Hydrodynamic Instability Investigation In Channel Flow With Periodical Spanwise Variation

The baseflow in the local instability theory can be described by a function with only one coordinate, or an approximate one. Therefore, the local instability theory can be used in Channel flow for it only relates with normal coordinate. On the contrary, if the flow also relates with another spatial coordinate or time, the global instability theory is often used to describe the unstable wave. For a special situation, whether and how the local instability theory can predict global instability characteristic is need to investigate.This paper investigates the local and global instability in the Channel Flow with spanwise variation by LST theory and numerical simulation, in order to find the relationship in the two theories.The numerical investigation is made for the global instability with the linear disturbances equation in temporal mode. Fourier spectral method in streamwise and spanwise, Chebyshev collocation point in normal discretization and two-order difference schemes in temporal discretization are adopted.At first , some linear perturbation is given to the flow field to calculate the time evolution, the eigenvalue is received after full development. Especially, in the global cases of neutral ones, the relationship between the variation of b and the Reynolds number, the eigenfunction in x-z plane perturbation contours and the comparison with the results by LST are all received. The following conclusions are found by calculation and analysis:(1) The instability waves of the global stability have exponential growth.(2) The spanwise variation in the base flow stabilize the disturbances. With b increasing, the global growth rates are smaller than the results by local LST.(3) In the lower branch of the global neutral curve ,with the increasing of b, the reynolds number begins to increase from 5772.3; In the upper one, with the increasing of b, the reynolds number begins to reduce from 26396.8. Also, the slope of Recr-b cuve will reduce.(4) From the results of the comparison between the calculation for global instability and the analysis for local instability, the results of global stability is neutral, while the forecast growth rates of local stability are greater than zero for a given Reynolds number. Therefore, for the channel flow with periodical spanwise variation, it can't be predict the global instability parameter only by the local linear stability theory.