Dissipative Flow Research Articles

Numerical Investigation of Pressure Measurement by Pitot Tubes in Micro-scale Taylor Couette Flow with Hyper Rotational Speed and its Correction

Abstract As the most classic technique in fluid mechanics, the pitot tube has been employed for hundreds of years. Its feasibility is based on the premise that the measured fluid is ideal, irrotational, and incompressible, and the presence of pitot tube does not interfere with the measured flow field. However, the development of rotating machinery, especially Dynamic Gas Bearing, has brought about strong dissipation, large gradients, and scale effects in the shear flow between the rotor and stator, which urgently require pressure correction as conventional pitot tube has been utilized. The physical mechanism of the influence of the existence of pitot tubes on micro-scale shear flow has been revealed. The cavity formed by the pitot tubes interferes with the shear flow in the transition static gap, causing instability in the shear flow. Compared to the absence of a pitot tube, the pitot tube has brought 3.5% -10% of underprediction of mean pressure. the effects of rotational speed, clearance has been performed in microscale hyper-rotational speed shearing flow. A pressure correction method using the pitot tube pressure was presented for strong dissipation, large gradients, and scale effects, verifying its effectiveness in microscale hyper-rotational speed Taylor Coulter flow.

Topology optimization for 3D fluid diode design considering wall-connected structures

This paper proposes a density-based topology optimization method for the three-dimensional design of fluid diodes considering wall-connected structures based on the fictitious physical modeling approach. The optimum design problem of fluid diodes is formulated as maximizing the energy dissipation in the reverse flow subject to the upper bound constraint of the energy dissipation in the forward flow. A fictitious physical model and a geometric constraint are constructed to detect and restrict the “floating” solid domains, which are not connected to the outer boundaries. The sensitivities of cost functions are derived and computed based on the continuous adjoint method. The finite volume method is employed to discretize the governing and adjoint equations to mitigate the huge computational costs of three-dimensional fluid analysis. Numerical investigations are presented to validate the fictitious physical model and the geometric constraint for excluding “floating” islands. Finally, topology optimization for fluid diodes with and without the geometric constraint is performed, and the result demonstrates that the proposed method is capable of generating fluid diodes with wall connectivity, while maintaining a good functional performance.

Global Transonic Solution of magnetized dissipative accretion flow around non - rotating black holes with thermal conduction

Global Transonic Solution of magnetized dissipative accretion flow around non - rotating black holes with thermal conduction

Thermal and Computational Analysis of MHD Dissipative Flow of Eyring-Powell Fluid: Non-Similar Approach via Overlapping Grid-Based Spectral Collocation Scheme

The aim of the present study is to numerically investigate the non-similar flow and heat transfer in a dissipative Eyring-Powell fluid (EPF) over a stretching surface. A constant magnetic field is applied perpendicular to the stretched surface to explore the impact of the Lorentz force. Both viscous and magnetic dissipation are considered to comprehensively examine their effects on heat transfer. The problem in hand does not admit self-similar solutions as the non-Newtonian fluid parameter varies with the spatial variable along the stream-wise direction. Consequently, the set of nonlinear partial differential equations, modeling the flow problem is nondimensionalized primarily by employing a pseudo-similarity variable and stream-wise coordinate. The non-dimensional set of nonlinear partial differential equations is solved by a newly developed and efficient “overlapping multi-domain bivariate spectral local linearization method (OMD-BSLLM)”. The current study includes residual error analysis and convergence tests to demonstrate the accuracy of the numerical method applied to the current mathematical model. Graphs show fluid flow and heat transfer results for different flow parameters, while tables display skin friction and Nusselt number values. The results indicate that the non-Newtonian fluid parameter enhances both the velocity profile and temperature distribution. The fluid decelerates with increasing the dimensionless stream wise coordinate and Hartmann number. Viscous dissipation and dimensionless stream-wise coordinate enhances the temperature profile.

