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Deriving the Vorticity Equation From the Conservation of Momentum Equation

Key Takeaways

  • Vorticity is the measure of the rate of local fluid rotation. 

  • If the vorticity at a point in the velocity field is non-zero, then the fluid element at that point is rotating. 

  • The first term on the right-hand side of the vorticity equation corresponds to the rate of change of vorticity due to the stretching and tilting of vortex lines. The second term represents the rate of change of vorticity due to diffusion. 

Vorticity and circulation

In fluid dynamics, vorticity and circulation are two terms often associated with rotational motion

In fluid dynamics, vorticity and circulation are two terms often associated with rotational motion. Vorticity is the measure of the rate of local fluid rotation, whereas circulation deals with global rotation. Vorticity is a first-ordered tensor in fluid dynamics and can be defined as the curl of the velocity vector. The vorticity equation is used to describe the changes in vorticity by various properties of the fluid flow. In this article, we will discuss vortices, vorticity, the vorticity equation, and the circulation of fluids. 

Vortices 

Vortex is a common phenomenon in smoke rings, winds, whirlpools, or any stirred fluids. It is an important factor in turbulent fluid flows. When there is vortex motion, the fluid motion leads to circular or nearly circular streamlines. In the presence of vortices, the velocity of fluid flow is at its maximum next to its axis and decreases as the distance from the axis increases. There is a circulation of fluid around the vortex. As the vortex is approached, speed increases and pressure decreases. If the angular velocity of the fluid flow is zero, then there is no rotating motion. 

Vorticity

Vorticity is a metric that indicates the local spinning of a fluid. The vorticity vector can be obtained as twice the angular velocity of the fluid particle. Both angular velocity and vorticity are first-ordered tensors. The vorticity vector is the curl of the velocity vector.  In terms of vorticity, a vortex is the concentration of co-directional or nearly co-directional vorticity.

In a fluid flow, vorticity is neither unidirectional nor steady. Vorticity is embedded in fluid elements such that re-orientation, diffusion, or concentration of vorticity occurs either due to motion and deformation of the fluid element or due to the torques applied to it by surrounding fluid elements.

Vorticity and velocity vectors are closely related. If the vorticity at a point in the velocity field is non-zero, then the fluid element at that point is rotating. If the vorticity is zero at a point in the velocity field, the fluid element is not rotating at that particular point. It can be summarized that vorticity is the measure of the local rotation of the fluid element. 

Vortex Lines

A curve in the fluid that is tangential to the vorticity vector is called a vortex line. The relationship between vortex lines and the vorticity vector is similar to the one between streamlines and the velocity vector. Vortex lines are absent in irrotational flows. 

Types of Vortices

In fluids, there are two types of vortices. They are:

  1. Forced vortex - In a forced vortex, the forces from external agencies are the cause. There are centripetal or centrifugal accelerations in the forced vortex. The fluid elements behave like a rotating solid body. 

  2. Free vortex - Free vortex is an irrotational vortex. In a free vortex, the velocity is inversely proportional to the distance from the center of the vortex. The center of the vortex is called the point of singularity, where the magnitude of the velocity approaches infinity. 

Circulation and Vorticity

Circulation is a quantity associated with vortices. In vortex dynamics, circulation can be obtained by taking the closed loop integral of the velocity vector field. Circulation is the flux of vorticity. Vorticity at a given point can be regarded as the circulation per unit area. When the vector quantity of vorticity corresponds to the local rotation of fluid elements, the scalar quantity of circulation gives the measure of global rotation. Vorticity is microscopic, whereas circulation is macroscopic. For a free vortex, the circulation is zero, excluding the point of singularity. The circulation becomes non-zero when including the point of singularity. In the case of the forced vortex,  the circulation is non-zero for any closed loop taken.

The Vorticity Equation  

Consider a fluid with constant density ⍴ so that the fluid flow is barotropic. Viscosity is considered a constant quantity; however, viscous effects are retained. As we have already discussed vorticity (ω) is the curl of velocity, the vorticity equation can be derived from the curl of conservation of the momentum equation given below: 

Curl of conservation of momentum equation

The vorticity equation of a fluid with constant density and conservation body forces can be given as:

Vorticity equation

The first term on the right-hand side of the vorticity equation corresponds to the rate of change of vorticity due to the stretching and tilting of vortex lines. The second term represents the rate of change of vorticity due to diffusion.

The vorticity equation given above is valid for a fluid with constant density and viscosity observed from an inertial frame of reference. The vorticity equation in the inertial frame can be generalized for a variable density fluid under a rotating frame of reference. Cadence can help you model fluid flows with non-zero and zero-vorticity vectors. Cadence offers you CFD solutions for fluid flows involving vorticity and circulations.

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