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1
Eldho, T. I., and D. L. Young. "Two-Dimensional Incompressible Viscous Flow Simulation Using Velocity-Vorticity Dual Reciprocity Boundary Element Method." Journal of Mechanics 20, no. 3 (September 2004): 177–85. http://dx.doi.org/10.1017/s1727719100003397.
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AbstractThis paper describes a computational model based on the dual reciprocity boundary element method (DRBEM) for the solution of two-dimensional incompressible viscous flow problems. The model is based on the Navier-Stokes equations in velocity-vorticity variables. The model includes the solution of vorticity transport equation for vorticity whose solenoidal vorticity components are obtained by solving Poisson equations involving the velocity and vorticity components. Both the Poisson equations and the vorticity transport equations are solved iteratively using DRBEM and combined to determine the velocity and vorticity vectors. In DRBEM, all source terms, advective terms and time dependent terms are converted into boundary integrals and hence the computational domain of the problem reduces by one. Internal points are considered wherever solution is required. The model has been applied to simulate two-dimensional incompressible viscous flow problems with low Reynolds (Re) number in a typical square cavity. Results are obtained and compared with other models. The DRBEM model has been found to be reasonable and satisfactory.
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ESCHER, JOACHIM, and MARCUS WUNSCH. "RESTRICTIONS ON THE GEOMETRY OF THE PERIODIC VORTICITY EQUATION." Communications in Contemporary Mathematics 14, no. 03 (June 2012): 1250016. http://dx.doi.org/10.1142/s0219199712500162.
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We prove that several evolution equations arising as mathematical models for fluid motion cannot be realized as metric Euler equations on the Lie group DIFF∞(𝕊1) of all smooth and orientation-preserving diffeomorphisms on the circle. These include the quasi-geostrophic model equation, cf. [A. Córdoba, D. Córdoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. 162 (2005) 1377–1389], the axisymmetric Euler flow in ℝd (see [H. Okamoto and J. Zhu, Some similarity solutions of the Navier–Stokes equations and related topics, Taiwanese J. Math. 4 (2000) 65–103]), and De Gregorio's vorticity model equation as introduced in [S. De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation, J. Stat. Phys. 59 (1990) 1251–1263].
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Lo, D. C., T. Liao, D. L. Young, and M. H. Gou. "Velocity-Vorticity Formulation for 2D Natural Convection in an Inclined Cavity by the DQ Method." Journal of Mechanics 23, no. 3 (September 2007): 261–68. http://dx.doi.org/10.1017/s1727719100001301.
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AbstractThe aim of this paper attempts to apply the differential quadrature (DQ) method for solving two-dimensional natural convection in an inclined cavity. The velocity-vorticity formulation is used to represent the mass, momentum, and energy conservations of the fluid medium in an inclined cavity. We employ a coupled technique for four field variables involving two velocities, one vorticity and one temperature components. In this method, the velocity Poisson equation, continuity equation, vorticity transport equation and energy equation are all solved as a coupled system of equations so as to we are capable of predicting four field variables accurately. The main advantage of present approach is that coupling the velocity and the vorticity equations allows the determination of the boundary values implicitly without requiring the explicit specification of the vorticity values at the boundary walls. A natural convection in a cavity with different angle of inclinations for Rayleigh number equal to 103, 104, 105 and 106 and H/L aspect ratios varying from 1 to 3 is investigated. It is shown that with the use of the present algorithm the benchmark results for temperature and flow fields could be obtained using a coarse mesh grid.
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Ponta, Fernando L. "Kinematic Laplacian Equation Method: A Velocity-Vorticity Formulation for the Navier-Stokes Equations." Journal of Applied Mechanics 73, no. 6 (February 4, 2006): 1031–38. http://dx.doi.org/10.1115/1.2198245.
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In this work, a novel procedure to solve the Navier-Stokes equations in the vorticity-velocity formulation is presented. The vorticity transport equation is solved as an ordinary differential equation (ODE) problem on each node of the spatial discretization. Evaluation of the right-hand side of the ODE system is computed from the spatial solution for the velocity field provided by a new partial differential equation expression called the kinematic Laplacian equation (KLE). This complete decoupling of the two variables in a vorticity-in-time/velocity-in-space split algorithm reduces the number of unknowns to solve in the time-integration process and also favors the use of advanced ODE algorithms, enhancing the efficiency and robustness of time integration. The issue of the imposition of vorticity boundary conditions is addressed, and details of the implementation of the KLE by isoparametric finite element discretization are given. Validation results of the KLE method applied to the study of the classical case of a circular cylinder in impulsive-started pure-translational steady motion are presented. The problem is solved at several Reynolds numbers in the range 5<Re<180 comparing numerical results with experimental measurements and flow visualization plates. Finally, a recent result from a study on periodic vortex-array structures produced in the wake of forced-oscillating cylinders is included.
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Wen, Jiangang, and Philip L. F. Liu. "Mass transport under partially reflected waves in a rectangular channel." Journal of Fluid Mechanics 266 (May 10, 1994): 121–45. http://dx.doi.org/10.1017/s0022112094000959.
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Mass transport under partially reflected waves in a rectangular channel is studied. The effects of sidewalls on the mass transport velocity pattern are the focus of this paper. The mass transport velocity is governed by a nonlinear transport equation for the second-order mean vorticity and the continuity equation of the Eulerian mean velocity. The wave slope, ka, and the Stokes boundary-layer thickness, k (ν/σ)½, are assumed to be of the same order of magnitude. Therefore convection and diffusion are equally important. For the three-dimensional problem, the generation of second-order vorticity due to stretching and rotation of a vorticity line is also included. With appropriate boundary conditions derived from the Stokes boundary layers adjacent to the free surface, the sidewalls and the bottom, the boundary value problem is solved by a vorticity-vector potential formulation; the mass transport is, in gneral, represented by the sum of the gradient of a scalar potential and the curl of a vector potential. In the present case, however, the scalar potential is trivial and is set equal to zero. Because the physical problem is periodic in the streamwise direction (the direction of wave propagation), a Fourier spectral method is used to solve for the vorticity, the scalar potential and the vector potential. Numerical solutions are obtained for different reflection coefficients, wave slopes, and channel cross-sectional geometry.
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Swaters, Gordon E. "A perturbation theory for the solitary-drift-vortex solutions of the Hasegawa-Mima equation." Journal of Plasma Physics 41, no. 3 (June 1989): 523–39. http://dx.doi.org/10.1017/s0022377800014069.
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A multiple-scales adiabatic perturbation theory is presented describing the adiabatic dissipation of the solitary vortex-pair solutions of the Hasegawa-Mima equation. The vortex parameter transport equations are derived as solvability conditions for the asymptotic expansion and are identical with the transport equations previously derived by Aburdzhaniya et al. (1987) using an energy- and enstrophy-conservation balance procedure. The theoretical results are compared with high-resolution numerical simulations. Global properties such as the decay in the enstrophy and energy are accurately reproduced. Local properties such as the position of the centre of the vortex pair, decay of the extrema in the vorticity and stream-function fields, and the dilation of the vortex dipole are also in good agreement. In addition, time series of vorticity–stream-function scatter diagrams for the numerical simulations are presented to verify the adiabatic ansatz.
