Experimental Data Based Reduced Order Model for Analysis and Prediction of Flame Transition in Gas Turbine Combustors
Shivam Barwey, Malik Hassanaly, Qiang An, Venkat Raman, Adam Steinberg
EExperimental Data Based Reduced Order Model for Analysis andPrediction of Flame Transition in Gas Turbine Combustors
Shivam Barwey ∗ , Malik Hassanaly , Qiang An , Venkat Raman , and Adam Steinberg Department of Aerospace Engineering, University of Michigan Institute for Aerospace Studies, University of Toronto
Abstract
In lean premixed combustors, flame stabilization is an important operational concern that can af-fect efficiency, robustness and pollutant formation. The focus of this paper is on flame lift-off andre-attachment to the nozzle of a swirl combustor. Using time-resolved experimental measurements, adata-driven approach known as cluster-based reduced order modeling (CROM) is employed to 1) isolatekey flow patterns and their sequence during the flame transitions, and 2) formulate a forecasting modelto predict the flame instability. The flow patterns isolated by the CROM methodology confirm someof the experimental conclusions about the flame transition mechanism. In particular, CROM highlightsthe key role of the precessing vortex core (PVC) in the flame detachment process in an unsupervisedmanner. For the attachment process, strong flow recirculation far from the nozzle appears to drive theflame upstream, thus initiating re-attachment. Different data-types (velocity field, OH concentration)were processed by the modeling tool, and the predictive capabilities of these different models are alsocompared. It was found that the swirling velocity possesses the best predictive properties, which givesa supplemental argument for the role of the PVC in causing the flame transition. The model is testedagainst unseen data and successfully predicts the probability of flame transition (both detachment andattachment) when trained with swirling velocity with minimal user input. The model trained with OH-PLIF data was only successful at predicting the flame attachment, which implies that different physicalmechanisms are present for different types of flame transition. Overall, these aspects show the greatpotential of data-driven methods, particularly probabilistic forecasting techniques, in analyzing and pre-dicting large-scale features in complex turbulent combustion problems.
Keywords: Reduced order modeling; Data-driven modeling; Flame transition; Probabilistic forecasting;Premixed combustion
Contents ∗ Contact: [email protected]. Preprint accepted to Combustion Theory and Modelling. a r X i v : . [ phy s i c s . f l u - dyn ] A p r .3.1 Flame Detachment Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3.2 Flame Attachment Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The issue of flame stabilization is critical for lean premixed flames with the perspective of controlling pol-lutant formation and ensuring robustness at various operating loads [1]. In swirl-stabilized combustors, theinteraction of multiple recirculation zones with flame propagation impacts the stabilization process. Twodifferent stabilization mechanisms are feasible [2, 3, 4]: a) a shear layer stabilized flame resulting in a flameattachment to some feature of the geometry, and b) lifted flames that are stabilized near the stagnationsurface formed between inflowing gases and an inner recirculation zone. For practical considerations, thetransition into a lifted state might serve as a precursor for flame blow-off [5]. Prior studies have focused onelucidating physical mechanisms that cause transition between the attached and lifted flame states, which aredue to either inflow or other operational variations. The focus of this work is to use one such experimentalstudy to develop a prognostic data-driven model with a twofold purpose: 1) to understand how the flameinstability occurs and 2) to predict the flame transition.Instabilities in complex systems such as gas turbines are subject to multi-scale mechanisms that require asimplified representation to be meaningful. A first technique is to quantify the growth rates of perturbations,which are obtained using system identification (SI) methods [6]. The perturbations are quantified for somequantity of interest, e.g. pressure at some location. The SI technique can be input-output based, wherethe response of the combustor to perturbations are used, or output based [7, 8, 6]. A second techniqueutilizes data decomposition to obtain a simplified representation of the governing equations. Proper orthog-onal decomposition (POD) [9, 10] projects the governing equations onto some meaningful flow modes, anddynamic mode decomposition (DMD) attempts to linearize the governing equations [11]. Such methodsdetermine instability modes from experimental data and the variation of their relative contribution to theflow field. These characteristic modes can be used to directly analyze physical system properties of interest,and also to construct a reduced-order model (ROM) of the system. These approaches have been used in thepast for combustion chambers [12, 13], and additional studies have utilized laser diagnostics to determinecharacteristic modes [14, 15].From a prognostic viewpoint, the above methods are derived from two different processes. SI and DMD-related techniques are purely data-driven approaches, where finite measurements of the system are usedto construct empirical tools which may then be used to predict combustor state transitions. While suchtools are easier to use in a practical setting, interpreting the model in terms of the physical processes isnot straightforward. Further, DMD related methods invoke linearization of the underlying dynamics, whichmay not be valid in many practical configurations. In contrast, POD-type techniques attempt to simplifythe theoretical governing equations to predict the future states of the system. First, data from the system isused to construct lower-order representations, or a ROM; second, this model can be propagated in time inorder to predict the future state of the system. Since the decomposition techniques can be related to physicalprocesses, the first step provides meaningful projection operators which can be used to analyze the physicsof the combustion. However, the projection applied to the governing equations leads to issues with closure2nd numerical discretization, similar to those encountered in reduced-fidelity models such as RANS/LES[16, 17, 18].Recently, a so-called cluster-based reduced-order modeling (CROM) framework has been proposed by Kaiseret al. [19]. Unlike prior decomposition tools that obtain a linear operator to describe the nonlinear dynamics,CROM retains the nonlinearity of the underlying system. Furthermore, CROM avoids a direct projectionof the governing equations, thereby alleviating the closure problem; instead, it constructs a discrete-timeMarkov process from a set of experimental or high-resolution simulation data. As a result, CROM providesa physically-meaningful decomposition of the dataset which allows to understand from the available datahow the flame instability occurs [20]. It also provides a probabilistic model for the forecast of the combustorstate. In the original work of Kaiser et al. [19], CROM was used to construct the model of a turbulent mixinglayer. This method has also been used to predict cycle-to-cycle variations in internal combustion engines viacluster discretization of data [21].In this study, CROM is used to develop a means for transition mechanism analysis and to model flametopology transition in a swirl combustor. Experimental stereoscopic particle image velocimetry (S-PIV)and OH planar laser induced fluorescence (PLIF) images are used to develop a predictive ROM for flametransition. The flow modes obtained are analyzed to illustrate the physical information that can be obtainedfrom the CROM methodology. The remainder of this paper is organized as follows: experimental data andoperating conditions are provided in Sec. 2; the CROM methodology is discussed in Sec. 3; outcomes of theCROM approach are used to gain physical intuition about the flame transition mechanism in Sec. 4; resultsof the CROM approach both in terms of the prediction horizon time and forecast capability for differentdata types are discussed in Sec. 5. Finally, conclusions and future directions are presented in Sec. 6.
