Proper Orthogonal Decomposition Method Research Articles

The structure and dynamics of the laminar separation bubble

A novel selective mode decomposition, proper orthogonal decomposition and dynamic mode decomposition methods are used to analyse large-eddy simulation data of the flow field about a NACA0012 airfoil at low Reynolds numbers of $5\times 10^4$ and $9\times 10^4$ , and at near-stall conditions. The objective of the analysis is to investigate the structure of the laminar separation bubble (LSB) and its associated low-frequency flow oscillation (LFO). It is shown that the flow field can be decomposed into three dominant flow modes: two low-frequency modes (LFO-Mode-1 and LFO-Mode-2) that govern an interplay of a triad of vortices and sustain the LFO phenomenon, and a high-frequency oscillating (HFO) mode featuring travelling Kelvin–Helmholtz waves along the wake of the airfoil. The structure and dynamics of the LSB depend on the energy content of these three dominant flow modes. At angles of attack lower than the stall angle of attack and above the angle of a full stall, the flow is dominated by the HFO mode. At angles of attack above the stall angle of attack the LFO-Mode-2 overtakes the HFO mode, triggers instability in the LSB and initiates the LFO phenomenon. Previous studies peg the structure, stability and bursting conditions of the separation bubble to local flow parameters. However, the amplitude of these local flow parameters is dependent on the energy content of the three dominant flow modes. Thus, the present work proposes a more robust bursting criterion that is based on global eigenmodes.

Investigation of the characteristics of the heaving lock-in region of the 10:1 π-shaped section

To gain a comprehensive understanding of the intrinsic characteristics of the lock-in region of the π-shaped section, a study was conducted to investigate the influence of the Angle of Attack (AOA), damping ratio, and mass ratio on the characteristics of the two lock-in regions using a 1:50 segment model test. The Proper Orthogonal Decomposition (POD) method was employed to analyze the characteristics of the two lock-in regions, and the flow field difference between the two lock-in regions was analyzed using Computational Fluid Dynamics (CFD). The findings indicate that both the Angle of Attack (AOA) and mass ratio have comparable effects on the lock-in region. However, it is noteworthy that the AOA appears to exert a more pronounced influence on the second lock-in region, whereas the damping ratio predominantly impacts the amplitude of the lock-in region. The analysis of the distribution and jumping degree of the surface time-averaged pressure enables us to infer the key-action zone number and action position of vortices present around the given section. In the two-lock-in region, there exist two different distribution forms for the energy ratio of POD modes. The first lock-in region is primarily affected by the Karman vortex street phenomenon, whereas the second lock-in region is mainly influenced by the long tail vortex shedding.

An efficient dimensionality reduction approach for modelling cryogenic hydrogen release

This study develops a precise and cost-effective method for reducing the complexity of cryogenic hydrogen unintended release. For this purpose, the dominant modes of a well-characterised cryogenic hydrogen jet (Sandia National Laboratories), with stagnation pressure and temperature fixed at 5 bar and 50 K, are extracted using the proper orthogonal decomposition (POD) method. Building on validating the large eddy simulation (LES) against experimental measurements, the velocity field within the LES results is analysed near the release point. The mass fraction half-width of cryogenic hydrogen jets grows linearly with a slope of 0.1115 against the downstream distances, resembling room-temperature jets. The first 10 extracted modes from 500 LES snapshots demonstrate dominance over the remaining modes. In addition, this study evaluates the flammable gas dispersion using LES and 10-mode POD, showing POD effectively identifies high-risk areas within hydrogen's flammability range. Minor deviations (2.1–3.3%) confirm POD's reliability for hazard assessment compared with LES. Furthermore, an in-depth analysis is performed on the radial distributions of hydrogen mass fraction and centreline hydrogen mole fraction.