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Hu, Running, Xinliang Li, and Changping Yu. "Effects of the Coriolis force in inhomogeneous rotating turbulence." Physics of Fluids 34, no. 3 (March 2022): 035108. http://dx.doi.org/10.1063/5.0084098.
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The effects of the Coriolis force in inhomogeneous rotating turbulence are studied in the paper. Linear analyses and numerical simulations both reveal that energy is transported to the slowly rotating fields, and the energy distribution is proportional to [Formula: see text]. The scale energy is almost spatially self-similar, and the inverse cascade is reduced by inhomogeneous rotation. The corresponding evolution equation of the scale energy, i.e., the generalized Kolmogorov equation, is calculated to study the scale transport process in the presence of inhomogeneity. The equation is reduced to twice the energy transport equation at sufficiently large scales, which is verified by numerical results. In addition, the results reveal the dominant role of the corresponding pressure of the Coriolis force in the spatial energy transport. An extra turbulent convention effect in r-space solely in slowly rotating fields is also recognized. It can be associated with the small-scale structures with strong negative vorticity, whose formation mechanism is similar to rotating condensates. Finally, by vortex dynamic analyses, we find that the corresponding pressure of the Coriolis force transports energy by vorticity tube shrinking and thickening. The effects of the Coriolis force can be divided into two components: one is related to the gradient of rotation, and the other is associated with the strength of rotation.
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Brown, Richard E., and Andrew J. Line. "Efficient High-Resolution Wake Modeling Using the Vorticity Transport Equation." AIAA Journal 43, no. 7 (July 2005): 1434–43. http://dx.doi.org/10.2514/1.13679.
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Marn, Jure, and Ivan Catton. "Analysis of Flow Induced Vibration Using the Vorticity Transport Equation." Journal of Fluids Engineering 115, no. 3 (September 1, 1993): 485–92. http://dx.doi.org/10.1115/1.2910164.
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Crossflow induced vibrations are the subject of this work. The analysis is two dimensional. The governing equations for fluid motion are solved using linearized perturbation theory and coupled with the equations of motion for cylinders to yield the threshold of dynamic instability for an array of cylinders. Parametric analysis is performed to determine the lowest instability threshold for a rotated square array and correlations are developed relating the dominant parameters. The results are compared with theoretical and experimental data for similar arrays and the discrepancies are discussed.
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Halpern, Federico D., Ronald E. Waltz, and Tess N. Bernard. "Drift-ordered fluid vorticity equation with energy consistency." Physics of Plasmas 30, no. 3 (March 2023): 032302. http://dx.doi.org/10.1063/5.0135158.
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Although drift-ordered fluid models are widely applied in tokamak edge turbulence simulations, the models used are acknowledged not to conserve energy or even electrical charge. The present paper aims to remove many of the existing pitfalls in drift-fluid models, however, with the objective of finding a solution simple enough to be implemented in numerical applications. Our main result is an improved version of the drift-Braginskii equations involving a generalized vorticity function. In the new drift-Braginskii system, the quasi-neutrality condition translates into a transport equation for a generalized vorticity, expressed in conservation form, and related to the total mass-weighted circulation. It is found that kinetic energy conservation can be achieved if the polarization flow is defined recursively. The resulting model conserves the kinetic energy associated with [Formula: see text] and diamagnetic flows and retains the associated perpendicular kinetic energy flux.
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Raul, R., and P. S. Bernard. "A Numerical Investigation of the Turbulent Flow Field Generated by a Stationary Cube." Journal of Fluids Engineering 113, no. 2 (June 1, 1991): 216–22. http://dx.doi.org/10.1115/1.2909483.
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The turbulent flow field generated by a stationary cube at Reynolds numbers 2000 and 14,000 is investigated numerically. A vorticity-vector potential formulation of the equations of motion is employed. Turbulence effects are accounted for through the use of a vorticity transport closure scheme in which dynamical equations for vorticity mean and covariance are supplemented by a kinematic equation for turbulent kinetic energy. Semi-implicit finite difference approximations to the equations of motion are solved iteratively by a vectorizable 8-color SOR algorithm. The numerical mesh is designed so that the turbulent flow field can be computed down to solid surfaces without the use of wall functions. The properties of the computed flow field, including drag, axial velocity, separation points, and three-dimensional flow structure show good agreement with experimental observations of similar bluff body flows.
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Ralph, M. E., and T. J. Pedley. "Flow in a channel with a moving indentation." Journal of Fluid Mechanics 190 (May 1988): 87–112. http://dx.doi.org/10.1017/s0022112088001223.
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The unsteady flow of a viscous, incompressible fluid in a channel with a moving indentation in one wall has been studied by numerical solution of the Navier-Stokes equations. The solution was obtained in stream-function-vorticity form using finite differences. Leapfrog time-differencing and the Dufort-Frankel substitution were used in the vorticity transport equation, and the Poisson equation for the stream function was solved by multigrid methods. In order to resolve the boundary-condition difficulties arising from the presence of the moving wall, a time-dependent transformation was applied, complicating the equations but ensuring that the computational domain remained a fixed rectangle.Downstream of the moving indentation, the flow in the centre of the channel becomes wavy, and eddies are formed between the wave crests/troughs and the walls. Subsequently, certain of these eddies ‘double’, that is a second vortex centre appears upstream of the first. These observations are qualitatively similar to previous experimental findings (Stephanoff et al. 1983, and Pedley & Stephanoff 1985), and quantitative comparisons are also shown to be favourable. Plots of vorticity contours confirm that the wave generation process is essentially inviscid and reveal the vorticity dynamics of eddy doubling, in which viscous diffusion and advection are important at different stages. The maximum magnitude of wall vorticity is found to be much larger than in quasi-steady flow, with possibly important biomedical implications.
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Coghi, Michele, and Mario Maurelli. "Regularized vortex approximation for 2D Euler equations with transport noise." Stochastics and Dynamics 20, no. 06 (June 5, 2020): 2040002. http://dx.doi.org/10.1142/s021949372040002x.
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We study a mean field approximation for the 2D Euler vorticity equation driven by a transport noise. We prove that the Euler equations can be approximated by interacting point vortices driven by a regularized Biot–Savart kernel and the same common noise. The approximation happens by sending the number of particles [Formula: see text] to infinity and the regularization [Formula: see text] in the Biot–Savart kernel to [Formula: see text], as a suitable function of [Formula: see text].
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Han, Chengzao, Yun Long, Xiaorui Bai, and Bin Ji. "Numerical Investigation of Unsteady Cavitation Flow around E779A Propeller in a Nonuniform Wake with an Insight on How Cavitation Influences Vortex." Shock and Vibration 2021 (February 20, 2021): 1–10. http://dx.doi.org/10.1155/2021/5577517.
