Vorticity transport equation

The vorticity transport equation can be obtained by taking the curl of Eq. (11.2) ... 

A vorticity-transport equation can be derived by taking the taking the vector curl of the full Navier-Stokes equations. For incompressible flows with constant viscosity, the vorticity-transport equation can be expressed in a form that is quite similar to the other transport equations. Begin with the full Navier-Stokes equations, which for constant viscosity can be written in compact vector form as (Eq. 3.61)... 

In this expression one term vanishes because V V = 0 for an incompressible flow and Vw = 0 because the divergence of the curl of a vector vanishes (vorticity is the curl of the velocity vector). For the same reason the last term on the right-hand side of the vorticity equation also vanishes. As a result the vorticity-transport equation is further reduced to... 

With the exception of a spatially dependent body force f, there is no source or sink term in the vorticity transport equation. Therefore the source of vorticity is usually at boundaries, with the shear at solid walls being the most common means to produce vorticity. To illustrate the behavior of vorticity generation at a wall, consider the axisymmetric flow in a circular channel as illustrated in Fig. 3.12. 

The pressure does not appear directly in the vorticity-transport equation. Thus, it is apparent that the convective and diffusive transport of vorticity throughout a flow cannot depend directly on the pressure field. Nevertheless, it is completely clear that pressure affects the velocity field, which, in turn, affects the vorticity. By taking the divergence of the incompressible, constant-viscosity Navier-Stokes equations, a relationship can be derived among the velocity, pressure, and vorticity fields. Beginning with the Navier-Stokes equations as... 

For the two-dimensional problem the body force must be purely in the two-dimensional plane. Therefore Vxf must be purely orthogonal to the plane for example, in the r-6 problem, it must point in the z plane. It can be shown that the vortex-stretching term vanishes under these conditions. As a result the vorticity-transport equation is a relatively straightforward scalar parabolic partial differential equation,... 

In addition to the vorticity transport equation, a relationship between vorticity and stream function can be developed for two-dimensional steady-state problems. Continuing to use the r-6 plane as an example, the stream function is defined to satisfy the continuity equation exactly (Section 3.1.3),... 

The stream-function-vorticity equation, taken together with the vorticity transport equation, completely replaces the continuity and momentum equations. The pressure has been eliminated as a dependent variable. The continuity equation has been satisfied exactly by the stream function, and does not need to be included in the system of equations. The... 

Deriving the vorticity-transport equation requires taking the curl of the momentum equation, which leads to a substantial derivative of vorticity. Develop a general expression for this substantial derivative, noting that in general... 

As discussed in Section 3.1.1, the two-dimensional vorticity-transport equation is given as... 

The vorticity-transport equation has convective and diffusive terms in both the axial and radial directions. For the stagnation flow these terms contribute in such a way that the... 

By direct substitution, verify that a velocity profile from the numerical solution satisfies the scaled-vorticity transport equation (Eq. 6.50). 

Derive the relationship between the vorticity-transport equation (Eq. 6.47) and the radial-momentum equation for semi-infinite stagnation flow. In other words, carefully fill in all the missing pieces in the derivation leading to Eq. 6.55. In the course of this exercise, review how to apply integration by parts. 

Similarly, the finite-difference form of the vorticity transport equation, i.e. Eq. (8.137), gives ... 

From this equation, it is possible to discuss further about the general properties of the near-held solution. Noticing that the kinematic equation when substituted in the governing vorticity transport equation... 

At the outflow of the domain, the traditional extrapolation based on = 0 was applied for stream function and vorticity transport equations. Equations (2.7.15) and (2.7.16) have been solved in Sengupta et al. (2002) first for the case of c = Uoa, in a domain for which Reynolds number varies from 165 to 1900. The results are shown in Fig. 2.34 at different indicated times. 

The two-dimensional Navier-Stokes equation is solved in stream function-vorticity formulation, as reported variously in Sengupta et al. (2001, 2003), Sengupta Dipankar (2005). Brinckman Walker (2001) also simulated the burst sequence of turbulent boundary layer excited by streamwise vortices (in X- direction) using the same formulation for which a stream function was defined in the y — z) -plane only. To resolve various small scale events inside the shear layer, the vorticity transport equation (VTE) and the stream function equation (SFE) are solved in the transformed — rj) —... 

The main value of the vorticity transport equation, in the present context, is that a direct analogy exists for 2D motions between this equation and the thermal energy equation of Chap. 9. Specifically, for a 2D flow,... 

In particular, let us start with the nondimensionalized vorticity transport equation (10-6) and attempt to obtain an approximate solution for Re 1. We expect an asymptotic expansion with Re 1 as the small parameter, but we restrict our attention here to the leading-order term in this expansion, which we can obtain by solving the limiting form of (10-6) lor Re —> oo, namely,... 

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