Implicit Finite Volume Method to Simulate Reacting Flow

Transcription

1 43rd I erospace Sciences Meeting and Ehibit, -3 Jan. 5, Reno, Nevada Implicit Finite Volume Method to Simulate Reacting Flow Masoud Darbandi and raz Banaeizadeh Sharif University of Technology, Tehran, P.O. Bo , Iran Gerry E. Schneider University of Waterloo, Waterloo, Ontario, NL 3G, Canada In this work, an efficient bi-implicit strategy is suitably developed within the contet 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 fleibility 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 approimate the advection flues 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 ɛ turbulence model and one step miture 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 eperimental data and other available numerical results. I. Nomenclature B body force C pi specific heat of species i at constant pressure D diffusion coefficient f miture fraction h enthalpy h F heat reaction N finite element shape function n total number of species P pressure R gas constant T temperature u, v velocity components ū friction velocity V velocity vector ssociate Professor, Department of erospace Engineering. Graduate Student, Department of erospace Engineering. Professor and Chair, Department of Mechanical Engineering, I Fellow. Copyright c 5 by M. Darbandi. Published by the merican Institute of eronautics and stronautics, Inc. with permission. of

2 ū friction velocity, y Cartesian coordinates Y i mass fraction of species i z a reative of the etensive quantities Υ dissipation rate of turbulent kinetic energy at nodes ɛ dissipation rate of turbulent kinetic energy κ turbulent kinetic energy µ laminar viscosity µ e effective viscosity µ t turbulent viscosity ρ miture density σ viscous shear tensor. Subscripts i p st chemical species the grid point adjacent to the solid wall stoichiometric condition B. Superscripts + non-dimensional magnitudes II. Introduction The reacting flow in combustion chambers mostly occurs in comple geometries which in turn need robust numerical tools to treat the resulting comple turbulent reactive flow fields. Finite-difference-based methods do not straightforwardly provide efficient means to solve the flow fields occurred in comple geometries. On the contrary, the finite element methods provide easy treatment of complicated domains. The past eperience has shown that a robust approach toward handling the comple geometries is the finitevolume-based finite-element one. 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. In this work, the primitive implicit strategy introduced by Darbandi and Schneider 3 is suitably etended in order to solve the turbulent reactive flow governing equations. Back to the primary problem with the continuity equation, Darbandi and Bostandoost 4 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 7-diagonal matri in treating the D Navier-Stokes equations. Darbandi, et al. 5 etended 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, -momentum and y-momentum are solved implicitly in one matri and κ, ɛ, miture fraction and energy equations in another matri. 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 of

3 equations are written as (ρ V ) = () (ρ V u) = p + (µ e u) () (ρ V v) = p y + (µ e v) (3) where µ e is the effective viscosity defined by µ e = µ + µ t. The eddy turbulent viscosity µ t is calculated from the turbulent kinetic energy κ equation and its dissipation rate ɛ equation. The transport equations for these two components are written as (ρ V κ) = ( µ e σ κ κ) + (Gκ ρɛ) (4) (ρ V ɛ) = ( µ e σ ɛ ɛ) + ɛ κ (c G κ c ρɛ) (5) In these equations, G κ and c G κ are the production terms and ρɛ and c ρɛ are the destruction terms. The epression for G κ is given by G κ = µ t {[( u ) + ( v ) ] + [ u y + v ] } (6) The derived κ and ɛ magnitudes from Eqs. (4)-(5) are used to calculate the eddy viscosity definition, i.e., µ t = c d ρ κ ɛ The constants in κ and ɛ equations are σ κ =., σ ɛ =.3, c =.44, c =.9, and c d =.9. s 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 etra considerations. Depending on the flow Reynolds number, there are different choices such as low Reynolds number model and wall function approimation to cure the problem. In the work, the wall function choice is used to describe the flow close to the wall. The elliptic nature of κ and ɛ transport equations directs the users to specify boundary conditions at all domain boundaries. s was mentioned, we apply wall function at solid walls. Indeed, the region close to the solid walls can be divided into two layers ) a laminar sublayer or viscous sublayer where viscous effects are dominant and ) 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 κ p = ɛ p = (7) ū cd (8) ū3 k y p (9) where c d and k constants are.9 and.4, respectively. y p is the actual distance between node p and wall and ū is the friction velocity. To specify the friction velocity at node p, the non-dimensional distance from the solid wall is defined as y p + = ρūy p /µ. Using this definition, the non-dimensional velocity u + is determined from { y p +.63 u + p = k ln y+ p () y p + >.63 u + p = y p + 3 of