Exploration on flutter mechanism of a damaged transonic rotor blade using high-fidelity fluid–solid coupling method

Structural damage in turbomachinery is a primary origin of aeronautic accidents, which is receiving increased attention. This study is thus focused on the aeroelastic analysis of damaged blades, including the onset of flutter and underlying mechanisms. First, a high-fidelity fluid–solid coupling system is established with computational fluid dynamics (CFD) and computational structural dynamics (CSD) technologies, via which the dynamic aeroelastic analysis is conducted based on static aeroelastic deformation. Second, a damaged rotor blade is parametrically modelled with variable damage levels, extents, and positions. Finally, the modal identification method of spectral proper orthogonal decomposition (SPOD) is applied to observe flow details and provide physical insight into the flutter mechanism for damaged blades. Numerical analysis finds that there is a critical damage level below which the aeroelastic stability is positively improved with increasing damage level; otherwise, a significant loss of stability is induced. The damage location and extent further affect this critical damage level and the change rate crossing the threshold. The simulation with CFD/CSD finds that the high pressure near the trailing edge induced from boundary layer separation suppresses vibrations in stable conditions, but motivates vibrations during flutter, which is because of the high-pressure spread to nearing blades. SPOD modes reveal that high-frequency disturbances with large scale are primary factors inducing flutter, which is further stimulated by the high-order disturbances with small scale. This study provides a crucial foundation for the fatigue prediction for rotor blades in service and the optimisation design for high-performance turbomachinery in the near future.

Analysis of Wind Power Fluctuation in Wind Turbine Wakes Using Scale-Adaptive Large Eddy Simulation

In large wind farms, the interaction of atmospheric turbulence and wind turbine wakes leads to complex vortex dynamics and energy dissipation, resulting in reduced wind velocity and subsequent loss of wind power. This study investigates the influence of vortex stretching on wind power fluctuations within wind turbine wakes using scale-adaptive large eddy simulation. The proper orthogonal decomposition method was employed to extract the most energetic contributions to the wind power spectra. Vertical profiles of mean wind speed, Reynolds stresses, and dispersive stresses were analyzed to assess energy dissipation rates. Our simulation results showed excellent agreement when compared with wind tunnel data and more advanced numerical models, such as the actuator line model and the actuator line model with hub and tower effects. This highlights the important role of coherent and energetic flow components in the spectral behavior of wind farms. The findings indicate a persistent energy cascading length scale in the wake of wind turbines, emphasizing the vertical transport of energy to turbine blades. These results complement existing literature and provide new insights into the dynamics of wind turbine wakes and their impact on wind farm performance.

The Reduced-Dimension Method for Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors of the Extended Fisher–Kolmogorov Equation

A reduced-dimension (RD) method based on the proper orthogonal decomposition (POD) technology and the linearized Crank–Nicolson mixed finite element (CNMFE) scheme for solving the 2D nonlinear extended Fisher–Kolmogorov (EFK) equation is proposed. The method reduces CPU runtime and error accumulation by reducing the dimension of the unknown CNMFE solution coefficient vectors. For this purpose, the CNMFE scheme of the above EFK equation is established, and the uniqueness, stability and convergence of the CNMFE solutions are discussed. Subsequently, the matrix-based RDCNMFE scheme is derived by applying the POD method. Furthermore, the uniqueness, stability and error estimates of the linearized RDCNMFE solution are proved. Finally, numerical experiments are carried out to validate the theoretical findings. In addition, we contrast the RDCNMFE method with the CNMFE method, highlighting the advantages of the dimensionality reduction method.

Numerical investigation of serration angle on trailing edge noise reduction

The influence of serration angle on the three-dimensional flow characteristics and noise attenuation mechanisms of a NACA6512-63 airfoil is explored in this research using large eddy simulation (LES) and spectral proper orthogonal decomposition (SPOD) methods. Analysis of the mean flow field reveals the formation of counter-rotating vortex pairs, resulting from the outward flow towards the serration edges and the downwash flow in the valleys. This flow modification alters the effective angle of the serration, which in turn impacts its noise reduction performance. Furthermore, the noise reduction mechanism is investigated by comparing the general solution of Lyu's semi-analytical equation with the SPOD mode results, and the frequency range associated with noise attenuation is analyzed. The findings indicate that the case with λ/h=0.15 demonstrates the most significant noise reduction effect. This research offers valuable insights into the interplay between serration geometry and the resulting flow characteristics and noise reduction mechanisms.

