2.2 Velocity field

The velocity field determines the motion of the vorticity field. On the other hand, it turns out that we can find the velocity field from the vorticity field; in the following, we describe this.

The mass conservation equation (2.1)

$$\nabla \cdot \vec{u}\;=\;0 \quad \ ,$$ (2.8)

can be satisfied using a scalar function $\psi(\vec{x},t)$ called the stream function [14] such that

$$\nabla\times\psi \hat{z} = \vec{u} \quad \ .$$ (2.9)

Equivalently, (2.9) implies that the velocity components $(u,v)$ are given by,

$$u = \frac{\partial \psi}{\partial y}$$ (2.10)

$$-v = \frac{\partial \psi}{\partial x}\quad \ .$$ (2.11)

We substitute (2.10) and (2.11) in the definition of vorticity (2.3) and obtain the following Poisson equation for the stream function,

$$\nabla^2 \psi \; = \; - \omega \quad \ .$$ (2.12)

We can solve (2.12) to find the stream function and then the velocity field using (2.9). A standard approach to solve the Poisson equation (2.12) is the Green's function method [13,95]. Using this method, for flows in free space (no boundaries) we can obtain $\psi$ from (2.12) as,

$$\psi(\vec{x},t) = \int\int\, G(\vec{x};\vec{x}')\: \omega(\vec{x}',t) \; dx'\,dy'$$ (2.13)

$$\equiv G\ast \omega \quad \ ,$$

where $\ast$ denotes the convolution operation [13] and $G$ is the free space Green's function, also known as fundamental solution, of the Laplace equation. This Green's function satisfies

$$\nabla^2_{\vec{x}} G(\vec{x};\vec{x}') \; = \; \delta(\vec{x}-\vec{x}') \quad \ ,$$ (2.14)

where $\delta(\cdot)$ is the two-dimensional Dirac delta function; and $\nabla^2_{\vec{x}}$ is the Laplacian operator in which the derivatives are with respect to $\vec{x}$. The actual form of $G$ is [13,95],

$$G(\vec{x};\vec{x}') \; = \; \frac{1}{2 \pi}\; {\rm ln}(\vert\,\vec{x}-\vec{x}'\,\vert) \quad \ .$$ (2.15)

Using (2.13) in (2.9), we obtain the velocity field

$$\vec{u}(\vec{x},t) = \int\int\; \vec{K}(\vec{x};\vec{x}')\; \omega(\vec{x}',t)\; dx'\,dy'$$ (2.16)

$$\equiv \vec{K}\ast\omega \quad \ ,$$

where $\vec{K}$ is given by

$$\vec{K}(\vec{x};\vec{x}') = \nabla_{\vec{x}}\times G(\vec{x};\vec{x}')\hat{z}$$ (2.17)

$$= \frac{1}{2\,\pi\,\vert\,\vec{x}-\vec{x}'\,\vert^2 }\: \left( \begin{array}{c} y-y'\\ x'-x \end{array}\right) \quad \ .$$ (2.18)

The equations (2.16) and (2.18) for the velocity are known as the Biot-Savart law [109,126]. The function $\vec{K}$ is also called the kernel [11], the Biot-Savart Kernel [109] or the velocity kernel [17].

The velocity given by (2.16) is for free space flows since we used the free space Green's function in (2.13). For flows over solid boundaries, this velocity can be corrected to satisfy the boundary condition (2.5); we will describe that in section 6.3.

The vorticity equation (2.7), the velocity equation (2.16), the initial condition and the boundary conditions together describe the evolution of the vorticity. Next we review the conservation laws derived from the vorticity equation.