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The design optimization taking into account the impact of uncertainties favors improving the robustness of the design. A Surrogate-Assisted Gradient-Based (SAGB) method for the robust aerodynamic design optimization of turbomachinery blades considering large-scale uncertainty is introduced, verified and validated in the study. The gradient-based method is employed due to its high optimization efficiency and any one surrogate model with sufficient response accuracy can be employed to quantify the nonlinear performance changes. The gradients of objective performance function to the design parameters are calculated first for all the training samples, from which the gradients of cost function can be fast determined. To reveal the high efficiency and high accuracy of SAGB on gradient calculation, the number of flow computations needed is evaluated and compared with three other methods. Through the aerodynamic design optimization of a transonic turbine cascade minimizing total pressure loss at the outlet, the SAGB-based gradients of the base and optimized blades are compared with those obtained by the Monte Carlo-assisted finite difference method. Moreover, the results of both the robust and deterministic aerodynamic design optimizations are presented and compared to demonstrate the practicability of SAGB on improving the aerodynamic robustness of turbomachinery blades.

A Reduced Order Model (ROM) based analysis method for turbomachinery cascade coupled mode flutter is presented in this paper. The unsteady aerodynamic model is established by a system identification technique combined with a set of Aerodynamic Influence Coefficients (AIC). Subsequently, the aerodynamic model is encoded into the state space and then coupled with the structural dynamic equations, resulting in a ROM of the cascade aeroelasticity. The cascade flutter can be determined by solving the eigenvalues of the ROM. Bending-torsional coupled mode flutter analysis for the Standard Configuration Eleven (SC11) cascade is used to validate the proposed method.

Modern turbomachine analysis seeks designs that maximize efficiencies and push operational extremes, highlighting a continual need for accurate and efficient component level performance prediction methods. Annular seals are particular components found in a wide variety of turbomachinery applications that serve to reduce leakage flow across a region with a large pressure differential. Circumferentially grooved seals further reduce leakage through the use of grooved sections on the rotor or stator surfaces that serve to dissipate kinetic energy through the formation of vortices, though the addition of grooves adds significant complexity to the fluid dynamic response of the sealing component. Computational analysis of grooved seals includes simplified, one-dimensional bulk flow models and full Navier-Stokes computational fluid dynamics (CFD) studies. Bulk flow models, while efficient and easy to use, lack accuracy due to numerous assumptions, while full CFD studies display higher accuracy but are expert knowledge driven and computationally intensive. This work seeks to address the limited accuracy of bulk flow models for circumferentially grooved seals through the use of a novel modeling approach based on an effective film thickness, a physical flow boundary that separates the jet flow in the seal clearance region from the recirculation flow within the seal grooves. A simplified, single groove CFD model is employed to establish the effective film thickness analysis framework, providing insight into the flow mechanisms dictating leakage performance and illustrating the potential for reduced empiricism in bulk flow modeling. This framework is then applied to shear stress modeling within the groove region, where the additional shear stress contribution is isolated and directly quantified as a correction to a traditional bulk flow shear stress definition. These shear stresses are termed here as form shear stresses (FSS) based on the close relationship between their behavior and the expansion and contraction of the effective film thickness. Models for the FSS are developed as functions of local Reynolds number and implemented into a simplified bulk flow method, demonstrating the ability of the novel modeling approach to capture physical flow behavior and eliminate the need for an empirical groove loss coefficient. Finally, a rigorous sensitivity study examines the impact of the upstream and downstream regions, rotor centrifugal growth, and modeling whirl amplitude on the prediction of physical flow phenomena and rotordynamic coefficients from a quasi-steady full CFD method. The results highlight the careful consideration needed during model setup in order to realize the accuracy advantages assumed by the use of these higher fidelity methods. This work presents the first use of an effective film thickness in bulk flow analysis and shear stress modeling for grooved seals. The bulk flow developments of this dissertation set forth a new modeling approach that can be applied to many sealing scenarios for reduced empiricism and increased prediction accuracy, while the contributions to the knowledge base of full CFD rotordynamic prediction enable more widespread and appropriate use of higher fidelity methods for seal analysis.

Unsteady flow in the hub endwall region has long been a hot topic in the turbomachinery community. However important it is to the performance of the whole engine, the coherent unsteady flow phenomena are still not well understood. In this paper, the complex flow field in the hub endwall of a cantilevered compressor cascade has been investigated through numerical approach. The predicted results were validated by experimental data. To highlight the dominant flow structures among irregular and chaotic motions of various vortices, a Dynamic Mode Decomposition (DMD) method was utilized. The results show that there exist three dominant periodic flow structures: the oscillation of the leakage vortex, a circumferential migration of a Breakdown Induced Vortex (BIV) and the fluctuation of the passage vortex. These three coherent structures all together form a self-sustained closed loop which accounts for the flow unsteadiness of the studied cascade. During this process, the BIV plays a key role in inducing the flow unsteadiness. Only if the BIV is strong enough to affect the passage vortex, the flow unsteadiness occurs. This study expands current knowledge base of flow unsteadiness in a compressor environment, and shows the efficacy of the DMD method for revealing the origin of flow unsteadiness.

Brush seals are leakage performance systems that are operating against turbine shafts and bucket tips. Flexible bristle structure can compensate the rotor interference during transients, which enables closer clearance control in turbine and turbo-engine applications. During turbomachinery transients, contacting bristles exert force on turbine shaft, which in turn result in wear, frictional heat generation, counter torque and power loss. Brush seal design with proper stiffness has crucial importance since high bristle tip force levels during rotor contact may result in rotor instability, bristle tip melting or thermo-mechanical fatigue due to large amount of local frictional heat input. Counter torque applied by the contacting brush seal with improperly excessive tip force may result in turbine stall and overall efficiency losses. High levels of bristle contact forces may induce elevated wear rates, which degrades the leakage performance of the brush seal during its life cycle. Brush seal contact force evaluation under operating conditions has critical importance and has to be conducted before turbine integration to avoid performance losses and possible turbine failure in some extreme cases. In this paper, ‘Bristle Contact Force Theory’ is developed and closed form bristle tip force (BTF) function is derived where rotor-bristle, inter-bristle and bristle-back plate interactions are included. BTF measurements under pressurized and unpressurized air environment are conducted for the selected test seals by using the custom ‘Rotary Test Rig’. BTF function of bristle contact force theory show high level of correlation with the test data. The effect of cant angle manufacturing tolerance on bristle tip force levels is also examined through bristle tip contact force theory, and compared with the test seal measurements. Methodology for number of bristle row calculation and comparison with the geometric inspection of test seals are also detailed within the content of this paper.