Large eddy simulation of micro vortex generator-controlled cavitation across multiple stages

Micro vortex generators (mVGs) control cavitation by altering the boundary layer flow structure. This study employs the wall-adapting local eddy-viscosity large eddy simulation (WALE-LES) turbulence model combined with the Zwart–Gerber–Belamri cavitation model to conduct transient numerical simulations on the National Advisory Committee for Aeronautics 0015 baseline hydrofoil and the hydrofoil equipped with mVGs under various cavitation numbers. The proper orthogonal decomposition method and experiments verify the accuracy and consistency of these simulations regarding cavity scale. The study elucidates mechanisms by which mVGs suppress cloud cavitation at low cavitation numbers and induce vortex cavitation at high cavitation numbers. Results indicate that mVGs maintain sheet cavitation characteristics at low cavitation numbers, reducing wall pressure fluctuations and enhancing flow stability. During cavitation inception, mVG-induced vortex cavitation leads to early cavitation formation. In the sheet cavitation phase, modal energy distribution is more dispersed, while in the inception phase, energy is concentrated with significant dominant modes. Moreover, the counter-rotating vortices generated by mVGs mitigate flow separation, enhance leading-edge flow attachment stability, and reduce high-frequency vibrations caused by bubble shedding. This study significantly advances the understanding of cavitation control by accurately simulating and revealing the cavitation control mechanisms of mVGs across different stages using the WALE-LES model. The findings demonstrate that mVGs can effectively stabilize cavity structures at low cavitation numbers, reducing flow instabilities and enhancing overall hydrofoil performance. These insights will have a significant impact on the design of hydrofoils and the development of cavitation control strategies.

Analyzing the inspiratory airflow unsteadiness in the respiratory tract using dynamic mode decomposition method

To enhance the understanding of airflow characteristics in the human respiratory tract, the inspiratory airflow field was simulated under both tidal and quasi-steady inspiratory flow rates at the mouth inlet using the large eddy simulation method. Special attention was paid on analyzing the inspiratory airflow unsteadiness using the dynamics mode decomposition (DMD) method based on the vorticity field and comparing it with the proper orthogonal decomposition (POD) method. The following novel findings were obtained. (1) Power spectral density indicates that the inspiratory airflow is highly turbulent in the pharynx–larynx region. The vorticity field in the upper airway is more affected by inspiratory patterns compared to turbulence fluctuations. (2) The DMD results indicate that the shear flow in the pharynx–larynx region is mainly caused by flow under low-frequency modes, while the disturbances of the jet flow are caused by flow under multiple frequency modes. Steady-state inspiratory pattern demonstrates the decay characteristics different from the tidal inspiratory pattern. (3) Compared to the POD method, which may contain multiple frequency components, the DMD decomposition yields modes with a single frequency, enabling a more accurate capture of the frequency and decay characteristics of the respiratory flow under each mode. In conclusion, this study demonstrates that the DMD method is more suitable for studying the respiratory airflow unsteadiness and further confirms the necessity of adopting clinically measured inspiratory data to investigate airflow unsteadiness.

Enhancing gas-liquid mixing effect in pipelines: Experimental and numerical investigations of blind tee and traditional elbow

Blind tees, as typical upstream component for multiphase flow measurements, can promote the gas-liquid mixing in pipelines. However, detailed investigations on the mixing mechanism in blind tees are still scarce. This study employs a combination of experiments and numerical simulations on bubble flow in bend pipes to analyze two-phase flow dynamics in both the traditional elbow and blind tee. Firstly, the swirling phenomena at the outlet of bending structures are revealed using the measured flow images, and the dominant modes of gas distribution are extracted with proper orthogonal decomposition method to conduct a low-dimensional spatiotemporal analysis. Then, the mixing mechanism and vorticity distribution in different bending structures are compared using numerical results. Finally, the influences of bending structures and gas-liquid flow conditions on the mixing performance are comprehensively assessed. Results indicate that blind tee has a better mixing performance than traditional elbow, which significantly improves the gas distribution uniformity at the bend outlet. With the increase of volume gas content, the vortex intensity and frequency at the blind tee outlet increase, resulting in an increase in the uniformity of gas distribution. Moreover, the blind tee exhibits a shorter distance of turbulence dissipation than the traditional elbow. Therefore, a more uniform and stable bubble flow can be obtained within the range of 6D-8D downstream of the blind tee.

