Conference on Turbulence and Interactions TI2006, May 29 - June 2, 2006, Porquerolles, France
MODELING INTERACTION OF TURBULENCE WITH PREMIXED COMBUSTION USING A JOINT PDF APPROACH P. Jenny, B. Rembold Institute of Fluid Dynamics, ETH Z¨urich, Switzerland Email:
[email protected] ABSTRACT In this paper, a new model for premixed turbulent combustion is presented. The approach is based on a joint velocity probability density function (PDF) method and a progress variable, which is transported by notional particles. Compared with other methods employing progress variables, the advantage here is that turbulent mixing of the progress variable requires no modeling. Moreover, by applying scale separation, the Lagrangian framework allows to account for the embedded, quasi laminar flame structure in a very natural way. The numerical results presented here are based on a simple closure of the progress variable source term and it is demonstrated that the new modeling approach is robust, internally consistent and shows the correct qualitative behavior.
I NTRODUCTION
Premixed turbulent combustion is central for many engineering applications like internal combustion engines and gas turbines. It is also recognized that accurate and reliable modeling of the related physical processes and their interaction is extremely difficult and currently, there exists no satisfactory approach which is general. In many cases, the probability density function (PDF) of a progress variable c can be approximated by a distribution, which consists of two Dirac peaks. Such a PDF is used by the Bray-Moss-Libby (BML) model [1] and implies that the flame is extremely thin. With such a presumed shape the PDF is specified by the Favre mean value of c, which can be obtained by solving ′′ ′′ c 1 ∂hρiug 1 ∂˜ c ˜ ∂˜ ic + Ui =− + hωc i, ∂t ∂xi hρi ∂xi hρi
(1)
where the operators ˜·, h·i and ·′′ denote Favre averaged, Reynolds averaged and Favre fluctuating quantities, respectively. Moreover, U is the fluid velocity, ρ the density and hωc i the mean progress variable source term. A physical interpretation of Eq. (1) is given in [7]. The difficulties are the closures of the two terms on the right-hand side, i.e. of turbulent transport and production of c. More recently, flamelet models were introduced for premixed combustion [7]. This approach is based on a level set method, in which an equation for the scalar G is solved. The role of G is to determine the flame position and together with a 1D flamelet solution the embedded, quasi laminar profile can be resolved. For turbulent combustion, however, the closure of the G-equation remains a major challenge. Another approach is based on solving a joint PDF transport equation [9]. Crucial advantages are that the reaction source term and turbulent convection, which includes turbulent mixing, appear in closed form. However, ac-
curate and general modeling of molecular mixing and its coupling with chemical reactions proved to be extremely difficult. Here, we present a new approach, which is a combination of a hybrid joint PDF method [3,4,6,10], a transported progress variable and the flamelet model. In the following two sections the approach is described and numerical results are presented.
M ODELING A PPROACH
Each computational particle used by the joint PDF solution algorithm has a property c∗ ∈ {0, 1}, indicating whether it has been reached by an embedded quasi laminar flame or not. The relative position of the particle in a pre-computed flame profile can be determined based on the time τ ∗ , which is the time elapsed since c∗ switched from zero to one (see Fig. 1). Note that unlike in T Tb
Tu c=0
c=1 τ> 0
x
Fig. 1. Sketch of a 1D quasi laminar premixed flame profile; shown are the critical temperature Tc below and above which c is zero and one, respectively.
the BML model it is not assumed that the flame is infinitely thin. We denote the probability that c∗ switches from zero to one by F dt, where dt is an infinitesimal time period and F has to be modeled. One then can derive the mean transport equation
′′ ′′ ∂˜ c ˜ ∂˜ c 1 ∂hρiug ic + Ui =− + (1 − c˜)F, ∂t ∂xi hρi ∂xi
(2)
which is equivalent to Eq. (1) with hωc i = hρi(1 − c˜)F . However, a crucial advantage compared with the original BML model [2,5] is that ′′ ′′ no modeling for the turbulent flux, ug i c , is reg ′′ quired. This is due to the fact that ui c can directly be obtained from the mass weighted PDF of U and c, for which a transport equation is solved (where no model is required for spatial transport due to velocity fluctuations). Closure of the source term is achieved by specifying the transition probability F dt, e.g. as
F dt = hci
sL I0 dt, ˆ L
(3)
where sL is the propagation velocity of the unstretched laminar flame, I0 = 0.117Ka−0.784 hu /hb the stretch factor and ˆ = cl lt (sL /urms )n the wrinkling length scale. L This specification for F is consistent with the BML closure given in [8]. Ka is the Karlovitz number, lt the integral turbulent length scale, urms = (2k/3)1/2 , and hu /hb the enthalpy ratio of the burned and unburned stochiometric mixture. The constants n and cl are of order unity and k is the turbulent kinetic energy. Note that the Karlovitz number can be estimated from the dissipation rate, viscosity and flame thickness. Note that other closures for F (i.e. for hωc i) can be found in the literature.