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In the current study, the turbulent cavitation flow around a marine propeller in a nonuniform wake is simulated with the shear stress transport (k−ω SST) turbulence model combining Zwart–Gerber–Belamri (ZGB) cavitation model. The predicted cavity evolution shows a fairly well agreement with the available experimental results. Important mechanisms of propeller cavitation flow, including side-entrant jet and cavitation-vortex interaction, are analyzed in this paper. Vorticity is found to be mainly located in cavitation regions and the propeller wake during propeller rotating. The unsteady behavior of cavitation and side-entrant jet can both promote local vorticity generation and flow unsteadiness. In addition, it is indicated with the relative vorticity transport equation that the stretching term plays a major role in vorticity transportation, while baroclinic torque and Coriolis force term mainly influence the vorticity distribution along the liquid-vapor interface.
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ONG, LAWRENCE, and JAMES M. WALLACE. "Joint probability density analysis of the structure and dynamics of the vorticity field of a turbulent boundary layer." Journal of Fluid Mechanics 367 (July 25, 1998): 291–328. http://dx.doi.org/10.1017/s002211209800158x.
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An experimental study of a turbulent boundary layer at Rθ≈1070 and Rτ≈543 was conducted. Detailed measurements of the velocity vector and the velocity gradient tensor within the near-wall region were performed at various distances from the wall, ranging from approximately y+=14 to y+=89. The measured mean statistical properties of the fluctuating velocity and vorticity components agree well with previous experimental and numerically simulated data. These boundary layer measurements were used in a joint probability density analysis of the various component vorticity and vorticity–velocity gradient products that appear in the instantaneous vorticity and enstrophy transport equations. The vorticity filaments that contribute most to the vorticity covariance Ω[bar]xΩ [bar]y in this region were found to be oriented downstream with angles of inclination to the wall, when projected on the streamwise (x, y)-plane, that decrease with distance moving from the buffer to the logarithmic layer. When projected on the planview (x, z)- and cross-stream (y, z)-planes, the vorticity filaments that most contribute to the vorticity covariances Ω [bar]xΩ [bar]z and Ω [bar]yΩ [bar]z have angles of inclination to the z-ordinate axis that increase with distance from it. All the elements of the ΩiΩj ∂Ui/∂xj term in the enstrophy transport equation, i.e. the term that describes the rate of increase or decrease of the enstrophy by vorticity filament stretching or compression by the strain-rate field, have been examined. On balance, the average stretching of the vorticity filaments is greater than compression at all y+ locations examined here. However, some individual velocity gradient components compress the vorticity filaments, on average, more than they stretch them.
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Hazra, Gopal, and Arnab Rai Choudhuri. "Explaining the variation of the meridional circulation with the solar cycle." Proceedings of the International Astronomical Union 13, S340 (February 2018): 313–16. http://dx.doi.org/10.1017/s1743921318001357.
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AbstractThe meridional circulation of the Sun is observationally found to vary with the solar cycle, becoming slower during the solar maxima. We explain this by constructing a theoretical model in which the equation of the meridional circulation (the φ component of the vorticity equation) is coupled with the equations of the flux transport dynamo model. We find that the Lorentz force of the dynamo-generated magnetic fields can slow down the meridional circulation during the solar maxima in broad conformity with the observations.
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WEI, LIANG, and ANDREW POLLARD. "Interactions among pressure, density, vorticity and their gradients in compressible turbulent channel flows." Journal of Fluid Mechanics 673 (February 14, 2011): 1–18. http://dx.doi.org/10.1017/s0022112010006166.
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The interactions among pressure, density, vorticity and their gradients in compressible turbulent channel flows (TCF) are studied using direct numerical simulations (DNS). DNS of three isothermal-wall TCF for Mach number Ma = 0.2, 0.7, and 1.5, respectively are performed using a discontinuous Galerkin method (DGM). The Reynolds numbers of these three cases are ≈2800, based on the bulk velocity, bulk density, half channel width and dynamic viscosity at the wall. A high cross-correlation between density and spanwise vorticity occurs at y+≈4, which is coincident with the peak mean spanwise baroclinicity. The relationship between the spanwise baroclinicity and the correlation is analysed. The difference between the evolution of density and spanwise vorticity very near the wall is discussed. The transport equation for the mean product of density and vorticity fluctuations 〈ρ′ω′i〉 is presented and the distributions of terms in the 〈ρ′ω′z〉 transport equation indicate that the minima and maxima of the profiles are located around y+≈5. The connection between pressure gradients and vorticity fluxes for compressible turbulent flows with variable viscosity has been formulated and verified. High correlations (0.7–1.0) between pressure gradient and vorticity flux are found very close to the wall (y+<5). The correlation coefficients are significantly influenced by Ma and viscosity in this region. Turbulence advection plays an important role in destroying the high correlations between pressure gradient and vorticity flux in the region away from the wall (y+ > 5).
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Ravnik, J., and J. Tibaut. "Boundary-domain integral method for vorticity transport equation with variable viscosity." International Journal of Computational Methods and Experimental Measurements 6, no. 6 (January 1, 2018): 1087–96. http://dx.doi.org/10.2495/cmem-v6-n6-1087-1096.
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Arul Prakash, M., K. Mayilsamy, and P. Rajesh Kanna. "Numerical Simulation of Two Dimensional Laminar Wall Jet Flow over Solid Obstacle." Applied Mechanics and Materials 592-594 (July 2014): 1935–39. http://dx.doi.org/10.4028/www.scientific.net/amm.592-594.1935.
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A Computational Fluid Dynamics code was developed to study the flow characteristics of two dimensional laminar incompressible flow. Stream function-vorticity formulation was used for solving two dimensional continuity and momentum equations. The unsteady vorticity transport equation is solved by alternate direction implicit scheme. The stream function equation is solved by the successive over relaxation method. A computational code in c-language was developed to solve the tridiagonal system of algebraic equations. Two dimensional flow through a channel with rectangular block at the bottom wall was considered for the validation. The streamline patterns obtained for different Reynolds number shows good agreement with published results. The code was modified to simulate an incompressible laminar wall jet flow around a solid obstacle. Simulations were carried out for different Reynolds numbers. Contour plots of Stream line, u-velocity and v-velocity were obtained. The variations of flow patterns and the development of vortices were studied and reported.
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Myong, Hyon Kook. "A Numerical Study on the Generation Mechanism of Turbulence-Driven Secondary Flow in a Square Duct." Journal of Fluids Engineering 115, no. 1 (March 1, 1993): 172–75. http://dx.doi.org/10.1115/1.2910103.
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The generation mechanism of turbulence-driven secondary flows in a square duct is numerically investigated in the present study by using an anisotropic low-Reynolds-number k–ε turbulence model. Special attention is directed to the distributions of turbulence quantities, which are responsible for the secondary flow generation, such as the anisotropy of normal Reynolds stresses and the secondary Reynolds shear stress acting on the cross-sectional plane. The vorticity transport process is also discussed in detail, based on the numerical evaluation of the individual terms which appear in the streamwise vorticity transport equation.
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NWOGU, OKEY G. "Interaction of finite-amplitude waves with vertically sheared current fields." Journal of Fluid Mechanics 627 (May 25, 2009): 179–213. http://dx.doi.org/10.1017/s0022112009005850.
