A HIGH ORDER POLYNOMIAL UPWIND SCHEME FOR … · the derivation of the TOPUS scheme, it is used the...

SIMMEC / EMMCOMP 2014XI Simposio de Mecanica ComputacionalII Encontro Mineiro de Modelagem ComputacionalJuiz de Fora, MG, 28-30 de maio de 2014

A HIGH ORDER POLYNOMIAL UPWIND SCHEME FOR NUMERICALSOLUTION OF CONSERVATION LAWS AND FLUID DYNAMICS

PROBLEMS

Rafael Alves Bonfim de Queiroz, Bernardo Martins [email protected], [email protected] Federal de Juiz de Fora, Instituto de Ciencias Exatas, Departamento de Ciencia daComputacaoRua Jose Lourenco Kelmer, Sao Pedro, 36036330, Juiz de Fora, MG, Brasil

Abstract. A high order polynomial upwind scheme, namely TOPUS, for numerical solution ofconservation laws and fluid dynamics problems is revisited in this paper. The scheme is based onTVD and CBC stability criteria. The performance of the scheme is investigated for solving thescalar convection equation with discontinuous initial data and 1D Riemann problem for Burgerequation. The scheme is then applied in the simulation of 2D incompressible flow over a backwardfacing step. The numerical results show good agreement with other numerical and experimentaldata.

Keywords: TOPUS scheme, High-resolution schemes, Convection modeling

Rafael A.B. de Queiroz, Bernardo M. Rocha

1 INTRODUCTIONThe development of numerical schemes for approximation of convection terms of conserva-

tion laws and transport equations in computational fluid dynamics (CFD) has presented a contin-uing challenge. The major problem is to prevent the unbounded growth of the unphysical spatialoscillations 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 andnumerical dissipation becomes dominant (Brandt and Yavneh, 1991). The cure for this has beento use conventional schemes, such as second-order upwind (SOU) (Warming and Beam, 1976),quadratic-upstream interpolation for convective kinematics (QUICK) (Leonard, 1979) and QUICKwith estimated stream terms (QUICKEST) (Leonard, 1988) schemes, due to their intrinsic lack ofnumerical dissipation. However, they also inevitably can generate unphysical spatial oscillationsand 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 andmonotonic algorithm for realistic transport (SMART) (Gaskell and Lau, 1988), the simple highaccuracy resolution program (SHARP) (Leonard, 1987), the variable-order non-oscillatory scheme(VONOS) (Varonos and Bergeles, 1998), and convergent and universally bounded interpolationscheme 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 oscillationsand, meanwhile, to improve the rate of convergence. It should be also noted, however, that theseschemes, though performing well in some problems, cannot be bounded in situations such as shockphenomena in compressible flows (Kuan and Lin, 2000) and/or incompressible viscoelastic flowcalculations 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 developmentof 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 fluxfunction using a combination of low and high order schemes through some limiter function. Thisscheme 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 highestwave numbers that are resolved, and thus automatically decreases numerical dissipation at highergrid resolutions. The expectation is that the use of this upwind scheme will enable not merely tocapture the shock, but also to resolve the delicate features and structures of free surface flows. Inthe 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 offergreat flexibility in the construction of the higher-order bounded schemes.

In this paper, the TOPUS scheme is revisited for solving the following problems: 1D scalar

SIMMEC/EMMCOMP 2014XI Simposio de Mecanica Computacional e II Encontro Mineiro de Modelagem Computacional

Juiz de Fora, MG, 28-30 de maio de 2014


convection equation with discontinuous initial data, 1D Riemann problem for Burgers’ equationand 2D laminar flow over a backward facing step. The objective is to examine the utility andeffectiveness of the TOPUS scheme when incorpored into the CLAWPACK (Conservation LAWsPACKage) software (LeVeque, 2002; Clawpack, 2014) and Freeflow simulation system (Casteloet al., 2000).

