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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Prediction of Mixing and Reacting Flow Inside a Combustor
A. L. De Bortoli *
Department of Pure and Applied Mathematics, UFRGS, Porto Alegre, PO Box 15080, Brazil
Email: [email protected]
Abstract The aim of this work is the numerical solution of mixing and reacting flow inside a combustor where the fuel
and the air enter it in separate streams. The model approximates the overall and irreversible reaction between two
species and the reaction-rate is controlled by temperature-dependent Arrhenius kinetics. Numerical tests, for governing
equations discretized by the finite difference explicit Runge-Kutta five-stage scheme based on fourth order space and
third order time approximations, are carried out for Reynolds 10000, Damk¨ohler 300, Zel'dovich 10, Heat Release 10
and Prandtl and Schmidt values both equal to 0.7. The results contribute to obtain a better understanding of the
mixture-reaction behaviour which is common in combustors.
Key words: diffusion flame, combustor, finite difference, Runge-Kutta
INTRODUCTION
Chemical-reacting flows are characterised by rapid and exothermic reactions. Such rapid, exothermic and complex
convective-diffusive-reactive process presents many scales which are related to velocity, length, time, energy and
vorticity. While big scales are geometry dependent, the small scales depend on the energy dissipation process [1].
The field of combustion is usually divided into non-premixed and premixed combustion. The majority applications of
technical interest are classified as non-premixed and turbulent. Therefore, in practice, the fuel and the oxidiser are not
perfectly mixed before burning, then the combustion process turns more pollutant formation (less efficient).
Mixing is intensified by flame-vortex interactions [2]; the heat release distribution exerts a significant influence on the
flame evolution and on turbulence. To understand the turbulent field it is necessary an accurate prediction of the
turbulent velocity field.
Turbulent mixing plays an important role in non-premixed combustion. It changes the density, temperature, heat
capacity, molar mass and also the mixture transport properties [3]. The mixing can be divided in, at least, three levels:
The first can be considered that where dispersion and mixing are driven by the turbulent flow, so correct mixing is not
required to describe the flow dynamics. In the intermediate level the mixing is coupled to the fluid dynamics; therefore,
the mixing must be captured correctly. The third level occurs when the mixing produces changes to the fluid: at its
composition, density, enthalpy and pressure, like in the combustion phenomena; therefore, at this level the mixing
must be captured correctly [3].
Turbulence effects, for moderate to high Reynolds values, seems to be best modelled using Large-Eddy Simulation
(LES), where the eddy resolution is determined directly by the grid. Reynolds Averaged Navier-Stokes (RANS) is
employed to steady flows, while LES and Direct Numerical Simulation (DNS) are preferred to unsteady flows; the last
technique is the most expensive and the grid resolution is usually not fine enough to resolve the internal structures of
the flame. Using LES the big scales are captured while the unresolved subgrid scales need to be modelled. Usually
RANS combustion models are adapted to LES modelling.
Moreover, combustion models may turn very complex. The complete reaction mechanism of the methane-air, for
example, involves over 30 chemical species and more than 300 elementary reactions [4]. For iso-octane oxidation there
appear 3600 elementary reactions and 860 chemical species [5]. Therefore, reduced reaction mechanisms are usually
employed to analyse combustion due to difficulties to establish general chemical kinetics models.
This work develops a numerical technique for the solution of mixing and reacting flow inside a combustor where the
fuel and the air enter it in separate streams, as shown in Fig. 1. The numerical model is based on the finite difference
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explicit Runge-Kutta five-stage scheme for third order time and fourth order space approximations. Numerical tests are
carried out for Reynolds Re = 10000, Damköhler Da = 300, Heat Release He = 10, Zel'dovich Z = 10 and Prandtl and
Schmidt both equal to 0.7. Such non-dimensional values are assumed to be the same and constant for all species.
Figure 1: Burner geometry
GOVERNING EQUATIONS AND SOLUTION PROCEDURE
The model considers the following hypothesis: the Mach number is low, the pressure remains almost constant and the
heat losses to the walls are negligible [6, 1].
Simulations are based on the two-dimensional set of reacting Navier-Stokes equations. Although the flow is
threedimensional in nature, much insight is gained when analysing a two-dimensional situation, which allows to obtain
a better understanding of this complex mixing and reacting flow with reduced computational costs.
Moreover, the model assumes the finite rate Arrhenius kinetics hypothesis
Contribution arising from radiation is considered negligible; it is more important for large flames as in furnaces or in
building or wildland fires [7].
For large density variations it is convenient [8] to introduce the density weighted filtered quantities, which are defined
. The Favre filtered reacting Navier-Stokes equations are, for the problem of interest, given by [9]:
by
Mass Conservation
Momentum
Energy
Chemical Species
where Re is the Reynolds, Da the Damköhler, Sc the Schmidt, Pr the Prandtl, Z the Zel'dovich and He the heat release
is the averaged or filtered reaction rate of species,
the filtered
parameter;
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stress tensor and
velocities and the temperature
comes from the state relation
the mean resolved strain rate. The incognits are the mixture density, , the fluid
, the chemical specie mass fractions
and
and the pressure
which
Viscosity is assumed to be temperature dependent and is approximated as [8, 10].
The sub-grid terms are conveniently modelled with an eddy viscosity as
where
is the eddy viscosity and C =0:1 the Smagorinsky constant. There are many subgrid
models, but it seems that there is no established model for all mixing and reacting flow situations. The Smagorinsky
model with a Van Driest Damping function has some drawbacks, but helps to simplify the analysis.
