Fluid Rotation Research Articles

Mathematical foundation of turbulence generation—From symmetric to asymmetric Liutex

Vortices have been regarded as the building blocks and muscles of turbulence for a long time. To better describe and analyze vortices or vortical structures, recently a new physical quantity called Liutex (previously named Rortex) is introduced to present the rigid rotation part of fluid motion (Liu et al. 2018). Since turbulence is closely related to the vortex, it can be postulated that there exists no turbulence without Liutex. According to direct numerical simulations (DNS) and experiments, forest of hairpin vortices has been found in transitional and low Reynolds number turbulent flows, while one-leg vortices are predominant in full developed turbulent flows. This paper demonstrates that the hairpin vortex is unstable. The hairpin vortex will be weakened or lose one leg by the shear and Liutex interaction, based on the Liutex definition and mathematical analysis without any physical assumptions. The asymmetry of the vortex is caused by the interaction of symmetric shear and symmetric Liutex since the smaller element of a pair of vorticity elements determines the rotational strength. For a 2-D fluid rotation, if a disturbance shear effects the larger element, the rotation strength will not be changed, but if the disturbance shear effects the smaller element, the rotation strength will be immediately changed due to the definition of the Liutex strength. For a rigid rotation, if the shearing part of the vorticity and Liutex present the same directions, e.g., clockwise, the Liutex strength will not be changed. If the shearing part of the vorticity and Liutex present different directions, e.g., one clockwise and another counterclockwise, the Liutex strength will be weakened. Consequently, the hairpin vortex could lose the symmetry and even deform to a one-leg vortex. The one-leg vortex cannot keep balance, and the chaotic motion and flow fluctuation are doomed. This is considered as the mathematical foundation of turbulence formation. The DNS results of boundary layer transition are used to justify this theory.