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A computationally efficient numerical method is developed to investigate nonlinear interactions between steep surface gravity waves and depth-varying ocean currents. The free-surface boundary conditions are used to derive a coupled set of equations that are integrated in time for the evolution of the free-surface elevation and tangential component of the fluid velocity at the free surface. The vector form of Green's second identity is used to close the system of equations. The closure relationship is consistent with Helmholtz's decomposition of the velocity field into rotational and irrotational components. The rotational component of the flow field is given by the Biot–Savart integral, while the irrotational component is obtained from an integral of a mixed distribution of sources and vortices over the free surface. Wave-induced changes to the vorticity field are modelled using the vorticity transport equation. For weak currents, an explicit expression is derived for the wave-induced vorticity field in Fourier space that negates the need to numerically solve the vorticity transport equation. The computational efficiency of the numerical scheme is further improved by expanding the kernels of the boundary and volume integrals in the closure relationship as a power series in a wave steepness parameter and using the fast Fourier transform method to evaluate the leading-order contribution to the convolution integrals. This reduces the number of operations at each time step from O(N2) to O(NlogN) for the boundary integrals and O[(NM)2] to O(NlogN) for the volume integrals, where N is the number of horizontal grid points and M is the number of vertical layers, making the model an order of magnitude faster than traditional boundary/volume integral methods. The numerical model is used to investigate nonlinear wave–current interaction in depth-uniform current fields and the modulational instability of gravity waves in an exponentially sheared current in deep water. The numerical results demonstrate that the mean flow vorticity can significantly affect the growth rate of extreme waves in narrowband sea states.
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Chae, Dongho, and In-Jee Jeong. "Preservation of log-Hölder coefficients of the vorticity in the transport equation." Journal of Differential Equations 343 (January 2023): 910–18. http://dx.doi.org/10.1016/j.jde.2022.11.017.
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Mansfield, John R., Omar M. Knio, and Charles Meneveau. "A Dynamic LES Scheme for the Vorticity Transport Equation: Formulation anda PrioriTests." Journal of Computational Physics 145, no. 2 (September 1998): 693–730. http://dx.doi.org/10.1006/jcph.1998.6051.
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Thomas, Matthew D., Agatha M. De Boer, Helen L. Johnson, and David P. Stevens. "Spatial and Temporal Scales of Sverdrup Balance*." Journal of Physical Oceanography 44, no. 10 (October 1, 2014): 2644–60. http://dx.doi.org/10.1175/jpo-d-13-0192.1.
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Abstract Sverdrup balance underlies much of the theory of ocean circulation and provides a potential tool for describing the interior ocean transport from only the wind stress. Using both a model state estimate and an eddy-permitting coupled climate model, this study assesses to what extent and over what spatial and temporal scales Sverdrup balance describes the meridional transport. The authors find that Sverdrup balance holds to first order in the interior subtropical ocean when considered at spatial scales greater than approximately 5°. Outside the subtropics, in western boundary currents and at short spatial scales, significant departures occur due to failures in both the assumptions that there is a level of no motion at some depth and that the vorticity equation is linear. Despite the ocean transport adjustment occurring on time scales consistent with the basin-crossing times for Rossby waves, as predicted by theory, Sverdrup balance gives a useful measure of the subtropical circulation after only a few years. This is because the interannual transport variability is small compared to the mean transports. The vorticity input to the deep ocean by the interaction between deep currents and topography is found to be very large in both models. These deep transports, however, are separated from upper-layer transports that are in Sverdrup balance when considered over large scales.
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Viviano, Antonino, Rosaria Ester Musumeci, and Enrico Foti. "A NEW 3D ROLLER APPROACH FOR FACING ROTATIONAL SURF ZONE HYDRODYNAMICS." Coastal Engineering Proceedings 1, no. 32 (January 30, 2011): 50. http://dx.doi.org/10.9753/icce.v32.currents.50.
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A 2DH highly nonlinear Boussinesq-type of model for breaking waves has been developed in order to investigate surf zone hydrodynamics, also in the presence of complex bathymetries. The set of equations includes continuity and rotational momentum equations, coupled with the vorticity transport equation. An appropriate spatial definition of the 3D roller concept, along with an algorithm for accurately tracking the roller position, have been on purposely developed. Several numerical simulations have been carried out for the case of a submerged elliptic shoal. The results have been compared with both experimental data and with the results of other numerical models available in the literature. Finally, the vorticity dynamics under a breaking wave has been analyzed both in time and space, showing that a fairly correct interpretation of the wake effect in the rear part of the wave crest is obtained.
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Zhao, Xiaoyang, Tairan Chen, Biao Huang, and Guoyu Wang. "Numerical analysis of the cavitating flow in an axial flow waterjet pump with special emphasis on the tip leakage flow and tip leakage vortex." Journal of Physics: Conference Series 2217, no. 1 (April 1, 2022): 012018. http://dx.doi.org/10.1088/1742-6596/2217/1/012018.
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Abstract Due to the structural design requirements, tip leakage flow is a common phenomenon in the field of axial flow rotating machinery. The tip leakage flow interacts with the mainstream and induce complex tip leakage vortex. When the local pressure drops to the saturation pressure, cavitation may occur in flow passage and the vortex core. The tip leakage cavitating flow has a huge effect on the flow stability, which may reduce the efficiency of the waterjet pump. In the present study, Zwart cavitation model and SST k-ω turbulence model are employed to simulate the cavitating flow. The new proposed Liutex criterion has been used to capture the vortex structures in the flow passage and the tip gap. The vorticity transport equation in cylindrical coordinates is also used to discuss the variation tendency of the vorticity in the vicinity of the gap region under cavitation condition. The evolution of the cavitation is proved to be vital important for the formation and the breakdown of the tip leakage vortex. By analyzing the vorticity transport equation, the stretching term is shown to be the decisive factor.
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Chanyal, B. C., and Mayank Pathak. "Quaternionic Approach to Dual Magnetohydrodynamics of Dyonic Cold Plasma." Advances in High Energy Physics 2018 (August 14, 2018): 1–13. http://dx.doi.org/10.1155/2018/7843730.
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The dual magnetohydrodynamics of dyonic plasma describes the study of electrodynamics equations along with the transport equations in the presence of electrons and magnetic monopoles. In this paper, we formulate the quaternionic dual fields equations, namely, the hydroelectric and hydromagnetic fields equations which are an analogous to the generalized Lamb vector field and vorticity field equations of dyonic cold plasma fluid. Further, we derive the quaternionic Dirac-Maxwell equations for dual magnetohydrodynamics of dyonic cold plasma. We also obtain the quaternionic dual continuity equations that describe the transport of dyonic fluid. Finally, we establish an analogy of Alfven wave equation which may generate from the flow of magnetic monopoles in the dyonic field of cold plasma. The present quaternionic formulation for dyonic cold plasma is well invariant under the duality, Lorentz, and CPT transformations.