Complex flow field analysis in Multi-Shaft stirred Reactors: Dynamics of Wave-Vortex coupling revealed by POD and DMD methods

Insufficient understanding of complex flow structures in multi-shaft stirred reactors hinders industrial adoption. In this work, Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) methods were used to analyze the Large-Eddy Simulation (LES) results of the stationary and rotating zones for three types of stirred reactors, including multi-shaft configurations. The results indicate that the flexible spatial distribution of the impellers in multi-shaft stirred reactors enhances fluid interactions, leading to the superior performance in energy cascading, flow field self-regulation, and wave-vortex coupling. Frequency analysis of single modes underscores DMD’s efficacy in interpreting complex flow phenomena. Based on this, the DMD modal coefficients were fitted to equations, clarifying the development process of wave-vortex coupling within the stirred reactors. Additionally, by integrating fundamental fluid mechanics equations with the outcomes of the DMD method, a wave-vortex coupling model was developed, which is anticipated to yield a more accurate representation of the flow field.

A second-generation URANS model (STRUCT-$$\epsilon $$) applied to a generic side mirror and its impact on sound generation

AbstractA generic side mirror can be approximated to the combination of a half cylinder topped with a quarter of sphere. The flow structure in the wake of the side mirror is highly transient and the turbulence plays an important role affecting aeroacoustics through pressure fluctuation. Thus, this geometry is one of the test cases object of several numerical studies in recent years to assess the aerodynamic and aeroacoustic capabilities of the turbulence models. In this context, this study presents how the second-generation URANS closure STRUCT-$$\epsilon $$ ϵ is able to properly predict the expected stagnation, flow separation and vortex shedding phenomena. Besides, the predictive accuracy for the noise generation mechanism is evaluated by comparing the spectra of the sound pressure level measured at several static pressure sensors with the numerical results obtained with the STRUCT-$$\epsilon $$ ϵ . The response of this turbulence model has exceeded that from other hybrid methods and is in good agreement with the results from Large-Eddy Simulations or the experiments. To conclude the paper, the applicability of STRUCT-$$\epsilon $$ ϵ to construct a Spectral Proper Orthogonal Decomposition method that helps identifying the most energetic modes to appropriately capture the dominant flow structures is also introduced.

A new proper orthogonal decomposition method with second difference quotients for the wave equation

Recently, researchers have investigated the relationship between proper orthogonal decomposition (POD), difference quotients (DQs), and pointwise in time error bounds for POD reduced order models of partial differential equations. In a recent work (Eskew and Singler, Adv. Comput. Math., 49, 2023, no. 2, Paper No. 13), a new approach to POD with DQs was developed that is more computationally efficient than the standard DQ POD approach and it also retains the guaranteed pointwise in time error bounds of the standard method. In this work, we extend this new DQ POD approach to the case of second difference quotients (DDQs). Specifically, a new POD method utilizing DDQs and only one snapshot and one DQ is developed and used to prove ROM error bounds for the damped wave equation. This new approach eliminates data redundancy in the standard DDQ POD approach that uses all of the snapshots, DQs, and DDQs. We show that this new DDQ approach also has pointwise in time data error bounds similar to DQ POD and use it to prove pointwise and energy ROM error bounds. We provide numerical results for the POD ROM errors to demonstrate the theoretical results. We also explore an application of POD to simulate ROMs past the training interval for collecting the snapshot data for the standard POD approach and the DDQ POD method.

Application of proper orthogonal decomposition to flow fields around various geometries and reduced-order modeling

This study is focused on a reduced-order model (ROM) based on proper orthogonal decomposition (POD) for unsteady flow around a stationary object, which allows prediction with different object geometry as a parameter. The conventional POD method is applicable only to data with the same computational grid for all snapshots. This study proposed a novel POD methodology that performs on flow snapshots, including some time-series data of flow fields around objects of different shapes and numerically computed by different computational grids. The concept of the proposed POD involved mapping the flow fields computed on different grids in computational space. Consequently, the optimal POD basis for minimizing reconstruction errors in physical space was obtained in the computational space. The proposed POD was applied to the flow around ellipses and airfoils generated via conformal mapping to a cylinder. The ROM formulated using the proposed POD bases reconstructed the flow fields around the ellipses with different aspect ratios and airfoils with varying shapes. Using the modes obtained by the proposed POD, the ROM was demonstrated to stably predict the time evolution of the flow around objects, which is not included in the snapshots. In the ROM, the difference between the frequency of the flow field in the POD snapshot and that of the reconstructed flow field resulted in a phase error owing to the time evolution. The mean squared error between the flow fields obtained via the ROM and the directly solved Navier–Stokes equations was under 10−7 when the reconstructed flow and the flow included in the snapshot had the same frequency as that of Kármán vorticities behind the objects. Based on these observations, the proposed POD is suitable for constructing an ROM to reconstruct the flow around various geometries.

