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.

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.

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

scholarly journals Investigation of the Effects of Length to Depth Ratio on Open Supersonic Cavities Using CFD and Proper Orthogonal Decomposition

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Ibrahim Yilmaz ◽  
Ece Ayli ◽  
Selin Aradag
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Reduced Order Modeling ◽  
Orthogonal Decomposition ◽  
Navier Stokes ◽  
Reduced Order ◽  
Modeling Approach ◽  
Depth Ratio ◽  
Minimum Number ◽  
Proper Orthogonal

Simulations of supersonic turbulent flow over an open rectangular cavity are performed to observe the effects of length to depth ratio (L/D) of the cavity on the flow structure. Two-dimensional compressible time-dependent Reynolds-averaged Navier-Stokes equations with k-ωturbulence model are solved. A reduced order modeling approach, Proper Orthogonal Decomposition (POD) method, is used to further analyze the flow. Results are obtained for cavities with severalL/Dratios at a Mach number of 1.5. Mostly, sound pressure levels (SPL) are used for comparison. After a reduced order modeling approach, the number of modes necessary to represent the systems is observed for each case. The necessary minimum number of modes to define the system increases as the flow becomes more complex with the increase in theL/Dratio. This study provides a basis for the control of flow over supersonic open cavities by providing a reduced order model for flow control, and it also gives an insight to cavity flow physics by comparing several simulation results with different length to depth ratios.

scholarly journals Proper orthogonal decomposition method for Schrödinger equation

2013 ◽  
Vol 54 ◽  
Author(s):  
Raimondas Čiegis ◽  
Gerda Jankevičiūtė ◽  
Teresė Leonavičienė
Keyword(s):  
Schrödinger Equation ◽  
Proper Orthogonal Decomposition ◽  
Decomposition Method ◽  
Schrodinger Equation ◽  
Approximate Solutions ◽  
Orthogonal Decomposition ◽  
Optimal Basis ◽  
One Dimensional ◽  
Basic Ideas ◽  
Proper Orthogonal

In this paper we consider the proper orthogonal decomposition (POD) method for one-dimensional Schrödinger equation. We begin of the review of basic ideas of POD. Later this method is applied to study the linear Schrödinger equation. The generation of optimal basis using POD and model reduction questions are discussed. Also the errors between the POD approximate solutions and the exact problems solutions are calculated. The results of two numerical examples for standing and travelling Gaussian wave are presented and analyzed.

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.

Unsteady Flow Interactions Between a High- and Low-Pressure Turbine: Part 2 — Rotor-Synchronic Averaging and Proper Orthogonal Decomposition of the Unsteady Flow Fields

2019 ◽  
Author(s):  
M. Dellacasagrande ◽  
P. Z. Sterzinger ◽  
S. Zerobin ◽  
F. Merli ◽  
L. Wiesinger ◽  
...  
Keyword(s):  
Flow Field ◽  
Unsteady Flow ◽  
Proper Orthogonal Decomposition ◽  
Low Pressure ◽  
Fast Response ◽  
Orthogonal Decomposition ◽  
Pressure Probe ◽  
Low Pressure Turbine ◽  
Aerodynamic Pressure ◽  
Proper Orthogonal

Abstract This paper, the second of two parts, presents an experimental investigation of the unsteady flow field evolving in a two-stage two-spool test turbine facility. The experimental setup, which was designed to reproduce the operating condition of modern commercial aero-engines, consists of a high-pressure turbine (HPT) stage followed by a turbine center frame (TCF) with non-turning struts, and a co-rotating low-pressure turbine (LPT) stage. Measurements carried out with a fast-response aerodynamic pressure probe (FRAPP) were post-processed to describe the unsteady evolution of the flow downstream of the HPT rotor, through the TCF duct, and at the exit of the LPT stage. The time-resolved results presented in the first part of this paper show that deterministic fluctuations due to both rotors characterize the flow field downstream of the LPT. In order to characterize the deterministic unsteadiness induced by all the components constituting the turbine facility (HPT, TCF and LPT) and their interactions, measurements were carried out in three different planes located downstream of the HPT, at the exit of the TCF and downstream of the LPT stage. The unsteady results obtained in the plane located at the exit of the LPT are discussed in more details in this second part of this paper, providing information about the interactions between the two rotors. A proper phase-average procedure, known as rotor synchronic averaging (RSA), which takes into account the rotorrotor interaction, was adopted to capture the unsteadiness due to both rotors. Proper Orthogonal Decomposition (POD) was also applied to provide a characterization of the major contributors in terms of energy to the deterministic unsteadiness occurring in the test turbine facility. At the exit of the LPT rotor, the perturbations induced by the HPT stage and the interactions between the two rotors were found to dominate over the unsteadiness due to the LPT only.

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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