Implicit Finite Volume Method to Simulate Reacting Flow

43rd AIAA Aerospace Sciences Meeting and Exhibit, 10-13 Jan. 2005, Reno, Nevada

Implicit Finite Volume Method to Simulate Reacting Flow Masoud Darbandi∗ and Araz Banaeizadeh† Sharif University of Technology, Tehran, P.O. Box 11365-8639, Iran

Gerry E. Schneider‡ University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada In this work, an efficient bi-implicit strategy is suitably developed within the context of a finite volume element approach in order to solve turbulent reactive flow governing equations. Based on the essence of control-volume-based finite-element methods, the formulation retains the geometrical flexibility of the pure finite element methods while derives the discrete algebraic governing equations through using the conservation balance applied to discrete control volumes distributed all over the solution domain. The physical influence upwinding scheme is used to approximate the advection fluxes at all cell faces. While respecting the physics of flow, this scheme also provides the necessary coupling of velocity and pressure fields. The two-equation k −  turbulence model and one step mixture fraction chemistry equation are simultaneously solved in a semi-coupled manner in order to achieve a better prediction of both the transport of turbulent species and the transport of mass fraction species. The validation of the current numerical results is fulfilled by comparing them with experimental data and other available numerical results.

I. B Cpi D f h hF N n P R T u, v u ¯ V

Nomenclature

body force specific heat of species i at constant pressure diffusion coefficient mixture fraction enthalpy heat reaction finite element shape function total number of species pressure gas constant temperature velocity components friction velocity velocity vector

∗ Associate

Professor, Department of Aerospace Engineering. Student, Department of Aerospace Engineering. ‡ Professor and Chair, Department of Mechanical Engineering, AIAA Fellow. c 2005 by M. Darbandi. Published by the American Institute of Aeronautics and Astronautics, Inc. with Copyright permission. † Graduate

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u ¯ x, y Yi z Υ  κ µ µe µt ρ σ

A. i p st B. +

friction velocity Cartesian coordinates mass fraction of species i a representative of the extensive quantities dissipation rate of turbulent kinetic energy at nodes dissipation rate of turbulent kinetic energy turbulent kinetic energy laminar viscosity effective viscosity turbulent viscosity mixture density viscous shear tensor

Subscripts chemical species the grid point adjacent to the solid wall stoichiometric condition Superscripts non-dimensional magnitudes

II.

Introduction

he reacting flow in combustion chambers mostly occurs in complex geometries which in turn need roT bust numerical tools to treat the resulting complex turbulent reactive flow fields. Finite-difference-based methods do not straightforwardly provide efficient means to solve the flow fields occurred in complex geometries. On the contrary, the finite element methods provide easy treatment of complicated domains. The past experience has shown that a robust approach toward handling the complex geometries is the finitevolume-based finite-element one.1 This method guarantees the great benefits of both finite volume and finite element methods. Finite volume element method incorporated with a suitable physical upwinding influence scheme effectively enhances the capabilities of the current dual-based method.2 In this work, the primitive implicit strategy introduced by Darbandi and Schneider3 is suitably extended in order to solve the turbulent reactive flow governing equations. Back to the primary problem with the continuity equation, Darbandi and Bostandoost4 have developed a fully implicit finite volume method which eliminates the need for the pressure Poisson equation and takes the pressure role directly into the continuity equation. The approach is fully implicit and produces a 27-diagonal matrix in treating the 2D Navier-Stokes equations. Darbandi, et al.5 extended the formulation for solving the diffusive flame. In this work, the original approach is applied to discretize turbulence and fast chemistry equations in a way that continuity, x-momentum and y-momentum are solved implicitly in one matrix and κ, , mixture fraction and energy equations in another matrix. This approach leads to a bi-implicit treatment of the governing equations.

III.

The Governing Equations

The conservative equations for a reacting flow can be categorized into fluid flow and species transport equations. The radiation heat transfer effect is ignored in this study. The fluid flow governing equations consist of the conservation statements for mass, momentums, and energy. The incompressible steady governing

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equations are written as

~ · (ρV ~)=0 ∇

~ · (ρV ~ u) = − ∂p + ∇ ~ · (µe ∇u) ~ ∇ ∂x ~ · (µe ∇v) ~ ~ · (ρV ~ v) = − ∂p + ∇ ∇ ∂y