Analysis of flow and energy dissipation in pump turbine with small guide vane opening

In this study, a detailed flow state analysis of a reversible pump turbine (RPT) under small guide vane opening conditions was conducted using both numerical simulation and experimental methods on three analysis planes at 0.5 span. The entropy production rate (EPR) in entropy production theory was utilized to visualize the location and magnitude of energy losses in the flow process. The focus was placed on the coherent vortex structures in the volute, stay vane, guide vane, and runner flow path. The results demonstrate that, throughout the runner’s cycle, the leading edge of the guide vane experiences low-pressure regions with uneven pressure distribution within the runner channel. Velocity streamlines reveal flow separation and vortex generation at the typical guide vane trailing edge, with a maximum velocity reaching 49.7. Analysis of radial force and torque on typical guide vanes and the runner indicates that radial force and torque on the guide vanes do not significantly change, with the radial force coefficient CF r−x varying between 0.254 and 1.3. Major energy losses are concentrated behind the trailing edge of the guide vane, primarily in the form of jet wake. Coherent vortex structures are observed in the volute, stay vane, guide vane, and runner flow path, with slight turbulent wake at the trailing edge of the stay vane, clear shedding vortex streets, and vortex build-up in the bladeless area between the guide vane and the runner.

A granular thermodynamic framework-based coupled multiphase-substance flow model considering temperature driving effect

Based on the energy dissipation caused by consolidation deformation of the porous media under external force and migration of the internal suspended substances, a coupled multiphase-substance flow (CMF) model was established. This model introduced the new concepts, such as particle temperature and particle entropy, to describe energy dissipation at meso-level. This model used a potential energy density function and migration coefficients to establish the corresponding connection between the dissipative force and dissipative flow. This viewpoint unifies the deformation, seepage, and suspended substance migration of geotechnical materials under the framework of granular thermodynamics. It can reflect the evolution of effective stress in the solid matrix of multi-components in a particle-reorganized state, and considers the temperature driving effect. The proposed CMF model is validated using the experimental results under coupled migration of heavy metal ions (HMs) and suspended particles (SPs). The calculation results demonstrated that the CMF model can describe the flow process under the conditions of arbitrary changes in different suspended substance types, injection concentrations, and injection velocities.

Study on cross effect of mesh and turbulence model on separated flow prediction using a tandem cylinder case

Abstract Low dissipative separated flow prediction is still a great challenge in aeronautics engineering. An open-source CFD code named SU2 is employed to study the cross effect of mesh as well as turbulence model on the numerical resolution of separated flow for a tandem cylinder case. Firstly, a nonlinear variation of the numerical resolution for the separated flow is found by comparing the mean surface pressure distribution, mean streamwise velocity profile, the instantaneous vorticity magnitude as well as the aerodynamic forces on the rear cylinder. The resolution of separated flow from the medium mesh is the best compared to those getting from the coarse and fine mesh. Secondly, the reason causing the nonlinear variation of flow resolution is analyzed considering the influence of spatial discretization error and turbulent viscosity locally. A finer mesh would lead to a low spatial discretization error and then a high turbulent viscosity and then causes nonlinear variation of flow resolution. Finally, the existence of a “density boundary” of the mesh is proposed which could balance the spatial precision as well as turbulent viscosity and achieve the prediction of separated flow with a good resolution employing the RANS method.

Numerical simulation of magnetically driven nanomaterial rotating flow configured by convective-radiative cone with chemical reaction

Indeed, nanoliquids have acquired substantial consideration in heat transference field because of their inimitable thermal attributes and favorable application likelihoods. In contrast to orthodox liquids, the haphazard movement of nanoparticles within nanoliquid strengthens fluid turbulence, accomplishes superior thermal effectiveness and declines thermal resistance. Nanoliquids have ample utilization, for illustration, solar energy, electronic chips, automotive radiators and heat exchangers etc. This communication reports chemically reactive electro-magnetized nanomaterial dissipative flow confined by rotating cone. Flow expressions include thermo-solutal buoyancy, varying viscosity and magneto-hydrodynamics. Radiative heat, thermophoresis, viscous dissipation, Brownian diffusion, thermal source and first order chemical reaction are pondered to model transport expressions. Relevant variables are introduced to transfigure partial differential mathematical expressions to mathematical ordinary ones. Numerical outcomes for non-dimensional mathematical expressions are reported via bvp4c algorithm in MATLAB. The comprehensive results featuring dimensionless quantities are explored through graphs and arithmetic representations. It is evaluated that escalating values of variable viscosity, Prandtl number and unsteady parameter decline temperature but temperature is improved as a consequence of progressive variation in radiation parameter, Eckert number, thermophoresis parameter, heat generating and Brownian diffusive variables. The study is relevant to cooling industry, electroconductive, thermal collector and nano-materials processing.