Rheology of a suspension of conducting particles in a magnetic field

When a suspension of conducting particles is sheared in a magnetic field, the fluid vorticity causes particle rotation. Eddy currents are induced in a conductor rotating in a magnetic field, resulting in magnetic moment, and a magnetic torque due to the external field. In the absence of inertia, the angular velocity of a particle is determined from the condition that the sum of the hydrodynamic and magnetic torques is zero. When the particle angular velocity is different from the fluid rotation rate, the torque exerted by the particles on the fluid results in an antisymmetric particle stress. The stress is of the form $\unicode[STIX]{x1D748}^{(p)}=|\unicode[STIX]{x1D74E}|(\unicode[STIX]{x1D702}_{c}^{(1)}(\hat{\unicode[STIX]{x1D750}}\boldsymbol{ : }\hat{\unicode[STIX]{x1D74E}})+\unicode[STIX]{x1D702}_{c}^{(2)}\hat{\unicode[STIX]{x1D750}}\boldsymbol{ : }(\hat{\boldsymbol{H}}-\hat{\unicode[STIX]{x1D74E}}(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}))/(\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}})+\unicode[STIX]{x1D702}_{c}^{(3)}(\hat{\unicode[STIX]{x1D74E}}\hat{\boldsymbol{H}}-\hat{\boldsymbol{H}}\hat{\unicode[STIX]{x1D74E}})/\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}})$, where $\unicode[STIX]{x1D74E}$ is the fluid vorticity at the centre of the particle, $\hat{\unicode[STIX]{x1D74E}}$ and $\hat{\boldsymbol{H}}$ are the unit vectors in the direction of the fluid vorticity and the magnetic field, $\hat{\unicode[STIX]{x1D750}}$ is the third order Levi-Civita antisymmetric tensor and $\unicode[STIX]{x1D702}_{c}^{(1)}$, $\unicode[STIX]{x1D702}_{c}^{(2)}$ and $\unicode[STIX]{x1D702}_{c}^{(3)}$ are called the first, second and third couple stress coefficients. The stress proportional to $\unicode[STIX]{x1D702}_{c}^{(1)}$ is in the plane perpendicular to $\hat{\unicode[STIX]{x1D74E}}$, that proportional to $\unicode[STIX]{x1D702}_{c}^{(2)}$ is in the plane perpendicular to the unit normal to $\hat{\unicode[STIX]{x1D74E}}$ in the $\hat{\unicode[STIX]{x1D74E}}{-}\hat{\boldsymbol{H}}$ plane, and that proportional to $\unicode[STIX]{x1D702}_{c}^{(3)}$ is in the $\hat{\unicode[STIX]{x1D74E}}{-}\hat{\boldsymbol{H}}$ plane. A relation $\unicode[STIX]{x1D702}_{c}^{(2)}=-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}\unicode[STIX]{x1D702}_{c}^{(1)}/\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}})$ results from the condition that the component of the eddy current torque along the magnetic field is zero. The couple stress coefficients are obtained for two geometries, a uniform spherical particle of radius $R$ and a thin spherical shell of radius $R$ and thickness $\unicode[STIX]{x1D6FF}R$ with $\unicode[STIX]{x1D6FF}\ll 1$, in the dilute (non-interacting) limit in the absence of fluid inertia. These couple stress coefficients are functions of two dimensionless parameters, $\unicode[STIX]{x1D6F4}=(\unicode[STIX]{x1D707}_{0}H_{0}^{2}/4\unicode[STIX]{x03C0}\unicode[STIX]{x1D702}|\unicode[STIX]{x1D74E}|)$, the ratio of the characteristic magnetic and hydrodynamic torques, and $\unicode[STIX]{x1D6FD}$, the product of the vorticity and current relaxation time. Here $\unicode[STIX]{x1D707}_{0}$ is the magnetic permeability, $H_{0}$ is the magnetic field and $\unicode[STIX]{x1D702}$ is the fluid viscosity. The parameter $\unicode[STIX]{x1D6FD}$ has the form $\unicode[STIX]{x1D6FD}_{p}=(|\unicode[STIX]{x1D74E}|\unicode[STIX]{x1D707}_{0}R^{2}/2\unicode[STIX]{x1D71A})$ for a uniform particle and $\unicode[STIX]{x1D6FD}_{s}=(|\unicode[STIX]{x1D74E}|\unicode[STIX]{x1D707}_{0}R^{2}\unicode[STIX]{x1D6FF}/2\unicode[STIX]{x1D71A})$ for a thin shell, where $\unicode[STIX]{x1D71A}$ is the electrical resistivity. Scaled couple stress coefficients are defined, $\unicode[STIX]{x1D702}_{1}^{\ast }=(\unicode[STIX]{x1D702}_{c}^{(1)}/((3\unicode[STIX]{x1D702}\unicode[STIX]{x1D719}/2)(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})))$ and $\unicode[STIX]{x1D702}_{3}^{\ast }=(\unicode[STIX]{x1D702}_{c}^{(3)}/((3\unicode[STIX]{x1D719}\unicode[STIX]{x1D702}/2)\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}))$, which are independent of the fluid viscosity and the particle volume fraction, and which do not depend on $\hat{\unicode[STIX]{x1D74E}}$ and $\hat{\boldsymbol{H}}$ in the limits $\unicode[STIX]{x1D6F4}\ll 1$ and $\unicode[STIX]{x1D6F4}\gg 1$. Here, $\unicode[STIX]{x1D719}$ is the volume fraction of the particles. Asymptotic analysis is used to determine the couple stress coefficients in the limits $\unicode[STIX]{x1D6F4}\ll 1$ and $\unicode[STIX]{x1D6F4}\gg 1$, and a numerical solution procedure is formulated for $\unicode[STIX]{x1D6F4}\sim 1$. For $\unicode[STIX]{x1D6F4}\ll 1$, the particle angular velocity is aligned close to the fluid vorticity, and the scaled couple stress coefficients are $\unicode[STIX]{x1D6F4}$ times a function of $\unicode[STIX]{x1D6FD}$. For $\unicode[STIX]{x1D6F4}\gg 1$, the particle angular velocity is aligned close to the magnetic field, $\unicode[STIX]{x1D702}_{1}^{\ast }\rightarrow 1$ and $\unicode[STIX]{x1D702}_{3}^{\ast }\propto \unicode[STIX]{x1D6F4}^{-1}$. When the magnetic field is perpendicular to the fluid vorticity, $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}=0$, the particle angular velocity is aligned along the vorticity, and only the first couple stress coefficient is non-zero. For high $\unicode[STIX]{x1D6FD}$, there are multiple solutions for the couple stress coefficient. Multiple steady states are also observed for a near perpendicular magnetic field, $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}<(1/3)$, for a reason different from that for a perpendicular magnetic field. Asymptotic analysis is used to explain the existence of multiple steady states in both cases.