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Paolino, M. A., R. B. Kinney, and E. A. Cerutti. "Numerical Analysis of the Unsteady Flow and Heat Transfer to a Cylinder in Crossflow." Journal of Heat Transfer 108, no. 4 (November 1, 1986): 742–48. http://dx.doi.org/10.1115/1.3247007.
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The unsteady, two-dimensional viscous flow of an incompressible, constant-property fluid flowing over a cylinder is numerically analyzed by integrating the vorticity transport equation and the energy equation. Departing from the usual stream function approach, the velocity distribution is obtained from the vorticity distribution by integrating the velocity induction law. Calculations start with the impulsive motion of the free stream and a step change in the surface temperature of the cylinder. The solution is advanced in time until steady-state conditions are achieved. Results are obtained for a Prandtl number of 0.7 and Reynolds numbers of 3000 and 70,800. Local and average Nusselt numbers and force coefficients are presented and compared to available experimental data.
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GREGA, L. M., T. Y. HSU, and T. WEI. "Vorticity transport in a corner formed by a solid wall and a free surface." Journal of Fluid Mechanics 465 (August 25, 2002): 331–52. http://dx.doi.org/10.1017/s0022112002001088.
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There is a growing body of literature in which turbulent boundary layer flow along a mixed-boundary corner formed by a vertical solid wall and a horizontal free surface has been examined. While there is consensus regarding the existence of weak secondary flows in the near corner region, there is some disagreement as to the exact nature and origin of these flows. In two earlier works by the authors, evidence was presented supporting the existence of a weak streamwise vortex which rotates in toward the wall at the free surface and down away from the surface along the wall. This ‘inner secondary vortex’ is accompanied by an ‘outer secondary flow’ which transports low-momentum boundary layer fluid up along the wall and outward at the free surface. The magnitudes of the cross-stream velocities associated with these secondary flows were measured to be on the order of 1% of the free-stream speed. In this paper, high-resolution DPIV measurements made in the cross-stream plane are presented. These clearly show the inner and outer secondary flows. The cross-stream vector fields allow computation of terms in the turbulent streamwise vorticity transport equation. These terms indicate mean vorticity transport at the free surface associated with the outer secondary flow. In addition there appears to be a balance between the wall-normal and free-surface-normal fluctuating vorticity reorientation terms.
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Spall, Michael A., Robert S. Pickart, Paula S. Fratantoni, and Albert J. Plueddemann. "Western Arctic Shelfbreak Eddies: Formation and Transport." Journal of Physical Oceanography 38, no. 8 (August 1, 2008): 1644–68. http://dx.doi.org/10.1175/2007jpo3829.1.
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Abstract The mean structure and time-dependent behavior of the shelfbreak jet along the southern Beaufort Sea, and its ability to transport properties into the basin interior via eddies are explored using high-resolution mooring data and an idealized numerical model. The analysis focuses on springtime, when weakly stratified winter-transformed Pacific water is being advected out of the Chukchi Sea. When winds are weak, the observed jet is bottom trapped with a low potential vorticity core and has maximum mean velocities of O(25 cm s−1) and an eastward transport of 0.42 Sv (1 Sv ≡ 106 m3 s−1). Despite the absence of winds, the current is highly time dependent, with relative vorticity and twisting vorticity often important components of the Ertel potential vorticity. An idealized primitive equation model forced by dense, weakly stratified waters flowing off a shelf produces a mean middepth boundary current similar in structure to that observed at the mooring site. The model boundary current is also highly variable, and produces numerous strong, small anticyclonic eddies that transport the shelf water into the basin interior. Analysis of the energy conversion terms in both the mooring data and the numerical model indicates that the eddies are formed via baroclinic instability of the boundary current. The structure of the eddies in the basin interior compares well with observations from drifting ice platforms. The results suggest that eddies shed from the shelfbreak jet contribute significantly to the offshore flux of heat, salt, and other properties, and are likely important for the ventilation of the halocline in the western Arctic Ocean. Interaction with an anticyclonic basin-scale circulation, meant to represent the Beaufort gyre, enhances the offshore transport of shelf water and results in a loss of mass transport from the shelfbreak jet.
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Loc, Ta Phuoc, and R. Bouard. "Numerical solution of the early stage of the unsteady viscous flow around a circular cylinder: a comparison with experimental visualization and measurements." Journal of Fluid Mechanics 160 (November 1985): 93–117. http://dx.doi.org/10.1017/s0022112085003408.
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Early stages of unsteady viscous flows around a circular cylinder at Reynolds numbers of 3 × 103 and 9.5 × 103 are analysed numerically by direct integration of the Navier–Stokes equations – a fourth-order finite-difference scheme is used for the resolution of the stream-function equation and a second-order one for the vorticity-transport equation. Evolution with time of the flow structure is studied in detail. Some new phenomena are revealed and confirmed by experiments.The influence of the grid systems and the downstream boundary conditions on the flow structure and the velocity profiles is reported. The computed results are compared qualitatively and quantitatively with experimental visualization and measurements. The comparison is found to be satisfactory.
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Xing, Tao, Zhenyin Li, and Steven H. Frankel. "Numerical Simulation of Vortex Cavitation in a Three-Dimensional Submerged Transitional Jet." Journal of Fluids Engineering 127, no. 4 (April 7, 2005): 714–25. http://dx.doi.org/10.1115/1.1976742.
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Vortex cavitation in a submerged transitional jet is studied with unsteady three-dimensional direct numerical simulations. A locally homogeneous cavitation model that accounts for non-linear bubble dynamics and bubble/bubble interactions within spherical bubble clusters is employed. The velocity, vorticity, and pressure fields are compared for both cavitating and noncavitating jets. It is found that cavitation occurs in the cores of the primary vortical structures, distorting and breaking up the vortex ring into several sections. The velocity and transverse vorticity in the cavitating regions are intensified due to vapor formation, while the streamwise vorticity is weakened. An analysis of the vorticity transport equation reveals the influence of cavitation on the relative importance of the vortex stretching, baroclinic torque, and dilatation terms. Statistical analysis shows that cavitation suppresses jet growth and decreases velocity fluctuations within the vaporous regions of the jet.
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Sellountos, E. J., Jorge Tiago, and Adelia Sequeira. "Meshless velocity – vorticity local boundary integral equation (LBIE) method for two dimensional incompressible Navier-Stokes equations." International Journal of Numerical Methods for Heat & Fluid Flow 29, no. 11 (November 4, 2019): 4034–73. http://dx.doi.org/10.1108/hff-06-2018-0310.
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Purpose This paper aims to describe the 2D meshless local boundary integral equation (LBIE) method for solving the Navier–Stokes equations. Design/methodology/approach The velocity–vorticity formulation is selected to eliminate the pressure gradient of the equations. The local integral representations of flow kinematics and transport kinetics are derived. The integral equations are discretized using the local RBF interpolation of velocities and vorticities, while the unknown fluxes are kept as independent variables. The resulting volume integrals are computed using the general radial transformation algorithm. Findings The efficiency and accuracy of the method are illustrated with several examples chosen from reference problems in computational fluid dynamics. Originality/value The meshless LBIE method is applied to the 2D Navier–Stokes equations. No derivatives of interpolation functions are used in the formulation, rendering the present method a robust numerical scheme for the solution of fluid flow problems.
