2.1 Navier-Stokes equations

Consider the two-dimensional flow of a homogenous and incompressible fluid. The density and the viscosity of the fluid are both assumed to be uniform. We assume that any body forces on the fluid are derived as a gradient of a scalar function. The governing equations for the motion of the fluid are the conservation of mass and linear momentum [14].

The mass conservation equation is

$$\nabla \cdot \vec{u}\;=\;0 \quad \ ,$$ (2.1)

where $\vec{u}$ is the velocity and $\nabla$ is the gradient operator. We also denote $\vec{x}=(x,y)$ to be any point in the plane and $\hat{x}$ and $\hat{y}$ to be the unit vectors along the axes.

The linear momentum conservation for a Newtonian fluid is given by the Navier-Stokes equations [14],

$$\frac{\partial \vec{u}}{\partial t} \; + \; \vec{u} \cdot \... ...}{\rho}\,\nabla p \;+\; \nu\,\nabla^2 \vec{u} \;+\; \vec{F}\ ,$$ (2.2)

where $t$ is time; $p$ is mechanical pressure; $\vec{F}$ is body force per unit mass of the fluid; $\nu$ is kinematic viscosity, defined as the ratio of the dynamic viscosity and the density of the fluid and $\nabla^2$ is the Laplacian operator.

The equation for the the evolution of vorticity can be derived from the Navier-Stokes equations (2.2). To do that, we first define the vorticity $\vec{\omega}$ to be the curl of the flow velocity,

$$\vec{\omega} \equiv \nabla \times \vec{u} \quad \ .$$ (2.3)

For two-dimensional flows, the vorticity vector is normal to the plane of the flow; that is, $\vec{\omega}=\omega \hat{z}$, where $\hat{z}=\hat{x}\times\hat{y}$ is an unit vector normal to the plane. The vorticity equation is obtained by taking the curl of (2.2) and it is given by:

$$\frac{\partial \omega}{\partial t} \; = \; - \vec{u} \cdot \nabla \omega + \nu\,\nabla^2 \omega \ .$$ (2.4)

The physical interpretation of each of the terms in the vorticity equation (2.4) is the basis for the formulation of vortex methods. On the right hand side of (2.4) the first term represents the transport of vorticity due to the velocity (convection process), and the second term represents the change in vorticity due to viscosity (diffusion process) [14]. Truesdell [225] has described the convection and diffusion processes in detail from a kinematic point of view.

To solve (2.4) for a particular problem, initial and boundary conditions must be specified. The initial vorticity field may be prescribed or it may also be derived as the curl of a specified initial velocity field [179]. Boundary conditions must be specified when there are boundaries in a flow. On a solid impermeable boundary, the velocity of the fluid on the boundary must be the same as the velocity of the boundary itself [14],

$$\vec{u}(\vec{s},t)\;=\;\vec{u}_s(\vec{s},t) \quad \ ,$$ (2.5)

where $\vec{s}$ is any point on the boundary and $\vec{u}_s$ is the velocity of the boundary. Notice that the boundary condition (2.5) is in terms of velocity and not vorticity; we will discuss the handling of this boundary condition in section 6.3. Further, in many applications the flow domain is unbounded and at large distances the velocity is either uniform or vanishes; hence the vorticity vanishes at large distances also.

A number of theoretical studies have investigated the validity of the vorticity formulation of the Navier-Stokes equations. McGrath [150] showed that for flows in free space the vorticity equation has an unique solution for any finite time if the initial vorticity is smooth (twice differentiable). A similar result for singular initial vorticity distributions (that are absolutely integrable) has been established by Benfatto, Esposito & Pulvirenti [21], Giga, Miakawa & Osada [87], Ben-Artzi [20,31] and Kato [114].

Guermond & Quartapelle [102] and Quartapelle [179] have shown that the the vorticity formulation is equivalent to the velocity-pressure form of Navier-Stokes equations. Gresho [98,99,100] discusses a number of theoretical and computational issues for the vorticity formulation for incompressible flows.

Vortex methods are based on the Lagrangian approach in which the ``fluid particles" are used as the basic computational elements [14]. Here the fluid particles are understood to be small volumes of fluid. To be precise, particles are volumes of fluid that are much smaller than all relevant length scales of the flow but still much larger than the molecular size and mean free-path length. The time derivative following a fluid particle is defined as

$$\frac{D}{Dt} \equiv \frac{\partial}{\partial{t}} + \vec u \cdot \nabla \quad \ .$$ (2.6)

In terms of this ``Lagrangian time-derivative", we can rewrite the vorticity equation (2.4) as [14],

$$\frac{D \omega}{Dt} \; = \; \nu\,\nabla^2 \omega \quad \ .$$ (2.7)

According to (2.7), the vorticity of a fluid particle changes only due to diffusion. In inviscid flows $(\nu=0)$ the vorticity of a fluid particle does not change [14,137]; this result is very useful in formulating vortex methods described in the next chapter.

However, the vorticity equation is only one equation for three unknowns, $\omega$, $u$, and $v$, and we need equations to determine the velocity field also; we will formulate the equations for the velocity field next.