Spectrum of Turbulence in a Contracting Stream

The spectrum concept is employed to study the selective effect of a stream contraction on the longitudinal and lateral turbulent velocity fluctuations of the stream. By a consideration of the effect of the stream contraction on a single plane sinusoidal disturbance wave, mathematically not dissimilar to a triply-periodic disturbance treated by G.I. Taylor, the effect on the spectrum tensor of the turbulence and hence on the correlation tensor are determined. Lack of interference between waves follows from the postulation of a low level of turbulence; this and the assumption of an inviscid fluid imply neglect of decay effects. The compressibility of the main stream is taken into account, but the density fluctuations associated with the turbulence is assumed to be negligible; this would be the case if the turbulence originated from wakes and boundary layers in the very low speed portion of the flow. For an axisymmetric contraction and a particular isotropic initial turbulence some explicit results are obtained. The one-dimensional longitudinal spectrum is found to be distorted (as well as reduced in amplitude) with its peak shifted well to the right of the initial position above the zero of the wave-number scale. The selective effect of the contraction on the mean square longitudinal and lateral components of turbulent velocity is found to be given uniquely when the initial turbulence is isotropic, regardless of the details of the spectrum. If the initial spectrum is anisotropic, as, for instance, that produced by a damping screen, then the selective effect is altered. In a crude extension, decay effects outside the scope of the theory are allowed for in first approximation. With this extension, a comparison with experiment is made of the selective effect on turbulent intensity where the estimated decay effects are comparable with the contraction effects.