Decomposition Of Vorticity Research Articles

Dynamic characteristics of the tip leakage vortex in an oil–gas multiphase pump based on vorticity decomposition theory

AbstractThe paper combines experimental and numerical simulation methods to investigate the tip leakage vortex (TLV) in an oil–gas multiphase pump. The study aims to understand the evolution and dynamic characteristics of the TLV. The vorticity decomposition theory is used to analyze the spatial–temporal evolution, transient behavior, and rigid vorticity of the TLV. The findings reveal that the TLV is formed near the pressure surface when the tip clearance jet passes through the shear layer. In one impeller rotation period, transient fluctuations of the TLV are attributed to pressure differences and velocity changes across the tip clearance, and the TLV undergoes two split–dissipation–recurrence processes within one impeller rotation cycle. The presence of the gas phase enhances the scale and strength of the TLV while prolonging its existence in the flow channel. Analysis based on the rigid vorticity transport equation shows that the growth of the TLV and collapse are primarily governed by the rigid vortex generation term. The Coriolis force term contributes to stabilizing the TLV, while the rigid vorticity stretching term affects its shape. Furthermore, the gas phase significantly increases the value of the Rigid vorticity Curl of pseudo‐Lamb vector Term (RCT), which plays a crucial role in the prolonged existence of the TLV under gas–liquid two‐phase conditions. From an inlet gas void fraction of 0%–20%, the RCT value increases by 4.8 × 106/s2, the entropy production value increases by 75.55%, and the hydraulic efficiency decreases by 16.29%.

Self-similar hierarchy of coherent tubular vortices in turbulence.

Energy transfers from larger to smaller scales in turbulence. This energy cascade is a process of the creation of smaller-scale coherent vortices by larger ones. In our recent study (Yoneda, Goto and Tsuruhashi 2022 Nonlinearity 35, 1380-1401), we reformulated the energy cascade in terms of this stretching process and derived the law of the energy spectrum under physically reasonable assumptions. In the present study, we provide a quantitative verification of these assumptions by using direct numerical simulations. We decompose developed turbulence in a periodic cube into scales by using the band-pass filter and identify the axes of coherent tubular vortices by the low-pressure method. Even when the turbulent kinetic energy and its dissipation rate temporally fluctuate about their temporal means, the total length of the vortices at each scale varies little with time. This result is consistent with our assumption of the temporal stationarity on the vorticity decomposition. The present numerical analysis also shows that the hierarchy of vortex axes is self-similar in a wide range of scales, i.e. in the inertial range and a lower part of the dissipation range and that the volume fraction occupied by the tubular vortices at each scale is independent of the scale.This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.

Some general solutions for linear Bragg–Hawthorne equation

Linear cases of Bragg–Hawthorne equation for steady axisymmetric incompressible ideal flows are systematically discussed. The equation is converted to a more convenient form in a spherical coordinate system. A new vorticity decomposition is derived. General solutions for 16 linear cases of the equation are obtained. These solutions can be specified to gain new analytical vortex flows, as examples in the paper demonstrate. A lot of well-known solutions like potential flow past a sphere, Hill's vortex with and without swirl, are included and extended in these solutions. Special relations between some vortex flows are also revealed when exploring or comparing related solutions.

On the Vortical Characteristics of Horn-Like Vortices in Stator Corner Separation Flow in an Axial Flow Pump

Abstract The phenomenon of horn-like vortex in stator corner separation flow in an axial flow pump was first reported by Wang et al. (2020, “Investigation on the Horn-Like Vortices in Stator Corner Separation Flow in an Axial Flow Pump,” ASME J. Fluids Eng., 142(7), p. 071208), and the associated external features were preliminarily presented. However, internal vortical characteristics of horn-like vortices, including the distributions of swirling strength, the deformation mechanism of vortex tube and the correlation with pressure fluctuation surge, are not revealed. In this paper, the newly developed vorticity decomposition approach is introduced, and thus more novel quantitative results are provided for the physics of horn-like vortex evolution in an axial flow pump. First, the distributions of absolute swirling strength, relative swirling strength and Liutex spectrum are presented to outline the vortical features of the horn-like vortex fields. Second, the deformation mechanism of the horn-like vortex tube is revealed. It is found that the horn-like vortex spatial evolution can be described by the deformation terms (Liutex stretching term, Liutex dilatation term, and curl term of the pseudo-Lamb vector) controlling the Liutex transport process. These terms constantly act on the horn-like vortex tube in an almost independent way, causing its continuous deformations in the transit process. Third, the quantitative correlation between horn-like vortex transit and pressure fluctuation surge is given. It is proved that periodic vortex transit can cause severe pressure fluctuation that is much larger than that induced by rotor–stator interaction. From multiple perspectives, a clearer evolution process of the horn-like vortex is outlined, which is conducive to controlling the corner separation flows and improving the stability of large-capacity and low-head pumping stations.

