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Paper 03F-52 Bifurcation diagrams for a discrete dynamical system model of reduced-mechanism H2 {O2 combustion J. C. Holloway , M. G. Chong and J. M. McDonough Department of Mechanical Engineering University of Kentucky, Lexington, KY 40506-0503 Abstract We present a turbulence modeling formalism similar to large-eddy simulation but distinct in the sense that solutions, rather than equations, are ltered, and physical variables, instead of their statistics, are modeled on the sub-grid scales. We argue that this approach yields specic advantages in the context of combustion, but we also show that the associated discrete dynamical systems (DDSs) contain numerous bifurcation parameters (hence, very high codimension) which must be calculated. It is thus useful to understand details of the DDS bifurcation sequences in order to complete a subgrid-scale model formalism. In the present work we provide 2-D regime maps (bifurcation diagrams) corresponding to selected pairs of parameters appearing in a simple reduced mechanism for H2 {O2 combustion due to Mauss et al. (in Reduced Kinetic Mechanisms for Applications in Combustion Systems, Peters and Rogg, Eds., 1993). We discuss bifurcation sequences appearing in the regime maps and their relevance to physical phenomena, as well as the ability of the DDS to model a wide range of H2 {O2 combustion scenarios. In addition, we illustrate the prodigious complexity of the DDS model by presenting evidence suggestive of fractal behavior. Keywords: turbulent combustion, reduced mechanism, discrete dynamical system, bifurcation 1. Introduction . [1] and Hylin and McDonough [2] used results of modeled uctuating quantities to augment the resolved-scale solution, unlike most approaches to LES which simply discard them. LES involving chemical kinetics has yet more challenges beyond that of the usual closure problem and the associated subgrid-scale (SGS) models, particularly the disparate time scales of the chemical kinetics and velocity and temperature uctuations. This suggests the usefulness of employing SGS models as outlined by McDonough and Huang [3]. McDonough and Zhang [4], [5] used the ideas of [3] in constructing LES SGS models for six and nine-step reduced mechanisms of H2 {O2 combustion. Results from these models compared qualitatively well with experiment. Furthermore, Zhang et al. [6] were able to produce temporal uctuations of quantities which resulted in relatively good al Large-eddy simulation (LES) has begun to be useful in simulating a wide range of uid ows for engineering purposes. However, it is well known that LES has deciencies, the rst and foremost of which is the closure problem resulting from ltering of governing equations. An alternative approach to LES as explained by McDonough et al. [1] alters the closure problem by ltering solutions to the governing equations rather than the equations themselves. This ltering technique allows modeling of uctuating quantities directly, rather than their statistics, because no correlation quantities are produced in the process of ltering solutions. To further improve LES, McDonough et Corresponding author. Tel: 859.269.1480 johncholloway@yahoo.com E-mail address: 1 quantitative agreement with experiments by Meier et al. [7] by using an SGS model for a two-step reduced mechanism for H2 {O2 combustion similar to those in [3], [4] and [5]. In all of these investigations, discrete dynamical systems (DDSs) were employed to model uctuating quantities on the sub-grid scales. But due to the large number of bifurcation parameters (and thus very high codimension) associated with these models, a suÆcient understanding of how to implement the models into a complete LES of combustion requires further investigation. In the present work we continue analyses begun of the two-step reduced mechanism for H2 { O2 combustion in [6] in order to understand the bifurcation sequences that can be observed in the DDS model as the numerous bifurcation parameters are changed. Furthermore, we seek an understanding of limiting values of several key parameters that may be used in LES. This understanding may then be employed to construct accurate SGS models for LES involving turbulent H2 {O2 combustion. 