Computational analysis of unsteady oscillatory flow of nanofluid with variable electric conductivity: gear-generalized differential quadrature approach

Abstract This study numerically investigated the entropy production in nanofluids’ dissipative unsteady oscillatory flow characterized by variable electric conductivity and magnetic heating effects. The imposition of the non-isothermal boundary condition on the oscillatory stretching sheet plays a crucial role in establishing the self-similar solution in the presence of viscous heating. An external magnetic field (uniform in space and time) is imposed perpendicular to the plane of the oscillating stretched boundary. The energy equation, incorporating viscous dissipation effects and momentum equation, is reduced to nonlinear coupled partial differential equations and numerically solved using the Gear-generalized differential quadrature scheme.Additionally, to ensure the precision and reliability of the outcomes, the numerical code undergoes a thorough validation process that involves comparing its outputs to the findings of previous available studies. The Corcione model is implemented to describe the nanofluid’s effective viscosity and thermal conductivity. Furthermore, expressions for entropy production and relative irreversibility parameter (Bejan number), considering variable electric conductivity, are derived and computed based on solutions obtained from momentum and energy equations. The impacts of parameters such as magnetic parameter, variable electric conductivity parameter, Eckert number, Strouhal number, Prandtl number and temperature difference parameters on flow, heat transfer, entropy generation, and Bejan number are systematically illustrated and examined. We observed that increasing the variable electric conductivity parameters reduces the velocity profiles while improving the thermal fields. Similar behavior is found when the strength of a magnetic field is increased. The skin friction coefficient exhibits an augmentation in response to the Eckert number, dimensionless time, Strouhal number, nanoparticle volume fraction, magnetic parameter, and variable thermal conductivity parameter. Conversely, the Nusselt number increases concerning the Strouhal number and nanoparticle volume fraction. At the same time, it declines in association with the magnetic parameter, dimensionless time, Eckert number, and variable electric conductivity parameter.This comprehensive investigation enhances our understanding of nanofluid dynamics and provides valuable insights for optimizing thermal management systems across various engineering disciplines.

Computational analysis of heat transport dynamics in viscous dissipative blood flow within a cylindrical shape artery through influence of autocatalysis and magnetic field orentation

Nanofluids consisting of tetra nanoparticles are crucial in bio medical sciences due to their improved thermal transport characteristics. The advanced tailored properties of tera nanoparticles make them useful in several medical interventions, such as hyperthermia treatment, where the targeted tissue can be heated more efficiently, leading to better treatment outcomes. The current study investigates the heat transfer enhancement in a hemodynamic system using tetra nanoparticles. The physical configuration of the blood flow is assumed with in a permeable cylindrical shape stenosed artery. The model incorporates the Carreau model with inclusion of diverse factors such as, exponential space-based heat source, viscous dissipation, infinite shear rate and permeability of surface. Additionally, impact of chemical reaction (autocatalysis) and magnetohydrodynamic (MHD) consequences is also integrated into the system. The framed partial differential equations (PDEs) generated by physical problem are converted into new dimensionless form of an ordinary differential system (ODEs). Bvp4c MATLAB procedure is fetched for numerical investigation. It is observed that, velocity profile of the fluid is reduced due to intensification in inclined magnetic effect, whereas autocatalysis effect promotes the concentration of nanoparticles in blood flow mixture, which increases the temperature field of fluid. Furthermore, augmentation in the values of Wassenberg number increased the elasticity in blood which enables it to deform and stretch more readily in reaction to alterations in flow conditions and hence reduction is seen in overall blood flow rate. The results revealed the significance of these integrated factors for accurate modelling of blood flow passing through a stenosed artery, which is crucial in medical interventions.