An efficient reduced order model for nonlinear transient porous media flow with time-varying injection rates

An intrusive Reduced Order Model (ROM) is developed for nonlinear porous media flow problems with transient and time-discontinuous fluid injection rates. The proposed ROM is significantly more computationally efficient than the Full Order Model (FOM). The training regime is generated using the FOM with constant injection rates during the offline stage. The trained ROM exhibits high accuracy for complex pumping schedules (rate vs time) simulated online. The proposed ROM uses the combination of Proper Orthogonal Decomposition and Discrete Empirical Interpolation Method (POD-DEIM), which is compared with the classical POD-Galerkin. The use of an approximated column-reduced Jacobian is shown to be vital to achieving a substantial speedup of ROM vs FOM run-times. An analysis of the trade-off between accuracy and run-time is conducted for ROMs of different sizes and hyper-parameters. The impact of the training regime on the performance of the presented ROM is assessed. The performance of the ROM is studied in the context of a two-dimensional analysis of time-varying injection into a two-well system in a layered porous media reservoir. The accuracy and efficiency of POD-DEIM motivate its potential use as a surrogate model in the real-time control and monitoring of fluid injection processes.

Application of deep learning reduced-order modeling for single-phase flow in faulted porous media

AbstractOur research is positioned within the framework of subsurface resource utilization for sustainable economies. We concentrate on modeling the underground single-phase fluid flow affected by geological faults using numerical simulations. The study of such flows is characterized by strong uncertainites in the data defing the problem due to the difficulty of taking precise measurements in the subsoil. We aim to demonstrate the feasibility of a reduced order model that is both reliable and computationally efficient, thereby facilitating the incorporation of uncertainties. We account for the uncertainities of the properties of the rock and the geometry of the fault. The latter is achieved by using a radial basis function mesh deformation method. This approach benefits from a mixed-dimensional framework to model the rock matrix and faults as n and $${n-1}$$ n - 1 dimensional domains, allowing for non-conforming meshes. Our primary focus is on a reduced-order model capable of reproducing flow variables across the entire domain. We utilize the Deep Learning Reduced Order Model (DL-ROM), a nonintrusive neural network-based technique, and we compare it against the traditional Proper Orthogonal Decomposition (POD) method across various scenarios. The most relevant contributions of this work are: the proof of concept of the use of neural network for reduced order models for subsoil flow, dealing with non-affine problems and mixed dimensional domain. Additionally, we generalize an existing mesh deformation method for discontinuous deformation maps. Our analysis highlights the capability of reduced order model, highlighting DL-ROM’s capacity to expedite complex analyses with promising accuracy and efficiency, making multi-query analyses with various quantities of interest affordable.

Equivalent static wind load based on displacement mode and load combination for high‐rise buildings

AbstractUnder the action of the fluctuating wind load, the low frequency part produces background response to the structures, while the high frequency part produces resonance response to the structures. In structural design, equivalent static wind load is usually used to equate the fluctuating wind load. Although there are various methods of evaluating the equivalent static wind load, they did not consider the correlation between modal responses or considered them insufficiently. Therefore, in this paper, the correlation between modal response is considered to evaluate the equivalent static wind load, the displacement response is decomposed by proper orthogonal decomposition (POD) method, the correlation between modal displacement is removed, the equivalent static wind load is expressed in the form of displacement mode, and then the correlation between modal response is fully considered in the extreme value combination. This paper combines the equivalent static wind loads in order to make the wind resistance design more reasonable for high‐rise buildings. First, the formulas of equivalent static wind loads expressed by displacement modes of background and resonant response are deduced based on modal decomposition and POD method. Second, the combination formulas of square‐root‐of‐sum‐square (SRSS) and complete‐quadratic‐combination (CQC) rules for the equivalent static wind loads considering the mean wind loads are proposed. Both the linear combination formula of SRSS for the equivalent static wind load and the weighting factor expressions of background and resonance equivalent static wind load are given. Third, the accuracy and validity of the formulas of equivalent static wind load are verified by a wind tunnel pressure test of a high‐rise building. Finally, the simplified combination coefficient formulas for the equivalent static wind loads are proposed, and the combination of the high‐rise building base's fluctuating equivalent static wind loads of along‐wind direction, across‐wind direction, and torsional direction is analyzed.