This study investigates the instabilities of the interstage flow between a rotor and a stator in turbomachinery and evaluates their impact on the fan noise. By viewing the interstage flow as the so...

Distributed aircraft propulsion has renewed the interest in power-dense, high-efficiency power packs. Ceramic turbomachinery could be a major enabler, although no successful design has been achieve...

Due to the high Reynolds numbers together with the complex turbulent motions in turbomachinery, selecting a proper turbulence-modeling method is of vital significance in interpreting the real flow ...

The performance of turbomachinery is largely affected by the nonuniform boundary conditions caused by the coupling between neighboring parts, such as the inlet distortion and hot streak. Existing works study this problem by comparing the flow fields with uniform and nonuniform boundary conditions, which is cost extensive. In this work a new and efficient method is developed by computing the sensitivities of arbitrary performance metric relative to the boundary conditions and this method quantitatively describes how the disturbance on the boundary surface affects the performance. The adjoint method popular in the design optimization is extended to current field for the efficient computation of the sensitivity. The extra cost is independent of the number of parameters describing the nonuniform distribution. Accuracy of the adjoint boundary sensitivity is validated against analytical results for a converging–diverging nozzle case. Its applications in a representative case are detailed and discussed in this paper. Current method provides a new tool, also direct guides to minimize the negative effect and improve the performance utilizing the adjoint boundary sensitivity.

Hybrid RANS-LES is a promising method for analyzing the complex flow structure in turbomachinery due to its excellent compromise between accuracy and computational cost. However, it is challenging to employ the existing URANS and LES grid resolution evaluation approaches in a hybrid RANS-LES simulation, thus commonly the required grid resolution is unknown, increasing the risk that LES relies on overly coarse grids and leading to unreliable predictions. Hence, spurred by this deficiency and driven by the aspects of coupling grid resolution evaluation with the interest quantity during turbomachinery design and mechanism analysis, this work proposes a novel grid resolution evaluation method suitable for both LES and URANS. The suggested technique considers three grid resolution criteria: no effects on the time-averaged flow field, major unsteadiness, and minor unsteadiness. The developed method is verified employing a classic scenario, i.e. Pitz-Daily backward-facing step. Furthermore, we demonstrate our method’s applicability through a T106A turbine cascade example, confirming the feasibility and reliability of the proposed scheme. When applied to hybrid RANS-LES turbomachinery simulations, the research results highlight our method’s capability to reduce uncertainty and improve reliability during gridding.

The current generation of axial turbomachines are the culmination of decades of experience, and detailed understanding of the underlying flow physics has been a key factor for achieving high efficiency and reliability. Driven by advances in numerical methods and relentless growth in computing power, computational fluid dynamics has increasingly provided insights into the rich fluid dynamics involved and how it relates to loss generation. This article presents some of the complex flow phenomena occurring in bladed components of gas turbines and illustrates how simulations have contributed to their understanding and the challenges they pose for modeling. The interaction of key aerodynamic features with deterministic unsteadiness, caused by multiple blade rows, and stochastic unsteadiness, i.e., turbulence, is discussed. High-fidelity simulations of increasingly realistic configurations and models improved with help of machine learning promise to further grow turbomachinery performance and reliability and, thus, help fluid mechanics research have a greater industrial impact. Expected final online publication date for the Annual Review of Fluid Mechanics, Volume 54 is January 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Regenerative flow pumps and compressors are low specific speed roto-dynamic turbomachines capable of producing high heads at low specific speeds. The present study focuses on the flow characteristics of both the turbomachines viz. a compressor and a pump and extended to investigate the structural integrity for the considered flow condition. CFD solver, ANSYS FLUENT was used to perform the flow simulation. The simulation is performed with pump running at 2900 rpm and compressor with 100,000 rpm to achieve a pressure ratio of about 3 in both cases. The influence of centrifugal force, pressure load and fluid temperature from the aerodynamic analysis are applied to the impeller blade and hub to perform the Fluid-Structure Interaction (FSI) in the FEA module of the software. From the CFD simulation it is observed that a maximum pressure of about 0.29 MPa in pump, 0.24 MPa in compressor and a maximum temperature of 562 °C in the case of compressor occur close to the outlet for both the turbomachinery. The Mach number is found to be around 2.5 in the compressor and 0.01 in the pump. For the stress assessment, the maximum von Mises stress in the blades is compared to the maximum allowable stress of grey cast Iron. Radial displacements were also considered to determine the structural integrity from the stress analysis. It is predicted that maximum stresses are induced on the blades near the outlet and hence the failure is expected to occur on the blades. The combined effect of varying pressure load, fluid temperature, and centrifugal forces with deformation and stress in both pump and compressor were also presented. This paper highlights the aero-mechanical features of the regenerative flow compressor and pump obtained from numerical simulations, which are expected to provide a sound basis for further investigations.

In order to improve the gas energy conversion efficiency at the outlet of the pulse detonation combustor, numerical calculation methods were used to study the influence of three different cross-section obstacle structures of square, circle and trapezoid on the distribution characteristics of gas pressure potential energy, kinetic energy and internal energy in a single cycle at the outlet of pulse detonation combustor. The results show that: the expansion of pulse detonation gas mainly includes three stages: primary expansion, secondary expansion, and over-expansion. Gas pressure potential energy, kinetic energy and internal energy increase during the primary and secondary expansion stages, and decrease during the over-expansion stage; Pulse detonation combustor with trapezoidal cross-section obstacle structure has the smallest proportion of gas energy at the outlet of the combustor during the over-expansion stage and the highest proportion during the secondary expansion stage; compared with square and circle cross-section obstacle structures, the gas energy distribution at the outlet of the pulse detonation combustor with trapezoidal cross-section obstacle structure is the easiest for turbomachinery to convert the gas energy.