2.1 Alternative Approach to LES For simulation of chemically reacting ows, we seek solutions to the conservation equations of mass, momentum, energy, and species, which may be represented as the general transport equation, Qt + r F (Q) = r G(Q) + S (Q) : (1) Here, the subscript t denotes partial dierentiation with respect to time, and r is the divergence operator. F and G are advective and diusive uxes, respectively, and S is a general nonlinear source term. To solve (1), we decompose dependent variables Q in the usual way to obtain Q(x; t) = q (x; t) + q (x; t) ; x 2 Rd ; d = 2; 3 ; (2) with q and representing the large and sub-grid scale contributions, respectively. After substitution of (2) into (1) leading to q (q + q )t + r F (q + q ) = r G(q + q ) + S (q + q ) ; The remainder of the paper proceeds as follows. We begin with an analysis section outlining the main ideas used in deriving the DDS SGS model as proposed in [3]{[6]. Next, we describe the relationships of the DDS bifurcation parameters to physical phenomena. We then present 2-D bifurcation diagrams illustrating the relationships between numerous pairs of these parameters, as well as their limiting values and the observed bifurcation sequences of the DDS SGS model. Finally, we discuss results and conclude the paper with a summary and a suggestion of plans for future work. (3) the large scale solution q may be calculated directly once q is set, much like in direct numerical simulation (DNS). As current hardware does not allow complete resolution of all ow scales (as in DNS) for most real-world engineering problems investigated, a more eÆcient means of calculating q , and thus a modeling approach, is required. We follow the approach used in [8] and model q as qi = Ai Mi ; i = 1; 2; : : : ; Nv ; (4) where Nv is the number of dependent variables. The Ai s are amplitude factors derived from Kolmogorov scalings [8], and the Mi s are algebraic chaotic maps derived from governing equations associated with Q in (2), which compose a discrete dynamical system. Detailed teatment has been given the Ai s in [8] and [9]. In the present work we investigate the chaotic behaviors associated with the Mi s. 2. Analysis For the sake of brevity we only outline the main steps of the subgrid-scale model derivation. We begin with an overview of the LES formalism and present the general governing equations. We then sketch the procedure that leads from these to a corresponding generalized DDS for reacting ows. We conclude the section by introducing the two-step mechanism that is the subject of the current inves- 2.2 Discrete Dynamical System tigation and provide a specic DDS corresponding To derive the Mi s we begin with the governto this mechanism. ing equations for chemically reacting ows, written 2 here using a conventional notation: crete dynamical system: a(n+1) = u a(n) 1 a(n) u a(n) b(n) ; (6a) 0 = t + r (U ) ; (5a) b(n+1) = b(n) 1 b(n) v a(n) b(n) + T c(n) ; v DU (6b) = rp + r (rU ) + g ; (5b) Dt " N X N X DT Yi T d d(in+1) uT a(n+1) cp =r (rT ) + cp Di Wi r rT c(n+1) = s s Dt Wi i i=1 Ns X i i=1 hi w_ i ; vT b(n+1) c(n) (5c) i=1 D(Yi ) =r (Di rYi) + w_ i ; Dt i = 1; : : : ; N s : (n+1) di (5d) = !_ i = r j =1 00 (i;j 0 i;j )wj ; wj = kf;j s `=1 W` 0 `;j kb;j N Y Y` s `=1 00 `;j W` # Hi !_ i =(1 + T ) (6c) (n) i N X r " Cf;ij i N Y s `=1 d i 0 j;` ` Cb;ij N Y s `=1 d 00 j;` ` # : Thus the as, bs, cs and di s are quantities related to the Fourier coeÆcients of the Fourier series representing the two velocity components, thermal energy, and chemical species concentrations, respectively. The superscripts (n) denote time steps (or map iterations). Note that (6) will apply to any chemically reacting ow; a DDS pertaining to a particular mechanism simply requires a di for each product species in that mechanism. The reader is encouraged to refer to detailed derivations as found in the extant literature, particularly [3] and [4]. with N Y Y` s Y + uY a(n+1) + vY b(n+1) di + !_ i + di;0 ; i = 1; 2; : : : ; Ns ; (6d) j =1 N X N X i=1 + c0 ; with Here, Ns is the total number of species and w_ i = Wi : We assume dependent variables may be accurately represented as generalized Fourier series. To simplify analysis, we further assume the basis functions f'k g used in constructing the series are i) complete, ii) orthnormal, and iii) like complex exponentials with regard to dierentiation. After substituting the Fourier representations into (5), we then carry out the Galerkin procedure as in [3]{[6], producing an innite number of ordinary dierential equations for the Fourier coeÆcients. For computational eÆciency, each series is decimated to