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Mizuta, Genta. "Role of the Rossby Waves in the Broadening of an Eastward Jet." Journal of Physical Oceanography 42, no. 3 (March 1, 2012): 476–94. http://dx.doi.org/10.1175/jpo-d-11-070.1.
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Abstract To investigate the effect of the Rossby waves on an eastward jet such as the Kuroshio or Gulf Stream Extensions, a series of numerical experiments is conducted using a primitive equation model. In these experiments, an inflow and an outflow imposed on the western and eastern boundaries drive an unstable narrow jet and a broad interior flow in the western and eastern regions of the model domain, respectively. The barotropic Rossby waves are radiated from the transient region between the two regions. The eddy potential vorticity flux by the waves tends to compensate for the difference in the mean potential vorticity along mean streamlines between both sides of the transient region. Instability of the jet is insufficient for this compensation and weakens the mean potential vorticity gradient too much. Moreover, as the potential vorticity of the outflow is increased, the Rossby waves are intensified in order to compensate for the increase in the difference in the mean potential vorticity. These features strongly suggest that the Rossby waves are substantial in matching a jet with an interior flow. The speed of the waves and properties of eddies in recirculations of the jet are consistent with a two-layer analytic model, which indicates that the Rossby waves are radiated from eddies in recirculations. These eddies as well as the Rossby waves increase in amplitude with the transport of the recirculation near the surface presumably because of mean advection. Therefore, the mean potential vorticity of the interior flow, the intensity of the Rossby waves, and the transport of the recirculation change consistently with one another.
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Balonishnikov, Alexander, Jeanna Balonishnikova, and Julia Kruchkova. "Derivation of the simplest transport equations for dissipation and vorticity from a modified equation for small-scale velocity." IOP Conference Series: Materials Science and Engineering 919 (September 26, 2020): 052052. http://dx.doi.org/10.1088/1757-899x/919/5/052052.
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Ong, Hing. "Comments on “On the Structure and Formation of UTLS PV Dipole/Jetlets in Tropical Cyclones by Convective Momentum Surges”." Monthly Weather Review 148, no. 11 (November 2020): 4693–95. http://dx.doi.org/10.1175/mwr-d-20-0156.1.
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AbstractThis comment on Hitchman and Rowe first deepens their introduction by distinguishing adiabatic and diabatic tilting of vorticity. Then, it strengthens their interpretation by emphasizing that momentum must be vertically transported with reference to isentropic levels to yield the potential vorticity (PV) dipoles. Moreover, it points out a flaw in their PV budget analysis and proposes a remedy for the flaw. Their convective momentum transport paradigm and the vorticity tilting paradigm reinterpret the same physical process. However, they counted one physical process twice by associating the two paradigms with two different terms. As an attempt to remedy the flaw, this comment associates the reinterpretation of the two paradigms with a transformation of the PV equation; their paradigm corresponds to a flux form. With the proposed remedy, their paradigm can be more easily translated to advances in convective parameterization because of its horizontal locality.
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Lv, Shujian, Xincheng Wang, Huaiyu Cheng, and Bin Ji. "Numerical Investigation of Tip Leakage Vortex Cavitating Flow in a Waterjet Pump with Emphasis on Flow Characteristics and Energy Features." Energies 15, no. 19 (September 21, 2022): 6916. http://dx.doi.org/10.3390/en15196916.
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The Delayed Detached Eddy Simulation (DDES) turbulence model was coupled with a homogeneous cavitation model to analyze the tip-leakage vortex (TLV) cavitating-flow characteristics in a waterjet pump. The numerical results agree well with experimental data. The results show that the vortex evolution in the waterjet pump has three stages, which is similar to that around a hydrofoil, but the vorticity variations in the waterjet pump are more complicated. The relative-vorticity-transport equation was then applied to find the reason for the differences between the vorticity variation observed in the waterjet pump and that around a hydrofoil. The results indicate that the drastic fusion process of the TSV cavity and the TLV cavity in the waterjet pump resulted in the formation of triangular cavitation region near the blade tip that is difficult to reproduce by stationary hydrofoil simulation. This fusion process caused the local variation of fluid volume and further affected the vorticity transport. The entropy-production evaluation method considering the phase transition was then used to analyze the dissipation losses in the complex cavitation region. The results indicate that the drastic fusion process of the TSV cavity and the TLV cavity significantly influenced the entropy production rate distributions and enhanced the disturbance of the flow field. In addition, severe phase transition occurs in the drastic fusion region accompanied by huge phase-transition losses.
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Klewicki, J. C. "A description of turbulent wall-flow vorticity consistent with mean dynamics." Journal of Fluid Mechanics 737 (November 20, 2013): 176–204. http://dx.doi.org/10.1017/jfm.2013.565.
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AbstractA depiction of the mean and fluctuating vorticity structure in turbulent wall flows is presented and described within the context of the self-similar properties admitted by the mean dynamical equation. Data from a relatively wide range of numerical and physical experiments are used to explore and clarify the structure postulated. The mean vorticity indicator for the onset of the four-layer regime of the mean dynamics is revealed. With increasing Reynolds number, the mean vorticity is shown to segregate into two increasingly well-defined domains. Half of the mean vorticity concentrates into a near-wall region of width (relative to the overall flow width) that diminishes proportionally to the inverse square root of Reynolds number. The remainder of the mean vorticity is spread, with diminishing amplitude, over an outer domain that approaches the overall flow width at high Reynolds number. Vorticity stretching and reorientation are surmised to be the characteristic mechanisms accounting for the inner domain behaviour of both the mean and fluctuating vorticity. Vorticity dispersion via advective transport is surmised to be the characteristic mechanism in the outer domain. In this domain, the fluctuating enstrophy approaches that of the instantaneous enstrophy with increasing Reynolds number. This underpins an emerging self-similarity between the mean and r.m.s. vorticity in the domain where the mean velocity profile is logarithmic. The Reynolds number dependence of a number of properties associated with the vorticity field is explored and quantified. The study closes with brief account of the combined vortical and mean dynamical structure of turbulent wall flows.
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SCHUTTELAARS, H. M., and H. E. DE SWART. "Initial formation of channels and shoals in a short tidal embayment." Journal of Fluid Mechanics 386 (May 10, 1999): 15–42. http://dx.doi.org/10.1017/s0022112099004395.
