Up to now, empirical modes, mostly linear combinations of the snapshot data, are the most widely used. DMD can approximate stability eigenmodes from transient data [45] and Fourier modes from post-transient data. In this method, the unsteady data are represented by a time-resolved sequence of n snapshots V These snapshots are assumed to be linearly dependent:. Here, the companion matrix S is given by. The eigenvectors bi of matrix S are projected onto the input snapshots in order to obtain the complex DMD modes vi, while the eigenvalues ki determine modal growth ratios real part and angular frequencies imaginary parts :.

The employed modes ui in 2 are Gram-Schmidt orthonormalized. We use the snapshots of six post-transient periods for the dynamic mode decomposition DMD of the snapshots taken from 6 cycles of fully periodic flow that has been done Fig. The resulting DMD modes are sorted by magnitude as discussed in Sect.

- Jackie Wilson: Lonely Teardrops!
- Pour la sauvegarde de la Terre : Chroniques dune Invitation à la Vie Tome 9 (French Edition).
- Breadcrumb.
- 2. EARLY YEARS IN POLAND.

Figure 4 displays the corresponding eigenvalues and norms. The eigenfrequencies and growth rates are resolved in Fig.

Higher-order modes exhibit multiples of this frequency in a monotonically increasing order. The growth rates almost vanish as predicted by Koopman theory for long-term post-transient data [39]. The first three DMD modes are presented in Figs. The structures slowly grow with dominant spatial periodicity in streamwise direction. The streamwise wave number increases in proportion to the frequency.

While DMD modes yield the same subspace as POD modes, they point in different directions—what is observed as the rotation of the modes around x-axis.

The mode amplitudes of DNS are well reproduced by the ROM within few percent error, as expected from earlier similar studies starting with Deane et al. The fluctuation energy is slightly overpredicted as the energy flow to neglected higher-order modes is not accounted by an eddy viscosity model. The flow is visualized by iso-surfaces of transverse velocity v top and w bottom. Next, the ROM is test for off-design conditions, the transient from steady solution to limit cycle Sect.

The initial condition of the mode amplitudes is obtained from straightforward Galerkin approximation:. For brevity, only odd modes 1, 3, 5 are presented. Evidently, the ROM underpredicts the growth rate. The limit cycle is researched after about units as opposed to This was to be expected as the onset of oscillations is described by the dominant stability eigenmodes which have, by construction, a larger growth rate than DMD modes. Velocity magnitude depicted on x — y plane top and iso-surfaces of velocity magnitude on x — z plane bottom.

One reason of GM-A poor reproduction of the transients is the unresolved base flow change. This deficit is cured by augmenting the model with the shift mode [30], the difference between averaged and steady solution orthonormalized with respect to 6 modes. The shift mode is also known as zeroth mode [34] and depicted in Fig. The resulting seven-dimensional model, called GM-B for brevity, exhibits a larger growth rate at the expense of a significant overshoot Fig.

The overshoot also be observed in cylinder wake flows for the analog ROM and is explained in [51]. Our study corroborates a frequent observation that empirical Galerkin models can well reproduce periodic flows but are challenged for off-design conditions, like transients. In this section, we construct a ROM, referred to as Galerkin model C, which is optimized for the linear dynamics near the fixed point. The training data comprise the first snapshots of the transient discussed in Sect. The linear dynamics is governed by one exponentially growing oscillatory stability mode in accord with the literature.

The direct computation of this mode requires the solution of huge, non-Hermitian eigenvalue problems. Global stability analysis is still hardly performable for three-dimensional flows, even for simple geometries without symmetries [53].

## On long-term boundedness of Galerkin models | Journal of Fluid Mechanics | Cambridge Core

DMD offers a convenient data-driven alternative to distill the most dominant stability modes. The corresponding eigenvalues and norms of the DMD modes are depicted in. Intriguingly, the first modes have positive growth rate. The first 3 DMD mode pairs define the six-dimensional expansion as discussed in Sect. These modes are depicted in Figs.