4 Then, the friction velocity is derived from ū p = u p u + p 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 = (Kg/s) for the air/fuel miture, it consists of a fuel rate of f(kg/s) and an air rate of ( f)(kg/s). Considering the above mass flow rates, any etensive property, such as z, of the miture resulting from the mi of these two streams can be written as () f = z M z z F z () If F indicates the fuel and indicates the oidizer, the miture fraction yields f = [Y F (F/O) st Y O ] M + (F/O) st Y O, + (F/O) st Y O, (3) From the chemical equilibrium assumption, no oidant s if a miture fraction richer than stoichiometric condition is employed. dditionally, no fuel s if the miture 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 miture fraction equation and the enthalpy can be calculated directly from the miture fraction and the inlet enthalpy value. Thus, for all combustion related variables such as enthalpy, mass fraction of fuel, oygen, and the products, it is sufficient to solve the transport equation for miture fraction f. 7 The transport equation for f is then given by (ρ V f) + (D f) = (4) ssuming a unit Lewis number, the reactive energy equation is reduced to where the enthalpy is defined by h = Y F h F + (ρ V h) = (µ h) (5) n i= T T ref Y i C pi (T )dt + V (6) Eventually, the density is calculated from the equation of state which is written as P = ρrt n Y i W i (7) IV. The Domain Discretization The solution domain is broken into a huge number of quadrilateral elements. The elements fully cover the solution domain with no overlapping. Figure 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. 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. s 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

5 Integration Point Control Volume Element Node t s y Figure. part of the solution domain illustrating four elements, one complete finite volumes, siteen 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.. The employment of Gauss divergence theorem to the governing equations leads to ρv d = (8) u(ρv ) d = v(ρv ) d = p d + p d y + µ e ( u d) (9) µ e ( v d) () The above integrals are evaluated over the surface which encloses each cell. The surface area is indicated by. The above equations are suitably discretized using finite difference scheme and finite element interpolations. In the above epressions, d=d i d y 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 ns ns ns ns [ρû u d + ρˆv u d y )] i = (p d ) i + [ρ(u d + v d y )] i = () ns ns [ρû v d + ρˆv v d y )] i = (p d y ) i + [ ( u µ e d + u )] y d y i [ ( v µ e d + v )] y d y where i counts the number of cell faces from to ns. There are 8 cell faces around non-boundary cells. To linearize the governing equations, the hat over û and ˆv indicates that these velocity components are approimated from the known magnitudes of the preceding iteration. Such approimation is essential to linearize the nonlinear momentum convection terms. The rest of procedure is to relate the cell face magnitudes i () (3) 5 of