Comparative assessment and analysis of Rortex vortex in swirling jets

Recently, a new definition, called rortex, has been proposed to quantify the purely rotational motion of fluids. In this work, based on the DNS data we employed the rortex to assess and visualize the rotational motion and structure of vortex in swirling jets in comparison with other kinds of vortex criteria of Q, λ2, vorticity and Omega. The rortex vector, Omega, Q, and λ2 criteria are found to be better than vorticity for vortex core identification. The vector triangle formed by the Rortex R, nonrelational shear S, and vorticity ΩV is analyzed to give mechanical explanations, especially on the effect of non-rotational shear on rotation of fluids. In addition, the probability density distributions (PDF) of Rortex R, nonrotational shear S, and vorticity ΩV have been computed. The peak value of PDF of vorticity could be used to explain the pure rigid rotation effect and combination effects of rigid rotation and non-rotational shear.

A variational principle for fluid sloshing with vorticity, dynamically coupled to vessel motion.

A variational principle is derived for two-dimensional incompressible rotational fluid flow with a free surface in a moving vessel when both the vessel and fluid motion are to be determined. The fluid is represented by a stream function and the vessel motion is represented by a path in the planar Euclidean group. Novelties in the formulation include how the pressure boundary condition is treated, the introduction of a stream function into the Euler–Poincaré variations, the derivation of free surface variations and how the equations for the vessel path in the Euclidean group, coupled to the fluid motion, are generated automatically.

Rotational and axial flow of pseudoplastic fluids

Effects of a controllable axial flow on the stability of rotational flow of pseudoplastic fluids in the gap between concentric cylinders are studied. It is assumed that the outer cylinder is fixed while the inner one has simultaneous and independent rotational and translational motions. The fluid follows the Carreau model and mixed boundary conditions are imposed. The four-dimensional low-order dynamical system resulting from Galerkin projection of the conservation of mass and momentum equations includes highly non-linear terms in the velocity components originating from the shear-dependent viscosity. In the absence of the axial flow as the pseudoplasticity effect increases, the critical Taylor number at which the rotational flow loses its stability to the vortex structure decreases. The emergence of the vortices corresponds to the onset of a supercritical bifurcation which is also seen in the flow of a Newtonian fluid in rotation. The existence of an axial flow, induced by the translational motion of the inner cylinder, appears to further advance the emergence of the vortex flow. Complete flow analysis together with viscosity maps are given for the entire flow field.

Numerical investigation of the back-mixing and non-uniform characteristics in the three-dimensional full-loop circulating fluidized bed combustor with six parallel cyclones

The design and operation of circulating fluidized bed (CFB), which offers excellent heat and mass transfer, remain challenging because of the lack of a comprehensive understanding. CFD-DEM is used here for the high-fidelity simulation of the three-dimensional full-loop CFB with six parallel cyclones. Detailed information on the particle-scale information (i.e., solid dispersion, particle rotational speed, fluid and collision forces, mass distribution, and backing-mixing intensity) throughout the CFB are presented for the first time. Interesting results include: (i) the solid dispersion behavior is significantly different among the three directions; (ii) the solid loading in the middle cyclone on one side of the riser is twice that of the two adjacent ones, and this non-uniform distribution underscores the need for full-loop simulation for the CFB with multiple cyclones; (iii) the solid dispersion coefficients in all three directions, fluid force, collision force and particle rotational speed are similar among the six cyclones and correspondingly the standpipes; and (iv) approximately 50% and 48% of the total material are distributed respectively in the riser and standpipes. The results obtained provide valuable insights regarding the mass loading, particle-scale characteristics and non-uniform distribution of the solid phase throughout the entire CFB.

Delme Hızının Akışkan Ortam Basıncı ve Dönme Hızına Bağlı Değişiminin Araştırılması

Bir maden işletmesinde maliyet teşkil eden parameterelerin başında hammadde temini gelmektedir. Hammadde üretim maliyetlerinin en önemli parametrelerinden biri de patlatma deliklerinin delinmesi işidir. Delme işlemlerinde istenen verimde çalışılmaması durumunda hem işletmede iş organizasyonu hem de üretim maliyeti önemli ölçüde olumsuz yönde etkilenecektir. Bu çalışmasında DSİ 18. Bölge Müdürlüğü’ne ait kireçtaşı ocağında kullanılan Atlas Copco Flexi ROC T35 model delici makineye ait akışkan ortam basıncı ve dönme hızı parametrelerinin değiştirilmesi sonucunda delik delme verimlerine etkisi incelenmiştir. Delik delme hızının belirlenmesi amacıyla Burdur ili, Çavdır ilçesinde bulunan kireçtaşı ocağında çalışılan ortamdaki delik delme süreleri ölçülmüş ve net ve brüt delik delme hızları hesaplanmıştır. Laboratuvar ortamında da kayaçların fiziko-mekanik özellikleri ile kimyasal özellikleri belirlenmiştir. Elde edilen bütün veriler birlikte değerlendirilerek akışkan ortam basıncı ve dönme hızının delme hızı üzerindeki etkileri incelenmiştir. Sonuç olarak tüm akışkan ortam basınç değerlerinde ve her iki dönme hızında en iyi delik delme hızı 103 rpm dönme hızı değerinde elde edilmiştir.