(1) (2) (3)

where µe is the effective viscosity defined by µe = µ + µt . The eddy turbulent viscosity µt is calculated from the turbulent kinetic energy κ equation and its dissipation rate  equation. The transport equations for these two components are written as ~ · (ρV ~ κ) = ∇ ~ · ( µe ∇κ) ~ + (Gκ − ρ) ∇ σκ

(4)

~ · (ρV ~ ) = ∇ ~ · ( µe ∇) ~ +  (c1 Gκ − c2 ρ) ∇ σ κ

(5)

In these equations, Gκ and c1 Gκ are the production terms and ρ and c2 ρ are the destruction terms. The expression for Gκ is given by Gκ = µt {2[(

∂u 2 ∂v ∂u ∂v 2 ) + ( )2 ] + [ + ] } ∂x ∂x ∂y ∂x

(6)

The derived κ and  magnitudes from Eqs. (4)-(5) are used to calculate the eddy viscosity definition, i.e., µt = c d ρ

κ2 

(7)

The constants in κ and  equations are σκ =1.0, σ =1.3, c1 =1.44, c2 =1.92, and cd =0.09. As is known, the calculated turbulent energy and its dissipation rate are valid sufficiently far from the solid walls and mainly in the interior region. The viscous effects dominate near the solid walls which subsequently require extra considerations. Depending on the flow Reynolds number, there are different choices such as low Reynolds number model and wall function approximation to cure the problem. In the present work, the wall function choice is used to describe the flow close to the wall. The elliptic nature of κ and  transport equations directs the users to specify boundary conditions at all domain boundaries. As was mentioned, we apply wall function at solid walls. Indeed, the region close to the solid walls can be divided into two layers 1) a laminar sublayer or viscous sublayer where viscous effects are dominant and 2) turbulent sublayer. In our model, it is supposed that the grid node p adjacent to the solid wall is far enough from the wall to be in turbulent sublayer region where the velocity is parallel to the wall at this node. The condition near the wall is calculated from the known logarithmic law. The turbulent kinetic energy and its dissipation rate at node p are then calculated from u ¯2 (8) κp = √ cd p =

u ¯3 k yp

(9)

where cd and k constants are 0.09 and 0.41, respectively. yp is the actual distance between node p and wall and u ¯ is the friction velocity. To specify the friction velocity at node p, the non-dimensional distance from the solid wall is defined as yp+ = ρ¯ uyp /µ. Using this definition, the non-dimensional velocity u+ is determined from ( 1 + yp+ ≤ 11.63 u+ p = k ln yp + 5.5 (10) + + + yp > 11.63 u p = yp 3 of 11 American Institute of Aeronautics and Astronautics Paper 2005-0574

Then, the friction velocity is derived from u ¯p =

up u+ p

(11)

For the combustion model, we consider a diffusion flame and a fast chemistry with one step irreversible chemical reaction. It is assumed that the chemistry is sufficiently fast and that the intermediate species do not play a significant role. Considering a mass flow rate of M = 1(Kg/s) for the air/fuel mixture, it consists of a fuel rate of f (Kg/s) and an air rate of (1 − f )(Kg/s). Considering the above mass flow rates, any extensive property, such as z, of the mixture resulting from the mix of these two streams can be written as zM − z A zF − z A

f=

(12)

If F indicates the fuel and A indicates the oxidizer, the mixture fraction yields f=

[YF − (F/O)st YO ]M + (F/O)st YO,A 1 + (F/O)st YO,A

(13)

From the chemical equilibrium assumption, no oxidant presents if a mixture fraction richer than stoichiometric condition is employed. Additionally, no fuel presents if the mixture fraction is lower than the stoichiometric condition.6 For any adiabatic operation under the assumption of unit Lewis number, the enthalpy equation is similar to the mixture fraction equation and the enthalpy can be calculated directly from the mixture fraction and the inlet enthalpy value. Thus, for all combustion related variables such as enthalpy, mass fraction of fuel, oxygen, and the products, it is sufficient to solve the transport equation for mixture fraction f .7 The transport equation for f is then given by ~ · (ρV ~ f) + ∇ ~ · (D ∇f ~ )=0 ∇

(14)

Assuming a unit Lewis number, the reactive energy equation is reduced to ~ · (ρV ~ h) = −∇ ~ · (µ∇h) ~ ∇

(15)

where the enthalpy is defined by h = Y F hF +

n Z X i=0

T Tref

1 Yi Cpi (T )dT + V 2 2

(16)

Eventually, the density is calculated from the equation of state which is written as P = ρRT

IV.

n X Yi W i i=1

(17)

