Dynamic Similarity

Dynamic Similarity exists between the model and the prototype when forces at corresponding points are similar (Fig. 5.6).

Figure 5.6: Dynamic Similarity

$$\left ( FA_v \over FA_h \right)_{prototype}~=~\left( Fa_v \over F... ...over FB_h \right)_{prototype}~=~\left ( Fb_v \over Fb_h \right)_{model},~~etc.$$ (5.18)

Of the three similarities, the dynamic similarity is the most restrictive.

The Dimensional Analysis that we have carried out in this chapter clearly indicates that to achieve dynamic similarity the all the non-dimensional numbers relevant to the flow must be preserved between the model and the prototype. In other words,

$${\Pi_1} _{model}~=~ {\Pi_1} _{prototype},~~ {\Pi_2} _{model}~=~ {\Pi_2} _{prototype},~~etc.$$ (5.19)

But it is common experience that this is something not easy to achieve. There seem to be many problems. One such is in the field of aerodynamics. In order to obtain total similitude between model and prototype it comes out that Reynolds Number and Mach Number should be the same between the model and the prototype. When one considers the inflight Reynolds numbers of today (10⁸ or near about) it appears that this is a difficult target with the wind tunnels in existence. A tradeoff is necessary. But nature comes to our rescue. It turns out that at lower speeds (Mach Number 0.3 and below) it is the Reynolds Numbers that are important and it is simply enough to preserve Reynolds number alone. At higher speeds where compressibility is important only Mach Number need be preserved.

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