On two-dimensional self-similar axisymmetric vortices

Superpositions of Lamb-Oseen axisymmetric vortices with Gaussian vorticity having zero net circulation and finite kinetic energy in unbounded domain are considered. Their evolution is described by self-similar solutions depending on a certain combination of space, time, and viscous diffusion. It is shown that the structure of a popular self-similar solution for a shielded vortex with Gaussian fluid rotation rate corresponds to two Lamb-Oseen vortices with opposite sign and nearly the same spatial scale. The radial structure of a combined vortex with different spatial scales is well suited for characterization of realistic vortical structures like atmospheric hurricanes. The linear stability of the combined vortex is investigated. The results have important implications for better understanding of vortex structures in two-dimensional flow.

Third generation of vortex identification methods: Omega and Liutex/Rortex based systems

A vortex is intuitively recognized as the rotational/swirling motion of fluids, but a rigorous and universally-accepted definition is still not available. Vorticity tube/filament has been regarded equivalent to a vortex since Helmholtz proposed the concepts of vorticity tube/filament in 1858 and the vorticity-based methods can be categorized as the first generation of vortex identification methods. During the last three decades, a lot of vortex identification methods, including Q, Δ, λ2 and λci criteria, have been proposed to overcome the problems associated with the vorticity-based methods. Most of these criteria are based on the Cauchy-Stokes decomposition and/or eigenvalues of the velocity gradient tensor and can be considered as the second generation of vortex identification methods. Starting from 2014, the Vortex and Turbulence Research Team at the University of Texas at Arlington (the UTA team) focus on the development of a new generation of vortex identification methods. The first fruit of this effort, a new omega (Ω) vortex identification method, which defined a vortex as a connected region where the vorticity overtakes the deformation, was published in 2016. In 2017 and 2018, a Liutex (previously called Rortex) vector was proposed to provide a mathematical definition of the local rigid rotation part of the fluid motion, including both the local rotational axis and the rotational strength. Liutex/Rortex is a new physical quantity with scalar, vector and tensor forms exactly representing the local rigid rotation of fluids. Meanwhile, a decomposition of the vorticity to a rotational part namely Liutex/Rortex and an anti-symmetric shear part (RS decomposition) was introduced in 2018, and a universal decomposition of the velocity gradient tensor to a rotation part (R) and a non-rotation part (NR) was also given in 2018 as a counterpart of the traditional Cauchy-Stokes decomposition. Later in early 2019, a Liutex/Rortex based Omega method called Omega-Liutex, which combines the respective advantages of both Liutex/Rortex and Omega methods, was developed. And a latest objective Omega method, which is still under development, is also briefly introduced. These advances are classified as the third generation of vortex identification methods in the current paper. To elaborate the advantages of the third-generation methods, six core issues for vortex definition and identification have been raised, including: (1) the absolute strength, (2) the relative strength, (3) the rotational axis, (4) the vortex core center location, (5) the vortex core size, (6) the vortex boundary. The new third generation of vortex identification methods can provide reasonable answers to these questions, while other vortex identification methods fail to answer all questions except for the approximation of vortex boundaries. The purpose of the current paper is to summarize the main ideas and methods of the third generation of vortex identification methods rather than to conduct a comprehensive review on the historical development of vortex identification methods.

Angular-momentum conservation in discretization of the Navier-Stokes equation for viscous fluids.

Although the Navier-Stokes equation (NSE) is derived under angular-momentum conservation (AMC), numerical simulation methods often lack it. Here we reveal that AMC violations result from implementation of the degenerated viscous terms of NSE. To maintain AMC, these degenerated terms must be separately integrated in accordance with their stress origins. As observed in particle-based hydrodynamics methods, the violation causes artificial rotations in multicomponent fluids with different viscosities. At the interface between two fluids or with a mobile solid object, AMC must be satisfied, whereas AMC can be neglected in bulk fluids. We also clarify that the condition for constant fluid rotation as a rigid body in a container rotating at a constant speed is not the AMC of the stresses, but the invariance of the viscous forces under a global rotation. To confirm our theory, we simulated the circular laminar flows of single- and binary-component fluids using two-dimensional Lagrangian finite-volume methods. The results show excellent agreement with the analytical predictions for fluids with and without AMC.