retain a single term corresponding to an arbitrary wavevector k. The resulting equations can be integrated with rst-order Euler methods and then rearranged to produce the following dis- 2.3 Bifurcation Parameters Derivation of (6) leads to the construction of numerous bifurcation parameters, all of which are related to physical quantities associated with the particular ow scenario investigated. It should be observed that in a complete LES, these parameters are calculated automatically \on the y," using results from the resolved-scale solution. In the current investigation, however, parameters are set to xed values manually. As discussed in [10], u and v in the momentum equations are related to Reynolds number via jkj2 i = 4 1 ; i = u; v : (7) Re 3 Here, is a time scale computed as the reciprocal of the norm of the strain rate tensor, and k is a single wavevector. The components of k, kj , should be treated as reciprocals of the Taylor micoscale length in each separate coordinate direction. In [11] it is shown that T is related to Peclet number, jkj2 ; Pe and this is true for each Y as well [8], T = be modeled by using reduced mechanisms [4]|[6]. We follow such an approach and model concentrations with a two-step reduced mechanism. As illustrated in [12], under steady-state assumptions, the 17 elementary reactions comprising H2 {O2 combustion may be reduced to the two-step, global mechanism 3H2 + O2 ! 2H + 2H2 O 2H + M ! H2 + M: (8) i (13a) (13b) jkj2 ; i = 1; 2; : : : ; Ns : (9) P ei The u , v , uT , vT , uY , and vY arise from the In our investigations, the third body M is taken to be H2 O , which has shown to be a reasonable assumption in previous studies [4]{[6]. As mentioned above, a discrete dynamical system may be Galerkin triple products [8], used to model reacting species concentrations in (1) the form of (6). Each species concentration to be j = j kj Ckkk ; j = u; v; uT; vT; uYi ; vYi ; (10) modeled, that is, each product species appearing in (13), requires calculation of a corresponding di . (1) with Ckkk representing the Galerkin triple product We set of momentum Fourier basis functions with passive scalar Fourier basis functions, and and kj comd1 H2 ; d2 O2 ; d3 H2 O ; d4 H : (14) puted as above. We note, however, that kj is the Fourier space representation of a partial derivative, Although d2 can only be a product in (13) if we and in implementations we usually construct this assume the reverse reaction holds as well as the forin physical space. Each T is related to Galerkin ward, we nd in [12] that calculations of other di require calculation of d2 (as will be shown below). triple products similar to the s [8]: In the current investigation we consider only the (2) T = jkj2 cdi Ckkk ; i = 1; 2; : : : ; Ns ; (11) forward reaction in (13a); d2 , then, is set as constant for all time steps. All other product species (2) with the exception that the Ckkk are computed in (13) are modeled in the form of di s in (6). Thus from derivatives of basis functions. Finally, T is for H2 , simply (n) (n+1) (n+1) (n+1) Gr d = + a + b d1 Y uY vY 1 1 1 1 T = 2 ; (12) Re + !_ 1(n) + d1;0 ; where Gr is a grid-scale Grashof number. In the (15) current investigation we set T = 0 to treat a nonwith buoyant jet ame. Y = i i i Yi Yi !_ 1(n) = Cf;1;2 d(4n+1) ; 2.4 Reduced Mechanism Cf;1;2 = 100;2 W1 k : W4 f;2 According to Mauss et al. [12], combustion of For O2 we have simply H2 {O2 involves eight chemically reacting species (16) d(2n+1) = d2;0 ; and approximately 17 elementary reactions. Thus an attempt to accurately model species concentrafor all time steps (n). We model H2 O with tions involved in the phenomenon could prove to (n) be quite exhaustive for current hardware, at least d(n+1) = (n+1) (n+1) b d3 a + + vY uY Y 3 3 3 3 in the context of 3-D LES or DNS. But there is evidence that suggests reasonably good approx+ !_ 3(n) + d3;0 ; (17) imations of primary species concentrations may 4 and !_ 3(n) = Cf;3;1 d(1n) d(2n) ; Cf;3;1 = 300;1 Divergent Broadband w/o fundamental Broadband w/ different fundamental Broadband w/ fundamental Noisy quasiperiodic w/o fundamental Noisy quasiperiodic w/ fundamental Noisy phase lock Noisy subharmonic Quasiperiodic Phase lock Subharmonic Periodic w/ different fundamental Periodic Steady W3 k : W1 W2 f;1 Finally, H is modeled by d(4n+1) = Y4 + uY4 a(n+1) + vY4 b(n+1) d(4n) + !_ 4(n) + d4;0 ; (18) with !