Utilizing numerical techniques for modeling radiating Casson nanofluid flow with thermophoretic phenomenon on a heated stretching surface

This work aims to use mathematical modeling to investigate the mechanics of heat and mass transport in dissipative Casson nanofluid flows over a linear rough sheet. This study considers various elements such as thermal radiation, magnetic fields, heat generation, the varied properties of porous media, and the thermophoretic impact. Besides that, it looks into the variations in viscosity, diffusivity, and thermal conductivity, as well as the energy dissipation from viscous internal friction and fluid temperature modifications. The method involves coming up with and changing the boundary layer equations into a group of linked nonlinear ordinary differential equations that use variables that do not have any dimensions. The foundation of the solution strategy for these equations is the Hermite collocation method (HCM), which is renowned for its precision and adaptability. It offers an organized approach to solving the complex differential equations, allowing for accurate numerical solutions. The use of graphical representations ensures thorough data analysis and clarity, while also providing insightful information about the computed outcomes. The code validation method uses numerical comparisons with recent research to confirm the algorithm’s accuracy and dependability, as well as its resilience in comparison to the existing literature. Important conclusions from the study show that thermal boundary layers and nanofluid velocity decrease with increases in the porosity parameter, slip parameter, and magnetic field parameter.

Using the Herschel-Bulkley Consistency Index to Characterise Complex Biopolymer Systems-The Effect of Screening.

The use of the consistency index, as determined from fitting rheological data to the Herschel-Bulkley model, is described such that it may yield systematic trends that allow a very convenient description of the dissipative flow properties of linear and branched (bio)polymers in general, both in molecular and weakly associated supramolecular solutions. The effects of charge-mediated interactions by the systematic variation of the ionic strength and hydrogen bonding by a systematic variation in pH, using levels that are frequently encountered in systems used in practice, is investigated. These effects are then captured using the associated changes in the intrinsic viscosity to highlight the above-mentioned trends, while it also acts as an internal standard to describe the data in a concise form. The trends are successfully captured up to 100 times the polymer coil overlap and 100,000 times the solvent viscosity (or consistency index). These results therefore enable the rapid characterization of biopolymer systems of which the morphology remains unknown and may continue to remain unknown due to the wide-ranging monomer diversity and a lack of regularity in the structure, while the macromolecular coil size may be determined readily.

Energy dissipation and power flow analysis based on acoustic black hole laminated beams

To enhance the understanding of the dynamic performance of composite beams, assessing their energy accumulation and dissipation capabilities is crucial. The purpose of this paper is to establish a dynamic analysis model of acoustic black hole (ABH) laminated beams based on the Timoshenko beam theory and isogeometric method and to analyze structural intensity and power flow. The vibration governing equation for the ABH laminated beam is derived by calculating its potential and kinetic energy. The convergence of the results is confirmed through multiple mesh refinements, while the model's accuracy and applicability are substantiated through comparisons with the literature, traditional finite element simulations, and experimental data. This investigation into the power flow and structural intensity under varying parameter conditions elucidates the energy transfer mechanisms within ABH laminated beams, offering valuable insights for the deployment of ABH technologies in practical engineering applications.