6 (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 re the locations for the unknown variables in the current algorithm. simple idea for treating the right-hand-side terms is to use the finite element shape functions N j=...4. The treatment results in p i = φ ξ = i 4 N P j (4) j= 4 j= N ξ Φ j (5) where p i 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. dditionally, the variable ξ res either or y coordinates and φ (and Φ) res either u (and U) or v (and V ) velocity components. s was mentioned, lower and upper case letters are used to re cell face and nodal magnitudes, respectively. The above approimations end the pressure and diffusion term treatments at all cell faces. However, more sophisticated epressions 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, Reference 5, 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 -momentum governing equation can result in 4 4 φ i = α Φ j + β P j + γ i (6) j= where α, β, and γ re matri, matri, and vector coefficients, respectively. The above statement indicates that φ ( u, v) at cell face can be approimated by the proper assemblage of Φ ( U, V ) and P influences. In fact, this approimation can be regarded as a pressure-weighted upwind scheme. s is observed in Eq.(6), 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 flu statements at the cell faces. The substitution of Eqs. (4)-(6) in Eqs. ()-(3) provides a set of algebraic equations for each cell. It is given by c pp c up c vp c pu c uu c vu c pv c uv c vv P j U j V j j= = where i and j count the global node numbers, i.e., i, j =... Nnode. It should be mentioned that the matri 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 matri is identified by c. The first letter in each superscript depicts the type of equation, i.e., p, u and v indicate continuity, -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 eplicitly 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 etended 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 d p i d u i d v i (7) 6 of

7 chemical species, Eq.(5), and mass fraction of chemical species, Eq.(4). Similar to the fluid flow governing equations, the aforementioned four governing equations are solved simultaneously. This implicit procedure enhances the stability of the etended 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 matri is given by c κκ c ɛκ c eκ c fκ c κɛ c ɛɛ c eɛ c fɛ c κe c ɛe c ee c fe c κf c ɛf c ef c ff K j Υ j H j F j = s was mentioned in the Introduction section, the 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. d κ i d ɛ i d e i d f i (8) VI. The Results In this section, the new developed procedure is tested in two stages. t the first stage, the chemical reaction switch is off and only the transport of turbulence species is investigated using our new convectiondiffusion model. To eamine the etended 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 eperimental 9 or numerical solutions for this test case. Figure shows the geometry (not scaled) of the calculation domain and the specified boundary conditions. Following, et al. 9 and, a uniform unit velocity is considered at the inlet. The flow Reynolds number based on the step height and inlet velocity is 696. t the eit, 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..5. Wall Function Wall Function 4. Figure. Turbulent flow in backward-facing step. Figure 3 shows the turbulent kinetic energy distributions at two different locations downstream of the step, i.e., =. and 4.. The current results are compared with those of, et al. 9 and. Despite using general κ ɛ model, there are good agreement among the ed results. Figure 4 illustrates the mean aial velocity profiles at =.33 and There are ecellent agreements between the current solutions and those of the references. The mean velocity profiles at =.33 and 5.33 indicate that there is a recirculation zone behind the step., et al. 9 report that the length of this recirculation is about 6 to 8 following their eperimental study. The numerical solution of predicts a length of 5.59 using 7 of

8 .5.5 Y (X=) Y (X=4.) K Figure 3. Turbulent kinetic energy distributions at =. and 4. and comparing them with those of, et al. 9 and. K.5 Y (X=.33) Y (X=5.33).75 - U Figure 4. The velocity profiles at =.33 and 5.33 and comparing them with those of, et al. 9 and. U standard κ ɛ model. The method calculates a length of Finally, Fig. 5 shows both mean aial velocity profile and turbulent kinetic energy distribution far downstream of the step at =7.67 and 8., respectively. In this figure, the behavior and conclusions are very similar to those discussed over Figs. 3 and 4. t 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 eperimentally or numerically in cylindrical coordinates. Unfortunately, the authors did not find any investigation in Cartesian coordinates for the non-premied 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

9 .5.5 Y (X=7.67) Y (X=8) K.75 U Figure 5. Turbulent kinetic energy distributions at =7.67 and the velocity profiles at =8. and comparing them with those of, et al. 9 and. y IR D d=8 mm d=. mm d3=8.8 mm R=.6 mm L=54 mm FUEL d3 d d L 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 ) y < d3 is the fuel (which is pure methane) inlet with a uniform velocity at this port equal to.3 m/s and a uniform temperature of 3 K and ) d < y < d is the air inlet with Y O =., Y N =.79, a uniform velocity of 34.3 m/s, and an average temperature of 589 K. t this stage of our investigations, we the results obtained after solving the above confined turbulent methane diffusion flame. Figures 7-9 the miture fraction distributions and molar concentrations at three different aial locations of =.95,.75, and.46 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 aisymmetry 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 miture fraction quantities. Generally speaking, s moving forward in the burner channel, the miture fraction decreases; thus, resulting in a decrease in fuel concentration. 9 of