## Cookies on our website

A Galerkin model based on these DMD modes diverges. Hence, we add the stabilizing shift mode to obtain a seven-dimensional Galerkin model C, the analog of model B for the limit-cycle data. The transient of GM-C see Fig. Both shortcomings are also observed for the analog mean-field model of the cylinder wakes with stability and shift modes [30]. In this section, we integrate the good performances of Galerkin model B and C in their respective ranges by a continuous mode interpolation CMI of the 6 oscillatory expansion modes.

This interpolation has been applied to a transient of a cylinder wake [27,48] and to a flow around an airfoil with changing Reynolds number and varying angles of attack [46]. Starting point of CMI is the Fredholm eigenproblem in space domain.

### Publications

The corresponding kernels are denoted by R0 and R1. Now, we introduce the linearly interpolated kernel. We "purify" the kernel from neglected eigenmodes by reconstructing the initial and final functions from. K0 and Ki are the initial and post-transient levels of fluctuation energy. K measures the magnitude of the Reynolds stress and thus, to first order, also the mean-field deformation from the fixed point or, equivalently, the shift-mode amplitude.

If k exceeds 1, its value is averaged using the previous value of k. The resulting Galerkin expansion starts with the unstable steady solution as base flow, contains the interpolated oscillatory modes and includes the shift mode as seventh mode:. The oscillatory modes i — 1, The steady solution us and shift mode u7 span the base flow approximation. The shift mode is exactly orthogonal to antisymmetric modes and approximately orthogonal to the remaining oscillatory modes.

The corresponding Galerkin system 4 is obtained by straightforward Galerkin projection of the Navier-Stokes equation on the basis The bottom figure shows the interpolation parameter. Figure 20 shows the corresponding phase portraits in good agreement with the DNS data. This study concerns model order reduction for an incompressible, three-dimensional, viscous flow past a sphere at Reynolds number The flow around sphere is a well-investigated textbook prototype of 3D bluff-body wakes.

Focus is placed on the transient dynamics from an unstable steady solution to the post-transient limit cycle. This transient flow constitutes a challenge for reduced-order models ROM as the base flow, and the associated coherent structures change considerably. The snapshot data are obtained by a direct numerical solution of the Navier-Stokes equations. One motivation for this transient comes from need of a control-oriented ROM to describe both unforced and forced transients [8]. Starting point are low-order Galerkin models based on the most dominant DMD modes.

- Spirits...They Are Present?
- String Quintet G Major - Viola 1.
- Deep in the Heart of Texas.

We emphasize that the dynamical system is derived from first principles, while the expansion is extracted from snapshot data. As expected from other wake studies, the DMD model incorporating the first 3 harmonics 6 modes describes well the limit cycle with corresponding training data but underpredicts the transient time by roughly a factor of 5.

Conversely, the DMD model optimized for the initial linear dynamics is far more accurate but diverges in the long term. Even the inclusion of a shift mode does not cure a significant overshoot. These results are well aligned with earlier cylinder wake studies starting with the pioneering POD model of Deane et al.

In this study, we address the coherent structure change from linear to nonlinear dynamics with a continuous mode interpolation [27] between the initial and final DMD expansion of sixth order. This interpolated expansion resolves the first, second and third harmonics. In addition, the shift mode is added as seventh mode to resolve changes in the zeroth harmonics or, equivalently, the base flow.

The resulting Galerkin system is derived, again, from the Navier-Stokes equation.

The effect of mode interpolation on the model accuracy is quite dramatic. Both the initial exponential growth and the limit cycle are well resolved, i. The bold curves represent limit cycle.

We conjecture that the presented model order reduction with shift mode s for the base flow variation and mode interpolations for coherent structure changes is applicable for a large range of transient flow dynamics— in particular, oscillatory coherent structures. There is no algorithmic restriction.

## Model order reduction for a flow past a wall-mounted cylinder

For multiple or broadband frequency dynamics, one should keep in mind that the interpolation between two states is far from trivial. An oscillation at one state may slowly deform into another oscillation during a transient, as in our study. The transient may also display frequency cross talk, e.

In this case, the modes of both frequencies need to be included during the transient [25]. Otherwise, the continuous mode interpolation might lead to a sudden jump between both oscillations by mode switching in the corresponding interpolated eigenvalue problem. This would be a very coarse approximation of the transient behavior.