Interaction of two equal vortices on a β plane
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The interaction of two equal vortices under the influence of a gradient of background vorticity (β) is studied numerically and experimentally. If the initial shape and vorticity distribution of the vortices is fixed, two parameters determine the evolution: the normalized intercentroid distance d*=d/R, where R is the radius of the vortex; and the normalized gradient of background vorticity β*=βR/ω, where ω is the peak vorticity of the vortex. Alternate ways of identifying regimes of behavior in the parameter plane (d*,β*) are presented. These are applied to numerical simulations of interaction of vortices with steplike, steep and smooth vorticity profiles. It is found that the critical distance for merger decreases with increasing β* for all vortex types, and that vortices with smooth vorticity profile are the most merger-prone vortices. Laboratory experiments were done in a rotating water tank with a flat sloping bottom providing the β effect. The vortices produced have a smooth vorticity profile and show the same behavior observed in the simulations, except that, as a result of viscous effects, the critical merger distance is shifted towards larger values of d*.Keywords:
Burgers vortex
Vortex stretching
Vortex stretching
Burgers vortex
Enstrophy
Positive vorticity advection
Starting vortex
Vortex Tube
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A three-dimensional theory of vorticity dynamics on an incompressible viscous and immiscible fluid–fluid interface, or interfacial vorticity dynamics for short, is presented as a counterpart of the vorticity dynamics on an arbitrarily curved rigid wall [J. Fluid Mech. 254, 183 (1993)]. General formulas with arbitrary Reynolds numbers Re are derived for determining (1) how much vorticity exists on an interface S, (2) how much vorticity is created from S and sent into the fluid per unit area in per unit time, and (3) the force and moment acted on a closed interface by the created vorticity thereon. The common feature and fundamental difference between interfacial vorticity dynamics and its rigid-wall counterpart are analyzed. In particular, on a free surface, the primary driving mechanism of vorticity creation is the balance between the shear stress (measured by tangent vorticity) and the tangent components of the surface-deformation stress alone, which results in a weak creation rate of O (Re−1/2) at large Re. Therefore, the exact form of the theory with its full complexity is of importance mainly at low Reynolds numbers, especially in understanding the small-scale coherent structures of interfacial turbulence. The vorticity creation rate at high-Re approximations, including an interfacial boundary layer of finite thickness and the limit of Re→∞ (the so-called Euler limit), is also studied, both allowing for a rotational inviscid outer flow. While for the former this leads to a generalization of Lundgren’s theory [in Mathematic Aspects of Vortex Dynamics, edited by R. E. Caflish (SIAM, Philadelphia, PA, 1989), pp. 68–79] and amounts to solving a linear boundary-layer problem, for the latter the creation rate can be directly obtained from an inviscid solution, leading to a dynamic evolution equation of interfacial vortex sheet. In three dimensions, a vortex sheet may bifurcate into a normal vorticity field, upon which the dependence of the sheet velocity is determined. A few examples are examined to illustrate different aspects and approximation levels of the general theory.
Burgers vortex
Vortex stretching
Inviscid flow
Reynolds stress
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A focus is presented on the investigation of vorticity generation and development and the vorticity structures inside/outside of a vortex. A vortex cannot be directly represented only by the vorticity, but is a region where the vorticity overtakes deformation. Except for those vorticity lines which come from and end at side boundaries, another type, self-closed vorticity lines named vorticity rings, is numerously generated inside the domain during flow transition. Vorticity and vortex are two different but closely related concepts, and both new vorticity and new vortices are generated during flow transition. According to our direct numerical simulation result, the generation and growth of the vorticity rings are produced by the buildup of the vortex; on the other hand, the vortex buildup is a consequence of the lengthening (stretching, tilting, and twisting) of the vorticity lines. According to the vorticity flux conservation law, a vorticity line cannot be interrupted, started, or ended inside the flow field; the newly generated vorticity has only one form which is a vorticity ring. In addition, for a single hairpin vortex or a ring-like vortex, it may consist of several types of vorticity lines: some could come from the side boundaries and some could be full of vorticity rings or part of vorticity rings.
Vortex stretching
Burgers vortex
Positive vorticity advection
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Secondary (negative) vorticity is frequently observed at the up-wash side of the lateral vortex interacting with a wall. The generation of the secondary vorticity is found to be due to the process term of the vorticity equation in which transversal vorticity component of the lateral vortex is converted into negative vorticity by the combination of velocity gradient. In the secondary vortex region, the process term provides positive vorticity which attenuate the vorticity of the secondary vortex.