A comparison of tonal-broadband decomposition algorithms for propeller noise

The usage of propellers is increasing as an alternative to traditional aircraft and has become the primary option for new urban mobility vehicles. However, noise emissions from rotating blades can be a challenge for the certification of these new vehicles. The acoustic sources from a propeller can be decomposed into deterministic and random components. Therefore, academic research has focused on developing methods to separate these components. This study compares the performance of several methods available in the literature to decompose propeller noise signals into tonal and broadband components. Initially, the methods were applied to synthetic data that represents the main characteristics of rotating blade noise. Later, the methods were tested on realistic acoustic data collected from an isolated propeller test rig. The results showed that methods based on phase-averaging recorded signals satisfactorily detected harmonic components. The method based on cross-correlation between data blocks effectively extracted the broadband spectra, but it performed poorly on harmonic detection. Additionally, the wavelet transform and Proper Orthogonal Decomposition methods were capable of reconstructing time series related to both deterministic and random parts. However, further investigation is required regarding the separation of POD modes, as unexpected trends were observed with the wavelet-based method.

Spray swirling flame dynamic under combined modulation of air and fuel

Combustion instability is still a main challenge in developing advanced gas turbine combustors. It is essential to conduct instability control once it occurs. In this paper, liquid fuel modulation by a high-speed on/off valve is used to control flame oscillation generated by loudspeakers. Spray swirling flame dynamic under the combined modulation of air and fuel is experimentally studied. Flame dynamical characteristics and coherent structures at different excitation phase delays are discussed with nonlinear time-series analysis and proper orthogonal decomposition method. Experimental results show flame heat release rate fluctuation is decreased by 86% at 40° while flame instability is intensively enhanced at 220°. Phase portraits and recurrence plots show the flame is in a chaotic oscillation with low amplitude aperiodic oscillations at 40° while it suffers from the largest fluctuation at 220°. Difference in flame dynamic is caused by the destructive (out of phase) and constructive (in phase) interference of heat release rate fluctuations induced by air excitation and fuel modulation. Flame returns back to its nature dynamical behavior under out-of-phase modulation. Longitudinal modes of the spray flame are enhanced while the spiral modes are inhibited under in-phase modulation.

Temperature history reconstruction in steel box girders using limited data and proper orthogonal decomposition-based dimension reduction representation

The monitoring temperature history of steel box girders inevitably contains gaps or anomalies due to the malfunctions of the structural health monitoring system. Traditional methods for reconstructing temperature histories face the challenge in accounting for the randomness of temperature variations caused by the uncertainty of complex environmental factors. This research proposes a framework for reconstructing the long-term temperature of steel box girders by combining the limited measurements with a proper orthogonal decomposition (POD) based dimension reduction representation approach, to account for the stochastic nature of daily temperature variations. Unlike the conventional Monte Carlo-based POD method, the developed POD-based dimension reduction representation approach effectively reduces the number of elementary random variables from thousands to two by introducing random functions serving as constraints, overcoming the challenge of high-dimensional random variables inherent in the Monte Carlo methods. The proposed approach divides the measured temperature histories into random high-frequency (HF) and deterministic low-frequency (LF) components and establishes the theoretical models of power spectral density and coherence functions of HF temperature components to accommodate the generation of HF temperature component samples, and finally reconstructs temperature samples by superimposing the LF components and generated HF component samples. The results from a practical example demonstrate that the statistical characteristics of representative HF temperature component samples generated by the POD-based dimension reduction representation align well with the corresponding targeted values, and the proposed method outperforms the traditional POD method, yielding a 60% efficiency enhancement without compromising computational accuracy. The developed framework owns apparent superiority in accuracy compared to the traditional POD and the long short-term memory methods, particularly in continuous and extensive missing data. Moreover, the reconstructed temperature samples with assigned probabilities present complete probability information from the level of total probability. These results advance the probabilistic methods in tackling long-term temperature history reconstruction.