The data used for this analysis was acquired in the experiments of Ref. [22], which contains additional detailson the combustor and diagnostic techniques. A brief summary is provided here.Figure 1a shows the gas turbine model combustor, which has been used for a number of previous studies, bothexperimental [23, 24, 25] and numerical [26]. A review addressing progress and key findings in experimentalstudies on hydrodynamic instabilities, including additional perspective on the advancements in numericalsimulations of instabilities in swirling combustors is provided in Ref. [27]. The schematic shown in Figure 1cdepicts the type of flame shape geometry observed in the attached and lifted states as seen by this particularswirl combustor.Premixed fuel and air are fed through a plenum to the radial swirler before entering the combustion chamber,where vortex breakdown generates a strong inner recirculation zone. In the present case, a fuel-air mixtureof equivalence ratio of φ = 0 .
60 is fed to the combustor at a preheated temperature of 400 K. The fuel ismade of 80% CH and 20% CO by volume. The air flow rate is 400 SLPM. This case was selected fromthe test matrix in Ref. [22] because it exhibits a high number of transitions between clearly defined attachedand detached flame states (8 observed in a span of 1.5s), while operating at fixed equivalence ratio and flowrates. Thus, the flow conditions were not changed to force a transition, and the inherent system can beconsidered ergodic; the flame experienced intermittent and spontaneous (in the sense of being apparentlyrandom in time) transitions between the detached and lifted states.Data was collected using 10 kHz repetition-rate OH PLIF and S-PIV, providing simultaneous time-resolved2D measurements of the OH radical distribution and of the three velocity components over a time spanof 1.5 s. Figure 1b shows typical instantaneous OH PLIF images in attached and detached states. In theattached state, high OH concentrations are present in the shear layers that separate the inner and outerrecirculation zones, while in the detached state, a rotating helical vortex core generates a highly asymmetricOH field. Attached and detached flame configuration shapes are shown in the drawings in Fig. 1c – theflame in the attached state takes on the characteristic V-shape, whereas the M-shape is observed in thedetached. There were roughly 8 total transitions captured in the 1.5 s dataset (combining attached-to-detached transitions and the detached-to-attached transitions). For the specified operating conditions, the3 igure 1: (a) DLR combustor schematic. The fuel consists of 80% CH and 20% CO by volume. (b) OH-PLIFsnapshots of detached (top) and attached (bottom) flames in units of relative pixel intensity. (c) Sketches of M-shapeddetached flame (top) and V-shaped attached flame (bottom) (from Ref. [1]). flow-through time is 14.6 ms (from Ref. [22]) and the flame transition timescale (determined by manuallyobserving the 8 detached-to-attached and attached-to-detached transition times in the OH-PLIF images) ison the order of 10 ms.In order to reduce the computational overhead, the PLIF images were reduced in size from 832 ×
504 pixelsto 104 ×
63 pixels via nearest-neighbor interpolation based filtering, which preserves the large-scale flamefeatures. The effective resolution of the down-sampled OH PLIF was 0.71 mm. The full-resolution PIV datawas used (79 × In this section, the CROM methodology is described. Essentially, it processes time-resolved data of anykind, identifies recurrent patterns in the system dynamics, and creates a probabilistic predictive model forthese patterns. Here, the data are the 2D experimental OH-PLIF and PIV images. The set of experimentalimages is denoted by S = { S , S , · · · , S N } , where N is the number of snapshots available and each subscriptdenotes a different snapshot. The snapshots are ordered in time. This time-ordered quality is importantto create the predictive model, but not important when classifying the data. Each element S i of the set S is a vector of pixel values associated with PIV and/or OH-PLIF images, and is of size N p (i.e, S i ∈ R N p ).Snapshots are separated by a time-step ∆ t that is determined by the 10 kHz sampling rate identical across allmeasurement types. In this section, the data classification (clustering) and the probabilistic model (transitionmatrix) obtained from the CROM methodology are described. Further, a procedure to relate the clusteroutputs to the individual swirler states is also presented (a systematic labeling of the clusters as detachedflame, attached flame or transitioning flame). 4 .1 Clustering The first step of the CROM methodology is to map the set of snapshots S to a smaller set of so-called centroids C = { C , C , · · · , C N k } , where C i ∈ R N p . The centroids are also images made of N p pixels thatrepresent some pattern of the flow field. Centroids can be interpreted as delta functions that discretizethe probability density function (PDF) of the states. States nearby one another are represented by thesame delta function. This step is reminiscent of classification methods used in language interpretation: thesame word can be pronounced differently, but a language interpreter finds common characteristics of thesedifferent signals to assign them the same meaning. The number of delta functions (centroids) N k chosen todiscretize the state-space is defined by the user, varying within the bounds of 1 to N ( N being the number ofsnapshots). The value of the parameter N k should be set depending on the purpose of the study, as discussedin Sec. 4.1 and Sec. 5.2. Each centroid C i represents a region of phase space that contains some proportionof the total number of snapshots. This region of phase space, denoted C i , is called a cluster . Since there isone centroid per cluster, N k defines both the number of centroids and the number of clusters.The N k centroids are chosen such that each snapshot is represented by only one centroid. The centroid thatrepresents each snapshot is the one that is the closest to the snapshot based on a distance measure. Here,the L -norm in the R N p space is used to compute this distance, which is given by d i,k = (cid:118)(cid:117)(cid:117)(cid:116) N p (cid:88) j =1 ( S ji − C jk ) , (1)where d i,k represents the L -norm between the i -th snapshot and k -th centroid. The superscript j in Eq. 1indexes the number of dimensions N p , or pixels, in the centroid/snapshot vectors.This snapshot-centroid assignment is represented in an association matrix, T i,k = (cid:40) S i ∈ C k , C i should be as close as possible to its assigned snapshots suchthat the centroid actually represents these snapshots. Note that the centroids are not flow fields that can beobserved, but statistical patterns that approximate what can be observed. An illustration of this statisticaleffect is shown in Fig. 2, which juxtaposes an example PLIF snapshot with its corresponding centroid. Here,the centroids are chosen using a k-means clustering algorithm [28], a very common classification techniqueused in many unsupervised machine learning frameworks.In any k-means algorithm, centroid convergence rates are highly dependent on initial centroid locations aswell as the input number of centroids, N k . Here, the initial centroid locations are determined using thek-means++ routine, a variant of k-means as used in Ref. [19] and described in Ref. [28]. Instead of pickingthe initial locations arbitrarily, which may lead to some centroids occupying very similar regions of thephase space initially, k-means++ initializes with the goal of achieving a large amount of separation betweenthe centroids. The algorithm is summarized as follows. The first centroid, C , is randomly assigned to asnapshot S i in S . Then, C is assigned to another snapshot with probability proportional to the squareddistance from its closest centroid. This initialization progresses for all centroids up to C N k , and ensures thatthe chance of choosing an initial centroid location far from an already existing centroid is high. Note thatthe k-means++ initialization is a stochastic algorithm. Therefore, multiple realizations of the algorithm arenecessary to assess the predictive capabilities of the method in a statistically meaningful way.With the centroid initialization method outlined, the full k-means algorithm as detailed in Ref. [28] issummarized as follows:1. Determine the initial distribution of N k centroids C with k-means++.2. Assign all N snapshots to the nearest centroid as per Eq. 1, accumulating the association matrix T i,k as per Eq. 2. 5 igure 2: An example of a PLIF snapshot of an attached flame with its corresponding centroid.