Rigid vorticity transport equation and its application to vortical structure evolution analysis in hydro-energy machinery

Vortex is the dominant flow structure in hydro-energy machinery, and its swirling features are mainly determined by the rigid rotation part of vorticity. In this study, the rigid vorticity transport equation is proposed based on the vorticity decomposition. The vortex spatial evolution is described by the stretching term RST, the dilatation term RDT, the Lamb term RCT and the viscous term RVT. The shape change of a vortex tube is mainly affected by RST and RDT, while vortex production and dissipation are mainly reflected by RCT and RVT. Compared with the traditional vorticity transport equation, this new theoretical tool can distinguish between rigid rotation and shearing motion of local fluids, and it can better reveal the intuitive evolution characteristics of vortical structures. Two cases with clear engineering demands are analysed by using the transport equation, and the results show that it can provide a practical method for the analysis and control of vortical flows in hydro-energy machinery.

Concentration-Based Decomposition of the Flow around a Confined Cylinder in a UV Disinfection Reactor

The proper orthogonal decomposition (POD) is applied to spatial concentration data, obtained using laser-induced fluorescence (LIF), for the flow around a confined circular cylinder (R=4,900 R=4,900) and compared with their corresponding velocity and vorticity fields. The velocity, vorticity, and concentration modes identify the dominating coherent structures, i.e., shear layer rollup, instantaneous recirculation and von Karman vortex shedding. In the velocity and vorticity decomposition, instantaneous recirculation, and von Karman vortex shedding are represented in the highest energy modes. However, in the concentration decomposition, the most energetic modes represent the regions with strong concentration intermittency, i.e., where no dye is present, whereas the coherent structures, which are identified in the highest modes in the velocity and vorticity decomposition, are only found in the less energetic modes. The concentration-based decomposition identifies the shear layer around the cylinder, which is only partially captured by its vorticity-based counterpart.

Wavelet analysis of coherent vorticity near the turbulent/non-turbulent interface in a turbulent planar jet

Coherent vorticity near the turbulent/non-turbulent (T/NT) interface is investigated by using direct numerical simulation of a planar jet. The coherent vorticity extraction (CVE) method based on the orthogonal wavelet decomposition of vorticity is applied to the planar jet for extracting the coherent vorticity. We analyze the conditional statistics conditioned on the distance from the T/NT interface. The coherent vorticity is reconstructed from small number of wavelet coefficients. Nevertheless, the coherent vorticity contains most of enstrophy in the planar jet. Furthermore, the characteristics of the vorticity field are well captured even near the T/NT interface by the coherent vorticity. The coherent velocity obtained by the Biot–Savart relation shows that the large-scale motions, such as induced flow and engulfing motion in the non-turbulent region, are also well represented by the coherent field. The enstrophy transport equation is decomposed into coherent and incoherent parts by the CVE for investigating the role of the coherent vorticity in the enstrophy transport mechanism. The conditional average of the enstrophy transport equation shows that the enstrophy production and dissipation associated with the incoherent vorticity are small compared with the coherent contributions. The enstrophy diffusion near the T/NT interface, which causes the local entrainment of non-turbulent fluids, arises from the coherent vorticity. Thus, most of the enstrophy transport mechanism near the T/NT interface is well captured by the coherent vorticity extracted by the CVE method.

Flow and finite deformation of surface elements in three dimensional homogeneous progressive deformations

Rocks are heterogeneous materials. Deformation structures and fabrics in the ductile lithosphere can result from the history of the three dimensional flow field, or the progressive deformation of some rheologically distinct elements embedded in the 3D flow field, or a combination of both. The flow and progressive deformation of a rheologically distinct element is generally non-steady even if the hosting 3D flow field is steady. Vorticity decomposition is the key to following the progressive history of the deformation of specific elements. A general theoretical background to the problem is presented. Flow and progressive deformation of surface (and layer-like) elements in 3D homogeneous flows are investigated specifically. Analytical procedures to calculate the sectional vorticity number and the sectional dilatancy number on a general section in a 3D flow field are introduced. The theoretical analysis is applied to various examples of sectional flow embedded in different types of 3D homogeneous and steady flow fields. Implications of this investigation for kinematic analysis of natural rock deformation are discussed. All types of material line histories are expected in sectional flows of layer-like elements. Shortened boudins can be formed in a single progressive deformation. In sequences of rocks with a well-developed transposition foliation common in large scale high-strain zones, each layer unit may have undergone a unique deformation history during its rotation to the final, shear zone boundary parallel orientation. In kinematic analysis of such areas, it is important to ascribe the correct kinematic significance to structural associations and fabrics.

