Dynamic Mode Decomposition for the Comparison of Engine In-Cylinder Flow Fields from Particle Image Velocimetry (PIV) and Reynolds-Averaged Navier–Stokes (RANS) Simulations

Experiments were run by motoring (i.e. running using a dynamometer without fuelling) an optically-accessible single-cylinder SI engine. Table 1 contains engine specifications along with details of the operating conditions for the test point studied. The cylinder liner consists of a fused silica upper section to provide optical access, and has a variable height of between 25 and 39 mm. The lower part of the liner is metal with internal water cooling required by the Torlon piston rings. The optical engine experiences piston ring ageing which affects the quality of the seal with the wall, so there are unknown levels of blow-by losses.

The experimental data were recorded under motored conditions, without fuel injection or ignition. Crank angle-resolved pressure data were recorded with a Kistler 6043A60 pressure transducer. Flow field data were obtained using the particle image velocimetry (PIV) technique. A Photonics Industries DM20-527-DH laser was used with a set of optics to create a light sheet approximately 1 mm thick in the measurement region. A 45\(^{\circ }\) mirror was used to reflect the light beam upwards through a fused silica window insert in the piston and illuminate seeded oil droplets on the desired planes in the cylinder. Images were then taken through the fused silica liner with a Vision Research Phantom VEO 710 L camera. The schematic for the set-up is shown in Fig. 1. The piston window had a diameter of 46 mm, and the width \(\times\) height dimensions of the tumble and cross-tumble plane fields of view after processing the PIV data were 46\(\times\)17 mm and 48\(\times\)38 mm, respectively.

Two separate PIV experiments were run at the same test point, one for each of the planes in the cylinder. The two planes of interest are illustrated together in Fig. 2 with the tumble (\(x-z\) with 1 mm offset away from the flywheel) plane shown in purple, and the cross-tumble (\(y-z\), central) plane shown in cyan. Each PIV experiment consisted of a total of 300 cycles of data, with each set of 300 comprising three runs of 100 consecutive cycles. Within each engine cycle, PIV data were gathered every 5 crank angle degrees (CAD) in the range \(-330\) to \(-30\) CAD after the firing top dead-centre (aTDCf), or equivalently, 330 to 30 CAD before the firing top dead-centre (bTDCf). Vector fields were generated from pairs of images that were processed using the DaVis software (LaVision, V. 8.4.0). More details on the experimental set-up and the PIV data post-processing methods can be found in previous publications (Shen et al. 2021; Fang et al. 2022).

Engine simulations were run in Simcenter STAR-CCM+ In-cylinder solution v2021.2 in a RANS framework. The turbulence model was the realizable \(k-\epsilon\) two-layer model with default constants (Rodi 1991), and the heat transfer model was the Grumo-UniMORE model (Berni et al. 2017). A trimmed mesh was used with hexahedral cells throughout most of the domain and prismatic cells next to wall surfaces. The base mesh size was 0.7 mm with volumetric refinements around the valves set to a minimum 0.35 mm, for approximately 3.4 million cells at bottom dead-centre (BDC). A picture of the mesh is shown along with the tumble and cross-tumble planes in Fig. 2. The available geometry for the CFD included a spark plug and an injector, although the spark plug was replaced by a metal blank in the physical experiments to reduce the background scatter for the PIV measurements. The PISO algorithm was used with a variable time-step reaching a minimum of 1.0E−7 s during low valve lifts. The crevice length (defined as the distance between the top piston ring and the piston surface) was increased to match the ensemble-average experimental pressure trace. This accounts for the unknown amount of blow-by losses in the physical engine. There was no further adjustment of the models. Ensemble-averaged measured intake and exhaust valve lift profiles were used as inputs to the simulation. Boundary and initial conditions were as follows:The simulations were initialised during the exhaust stroke of the previous cycle (at \(-600\) CAD aTDCf in order to take measurements from \(-330\) CAD aTDCf onwards) in order to improve the convergence of the simulation. Multi-cycle analyses were also conducted to study the effect of the initial conditions; the differences in peak pressure were all within 0.2%, so results from the first cycles were retained for this work.

Note that the results in this paper are from two separate CFD simulations that were run in order to reflect the separate tumble and cross-tumble experiments. This is because the experimentally-measured inlet and outlet pressures differed between the two experiments, despite efforts to control the conditions at the same test point. This resulted in two different sets of pressure boundary conditions for the CFD, which manifested as two slightly different peak pressures. This difference is quantified as part of the results discussed in Sect. 3.1.

