One, Two and Three Dimensional Flows

Term one, two or three dimensional flow refers to the number of space coordinated required to describe a flow. It appears that any physical flow is generally three-dimensional. But these are difficult to calculate and call for as much simplification as possible. This is achieved by ignoring changes to flow in any of the directions, thus reducing the complexity. It may be possible to reduce a three-dimensional problem to a two-dimensional one, even an one-dimensional one at times.

Figure 3.2 : Example of one-dimensional flow

 

Consider flow through a circular pipe. This flow is complex at the position where the flow enters the pipe. But as we proceed downstream the flow simplifies considerably and attains the state of a fully developed flow. A characteristic of this flow is that the velocity becomes invariant in the flow direction as shown in Fig.3.2. Velocity for this flow is given by

$$u~=~u_{max} \left[1~-~\left( r \over R \right)^2 \right]$$ (3.6)

It is readily seen that velocity at any location depends just on the radial distance $r$ from the centreline and is independent of distance, x or of the angular position $\theta$. This represents a typical one-dimensional flow.

Now consider a flow through a diverging duct as shown in Fig. 3.3. Velocity at any location depends not only upon the radial distance $r$ but also on the x-distance. This is therefore a two-dimensional flow.

Figure 3.3: Example of a two-dimensional flow

 

Concept of a uniform flow is very handy in analysing fluid flows. A uniform flow is one where the velocity and other properties are constant independent of directions. we usually assume a uniform flow at the entrance to a pipe, far away from a aerofoil or a motor car as shown in Fig. 3.4.

Figure 3.4 : Uniform Flow

 

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney