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**A hodograph-based method for the design of shock-free cascades.**
*(English)*
Zbl 0617.76064

A hodograph-based method, originally developed by the first author [Commun. Appl. Numer. Methods 2, 37-45 (1986; Zbl 0604.76043)] for the design of shock-free aerofoils, has been modified and extended to allow for the design of shock-free compressor blades. In the present procedure, the subsonic and supersonic regions of the flow are decoupled, allowing the solution of either an elliptic or a hyperbolic-type partial differential equation for the stream function. The coupling of both regions of the flow is carried out along the sonic line which adjoins both regions. For the subcritical portion of the flow considered here, the pressure distribution is prescribed in addition to the upstream and downstream flow conditions.

For the supercritical portion of the flow, the stream function on the sonic line is given instead of the supercritical pressure distribution which is found as part of the solution. In the special hodograph variables used, the equation for the stream function is solved iteratively using a second-order accurate line relaxation procedure for the subsonic portion of the flow. For the supercritical portion of the flow, a characteristic marching procedure in the hodograph plane is used to solve for the supersonic flow. The results are then mapped back to the physical plane to determine the blade shape and the supercritical pressures. Examples of shock-free compressor blade designs are presented. They show good agreement with the direct computation of the flow past the designed blade.

For the supercritical portion of the flow, the stream function on the sonic line is given instead of the supercritical pressure distribution which is found as part of the solution. In the special hodograph variables used, the equation for the stream function is solved iteratively using a second-order accurate line relaxation procedure for the subsonic portion of the flow. For the supercritical portion of the flow, a characteristic marching procedure in the hodograph plane is used to solve for the supersonic flow. The results are then mapped back to the physical plane to determine the blade shape and the supercritical pressures. Examples of shock-free compressor blade designs are presented. They show good agreement with the direct computation of the flow past the designed blade.

### Keywords:

hodograph-based method; shock-free aerofoils; shock-free compressor blades; supersonic regions; hyperbolic-type partial differential equation; stream function; sonic line; downstream flow conditions; hodograph variables; second-order accurate line relaxation procedure; marching procedure; hodograph plane; shock-free compressor blade designs### Citations:

Zbl 0604.76043### Software:

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\textit{A. A. Hassan} and \textit{G. S. Dulikravich}, Int. J. Numer. Methods Fluids 7, 197--213 (1987; Zbl 0617.76064)

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### References:

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