Various methods of quantitatively comparing vector flow field directions are available in the literature, such as using the root mean square (Enaux et al. 2011; Ameen et al. 2017), point-to-point metrics (Zhao et al. 2019), as well as more detailed weighted indices (Willman et al. 2020). This work is concerned with the overall alignment of vector fields, so the widely-used relevance index (RI) defined by Liu et al. (2011) is chosen for the analysis. Each vector field is re-organised into a column vector, and the RI is given for two fields \({\textbf{q}}_{A}\) and \({\textbf{q}}_{B}\) as:where \(\langle \cdot , \cdot \rangle\) is the inner product, and \(\left\| \cdot \right\| _2\) is the \({\mathbb {L}}^{2}\) norm. By arranging the vector fields into columns prior to the calculation, the RI returns a single value that measures the overall alignment of the fields, ranging between \(+1\) for a perfectly aligned field and \(-1\) for a perfectly opposite field.

Histograms are used in this study to give overall pictures of the velocity magnitudes in a flow field. The velocity magnitude distributions can then be quantitatively compared between flow fields by comparing the histograms and calculating the histogram intersection (also known as the histogram distance). This method has been widely used in image processing over the past few decades; further details can be found in the work by Swain and Ballard (1991). The histogram intersection D is defined as:for two histograms \(h_1\) and \(h_2\) evaluated at each bin i. The value of D varies between 1 and 0 for histograms that overlap perfectly and do not overlap at all respectively.

One of the objectives of proper orthogonal decomposition (POD) is to represent a complex high-dimensional dataset in terms of a reduced low-dimensional approximation. This is done by finding a minimal number of basis functions that preserve as much of the variance in the original dataset as possible (Taira et al. 2017). POD takes snapshots of data as inputs and returns a set of new orthogonal variables as outputs that best capture the variance in the original data. POD can be calculated in discrete form using the singular value decomposition (SVD), which is the method outlined below. Following the example of Brunton and Kutz (2022), we begin by analysing a mean-centred rectangular \(N \times M\) data matrix \({\textbf{Y}}\) with N variables (velocity components at each location in the PIV grid) and M sets of measurements (snapshots). Each column of \({\textbf{Y}}\) contains a separate set of measurements arranged as a column vector, denoted as \(y_{:,m}\) for the mth snapshot where \(m = 1,2,\dots ,M\). The SVD of the matrix \({\textbf{Y}}\) is given as:where \(^{*}\) denotes the matrix transpose. \({\textbf{S}}\) is an \(M \times M\) diagonal matrix containing the singular values as entries, \(s_{m,m}\). The singular values are the square roots of the eigenvalues of the \(N \times N\) row-wise correlation matrix \({{\textbf {Y}}}{{\textbf {Y}}}^{*}\), which represents the correlation between the variables. The \(N \times M\) matrix \({\textbf{L}}\) contains the left-singular vectors as columns, which are the eigenvectors of \({{\textbf {Y}}}{{\textbf {Y}}}^{*}\). Each column of the \({\textbf{L}}\) matrix, \(l_{:,m}\), can then be interpreted as a set of ‘eigen-variables’, scaled by the relevant entry in the singular value matrix, \(s_{m,m}\). The \(M \times M\) matrix \({\textbf{R}}\) consists of the right-singular vectors as columns given by \(r_{:,m}\), which contain the temporal information and give the correct combinations of \(l_{:,m}\) and \(s_{m,m}\) in order to reconstruct the original snapshot of data \(y_{:,m}\). Each combination \(l_{:,m}\times s_{m,m}\times ({r_{:,m}}^{*})\) can be regarded as a ‘POD component’, with combinations of POD components being used to create the POD-reconstructed flow fields.

The columns of \({\textbf{L}}\) are ordered by the amount of variance they capture, and are known as the POD modes. They, therefore, represent an optimal hierarchy, and the matrix \({\textbf{Y}}\) can be approximated by retaining the first few columns of \({\textbf{L}}\) and \({\textbf{R}}\) and the corresponding singular values. For a reduced rank p such that \(p < min(N,M)\), we have the truncated SVD:where \(\tilde{{\textbf{L}}}\) now has dimensions \(N \times p\), \(\tilde{{\textbf{S}}}\) is now \(p \times p\), and \(\tilde{{\textbf{R}}}^{*}\) is \(p \times M\). An interpretation of the truncated SVD is that it represents the dominant, coherent structures in a fluid flow (Chen et al. 2013). The discarded higher-order POD modes are associated with measurement noise (Epps and Techet 2010; Epps and Krivitzky 2019), smaller-scale turbulence or random Gaussian fluctuations (Roudnitzky et al. 2006). It should be noted that the choice of p is subjective, and remains a point of discussion in the literature (Rulli et al. 2021).