Rortex based velocity gradient tensor decomposition

Recently, a vector named Rortex was proposed to represent the local fluid rotation [C. Liu et al., “Rortex—A new vortex vector definition and vorticity tensor and vector decompositions,” Phys. Fluids 30, 035103 (2018)]. In this paper, a universal Rortex based velocity gradient tensor decomposition is proposed and the relevant local velocity increment decomposition is provided. Vortex structures in boundary layer transition on a flat plate are analyzed to quantify the local rotational, compression-stretching, and shearing effects. The results demonstrate that vorticity is shearing-dominant, while the rotational part or Rortex in general occupies a small part of vorticity in most areas of this case. In other words, vorticity is a quality representing shearing rather than rotation or vortex in most regions of this case.

Response to Rotation Fluid Shear Stress of Osteoblast-like Cell Line

Response to Rotation Fluid Shear Stress of Osteoblast-like Cell Line

Process of coating the inner surface of an inclined pipe and its mathematical modeling

A new method is proposed for coating the inner surface of pipes in a confined space in which a sloping position of the pipe is expected during the process. Mathematical modeling of non-pressure fluid flow in an inclined pipe was carried out. The method implies controlled outflow of the material being applied and simultaneous rotation of the pipe being worked. The thickness of the coating depends on the velocity of the material flow along the inner surface of the pipe. The dependence of the fluid velocity on the radius of the pipe being worked and the level height in the filling column is calculated. The results of calculating the analytical dependence in the Mathcad application are presented. Dependence diagrams of material flow are presented for various design parameters that make it possible to determine the operating practices in the process of applying the coating. Uniform coating is obtained due to constant velocity of the material flow along the full length of the pipe. The obtained graphs were analyzed and the direction of further research connected with the influence of rotational velocity and fluid viscosity was determined. The use of the proposed method makes it possible to increase labor productivity and improve the quality of protective coating.

Numerical analysis of micropolar hybrid nanofluid

Hybrid nanofluid has been streamlined as a new class of nanofluid, marked by its thermal properties and potential utilities which serve the purpose to enhance the rate of heat transfer. The main aim of the current analysis is to present a comparison between the behavior of traditional nanofluid and emerging hybrid nanofluid in the presence of micropolar fluid theory, rotation and porous medium over an exponentially stretched surface. The constructed mathematical differential system is solved numerically by means of the BVP-4C technique. The comparison between behavior of pure water, $$\text{C}\text{u}/\text{w}\text{a}\text{t}\text{e}\text{r}$$ nanofluid, $$\text{C}\text{u}-\text{T}\text{i}{\text{O}}_{2}/\text{w}\text{a}\text{t}\text{e}\text{r}$$ hybrid nanofluid over velocity, microrotation and temperature distribution has been visualized graphically. For better comprehension of flow characteristics and heat transfer rate, variation in skin friction coefficients in addition to the Nusselt number of nanofluid along with hybrid nanofluid is scrutinized. We perceive from the present study that the rate of heat transfer of nanofluid is lower than that of hybrid nanofluid even in the presence of micropolar effects, rotation and porosity.

Unsteady MHD Flow of Radiating and Rotating Fluid with Hall Current and Thermal Diffusion Past a Moving Plate in a Porous Medium

The paper examines the combined effects of Hall current, buoyancy forces, thermal radiation, thermo-diffusion and fluid rotation on an unsteady hydromagmetic boundary layer flow with heat and mass transfer over an impulsively moving vertical plate embedded in a porous medium. Base on some realistic simplified assumptions, the governing equations of momentum, energy and chemical species concentration are obtained and tackled analytically using Laplace transform technique. The numerical values of primary and secondary fluid velocities, fluid temperature and species concentration are displayed graphically while those of skin friction coefficient, Nusselt number and Sherwood number are presented in tabular form for different values of pertinent flow parameters.

