3D Mean-Stream-Line Method: A New Engineering Approach to the Inverse Problem of 3D Cascade

Abstract

The Mean-Stream-Line Method proposed by Wu and Brown (1952) for 2D cascade is developed and extended to solve the 3D inverse problem of turbomachine flow. The equations suitable to the 3D case are derived and the well-posed design conditions are discussed within the scope of the annular wall constraint of turbomachines mentioned by Cai (1983). A computer code has been programmed for this method and computation results compare well with 3D incompressible potential analytical solutions. These solutions, discovered by Cai et al. (1984), are similar to turbomachine flow. A comparison of computation results between this method and the conventional 3D potential function method is also presented. However, in solving the inverse problem, the 3D Mean-Stream-Line Method is much more simple and faster than all other methods already available. In addition, some ideas to solve direct and hybrid problems are also suggested.

Discussion on the convective heat transfer and field synergy principle

“…(12)- (16) with all previous results in this paragraph, the final solution can be expressed as follows. Because c 3 + c 4 always appears together in the final result, c 3 is chosen on behalf of c 3 + c 4 .…”

“…For example, several analytical solutions that can simulate the 3D potential flow in turbomachine cascades were obtained by Cai et al [14]. And they were successfully utilized by some investigators in their numerical calculation to check their computational techniques and computer codes [14][15][16][17].…”

“…In the following derivation procedure, the above-mentioned equations (12)- (16) are frequently applied to obtain algebraically explicit exact solutions for field synergy condition. In addition, a hybrid approach with both separating method is also applied in this paper.…”

Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

“…Exact solutions are therefore very useful even for the newly rapidly developing computational fluid dynamics and heat transfer. For example, several exact solutions which can simulate the 3-D potential flow in turbomachine cascades were obtained by Cai et al [17], and were successfully used by some investigators in their numerical calculation to check their computational techniques and computer codes [17][18][19][20]. For other related important publications concerning the importance of exact solutions, please refer to [21][22][23][24][25].…”

Algebraically explicit analytical solutions of unsteady 3-D nonlinear non-Fourier (hyperbolic) heat conduction

“…The analytical solutions are therefore very useful even for the newly rapidly developing computational heat conduction. For example, in the fluid dynamics field, several analytical solutions that simulate the 3-D potential flow in turbomachine cascades were given by one of the authors [3]; these solutions have been used successfully by independent researchers to verify the accuracy and exactness of their computational methods [4][5][6].Several algebraically explicit analytical solutions of unsteady nonlinear non-Fourier heat conduction are derived in this paper, both to develop the theory and to serve as benchmark solutions for numerical calculations. …”