The organization of the paper is as follows. In Section 2, the mathematical formulation ofhigh-resolution schemes is revised. A short outline of the governing equations of the problemsconsidered in this paper is presented in Section 3. The versatility and robustness of the TOPUSscheme is considered in Section 4. Section 5 contains a few concluding remarks and guidelines forfuture 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 variationof a convected property through, for example, the boundary face f between two control volumescan 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 tothe mass flux Vf 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 thenumerical flux at the interface can be determined. To this end, the original variable u is transformedinto NV by

u(x, t) =u(x, t)− unRunD − unR

. (1)

The advantage of this normalization is that the interface value uf depends on unU and θ only, sinceunD = 1 and unR = 0.

V

R U D

f

g f

Figure 1: The f face, together with its mass flux Vf > 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 context of the NV while enforcing the TVD property (Harten, 1983)and CBC conditions (Gaskell and Lau, 1988). According to Ferreira et al. (2009b), this schemecombines 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 thetests (Ferreira et al., 2009b), but only first-order near regions with large gradients (discontinuities).

The ADBQUICKEST scheme in NV is given by

uf =

(2− θ)uC , uC ∈ (0,a),

uC + 12(1− |θ|)(1− uC)− 1

6(1− θ2)(1− 2uC), uC ∈ [a, b],

1− θ + θuC , uC ∈ (b,1),

uC , uC /∈ (0, 1),

(2)

where θ = uδtδx

is the Courant number; and the a and b parameters are given by

a =2− 3|θ|+ θ2

7− 6θ − 3|θ|+ 2θ2and 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)rf3− 3θ

, 2

]}, (3)

where rf for uniform meshes is defined by

rf =uC

1− uC. (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 andFerreira, 2010; Ferreira et al., 2012a,b). In the following, we recapitulate the main features of thisscheme for ease of reference.

The TOPUS scheme in NV is given by

uf =

αu4U + (−2α + 1) u3U +(5α−10

4

)u2U +

(−α+10

4

)uU , uU ∈ [0, 1],

uU , uU /∈ [0, 1],(7)

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).




In the case of α = 2, TOPUS is entirely contained into the TVD region of Harten (1983), andits corresponding flux limiter ψ(rf ), with rf a sensor, satisfies Sweby’s monotonicity preservationcondition when rf tends to 0. The flux limiter ψ(rf ) for the TOPUS scheme is represented asfollows

ψ(rf , α) =0.5 (|rf |+ rf )

[(−0.5α + 1)r2f + (α + 4)rf + (−0.5α + 3)

](1 + |rf |)3

. (8)

3 1D/2D Problems

A short outline of the governing equations of the problems considered in this paper is presentedbelow.

3.1 1D unsteady linear advection equation

In this problem, the advection of a quantity u(x, t) by a constant convecting velocity a isinvestigated. The equation for the unsteady linear advection problem is given by

∂u

∂t+ a

∂u

∂x= 0. (9)

The exact solution is given by u(x, t) = u0(x−at), which states that the initial data u0(x) is simplytranslated with velocity a.

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 todemonstrate 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 doesnot change during its movement and the transport properties are constant. Considering numericalsimulation of a laminar 2D flow over a backward facing step, the mathematical modeling of theconservation laws are the instantaneous Navier-Stokes and continuity equations, respectively, givenby

∂vi∂t

+∂(vivj)

∂xj= − ∂p

∂xi+

1

Re

∂

∂xj

(∂vi∂xj

)+

1

(Fr)2gi, i = 1, 2, (11)

∂vi∂xi

= 0, (12)




where t is the time, vi is the ith component of velocity, p is the kinematic pressure (pressure dividedby density) and gi is the ith component of gravitational acceleration. The non-dimensional param-eters Re = LV0/ν and Fr = V0/

√L|g| denote the associated Reynolds and Froude numbers,

respectively. The constant parameter ν is the kinematic viscosity coefficient of the fluid given byν = µ/ρ (µ is the dynamic viscosity coefficient of the fluid), and V0 is characteristic velocity andL is 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 Dirichletcondition 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)

where un and uτ are the normal and tangential velocities to the boundary, respectively;