Exploring integrated heat and mass transfer in von-Kármán dynamics involving Reiner-Rivlin fluid with regression models

The rotating disk system is widely used in several important applications, including turbomachinery design, mixing and stirring processes, centrifugal blood pumps development, optimizing electronic component, cooling efficiency, and various others. Moreover, regression models offer comprehensive and reliable prediction for important parameters in flows developed by revolving disks. This article presents similarity solutions for coupled heat and mass transfer with viscous dissipation in an inelastic non-Newtonian fluid flow across a permeable revolving disk. The inclusion of diffusion terms due to Dufour and Soret effects establishes coupling between energy and concentration equations. Temperature and concentrations at disk surface are supposed to vary quadratically along the radial direction. The impacts of MHD, heat source/sink dynamics and Joule heating effects on thermal boundary layer are also scrutinized. MATLAB's bvp5c tool, based on collocation scheme, generates numerical results. Homotopy analysis method, a widely accepted analytical solution procedure, is then applied to produce the series solution. Selection of the optimal auxiliary parameter values featured in the series solution is also included. The impact of Reiner-Rivlin fluid assumption on the torque required by the disk is assessed. Furthermore, linear and quadratic regression models are developed for the skin friction factors, Nusselt number and Sherwood number.

Numerical investigation of an axisymmetric, dissipative, and unstable micropolar fluid flow over a stretched Riga plate incorporating mixed convection and chemical reaction

AbstractThe demand to improve the thermal performance of industrial working fluids and materials for better quality in engineering and household devices has spurred numerous studies on non‐Newtonian viscous fluids under various conditions. Researchers have explored different structures and geometries to understand and enhance the thermal propagation of fluid particles. As such, this research aims to investigate the effects of two‐dimensional unsteady boundary layer flow phenomena of a dissipative micropolar fluid over a radially stretching Riga plate when first‐order slip, thermal and solutal boundary conditions, mixed convection, and thermal radiation are all considered. Ordinary differential equations are the governing equations resulting from converting partial differential equations. The boundary layer conditions are altered and numerically solved using the Runge–Kutta fourth‐order shooting approach. Graphs and tables are used to visually analyze the physical attributes of temperature, concentration, velocity distributions, skin friction, Nusselt number, and Sherwood in relation to various factors. The results reveal significant insights into the impact of mixed convection and chemical reactions on the flow characteristics and dimensions. These findings have important implications for various engineering applications, particularly in enhancing industrial systems' heat and mass transfer processes, providing practical knowledge for improving thermal performance in real‐world applications.

Enhanced Dissipation for Two-Dimensional Hamiltonian Flows

Let H∈C1∩W2,p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H\\in C^1\\cap W^{2,p}$$\\end{document} be an autonomous, non-constant Hamiltonian on a compact 2-dimensional manifold, generating an incompressible velocity field b=∇⊥H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b=\ abla ^\\perp H$$\\end{document}. We give sharp upper bounds on the enhanced dissipation rate of b in terms of the properties of the period T(h) of the closed orbit {H=h}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{H=h\\}$$\\end{document}. Specifically, if 0<ν≪1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\ u \\ll 1$$\\end{document} is the diffusion coefficient, the enhanced dissipation rate can be at most O(ν1/3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\ u ^{1/3})$$\\end{document} in general, the bound improves when H has isolated, non-degenerate elliptic points. Our result provides the better bound O(ν1/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\ u ^{1/2})$$\\end{document} for the standard cellular flow given by Hc(x)=sinx1sinx2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{\ extsf{c}}(x)=\\sin x_1 \\sin x_2$$\\end{document}, for which we can also prove a new upper bound on its mixing rate and a lower bound on its enhanced dissipation rate. The proofs are based on the use of action-angle coordinates and on the existence of a good invariant domain for the regular Lagrangian flow generated by b.

Numerical assessment of transient flow and energy dissipation in a Pelton turbine during startup

