Analysis of transonic buffet using dynamic mode decomposition

A typical jet powered aircraft flies at transonic speed such that a supersonic region on the suction side of the airfoil develops which is terminated by a shock wave. For increasing Mach numbers or angles of attack, the pressure rise across the shock wave might become high enough to lead to a shock-induced separation of the boundary layer downstream of the shock wave. Within this transonic flight regime, highly unsteady flow including self-sustained periodic shock-wave oscillations, i.e., transonic buffet, might occur. The shock wave movement is a low frequency/large amplitude phenomenon. The resulting unsteady pressure distribution leads to unsteady loads acting on the wing structure. The aeroelastic response of the wing structure, the so-called buffeting, might lead to a critical state for the wing structure. To date, the mechanisms sustaining the shock-wave oscillations during buffet are not fully understood, yet. A detailed understanding of the buffet mechanisms are necessary to develop precise prediction methods for the onset of buffet, to shift the buffet boundary to higher Mach numbers or angles of attack, and to find strategies to damp or even suppress buffet, which will enhance the operational performance of an aircraft. A comprehensive review on self-sustained shock wave oscillations on airfoils at transonic speeds has been given by Giannelis et al. (2017).

Despite intensive research, the mechanisms leading to buffet are still discussed controversially. Furthermore, the existing theories and models are usually restricted to special airfoils and flow characteristics.

For supercritical airfoils, a widely recognized description of the self-sustaining shock wave oscillation has been given by Lee (1990). His theory gives an explanation for self-sustained shock wave oscillations and a method to estimate the shock-wave-oscillation frequency for buffet flows around supercritical airfoils where the shock wave oscillates sinusoidally on the upper airfoil surface and induces a complete boundary-layer separation downstream. The basic idea is that disturbances which propagate up- and downstream within the flow field downstream of the shock wave form a feedback loop. According to Lee, the oscillating shock wave generates large-scale turbulent structures that propagate downstream and generate pressure waves while passing over the sharp trailing edge of the airfoil. These pressure waves travel also upstream and exchange energy with the shock wave, enhancing its oscillatory motion.

Xiao et al. (2006), Deck (2005), and Hartmann et al. (2012, 2013a, b), found excellent agreement of their numerical and experimental results with Lee’s theory.

Hartmann et al. (2013a) refined the feedback model proposed by Lee (1990). The experiments by Hartmann et al. (2013a) revealed that the sound waves generated at the trailing edge and presenting the upstream propagating part of the feedback loop possess a high frequency which is about ten times higher than the shock-wave-oscillation frequency. Therefore, there has to be another low-frequency mechanism present in the flow field which forces the shock wave to oscillate at a frequency much lower than the trailing-edge noise. It is expected by Hartmann et al. (2013a) that the sound pressure level (SPL) of the sound waves originating at the trailing edge varies with a frequency that corresponds to the buffet frequency, i.e., the shock oscillation frequency. On the one hand, the relative velocity between the incoming flow and the oscillating shock wave is higher when the shock wave moves upstream. It triggers stronger disturbances which convect downstream towards the trailing edge. Hence, acoustic waves of an elevated SPL are generated which force the shock wave to move upstream while interacting with it. On the other hand, the shock wave is weaker when it moves downstream and excites weaker disturbances which convect downstream towards the trailing edge. As a consequence, acoustic waves with a lower SPL are generated which allow the shock wave to move back to its downstream position while interacting with it. In other words, the SPL of the trailing-edge noise and the frequency of the shock movement are supposed to be coupled. The findings of Feldhusen-Hoffmann et al. (2018) obtained from wind-tunnel experiments confirmed the expected variation of the SPL in the trailing-edge region of the airfoil during buffet. Crouch et al. (2009) performed URANS simulations of the transonic flow around a symmetrical NACA 0012 profile for varying freestream Mach numbers and angles of attack, and a chord-based Reynolds number of \({\mathrm{Re}}_{c} = 10^7\). A global stability analysis of the transonic airfoil flow was carried out to predict the onset of flow instabilities. A set of linearized equations was deduced from the Navier–Stokes equations which forms an eigenvalue problem governing the complex frequency and the shape of the global modes. The origin of buffet was predicted by the onset of an instability. The stability boundary as a function of the Mach number and angle of attack is in very good agreement with the boundary obtained from experimental data by McDevitt and Okuno (1985) for Mach numbers below 0.8. The shape of the fluctuating streamwise velocity component of the unstable mode shows a coupled movement of the shock wave and the boundary layer downstream of the shock wave. On the one hand, the shape of the pressure fluctuations of the unstable mode reveals pressure perturbations originating near the shock wave foot moving upward along the shock wave and finally forward into the sonic zone. On the other hand, pressure fluctuations move downstream behind the shock wave, intensify, spread around the trailing edge, and propagate along the airfoil’s pressure side until they enter into the sonic zone upstream of the shock wave. The observed pressure wave propagation is qualitatively different from the buffet model proposed by Lee (1990).

