Vorticity definition

Non-Newtonian Flow instabilities Chaotic Vortices Definition... 

The lack of hydrodynamic definition was recognized by Eucken (E7), who considered convective diffusion transverse to a parallel flow, and obtained an expression analogous to the Leveque equation of heat transfer (L5b, B4c, p. 404). Experiments with Couette flow between a rotating inner cylinder and a stationary outer cylinder did not confirm his predictions (see also Section VI,D). At very low rotation rates laminar flow is stable, and does not contribute to the diffusion process since there is no velocity component in the radial direction. At higher rotation rates, secondary flow patterns form (Taylor vortices), and finally the flow becomes turbulent. Neither of the two flow regimes satisfies the conditions of the Leveque equation. 

By definition, the dissipation range is dominated by viscous dissipation of Kolmogorov-scale vortices. The characteristic time scale rst in (2.74) can thus be taken as proportional to the Kolmogorov time scale rn, and taken out of the integral. This leads to the final form for (2.70),... 

We have determined the components of the rotation-rate vector dQ/dt for a general velocity field. However, it is conventional in fluid mechanics to represent rotation in the form of a derived variable called vorticity, which is denoted as the vector u. By definition,... 

In the jargon of vector calculus, the vorticity field is said to be soleniodal. A flow for which the vorticity is exactly zero, u = 0, is, by definition, called irwtational. 

One important use of the stream function is for the visualization of flow fields that have been determined from the solution of Navier-Stokes equations, usually by numerical methods. Plotting stream function contours (i.e., streamlines) provides an easily interpreted visual picture of the flow field. Once the velocity and density fields are known, the stream function field can be determined by solving a stream-function-vorticity equation, which is an elliptic partial differential equation. The formulation of this equation is discussed subsequently in Section 3.13.1. Solution of this equation requires boundary values for l around the entire domain. These can be evaluated by integration of the stream-function definitions, Eqs. 3.14, around the boundaries using known velocities on the boundaries. For example, for a boundary of constant z with a specified inlet velocity u(r),... 

Taking the vector curl of the right-hand side causes the first and last terms to drop out, since the curl of the gradient vanishes. However, for variable density, the left-hand side expands to long, complex, and not-too-useful expression (see Section A.14). Therefore let us restrict attention to incompressible flows, namely constant density. The curl of the incompressible Navier-Stokes equation, incorporating the definition of vorticity u = VxV, yields... 

Recognizing the definition of the substantial derivative, the vorticity equation can be written compactly as... 

In two-dimensional, incompressible, steady flows, there is a relatively simple relationship between the vorticity and the stream function. Consider the axisymmetric flow as might occur in a channel, Fig. 3.12. Beginning with the axisymmetric stream function as discussed in Section 3.1.2, substitute the stream-function definition into the definition of the circumferential vorticity u>q ... 

Substituting these definitions into the definition of vorticity in a two-dimensional velocity field yields... 

The boundary conditions provide a tight coupling between the vorticity and stream-function fields. Also velocities still appear in the convective terms. Given the stream-function field, velocity is evaluated from the definition of stream function. That is, velocity is computed from stream-function derivatives. 

In this case the pressure is eliminated altogether, since by vector identity, the curl of the gradient of a scalar field vanishes. From the definition of vorticity, Eq. 2.103, a simple diffusion equation emerges for the vorticity... 

The viscous shearing at the stagnation surface is a source of vorticity that is transported into the flow. One way to characterize the boundary layer is in terms of its vorticity distribution. By definition, the circumferential component of the vorticity vector is given as... 

Common examples of pseudo-vectors that will be relevant later include the angular velocity vector f2, the torque T, the vorticity vector co (or the curl of any true vector), and the cross product of two vectors. The inner scaler product of a vector and a pseudo-tensor or a pseudo-vector and a regular tensor will both produce a pseudo-vector. It will also be useful to extend the notion of a pseudo-vector to scalers that are formed as the product of a vector and a pseudo-vector. The third-order, alternating tensor e is a pseudo-tensor of third order as may be verified by reviewing its definition... 

The term at O(Re) in (9 119) represents a wakelike solution for the vorticity, which is analogous to the thermal wake in Eq. (9 51). To see this, we can calculate the vorticity associated with (9-119) by means of the definition... 

It is shown in Fig. 3.24, that the vortical fluidized bed exists for definite ratios between the pressure differential and the gas flow velocity [48]. At small differential and velocity values (AB) powder particles are motionless. The gas flow velocity v corresponds to the onset of transfer to the fluidized bed state (B) and is called the first critical velocity. On reaching the second critical velocity, V2, powder particles are carried away from the vessel by the gas flow (C). 

Fig. 2.2 Schematic of the/i-vortex formation and definition scotch Initially the vorticity is organized spanwise at it is uniformly distributed in flow direction. By an instability process, the vorticity starts to wrap up into a vortex which by stretching takes the idealized form of a A composed mainly of two side-vortex-rods with an internal flow behaviour as described by eq.(2.2). This process ends when the vorticity cannot be concentrated anymore and viscous diffusion processes start to dominate the flow field in the event, which starts from thereon to decay. Since this model also has to account for the non-slip condition, the wall near flow can be described by a viscous tornado. Therefore the question arises whether by incorporating the model of the viscous tornado into the 1-vortex model it would be possible to describe the flow field completely.

The zero-temperature phases can be observed using Bragg scattering with optical light, which allows for probing the crystalline phase. The detection of vortices can be used as a definitive signature of superfluidity. We notice that the two-dimensional (quasi) condensate involves a fraction of the total density only, and therefore we expect only small coherence peaks in a time-of-flight experiment. 

However, the same definition would apply to the case of 7t-electron currents in the same molecules, in which the separatrix coincides with the single vortical line through the centre of the molecule. It would be also applicable to diamagnetic atoms, in which the delocalized flow beyond the nucleus consists of concentric circular streamlines about a vortical stagnation axis identifiable with the separatrix [60]. 

At a particular range of Reynolds Number (<2,500), vortices are shed alternately from opposite sides of a body at a definite frequency as shown in Fig. 6.9. The vortices in one row are staggered with respect to those in the other. Vortex shedding may also occur at high values of Reynolds Number. 

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