Letter: Galilean invariance of Rortex

A new vector named Rortex [C. Liu et al., “Rortex—A new vortex vector definition and vorticity tensor and vector decompositions,” Phys. Fluids 30, 035103 (2018)] was proposed to represent the local fluid rotation in our previous work. However, the Galilean invariance of Rortex is yet to be elaborated. In the present study, we prove that Rortex is invariant under the Galilean transformation and several examples are provided to confirm the conclusion.

Rortex and comparison with eigenvalue-based vortex identification criteria

Most of the current Eulerian vortex identification criteria, including the Q criterion and the λci criterion, are exclusively determined by the eigenvalues of the velocity gradient tensor or the related invariants and thereby can be regarded as eigenvalue-based criteria. However, these criteria will be plagued with two shortcomings: (1) these criteria fail to identify the swirl axis or orientation; (2) these criteria are prone to severe contamination by shearing. To address these issues, a new vector named Rortex which represents the local fluid rotation was proposed in our previous work. In this paper, an alternative eigenvector-based definition of Rortex is introduced. The direction of Rortex, which represents the possible axis of the local rotation, is determined by the real eigenvector of the velocity gradient tensor. And then the rotational strength obtained in the plane perpendicular to the possible axis is used to define the magnitude of Rortex. This new equivalent definition allows a much more efficient implementation. Furthermore, a systematic interpretation of scalar, vector, and tensor versions of Rortex is presented. By relying on the tensor interpretation, the velocity gradient tensor is decomposed to a rigid rotation part and a non-rotational part including shearing, stretching, and compression, different from the traditional symmetric and anti-symmetric tensor decomposition. It can be observed that shearing always manifests its effect on the imaginary part of the complex eigenvalues and consequently contaminates eigenvalue-based criteria, while Rortex can exclude the shearing contamination and accurately quantify the local rotational strength. In addition, in contrast to eigenvalue-based criteria, not only the iso-surface of Rortex but also the Rortex vectors and the Rortex lines can be applied to investigate vortical structures. Several comparative studies on simple examples and realistic flows are studied to confirm the superiority of Rortex.

The Onset of Fluid Rotation in a Thermogravitational Boundary Layer with Local Cooling of the Free Surface

Axisymmetric regimes of flows of an inhomogeneous fluid in the boundary layer near a free surface are calculated for a nonuniform temperature distribution on this surface. For the fluid motion equations written in the Oberbeck-Boussinesq approximation, the leading terms of asymptotic expansions of solutions of a steady-state problem are constructed. It is shown that in the presence of local cooling of the free surface and a rising outer fluid stream, as a result of a bifurcation, a pair of rotational regimes may develop in a thin boundary layer near the free surface, with no rotation observed outside this layer. No bifurcation of rotation was detected in the case of local heating of the free surface.

Definitions of vortex vector and vortex

Although the vortex is ubiquitous in nature, its definition is somewhat ambiguous in the field of fluid dynamics. In this absence of a rigorous mathematical definition, considerable confusion appears to exist in visualizing and understanding the coherent vortical structures in turbulence. Cited in the previous studies, a vortex cannot be fully described by vorticity, and vorticity should be further decomposed into a rotational and a non-rotational part to represent the rotation and the shear, respectively. In this paper, we introduce several new concepts, including local fluid rotation at a point and the direction of the local fluid rotation axis. The direction and the strength of local fluid rotation are examined by investigating the kinematics of the fluid element in two- and three-dimensional flows. A new vector quantity, which is called the vortex vector in this paper, is defined to describe the local fluid rotation and it is the rotational part of the vorticity. This can be understood as that the direction of the vortex vector is equivalent to the direction of the local fluid rotation axis, and the magnitude of vortex vector is the strength of the location fluid rotation. With these new revelations, a vortex is defined as a connected region where the vortex vector is not zero. In addition, through direct numerical simulation (DNS) and large eddy simulation (LES) examples, it is demonstrated that the newly defined vortex vector can fully describe the complex vertical structures of turbulence.

Spinning superfluidHe4nanodroplets

We have studied spinning superfluid $^4$He nanodroplets at zero temperature using Density Functional theory. Due to the irrotational character of the superfluid flow, the shapes of the spinning nanodroplets are very different from those of a viscous normal fluid drop in steady rotation. We show that when vortices are nucleated inside the superfluid droplets, their morphology, which evolves from axisymmetric oblate to triaxial prolate to two-lobed shapes, is in good agreement with experiments. The presence of vortex arrays confers to the superfluid droplets the rigid-body behavior of a normal fluid in steady rotation, and this is the ultimate reason of the surprising good agreement between recent experiments and the classical models used for their description.