The Domain Discretization

he solution domain is broken into a huge number of quadrilateral elements. The elements fully cover T the solution domain with no overlapping. Figure 1 shows a small part of the solution domain. Nodes are located at the corners of elements and are shown by circles. The nodes are the locations of the unknown variables. Each node belongs to four neighboring elements. There are four quadrilaterals which enclose node P in figure 1. To utilize the benefits of cell-centered schemes, each element is divided into four quadrilaterals by the help of its medians. The median is demonstrated by dashed-line in this figure. The cells are then constructed from the proper assemblage of these sub-quadrilaterals. As is seen, irrespective of the shape and distribution of the elements, each node is surrounded by a number of sub-quadrilaterals. The proper assemblage of neighboring sub-quadrilaterals around any non-boundary node creates a complete cell. 4 of 11 American Institute of Aeronautics and Astronautics Paper 2005-0574

x Integration Point Control Volume Element t Node

x

x

x x

y

s

x x

x

x

x

Figure 1. A part of the solution domain illustrating four elements, one complete finite volumes, sixteen sub-volumes, and eight cell faces.

V.

The Computational Modelling

To utilize the advantages of finite element volume methods, the governing equations are initially integrated over an arbitrary volume, e.g., the shaded area or the cell face shown in Fig. 1. The employment of Gauss divergence theorem to the governing equations leads to Z ~ =0 ~ · dA ρV (18) A

Z

A

Z

A

~ =− ~ ) · dA u(ρV ~ =− ~ ) · dA v(ρV

Z

p dAx + A

Z

p dAy + A

Z

Z

~ ~ · dA) µe (∇u

(19)

A

~ ~ · dA) µe (∇v

(20)

A

The above integrals are evaluated over the surface which encloses each cell. The surface area is indicated by A. The above equations are suitably discretized using finite difference scheme and finite element interpolations. ~ ~ ~ In the above expressions, dA=dA x i − dAy j is a normal vector to the edges of cell. Using this definition, the above integrals can be evaluated by summation over the faces that enclose the cell center, i.e., ns X

[ρ(u dAx + v dAy )]i = 0

(21)

i=1

ns X

ns X

ns  X



 ∂u ∂u dAx + dAy ∂x ∂y i i=1 i=1 i=1    ns ns ns X X X ∂v ∂v [ρˆ u v dAx + ρˆ v v dAy )]i = − (p dAy )i + dAx + dAy µe ∂x ∂y i i=1 i=1 i=1 [ρˆ u u dAx + ρˆ v u dAy )]i = −

(p dAx )i +

µe

(22) (23)

where i counts the number of cell faces from 1 to ns. There are 8 cell faces around non-boundary cells. To linearize the governing equations, the hat over u ˆ and vˆ indicates that these velocity components are approximated from the known magnitudes of the preceding iteration. Such approximation is essential to linearize the nonlinear momentum convection terms. The rest of procedure is to relate the cell face magnitudes

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(identified by lower case letters such as u, v and p variables) directly to the nodal magnitudes (identified by upper case letters such as U , V and P variables) which represent the locations for the unknown variables in the current algorithm. A simple idea for treating the right-hand-side terms is to use the finite element shape functions Nj=1...4 . The treatment results in pi =

4 X

Nij Pj

(24)

j=1

4 X ∂φ ∂Nij Φj = ∂ξ i j=1 ∂ξ

(25)

where pi identifies the magnitude of p at the mid-point of ith edge of the cell face. The j notation counts the node numbers of an element where the ith cell face is located inside it. Additionally, the variable ξ represents either x or y coordinates and φ (and Φ) represents either u (and U ) or v (and V ) velocity components. As was mentioned, lower and upper case letters are used to represent cell face and nodal magnitudes, respectively. The above approximations end the pressure and diffusion term treatments at all cell faces. However, more sophisticated expressions are required to treat the convection terms. In fact, the treatment should not disregard the convection-diffusion physics and concept. To respect the correct physics of the convection, Reference5, 8 employ an upwind-based scheme (known as a physical influence scheme) within quadrilateral and triangular elements, respectively. The references show that a physical-based treatment of the x-momentum governing equation can result in 4 4 X X βij Pj + γi (26) αij Φj + φi = j=1