Burgers vortex
Vortex stretching
Positive vorticity advection
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Burgers vortex
Conservative vector field
Inviscid flow
Vortex stretching
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In the momentum equations for incompressible flow, this chapter focuses on velocity and pressure. In many instances it is advantageous to interpret the flow events in terms of the vorticity and the dynamic events that are interacting to give a certain vorticity distribution. The existence of vorticity generally indicates that viscous effects are important. This occurs because fluid particles can only be set into rotation by an unbalanced shear stress. The vorticity equation indicates that as one follows a material particle, vorticity is intensified by vortex line stretching and turning and is slowly diffused by viscosity. Pressure does not play a direct role in the equation governing vorticity dynamics. This viewpoint emphasizes viscous effects. Helmholtz's laws, allows to think of vortex lines as stringing together a set of material particles, are applicable whenever viscous diffusion is negligible.
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Vortex stretching
Positive vorticity advection
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We examine the possibility of a blowup of the vorticity due to self-stretching and mechanisms for its prevention. We first estimate directly from the Navier-Stokes equations the length scale of coherence in the direction of the vortex lines to be of the order of the Kolmogorov length. Alignment of vortex lines is seen to be a viscous phenomenon and may prevent some scenarios for blowup. Next we derive equations for the curvature and torsion of vortex lines. We show that the same stretching that amplifies the vorticity also tends to straighten out the vortex lines. Then we show that in well-aligned vortex tubes, the self-stretching rate of the vorticity is proportional to the ratio of the vorticity and the radius of curvature. Thus blowup of the vorticity in such tubes can be prevented by the growth of the vorticity being balanced by the straightening of the vortex lines. Implications for vorticity-strain alignment and the scaling theory of turbulence are noted. Finally, we examine the effects of viscous diffusion on the vorticity field and see how viscosity can lead to organization and alignment of vortex lines.
Vortex stretching
Burgers vortex
Vortex Tube
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The strain-induced evolution of shielded monopolar vortices has been investigated in a stratified fluid. A steady strain flow was generated by an arrangement of four rotating horizontal discs, whereas the monopolar vortex was created by a small spinning sphere. Quantitative information about the flow field was obtained by tracking passive tracer particles. The vortex was observed to deform into a tripolar-like structure, followed by the shedding of the accompanying satellites. During this stage, the remaining vortex core evolved quasi-steadily, which was evident from the functional relationship between the vorticity and the stream function. Furthermore, it was shown that the removal of the surrounding ring of oppositely signed vorticity induces an accelerated horizontal growth of the vortex core. Owing to the diffusive decay of vorticity, the vortex was finally torn apart along the horizontal strain axis. The dynamics of the vortex core appeared to be very similar to that of an elliptic patch of uniform vorticity. The instantaneous vorticity contours at high vorticity levels were close to ellipses with nearly the same aspect ratios and orientations. Moreover, the observed vortex evolution was in qualitative agreement with the calculated motion of an elliptic patch of uniform vorticity. As a second approach, the full two-dimensional vorticity equation was solved numerically by a finite-difference method in order to account for both the non-uniform spatial vorticity distribution of the laboratory vortex and the diffusion of vorticity in the horizontal directions. The numerically obtained vortex evolution was in good agreement with that observed in the laboratory.
Vortex stretching
Burgers vortex
Starting vortex
Circulation (fluid dynamics)
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Cavitation inception of a vortex is difficult to predict. This is due in a large part to a confusion in the type of cavitation occurring, i.e., vaporous versus nonvaporous cavitation. In addition, the vortex structure is poorly defined in many cases. These two problems are particularly important for the prediction of cavitation inception in a vortex created in the low momentum fluid near the inner wall of a rotor. The purpose of this paper is to present the results of a vortex cavitation investigation which are both experimental and theoretical. A vorticity flow analysis is developed and employed to assess the effect of vorticity on cavitation inception of a vortex. Previous investigations have shown that the minimum pressure coefficient of a vortex depends upon the vorticity associated with the vortex. Employing secondary vorticity equations, the vorticity is calculated in the blade passage. Changes in passage vorticity are used in a simple vortex model to predict trends in cavitation inception of a vortex. Theoretical results indicate that small changes in vorticity distribution near the inner wall of the rotor create rather large differences in the cavitation inception of the vortex. These small changes are primarily due to changes in the secondary vorticity. This secondary vorticity dominates the vortex structure. Comparisons are presented between the predicted and measured cavitation inception and good agreement is shown when the effects of gas on cavitation inception are reduced. Experimental data confirms that secondary vorticity dominates the vortex structure. In addition, experimental cavitation data are presented which show the dramatic influence of a gas on cavitation inception of a vortex.
