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An exact representation of the nonlinear triad interaction terms in spectral space. (English) Zbl 1309.76101
Summary: Spectral analysis of the Navier-Stokes equations requires treatment of the convolution of pairs of Fourier transforms \(\hat{f}\) and \(\hat{g}\). An exact, tractable representation of the nonlinear terms in spectral space is introduced, and relies on the definition and manipulation of a combination matrix. A spectral energy equation is derived where the nonlinear triad interactions are expressed using the combination matrix. The formulation is applied to the problem of homogeneous, isotropic turbulence. By finding the solution in an appropriate canonical basis, the energy spectrum in the inertial range \(E(k)\sim \epsilon^{2/3}k^{-5/3}\) is derived from the Navier-Stokes equations without invoking dimensional scaling arguments.

MSC:
76F05 Isotropic turbulence; homogeneous turbulence
76D05 Navier-Stokes equations for incompressible viscous fluids
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