Navier-Stokes equation Fourier-transformed

Few articles on receptivity present a qualitative view of particular transition routes created by not so well-defined excitation field (see e.g. Saric et al. (1999)). Such approaches do not demonstrate complete theoretical and /or experimental evidence connecting the cause (excitation field) and its effect(s) (response field). Here, a model based on linearized Navier- Stokes equation is presented to show the receptivity route for excitation applied at the wall. This requires a dynamical system approach to explain the response of the system with the help of Laplace-Fourier transform. 

The summation is over all the spanwise modes. One can use the above ansatz in three-dimensional Navier-Stokes equation and linearize the resultant equations after making a parallel flow approximation to get the following Orr-Sommerfeld equation for the Fourier- Laplace transform f of v ... 

Further contributions to the subject were made by Taylor in 1938. Two important consequences of the non-linearity of the Navier-Stokes equations were identified First, the skewness of the probability distribution of the difference between the velocities at two points, and the existence of an interaction or modulation between components of turbulence having different length scales. Secondly, the Fourier transform of the correlation between two velocities is an energy spectrum function in the sense that it describes the distribution of kinetic energy over the various Fourier wave-number components of the turbulence [164]. Taylor expressed in mathematical form the relation between the correlation function and the ID spectrum function. 

However, the theoretical interpretation of LES is that we are simulating the Fourier-transformed Navier-Stokes equation with its wavenumber representation truncated to the interval 0 < k < kc In accordance with the theory of... 

We present a brief introduction to coupled transport processes described macroscopically by hydrodynamic equations, the Navier-Stokes equations [4]. These are difficult, highly non-linear coupled partial differential equations they are frequently approximated. One such approximation consists of the Lorenz equations [5,6], which are obtained from the Navier-Stokes equations by Fourier transform of the spatial variables in those equations, retention of first order Fourier modes and restriction to small deviations from a bifurcation of an homogeneous motionless stationary state (a conductive state) to an inhomogeneous convective state in Rayleigh-Benard convection (see the next paragraph). The Lorenz equations have been applied successfully in various fields ranging from meteorology to laser physics. 

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