In this study, dynamic mode decomposition (DMD) is used as an alternative dimensionality reduction method to extract spatio-temporal coherent structures from an ensemble of PIV data. Following Kutz et al. (2016), we start with a discrete-time dynamical system that is locally linear:where \({\textbf{x}}\) is an N-dimensional state vector defined for up to \(M-1\) observations at instances k for \(k=1,2,\dots ,M-1\), and \({\textbf{A}}\) is the linear operator that maps \({\textbf{x}}_k\) onto the next time-step \({\textbf{x}}_{k+1}\). The matrix \({\textbf{A}}\) controls the temporal evolution of the data; its eigenvectors represent coherent structures that evolve in time according to a frequency and a growth/decay rate given by its eigenvalues. The objective of DMD is to optimally calculate the eigendecomposition of \({\textbf{A}}\). To this end, the problem is constructed as follows. The ensemble of \({\textbf{x}}_{k+1}\) and \({\textbf{x}}_{k}\) vectors can be represented as two \(N \times (M-1)\) data matrices \({\textbf{X}}^{\prime }\) and \({\textbf{X}}\) respectively, where \({\textbf{X}}^{\prime }\) and \({\textbf{X}}\) are simply composed of the last and the first \((M-1)\) columns of the full data matrix \({\textbf{Y}}\): This implementation of DMD requires the columns of the data matrices to be separated by a constant timestep \(\Delta t\). Despite having 300 cycles of PIV data available in total for each plane, the data were gathered as three runs of 100 cycles each. Therefore, in this study, a separate DMD analysis is conducted on each 100 cycle subset, with a constant \(\Delta t = 0.08\) s. With the data matrices established, the linear operator can be approximated as:where \(^{\dagger }\) denotes the Moore-Penrose pseudo-inverse. The pseudo-inverse is a least squares regression algorithm, minimising \(\left\| {\textbf{X}}^{\prime }-{\textbf{A}} {\textbf{X}}\right\| _{F}\), where the subscript \(_{F}\) indicates the Frobenius norm. The matrix \({\textbf{A}}\) is therefore interpreted as the best-fit linear operator that maps the columns of \({\textbf{X}}\) onto the columns of \({\textbf{X}}^{\prime }\). To aid computational efficiency and also reduce the sensitivity to noise, the calculation of Eq. 7 begins by taking the rank-reduced SVDs of \({\textbf{X}}\) and \({\textbf{X}}^{\prime }\), assuming that there is low-rank structure in the dataset: truncated at a rank p, where \(\tilde{{\textbf{U}}}\) is the \(N \times p\) matrix containing the leading left-singular vectors of \({\textbf{X}}\), \({\tilde{\Sigma }}\) is the diagonal \(p \times p\) matrix containing the largest p singular values of \({\textbf{X}}\), and \(\tilde{{\textbf{V}}}\) is the \((M-1) \times p\) matrix containing the leading right-singular vectors of \({\textbf{X}}\). In this case, the hard thresholding method proposed by Gavish and Donoho (2014) was used to provide the truncation value p. For a rectangular matrix assumed to be comprised of entries containing signal and unknown levels of white noise, the optimal hard threshold is given by:where \(\beta\) is the aspect ratio of the input matrix, \(\omega\) is the hard threshold coefficient which can be evaluated numerically following Gavish and Donoho (2014), and \(\sigma _{\text{ median }}\) is the median singular value of the input matrix. The next stage of the DMD algorithm makes use of Eqs. 7 and 8a to give the linear operator:noting that both \(\tilde{{\textbf{U}}}\) and \(\tilde{{\textbf{V}}}\) are unitary matrices. Rather than calculating the full \(N \times N\) matrix \({\textbf{A}}\), the \(p \times p\) matrix \(\tilde{{\textbf{A}}}\) can be found by projecting \({\textbf{A}}\) onto the left-singular vectors:The eigendecomposition of \(\tilde{{\textbf{A}}}\) is then:As the matrices \({\textbf{A}}\) and \(\tilde{{\textbf{A}}}\) are similar, they have the same eigenvalues \(\Lambda\). Finally, Tu et al. (2013) proved that the eigenvectors of \({\textbf{A}}\) (known as DMD modes) are given by:specifying this implementation as the ‘exact DMD’.