In this study, it is attempted to give further insight into the mechanisms of the buffet phenomenon. The buffet flow field around a supercritical airfoil is analyzed by dynamic mode decomposition (DMD). The velocity data are determined in wind tunnel experiments.

The dynamic mode decomposition (Schmid 2010) is a data-based technique which decomposes the underlying sequence of flow data into spatio-temporal coherent structures, i.e., dynamic modes. Each dynamic mode is associated with a single characteristic frequency, a growth or decay rate and an amplitude. On the one hand, DMD can be used for a reduced-order representation of the flow field, since the full dynamic system can be projected onto a subspace spanned by several extracted modes. On the other hand, it is an effective tool to search for physical mechanisms describing the underlying flow field evolution. There are many decomposition methods, the most common of which are the proper orthogonal decomposition (POD) method (Lumley 1967) and the global stability analysis. The POD modes generally are multi-frequential and the modes are sorted by their energy content, which in general does not allow any conclusion about the dynamical importance. The dynamic modes, however, are orthogonal in time and contain therefore only a single temporal frequency. Since the dominant features of the buffet phenomenon are related to characteristic frequencies, the decomposition by DMD is more suitable than the decomposition by POD. Furthermore, the decay rate of the dynamic modes reveals transient modes which are less dynamically important. The global stability analysis relies on the knowledge of the governing equations of the dynamical system under investigation. It is therefore not appropriate for the decomposition of experimental data. DMD, however, relies on the evaluation of time-discrete snapshot flow data and is therefore suitable to analyze experimental data. The dynamic modes can be regarded as a generalization of the global stability modes for nonlinear flow dynamics (Schmid 2010). If the underlying flow is linear or linearized, the dynamic modes are equivalent to the global modes resulting from the global stability analysis (Schmid 2010). This discussion shows the DMD technique to be an appropriate means for analyzing nonlinear dynamics of flow fields which are available as velocity snapshots. The assignment of dynamic modes to a single frequency will allow the revelation of possible couplings of flow structures. Since the dynamic modes are associated with a growth or decay rate, transient quickly decaying modes, which are of less dynamical importance, can be identified. The formalisms of DMD will be given in Sect. 3.

Since its introduction, DMD has extensively been used for the investigation of flow fields comprising jets, shock-wave/turbulent boundary layer interaction, wakes of airfoils and high-speed trains, wind turbine flow, and thermo-acoustic instabilities. These and other examples can be found in the comprehensive list of Rowley and Dawson (2017) compiled in their review on model reduction for flow analysis and control.

Several researchers applied DMD to the transonic buffet airfoil flow (Masini et al. 2018; Ohmichi et al. 2018; Kou and Zhang 2017; Kou et al. 2018; Gao et al. 2017; Poplingher et al. 2019). In these studies, the input data are pressure snapshots obtained numerically by unsteady Reynolds-averaged Navier–Stokes equations simulations (URANS) using the Spalart-Allmaras turbulence model. Masini et al. (2018) and Ohmichi et al. (2018) analyzed the three-dimensional shock buffet flow of swept wings. Only a few studies applied DMD to the two-dimensional shock buffet flow (Kou and Zhang 2017; Kou et al. 2018; Gao et al. 2017; Poplingher et al. 2019). The investigations comprise flows around a NACA 0012 airfoil (Kou and Zhang 2017; Kou et al. 2018; Gao et al. 2017) and around a supercritical RA16SC1 airfoil (Poplingher et al. 2019).