10 =.95 m Miture Fraction =.95 m Present Elkaim et al. Eperimental Molar Concentration CH4 O CO HO N Figure 7. The miture fraction distributions and the distributions of molar concentration of species and a comparison with those reported in Smoot and Lewis and Elkaim, et al., =.95 m =.75 m Miture Fraction =.75 m Present Elkaim et al. Eperimental Molar Concentration CH4 O CO HO N Figure 8. The miture fraction distributions and the distributions of molar concentration of species and a comparison with those reported in Smoot and Lewis and Elkaim, et al., =.75 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 contet improves the accuracy and performance of the etended 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 etended algorithm. The results perform very good agreement with the eperimental 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. of

11 .9.9 Miture Fraction =.46 Present Elkaim et al. Eperimental Molar Concentration =.46 CH4 O CO HO N Figure 9. The miture fraction distributions and the distributions of molar concentration of species and a comparison with those reported in Smoot and Lewis and Elkaim, et al., =.46 m. References Schneider, G.E., and Raw, M.J., Control Volume Finite Element Method for Heat Transfer and Fluid Flow Using Colocated Variables -.Computational Procedure, Numerical Heat Transfer, Vol., No.4, 987, pp Darbandi, M. and Banaeizadeh,., Parallel Computation of the Navier-Stokes Equations Using Implicit Finite Volume Method, Proceedings of the 4th European Congress on Computational Methods in pplied Sciences and Engineering, ECCOMS 4, Jyvaskyla, Finland, July 4-8, 4. 3 Darbandi, M., and Schneider, G.E., nalogy-based Method for Solving Compressible and Incompressible Flows, Journal of Thermophysics and Heat Transfer, Vol., No., 998, pp Darbandi, M., and Bostandoost S.M., New Formulation Toward Unifying the Velocity Role in Collocated Variable rrangement, Numerical Heat Transfer, Part B Vol.47, 5, in press. 5 Darbandi, M., Banaeizadeh,., and Schneider, G.E., Parallel Computation of a Mied Convection Problem Using Fully- Coupled and Segregated lgorithms, Paper no. SME-IMECE , SME International Mechanical Engineering Congress, naheim, California, US, Nov. 4-9, 4. 6 Kuo, K.K., Principles of Combustion John Wiley & Sons, New York, 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., 99, pp Darbandi, M., Mazaheri-Body, K., and Vakilipour, S., Pressure-Weighted Upwind Scheme in Unstructured Finite- Element Grids, in Numerical Mathematics nd dvanced pplications, Editted by M. Feistauer, V. Dolejsi, P. Knobloch, and K. Najzar, Springer-Verlag, 4, pp , J., Kline, S.J., and Johnston, J.P., Investigation of a Reattaching Turbulent Shear Layer: Flow over a Backward- Facing Step, Journal of Fluids Engineering, SME Trans., Vol., 984, pp.7-74., J.L., Evaluation of FIDP on Some Classical Laminar and Turbulent Benchmarks, International Journal for Numerical Methods in Fluids, Vol.8, 988, pp Smoot, J.L., and Lewis, H.M., Turbulent Gaseous Combustion: Part I, Local Species Concentration Measurements, Combustion and Flame, Vol. 4, 98, pp Elkaim, D., Reggio, M., and Camarero, R., Control Volume Finite-Element Solution of Confined Turbulent Diffusion Flame, Numerical Heat Transfer, Vol. 3, 993, pp of