- freely available
Math. Comput. Appl. 2019, 24(2), 45; https://doi.org/10.3390/mca24020045
2. Theoretical Framework
2.1. POD-Galerkin Reduced Order Modeling Applied to the Unsteady and Incompressible Navier–Stokes Equations
2.2. Physical Stabilization by Satisfying the Kinetic Energy Budget
2.2.1. Enrichment of the POD-Galerkin ROM with the Flow Rate Driving Forces
2.2.2. Enrichment of the POD-Galerkin ROM with the Most Dissipative Scales Based on the Velocity Gradient
- Compute the POD velocity modes and truncate at these POD modes. We note that N is intentionally chosen to be less than the needed number of the POD modes to represent all the features of the coherent energetic scales of the kinetic energy.
- Compute the fluctuating POD gradient modes and truncate at . Where for , a set of orthonormal eigenvectors of the temporal correlations matrix on the fluctuating velocity gradient: (W the mean velocity gradient being removed from these correlations), and is the sequence of the eigenvalues of this latter matrix.
- Compute the following velocity basis functions: .
- Perform the Gram–Schmidt orthonormalization process for the enriched set with respect to the energy-based inner product . This step is the key of the enforcement of dissipative energy modes with high singular values in early ranks of the reduced order basis, which is the opposite case when considering only the classical velocity-based POD modes (dissipative energy modes are classified respectively with very small singular values).
3. Application of the Stabilization Approach to a Typical Aeronautical Injector
3.1. Flow Solver
3.2. Typical Aeronautical Injector of Re = 45,000 Lean Preccinsta Burner
3.2.1. Test Case Presentation
3.2.2. POD Modes Computation for the Preccinsta
3.2.3. The Enhanced Reduced Order Basis
- We choose and start the enforcement by the new velocity modes from the 5th rank. This choice is made because we want to limit the number of classical global POD modes which do not exhibit at the end very large features of spatial scales, as we can see on the modes , and .
- We choose because, as already discussed, we need a large number of velocity gradient-based POD modes in order to reproduce of the small and dissipatives scales of the TKE as shown on Figure 19.
- We perform the Gram–Schmidt orthonormalization process for the enriched set with respect to the energy-based inner product .
- We recall that the choice is done intentionally in order to retrieve some dissipative modes at earlier stages than in the classical POD technique where we can see that even after 12 modes we do not have any modes of large scale’s features.
- The fact that the dissipative energy modes appear at late stages in the classical POD technique with very small singular values is the reason why we are not able to exploit their physical significance even if we increase the dimension of the classical POD reduced order model.
- We add starting velocity-based modes of high singular values and large features of scales.
- This enrichment by small scale enforcement and separation is the key to multi-scale reproduction within the reduced order modeling. We precise once more that this approach is very different than the ones based on the change of the inner product that defines the matrix of the correlations between the instanteneous snapshots, typically the approach where the inner product is considered instead of the inner product. By our approach we enable scale separation, then small scale’s enforcement, which is very hard to distinguish when performing a correlations matrix and then retrieving a complete POD basis: the small scales will remain dominated by the correlated large scales even if we perform this inner product change. Some authors use mathematical calibration in order to retrieve the small scales .
3.2.4. The Temporal Coefficients and Kinetic Energy of the Enriched Reduced Order Model and the Comparaison with the Classical POD-Galerkin Reduced Order Model
3.2.5. 3D Time Fields Obtained by the ROM and the High-Fidelity Model
3.2.6. CPU Time for Offline and Online Computation
4. Temporal Extrapolation of the Dissipative ROM
5. Conclusions and Prospects
Conflicts of Interest
|ROM||Reduced order modeling|
|POD||Proper orthogonal decomposition|
|PVC||Precessing vortex core|
|SVD||Singular value decomposition|
|LES||Large eddy simulation|
|FTT||Flow through time|
|TKE||Turbulent kinetic energy|
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|Operation||Wall Clock Time|
|High-fidelity YALES2 solver (512 cores)||5 days|
|Velocity-based POD + Disipative modes computation (768 cores)||15 h|
|Stabilization by Gram–Schmidt (768 cores)||3 h|
|Galerkin projection (768 cores)||3 min|
|Time python ROM-POD solver (1 core)||s|
|Speed up factor|
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