Finally, the solution to the system 5 may be expressed in the eigenvector basis:where the matrix \(\varvec{\Phi }\) has columns containing \(\phi _m\) the eigenvectors of \({\textbf{A}}\), \(\Lambda\) is a diagonal matrix containing \(\lambda _m\) the eigenvalues of \({\textbf{A}}\), and \({\textbf{b}}\) is a vector consisting of the mode amplitudes \(b_m\). By considering the initial conditions with \(k=0\), \({\textbf{b}}\) can be calculated from \({\textbf{b}}=\varvec{\Phi }^{\dagger } {\textbf{x}}_1\), where \({\textbf{x}}_1\) is the first snapshot of data. This definition of mode amplitudes relies heavily on the assumption that snapshots evolve linearly in time from the initial condition, which may be only approximately valid for experimental measurements of non-linear flows where there can be strong CCVs and measurement noise may be present (Kutz et al. 2016; Schmid 2022). Outlier flow structures and anomalous measurements may suddenly appear in one snapshot but be absent from the other measurements, causing very high decay rates and large amplitudes (Schmid 2022). High-amplitude modes defined in this way, therefore, do not necessarily contribute to the full time series of measurements or reflect the most important flow dynamics.

An alternative approach to the amplitude definition was proposed by Jovanovic et al. (2014) as part of the sparsity-promoting DMD (SPDMD) algorithm, where an optimal set of amplitudes is calculated by finding the DMD modes that have the highest contribution to the dynamics across the full dataset. This is cast as an optimisation problem, where a solution is sought that minimises the data sequence reconstruction error while retaining as few DMD modes as possible. The resultant optimisation problem is given as:where the first term in the minimisation argument represents the reconstruction error and the second term represents the sparsity of the amplitude vector via the \({\mathbb {L}}^{1}\) norm. The parameter \(\gamma\) controls the trade-off between sparsity and the reconstruction error. Equation 15 is solved via the alternating direction method of multipliers (ADMM), which switches between optimising for reconstruction error at a fixed sparsity and optimising for sparsity at a fixed reconstruction error. More details can be found in Jovanovic et al. (2014).

For the initial validation of the RANS model, the simulated pressure trace was plotted against the experimental average pressure as shown in Fig. 3. A good overall agreement between the model and the experiments can be seen. There is a slight difference in the peak pressure, with the CFD model under-predicting the experimental average by 2.8% for the tumble plane experiment, and over-predicting the experiments by 0.8% for the cross-tumble plane.

After looking at the global pressure, the goal would be to validate the simulated in-cylinder flow fields against experimental data from PIV to ensure that the predictions of local variables are also reliable. However, this is not a straightforward task, as was also pointed out in a previous publication (Shen et al. 2021); significant CCVs in internal combustion engines can result in the ensemble mean becoming unrepresentative of the original PIV dataset. This casts doubt as to the suitability of using the ensemble mean as a validation target for RANS simulations. As previously mentioned, the engine studied in this work experiences strong CCVs in the form of a flapping intake jet. Therefore, the crank angles during the intake stroke while the intake valves are open (between \(-360\) and \(-260\) CAD aTDCf) are of interest. \(-285\) CAD aTDCf is chosen as the analysis crank angle in this study as the intake jet is fully established, and the piston has moved far enough away from TDC to provide a larger field of view for the PIV measurements.

First, the adequacy of using 300 cycles for the PIV ensemble mean at the analysis crank angle of \(-285\) CAD aTDCf is tested using the RI and the histogram distance (see Swain and Ballard for more details (Swain and Ballard 1991) and Fig. 9 of this paper for an illustration), as shown in Fig. 4, for both the tumble plane (left) and the cross-tumble plane (right). The two similarity metrics were used to compare consecutive ensemble-average fields composed of incrementally-increasing numbers of PIV snapshots, increasing from the first snapshot. For example, the RI and the histogram distance between the average of the first 49 PIV snapshots and the first 50 were 1.00 and 0.97, respectively. In this case, the RI converged more quickly than the histogram distance, but a visual inspection of the flow fields supports the conclusion that the ensemble average only changes by small amounts when adding more cycles beyond the first 100, implying that 300 cycles is sufficient for this application. Figure 5 illustrates some example ensemble averages for the cross-tumble plane; note that the speeds in all flow field images in this paper are given in normalised units, denoted as ‘n.u.’. 0 mm in the flow field images corresponds to the centre of the cylinder in the radial directions, and the firing deck in the vertical direction.

