A wave theory of viscous
flows

L. S. Yao, Arizona State University

Analysis shows that a solution of the Navier-Stokes equations can be expressed solely in terms of waves. If the definition of waves is enlarged to include the mean flow as a wave of infinitely long wavelength as a special case of the Goldstone theorem, a laminar flow contains only one wave, i.e., the mean flow. With a supercritical instability, there is a mean flow as well as a dominant wave and its harmonics. For turbulence, there are many waves. The difference between turbulent and laminar flows is simply the number of excited waves whose amplitudes are appreciable. In this scenario, the amplitude density function of the waves is determined by linear and non-linear terms. The linear case is the target of flow-instability study and is the foundation of rapid distortion theories of turbulence. The nonlinear case involves energy transfer among the waves satisfying resonance conditions so that it is a discrete process. This implies that no wave from a continuous spectrum can exist, which is an unexpected result for viscous flows. Since the discrete wave numbers form a denumerable set, infinitely many sets can exist. The initial conditions determine which set will be excited in any particular situation. This means that viscous flows above the critical Reynolds number have multiple solutions. This phenomenon has been observed for many flows, such as Taylor-Couette instability, turbulent mixing layers, wakes, jets, pipe flows, etc.

The conclusions that traveling waves form the fundamental solutions and non-linear terms represent energy transfer resonantly are not limited to the Navier-Stokes equations and can be extended to other non-linear partial differential equations.

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