Observe that the governing equations and the boundary conditions need to be high order approximated; it is common
practice to employ fourth to six order for doing that. When the central finite difference scheme is employed, for a
general finite difference grid point (i, j) in the computational domain, results the conventional fourth order space
derivative approximations for a generic variable
and similarly for other derivatives [11].
Integration in time is performed using a third order low storage Runge-Kutta time-stepping scheme. After these
approximations results for the species mass fraction (Eq. 5), for example,
with
and in a similar manner for the other terms and derivatives.
The use of fourth order space approximation is normally sufficient for many mixing and reacting flow situations.
Certainly, such order is required not only for the interior domain, but also for the boundary conditions.
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BOUNDARY CONDITIONS
For the geometry definition the virtual boundary technique is employed. The great advantage of using this technique is
its ability to model geometries without the necessity of coordinates transformation, using simple Cartesian grids [12].
Although the uniform Cartesian grid allow considerable simplification compared to the boundary fitted technique,
there remains the necessity to find the best quantity of grid nodes to represent a great variety of length and time scales
present in the flow near to the walls of the combustor.
Consider the burner as shown in Fig. 1. The fuel and the air enter the domain in separate streams and the heat releasing
reactions are considered to be fast. Observe that the physical boundary conditions can only be related to incoming
waves while waves travelling from the inside of the domain to the outside are completely determined by the interior
field [13]. LES computations have already shown that the flow can be sensitive to the outflow conditions [14].
Therefore, the boundary conditions can be summarised in non-dimensional form as follows:
For south and north boundaries (for n the normal to the combustor surface):
For west boundary:
except at fuel injection place where
= 1 (parabolic),
and
.
For east boundary:
; the pressure
is set equal to 1. Perfect
where uc is the convective velocity and
non-reflecting boundary ctonditions at the outlet are not possible because a reflected wave is produced to bring p to
at the outlet [1].
when p is different from
NUMERICAL RESULTS
A jet is formed by admitting that a fuel stream enters an aired environment with a parabolic velocity profile, as
indicated in Fig. 1. The computational domain corresponds to a rectangle and the grid contains 133×69 points; the
time-step was chosen to be 10−6. Before solving the flow inside this combustor, some results were obtained for a
simplified combustor geometry [9].
Fig. 2 left shows the velocity map based on Cp formation for Z = 10, Da = 300, He = 10 and Re = 10000. As it is known,
the structure of the diffusion flame depends on the time needed to consume the reactants [15], that means on the
Damköhler value.
This flow turns rapidly asymmetric; such turns evident when analysing the velocity profiles for each combustor cross
section. After 0:4s travelling waves are seen reflecting inside the combustor; between 1.6 and 2.8 s the central jet
diffuses and oscillates, increasing the mixing, and after 4 s the flow turns turbulent. Fig. 2 right shows the product
formation contours till 4 seconds. Observe that the central jet turns unstable, increasing the mixing rapidly. This figure
confirms that the inflow and outflow boundary conditions do not introduce any significant spurious perturbation to the
interior domain, as desired.
Fig. 3 left presents the velocity profiles inside the combustor at position x =3L=4 and Fig. 3 right at position x =L =1.
Observe that after 2s the jet of fuel reaches the combustor position x = 3L=4 and the component u remains growing till
3.0 s; after it the velocity profile tends to oscillate considerably.
At position x = L, on the other hand, the gas expansion is best seen. During the first 1.0 s the gas expand almost 50%
and till 4s this mass expand more than 100%; the velocity profile changes considerably from 1.0 to 4.0 s. Such
behaviour turns interesting, but time-consuming to be predicted with the increase of the Reynolds value. It is clear that,
after 1.0s, the principal jet distorts inside the combustor because the vortices are able to change the mean flow
configuration.
Finally, Fig. 4 shows the velocity components fluctuations at position (x = L=2; y = H=4) inside the combustor. This
picture clearly indicate the flow unpredicability, as because such components change value not obviously in a short
period of time.
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Figure 2: Velocity map (left) and product contours (right) for the flow inside the combustor till 4s
Re = 10000, Da = 300, Z = 10 and He = 10
Figure 3: Velocity component u at position x = 3L=4 (left), at position x = L (right) till 4 s
Figure 4: Velocity component fluctuations at position x = L=2, y = H=4
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CONCLUSIONS
A computational model is developed for coupling fluid dynamics, chemistry and heat transfer for the mixing and
reacting flow inside a combustor. Computations are performed to obtain a better physical understanding of this
mixture/reaction behaviour. Consistent results are obtained showing that the model is able to follow the non linear
behaviour of the mixing and reacting progress for reasonable values for gaseous hydrocarbon chemistry.
Besides, it is shown that the mixture increases when starting the reaction. After around 1 s the jet turns unstable and the
recirculations diffuse the central jet. These vortices are small and they decrease size with the increase of the Reynolds
value. It is clear that the asymmetry of the axial velocity profile along the duct centerline is a consequence of the
diffusion which is by the flow expansion accelerated.
In conclusion, the numerical results obtained here indicate that there are many interesting physical phenomena
associated with mixing and reacting flows which are untill now not totally understood. Such understanding is essential
to have real progress in the solution of turbulent combustion whose applications in aircraft engines or jet engines are
obvious.
Acknowledgements
The author gratefully acknowledge financial support from CNPq (Conselho Nacional de Desenvolvimento Científico e
Tecnológico) under process 310010/2003-9.
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