_ 4(n) = Cf;4;1 d(1n) d(2n) ; Cf;4;1 = 400;1 W4 k : W1 W2 f;1 Thus to model H2 {O2 combustion primary species concentrations with this two-step reduced mechanism requires a total of only three iterated maps, which may be calculated with ease. The entire DDS used to model momentum, thermal energy, and species concentrations is simply (6) with the di s simplied as in (15){(18). Figure 1: Regime map color legend. Each color represents a distinctive classication of temporal uctuations that may be observed in the time series of components in (6). 3. Results here are not meant to be representative of all possible behaviors of the components of (6). Each map represents time series behavior for only one combination of bifurcation parameters (except, of course the two parameters varied within the regime map), the slightest perturbation of which can potentially alter the map signicantly. In most cases, we will omit reporting specic values of bifurcation parameters employed as such is not neccessary to illustrate the potential of the DDS to model H2 { O2 combustion or to comprehend its inherent complexity. Finally, recall in a complete LES, all bifurcation parameters are calculated automatically by using results from the resolved-scale solution. In the current investigation we simply seek an understanding of the possible behaviors that could be observed in such an LES. We present 2-D bifurcation diagrams corresponding to various pairs of bifurcation parameters in (6) to illustrate the wide range of temporal behaviors that can be observed in the components of (6). These diagrams, which we call regime maps, are produced by a Fortran 77 program that iterates (6) for a specied number of time steps (producing a time series for each component in the DDS), calculates power spectral densities (PSDs) based on these time series, and then essentially \looks" at each PSD to classify the temporal behavior produced by the particular combination of bifurcation parameters employed. Temporal behavior of the components of (6) can normally be classied into one of several generalized groups of temporal uctuations, which to each we assign a color as shown in Fig. 1. On a regime map for a particular component of (6), each point corresponds to a unique combination of bifurcation parameters. Thus for each point on a map the color represents the type of temporal uctuations observed in the time series for the component of (6) portrayed in the map when this combination of bifurcation parameters is set. Regarding discussions that follow, it is important to understand that the regime maps presented 3.1 Regime Maps for H2 O In this section we investigate the temporal behavior of the uctuating H2 O concentration calculated from (17) as the parameters v and uY3 of (6) are changed. As the main product of H2 {O2 combustion, H2 O is of particular interest in our studies. Recall that uY3 is related to strain rate and a spatial derivative for H2 O concentration via 5 (10), and v is directly related to Re (7). Thus increasing v in (6) is analagous to physically increasing the Re in the vertical direction. In Fig. 2 we can see a common bifurcation sequence leading to turbulence as v is increased and uY3 is held constant. Note the transition from subharmonic to phase lock (v 2:8), which then bifurcates to noisy quasiperiodic with a distinctive fundamental mode. We also observe reverse bifurcation sequences, such as where the quasiperiodic uctuations bifurcate back into phase lock as v is further increased (v 3:2). Note as v ! 4, the time series diverges, symbolizing a mathematical irregularity and a non-physical solution. If instead v is held constant as uY3 is changed, we notice little change in time series behavior, and few clear bifurcation sequences. The main visible transitions as uY3 is varied occur around uY3 1:5, where subharmonic H2 O uctuations become phase locked as uY3 is increased before bifurcating back into subharmonic. Other transitions are less distinctive, mainly comprised of changes between broadband with, without, or with dierent fundamentals. 