This paper presents a first investigation into the aerodynamics and the performance breakdown of a distributed aft-fuselage Boundary Layer Ingesting (BLI) tube-and-wing aircraft using fully coupled unsteady RANS calculations that resolve the complete fan and installation geometries. Through the URANS simulations, the interaction between the turbulence from the fuselage Boundary Layer (BL) and the BLI propulsor is identified as an area that warrants further research. Using the URANS approach, the ingested turbulence leads to a 4.5% reduction in propulsor stage total-total isentropic efficiency. A mechanical power balance has been drawn up to compare the power sources and sinks throughout the installation and propulsor for two test cases with different thicknesses of ingested BL. The test case with a thinner BL was found to generate significantly more dissipation in the BL development upstream of the propulsor, the flow separation over the outer cowl, and the interaction between reversed flow over the cowl and the propulsor exhaust jet. Due to this increase in dissipation, the case with thinner ingested BL consumes 7% more power relative to a baseline case with thicker BL, representative of the upstream fuselage at cruise. This demonstrates the importance of matching the installation with the incoming fuselage BL., This research was funded by Rolls-Royce plc and EPSRC under the Centre for Doctoral 503 Training in Gas Turbine Aerodynamics (EP/L015943/1) with a reference number 1799072.

Blade mistuning in blisks arises primarily from the scatters of blade geometry profiles caused by manufacturing tolerance, in-service wear, blade repairs, etc. There is a recent trend to capture the blade-to-blade geometry variances through precise geometry measurements by a 3D optical scanning system in order to obtain an improved blade geometric mistuning evaluation capability. However, this usually leads to prohibitive computational costs due to the large-scale, high-fidelity industrial blisk finite element models. This paper develops an original model reduction approach, Sector Mode Assembling Reduction Technique (SMART), specifically for the high-fidelity blisk model fully mistuned by blade geometric variances, with either topologically compatible or incompatible blade meshes. The basic idea ofSMART is to construct the sector-level reduction mode basis by strategically assembling the truncated cyclic modes independently computed for each “isolated” sector with assumed cyclic symmetry at the sector interfaces. Benefiting from the block structure of the SMART mode basis, the reduced-order models are derived by a series of sector-level projections with a relatively low memory requirement and computational cost. Another hidden benefit is that the SMART approach enables efficient structural modification predictions of the global blisk modes because only the modes of the sectors undergoing blade modification need to be re-evaluated and replaced in theSMARTmodebasis. TheSMARTapproach is applied into a high-fidelity “as-measuredmodel” ofa blended blisk, constructed upon the geometry measurement by the state-of-the-art 3D optical scanning technology. It is fully demonstrated that the reduced-order model derived by SMART, featured by a minimal size, is able to reproduce the dynamics of the full-order as-measured blisk model with high accuracy.

At a low mass ratio of structure to air, the work-per-cycle approach, or better known as the energy method, will lead to non-conservative results as aerodynamic coupling of modeshapes acts destabilizing. Using the p-k method to solve the aeroelastic stability equation, the effects of various structural aspects are investigated for a two-dimensional compressor cascade in subsonic and transonic flow conditions. The investigated key parameters are frequency separation, mass ratio and solidity. Furthermore, the effect of a high frequency dependency of the aerodynamic forces is presented. Such phenomena can happen in case of aerodynamic or acoustic resonances. If the resonance peaks are close to the aeroelastic frequency, a discontinuous behavior of the frequency or damping solution can lead to a rapid destabilization of the system, once the aeroelastic frequency moves from one side to the other of the peak. In these regimes, it is crucial to have a high quality representation of the frequency-dependent generalized aerodynamic forces for an accurate prediction of the flutter onset.

During the operation of supercritical CO $ _2 $ (sCO $ _2 $ ) turbomachineries, large pressure ratios, supersonic conditions and non-ideal gas fluid dynamic phenomenon may happen, which will decrease the whole cycle efficiencies. Hence non-standard designs for the turbomachineries geometry are needed. Successfully simulations in capturing non-ideal gas fluid dynamics and coupled with optimization algorithm are hard and currently not available in any commercial CFD software. Therefore, the problem of optimizing sCO $ _2 $ turbomachinery is difficult to solve, which has become a major obstacle to obtain compact and high-efficiency sCO $ _2 $ turbomachineries. In this study, a modularized geometry optimizer is developed to obtain the non-standard geometric designs for sCO $ _2 $ turbomachineries. Multiple techniques are applied to this optimizer, include the Nelder–Mead algorithm, Mahalanobis distance and stochastic algorithm. The newly developed optimizer can successfully find the optimum satisfying the objective function under given weighting factors. The computational cost can be reduced through a stochastic algorithm. To validate the optimizer, a convergent–divergent nozzle for air with a target Mach number equal to 2.4 is optimized. Different starting points and combinations of weighting factors are used to create a Pareto front. Adjust the weighting factors for different terms of the objective function leading the optimizer to go to different directions in n-dimensional spaces. Three optimized cases, one is Mach number optimized, one is outlet flow uniformity optimized and the other is compromised case, are picked out and analyzed. The results show that the optimizer can successfully find optimized geometry than the reference case and potentially save computational cost. Due to the modularized characteristics, the components of this optimizer can be replaced with any available techniques, which mean the optimizer can be applied to solve different types of optimization problems.

Modern turbomachinery relies on accurate prediction of the flow and especially the state of turbulence to achieve the required level of performance. Transition, relaminarization, wake interactions,...

For traditional 12/8 bearingless switched reluctance motors (BSRMs), the coupling between torque and levitation increases the complexity of control. For the turbomachinery application, 2 degrees of freedom (DOF) BSRMs stabilized by extra 3-DOF active magnetic bearings are required. In this article, an improved BSRM with 4 rotor poles and 12 stator poles is proposed to realize 4-DOF radial levitation, and the torque dead zone is compensated because that two rotors and these poles are assemble with 15 mechanical degree shifted. The decoupling of torque and radial levitation is realized simultaneously. Moreover, the operation principle of axial reluctance force is analyzed for passive levitation in the axial direction. Various experiments were completed to verify the performance of the proposed BSRM, and the prototype achieved stable levitation of the rotor in 10 000 r/min.