j=1

where α, β, and γ represent matrix, matrix, and vector coefficients, respectively. The above statement indicates that φ (≡ u, v) at cell face can be approximated by the proper assemblage of Φ (≡ U, V ) and P influences. In fact, this approximation can be regarded as a pressure-weighted upwind scheme. As is observed in Eq.(26), contrary to the problems raised in the Introduction section, the pressure field is not obtained through solving the pressure Poisson equation in this study. In fact, the continuity preserves its original identification and the direct contribution of the pressure terms are forced through the mass flux statements at the cell faces. The substitution of Eqs. (24)-(26) in Eqs. (21)-(23) provides a set of algebraic equations for each cell. It is given by     p  cpp cpu cpv Pj  di    ij ij ij                        up  uu uv u  c  = (27) c c U d j ij ij    ij i                        v   vp vu vv V d cij cij cij j i

where i and j count the global node numbers, i.e., i, j = 1 . . . N node. It should be mentioned that the matrix is a diagonal one which is normally encountered in implicit-based finite-element methods. Therefore, it is strongly sparse and need sparse solution strategies. The coefficients in the global assembled matrix is identified by c. The first letter in each superscript depicts the type of equation, i.e., p, u and v indicate continuity, x-momentum and y-momentum equations, respectively. The second letter in the superscripts indicates which unknown the coefficient belongs to. The right-hand-side vector is shown by d. Since all U , V , and P unknowns explicitly appear in all the three governing equations, there is strong coupling between the pressure and velocity fields and the pressure checkerboard problem may not occur. In a similar manner, the procedure extended for the fluid flow governing equations can be repeated for the turbulent kinetic energy, Eq.(4), the dissipation rate of turbulent kinetic energy, Eq.(5), enthalpy of 6 of 11 American Institute of Aeronautics and Astronautics Paper 2005-0574

chemical species, Eq.(15), and mass fraction of chemical species, Eq.(14). Similar to the fluid flow governing equations, the aforementioned four governing equations are solved simultaneously. This implicit procedure enhances the stability of the extended method. The consideration of this latter point is so crucial in solving turbulent reacting flow using the current pressure-weighted approach. If we follow the preceding procedure, another set of algebraic equations is derived. The new derived matrix is given by     κκ κ κe κf   cij cij cij cij   Kj  dκi                            κ        f      e  cij cij cij cij      Υ d  j   i     = (28)       eκ   ef   e ee e      c c c c H d  ij  j  ij ij i  ij                               f   fκ f fe ff F d cij cij cij cij j i As was mentioned in the Introduction section, the present algorithm is a bi-implicit one and solves two sets of algebraic equations in each iteration, one for the fluid flow governing equations and the other one for energy, turbulence, combustion, and mass fraction of species.

VI.

The Results

In this section, the new developed procedure is tested in two stages. At the first stage, the chemical reaction switch is off and only the transport of turbulence species is investigated using our new convectiondiffusion model. To examine the extended model, turbulent flow over a backward-facing step is chosen as the test case. The chosen test case is known as a benchmark case which is mainly used to test newly developed numerical algorithms. There is a wide range of either experimental9 or numerical10 solutions for this test case. Figure 2 shows the geometry (not scaled) of the calculation domain and the specified boundary conditions. Following Kim, et al.9 and Sohn,10 a uniform unit velocity is considered at the inlet. The flow Reynolds number based on the step height and inlet velocity is 69610. At the exit, the flow is assumed to be fully developed. The specified wall functions are applied at all solid walls. Different fine grid resolutions have been used to ensure the grid independency of the solution. 0.05 Wall Function 2.0 1.0

Wall Function 0.05

4.0

24.0

Figure 2. Turbulent flow in backward-facing step.

Figure 3 shows the turbulent kinetic energy distributions at two different locations downstream of the step, i.e., x=1.0 and 4.1. The current results are compared with those of Kim, et al.9 and Sohn.10 Despite using general κ −  model, there are good agreement among the presented results. Figure 4 illustrates the mean axial velocity profiles at x=1.33 and 5.33. There are excellent agreements between the current solutions and those of the references. The mean velocity profiles at x=1.33 and 5.33 indicate that there is a recirculation zone behind the step. Kim, et al.9 report that the length of this recirculation is about 6 to 8 following their experimental study. The numerical solution of Sohn10 predicts a length of 5.59 using 7 of 11 American Institute of Aeronautics and Astronautics Paper 2005-0574

2

2

present Sohn Kim

1.5

Y (X=4.1)

Y (X=1)

1.5

1

0.5

0

present Sohn Kim

1

0.5

0

0.01

0.02

0.03

K

0

0.04

0

0.01

0.02

0.03

0.04

K

0.05

0.06

Figure 3. Turbulent kinetic energy distributions at x=1.0 and 4.1 and comparing them with those of Kim, et al.9 and Sohn.10

