Low Rank Education of Cascade Loss Sensitivity to Unsteady Parameters by Proper Orthogonal Decomposition

2021 ◽  
Author(s):  
Daniele Simoni ◽  
Davide Lengani ◽  
Vianney Yepmo ◽  
Francesco Bertini ◽  
Pietro Zunino ◽  
...  
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Orthogonal Decomposition ◽  
Low Rank ◽  
Proper Orthogonal

Low Rank Education of Cascade Loss Sensitivity to Unsteady Parameters by Proper Orthogonal Decomposition

2020 ◽  
Author(s):  
D. Lengani ◽  
D. Simoni ◽  
V. Yepmo ◽  
M. Ubaldi ◽  
P. Zunino ◽  
...  
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Least Squares Method ◽  
Orthogonal Decomposition ◽  
Flow Coefficient ◽  
Low Rank ◽  
Optimal Basis ◽  
Loss Distribution ◽  
Pressure Loss Coefficient ◽  
Smooth Kernel ◽  
Proper Orthogonal

Abstract In the present work, Proper Orthogonal Decomposition (POD) has been applied to a large dataset describing the profile losses of Low Pressure Turbine (LPT) cascades, thus allowing: i) the identification of the most influencing parameters that affect the loss generation; ii) the identification of the minimum number of requested conditions useful to educate a model with a reduced number of data. The dataset is constituted by the total pressure loss coefficient distributions in the pitchwise direction. The experiments have been conducted varying the flow Reynolds number, the reduced frequency and the flow coefficient. Two cascades are considered: the first for tuning the procedure and identifying the number of really requested tests, and the second for the verification of the proposed model. They are characterized by the same axial chord but different pitch-to-chord ratio and different flow angles, hence two Zweifel numbers. The POD mode distributions indicate the spatial region where losses occur, the POD eigenvectors provide how such losses vary for the different design conditions and the POD eigenvalues provide the rank of the approximation. Since the POD space shows an optimal basis describing the overall process with a low rank representation (LRR), a smooth kernel is educated by means of Least-Squares method (LSM) on the POD eigenvectors. Particularly, only a subset of data (equal to the rank of the problem) has been used to generate the POD modes and related coefficients. Thanks to the LRR of the problem in the POD space, predictors are low order polynomials of the independent variables (Re, f+ and ϕ). It will be shown that the smooth kernel adequately estimates the loss distribution in points that do not participate to the education. Additionally, keeping the same steps for the education of the kernel on another cascade, loss distribution and magnitude are still well captured. Thus, analysis show that the rank of the problem is much lower than the tested conditions, and consequently a reduced number of tests are really necessary. This could be useful to reduce the number of hi-fidelity simulations or detailed experiments in the future, thus further contributing to optimize LPT blades.

A Dual-Weighted Approach to Order Reduction in 4DVAR Data Assimilation

2008 ◽  
Vol 136 (3) ◽  
pp. 1026-1041 ◽  
Author(s):  
D. N. Daescu ◽  
I. M. Navon
Keyword(s):  
Data Assimilation ◽  
Proper Orthogonal Decomposition ◽  
Initial Conditions ◽  
Order Reduction ◽  
Orthogonal Decomposition ◽  
Low Rank ◽  
Water Model ◽  
Reduced Order ◽  
Model Dynamics ◽  
Proper Orthogonal

Abstract Strategies to achieve order reduction in four-dimensional variational data assimilation (4DVAR) search for an optimal low-rank state subspace for the analysis update. A common feature of the reduction methods proposed in atmospheric and oceanographic studies is that the identification of the basis functions relies on the model dynamics only, without properly accounting for the specific details of the data assimilation system (DAS). In this study a general framework of the proper orthogonal decomposition (POD) method is considered and a cost-effective approach is proposed to incorporate DAS information into the order-reduction procedure. The sensitivities of the cost functional in 4DVAR data assimilation with respect to the time-varying model state are obtained from a backward integration of the adjoint model. This information is further used to define appropriate weights and to implement a dual-weighted proper orthogonal decomposition (DWPOD) method for order reduction. The use of a weighted ensemble data mean and weighted snapshots using the adjoint DAS is a novel element in reduced-order 4DVAR data assimilation. Numerical results are presented with a global shallow-water model based on the Lin–Rood flux-form semi-Lagrangian scheme. A simplified 4DVAR DAS is considered in the twin-experiment framework with initial conditions specified from the 40-yr ECMWF Re-Analysis (ERA-40) datasets. A comparative analysis with the standard POD method shows that the reduced DWPOD basis may provide an increased efficiency in representing an a priori specified forecast aspect and as a tool to perform reduced-order optimal control. This approach represents a first step toward the development of an order-reduction methodology that combines in an optimal fashion the model dynamics and the characteristics of the 4DVAR DAS.