On the structure and dynamics of sheared and rotating turbulence: Direct numerical simulation and wavelet-based coherent vortex extraction

The influence of rotation on the structure and dynamics of sheared turbulence is investigated using a series of direct numerical simulations. Five cases are considered: turbulent shear flow without rotation, with moderate rotation, and with strong rotation, where the rotation configuration is either parallel or antiparallel. For moderate rotation rates an antiparallel configuration increases the growth of the turbulent kinetic energy, while the parallel case reduces the growth as compared to the nonrotating case. For strong rotation rates decay of the energy is observed, linear effects dominate the flow, and the vorticity probability density functions tend to become Gaussian. Visualizations of vorticity show that the inclination angle of the vortical structures depends on the rotation rate and orientation. Coherent vortex extraction, based on the orthogonal wavelet decomposition of vorticity, is applied to split the flow into coherent and incoherent parts. It was found that the coherent part preserves the vortical structures using only a few percent of the degrees of freedom. The incoherent part was found to be structureless and of mainly dissipative nature. With increasing rotation rates, the number of wavelet modes representing the coherent vortices decreases, indicating an increased coherency of the flow. Restarting the direct numerical simulation with the filtered fields confirms that the coherent component preserves the temporal dynamics of the total flow, while the incoherent component is of dissipative nature.

A Note on Natural Coordinates and Frenet Frames

This paper discusses the relationship between natural coordinates in fluid mechanics and orthogonal curvilinear coordinates. Since orthogonal curvilinear coordinates have some excellent mathematical properties, natural coordinates can be applied more widely if they can be transformed to orthogonal curvilinear coordinates. Frenet formulas which describe the differential properties of natural coordinates were compared with the derivative formulas of orthogonal curvilinear coordinates to show that natural coordinates are not generally orthogonal curvilinear coordinates. A method was introduced to transform natural coordinates into orthogonal curvilinear coordinates by rotating the normal planes of the natural coordinates about the streamlines. The transformation is true as long as the natural coordinates satisfy several equations. Vorticity decomposition in the natural coordinates is used to show that these conditional equations are satisfied only if the streamlines are perpendicular to the vortexlines on every given point in the flow field. These equations apply in both planar flows and axisymmetric flows without a circumferential velocity component, but do not apply in some 3-D flows such as Beltrami flow.

A comparison between snapshot POD analysis of PIV velocity and vorticity data

Proper orthogonal decomposition (POD) was performed on both the fluctuating velocity and vorticity fields of a backward-facing step (BFS) flow at Reynolds numbers of 580 and 4,660. The data was obtained from particle image velocimetry (PIV) measurements. The vorticity decomposition captured the fluctuating enstrophy more efficiently than the equivalent velocity field decomposition for a given number of modes. Coherent structures in the flow are also more easily identifiable using vorticity-based POD. A common structure of the low-order vorticity POD modes suggests that a large-scale similarity, independent of the Reynolds number, may be present for the BFS flow. The POD modes obtained from a vorticity-based decomposition would help in determining a basis for constructing simplified vortex skeletons and low-order flow descriptions based on the vorticity of turbulent flows.

Wavelet filtering to study mixing in 2D isotropic turbulence

This paper presents the application of coherent vortex simulation (CVS) filtering, based on an orthogonal wavelet decomposition of vorticity, to study mixing in 2D homogeneous isotropic turbulent flows. The Eulerian and Lagrangian dynamics of the flow are studied by comparing the evolution of a passive scalar and of particles advected by the coherent and incoherent velocity fields, respectively. The former is responsible for strong mixing and produces the same anomalous diffusion as the total flow, due to transport by the coherent vortices, while mixing in the latter is much weaker and corresponds to classical diffusion.

Vorticity decomposition and its application to sectional flow characterization: Comment

Vorticity decomposition and its application to sectional flow characterization: Comment

Vorticity decomposition and its application to sectional flow characterization: Reply

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Vorticity decomposition and its application to sectional flow characterization

Great insights have been gained by applying the concept of vorticity to the study of the natural flow of rocks. Geologists have used various schemes of vorticity decomposition, depending on the purpose of their studies. The fundamental decomposition divides the overall vorticity into that of the material with respect to the instantaneous stretching axes and the spin of these axes in an external reference frame in which the flow is defined. This represents a new contribution to continuum mechanics, material science and structural geology. Its implications for the kinematic analysis of geological structures and fabrics are more significant than have been realized. In virtually every geological situation the flow has a spin component, and therefore vorticity decomposition needs to be considered. The decomposition of vorticity is applied to the characterization of the two-dimensional flow on a material plane (sectional flow) in a general three-dimensional flow field. The analysis of the sectional flow and sectional non-coaxiality is a precondition for a determination of the initial orientation of structural elements (such as the orientation of a fold developed in a competent layer that is embedded in a high-strain zone) and their temporal evolution. Theoretically, it may also be possible to derive the symmetry of a natural deformation path by examining fabric symmetries on differently oriented sections from deformed rock samples.