To illustrate what might happen in an unsteady one-dimensional compressible flow, figure 12.18 shows a possible simple unsteady flow of a gas within a very long pipe. The gas next to the piston moves at the same velocity as the piston, but gas further to the left of the piston moves more slowly because it has not yet experienced the full drop in pressure needed to accelerate it to the speed . This initiating sound wave reaches -x at the time . At any time t , the gas that lies in the range is in a state of unsteady flow, as shown in figure 12.18 . For this inviscid compressible flow, there is only one velocity component, .
All flow variables depend upon only two independent variables, x and t .
It is preferable to introduce the sound speed a as a variable in place of the density . Just as the time derivative following a material particle is , the time derivative following a right-running sound wave is . By the same token, the differential operator of equation 12.70 is the time derivative along a left-running sound wave. In the plane, the directions are the local characteristic directions . To illustrate how equations 12.72 are used to solve an unsteady compressible flow problem, let us consider the flow illustrated in figure 12.18 where a moving piston accelerates an initially stationary gas.