Dry gas seals represent a significant advancement in turbo machinery due to their ability to handle high pressures and speeds without the use of external sealing fluids, such as oil or water, thus reducing contamination and increasing reliability. Despite their widespread use, internal working mechanisms are not fully understood to date, in particular regarding fluid film thickness prediction, which is an essential component of the seal design. The axial deflection of the rotating and stationary rings in a dry gas seal affects the development of the fluid film formed between the ring faces of the seal, influencing the performance of the seal during its operation, as well as leakage of the seal when it is at rest. The hydrodynamic and hydrostatic pressure fields of the fluid film, together with temperature gradients in the rings, induce axial deflection of these components. This in turn modifies the pressure field developed in the film. This paper focuses on establishing a methodology to couple the deformation field and the dynamic behavior of the fluid film (pressure and temperature fields) through numerical computations. Analytical relationships are employed to obtain the thermo-elastic deflection of the seal rings in the axial direction and this distortion is used in the numerical methodology to accelerate the prediction of the seal behavior. The coupled seal ring-fluid film dynamic system with 11° and 15° spiral angle is stable because the axial deflection calculated from numerical analysis produces a converging radial taper in the direction of the flow (producing a net opening force). An important result of this work is that the predicted magnitude of the axial deflection (as a result of pressure and temperature effects) under thermal and pressure loads on the stationary and rotating rings is smaller but of the same order of magnitude as the fluid film thickness.

Purpose The present paper aims at evaluating the lattice Boltzmann method (LBM) on a high-subsonic high-pressure compressor stage at nominal regime. Design/methodology/approach The studied configuration corresponds to the H25 compressor operated in a closed-loop test rig at the von Karman Institute. Several operating points are simulated with LBM for two grids of successive refinements. A detailed analysis is performed on the time-averaged flow predicted by LBM, using a comparison with experimental and existing RANS data. Findings The finest grid is found to correctly predict the mean flow across the machine, as well as the influence of the rotor tip gap size. Going beyond time-averaged data, some flow analysis is performed to show the relevance of such a high-fidelity method applied to a compressor configuration. In particular, vortical structures and their evolution with the operating points are clearly highlighted. Spectral analyses finally hint at a proper prediction of tonal and broadband contents by LBM. Originality/value The application of LBM to high-speed turbomachinery flows is very recent. This paper validates one of the first LBM simulations of a high-subsonic high-pressure compressor stage.

Pylons are commonly used for the mounting of engines in the aircraft industry. On the other hand, the installation of a pylon influences the noise generation mechanisms and therefore alters the broadband noise characteristics of a given turbomachinery setup. In this investigation, a counter-rotating open rotor with and without a pylon is investigated in order to determine its effects on broadband noise sources. The various broadband noise sources and their typical frequency ranges have been determined using beamforming maps and spectral analysis. In order to attain a clear impression regarding the broadband noise sources, the Double Filtering beamforming method has been utilized in the investigation. This method removes the tonal components from the recorded signal of a microphone array, resulting in a purely broadband signal. Using beamforming maps, the dominant broadband noise source amplitudes and locations can therefore be investigated in great detail. Compared to other methods, the investigation of measurement data and beamforming maps helps determine the amplitude, the frequency range, and the significance of the various types of broadband noise sources that are truly present in the emitted noise. It has been found for lower frequencies, that the broadband noise sources at the blade root of the aft rotor are dominant, while for higher frequencies, the significant broadband noise sources are localized to the trailing edge region of the forward rotor and the leading edge of the aft rotor. The installation of a pylon has resulted in an additional broadband noise source appearing at the blade tip of the aft rotor.

The lack of established optimal design guidelines for turbomachinery operating in the nonideal flow regime (e.g., organic Rankine cycle turbines, CO2 compressors, compressors for refrigeration systems) demands for effective and efficient automated design methods. Past research work focused on gradient-free methods applied to computational fluid-dynamic simulations. The application of the adjoint method is a cost-effective alternative as it enables gradient-based optimization irrespective of the number of design variables. This paper presents the application of a fully turbulent unsteady adjoint method for the automated design of multirow turbomachinery partly operating in the nonideal flow regime. The method therefore allows for the solution of constrained unsteady fluid-dynamic optimization problems, in which the thermodynamic properties of the working fluid need to be modeled by means of complex equations of state. The optimal designs computed with unsteady simulations obtained with the harmonic balance method are then compared with optimal design resulting from mixing-plane simulations. The method is applied to the optimization of 1) a two-dimensional turbine cascade subject to time-varying inlet conditions, and 2) a two-dimensional turbine stage of an organic Rankine cycle power system. The results demonstrate the importance of computing fluid properties using accurate thermodynamic models and of using unsteady simulations for shape optimization of these machines.

THE PURPOSE. Determination of the optimal law of swirling of the blades of the last stage of a stationary GTU. Due to the specificity of its operating conditions - in a system with a diffuser - the traditional laws of swirling lead to a non-optimal flow in the diffuser and, consequently, reduce the efficiency of the entire unit and the power plant as a whole. In this paper, we used numerical and experimental methods for studying a three-dimensional flow. Two stages with different laws of swirling were investigated - with the traditional law of constancy of the angle of flow out of the guide vanes along the radius, and with reverse swirling. The same diffuser was used in both cases. METHODS. Experimental studies were carried out using pneumometric five-channel probes of an original design on an ET-4 aerodynamic stand in the Turbomachinery laboratory of SPbPU. Numerical studies were carried out in the CFX gas dynamic calculation package; the parameters in the corresponding sections, obtained during the physical experiment, were used as boundary conditions.RESULTS. Integral characteristics of the stage, the vector of flow velocities in various sections were obtained. The experiment was compared with the numerical calculation and showed satisfactory convergence of the results. CONCLUSION. The optimal swirling law for the last stage operating in a system with a diffuser is forced vortex flow.

Cavitation instability is a common phenomenon that causes vibration and noise of turbomachinery. In this study, an attempt is made to suppress the cavitation instability. A high-speed centrifugal pump with inducer is taken as the research objective. Four baffles are evenly arranged at the inlet of the inducer as a hydrodynamic improvement. The energy characteristics of the pump are measured on a closed hydraulic test rig. The pressure, vibration, and noise under different flow rates and different cavitation number are acquired for comparative analyses. Experimental results show that the energy characteristics changed after hydrodynamic improvement. The original pump is mainly affected by y-direction vibration and is clearly suppressed in the new pump. The low-frequency pressure pulsation under partial flow rate condition can be effectively suppressed. The baffles can also reduce the broadband center frequency at the pump outlet and change the relationship between center frequency and cavitation number. These results show that the hydrodynamic improvement at the inlet helps the suppression of cavitation instability of the high-speed centrifugal pump.

This work presents an experimental and computational investigation of a rotating bladed disk excited with air jets. The bladed disk is accelerated through critical speeds corresponding to specific ...