3. Update the k-th centroid by computing the center of mass of all the snapshots in the k-th cluster, C k = N (cid:80) i =1 S i T i,kN (cid:80) i =1 T i,k , k = 1 , ..., N k , i = 1 , ..., N. (3)4. Repeat (2) and (3) until convergence, where convergence is defined as the point at which an additionaliteration would cause all centroids to change by a negligible distance. The prediction tool of the CROM approach is the transition matrix, which is the practical ROM necessaryto make state forecasts. The transition matrix provides the probability of a snapshot transitioning from onecluster to another within a given forward time-step ∆ t . In a mathematical sense, this matrix represents aMarkovian discrete time-step mapping. The transition probability of an image in cluster j transitioning toan image in cluster k is obtained based on the association matrix T as: P j,k = (cid:80) N − m =1 T m,j T m +1 ,k (cid:80) N − m =1 T m,j , for k = 1 , ..., N k . (4)The transition matrix P defined above is valid only for a finite time-step ∆ t . A key assumption regardingthe cluster-based ROM framework is that of Markovianity, i.e., that the finite time-step transport of prob-ability distributions using the transition matrix generated by the snapshots in set A is memoryless. Anyfinite-dimensional spatial discretization of Markovian governing equations will naturally yield a Markoviandynamical system. Here, it has only been assumed that the governing equations for the field measuredare themselves memoryless. This is reasonable since fluid flow equations are memoryless. Moreover, giventhe large density of data points, it is reasonable to consider the system experimentally observed to followMarkovian dynamics as well. Model validation techniques for forecasting purposes are presented in Sec. 5.More details are provided in Ref. [19] (Sections 2.2.3 and 5).Additionally, before centroid analysis and implementation of the transition matrix P in a prediction setting,it is useful to re-arrange the clusters in some probability-based order. One method (used in these results)is to order the clusters by descending eigenvalue modulus of P for clearer identification of cluster groupswithin the matrix structure [19]. The centroid set C combined with the transition matrix P now serve as the prognostic model. From aninitial PDF of the clusters P ∈ R N k the forward model determines the probability distribution of centroids6t a future time t : P t = P n s P , (5)where n s = t/ ∆ t . It is important to note that in the infinite time limit, the probability distribution representsthe statistically stationary state that can be obtained from all the snapshots. This is the ergodic behaviorof the transition matrix: lim t →∞ P t = e ≈ q , q k = (cid:80) Nm =1 T m,k N , for k = 1 , ..., N k . (6)In Eq. 6, e represents the PDF in the ergodic limit of the transition matrix, and q represents the initial clusterdistribution of the snapshots. Furthermore, as P is a Markov matrix, its first eigenvector v correspondsto the distribution e ( e = v ). Simply put, this feature states that the model gradually loses its predictivecapability as P is raised to successively higher powers. Therefore, the transition matrix does not giveinformation about any upcoming flame transition or flame state after a certain finite number of time-steps.This point can be defined as a finite critical time, τ h , after which the probability distribution remainsstationary regardless of the initial PDF P ; this is referred to as the prediction horizon time [19]. As willbe discussed in the results section, τ h is highly sensitive to the type of data used to apply the CROMmethodology, and is a key metric in evaluating the forecast power of one particular dataset over another.This will be the object of the discussion in Sec. 5. The clusters obtained from the CROM methodology isolate flow patterns in an unsupervised manner, butneed to be assigned to states of interest for meaningful interpretation and forecasts. Here, it is explained howeach cluster is labeled with the “detached flame”, “attached flame” or “transitioning flame” category. Thecentroid classification procedure is presented using a model generated with a combined dataset (OH-PLIF+ PIV-x/y/z components).Figure 3a displays the transition matrix with probabilities appearing as elements of the heatmap color-codedin log scale for N k = 16. Three distinct structures are identified within the transition matrix as indicatedby the blue boxes; further detail regarding the assignment of each structure to a physical state follows thislist:1. Clusters 1-7 represent a periodic step-like probability structure, where the most likely path (aside fromremaining in the same cluster) is to move on to the next cluster.2. Clusters 12-16 represent a highly interconnected probability structure (i.e. a snapshot in one clusterin this group has a roughly equally probable chance of moving on to any other cluster in this group,not just the next cluster).3. Clusters 8-11 are also step-like and periodic, similar to 1-7, but are probabilistically connected to bothof the regions described in (1) and (2).Centroid groups are further identified via the cluster distance matrix shown in Fig. 3b, which displays the L2distance between each centroid [19]. The distance matrix is symmetric with diagonal equal to zero.Centroids corresponding to the attached state are expected to be relatively similar in the phase space, asthere should be far fewer possible phase space realizations of an attached flame in the given domain than adetached flame. This coincides with centroids 12-16 as indicated by the distance matrix, which is a groupingof centroids markedly close together. A clear cluster grouping is also visible when observing centroids 12-16in the transition matrix – the probabilistic interconnection of these centroids is related directly to their phasespace similarity. Based on the above analysis, centroids 12-16 are associated with the attached flame state.Visualizing these centroids confirms the classification, as seen in Fig.6.Of the two remaining groups (1-7 and 8-11), one must be assigned a “detached” label and the other a“transition” label. Centroids 8-11 in the transition matrix are connected to the other two groups: forexample, a snapshot starting in clusters 8-11 has some probability of entering either clusters 1-7 or clusters7 a) (b) Figure 3: (a) Transition matrix with substructures boxed. (b) Distance matrix with substructures boxed.Figure 4: CROM workflow. The k-means clustering procedure occurs between the first two steps. Centroids andtransition matrix shown here are arbitrary. P does not allow transition to the attached centroids (12-16). Centroids 1-7 are plottedin Fig. 8, confirming the detached assignment. Note that there are more detached clusters than there areattached clusters, as is expected – the detached flame state should occupy a larger region of the phase spacethan the attached state based on the experimental dataset.CROM then provides the labeled centroids and the transition matrix, which can be used for a) analysis ofthe centroids and transition mechanism (Sec. 4), and b) prediction of flame transition and analysis of horizontime (Sec. 5). In context of the current experimental setup, this approach is valid only when the system isergodic. For additional perspective, a summary schematic of the model procedure is shown in Fig. 4.8 Analysis of the Transition Mechanism