Kou and Zhang (2017) and Kou et al. (2018) performed flow reconstruction and prediction of periodic dynamics from the initial unstable transient flow solution based on dynamic modes. Besides the time-invariant mode, they found dominant modes related to the shock wave movement and its coupling to the boundary layer. However, the buffet mechanism was not further examined.

Gao et al. (2017) realized an active control to suppress buffet in an unstable steady airfoil flow being perturbed by trailing edge flap oscillations. DMD was used as a complementary tool to detect dominant frequencies and coherent structures. They found the dominant periodic dynamic modes to result from the shock wave movement and the respective frequency to be the shock-wave-oscillation frequency or multiples of it. Since the dominant buffet mode is damped but present under active flow control, the authors stated that buffet occurs due to a global instability, supporting the findings of Crouch et al. (2009).

Poplingher et al. (2019) used DMD of the transonic supercritical RA16SC1 airfoil flow to find dominant dynamic modes describing the buffet flow, to reconstruct the flow field, and to investigate the response of the pre-buffet unstable flow to excitations with the vertical velocity component. The frequencies of the dominant periodic dynamic modes are multiples of the shock-wave-oscillation frequency and the mode shapes present pressure gradients in the shock travel region and in the boundary layer downstream of the shock wave. The authors stated, however, that trailing edge vortex shedding cannot be identified with certainty due to restrictions given by the URANS modeling and the grid resolution in the wake area. The time–space modal history of the buffet mode revealed similar pressure propagation as found by Crouch et al. (2009). The investigation of the modal excitation showed buffet-like mode shapes oscillating with a frequency close to that during buffet onset. The dominant buffet mode is damped but this damping decreases to zero at buffet onset conditions.

The present study applies DMD to experimental velocity data of the transonic buffet flow around the supercritical DRA 2303 airfoil. The analysis evidences individual dominant periodic dynamics of the buffet flow, comprising not only the shock wave oscillation but also the sharp trailing edge vortex shedding. The dominant modes are extracted by combining a priori knowledge of dominant frequencies resulting from preceding investigations with results from sparsity-promoting DMD (SP-DMD). Since the transonic buffet flow exhibits some periodic features, DMD will help to capture the dominant frequency information and stability characteristics. In this context, DMD is not used to reconstruct or predict the flow behavior but to gain further insight into the mechanisms of buffet.

In contrast to the previously mentioned studies, in this work measurement data determine the input snapshots for the DMD analysis of a two-dimensional transonic buffet flow field. The results will show that experimentally obtained velocity distributions are suitable as input data for the DMD analysis of this particular flow case. An a priori investigation is performed to show the sensitivity of the SP-DMD results with regard to the number of snapshots as input data. Since the velocity data are recorded by a high sampling rate, it is possible unlike in the study of Poplingher et al. (2019) to detect the trailing-edge vortex shedding by DMD. Other than in the studies (Masini et al. 2018; Ohmichi et al. 2018; Kou and Zhang 2017; Kou et al. 2018; Gao et al. 2017; Poplingher et al. 2019), in which only indications of the theory of Crouch et al. (2009) were found, the current measurement data based DMD results corroborate Lee’s theory (Lee 1990).

The paper is structured as follows. The Introduction is followed by the description of the experimental setup in Sect. 2. In Sect. 3, the DMD approach is concisely presented. In the results Sect. 4, some characteristics of the DRA 2303 buffet flow field, which are known from previous investigations, are described in Sect. 4.1 to allow a comparison of the former findings with the results determined by the DMD analysis. The DMD results of the DRA 2303 airfoil buffet flow are given in Sect. 4.2, followed by concluding remarks given in Sect. 5.