Adjoint methods are widely used for turbomachinery aerodynamic shape optimization. However, for industrial applications, the degradation of robustness and efficiency of adjoint solvers for edge-of-...

Losses induced by tip clearance limit decisive improvements in the system efficiency and aerodynamic operational stability of aero-engine axial compressors. The tendency towards even lower blade heights to compensate for higher fluid densities aggravates their influence. Generally, it is emphasised that the tip clearance should be minimised but remain large enough to prevent collisions between the blade tip and the casing throughout the entire mission. The present work concentrates on the development of a preliminary aero-engine axial compressor casing design methodology involving meta-modelling techniques. Previous research work at the Institute for Turbomachinery and Flight Propulsion resulted in a Two-Dimensional (2D) axisymmetric finite element model for a generic multi-stage high-pressure axial compressor casing. Subsequent sensitivity studies led to the identification of significant parameters that are important for fine-tuning the tip clearance via specific flange design. This work is devoted to an exploration of the potential of surrogate modelling in preliminary compressor casing design with respect to rapid tip clearance assessments and its corresponding precision in comparison with finite element results. Reputed as data-driven mathematical approximation models and conceived for inexpensive numerical simulation result reproduction, surrogate models show even greater capacity when linked with extensive design space exploration and optimisation algorithms.Compared with high-fidelity finite element simulations, the reductions obtained in computational time when using surrogate models amount to 99.9%. Validated via statistical methods and dependent on the size of the training database, the precision of surrogate models can reach down to the range of manufacturing tolerances. Subsequent inclusion of such surrogate models in a parametric optimisation process for tip clearance minimisation rapidly returned adaptions of the geometric design variables.

The topology optimization can be used to obtain preliminary turbomachinery disk designs which meet strength requirement. In order to eliminate enclosed holes which challenge manufacturing processes and to ensure distinct solid-void interface in the optimal result obtained by the topology optimization, a density distribution function is introduced for each element column in the design domain. Then, a parameter in each function is used to determine the disk’s thickness at corresponding radial position by controlling element densities. Once thicknesses at all radial positions are optimized, the shape of disk is thus determined. In this way, the optimization problem can be simplified by using these parameters as design variables. Illustrative examples are carried out to demonstrate the effectiveness of the proposed method in designing both compressor disks and turbine disks in comparison to the software T-Axis Disk and shape optimization method.

Blade tip-timing is one of the most effective methods for blade vibration parameters identification of turbomachinery. Once-per revolution (OPR) sensor is usually used to determine the rotating speed and as a time reference. However, the OPR sensor leads to a large measurement error or even failure, or it is difficult to install. A new improved two-parameter method without the OPR sensor is proposed to identify the synchronous vibration parameters of variable rotating speed blades. Three eddy current sensors are required to identify the excitation order, vibration amplitude, resonance rotating speed frequency, resonance frequency, and the initial phase of the blades. Numerical simulation of blade synchronous vibration parameters identification is presented, and the identification error of the method is investigated. The simulation results show that the identification accuracy of this method is better than that of the traditional two-parameter method and the improved method in reference, especially in the identification of the vibration initial phase. Experiments are conducted based on the blade tip-timing vibration measurement system. The results indicate that the standard errors of vibration parameter identification results between the new method and the method in reference are smaller, except for the initial phase. It is consistent with the results of the simulation identification. The synchronous vibration parameter identification of variable rotating speed blades without the OPR sensor is achieved based on the new improved two-parameter method.

Shock loss is the primary source of total pressure loss of transonic axial compressors. Reducing the shock by redesigning the geometry of rotor is of great interest for turbomachinery designers. However, the complex flow field involving shock waves, shock-boundary interaction, intense secondary flows, etc., in the compressor makes the design of rotor difficult. The conventional method of design and optimization is computationally intensive and time-costly. This study introduces an inverse design method to design rotor blades corresponding to prescribed isentropic Mach number distributions with no modification of flow-governing equations. An artificial neural network is trained to predict the isentropic Mach number distributions of any deformed blades. Then, with the pattern search optimization, the blade corresponding to the prescribed isentropic Mach number distributions can be achieved. When the aerodynamic parameter database is calculated and the neural network is obtained, this method can design large numbers of blades of changed isentropic Mach number distributions immediately, without modifying the computational fluid dynamics (CFD) flow solver. The design process is fully automatic and efficient. In this study, NASA Rotor 37 is redesigned and optimized as test cases. Some analysis on the influence of blade shape on aerodynamic characteristics of the rotor is represented in this study.

Gas Foil Bearings (GFBs) have fulfilled most of the requirements of novel oil-free turbomachinery. It has been considered as an alternative to traditional bearings in turbopumps, turbocompressors and turbochargers. However, they are prone to instabilities due to its low damping characteristics. Therefore, a hybrid bearing combining the GFBs and Active Magnetic Bearings (AMBs) has been in rigorous study due to its obvious advantages. The main aim of this paper is to provide a coupled time domain numerical model of the rotor supported on hybrid GFBs. The developed non-dimensional model has been used to investigate the capability of hybrid GFBs to mitigate the effect of unbalance force and in turn instability. Further, the ability of the hybrid GFB to withstand unbalance at an arbitrary time has also been investigated. The magnetic force generated is calculated using reluctance network method while fluid film forces are obtained by solving the governing Reynolds equation. The magnetic force of the AMB is non-dimensionalized in line with the fluid film forces of the conventional GFB. The non-dimensionalized fluid film forces from the GFBs and the electromagnetic forces from the EMAs are integrated into the equations of motion of the rotor. The response is then recorded for different operating parameters of the rotor. The sub-synchronous frequency which is the dominating frequency in case of the conventional GFB is eliminated due to the implementation of hybrid GFB. It has been demonstrated that the hybrid GFB has higher capability to withstand unbalance compared to conventional GFB. It has been observed that higher control current is required for higher unbalance eccentricity. Moreover, the hybrid GFB is also capable of adapting to the unbalance occurring at an arbitrary time.

A data-driven approach to predict dynamic responses of mistuned cyclic dynamic structures is presented. Nominally cyclic structures such as turbomachinery bladed disks, or blisks, contain deviation...

