofArmaly 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, Figure5graphically displays a plot of reattachment lengthsx1 against Reynolds
numbers using the data ofArmaly 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 for0< Re <400. Besides, one can see from this same figure that forRe≥400the numerical results give poor agreement; this may be explained by the three dimensional effects and, possibly, the turbulence transition in this high Reynolds number problem.
0.0 0.2 0.4 x 0.6 0.8 1.0 −0.5 0.0 0.5 1.0 1.5 u(x ,t) Minmod at t = 1 0.0 0.2 0.4 x 0.6 0.8 1.0 −0.5 0.0 0.5 1.0 1.5 u(x ,t) Minmod at t = 5 (a) 0.0 0.2 0.4 0.6 0.8 1.0 x −0.5 0.0 0.5 1.0 1.5 u( x, t) TOPUS at t = 1 0.0 0.2 0.4 0.6 0.8 1.0 x −0.5 0.0 0.5 1.0 1.5 u( x, t) TOPUS at t = 5 (b) 0.0 0.2 0.4 0.6 0.8 1.0 x −0.5 0.0 0.5 1.0 1.5 u( x, t) van Leer at t = 1 0.0 0.2 0.4 0.6 0.8 1.0 x −0.5 0.0 0.5 1.0 1.5 u( x, t) van Leer at t = 5 (c)
−8 −6 −4 −2 0x 2 4 6 8 −3 −2 −1 0 1 2 3 4 5 6 u(x ,t) Minmod at t = 0.1 N = 2000 N = 100 −8 −6 −4 −2 0x 2 4 6 8 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 u(x ,t) Minmod at t = 1 N = 2000 N = 100 (a) −8 −6 −4 −2 0 2 4 6 8 x −3 −2 −1 0 1 2 3 4 5 6 u( x, t) TOPUS at t = 0.1 N = 2000 N = 100 −8 −6 −4 −2 0 2 4 6 8 x −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 u( x, t) TOPUS at t = 1 N = 2000 N = 100 (b) −8 −6 −4 −2 0 2 4 6 8 x −3 −2 −1 0 1 2 3 4 5 6 u( x, t) van Leer at t = 0.1 N = 2000 N = 100 −8 −6 −4 −2 0 2 4 6 8 x −3 −2 −1 0 1 2 3 4 5 6 u( x, t) van Leer at t = 0.1 N = 2000 N = 100 (c)
Figure 4: Geometry of the backward facing step problem. Table 1: Reattachment lengthsx1against Reynolds numbers.
Experimental 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 and800 ×40cells. This is illustrated in Figure7, which shows how the reattachment length was estimated (the change in the sign of theuvelocity profile adjacent to the lower bounding wall) in the code, i.e,x1 = (xg−1)/0.1.
5
CONCLUDING REMARKS
In this paper, it was presented the TOPUS scheme for numerical solution of time depen-dent 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 nu-merical and analytical data. From 2D incompressible flow, the TOPUS provided result compatible with that obtained using the ADBQUICKEST scheme and experimental 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 experimental
2 4 6 8 10 12 14 16 100 200 300 400 500 600 700 800 x Re Exp. of Armaly et al.
Num. of Armaly et al. ADBQUICKEST TOPUS
Figure 5: Comparison between experimental data and numerical results.
(a) ADBQUICKEST
(b) TOPUS
Figure 6: Numerical solution of backward facing step flow at Reynolds numberRe= 800using ADBQUICK-EST and TOPUS schemes - velocityuin thexaxis direction
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 1 1.2 1.4 1.6 1.8 2 2.2 2.4 u x ADBQUICKEST mesh 200 x 10 mesh 400 x 20 mesh 800 x 40 u = 0 xg = 2.005 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 1 1.2 1.4 1.6 1.8 2 2.2 2.4 u x TOPUS mesh 200 x 10 mesh 400 x 20 mesh 800 x 40 u = 0 xg = 2.008
Figure 7: Convergence test of the numerical solution of backward facing step flow at Reynolds numberRe =
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 methodToro(1999).
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