(Fluid Mechanics-Lecture 6 (Quasi-One-Dimensional Flow

Quasi-One-Dimensional Flow
In quasi-one-dimensional ?ow, in addition to ?ow conditions, the area of duct also changes with x. The
governing equations for quasi-one-dimensional ?ow can be written in a differential form as follows
using an in?nitesimal control volume
Some very useful physical insight can be obtained from this area?velocity relation.
¥ For subsonic ?ow (0 £ M < 1), an increase in area results in decrease in velocity, and vice versa.
¥ For supersonic ?ow (M > 1), an increase in area results in increase in velocity, and vice versa.
¥ For sonic ?ow (M = 1), dA/A = 0, which corresponds to a minimum or maximum in the area
distribution, but it can be shown that a minimum in area is the only physical solution.
Nozzle Flow
Using the area relations, we can now plot the distributions of Mach number and pressure along a nozzle.
Figure 3.7.9 shows pressure and Mach number distributions along a given nozzle and the wave con?gurations
for several exit pressures. For curves a and b, the ?ow stays subsonic throughout and the exit
pressure controls the ?ow in the entire nozzle. On curve c, the throat has just become sonic, and so the
pressure at the throat, and upstream of it, can decrease no further. There is another exit pressure
corresponding to curve j (pj < pc) for which a supersonic isentropic solution exists. But if the pressure
lies between pc and pj, there is no isentropic solution possible. For example, for an exit pressure pd, a
shock will form in the nozzle at location s which will raise the pressure to pd¢ and turn the ?ow subsonic.
The pressure will then rise to pd as the subsonic ?ow goes through an increasing area nozzle. The
location, s, depends on the exit pressure. Various possible situations are shown in Figure 3.7.9. It is clear
that if the exit pressure is equal to or below pf , the ?ow within the nozzle is fully supersonic. This is