N UMERICAL E XPERIMENT
As a testcase we consider the simulation of a turbulent Bunsen flame with the intention to demonstrate that the modeling approach is robust and produces qualitatively correct results. A sketch of the Bunsen flame is shown in Fig. 2. The size of the computational domain (only one symmetry half) is 0.1 × 0.3 (all units are SI) and the
x2
3.9 × 105
0.51
grid consists of 44 × 88 cells. An average of 20 particles per cell was used. All simulations were performed with our PDF simulation plattform LEMSBK [10]. Peak values in the inflow velocity profile are 40 in the jet and 10 in the co-flow region. Moreover, a mixture fraction, Z, was introduced, in order to distinguish between jet (Z = 1) and co-flow (Z = 0) streams. Other quantities ˜1 /8| (i = defined at the inflow are ui,rms = |U ′′ ′′ 1, 2, 3), |ug 1 u2 | = 0.4(u1,rms u2,rms ) (negative for ˜ ∂ U1 /∂x2 > 0 and positive for ∂ U˜1 /∂x2 < 0) and the turbulence frequency, which is determined based on assuming equilibrium between turbulence production and dissipation. Since the goal of this numerical experiment was to investigate robustness and qualitative behavior of the model, a very simple formulation for F was employed, i.e. F = 5000˜ c. The flame was ignited by initially setting c˜ = 1 in a small area of the expected flame brush. Figs. 3a and b show the density and enthalpy distributions, respectively (dark denotes low values), where the V shaped flame can be ′′ ′′ observed. The Reynolds stress ug 1 u2 is shown in
(a)
(b)
Fig. 3. Results for Bunsen flame; (a): hρi (increment 0.14); (b): hhi (increment 9.9 × 104 ).
thermore, in Figure 5a, the turbulent frequency ω ˜ is depicted. Its peak values can be found in the incoming shear layer. Figure 5b finally shows the ] conditional mean mixture fraction, z| c=1 , conditioned on c = 1. This statistical quantity can be easily extracted due to the Lagrangian model framework.
c˜ = 0.5
exemplary particle trajectory
0 1 1 0 0 1 0 1 0 1 0 1
premixed gas Z=1, c=0
0.02
0 1 1 0 0 1 0 1 0 1 0 1
air Z=0, c=0
x1
0.17
air Z=0, c=0
−6.7
instantaneous flame position
Fig. 2. Sketch of a turbulent bunsen flame with flame separating burnt from unburnt region.
(a) (b) Fig. 4a. I can be seen how the turbulence inten′′ ′′ ] sity first decreases near the inflow before more Fig. 4. Results for Bunsen flame; (a): u1 u2 (increment 3.5); (b): c˜ (increment 0.17). turbulence is produced in the shear layer, which is a result of fluid expansion. The mean of the progress variable, c˜, is plotted in Figure 4b. Fur-
[4] P. Jenny, S. B. Pope, M. Muradoglu, and D. A. Caughey. A hybrid algorithm for the joint PDF equation of turbulent reactive flows. J. Comp. Phys, 166-2:218–252, 2001.
0.17
248.4
body stabilized flow. J. Comp. Phys, 169-1:1– 23, 2001.
[5] P. A. Libby and F. A. Williams. . In Turbulent Reacting Flows, pages 1–61. Academic Press, London, 1994.
(a)
(b)
Fig. 5. Results for Bunsen flame; (a): ω ˜ (increment ] 248.4); (b): z| (increment 0.17). c=1
C ONCLUSIONS
[6] M. Muradoglu, P. Jenny, S. B. Pope, and D. A. Caughey. A consistent hybrid finitevolume/particle method for the PDF equations of turbulent reactive flows. J. Comp. Phys, 154:342–371, 1999. [7] N. Peters. Turbulent Combustion. Cambridge University Press, 2000. [8] T. Poinsot and D. Veynante. Theoretical and Numerical Combustion. Edwards, Philadelphia, 2 edition, 2005.
A new modeling approach for premixed turbulent combustion is presented. It is a combination of a joint PDF method, a progress variable [9] S. B. Pope. Turbulent Flows. Cambridge and a flamelet model. Advantages are that turUniversity Press, 2000. bulent transport of the progress variable appears in closed form and it is very natural to account [10] B. Rembold and P. Jenny. A multiblock joint pdf finite-volume hybrid algorithm for for the embedded, quasi-laminar flame structure. the computation of turbulent flows in complex With a numerical test case it is demonstrated that geometries. J. Comp. Phys., 2006. to appear. the simulation approach is robust and that the results show the correct qualitative behavior. However, more investigations are required for a quantitative validation.
BIBLIOGRAPHY
[1] K. N. C. Bray and J. B. Moss. A unified statistical model of the premixed turbulent flame. Acta Astronautica, 4:291–319, 1977. [2] K. N. C. Bray and N. Peters. . In Turbulent Reacting Flows, pages 68–113. Academic Press, London, 1994. [3] P. Jenny, M. Muradoglu, K. Liu, S. B. Pope, and D. A. Caughey. PDF simulations of a bluff-