0 γvY (3) -0.5 -1 −1.5 −2 2 2.5 (3) γvY 0 −1 3 βv 4 H In addition to understanding the possible behavior of H2 O uctuations permitted by (6), we take interest in the temporal uctuations of products H and H2 . In Fig. 4 we choose (arbitrarily) to present a regime map for H concentration as the parameters uY4 and vY4 are varied. Recall that uY4 and vY4 are related to products of momentum Fourier basis functions with the Fourier basis functions for H as in (10). Here we see the wide range of uY4 and vY4 values that are permissible to produce non-divergent time series, a range which is much larger than allowable for many other parameters appearing in (6). This large range of permissible values has shown to be characteristic of most s in (6) for our investigations thus far. Also in Fig. 4 it is interesting to observe bifurcation sequences as uY4 or vY4 increase in magnitude. In particular, beginning one sequence we see that for small magnitudes of uY4 and vY4 (near the center of the regime in Fig. 4), there are quasiperiodic uctuations of H concentration. As uY4 and vY4 are both 1 2 4 that as we increase grid resolution of the regime map there are more visible bifurcations than at the coarser resolution. This illustrates the complexity of a DDS with such high codimension as (6). 3.2 Regime Map for 1 3.5 Figure 3: Regime map for H2 O; a magnication of Fig. 2, illustrating the possible fractal behavior of (6). 2 −2 3 βv 5 Figure 2: Regime map for H2 O illustrating relationships between H2 O concentration uctuations and changes in v and uY3 . It is also interesting to observe what appears to be fractal behavior in certain regions of Fig. 2. In Fig. 3 we magnify one of these regions. We see 6 when T < 0. This suggests the mathematics of (6) is, perhaps, more complex in nature than the actual physics described by it. 8 4 γvY (4) 4 0 3 βT −4 −8 −8 −4 4 0 γuY 8 2 1 (4) Figure 4: Regime map for H displaying changes in H uctuations as uY4 and vY4 are varied. 0 −1 0 1 γu 2 3 increased in (positive) magnitude, the H uctua- Figure 5: Regime map for thermal energy showing tions bifurcate to noisy subharmonic, then to noisy relationships between thermal energy uctuations quasiperiodic, to eventually broadband without a and changes in and . u T fundamental mode before nally diverging. In Fig. 6 we present regime maps for the thermal energy component of (6) where u and v are varied. For the combination of bifurcation parameters employed here, thermal energy uctuations are extremely sensitive to perturbations in either u or v . Observe, for example, in the lower middle region of the regime of Fig. 6(a) (v < 0:75; u 0), thermal energy uctuations bifurcate from periodic to subharmonic, and then further to noisy quasiperiodic with changes in u over a range as small as 0:2. In addition, notice the bifurcations to phase-locked uctuations littered throughout the quasiperiodic region (in the lower left quadrant of the regime) as well as bifurcations to noisy quasiperiodic with a fundamental mode in the noisy quasiperiodic region without a fundamental mode (in lower right quadrant of the regime). Such regions illustrate the extreme sensitivity to initial conditions in (6), which is a common characteristic of turbulence in general. Unlike in other gures presented thus far, in Fig. 6(a) we observe regions in the regime which produce periodic uctuations and even steady state behavior. This can be attributed to having set u = 3:2, 3.3 Regime Maps for Thermal Energy In this section we wish to demostrate the ability of (6) to model temporal uctuations of thermal energy as are often observed in H2 {O2 combustion systems. To do so we present regime maps for the c s in (6), which represent the thermal energy of the system, as u and T in (6) are varied. Recall from (8) that T is directly related to Peclet number; thus changing T is analagous to altering the thermal diusivity of the system. Also, from (10), u is related to products of Fourier basis functions for momentum, which has a less intutive relation to physical space than that of T , but it carries spectral information on velocity derivatives. An important characterisic of Fig. 5 is that as T is varied, there is little change to the temperature uctuation behavior, except at low values of T . Clearly, at least in the case shown in Fig. 5, temperature uctuations have a much larger dependence on the value of u . Finally, it should be noted that due to the nature of (8) we consider only positive values of T , although regime maps can be used to show that there exists non-divergent behavior for cases 7 which in our investigations normally produces less chaotic uctuations than s > 3:5. For example, in Fig. 6(b) we employ the same combination of bifurcation parameters as in Fig. 6(a), only we increase u to 3:6. Notice here the steady and periodic regions in Fig. 6(a) have now bifurcated to the more turbulent uctuations of noisy quasiperiodic and broadband. 