In the DMD flow field data, e.g., velocity data \({\varvec{v}}({\varvec{x}},t)\) are decomposed into n spatial modes \({\varvec{\phi }}_{n}({\varvec{x}})\) with the amplitudes \(a_{n}\), and the complex frequencies \(\lambda _{n}\)The flow field data are collocated from numerical simulations or measurements as a sequence of N snapshots, e.g., velocity fields, equispaced in time with the time step \(\Delta t\) in the time interval \(t = [0,(N-1)\Delta t]\). The snapshot sequence is column-wise stored in a data matrixwhere the subscript denotes the first entry and the superscript the last entry in the sequence. The spatial dimension M of the data is typically much larger than the number of snapshots N, i.e., \(M\gg N\). A linear mapping is assumed which is approximately the same over the full sampling intervalallowing the sequence to be described as a Krylov sequence (Greenbaum 1997; Trefethen and Bau III 1997)If the snapshots are derived from flow fields with nonlinear dynamics, this assumption is equal to a linear tangent approximation. The sought characteristics of the dynamical process described by the sequence \({\varvec{V}}_{1}^{N}\) can be extracted from the matrix \({\varvec{A}}\) by calculating its eigenvectors and eigenvalues. However, \({\varvec{A}}\) is usually hard to determine due to its vast size of \(M \times M\) with \(M\gg N\). Using DMD, a low-order representation of \({\varvec{A}}\) is determined which equally captures the dynamics of the snapshot sequence.

Schmid (2010) introduced a robust implementation by extracting the dynamic characteristics from a ’full and robust’ DMD matrix \({\varvec{F}}_{\mathrm{dmd}} \in {\mathbb{C}}^{N-1 \times N-1}\) being a low-dimensional representation of the linear intersnapshot operator \({\varvec{A}}\) on the subspace spanned by the basis \({\varvec{U}}\)with \((\cdot )^{*}\) denoting the conjugate transpose of a matrix. Robustness is achieved by determining \({\varvec{U}}\) via a singular value decomposition (SVD) of the snapshot sequenceThe quantities, \({\varvec{U}} \in {\mathbb{C}}^{M\times (N-1)}\) and \(\varvec{W} \in {\mathbb{C}}^{(N-1)\times (N-1)}\) are the left and right singular vectors and the matrix \({\varvec{\Sigma } \in {\mathbb{C}}^{(N-1)\times (N-1)}}\) contains the singular values of the snapshot sequence \({\varvec{V}}_{1}^{N-1}\).

The matrix \({\varvec{F}}_{\mathrm{dmd}}\) describes the dynamics of the snapshot sequence, which has originally been described by the matrix \({\varvec{A}}\) in Eq. (3), on the subspace spanned by the basis \({\varvec{U}}\) such thatThe eigenvalue decomposition of the DMD matrixyields the so-called Ritz eigenvalues \(\mu _{n}\) and the eigenvectors \({\varvec{y}}_{n}\).