• At the outlet section - homogeneous Neumann (fully developed flow) conditions are specifiedfor all variables:

∂vn∂n

= 0 and∂vτ∂n

= 0, (14)

where n and τ 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 tothe three problems, namely, 1D advection of scalars, 1D Riemann problem for Burgers equationand 2D incompressible flow over a backward facing step are shown. The 1D problems were solvedusing 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 with N = 100 cells (dx = 0.01) was adopted.Two final times of simulation are considered: tf = 1.0 and tf = 5.0. Figure 2 depicts the exactsolution and numerical results obtained with Minmod, TOPUS and van Leer limiters. It can be seenfrom 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 x = 0.3 at time tf = 5. Mainly,the Minmod which is symmetric flux limiter.

4.2 1D inviscid Burgers problem

It is adopted the equation (10) with x ∈ [−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 with N = 100 and N = 2000 computational cellsand final time simulation tf = 1 were adopted. Figure 3 shows the numerical results obtained withMinmod, TOPUS and van Leer limiters. From numerical results, which are very similar, one canconclude that the TOPUS scheme is as robust strategy to capture shock as high-resolution limiterswhich 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 facingstep. The geometry of this problem is illustrated in Figure 4. Flow separation and recirculationcaused 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, wesimulate numerically this fluid flow problem for a wide range of Reynolds numbers. These arebased on the maximum velocity Umax = 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 simulation100 s.

Table 1 shows values of the reattachment length x1 obtained by experimental/numerical dataof 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 x1 against Reynoldsnumbers 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 thedata for 0 < Re < 400. Besides, one can see from this same figure that forRe ≥ 400 the numericalresults 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 0.6 0.8 1.0x

−0.5

0.0

0.5

1.0

1.5

u(x,

t)

Minmod at t = 1

0.0 0.2 0.4 0.6 0.8 1.0x

−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 1.0x

−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 1.0x

−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 1.0x

−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 1.0x

−0.5

0.0

0.5

1.0

1.5

u(x,

t)

van Leer at t = 5

(c)

Figure 2: Numerical (blue symbol) and exact (black line) of the unsteady linear advection equation.




−8 −6 −4 −2 0 2 4 6 8x

−3

−2

−1

0

1

2

3

4

5

6

u(x,

t)

Minmod at t = 0.1

N = 2000N = 100

−8 −6 −4 −2 0 2 4 6 8x

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

u(x,

t)

Minmod at t = 1

N = 2000N = 100

(a)

−8 −6 −4 −2 0 2 4 6 8x

−3

−2

−1

0

1

2

3

4

5

6

u(x,

t)

TOPUS at t = 0.1

N = 2000N = 100

−8 −6 −4 −2 0 2 4 6 8x

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

u(x,

t)

TOPUS at t = 1

N = 2000N = 100

(b)

−8 −6 −4 −2 0 2 4 6 8x

−3

−2

−1

0

1

2

3

4

5

6

u(x,

t)

van Leer at t = 0.1

N = 2000N = 100

−8 −6 −4 −2 0 2 4 6 8x

−3

−2

−1

0

1

2

3

4

5

6

u(x,

t)

van Leer at t = 0.1

N = 2000N = 100

(c)

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 x1 against 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 number400 and on three uniform meshes consisting of 200 × 10, 400 × 20 and 800 × 40 cells. This isillustrated in Figure 7, which shows how the reattachment length was estimated (the change in thesign of the u velocity 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 incompressibleflow over a backward facing step. From 1D numerical results, one can conclude that the TOPUSscheme 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 compatiblewith 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 inves-




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.

ADBQUICKESTTOPUS

Figure 5: Comparison between experimental 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 x axis 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 10mesh 400 x 20mesh 800 x 40

u = 0xg = 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 10mesh 400 x 20mesh 800 x 40

u = 0xg = 2.008

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, andaero-thermodynamics of supersonic/hypersonic vehicles. Moreover, this problem is very useful forassessing the entropy satisfaction property of numerical method Toro (1999).

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