The view of the tumble plane at \(-285\) CAD aTDCf is shown in Fig. 6. The top row consists of arbitrary consecutive PIV cycles, and the bottom row consists of the PIV ensemble mean (left) and the RANS CFD results (right). Along the top row, differences in the PIV images can be seen due to turbulent fluctuations, but core flow features can still be identified. In both cycles, the flow is dominated by a strong cross-flow along the top of the cylinder. This creates an anti-clockwise tumble vortex, which has a centre located at approximately (\(x= -10, z= -22\) mm) in both cycles. The tumble vortex structure does not vary much from cycle to cycle as the intake jet is always aiming downwards and away from the intake valves. The ensemble mean (bottom-left) can therefore be a good representation of the individual cycles, constructing a clear tumble vortex and preserving realistic velocity magnitudes, while smoothing over more fluctuating structures. Although the RANS result (bottom-right) predicts a higher flow speed along the left of the cylinder and a slightly lower tumble vortex centre, it still presents a good match with the ensemble mean, with \(RI=0.91\).

However, the flow structures are more complex on the cross-tumble plane. In the optical engine cylinder, separate air streams from the intake valves collide with varying strengths, resulting in jet flapping. The jets collide near the cylinder’s central line of symmetry \((y=0)\), so this behaviour can be observed more clearly on the cross-tumble plane, illustrated in Fig. 7 at \(-285\) CAD aTDCf. The top row of the figure consists of arbitrary consecutive PIV cycles A (left) and B (right), where a high-speed intake jet can be observed in both flow fields. The motion of the jet flapping phenomenon can be seen, with the jet pointing in different directions between cycles. The ensemble mean is shown in the bottom-left of the figure, with a centrally-located intake jet and a vortex on either side. However, a qualitative comparison to the individual cycles reveals that the averaging process has diminished the magnitude of the intake jet. This motivates an investigation into whether the ensemble averaged flow field is a fair representation of the original PIV dataset. In the RANS CFD flow field (bottom-right), a central intake jet is also predicted, though the magnitude of the RANS CFD jet is larger than that of the ensemble mean.

Comparing the CFD results to the ensemble average PIV field in isolation would naturally lead one to the conclusion that the simulation over-predicted the speed of the intake jet, with a potential recommendation that the CFD should be adjusted to give a slower jet speed. Conversely, when looking at some of the individual PIV snapshots, it would seem that the CFD slightly under-predicted the intake jet speed. This opposite conclusion would imply that the average PIV flow field does not faithfully represent the nature of the physics in the cylinder for any given cycle in this case. This hypothesis, which was initially proposed in a previous publication (Shen et al. 2021), is explored further in the next section.

To quantify the directional similarity between the ensemble mean and each of the 300 PIV cycles, the RI was calculated between the ensemble mean and each cycle for both the tumble and cross-tumble planes, shown in Fig. 8. The RI is consistently high on the tumble plane, but it is substantially lower and more variable on the cross-tumble plane. Further information can be obtained by looking at the velocity magnitudes. To represent the overall distribution of velocity magnitudes for each flow field, the magnitudes at each point in the field are plotted as a histogram. The histogram for the ensemble mean is then quantitatively compared to the ‘base case’ histogram via the histogram intersection method (Swain and Ballard 1991). The base case histogram is taken here to be the average of the 300 histograms representing each PIV cycle; note that this average histogram does not suffer from artificial diminishing of velocity magnitudes, as the numbers of instances for each of the velocity bins (which is always non-negative) are being averaged rather than the velocity vectors themselves. This average histogram therefore contains contributions from each of the individual cycles without any cycles cancelling each other out. The plots comparing the base case average histogram to the histogram of the ensemble mean are shown in Fig. 9 for the tumble plane (left) and cross-tumble plane (right). There is a marked difference between the two planes, where the tumble plane has a histogram intersection of 0.88, as opposed to 0.59 for the cross-tumble plane.

