Two-dimensional transonic potential flows are expressed in the form of non-linear partial differential equation in the physical plane. This equation is a mixed type as elliptic and hyperbolic. Therefore, it is almost impossible to obtain analytical solutions of the partial differential equation. Furthermore, we can not know previously the sonic-line in the physical plane. This is one of the reasons why numerical calculations become difficult.
Following method is divided into two groups, calculations of subsonic flows and of supersonic flows.
At first, an artifical function of the Mach number M which can be determined arbitrarily is introduced in M>1 region of the differential equation. This function of M changes the partial differential equation from the mixed type into the elliptic type, however does not show natural law of the gaseous flows.
The converted partial differential equation is solved in the boundary value problem. A subsonic flow and an artifical flow which is due to the introduced artifical function of M around a given arbitrary aerofoil section can be obtained. A sonic-line is determined as boundary of the both flows.
Next, under initial conditions along the sonicline, the original partial differential equation which is the hyperbolic type in M>1 region is solved in the CAUCHY problem. A supersonic flow is obtained and the aerofoil section must be changed partially. The artifical flow is been superseded by this supersonic flow. The subsonic flow and the oupersonic flow must be continuous for the sonic-line.
As a result of those problems, shockless transonic flows around aerofoil sections which are partially different from a given arbitrary aerofoil section can be obtained.
At least, this method could be put into practice numerically. Some numerical examples were obtained in the way described above.
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