Not only does the CROM methodology allow for the construction of a predictive model, but it also helps inferthe physical mechanism through which particular events happen. In this case, CROM can be used to extractthe sequence of events leading to flame detachment or attachment. The outcomes of CROM (centroids andtransition matrix) are used in this section to perform this analysis.First, an ideal cluster number N k is chosen such that it is low enough to simplify the analysis but highenough to better identify all three flame states (Sec. 4.1). Second, the clustering process is applied to thefull dataset combining OH-PLIF and all the velocity components. As a result, each centroid obtained iscomposed of 4 different fields (OH and the three velocity directions). It will therefore be possible to describethe interaction between the flame and the velocity field during the flame transition process. Third, thistransition process (both from detached to attached flame and attached to detached flame) is analyzed bycombining the information provided by the transition matrix in the form of cluster transition probabilitieswith the centroids for the combined dataset. In particular, the most likely sequence of events leading toattachment or detachment will be extracted. For optimal mechanism analysis, it is critical to choose a cluster number that discretizes the phase spacein such a way that all states of interest are captured. In this case, the goal is to choose a lower bound on N k such that the attached, detached, and transition flame states are properly refined. Transition matrixstructures and cluster time series are compared in Fig. 5. The cluster time series plots (bottom row in Fig. 5)show the time evolution of the images represented by associating each snapshot to its closest centroid. Asseen from the data, the flame is initially in one state, but quickly transitions to another. Later in time(around t = 0 . N k = 16, as it bestcaptures the transition process and contains the necessary resolution of all three flame states of interest.The transition matrix structure shows clear groups that can be labeled as explained in Sec. 3.4 – clusters1-3 are associated with an attached flame, clusters 8-16 with detached, and clusters 4-7 with a transitionstate. Figure 5 also implies that the selection of N k is more complex than assigning simply one cluster toone state (i.e. for three states, N k = 3), as the cluster number is dependent on the time spent in each state.Because the N k requirement depends on the sufficient resolution of the transition process, and the flametransition between attached/detached states occurs very quickly compared to the flame residence time ineither attached or lifted states, in the end of the clustering process there should be more than one clusterper label. The transition matrix then has non-zero transition probability for several cluster changes, whichallows variations in the density of trajectories in each individual flame state to be better represented. Forexample, having only one centroid in the detached state would be a drastic oversimplification of the evolutionof trajectories corresponding to a detached flame, and would not capture the associated periodicity of theflame anchoring point.Besides facilitating the interpretation of the CROM output, using the smallest possible cluster number thatcaptures the states of interest decreases the statistical uncertainty for the entries of the transition matrix.Therefore, it should be noted that if the sole interest is the forecasting power of the model, the optimalnumber of clusters N k will be different as will be shown in Sec. 5. For the purposes of centroid analysis asit pertains to transition mechanism identification, in the following discussion, centroids were produced for acluster discretization of N k = 16. In this section, the centroids obtained for the value N k = 16 are analyzed to illuminate the behavior of theswirling flame during the attached, detached and transitioning phases.9 igure 5: (Top) Transition matrices for N k = 3, 9, and 16. (Bottom) Corresponding cluster time series for the first0.4 seconds. The complete set of attached, transition, and detached centroids are shown in Figs. 6, 7, and 8 respectivelyfor the full dataset. These images essentially show how the clustering algorithm organizes the raw snapshots.In the figures, OH isocontours are shown as black lines, and the arrows represent the velocity in the axial andtransverse directions (x and y). Out-of-plane velocity (z component) is shown by the colored contour.In the attached state, full flame symmetry is observed for centroids 15 and 16 which is expected – the OHisocontours indicate a similar flame shape to the instantaneous PLIF attached image seen in Fig. 1. However,in some attached centroids (12, 13, 14), there exists asymmetric alternating high-OH concentration regions(circled in red Fig. 6). For the same centroids, some slight velocity field asymmetries are also observed(”bending” of the out-of-plane velocity streaks and alternating recirculation zone presence), potentiallyindicating the initial formation of a PVC. This is also observed in experimental results [29, 1].Images corresponding to the transition centroids (8-11, Fig. 7) show more complex structures for the velocityand OH profiles. These centroids show strong flame and flow asymmetry, suggesting that the PVC gained instrength. All transition centroids clearly display alternating recirculation zones (zig-zag structure as observedin [29]) with coinciding high OH-concentration fields. It appears that the vortices creating vorticity normal tothe measurement plane serve as flame anchoring points. The transition centroids obtained can be associatedin pairs 8/10 and 9/11 which exhibit symmetry with respect to the swirling axis. The radial symmetryimplies that the detachment process happens independently of the swirling process, but is localized in theazimuthal direction.The detached centroids are shown in Fig. 8. The OH concentration and the flow field are similar to thoseof the transition centroids, and are more pronounced. The flame front and the recirculation zones are inparticular more lifted. One key difference that can be noted in terms of the flame topology is the existenceof hook-like structures for the OH contour in centroid 3 and 7 (red arrows). This suggests that the flame isexposed to stronger strain rates than during the transition.A key aspect from the above discussion is that a pure phase-space clustering process is able to isolateimportant flow features relevant to the transition phenomenon in an unsupervised manner.10 igure 6: Attached centroids of the combined dataset. OH-PLIF isocontours indicated in black lines. PIV-x and ycomponents given by arrow overlays, and PIV-z is given by the heatmap (colorbar units in m/s). Circled regionsenclose regions of increased OH-concentration.