2.25 2

2

Y (X=1.33)

1.75 1.5

Y (X=5.33)

present Sohn Kim

1.25 1

present Sohn Kim

1

0.75 0.5 0.25 0 -0.5

0 0

U

0.5

1

0

U

0.5

1

Figure 4. The velocity profiles at x=1.33 and 5.33 and comparing them with those of Kim, et al.9 and Sohn.10

standard κ −  model. The present method calculates a length of 5.75. Finally, Fig. 5 shows both mean axial velocity profile and turbulent kinetic energy distribution far downstream of the step at x=7.67 and 8.0, respectively. In this figure, the behavior and conclusions are very similar to those discussed over Figs. 3 and 4. At the second stage, the turbulent reacting flow and the transport of chemical species are investigated. The chosen test case is a confined diffusion flame. The solution domain geometry is observed in Fig. 6. This test case has been widely investigated either experimentally 11 or numerically12 in cylindrical coordinates. Unfortunately, the authors did not find any investigation in Cartesian coordinates for the non-premixed turbulent flame. Therefore, we compare our results with those obtained in cylindrical coordinates. Of course, this can cause some uncertainty in the evaluation of the current results in the validation part. Because of the symmetry, only one half of the physical domain is studied. The turbulent-combustion interaction and the 8 of 11 American Institute of Aeronautics and Astronautics Paper 2005-0574

2

2

present Sohn Kim

1.5

Y (X=8)

Y (X=7.67)

1.5

1

1

0.5

0.5

0

present Sohn Kim

0 0

0.01

0.02

0.03

K

0.04

0.05

0

0.25

0.5

U

0.75

1

Figure 5. Turbulent kinetic energy distributions at x=7.67 and the velocity profiles at x=8.0 and comparing them with those of Kim, et al.9 and Sohn.10

y

D AIR

d1=8 mm d2=11.1 mm d3=28.8 mm R=101.6 mm L=1524 mm

d1 FUEL

d3

d2 L

x

Figure 6. Confined turbulent diffusion flame.

fluctuation of species are not taken into account. Considering the dimensions given in Fig. 6, the boundary conditions for the chosen geometry are defined as 1) y < d3 is the fuel (which is pure methane) inlet with a uniform velocity at this port equal to 20.3 m/s and a uniform temperature of 300 K and 2) d2 < y < d1 is the air inlet with YO2 = 0.21, YN2 = 0.79, a uniform velocity of 34.3 m/s, and an average temperature of 589 K. At this stage of our investigations, we present the results obtained after solving the above confined turbulent methane diffusion flame. Figures 7-9 present the mixture fraction distributions and molar concentrations at three different axial locations of x=0.095, 0.175, and 0.246 m. The agreement between the current solutions and those of references is good. The current solutions are almost over-specified in all cases. This might be because of either considering a rectangular burner geometry instead of an axisymmetry one (which has been investigated by the other workers) or ignoring the turbulent-combustion interaction in our algorithm. Figures 7-9(left) show the molar concentration of species which are determined from the mixture fraction quantities. Generally speaking, As moving forward in the burner channel, the mixture fraction decreases; thus, resulting in a decrease in fuel concentration.

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0.9

0.9

0.8

0.8 0.7

Molar Concentration

Mixture Fraction

0.7

x = 0.095 m

0.6

0.6

CH4 O2 CO2 H2O N2

0.5

0.5 0.4

0.4

Present Elkaim et al. Experimental

0.3

0.3 0.2

0.2

0.1

0.1 0

x = 0.095 m

0

0.02

0.04

0.06

0.08

y (m)

0

0.1

0.025

0.05

y (m)

0.075

0.1

Figure 7. The mixture fraction distributions and the distributions of molar concentration of species and a comparison with those reported in Smoot and Lewis11 and Elkaim, et al.,12 x=0.095 m.

0.9

0.9

0.8

0.8

Molar Concentration

Mixture Fraction

0.7

x = 0.175 m

0.6 0.5

Present Elkaim et al. Experimental

0.4 0.3 0.2 0.1 0

x = 0.175 m

0.7 0.6 CH4 O2 CO2 H2O N2

0.5 0.4 0.3 0.2 0.1

0

0.02

0.04

0.06

y (m)

0.08

0

0.1

0.02

0.04

0.06

y (m)

0.08

0.1

Figure 8. The mixture fraction distributions and the distributions of molar concentration of species and a comparison with those reported in Smoot and Lewis11 and Elkaim, et al.,12 x=0.175 m.

