Vector, vorticity

Here (0 is the magnitude of the vorticity vector, which is directed along the z axis. An irrotational flow is one with zero vorticity. Irro-tational flows have been widely studied because of their useful mathematical properties and applicability to flow regions where viscous effects m be neglected. Such flows without viscous effec ts are called in viscid flows. 

Develop a general expression for the vorticity vector that characterizes this flow field. Based on the vorticity field, what can be said about this flow ... 

The term (ui V) V, which is called vortex stretching, originates from the acceleration terms (2.3.5) in the Navier-Stokes equations, and not the viscous terms. In two-dimensional flow, the vorticity vector is orthogonal to the velocity vector. Thus, in cartesian coordinates (planar flow), the vortex-stretching term must vanish. In noncartesian or three-dimensional flows, vortex stretching can substantially alter the vorticity field. 

A further reduction of the vorticity equation is possible by restricting attention to two-dimensional flows. Here, since the vorticity vector is orthogonal to the velocity vector, the term (u> V) V vanishes. To retain the two-dimensional flow, the body force f must remain two-dimensional. 

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... 

In a general hydrodynamic system, the vorticity w is perpendicular to the velocity field v, creating a so-called Magnus pressure force. This force is directed along the axis of a right-hand screw as it would advance if the velocity vector rotated around the axis toward the vorticity vector. The conditions surrounding a wing that produce aerodynamic lift describe this effect precisely (see Fig. 2). 

The vorticity vector, w, is a measure of rotational effects, being equal to twice the local rate of rotation (angular velocity) of a fluid particle (i.e., uj = curl(v) = rot(v) = Vx V = 2Q) [168]. Many flows have negligible vorticity, uj curl(v) 0, and are appropriately called irrotational flows. 

Problem 7-9. Motion of a Force- and Torque-Free Axisymmetric Particle in a General Linear Flow. We consider a force- and torque-free axisymmetric particle whose geometry can be characterized by a single vector d immersed in a general linear flow, which takes the form far from the particle y°°(r) = U00 + r A fl00 + r E00, where U°°, il, and Ex are constants. Note that E00 is the symmetric rate-of-strain tensor and il is the vorticity vector, both defined in terms of the undisturbed flow. The Reynolds number for the particle motion is small so that the creeping-motion approximation can be applied. 

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... 

We may note also that = — smniNi. In our system, the tensor Mis the velocity gradient tensor, T Hence, tr M = tr T = 0, so that S = 0, and H is just the rate-of-strain tensor E. The vector N is the vorticity vector go. Hence, expressing T in terms of irreducible tensors reduces to writing it in terms of E and go ... 

It turns out that there are significant differences in some aspects of turbulence between three- and two-dimensional systems. As the vorticity vector points in the direction perpendicular to the plane of the flow u> T v, fluid motion can be fully described by a scalar field u (x,y). A consequence of two-dimensionality is that the vortex stretching term u> Vv vanishes in the vorticity equation (1.11) that becomes... 

Equation (8-78) is also known as the equation of change for fluid angular velocity in the low-Reynolds-number limit for incompressible Newtonian fluids because V X V, which is twice the vorticity vector, yields twice the angular velocity vector of a solid that rotates at constant angular velocity. The summation representation of the Laplacian of the vorticity vector. 

For solid-body rotation at constant angular velocity, the vorticity vector, defined by I (V X v), is equivalent to the angular velocity vector of the solid. For two-dimensional flow in cylindrical coordinates, with Vr(r,0) and V0 r,9), the volume-averaged vorticity vector. 

Potential flow in liquids implies that there are no rotational tendencies within the fluid, especially near a boundary. The microscopic description of potential flow, given by (8-199), requires that the vorticity vector must vanish. The macroscopic... 

Obviously, the second-term on the right side of this identity vanishes for ideal fluid flow in which the vorticity vector vanishes. Hence, for incompressible fluids with constant density. 

One solves for the stream function by invoking no vorticity at the microscopic level. Since Vr and vg are both functions of r and 9, with = 0, the r and 9 components of the vorticity vector are ffivially zero. The (/>-component of (V X v) yields an equation that must be solved for ir(r,9). Hence, one combines the nontrivial component of the vorticity vector with the relations between Vr, vg and f, given by (8-239) ... 

It is left as an exercise for the student to verify that these relations between the two nonzero velocity components and ilr conserve overall fluid mass, and that streamlines intersect equipotentials at right angles in cylindrical coordinates. The stream function is obtained by invoking no vorticity at the microscopic level. Only the z-component of the fluid vorticity vector yields nontrivial information about ir. For example. 

The velocity vector for rigid-body rotation of a solid that spins at constant angular velocity is v = ft x r, where ft is the angular velocity vector and r is the position vector from the axis of rotation. Obtain an expression for the vorticity vector V x v for rigid-body rotation in terms of ft. 

Answer Consult the three scalar components of the vorticity vector shown in Bird et al. (2002, p. 836). For one-dimensional flow in the (p direction, as described in part (a), one obtains the following result ... 

Since there are two nonzero components of the vorticity vector, the r and 9 components of the Laplacian of the vorticity vector will yield... 

Answer Approach (i) requires that one must solve the ( -component of the equation of motion. Since is a function of both r and 9, there are two nonzero components of V x v (i.e., the r and 9 components are nontrivial). Hence, approach (ii) requires that one must consider the r and 9 components of the Laplacian of the vorticity vector to obtain an expression for the stream function via the low-Reynolds-number equation of change for fluid vorticity. The preferred approach is (1). 

Answer The 9 component. Consider all three components of the vorticity vector in cylindrical coordinates when the velocity vector field... 

There are no dimensionless numbers in this potential flow equation because convective forces per unit volume and dynamic pressure forces per unit volume both scale as pV /L. Furthermore, potential flow theory provides the formalism to calculate p and the dimensionless scalar velocity potential such that the vorticity vector vanishes and overall fluid mass is conserved for an incompressible fluid. Hence,... 

Another method consists in the exact solution of the problem of motion of two spherical particles at any value of a/l. The solution is derived in a special bipolar system of coordinates [5). As an illustration of the method, consider the motion of two identical solid spheres with constant and equal velocities along the line of centers. Introduce the vorticity vector... 

---