In this section, two different transition mechanisms are analyzed: the detachment process (attached todetached flame) and the attachment process (detached to attached flame). This section combines the physicalinformation gained from individual centroid analysis (Sec 4.2) with the probability transition informationfrom the transition matrix in Fig. 3a in order to interpret the dynamics of the transition mechanism. Theanalysis below is conducted for model generated with the full dataset (OH and all three velocity fieldcomponents).
Figure 9 displays the probability paths as outputted by the model for detachment. Colors associated withthe arrows are coded in the same scale as the probabilities in Fig. 3a – lighter colored arrows indicatehigher probabilities. In Fig. 9, an interesting distinction is that the attached centroids associated with slightasymmetry in the OH field (centroids 12, 13, 14 and 15) have a higher chance of moving on to the transitionclusters. Interestingly, the attached centroid that is most symmetric (centroid 16) has zero chance of movingon directly to a transition centroid, implying that asymmetry is a leading indicator of detachment. Thevelocity field asymmetry was also used as a marker of the beginning of flame detachment in the study ofOberleithner et al. [29]. Here, the results suggest that slight asymmetry of the flame itself could be atthe inception of the detachment. Further analysis will be required to clearly identify the causes of thedetachment. Nevertheless, this finding highlights the capabilities of CROM in assisting the interpretation ofexperimental data for transient flows. 11 igure 7: Transition centroids of the combined dataset. OH-PLIF isocontours indicated in black lines. PIV-x and ycomponents given by arrow overlays, and PIV-z is given by the heatmap (colorbar units in m/s). Recirculation zonecenters in indicated by markers.
The transition centroids are characterized by the tendency to evolve into a neighboring centroid. Eachcluster appears to resolve a particular phase of the periodic swirling motion. Centroid 8 tends to evolve into9, 9 into 10, 10 into 11, and 11 back into 8, etc. In the case of a detachment, both centroids 10 and 11are likely to evolve into the detached centroids 5 and 7. Centroids 8 and 9 can each evolve into differentdetached centroids: 1/2 and 3/4 respectively. Similar features are seen within transition centroid pairs in theevolution to the lifted state; for example, centroids 9 and 11 (one mirrored pair) each evolve into detachedcentroids which exhibit a hook-like feature on the same side of the symmetry plane. Furthermore, centroids8 and 10 (the other transition centroid mirrored pair) evolve into lifted centroids which depict the flameon the opposing side of the symmetry plane (i.e. centroid 8 is left-leaning and evolves into a right-leaninglifted flame – vice-versa for centroid 10). The detached centroids depict a similar periodic tendency as thetransition centroids, as the most probable path is to simply evolve into the next centroid. Assessment ofthe lower stagnation point in the detached regime gives insight to the PVC structure location, and it can beseen that this feature oscillates back and forth across the symmetry plane. An interesting feature to noteis that the detachment process does not include centroid 6, which may be a product of the chosen clusternumber.
In the flame attachment case as outlined in Fig. 10, it is seen that a snapshot in the first cluster has nochance to directly enter the transition region. The most likely detached-to-transition pathway is given bythe progression from centroid 2 (right-leaning detached flame) to centroid 9. Centroid 2 differs from the12 igure 8: Detached centroids of the combined dataset. OH-PLIF isocontours indicated in black lines. PIV-x and ycomponents given by arrow overlays, and PIV-z is given by the heatmap (colorbar units in m/s). In centroids 3 and7, the hook-like structures are indicated by red arrows and the recirculation zones near the burner exit are markedin red. other detached centroids in that there exists a strong recirculation zone on the left of the image (highlightedin red) almost at the same axial location as the flame anchoring point on the right. The flame can thereforebe entrained toward the nozzle by the local negative axial velocity.It is noted that centroids 1 and 2 exhibit a very similar flame front but have a very different velocity field.Centroid 1 does not play a direct role in the attachment mechanism while centroid 2 is key for the inceptionof the attachment. The velocity field is therefore not only at the root of the flame detachment, but alsoplays a key role in flame attachment. In the case that centroid 2 evolves into centroid 9, development of arecirculation zone very near the nozzle exit is seen. In fact, higher probability cases in which a flame entersthe transition state (e.g. centroids 4 to 10, 5 to 11) all depict the presence of a lower recirculation zone nearthe nozzle. The most probable path for transition-to-attached is the progression from centroid 8 to centroid13. Centroid 8 is the most symmetric centroid of all the transition clusters in terms of both the velocity andOH concentration. This appears to help the flame stabilize into the ”V-shape” profile characteristic of theattached configuration.