The dynamic modes are obtained by multiplying the right singular vectors of the snapshot sequence \({\varvec{V}}_{1}^{N-1}\) with the nth eigenvector \({\varvec{\phi }}_{n}:={\varvec{U}}{\varvec{y}}_{n}\) and their corresponding amplitude based on the first snapshot is given by \(a_{n}:={\varvec{z}}_{n}^{*}{\varvec{x}}_{1}\) with \(\{{\varvec{z}}_{1}^{*},\ldots ,{\varvec{z}}_{N-1}^{*}\}\) being the eigenvectors of \({\varvec{F}}_{\mathrm{dmd}}^{*}\) and \(\varvec{x}_{1}\) representing the initial condition. The snapshots are approximated by a linear combination of the dynamic modeswhich can be written in matrix form for the discrete snapshot sequenceThe matrix filled with the geometric progression of the Ritz eigenvalues is called Vandermonde matrix \({\varvec{V}}_{\mathrm{and}}\) and describes the temporal evolution of the modes. The matrix \({\varvec{D}}_{a}\) contains the optimized amplitudes based on all snapshots \(a_{n}\) of the modes which can be found by minimizing the objective functionwith \(\Vert \cdot \Vert _{F}\) being the Frobenius norm. For input data from experiments or numerical simulations contaminated by noise and/or other uncertainties, the matrix \(\varvec{\Sigma}\) usually has a full rank such that the DMD analysis yields the maximum number of \((N-1)\) modes and eigenvalues. The classical optimized DMD generally extracts \(r\le (N-1)\) modes with r being the rank of the matrix \(\varvec{\Sigma }\), which results from the SVD of the snapshot sequence given in Eq. (6). Knowing the complex quantities \(\{a_{n},\mu _{n},\varvec{\phi _{n}}\}\), it follows from Eqs. (1) and (9) thatThus, the real part \(\lambda _{\mathrm{r}}\) of the complex frequency contains information about the growth or decay rate of the DMD mode. The dynamic mode is stable for decaying \(\lambda _{\mathrm{r}}\), i.e., \(\lambda _{\mathrm{r}}<0\), and unstable for an excited \(\lambda _{\mathrm{r}}\), i.e., for \(\lambda _{\mathrm{r}}>0\). For \(\lambda _{\mathrm{r}}=0\), the mode is a stable mode which is present throughout the whole snapshot sequence. Transient modes, which do not contribute significantly to the data sequence, are characterized by a high decay rate \(\lambda _{\mathrm{r}}\ll 0\) and therefore by complex Ritz eigenvalues \(\mu _{n}\) with an absolute value of \(r_{n}\ll 1\) [see Eq. (12)]. Periodic stable Ritz eigenvalues are characterized by an absolute value of \(r_{n}= 1\). The imaginary part \(\lambda _{\mathrm{i}}\) of the complex frequency contains information about the temporal periodicity of the DMD mode, given in the unit [rad]. The frequency in the unit [Hz] is \(f=\mathfrak {I}(\ln (\mu _{n}))/(2\pi \Delta t)\). For \(\lambda _{\mathrm{i}}=0\), the respective frequency is zero meaning that the dynamic mode represents the mean mode. Due to real input data, the Ritz eigenvalues \(\mu _{n}\) with a non-zero frequency appear as pairs of complex conjugates of the same frequency and decay or growth rate. Detailed information on the DMD method and its application can be found in Schmid (2010) and in recent review papers (Rowley and Dawson 2017; Taira et al. 2017, 2020).

Identifying the dynamically relevant modes is usually not straightforward and makes the ’manual’ mode selection very tedious in practice. The standard approach to select the dominant dynamic modes by their amplitude \(a_{n}\) is often not reasonable since transient modes are likely to appear with artificially high amplitudes as will be shown in the following section. There exist, however, some pre- and post-processing approaches to identify dynamically relevant modes, which can be applied depending on the problem under investigation and the main objective of the analysis. In this study, DMD mode selection based on the comparison to the spectral analyses and on the sparsity-promoting DMD (SP-DMD) have been applied. They will be explained in the following.

A preceding spectral analysis by, e.g., FFT or the PSD function of the data, can help to find distinct frequencies, and thus eigenvalues, which are dominant in the flow field and should be further analyzed by DMD.

The SP-DMD, introduced by Jovanović et al. (2014), is an extension of the standard DMD algorithm which seeks a subset of DMD modes that have the most substantial influence on the quality of the approximation of the snapshot sequence. Note, however, that the SP-DMD also might select transient modes. The user defines a tradeoff between the approximation error and the number of extracted modes. A constraint consisting of a penalty term for the number of non-zero elements in the amplitude vector \(\varvec{a}\) is introduced to the least-squares problem given in Equation (11) and forces a sparse solutionThe parameter \(\gamma\) is the positive regularization parameter. The larger its value, the more zero elements exist in the amplitude vector \(\varvec{a}\). This sparsity structure is fixed in the following and the optimal amplitudes are computed for the resulting subset of dynamic modes. Jovanović et al. (2014) developed an efficient algorithm to solve the regularized least-squares problem given in Eq. (13) which uses the alternating direction method of multipliers (ADMM). The ADMM algorithm requests the user to choose a maximum number of iterations, to set the augmented Lagrangian parameter, or step-size parameter, \(\rho\), which also determines the rate of convergence of the algorithm, and to give the absolute and relative tolerances \(\epsilon _{\mathrm{abs}}\) and \(\epsilon _{\mathrm{rel}}\) for the minimization steps. The step-size parameter \(\rho >0\) influences the convergence speed of the algorithm (Ghadimi et al. 2012). Furthermore, large values of \(\rho\) diminish the primal residual and augment the dual residual of the optimization problem and the opposite is true for small values of \(\rho\). It has been shown, however, that the ADMM method converges for all values of \(\rho\) (Boyd et al. 2011). The primal and dual residuals of the optimization problem can be used to define stopping criteria \(\epsilon _{\mathrm{abs}}\) and \(\epsilon _{\mathrm{rel}}\) for the ADMM algorithm (Boyd et al. 2011). The details are given in Jovanović et al. (2014). Tu et al. (2014) addressed the appropriate selection of dominant dynamic modes and noted that the SP-DMD approach proposed by Jovanović et al. (2014) is well suited for the DMD analysis.