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It is demonstrated, by using a simple model, that bedforms in a short tidal embayment can develop due to a positive feedback between tidal currents, sediment transport and bedforms. The water motion is modelled by the depth integrated shallow water equations. The system is forced by a prescribed free-surface elevation at the entrance of the embayment. For the sediment dynamics a diffusively dominated suspended load transport model is considered. Tidal averaging is used to obtain the bottom profiles at the long morphological time scale.The stability of a constantly sloping equilibrium bottom profile is studied for various combinations of the model parameters. It turns out that without a mechanism that generates vorticity this equilibrium profile is stable. In that case small-scale perturbations can at most become marginally stable if no bedload term in the bottom evolution equation is incorporated. If vorticity is generated, in our model by bottom friction torques, the basic state is unstable. The spatial patterns of the unstable modes and their growth rates depend, among other things, on the strength of the bottom friction, the width of the embayment and the grain size: if the sediment under consideration consists of large particles, the equilibrium will be more stable than when smaller particles are considered. Without a diffusive term in the bed evolution equation, small-scale perturbations become unstable. To avoid this physically unrealistic behaviour bedload terms are included in the sediment transport. Furthermore, it is shown that using an asymptotic expansion for the concentration as given in earlier literature is only valid for small or moderate mode numbers and the technique is extended to large mode numbers. A physical interpretation of the results is also given.
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Sinha, Krishnendu. "Evolution of enstrophy in shock/homogeneous turbulence interaction." Journal of Fluid Mechanics 707 (August 8, 2012): 74–110. http://dx.doi.org/10.1017/jfm.2012.265.
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AbstractInteraction of turbulent fluctuations with a shock wave plays an important role in many high-speed flow applications. This paper studies the amplification of enstrophy, defined as mean-square fluctuating vorticity, in homogeneous isotropic turbulence passing through a normal shock. Linearized Navier–Stokes equations written in a frame of reference attached to the unsteady shock wave are used to derive transport equations for the vorticity components. These are combined to obtain an equation that describes the evolution of enstrophy across a time-averaged shock wave. A budget of the enstrophy equation computed using results from linear interaction analysis and data from direct numerical simulations identifies the dominant physical mechanisms in the flow. Production due to mean flow compression and baroclinic torques are found to be the major contributors to the enstrophy amplification. Closure approximations are proposed for the unclosed correlations in the production and baroclinic source terms. The resulting model equation is integrated to obtain the enstrophy jump across a shock for a range of upstream Mach numbers. The model predictions are compared with linear theory results for varying levels of vortical and entropic fluctuations in the upstream flow. The enstrophy model is then cast in the form of$k$–$\epsilon $equations and used to compute the interaction of homogeneous isotropic turbulence with normal shocks. The results are compared with available data from direct numerical simulations. The equations are further used to propose a model for the amplification of turbulent viscosity across a shock, which is then applied to a canonical shock–boundary layer interaction. It is shown that the current model is a significant improvement over existing models, both for homogeneous isotropic turbulence and in the case of complex high-speed flows with shock waves.
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ARAKI, Shingo, Itsuro HONDA, Tetsuya YAMADA, Hideki OHBA, and Mizue MUNEKATA. "High-Order Numerical Analysis of Two-Dimensional Square Driven Cavity Flow Using Vorticity Transport Equation." Transactions of the Japan Society of Mechanical Engineers Series B 62, no. 599 (1996): 2572–78. http://dx.doi.org/10.1299/kikaib.62.2572.
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M K, AL-MUTAIRI, and BASSET H ABDEL. "Diagnostic study of diabatic heating and potential vorticity during a case of cyclogenesis." MAUSAM 71, no. 2 (August 3, 2021): 255–74. http://dx.doi.org/10.54302/mausam.v71i2.24.
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On 16-17 November, 2015, north and middle regions of Saudi Arabia were hit by a case of cyclogenesisassociated with heavy rainfall. This work presents a diagnostic study of this heavy rainfallcase based on the analysis of diabatic heating and potential vorticity. The synoptic analysis investigate that the important dynamical factors that causes this case are the northward extension of Red Sea Trough, anticyclone over the Arabian Peninsula, a travailing midlatitude upper trough, moisture transport pathways and strong upward motion arising from tropospheric instability. The calculation of diabatic heating by the thermodynamic equation illustrate that the contribution of vertical temperature advection and the adiabatic term are opposite to each other during the period of study. The largest contribution of the horizontal cold advection occurs during the first two days while the largest contribution of the horizontal warm advection occurs during the maximum development days. The dynamics of the studied case are also investigated in terms of isobaric Potential Vorticity. It is found that the location of the low-level Potential Vorticity anomaly and the Potential Vorticity generation estimates coincides with the heating region, which implies that condensation supports a large enough source to explain the existence of the low-level Potential Vorticity anomaly.
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JAKIRLIĆ, S., and K. HANJALIĆ. "A new approach to modelling near-wall turbulence energy and stress dissipation." Journal of Fluid Mechanics 459 (May 25, 2002): 139–66. http://dx.doi.org/10.1017/s0022112002007905.
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A new model for the transport equation for the turbulence energy dissipation rate ε and for the anisotropy of the dissipation rate tensor εij, consistent with the near-wall limits, is derived following the term-by-term approach and using results of direct numerical simulations (DNS) for several generic wall-bounded flows. Based on the two-point velocity covariance analysis of Jovanović, Ye & Durst (1995) and reinterpretation of the viscous term, the transport equation is derived in terms of the ‘homogeneous’ part εh of the energy dissipation rate. The algebraic expression for the components of εij was then reformulated in terms of εh, which makes it possible to satisfy the exact wall limits without using any wall-configuration parameters. Each term in the new equation is modelled separately using DNS information. The rational vorticity transport theory of Bernard (1990) was used to close the mean curvature term appearing in the dissipation equation. A priori evaluation of εij, as well as solving the new dissipation equation as a whole using DNS data for quantities other than εij, for flows in a pipe, plane channel, constant-pressure boundary layer, behind a backward-facing step and in an axially rotating pipe, all show good near-wall behaviour of all terms. Computations of the same flows with the full model in conjunction with the low-Reynolds number transport equation for (uiui) All Overbar, using εh instead of ε, agree well with the direct numerical simulations.
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Moussiopoulos, N. "Numerical Simulation of Spray Cooling Pond Performance." Journal of Fluids Engineering 109, no. 2 (June 1, 1987): 179–85. http://dx.doi.org/10.1115/1.3242641.
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A mathematical model for predictions of the performance of spray cooling ponds is presented. In contrast to previous methods, the present model requires neither empirical information from field measurements nor an adaptation of model constants. The airflow is described by partial differential equations for the vorticity and the stream function. Turbulence is taken into account by a modified version of the k-ε model. Temperature and humidity of air are obtained by solving appropriate transport differential equations. The equation system is solved by means of a finite difference method. The utilized numerical algorithm has been proved to be reasonably accurate. Predicted distributions for the dependent variables are presented for a circular spray cooling pond and the case of zero wind velocity. Results for the thermal performance of this pond are in good agreement with observations.
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Wang, Like, Jinling Lu, Weili Liao, Yaping Zhao, and Wei Wang. "Numerical Simulation of the Tip Leakage Vortex Characteristics in a Semi-Open Centrifugal Pump." Applied Sciences 9, no. 23 (December 2, 2019): 5244. http://dx.doi.org/10.3390/app9235244.