Along with providing information about the transition mechanism, the transition matrix can also be usedas a forecasting tool. The quantification of model predictability (or predictive strength) is found in theprediction horizon time, τ h , defined in Sec. 3. It is the time after which the transition matrix converges tothe ergodic probability distribution e . This convergence is not a sudden process, since the transition matrix13 igure 9: Probability paths for the cluster transitions in the flame detachment process. Arrows indicate the pathsand are color-coded with the same colorbar as the transition matrix; darker colors are smaller probabilities, andbrighter colors are higher probabilities.Figure 10: Probability paths for the cluster transitions in the flame attachment process. Arrows indicate the pathsand are color-coded with the same colorbar as the transition matrix; darker colors are smaller probabilities, andbrighter colors are higher probabilities. igure 11: Example of second-largest eigenvalue convergence for some P . The horizon time, τ h , is the time at whichthe slope of λ falls below some threshold ε . continuously diffuses towards its stationary state as it is raised to successively higher powers. Therefore, anyinput probability distribution will eventually converge to e given a large enough number of time-steps.Similar to Sec. 4, it is necessary to choose a value for parameter N k , the number of clusters. Here, N k ischosen to optimize forecasting or predictive purposes . With different constraints than in Sec. 4, it will beseen that the optimal number of cluster will also be different. It should be noted that in this section, theprimary focus is to compare the predictive power of different types of data, where the data types in this caseare OH-PLIF, PIV-x, y, and z. This means that unlike in Sec. 4, which utilized a single model composed ofa combined dataset to assess the physical mechanisms of the transition, different models will be created foreach type of data to compare horizon times, uncertainty and predictive capability.This section is organized in the following manner. First, the process of extracting the horizon time τ h from thetransition matrix is demonstrated in Sec. 5.1. Then, the procedure used to set the optimal number of clustersis described in Sec. 5.2. In Sec. 5.3, horizon times obtained with different datasets are compared. Finally, inSec. 5.4, flame transition forecasts are compared across models constructed with different data-types (PIVmeasurements and OH-PLIF). Model validation tests are shown in the Appendix. The prediction capability of the transition matrix can be quantified by the prediction horizon time τ h (see Sec. 3). There are several methods outlined in Ref. [19] that can be used to obtain τ h for a giventransition matrix P (second eigenvalue convergence to zero, probability distribution convergence to ergodicdistribution, convergence of finite-time Lyapunov exponent, and Kullback-Leibler entropy convergence); here,the eigenvalue spectrum of P n is monitored as n increases. An inherent property of P is that the largesteigenvalue is unity and is stationary (does not change in time), meaning that an eigendecomposition of thetransition matrix raised to any discrete positive power always yields a maximum eigenvalue of one. Thecorresponding eigenvector of the unity eigenvalue is the statistically stationary state (the ergodic limit). Asthe system evolves in time, the remaining N k − λ is the last eigenvalue to reach zero. Its value is trackedas the transition matrix is raised to higher and higher powers. Figure 11 shows a typical evolution of thesecond-largest eigenvalue with respect to time. In practice, the prediction horizon time is defined as the timeat which dλ /dt < ε . Here, ε = 1e − Similar to what was done for the analysis of the transition mechanism in Sec. 4, the cluster number must becarefully chosen. In this section, the purpose of CROM is different than in Sec. 4, and as such the optimalnumber of clusters/centroids N k is determined differently. Here, the goal is to optimize the prediction horizon15 igure 12: Out-of-plane velocity field transition matrix uncertainties (left) and horizon times (right) versus N k .Maximum and minimum bounds are indicated by shaded boundaries derived from individual k-means++ runs, redlines indicate mean. time τ h while decreasing the statistical uncertainty of the transition matrix entries. The measure of thisstatistical uncertainty is explained below.Consider the square transition matrix P , where each entry of the matrix is P i,j and i, j = 1 , ..., N k . Onecan then associate a relative error to each element P i,j as δ P i,j = (cid:115) − P i,j N P i,j , (7)where N is the total number of snapshots in the dataset. The measure chosen for the uncertainty quantityis the Frobenius norm of the relative uncertainty matrix, || δ P|| = (cid:118)(cid:117)(cid:117)(cid:116) N N k (cid:88) i =1 N k (cid:88) j =1 − P i,j P i,j . (8)In the rest of the section, the transition matrices and the clusters are constructed with data in the range [0s,1s] of the available measurement. This training set covers 4 of the 8 flame transitions available in the data.The last two transitions (range [1s, 1.5s]) are kept to test the model performances (testing set). Transitionmatrices are constructed for different number of clusters in order to find the optimal N k sought here.Figure 12 displays the uncertainty measure and normalized prediction horizon time as a function of clusternumber, N k , for the PIV-z (out-of-plane velocity) dataset. Prediction horizon time τ h was normalized withthe combustor flow-through time τ f = 14 . N k = 6.The error metric appears to be increasing, albeit slowly, beyond a small relative maximum at N k = 10after the sharp increase at lower cluster numbers. A similar sharp increase trend is seen in the normalizedprediction horizon plot, where the mean stabilizes after N k = 10 ∼
11. The results of both the predictionhorizon and the matrix uncertainty suggest that the optimal cluster number for the PIV-z dataset could be N k = 11. 16 able 1: Optimal N k values with corresponding uncertainties and horizon times as extracted from Fig.13.Figure 13: Comparisons of transition matrix uncertainty (left) and horizon times (right) for the four tested datatypes. The curves are averaged quantities from 20 independent k-means++ runs on the dataset. Maximum andminimum bounds are indicated by shaded boundaries. The process of choosing an optimal cluster number based on horizon time and transition matrix uncertaintywas conducted for the other data-types in the same manner as shown for the PIV-z dataset in Sec. 5.2. Inthe end of the process, different optimal cluster numbers N k are found for different data types. The resultsare summarized in Table 1. Similar model validation test trends as shown for PIV-z in the Appendix wereobserved for models derived from all data-types.Figure 13 shows a more detailed comparisons of normalized horizon times and uncertainty measures for thefour different datasets (OH-PLIF, PIV-x, y, and z directions) as a function of cluster number – this is thesame plot as presented in Fig. 12, but with overlays for all data types for clearer trend visualization. Thecurves shown are average quantities of 20 runs for each data type with minimum and maximum bounds foreach curve displayed in the same corresponding color.The plot shows that the out-of-plane component of velocity (PIV-z) provides the highest horizon time fornearly all cluster numbers. In contrast, the OH-PLIF dataset interestingly gives the lowest horizon time formost of the tested cluster numbers. This is counter-intuitive since the OH concentration is a marker of theflame position and can be expected to be a good descriptor of the lift-off or attached states. This findingcan be interpreted by considering the cause and effect of flame transition. The OH signal could be an effectof the flame transitioning to a new state, while the out-of-plane velocity field could be a cause. As a results,the PIV-z data contains more detail about the future state of the combustor. This agrees with prior analysesin that the PIV-z data was actually found to be at the root of flame transitions [1, 30]. The flame lift-off,for example, is aided by high-strain rates in the inner recirculation zone, which causes flame extinction andformation of a PVC. Since a signature of the PVC can be found in the out-of-plane velocity component(PIV-z), it is unsurprising that the predictive power of this field is high compared to other measurements.17 igure 14: Schematic juxtaposing the procedure for finding P τ p ,exp (directly from data) and the procedure for forfinding P τ p (from CROM). In practice, a tool to forecast transitions between different macroscopic states is invaluable to anticipateand control flame instabilities. As discussed in Sec. 3, CROM can be used to predict the future probabilitydistribution of the combustor state in terms of the clusters. Using the cluster labeling method describedin Sec. 3.4, the predictions can be formulated in terms of the macroscopic flame states. Starting from astate where the flame is in either the detached or attached state, the goal is to understand if the model canaccurately predict the probability of exiting the initial state, or transitioning. In other words, the objectiveis to determine how accurately the model can predict a flame detachment (flame exiting the attached state)as well as a flame attachment (flame exiting the detached state). The data was partitioned in trainingand testing sets in the same way as indicated in Sec. 5.2. The state prediction process is conducted asfollows:(a) Pick one cluster labeled as attached (resp. detached). Start with a δ − distribution P in this particularcluster. P is a vector equal to 0 everywhere and 1 for the index of the cluster picked. This situationwould correspond to observing the flame as being in a state that corresponds to the cluster picked.