Low rank education of cascade loss sensitivity to unsteady parameters by Proper Orthogonal Decomposition

2021 ◽  
pp. 1-18
Author(s):  
Davide Lengani ◽  
Daniele Simoni ◽  
Vianney Yepmo ◽  
Marina Ubaldi ◽  
Pietro Zunino ◽  
...  
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Least Squares Method ◽  
Orthogonal Decomposition ◽  
Low Rank ◽  
Optimal Basis ◽  
Pressure Loss Coefficient ◽  
Low Pressure Turbine ◽  
Smooth Kernel ◽  
Minimum Number ◽  
Proper Orthogonal

Abstract Proper Orthogonal Decomposition POD) has been applied to a large dataset describing the profile losses of Low Pressure Turbine (LPT) cascades, thus allowing: i) the identification of the most influencing parameters that affect the loss generation; ii) the identification of the minimum number of requested conditions useful to educate a model with a reduced number of data. The dataset is constituted by the total pressure loss coefficient distributions in the pitchwise direction. Two cascades are considered: the first for tuning the procedure and identifying the number of really requested tests, and the second for the verification of the proposed model. Since the POD space shows an optimal basis describing the overall process with a low rank representation (LRR), a smooth kernel is educated by means of Least-Squares method (LSM) on the POD eigenvectors. Particularly, only a subset of data (equal to the rank of the problem) has been used to generate the POD modes and related coefficients. Thanks to the LRR of the problem in the POD space, predictors are low order polynomials of the independent variables (Re, f + and f ). It will be shown that the smooth kernel adequately estimates the loss distribution in points that do not participate to the education. Thus, analysis show that the rank of the problem is lower than the tested conditions, and consequently a reduced number of tests are really necessary. This could be useful to reduce the number of hi-fidelity simulations or detailed experiments in the future.

scholarly journals A tale of two airfoils: resolvent-based modelling of an oscillator versus an amplifier from an experimental mean

2019 ◽  
Vol 881 ◽  
pp. 51-83 ◽  
Author(s):  
Sean Symon ◽  
Denis Sipp ◽  
Beverley J. McKeon
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Resolvent Operator ◽  
Stokes Equations ◽  
Velocity Fields ◽  
Orthogonal Decomposition ◽  
Low Rank ◽  
Experimental Measurements ◽  
Mean Velocity ◽  
Proper Orthogonal ◽  
The Resolvent Operator

The flows around a NACA 0018 airfoil at a chord-based Reynolds number of $Re=10\,250$ and angles of attack of $\unicode[STIX]{x1D6FC}=0^{\circ }$ and $\unicode[STIX]{x1D6FC}=10^{\circ }$ are modelled using resolvent analysis and limited experimental measurements obtained from particle image velocimetry. The experimental mean velocity fields are data assimilated so that they are solutions of the incompressible Reynolds-averaged Navier–Stokes equations forced by Reynolds stress terms which are derived from experimental data. Resolvent analysis of the data-assimilated mean velocity fields reveals low-rank behaviour only in the vicinity of the shedding frequency for $\unicode[STIX]{x1D6FC}=0^{\circ }$ and none of its harmonics. The resolvent operator for the $\unicode[STIX]{x1D6FC}=10^{\circ }$ case, on the other hand, identifies two linear mechanisms whose frequencies are a close match with those identified by spectral proper orthogonal decomposition. It is also shown that the second linear mechanism, corresponding to the Kelvin–Helmholtz instability in the shear layer, cannot be identified just by considering the time-averaged experimental measurements as an input for resolvent analysis due to missing data near the leading edge. For both cases, resolvent modes resemble those from spectral proper orthogonal decomposition when the resolvent operator is low rank. The $\unicode[STIX]{x1D6FC}=0^{\circ }$ case is classified as an oscillator and its harmonics, where the resolvent operator is not low rank, are modelled using parasitic modes as opposed to classical resolvent modes which are the most amplified. The $\unicode[STIX]{x1D6FC}=10^{\circ }$ case behaves more like an amplifier and its nonlinear forcing is far less structured. The two cases suggest that resolvent-based modelling can be achieved for more complex flows with limited experimental measurements.

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