Surface roughness is a critical health parameter of a turbine blade due to its implications on blade surface heat transfer and structural integrity. This paper proposes a physics-based online framework for Gas Turbine Engines (GTE), in order to assess the blade surface roughness in a high-pressure turbine without engine shutdown. The framework consolidates Gas Path Analysis (GPA) based performance monitoring models and meanline turbomachinery analysis, using a novel GPA-meanline matching process. This extracts meaningful performance deviation trends from GPA, while addressing the uncertainties associated with the measurements and modelling. To relate efficiency loss to surface roughness severity, a meanline-based system-identification process has been developed to establish the meanline representation of the turbine stage, and to incorporate the empirical surface roughness loss correlations. The roughness loss correlations have been evaluated against recent transonic test data in the literature. A modification to the compressibility correction factor has been made according to the evaluation outcome, which improved loss predictions compared to the experimental measurements. The framework was tested on the three-year operational data of a cogeneration GTE, and the results verified the framework's potential for online surface roughness monitoring. The predicted surface roughness showed agreement in both trend and the magnitude-level with the measurements reported in the literature.

Rotating shaft–disk–blade (RSDB) system is one of the most important parts of turbomachinery, such as aero-engine, gas turbine and power plant. The coupling vibration of RSDB system with blade crack is vital for the blade health monitoring and crack detection of rotating blade. This study aims at addressing the dynamic modeling and steady-state coupling vibration mechanism of RSDB system with blade crack. First and foremost, on the basis of the stress state at crack section, an improved analytical breathing crack model (modified stress-based breathing crack model, MSBCM) for rotating blade is proposed. The validity of the proposed breathing crack model is verified by comparing the results obtained by MSBCM, finite element contact crack model and conventional analytical crack models. The comparative results suggest that MSBCM is of high fidelity and behaves best among the analytical crack models. Subsequently, a comprehensive dynamic model of the coupling vibration for RSDB system with blade crack is formulated on the basis of continuum beam theory and Lagrange equation. The shaft bending, shaft torsion, blade bending and blade radial deformation are comprehensively considered in this model. The validity of the proposed dynamic model is verified through comparison with finite element simulation and experimentation results. By introducing the proposed MSBCM, the dynamic coupling vibration model of the RSDB system with blade crack is formulated. At last, the steady-state coupling vibration mechanism of two typical structures for RSDB system is comprehensively investigated. It is suggested that the shaft torsional vibration is much more sensitive to blade crack than the shaft bending vibration be, which indicates that the vibration features of shaft torsional vibration may offer indicators for the presence of blade crack.

Porous gas bearings (PGBs) have a proactive application in aerospace and turbomachinery. This study investigates the gas lubrication performance of a PGB with the condition of velocity slip boundary (VSB) owing to the high Knudsen number in the gas film. The Darcy-Forchheimer laws and modified Navier-Stokes equations were adopted to describe the gas flow in the porous layer and gas film region, respectively. An improved bearing experimental platform was established to verify the accuracy of the derived theory and the reliability of the numerical analysis. The effects of various parameters on the pressure distribution, flow cycle, load capacity, mass flow rate, and velocity profile are demonstrated and discussed. The results show that the gas can flow in both directions, from the porous layer to the gas film region, or in reverse. The load capacity of the PGB increases with an increase in speed and inlet pressure and decreases with an increase in permeability. The mass flow rate increases as the inlet pressure and permeability increase. Furthermore, the simulation results using VSB are in agreement with the experimental results, with an average error of 3.4%, which indicates that the model using VSB achieves a high accuracy. The simulation results ignoring the VSB overrate the load capacity by 16.42% and undervalue the mass flow rate by 11.29%. This study may aid in understanding the gas lubrication mechanism in PGBs and the development of novel gas lubricants.

Turbine blade failures are among the leading causes of steam turbine failure. Failure types typically include cracking, rubbing, blade fouling, and foreign object damage. There is currently a range of non-destructive testing methods used to detect damage at the blade-disk attachment zone, all of which involve disassembling of the blade from the disk for periodic inspection. Evidence indicate that a method to detect damage at the blade-disk attachment zone using a non-contact, non-destructive in-situ off-line modal-based structural health monitoring technique could be useful under some circumstances. Such a technique would have the advantage of eliminating the necessity to disassemble blades during inspection. This would result in significant cost savings. Also, defects associated with the disassembly and reassembly of blades would be avoided. Thus, the aim of this study was to develop a modal-based turbomachinery blade disk attachment inspection technique. Modal parameters were acquired from a robust experimental modal analysis of freely supported low-pressure steam turbine blade-disk segment assemblies. Artificial single-location cracks were intentionally introduced into the turbine blades by cutting a 1 mm thickness notch at three probable damage locations, namely, at the upper pinhole on the leading-edge pressure side, above the root at the base of the aerofoil on the leading-edge and on the trailing-edge. In this work, a finite element analysis of the bladed disk segment assemblies was carried out with and without damage. To validate the reliability of the numerical models, the numerical results were correlated with the measured values, the results of which showed a strong correlation. Finally, a parametric study was conducted in which various healthy and damaged blade-disk cases were systematically investigated. This was done to examine the sensitivity of the blade natural frequency to damage. The artificial damage above the root was found to cause the largest changes in natural frequency. These changes were even more pronounced for assemblies with two blades. Receiver operating characteristic curves were used to assess the discriminatory ability of the results. Each damage case was found to be unique and therefore identifiable from its corresponding healthy case.