(b) Determine all snapshots associated with the chosen attached (resp. detached) centroid in (a) in thecluster time series for the testing dataset.(c) Advance P to P τ p using the transition matrix (Eq. 5), where τ p is the time in the future at whichpredictions are sought.(d) Use the experimental snapshots at time τ p relative to the original snapshots determined in (b) todetermine an experimentally obtained P τ p ,exp from the testing dataset.The computed (from model, P τ p ,exp ) and experimental (from data, P τ p ) probabilities are then compared todetermine the accuracy of the ROM. This is schematically shown in Fig. 14.Forecasts are compared for out-of-plane velocity field (PIV-z) and OH-PLIF measurements for conciseness.The number of cluster used for each data-type is indicated in Table 1. Two prediction scenarios were testedfor each data-type: 1) the snapshot initially resides in the attached state (100% chance of a snapshot inattached cluster), and 2) the snapshot initially resides in the detached state (100% chance of a snapshot indetached cluster). For each scenario, three prediction times were assigned to illustrate model performanceat various time scales: τ p = τ f = 14 . τ p = 0 . τ f = 1 .
46 ms (short-timeprediction), and τ p = 7 . τ f = 10 .
22 ms (long-time prediction, i.e. beyond the prediction horizons for bothPIV-z and OH-PLIF). The initial cluster distributions P were propagated through the transition matrix18 igure 15: Forecast results for τ p = τ f for out-of-plane velocity (left) and OH-PLIF (right) data, with the initialcondition in the attached state (flame detachment, or liftoff). Probability values indicated in bars, and relative precenterror e with respect to data-derived quantities is also shown. Y-axis is future probability at τ p . Results shown for τ p /τ f = 1. for these specified τ p values, and the resulting final distribution was then compared with that of the testingdataset.Because different models are generated when using PIV-z and PLIF data (different cluster numbers, differentclusters, different transition matrix), the comparison of performance between datasets is non-trivial. Here,the initial cluster for each dataset is chosen such that 1) the cluster appears to depict a similar flow state,and 2) the final probability distributions P τ p ,exp are similar for both model outputs. This ensures that bothclusters are representative of a similar combustor state. Therefore, the test compares the ability of bothdatasets to predict transitions that are as close as possible. This criterion becomes more apparent in theresults below. This method was used with different pairs of clusters showing similar results. Here it is shownfor one of these pairs.Detachment prediction results for τ p = τ f are shown in Fig. 15. Note that final probabilities are in termsof two categories: 1) ”attached” and 2) ”not-attached”. The category ”attached” is essentially a persistenceprobability for remaining in the attached state after time τ p . The category ”not-attached” includes probabil-ity of entering both transitioning and detached clusters from the attached state. As the quantity of interestin this case is the probability of exiting the attached state, or transitioning out of the attached state, onlythese two categories are considered. In Fig. 15, plots on the left reflect PIV-z results, and tables on the rightreflect OH-PLIF results. The results show predicted probabilities (red) and observed probabilities (green)for both data types. Note that, as alluded to above, the data-derived probabilities (green bars) across PIV-zand PLIF models are nearly equal. Relative percent errors e between model and data-derived quantitiesare shown above the corresponding bars. Note that the PLIF model severely overshoots the probability ofa snapshot leaving the attached state when compared to PIV-z (relative error of 129.47% versus 11.07%).It is clear that in forecasting flame detachment, PIV-z predictions are much more representative of thetesting data than PLIF counterparts. This is in line with previously observed experimental results showingOH-PLIF as a lagging indicator of PVC-induced strain rate, a direct catalyst for flame lift-off [1].When predicting flame attachment for τ p = τ f in Fig. 15, errors associated with the OH-PLIF models areagain expectedly higher. Note that the data-derived probabilities also match to a reasonable degree acrossthe different models. An interesting feature to note is the increased accuracy of models across the board forprediction of flame attachment (Fig. 16) versus flame detachment (Fig. 15): both PIV-z and PLIF modelrelative errors observe notable drops in relative error. Nevertheless, results show that in both detachment andattachment predictions, PIV-z produces more accurate forecasts when compared to the PLIF model.Prediction results for τ p = 0 . τ f and τ p = 7 . τ f are shown in Figs. 17 and 18, respectively. In each of these19 igure 16: Forecast results for τ p = τ f for out-of-plane velocity (left) and OH-PLIF (right) data, with the initialcondition in the detached state (flame attachment). Probability values indicated in bars, and relative percent error e with respect to data-derived quantities is also shown. Y-axis is future probability at τ p . Results shown for τ p /τ f = 1. figures, the flame detachment predictions (similar to Fig. 15) are shown in the upper row and attachment(similar to Fig. 16) in the lower row. As in the τ p = τ f predictions, the short-time predictions in Fig. 17 showa similar trend of higher PIV-z model performance in the detachment process. Because the forecasting timeis smaller by a factor of 10, this effect is not seen to the same level as in Fig. 9. Despite the slightly lower levelof performance here for OH-PLIF, there is high accuracy in both models in short-time predictions – theyboth forecast the expected state persistence with minimal relative errors. In fact, the predictions in Fig. 17for PIV-z in the attachment process gives zero relative error. Note that probabilities for the ”not-attached”and ”not-detached” labels are low in Fig. 17 because the prediction time is extremely small.The long-time predictions in Fig. 18 are provided to showcase CROM-based forecasting for timescales ator beyond model horizon times. Although trends of significantly lower OH-PLIF performance are seenhere in the attachment process again, in the long-time prediction setting, both PIV-z and OH-PLIF modelpredictions show similar patterns. For example, they overshoot transition probability in the detachmentprocess and undershoot in the attachment process. The model predictions (red bars) between PIV-z andPLIF are similar (especially in the detachment process) because the limit of the horizon time for bothmodels has been reached. In other words, in the long-time limit, the models will always approximate theinitial distribution of snapshots based on the training data (see Eq. 6). Additionally, if the initial distributiondoes not perfectly match the ergodic distribution of the Markov chains, and if there are differences betweenthe ergodic distributions in the training and testing data, the discrepancy in the CROM predictions at timesat or beyond the prediction horizon is expected to rise even higher.For a clear illustration of prediction time effect on model performance, the variation in relative errorsas a function of prediction time for the ”attached” label in the detachment process is shown for PIV-zand OH-PLIF in Fig. 19. The red vertical lines indicate the horizon times for the respective models. Asevidenced in the discussion above, the plot shows large performance advantage in the PIV-z model in theranges of τ p /τ f = 0 . τ p /τ f = 5 .