The Pelton turbine, known for its high application water head, wide efficient operating range, and rapid start-stop capability, is ideal for addressing intermittent and stochastic load issues. This study numerically analyzes the transient two-phase flow and energy dissipation during the startup of a Pelton turbine. Dynamic mesh technology controlled nozzle opening changes, and momentum balance equations managed runner rotation. Findings showed that the runner speed initially increased rapidly and then more slowly, and flow rate matched the nozzle opening variations. Runner torque first rose linearly, then decreased, with the fastest decline during nozzle closing. Hydraulic efficiency peaked early in nozzle reduction but then dropped sharply. Strong vortices formed due to upstream inflow and downstream backflow impact in the distributor pipe. The jet needle and guide vane improved flow in the converging section of nozzle, but flow began to diffuse with increased stroke. Initially, the jet spread fully on the bucket surface, but later only affected the bucket tips. Pressure fluctuations in the water supply mechanism were primarily due to jet needle motion, with higher amplitude during movement and lower when stationary. These fluctuations propagated upstream, weakening over distance. Reynolds stress work and turbulent kinetic energy generation, respectively, dominated energy transmission and energy dissipation, with their maximum contribution exceeding 96% and 70%. High-energy clusters corresponded to jet impact positions, highlighting jet-bucket interference as crucial for energy transport. This study established a performance evaluation method for Pelton turbine startups, supporting further investigation into characteristic parameters, flow evolution, and energy dissipation patterns.

Computation of Internal Heat Source, Viscous Dissipation and Mass Flow Effects on Mono-Diffusive Thermo-Convective Stability in a Horizontal Porous Medium

Computation of Internal Heat Source, Viscous Dissipation and Mass Flow Effects on Mono-Diffusive Thermo-Convective Stability in a Horizontal Porous Medium

Thermal dynamics of nanoparticle aggregation in MHD dissipative nanofluid flow within a wavy channel: Entropy generation minimization

This paper examines how nanoparticle aggregation and a consistent magnetic field influence the peristaltic movement of a dissipative nanofluid, which is caused by the sinusoidal deformation of the boundary. The viscosity of TiO2/H2O nanofluids is accurately determined by the Krieger-Dougherty model with nanoparticle aggregation, while thermal conductivity (TC) is estimated through the Bruggeman model. The set of governing equations are modeled in a fixed frame by utilizing the conservation laws of energy, mass and momentum. Galilean transformation is utilized to transform the system of equations into a wave frame, which is then converted into a dimensionless form. The assumption of a small Reynolds number and long wavelength serve to further simplify the set of equations, which are subsequently addressed through the implementation of the differential quadrature method (DQM), a highly effective numerical technique. Quantities of interest, namely velocity, pressure gradient, temperature, trapping phenomena, heat transfer, and volumetric entropy generation are analyzed across a range of physical parameters, including the solid volume fraction (Φ=0.01−0.04), Eckert number (Ec=0.0−0.1), Hartman number (Mh=0.2−2.2), Grashof number (Gr=1.0−3.0) and temperature ratio parameter (θd=0.5−2.5). A comparative analysis is conducted between the scenario involving aggregation and the one without aggregation. It is observed that nanoparticle aggregation significantly alters these quantities.

Analysis of unsteady wake flow in centrifugal pump volute by using mode decomposition method

To analyze the coherent structure of wake flow in volute and its corresponding frequency information, dynamic modal decomposition (DMD), classic and spectral proper orthogonal decomposition (SPOD), were employed to decompose the transient flow field of the centrifugal pump volute. The snapshot set was constructed by means of the velocity field data at different moments in volute based on the large eddy simulation (LES) approach. The basic principles of the DMD, POD, and SPOD methods were compared in detail, and the decomposition results of the three methods on the wake flow structure in volute were compared and analyzed. The analysis results show that DMD can decompose the wake flow into coherent structures with different frequencies, including the basic steady-state structure, the dynamic modal flow field structure characterizing rotor–stator interaction (the first three modes with frequencies of 145.81, 291.61, and 437.43 Hz, respectively), and dissipative modal flow field structure characterizing fragmentized vortex (the fourth mode with a frequency of 486.03 Hz) in the volute. The POD can decompose the wake flow into flow structures with different energy levels. The first four modal energies account for more than 66% of total energy, which represents the large-scale structure with higher energy, and its dominant frequencies correspond to the blade passing frequency (145 Hz) and its frequency multiplication (290 Hz). The SPOD can not only decompose the complex wake flow into structural features of different energy levels but also has single-frequency characteristics of its modal structures. Compared with DMD and POD methods, the SPOD has the advantages of both, and can reflect the evolution characteristics of the wake flow in the volute.