5. Discussion In the present work we have presented results from preliminary analysis of the complex behaviors associated with (6). Complete analysis for a DDS with such high codimension is slowed by the vizualization limitations of 3-D space. As we have shown, 2-D bifurcation diagrams can be used to deduce relationships between pairs of bifurcation parameters appearing in (6). Likewise, 3-D bifurcation diagrams would be capable of illustrating such relationships among three parameters at one time. Thus to predict all possible changes in temporal uctuations for a DDS with codimension of n requires an n-dimensional space in which to visualize it. Under these circumstances, an attempt at understanding all possible behaviors and bifurcations associated with such a system would prove to be an insurmountable task, indeed. Rather, a more realistic goal is to implement such a DDS SGS model into a complete LES to observe its strengths and weaknesses in simulations by simply comparing the results to those of experiment and DNS. 8 γv 4 0 −4 −8 −8 a) −4 0 γu 4 8 8 6. Conclusion γv It has been the goal of this work to present a possible solution to the computational diÆculties as4 sociated with accurately modeling turbulent ows involving chemical kinetics. We propose a simulation formalism similar to LES, but distinct in 0 that solutions are ltered rather than equations, dependent variables are modeled directly rather than their statistics, and subgrid-scale solutions are used to augment the resolved-scale solutions. −4 In this context we have presented a subgrid-scale model for H2 {O2 combustion in the form of a disb) crete dynamical system, which was derived based −8 8 −8 4 0 −4 on a two-step reduced mechanism constucted by γu Mauss et al. [12]. 2-D regime maps, or bifurcation diagrams, were used to illustrate relationFigure 6: Regime maps for thermal energy illus- ships between the numerous bifurcation parametrating relationships between thermal energy uc- ters in (6) and were able to demonstrate the comtuations and changes in u and v . plex temporal uctuations that can be produced in the components of this DDS. Additionaly, regime maps were able to model bifurcation sequences commonly observed in turbulent combustion phenomena. We conclude by calling attention to the 8 potential for this approach to accurately model H2 {O2 combustion in a complete LES. For future work, it will be worthwhile to implement (6) and compare results to those obtained by employing DDS SGS models as in [4] and [5], which were based on six and nine-step reduced mechanisms. [12] F. Mauss, N. Peters, B. Rogg and F. A. Williams, in Reduced Kinetic Mechanisms for Applications in Combustion Systems (Peters and Rogg, eds), Lecture Notes in Physics m15, Springer-Verlag, Berlin, 1993. 7. Acknowledgements The rst author would like to express his gratitude to the NASA Kentucky Space Grant Consortium for nancial support of this work. References [1] J. M. McDonough, Y. Yang and E. C. Hylin, Proc. of First Asian Computational Fluid Dynamics Conference, Hui et al. (Eds.), Hong Kong Uni- versity of Science and Technology, January 16{19, Hong Kong, 1995, pp. 747{752. [2] C. E. Hylin and J. M. McDonough, Int. J. Fluid Mech. Res., Vol. 26, 1999, pp. 539{567. [3] J. M. McDonough and M. T. Huang, paper ISSM3E8 in Proc. of Third Int. Symp. on Scale Modeling, Nagoya, Japan, Sept. 10{13, 2000. [4] J. M. McDonough and S. Zhang, presented at 32nd AIAA Fluid Dynamics Conference, St. Louis, MO, June 24{27, 2002. [5] J. M. McDonough and S. Zhang, presented at 37th Intersociety Energy Conversion Engineering Conference, Washington, D. C. July 29{31, 2002. [6] S. Zhang, J. D. Slade and J. M. McDonough, Proc. of Technical Meeting of Central States Section of the Combustion Institute, Knoxville, TN, April 7{ 9, 2002. [7] W. Meier, S. Prucker, M. H. Cao and W. Stricker, Combust. Sci. Technol. 118:293, 1996. [8] J. M. McDonough and J. C. Holloway, Proc. of Third Joint Meeting of U. S. Sections of the Combustion Institute, Chicago, IL, March 16{19, 2003. [9] J. M. McDonough, paper c3.4, Proc. of Technical Meeting of Central States Section of the Combustion Institute, Knoxville, TN, April 7{9, 2002. [10] J. M. McDonough and M. T. Huang, Int. J. Numer. Meth. Fluids, accepted 2003. [11] J. M. McDonough and D. L. Joyce, presented at 8 th AIAA Thermophysics and Heat Transfer Con- ference, St. Louis, MO, June 24{27, 2002. 9