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Tip leakage vortex has an important influence on the performance of semi-open centrifugal pumps. Simulations based on the three-dimensional Reynolds-Averaged Navier–Stokes were conducted to study the structural characteristics of tip leakage vortex and its effects on the internal flow field, and the Shear Stress Transport k-ω turbulence model was used to simulate the whole flow passage of centrifugal pumps with tip clearances of 0 mm and 1 mm. Then, the tip leakage vortex was analyzed using the relative vorticity transport equation. The numerical data and experimental results agreed well. The leakage vortex formed in the tip clearance led to 18.7% and 14.4% decrease in head and efficiency under design condition, respectively, and the bigger the flow rate, the fast the performance decreased. Tip leakage vortex formed at the leading edge of the blade moved along the suction surface. Whereas the tip leakage vortex formed near the middle of the blade extended to the pressure surface of the adjacent blade. This phenomenon deteriorated the flow field and induced passage vortex, thereby reducing the static pressure and blade load and changing the static pressure distribution law. The formation and development of leakage vortex could be attributed to the relative vortex stretching the term. The Coriolis force term could reflect the change of vorticity caused by leakage flow, and the viscous diffusion term served as the vorticity source.
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Кalashnik, М. V., and О. G. Chkhetiani. "Non-stationary vortex streets in shear flows." Известия Российской академии наук. Физика атмосферы и океана 55, no. 6 (December 21, 2019): 127–38. http://dx.doi.org/10.31857/s0002-3515556127-138.
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Spatially periodic vortex systems that form due to unstable shear flows are called vortex streets. A number of exact and asymptotic solutions of two-dimensional hydrodynamic equations describing nonstationary vortex streets have been constructed. It is shown that the superposition of the flow with a constant horizontal shear and its perturbations in the form of a nonmodal wave provides an exact solution that describes a nonstationary vortex street with rotating elliptic current lines. The width of the zone occupied by such a vortex street has been determined. The equation of separatrix separating vortex cells with closed current lines from an external meandering flow has been obtained. The influence of the quasi-two-dimensional compressibility and beta effect on the dynamics of vortex streets has been studied based on the potential vorticity transport equation. The solutions describing the formation of vortex streets during the development of an unstable zonal periodic flow and a free shear layer have been constructed using a longwave approximation.
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Wang, Chaoyue, Yongshun Zeng, Zhifeng Yao, and Fujun Wang. "Rigid vorticity transport equation and its application to vortical structure evolution analysis in hydro-energy machinery." Engineering Applications of Computational Fluid Mechanics 15, no. 1 (January 1, 2021): 1016–33. http://dx.doi.org/10.1080/19942060.2021.1938685.
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Han, Chengzao, Yun Long, Mohan Xu, and Bin Ji. "Verification and Validation of Large Eddy Simulation for Tip Clearance Vortex Cavitating Flow in a Waterjet Pump." Energies 14, no. 22 (November 15, 2021): 7635. http://dx.doi.org/10.3390/en14227635.
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In this paper, large eddy simulation (LES) was adopted to simulate the cavitating flow in a waterjet pump with emphasis on the tip clearance flow. The numerical results agree well with the experimental observations, which indicates that the LES method can make good predictions of the unsteady cavitating flows around a rotor blade. The LES verification and validation (LES V&V) analysis was used to reveal the influence of cavitation on the flow structures. It can be found that the LES errors in cavitating region are larger than those in the non-cavitating area, which is mainly caused by more complicated cavitating and tip clearance flow structures. Further analysis of the interaction between the cavitating and vortex flow by the relative vorticity transport equation shows that the stretching, dilatation and baroclinic torque terms have major effects on the generation and transport of vortex structure. Meanwhile the Coriolis force term and viscosity term also exacerbate the vorticity transport in the cavitating region. In addition, the flow loss characteristics of this pump are also revealed by the entropy production theory. It is indicated that the tip clearance flow and trailing edge wake flow cause the viscous dissipation and turbulent dissipation, and the cavitation can further enhance the instability of the flow field in the tip clearance.
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Felli, Mario, Massimo Falchi, and Giulio Dubbioso. "Tomographic-PIV Survey of the Near-Field Hydrodynamic and Hydroacoustic Characteristics of a Marine Propeller." Journal of Ship Research 59, no. 04 (December 1, 2015): 201–8. http://dx.doi.org/10.5957/jsr.2015.59.4.201.
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This article deals with a pioneering application of tomographic particle image velocimetry (tomographic PIV) for the hydrodynamic and hydroacoustic analysis of a marine propeller. The hydrodynamic study was mainly focused on the topological analysis of the propeller wake characteristics in the near field based on the vorticity field and on the tilting and stretching terms of the vorticity transport equation. Hydroacoustic analysis concerned the use of tomographic PIV in combination with the Powell's acoustic analogy. Tomographic PIV proved to be a valid tool for the detailed quantitative reconstruction of the complex vortex topology in the propeller wake and provided an accurate description of the source terms of the Powell's analogy. In particular, it was shown that the tip vortex perturbation represents the dominant nonlinear contribution to the radiated far-field noise in non-cavitating flow conditions.
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Su, Taoyong, Yang Lu, Jinchao Ma, and Shujun Guan. "Electrically Controlled Rotor Blade Vortex Interaction Airloads and Noise Analysis Using Viscous Vortex Particle Method." Shock and Vibration 2019 (November 6, 2019): 1–15. http://dx.doi.org/10.1155/2019/9678970.
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An electrically controlled rotor (ECR), also called a swashplateless rotor, replaces a swashplate with a trailing-edge flap system to implement primary rotor control. To investigate the aerodynamic characteristics of an ECR in blade-vortex interaction (BVI) condition, an analysis model based on the viscous vortex particle method, ECR blade pitch equation, and the Weissinger-L lifting surface model is established. In this model, the ECR wake flow field vorticity is discretized as multiple vortex particles, and the vorticity-velocity form of the Navier-Stokes equation is solved to simulate the transport diffusion of the vorticity. The flap motion-inducing blade-pitch movement is obtained by solving the ECR blade-pitch movement equation via the Runge–Kutta fourth-order method. On the basis, BVI noise radiation of an ECR is evaluated using the Ffowcs Williams and Hawkings (FW-H) equation. Based on the present prediction model, the aerodynamic and acoustic characteristics of a sample ECR in BVI condition are analyzed. The results show that since the BVI event of the ECR on the advancing side is mainly caused by the interaction between the flap tip vortex and the blade, the blade spanwise range of ECR BVI occurrence on the advancing side is smaller than that of the conventional rotor. In addition, the magnitude of the maximum sound pressure level on the advancing side as well as on the retreating side of the ECR is also different from that of the conventional rotor, which is consistent with the difference in the airloads between the ECR and conventional rotor. Furthermore, a study was performed to examine the effect of the pre-index angle on the BVI-induced airloads and noise. The amplitude of the impulsive airloads of the ECR on the advancing side is increased with the increase in pre-index angle, while the amplitude of the impulsive airloads of the ECR on the retreating side is decreased. Indeed, when the pre-index angle of the sample ECR is 8 degrees, the retreating-side noise radiation lobe is almost disappeared. In addition, the different intensity of wake vorticity is the main reason for the differences of the BVI-induced airloads and noise among the ECR with different pre-index angles.