In this section, the results of the experimental study of the DRA 2303 buffet flow field are presented. First, a general description of the flow field is given in Sect. 4.1, i.e., the main flow features and dominant frequencies are emphasized. This overview is used to justify the selection of relevant dynamic modes based on the comparison to spectral analyses as mentioned in Sect. 3. Furthermore, this overview allows to evaluate whether the dominant features of the buffet flow and the proposed feedback loop (Lee 1990) are correctly detected by the SP-DMD analysis.

The transonic buffet flow field around a supercritical DRA 2303 airfoil model has experimentally been investigated by particle-image velocimetry measurements. For the first time, the experimentally obtained velocity data of a buffet flow field are further processed to perform a sparsity-promoting dynamic mode decomposition (SP-DMD).

The physics of the buffet phenomenon around the DRA 2303 airfoil was thoroughly discussed, e.g., in Hartmann et al. (2013a) and in Feldhusen-Hoffmann et al. (2018), supporting the buffet model by Lee (1990). This model assumes that the shock wave oscillations are driven by an acoustic feedback loop consisting of a downstream propagating part represented by shock induced vortices and an upstream propagating part represented by sound waves which are generated when the downstream moving vortices pass over the trailing edge. The shock wave movement is supposed to be sustained by a varying SPL of the sound waves interacting with it.

From the previous studies, two dominant frequencies, the low frequency of the shock wave oscillation and the high frequency of the vortex shedding, are known. Using the present SP-DMD analysis these frequencies could be assigned to their spatial structures. The low-frequency buffet mode captures the coupling of the shock wave oscillation and the pulsation of the recirculation region. The downstream location of the shock wave is associated with a less pronounced separation of the boundary layer than the upstream location. Changes in the velocity of the flow in the recirculation region start at the shock foot and convect downstream towards the sharp trailing edge. This leads to a wave-like up- and down motion of the recirculation region, also visible in the w-velocity component of the buffet mode. Hartmann et al. (2013a) argued that the changing size of the recirculation region enhances the variation of the trailing-edge noise SPL. For a larger recirculation region, they found a lower wall-normal gradient of the streamwise velocity component indicating a lower vorticity which is the main driving mechanism of noise. Furthermore, the locally generated sound waves are weakened while propagating upstream since they undergo stronger interactions and refractions when the recirculation region is larger. This supports the idea of Lee (1990) and Hartmann et al. (2013a) that during buffet, the shock wave is forced to oscillate by the trailing-edge noise of varying SPL. Furthermore, the DMD substantiates the existence of the high-frequency vortex mode that excites the shock oscillations via the Lamb vector.

Four further marginally stable periodic modes have been identified. It is supposed that they are not related to additional singular physical features.

In brief, the study shows that experimentally obtained velocity distributions are suitable for the DMD analysis of the 2D buffet flow and can be used for the future analysis of 3D buffet flow. Since the velocity data are recorded by a high sampling rate, buffet generated trailing-edge vortex shedding is detected by measurement data based DMD, which was not discussed in the literature, yet. The experimental results confirm numerical findings, i.e., the dominant buffet and vortex modes are in good agreement with the feedback loop suggested by Lee (1990).