0, after which both errors appear to converge. Note that thisapparent convergence occurs before the PIV-z horizon time and after the OH-PLIF horizon time. Thisimportantly verifies that the prediction horizon time can indeed be used as indicator of forecasting strengthwhen considering prediction times smaller than the horizon times. An important result from Fig. 19 isthat although one can use horizon time as a metric for forecasting strength, the horizon time itself is nota prediction time at which one should necessarily expect accurate forecasts. Note that the OH-PLIF curveincreases until a peak near its horizon time, followed by a decrease until the convergence point. In contrast,the PIV-z curve appears to increase with heavy fluctuations. The intricacies of such behaviors require furtherstudy and are out-of-scope here, though this is an object of future work.20 igure 17: Short-time forecast results for τ p = 0 . τ f for out-of-plane velocity (left) and OH-PLIF (right) data. Upperrow is detachment forecast and lower row is attachment. Annotations made in same manner as Figs. 17 and 18. Ultimately, with the test conditions and model inputs used, out-of-plane velocity can be comfortably classifiedas the most potent dataset with regards to flame transition prognostics. This essentially hints that theprediction horizon time τ h is a good indicator to quantify the predictive strength of dataset. A cluster-based ROM was employed to analyze experimental data of flame transitions in a swirling com-bustor, and to create a model to anticipate such transitions. As an analysis tool, the method allows for theclassification of data into modes that are statistically representative of the flow field and extraction of themost probable path between these modes during flame transitions. An appealing property of this process isthat it can be done in an unsupervised manner, making the analysis as objective as possible. However, inorder to use CROM, the number of modes must be set directly by the user. Two different approaches to setthis number were presented in this work.For the swirl combustor analyzed here, the flame detachment and attachment process were both analyzedby means of the transition matrix and modes outputted by CROM. It was shown how the modes couldbe associated to a macroscopic flame states (attached, detached, transition) and thereby derive a physicalinterpretation from the CROM output. It was found that the flame detachment process stems from anasymmetry in the flow field and for the flame. This suggests that an asymmetric process causes the flamedetachment. This finding is in line with prior experimental analysis which identified PVC as the cause of21 igure 18: Long-time forecast results for τ p = 7 . τ f for out-of-plane velocity (left) and OH-PLIF (right) data. Upperrow is detachment forecast and lower row is attachment. Annotations made in same manner as Figs. 17 and 18.Figure 19: Relative error, e , as a function of normalized prediction time for the ”attached” label in the detachmentprocess. Vertical red lines indicate horizon times. References [1] Q. An, W.Y. Kwong, B.D. Geraedts, and A.M. Steinberg,
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A Model Validation
There are underlying assumptions in the predictive model obtained with the CROM method. First, inorder to use the model to predict unseen data, the model should have been trained on a dataset that isrepresentative of most of the combustor dynamics. Second, the transition from one time-step to the nextis assumed to be Markovian. Here, the validity of these assumptions is assessed. To this end, two testsare conducted and applied to the PIV-z dataset only for conciseness. The same trends hold for all otherdatasets.In the first test, the stationary PDF obtained from the transition matrix e (eigenvector of the unit eigenvalue)is compared to the PDF of clusters obtained from the dataset q [31]. In order to have good agreement betweenboth PDFs, the interplay between the fast and slow timescales of the combustor need to be accuratelyrepresented by the transition matrix. The acceptable agreement shown by Fig. 20 suggests that the trainingdata was sufficient to capture most of the relevant dynamics and allow to accurately describe the long termdynamics of the swirler.The second test consists in comparing short-term forecasts obtained from the transition matrix and the dataused to generate the model. Similar to what was done in Ref. [31], the persistence probability of the clustersis compared between the model and the data. The persistence probability is defined as the probability for asnapshot from some cluster to remain in this cluster after a certain number of time-steps. If the persistence25 igure 21: Test 2: persistence time plot from the PIV-z dataset, where y-axis is probability to remain in the samecluster and x-axis is time-step. probability given by the model does not deviate from that given by the data by a significant amount, theMarkov assumption is valid. It is reminded to the reader that the training data is defined for t = [0 . , . t = [1 . , .
5] seconds (5000 snapshots). Fig. 21shows the persistence probability for three clusters (one for each flame state) plotted against time. Notethat statistical uncertainties are included for probabilities extracted from data, with growing uncertaintiesfor shrinking probabilities. Interestingly, model-data agreement is nearly perfect in persistence probabilityfor an attached cluster, and is fairly well-captured for a transition cluster. In the detached case, data-basedprobabilities diverge quickly from the model counterparts, though uncertainties drastically increase as time-step increases. The severe drop in probability observed by the data-derived curve (in green) in the detachedcluster is expected, as a snapshot starting in a detached cluster has a high chance to move on to the nextdetached cluster in the data, leading to very small persistence probability with increasing time-step. Refiningthe detached state with a higher N kk