VII.

Conclusion

The Navier-Stokes equations were implicitly solved and the results were used to simulate both turbulent and combustion behavior in a reacting turbulent flow using a bi-implicit algorithm. The implementation of a new pressure-weighted upwinding scheme in a finite element volume context improves the accuracy and performance of the extended algorithm. The fast chemistry assumption is used in our combustion model. Two test cases including the turbulent flow in a backward-facing step and the confined turbulent diffusion flame were used to validate the extended algorithm. The results perform very good agreement with the experimental data and other numerical solutions. The results of chemical simulation indicates that the physical model has to be improved in order to enhance the overall accuracy of the algorithm.

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0.9

0.8

0.8

0.7

0.7

Molar Concentration

Mixture Fraction

0.9

x = 0.246

0.6 0.5 0.4

Present Elkaim et al. Experimental

0.3 0.2 0.1 0

x = 0.246

0.6 CH4 O2 CO2 H2O N2

0.5 0.4 0.3 0.2 0.1

0

0.02

0.04

0.06

y (m)

0.08

0

0.1

0.02

0.04

0.06

y (m)

0.08

0.1

Figure 9. The mixture fraction distributions and the distributions of molar concentration of species and a comparison with those reported in Smoot and Lewis11 and Elkaim, et al.,12 x=0.246 m.

References 1 Schneider, G.E., and Raw, M.J., “Control Volume Finite Element Method for Heat Transfer and Fluid Flow Using Colocated Variables - 1.Computational Procedure,” Numerical Heat Transfer, Vol.11, No.4, 1987, pp.363-390. 2 Darbandi, M. and Banaeizadeh, A., “Parallel Computation of the Navier-Stokes Equations Using Implicit Finite Volume Method,” Proceedings of the 4th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2004, Jyvaskyla, Finland, July 24-28, 2004. 3 Darbandi, M., and Schneider, G.E., “Analogy-Based Method for Solving Compressible and Incompressible Flows,” Journal of Thermophysics and Heat Transfer, Vol.12, No.2, 1998, pp.239-247. 4 Darbandi, M., and Bostandoost S.M., “A New Formulation Toward Unifying the Velocity Role in Collocated Variable Arrangement,” Numerical Heat Transfer, Part B Vol.47, 2005, in press. 5 Darbandi, M., Banaeizadeh, A., and Schneider, G.E., “Parallel Computation of a Mixed Convection Problem Using FullyCoupled and Segregated Algorithms,” Paper no. ASME-IMECE 2004-61879, ASME International Mechanical Engineering Congress, Anaheim, California, USA, Nov. 14-19, 2004. 6 Kuo, K.K., Principles of Combustion John Wiley & Sons, New York, 1987. 7 Elkaim, D., Reggio, M., and Camarero, R., “Numerical Solution of Reactive Laminar Flow by a Control-Volume Based Finite-Element Method and the Vorticity-Streamfunction Formulation,” Numerical Heat Transfer, Part B, Vol. 20, 1991, pp.223-240. 8 Darbandi, M., Mazaheri-Body, K., and Vakilipour, S., “A Pressure-Weighted Upwind Scheme in Unstructured FiniteElement Grids,” in Numerical Mathematics And Advanced Applications, Editted by M. Feistauer, V. Dolejsi, P. Knobloch, and K. Najzar, Springer-Verlag, 2004, pp.250-259. 9 Kim, J., Kline, S.J., and Johnston, J.P., “Investigation of a Reattaching Turbulent Shear Layer: Flow over a BackwardFacing Step,” Journal of Fluids Engineering, ASME Trans., Vol.102, 1984, pp.711-724. 10 Sohn, J.L., “Evaluation of FIDAP on Some Classical Laminar and Turbulent Benchmarks,” International Journal for Numerical Methods in Fluids, Vol.8, 1988, pp.1469-1490. 11 Smoot, J.L., and Lewis, H.M., “Turbulent Gaseous Combustion: Part I, Local Species Concentration Measurements,” Combustion and Flame, Vol. 42, 1981, pp.183-196. 12 Elkaim, D., Reggio, M., and Camarero, R., “Control Volume Finite-Element Solution of A Confined Turbulent Diffusion Flame,” Numerical Heat Transfer, Vol. 23, 1993, pp.259-279.

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