Compact axial compressors are of interest to the domestic appliance industry. The associated low Reynolds number leads to high losses compared to large-scale compressors due to a transitional flow field with large regions of separation. This thesis investigates how Reynolds number variations affect the three-dimensional and unsteady flow field in a compact axial compressor both pre-stall and in stall. An experimental study has been conducted using a scaled-up single-stage axial compressor across a Reynolds number range of 10⁴ to 10⁵. Steady and unsteady casing static pressure measurements, along with rotor upstream and downstream unsteady velocity measurements, have been used to observe the rotor flow field. As the Reynolds number is reduced below a critical value, 60,000 in the case of the compressor studied, the pressure rise coefficient of the compressor rapidly decreases. This fall-off in performance, and the critical Reynolds number, is determined by the point at which the compressor endwall separations begin to grow in size. Compact axial compressors have relatively large tip clearances, typically greater than 3% of the chord length. This results in unsteady movement of the tip leakage jet, which impinges on the subsequent blade, and causes leading edge spillage of the flow when operating towards stall. This is particularly prevalent when there is a large casing endwall separation, which can be caused by operating below the compressor’s critical Reynolds number. This spillage leads to a broadband hump and a pattern of equally spaced peaks appearing in the near-field casing static pressure spectra. The frequency spacing of these peaks has been found to equal the measured stall cell speed once rotating stall is established. With the use of an analytical model, and detailed unsteady measurements of the tip leakage jet, it is shown that these flow structures, which decay over a short circumferential distance and rotate at the subsequent stall cell speed, are capable of generating this spectral pattern. Previous research into axial compressor stall has mainly focused on stall inception and methods to extend the stable operating range. This thesis considers the performance of an axial compressor beyond stall and investigates how the characteristics of stall cells depend on Reynolds number. Detailed unsteady measurements have been used to measure the behaviour across a range of in-stall flow coefficients. These measurements have been used to extract the stall hysteresis and to determine the size, speed, number, and spanwise extent of the stall cells. The results show that for the stalled compressor, as the Reynolds number increases, the size of the minimum stall cell that can be sustained decreases. This means that a larger change in throttle area is needed to reduce the stall cell down to a size where the compressor can recover from stall. The stall cells increase in size and slow down as the Reynolds number decreases. The size and shape of the stall cells that form are related to the extent of the three-dimensional flow field present prior to stall. In all cases, as the number of stall cells increases, so do the speed and the total size of the stall cells. Further reductions in the flow coefficient cause an increase in the total size of the stall cells and a decrease in the speed. The value of the critical Reynolds number varies with the compressor blade geometry. A redistribution of the rotor spanwise loading away from the endwalls, with an increased loading at the midspan, can recover some of the loss in performance at the lower Reynolds numbers, and decrease the critical Reynolds number. Increasing the rotor clearance gap in a compact axial compressor results in a significant loss in pressure rise. However, when operating below the critical Reynolds number, this loss is lower, because the casing endwall flow is already separated at lower clearances. Increasing the rotor clearance reduces the stall hysteresis by increasing the minimum stable stall cell size. Below a critical value of Reynolds number, which is dependent on the specific geometry used, the pre-stall and in-stall flow features are highly sensitive to Reynolds number, becoming increasingly three-dimensional and unsteady., EPSRC and Dyson Ltd

Maintenance of adequate thermal properties is critical for correct operation of a gas foil bearing. In this work, the authors present the results of the experimentally conducted thermal characterization of a prototype installation of the bearing. A novel method of temperature identification, based on integrated thermocouples readings, has been employed to determine the thermal properties of the specialized sensing top foil mounted in the tested bearing. Two measurement campaigns have been subsequently completed, applying freely-suspended and two-node support configurations, to gather complementary knowledge regarding the bearing’s operation. Apart from the rotational speed and temperature field measurements, the authors have also studied the friction torque and the shaft’s journal trajectories based on its radial displacements. The temporal courses for the above-mentioned quantities have enabled inference on the effects present during run-up, run-out and stable state operation at a constant speed. As confirmed, the applied distribution of the integrated sensors allows for temperature readings on the entire outer surface of the foil, and therefore, provides useful data for the bearing characterization. The work is concluded with presentation of the recommended directions regarding future improvements of the proposed measurement technique and more comprehensive study of the bearing’s characteristics.

Fossil fuels are energy sources that supply a large part of the world's energy generation. However, they produce greenhouse gases such as carbon dioxide (CO2), nitrogen oxide (NOx) and particulates that increase global warming. For this reason, other forms of renewable energy such as hydropower have begun to be implemented through turbomachinery such as Pelton turbines, which significantly reduce these emissions since they are highly efficient turbines based on the use of natural resources (water). Pelton turbines are based mainly on three components for their operation, which are the Pelton injector, the bucket and the wheel. The injector is an important component in the energy transformation of Pelton turbines. Although to analyze its behavior, it is possible to use fluid dynamics (CFD) software to predict the trajectory of the flow through a solid or free surface. The objective of this work is to analyze by means of computational fluid dynamics (CFD) the incidence of the length and the needle tip angle of a Pelton turbine injector on the generated power. For this, an ANSYS 2020R2 computational fluid analysis software was used to study how the variation of the injector needle tip angle influences through the volume of fluid (VOF) method, starting from the generation of a commercial model with a tip angle of 60° and two (2) geometries of 55° and 75° respectively. Numerical results show a better performance for the 75° angle of 96 % and lower for the 55° and 60° with 94.1 % and 95.5 % respectively, whereby steeper angles achieve higher performances. In summary, the present study pretends to increase the power generation, in the face of phenomena occurred in the energy transfer. Although the performance of the injector in each angle configuration must be tested in practice

Turbomachines are still required for the efficient conversion of renewable, chemical or potential energy into propulsion, mechanical power, or electricity. The increasing demand for efficiency, availability, reduced footprint and cost of ownership poses fundamental challenges to design methods the accuracy of which needs to be constantly upgraded to keep the pace with the availability of new materials, type of fluids and fuels, manufacturing methods and technologies. This paper discusses recent trends in design methods that take advantage of both artificial intelligence and high-fidelity simulations techniques that guide the design process by harvesting design data from multiple sources and improve the accuracy of design verification respectively. Such approach has been successfully used in aero and thermal design as well as in the critical area of materials and fluids engineering. In the future the concerted use of machine learning and high-fidelity methods may allow researchers to conduct reliable virtual tests in the framework of design loops to cut down design time and risk.

As a promising metalwork processing technology, electromagnetic induction heating (EMIH) method has been applied in dealing with bolted flange joints in turbomachinery. In this study, a 3-D finite element model of electromagnetic induction heating system for the bolted flange joint is established, and the specific governing equations are derived based on Maxwell’s principle. The alternately-coupled magneto-thermal analysis is carried out considering temperature-dependent material properties to obtain the temperature distribution, followed with the uncoupled thermal-mechanical analysis to acquire the axial stress and deformation in EMIH process. The magnetic induction intensity mainly concentrates at the inner wall region, attenuates seriously along the radial direction, and reduces to almost zero at the outer wall. Due to the skin effect, the heat transfers radially and axially outward, indicating a diamondlike-shaped development from the center to the surrounding region. The axial stress with and without initial pretension are also discussed respectively. Then the corresponding experiments are introduced and carried out to validate the reliability of numerical simulation results. By comparing the results of the center point of inner surface and outer surface, the numerical simulation is proved reliable with a 5∼10% reasonable deviation. Further, the induction heating process has been improved through the optimization method based on pattern search algorithm. By adopting the stepped input current density optimized in the study, the optimal thermal stress tends to be constant and the final heating time reduces by 20.5% in the safe range of stress.

