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51-centimeter acoustic power airfoil analysis Annular duct axial velocity blade force blade passing frequency blade row BPF tone Branch cut cascade circumferential direction combined quadrupole-dipole model complex conjugate coordinate system defined by eq denotes dipole contribution dipole source distortion profile eigenfunctions entering the fan Equations 70 flow distortion noise force per unit Fourier coefficient Fourier series full-scale fan stages fundamental tone Gaussian profile Green's function harmonic Hence individual blade inlet flow distortion inlet turbulence inserting equations integration with respect interblade spacing Lewis Research Center lift force Marvin E measured model fan rotor NASA nonuniform number of blades potential-flow field predict pressure field quadrupole contribution quadrupole source quiet fans tested radial distortion radiated sound power ratio relative velocity result in equation Reynolds stress rotational plane rotor velocity field shown in figure shown in reference shows sound field source term square wave subsonic fan theory tion various quiet fans vortices width