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Burgers vortex
Starting vortex
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The flow around a circular cylinder mounted normal to the channel bed is essentially a three-dimensional one. The flow upstream undergoes a separation of the turbulent boundary layer and rolls up to form a flow system, known as horseshoe vortex, which is swept around the cylinder. This vortex system plays an important role, if the bed material is movable. The shear stress of the vortex system is responsible to the bed erosion. The shear stress depends on the velocity gradient, may be presented as the vorticity. This study is done to gain a better understanding of the vorticity around a cylinder, especially where the system of vortices exist. Flow measurements were used to study the vorticity fields on flow with a horse-shoe vortex system around a pile. The velocity vector plots show the presence of a primary vortex upstream of the cylinder. Based on these velocity vectors, the vorticity can be analyzed by using central finite difference approximations. Results of the vorticity calculation are presented as the vorticity contours. Results of study show the greatest value of the positive-vorticity on the horse-shoe vortex system is in the plane upstream of cylinder. This value decreases in the planes downstream, attaining the lowest value in the plane downstream, where the wake vortex established. The strength of the positive vorticity increases at larger Reynolds number. Underneath the field of positive vorticity, stretching around the cylinder, it appears a field of negative vorticity. This negative vorticity near the bed is high in front of the cylinder, proportional to the bed shear stress, which is responsible to the development of local scour at the movable bed. The line of zero vorticity is plotted originated at the bed, being as the separation point. It is also concluded that the maximum positive-vorticity is not necessary coincided with the center of the vortex.Abstrak. Pola aliran di sekitar silinder bulat yang dipasang tegak lurus dasar saluran merupakan aliran tiga dimensi Aliran di hulu silinder membentuk separasi pada lapis batas turbulen, membangkitkan terbentuknya pusaran tapal-kuda yang mengelilingi silinder. Sistem pusaran ini mempunyai peranan penting jika material dasar mudah bergerak. Tegangan geser pada sistem pusaran akan menyebabkan terbentuknya erosi lokal. Besarnya tegangan geser bergantung pada perbedaan kecepatan, yang dapat dipresentasikan dengan nilai vortisitasnya. Studi ini bertujuan untuk mendapatkan pemahaman yang lebih baik tentang vortisitas di sekitar silinder, terutama di lokasi dimana terbentuk system vortex. Data hasil pengukuran kecepatan aliran dipakai untuk mempelajari system vortisitas pada aliran dengan sistem pusaran di sekitar pilar. Gambar vektor kecepatan di sekitar pilar silinder memperlihatkan keberadaan pusaran utama di depan silinder. Berdasarkan vector kecepatan tersebut, dapat dianalisis vortisitasnya dengan menggunakan metode beda hingga. Hasil hitungan vortisitas digambarkan dalam bentuk kontur vortisitas. Hasil studi menunjukkan nilai maksimum dari vortisitas positif (searah jarum jam) dari sistem pusaran tapal kuda adalah pada bidang simetri di hulu silinder. Nilai vortisitas tersebut menurun, mencapai nilai minimum di hilir silinder. Besarnya vortisitas positif meningkat pada angka Reynold yang lebih tinggi. Di bawah bidang vortisitas positif, diperlihatkan terbentuknya system vortisitas negatif. Vortisitas negative di dekat dasar di depan silinder mempunyai nilai yang besar, proporsional dengan nilai tegangan geser di dasar, yang bertanggungjawab pada terbentuknya erosi lokal pada saluran dengan dasar bergerak. Garis yang memberikan nilai vortisitas sama dengan nol, merupakan garis separasi antara vortisitas positif dan negative. Dari hasil kajian juga disimpulkan bahwa posisi dari vortisitas positif maksimum tidak selalu berimpit dengan pusat pusaran tapal kuda.
Burgers vortex
Vortex stretching
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