This shows that while the ensemble mean is capable of fairly representing the flow on the tumble plane, it does not capture the flow dynamics as well on the cross-tumble plane with regards to both velocity direction and magnitude. The reason for this is due to the additional cyclic variability on the cross-tumble plane. A statistical analysis on the horizontal velocity components in both planes was conducted to illustrate this. Figure 10 shows histograms of horizontal velocity components taken at a representative point in each plane:The velocities in these histograms are normalised with reference to the maximum absolute velocities at those points on the corresponding planes. For the tumble plane, the horizontal velocity component is close to a normal distribution, with a large number of cycles at the centre (near-zero horizontal velocity component, as the vector is mostly vertical at the vortex edge). The dominance of the average velocity at this point explains why the ensemble mean is able to provide a good representation of the ensemble of cycles. On the cross-tumble plane, the horizontal velocity component is more indicative of a bi-modal distribution, and the fitted normal distribution is a poor approximation of the ensemble. This represents the fact that in a large number of cycles, the intake jet points to the left or the right rather than straight downwards. A simple ensemble average is therefore expected to be a poorer representation of this dataset, as the mean velocity is less prevalent in the ensemble. There is therefore potential for more advanced numerical methods to provide a more suitable validation target for RANS results on the cross-tumble plane, and two such methods are explored in the following sections.

As previously discussed, the POD method offers a statistical approach for analysing the flow fields. In this section, the ability of POD to extract core flow features for the analysis of RANS simulations is explored. Figure 11 shows the POD-reconstruction of PIV cycles A (left column) and B (right column) on the cross-tumble plane, with the number of retained POD components increasing from one (at the top row) to 299 (at the bottom row). As there are 300 cycles of PIV measurements, and the data were centred prior to the POD analysis, there are a total of 299 POD modes. The reconstruction containing all 299 POD components is therefore an identical recreation of the original cycle.

Beginning at the top of the figure, the first row shows the first POD component, which captures the most amount of energy in the flow field at each cycle. In both cycle A (left) and cycle B (right), a broad central intake jet structure can be seen with a vortex to either side, producing a flow field that is reminiscent of the ensemble mean. Already at this level, there are signs of jet flapping between cycles A and B. Moving down the figure, the influence of CCVs can be seen as successively higher-order POD components are included in the reconstructions.

For cycle A, there is a large increase in the velocity magnitude of the central intake jet when the largest five POD modes are retained (third row). Cycle B has retained a higher jet velocity on the first mode, but doesn’t show a large increase until the first ten modes are used in the construction (fourth row). There are two points worthy of note here: Cycle-dependency can pose problems when attempting to validate RANS simulations, which only produce flow fields representing average structures. For a 300 cycle dataset using POD, even if an objective cut-off point for the number of components can be identified and fairly applied to each cycle, there will be 300 validation targets for the RANS results. If the measured flow fields are significantly different from one another, such as an intake jet pointing in different directions, then the range of acceptable validation parameters may be excessively broad, making it excessively easy for the RANS model to match against. This effect was demonstrated in (Shen et al. 2021). The choice of cut-off is also an issue in itself; while the inclusion of more components allows for higher jet velocities to be retained, it may also take the reconstruction further away from the dominant flow structures, further increasing cycle-dependency. Deciding on cut-off criteria to optimally balance these aspects is a challenging task, and prone to subjectivity.

The POD-reconstructions are cycle-dependent because space-only POD does not search for correlations in time. POD modes are constructed and ordered according to variance, which is interpreted as energy in the context of velocity data, with the dominant POD component representing the flow structures that capture the largest amount of energy in the original dataset. Each POD mode, therefore, contains a mixture of different frequencies (Torregrosa et al. 2018); fluctuating structures may contain more energy than and be grouped together with steady structures, and as a result, the POD analysis does not guarantee a clear separation between the two (Dawson et al. 2016). This is of interest as the steady structures in their isolation could be a more relevant benchmark for RANS comparison, more closely mirroring the approach of the Reynolds decomposition. This concept is explored in the next section.

To summarise so far, the following observations have been made about the PIV measurements in the cross-tumble plane that present challenges for interpreting and using the data:Space-only, phase-dependent POD was used to investigate the flow fields by analysing the high-energy flow structures. While POD is capable of producing dominant flow fields that retain a higher intake jet velocity, the following limitations were discussed:This motivates an investigation into Dynamic mode decomposition (DMD), which may be thought of as an ideal combination of spatial and temporal dimensionality reduction techniques (Kutz et al. 2016). DMD modes are coherent in both space and time, with a single frequency attached to each mode, and the results do not exhibit cycle-dependency. The DMD technique, therefore, can produce target flow fields across a narrower validation range, which can be more appropriate for RANS model development, as shown in this section.