Mixed flow turbines are widely used in several industrial applications covering turbomachinery, automotive engineering and electricity production. For decades, it is well known that mixed flow turbines are a seat of several loss phenomena such as the volute to rotor interspace loss, subject of this paper. Commonly, the meanline approach is the first step solution for building a preliminary design of such turbines and estimating subsequent losses. The accuracy of the code used in the meanline modeling is crucial for building an optimized turbine design with a minimized loss generation. This paper presents an improved validated meanline code, written in the newest object-oriented version of the FORTRAN language, for turbomachinery performance prediction. Unlike commercially available codes, the code allows the calculation of the rotor passage loss coefficient given the turbine expansion ratio without the need for additional test data. The standard deviation value between the code and test data is less than 10%, for all studied cases which ensure the validity of the developed model. Then, the developed code is exploited to investigate the effect of the volute to rotor interspace geometry on the loss generation and performance of a mixed flow turbine. Indeed, a performance distribution over a wide range of rotational speed and an energy loss breakdown are depicted and discussed showing a significant impact of the volute to rotor interspace. The results revealed an improvement in the turbine efficiency up to 2.9% with a volute to rotor interspace radii ratio of 0.59 at 80% of the design speed.

Mesh generation and deformation are critical elements in gradient-based aerodynamic shape optimization (ASO). Improperly generated or deformed meshes may contain bad-quality cells that degrade the ...

Gas foil bearings are often used in high-speed turbomachinery such as turboexpanders and turbochargers due to their merits over simple gas-lubricated bearings. The merits of gas foil bearings include their ability to tailor dynamic parameters such as stiffness and damping. Gas foil bearings usually have low clearance and operate at a high rotational speed, which eventually leads to velocity slip at the solid–fluid interface. This article investigates the effect of slip flow on various parameters of gas foil thrust bearings. A numerical model is formulated to predict pressure, film thickness and temperature distribution of helium lubricated gas foil thrust bearing at high rotating speed. The Reynolds equation is modified by assuming first-order slip coupled with the structural (compliant) and energy equation. The temperature-dependent viscosity and density of the fluid are also considered in the Reynolds equation to predict the thermohydrodynamic behaviour of gas foil thrust bearings. The numerical model thus developed uses a finite-difference method and the Newton–Raphson method to solve the Reynolds equation,whereas the successive over-relaxation method is used to solve the energy equation. Various performance parameters are compared for slip and no-slip conditions for gas foil thrust bearings. The results show a considerable difference between the two phenomena. Also, the conventional Reynolds equation tends to overestimate the load-carrying capacity.

Восстановление надводного флота Российской империи в 1906–1917 гг. происходило в условиях революционных сдвигов в мировой кораблестроительной технике. Произошел переход от традиционных поршневых паровых машин в качестве главного двигателя к турбинным механизмам. Для самостоятельного изготовления судовых турбин машиностроительным заводам в империи не хватало мощного металлургического оборудования и технологического опыта. В условиях спешной подготовки к войне правительство пожертвовало традиционным официальным принципом — строить все у себя, из отечественных материалов, своими силами. Турбинные двигатели для всех классов кораблей царский флот получал из Германии, Англии, Швейцарии и США. В России же производство турбин в основном сводилось к механической обработке стальных заготовок, полученных от зарубежных поставщиков, и последующей сборки под руководством специалистов из авторитетных в данной области западных фирм. В историографии вопроса наблюдается систематическое противоречие: достоверное фактически описание кораблестроительной практики, как правило, опровергает обобщения, основанные на преувеличенных представлениях о достигнутом заводской техникой России научно-производственном уровне. The restoration of the Imperial Russian surface fleet in 1900–1917 took place during major technological shifts in world shipbuilding. It was the time of transition from traditional reciprocating steam engines to turbine mechanisms. The Empire’s machine-building plants lacked powerful metallurgical equipment and technological experience required for production of ship turbines. Within the circumstances of rapid war preparations, the government shifted from traditional principles of self-production, towards using international resources and powers. The Imperial Russian Navy was supplied by turbine engines for all classes of ships by Germany, England, Switzerland and the USA. Turbine engines in Russia were manufactured by processing steel templates provided by foreign suppliers. Further construction of the engines took place under the supervision of experienced western entities. A systematic contradiction in the historiography on this issue can be seen. The author reveals a verified factually based descriptions of the shipmanufacturing process, and disapproves past general conclusions, which were based on the exaggerated notions about the level of Russian scientific and technological development.

This work proposes a novel hybrid gas-magnetic bearing (HGMB) which has single bearing structure. The gas bearing and the active magnetic bearing (AMB) share the same airgap, and the closed magnetic poles are taken as the bushing of the gas bearing. Controllable electromagnetic forces are applied to adjust system dynamic characteristics. Numerical results demonstrate better rotating accuracy, suppressed sub-synchronous vibration, lower energy consumption compared with rigid gas bearings and AMB, and less air friction losses than conventional HGMBs, indicating the suitability of the proposed bearing for cryogenic turbomachinery.

In recent years, CFD has proven to be a very useful asset to help with predicting complex flows in a wide range of situations, including multiphase and gas-particle flows. On this track, numerical modelling of particle-laden flows in multistage turbomachinery has become an important step in helping to analyse the behaviour of a discrete phase in gas turbines. Furthermore, unsteady effects due, for example, to rotor–stator interaction may have an effect on trajectories and capture efficiencies of the discrete phase. Unfortunately, computational times for transient simulations can be exceedingly high, especially if a discrete-phase needs also to be simulated. For this reason, this work reports a new method for the efficient and accurate simulation of particle-laden flows in gas turbine engines components. The Harmonic Balance Method is exploited to gain orders of magnitude speedup exploiting the idea that once the flow field has been embedded in the spectral basis, it can be reconstructed at any desired time. In this way, not only can the computational time needed to reach convergence of the flow field be dramatically reduced, but there is also no need to keep simulating the flow field during particle tracking. On the contrary, the continuous phase field can be retrieved at any desired time through flow reconstruction. This technique is conceptually simple, but, to the authors’ knowledge, has never been applied so far in particle-laden flow simulations and represents a novelty in the field. First, the implementation of the method is described, and details are given on how phase-lagged boundary conditions can be applied to flow and particles to further speed up the calculation. Then, some relevant case studies are presented to highlight the performance of the method.