The plot of DMD mode frequencies against amplitudes is known as the DMD spectrum, which illustrates the dominant frequencies in the dataset. The spectra for each of the three 100 cycle subsets on the cross-tumble plane are plotted on the top row of Fig. 12, where the orange circles indicate the modes with the highest amplitude for each subset. However, as discussed previously, high-amplitude modes may represent outlier flow structures with fast decay rates, not necessarily representing the dataset as a whole (Schmid 2022; Jovanović et al. 2014). This can be investigated further by plotting the decay rates, given by the real parts of the DMD eigenvalues, against the mode amplitudes. This is shown on the bottom row of Fig. 12; modes towards the top-left of these plots indicate high amplitudes and fast decay rates. The high-amplitude modes are directed to the left of the figure in all three cases, with the third 100 cycle set (right) being the most extreme example with the largest amplitude and fastest decay rate relative to all the other modes.

The vector fields corresponding to each of the highest-amplitude modes are plotted along the top row of Fig. 13. Each field lacks a coherent structure, with the extreme case for the third cycle set (top-right) also saturating the colourmap due to the extreme amplitude. These fast-decaying modes are of less relevance to the purpose of this study, which is to construct a suitable validation target for RANS simulations. As RANS simulations cannot predict the stochastic nature of turbulence explicitly, it may therefore be more relevant to consider the 0 Hz background DMD modes and compare those steady features to the RANS results. The 0 Hz modes with the highest amplitudes are plotted as green circles in Fig. 12 and are presented as vector fields in the second row of Fig. 13. The 0 Hz modes for the first two cycle sets (left and middle) look similar; the overall structure is similar to the ensemble mean, but a higher intake jet velocity has been retained which is more representative of the individual cycles. The third cycle set, however, has an intake jet that is even more diminished than the ensemble mean. In addition, while there is a clear anti-clockwise rotational structure to the right of the intake jet, the left-hand counter-part appears to have shifted upwards slightly out of the field of view. An explanation for this can be given by observing some of the single cycle PIV snapshots, plotted in Fig. 14. In these cycles, the intake jet from the right-hand valve appears to be stronger than the left, causing the intake jet to swing round and up to the top-left. As the right-hand rotational structure is fairly consistent among the three 0 Hz modes, it would seem that it is the flow from the left intake valve that is more variable, periodically delivering less air flow into the chamber, contributing to a more extreme left-swing of the jet in some cycles.

However, the large drop in the intake jet velocity for the third cycle set presents a problem when attempting to fairly represent the individual PIV cycles with DMD modes. Due to the nature of comparing snapshots taken at the same crank angle across engine cycles in phase-dependent DMD, the linear mapping between the snapshots may only be approximately true, which calls for an alternative definition of the mode amplitudes that does not rely so heavily on this assumption and the initial conditions. The sparsity-promoting DMD (SPDMD) introduced by Jovanović et al. (2014) addresses this by solving for an optimal set of mode amplitudes that recreates the original dataset as closely as possible with as few modes as possible. Amplitudes defined in this way prioritise DMD modes that have the largest contributions to the entire dataset and resist being skewed by fast decay rates caused by outliers, which may be a more appropriate method of representing the full set of individual PIV cycles in an engine.

The spectra for the SPDMD modes for each set of 100 cycles are given in the top row of Fig. 15. Compared to the standard DMD spectra, the SPDMD spectra are much more similar across the three sets of cycles, with dominant background 0 Hz modes plotted in green giving similar amplitudes. The plots on the bottom row of Fig. 15 show that these dominant background modes also have near-zero decay rates, and contribute to the full datasets as a result.

Finally, the dominant SPDMD modes are plotted as flow fields in Fig. 16. The velocity magnitudes are now much more consistent across the three cycles sets, demonstrating the robustness of the SPDMD method. These flow speeds are also more representative of the individual PIV cycles, with the histogram intersections for each of the three background DMD modes given in Fig. 17. Note that the average ‘base case’ histogram is slightly different in each case, containing the relevant sets of 100 cycles for the DMD analysis. The histogram distances for the first, second, and third PIV subsets are 0.89, 0.87 and 0.89 respectively, showing consistent improvement from the histogram distance of 0.57 given by the ensemble mean.