Dynamics of the coherent structure for incompressible fluid flow in turbulent boundary layers

Abstract

We consider the nonlinear interaction system of waves to identify discrete clusters of resonant triads, which are classified on the basis of the resonance condition. This study is conducted to investigate the coherent structure of incompressible fluid flow in the turbulent boundary layer. The discrete wave turbulence is characterised by weakly nonlinear interaction modes for amplitude Tollmien Schlichting in a single-mode approximation. Within the framework of multiple-scale analysis, the coherent part of the amplitude equation is defined in the case of multiple three-wave resonance. The resonance condition is defined from the dispersion relation of these amplitudes, which are determined from solving the spectral problem of Orr–Sommerfeld equation by the Chebyshev collocation method. The spectral characteristics of these amplitudes are investigated to define the condition of multiple three-wave resonance. This condition is also defined for a resonant cluster made out of triads sharing a common mode, where all triads satisfy the resonance condition. The coherent amplitudes are represented dynamically by an autonomous system of ordinary differential equations. Non-integrable system of a single triad and a cluster of triads is noted, where the interaction coefficients in the given system do not have the same complex phase. The obtained dynamical system admits a number of invariants, similar to the classical Manley–Rowe invariants but of a different nature. One of these invariants is called the energy invariant manifold that represents the sum of modules square amplitudes of the dynamical system, this invariant is normalised to be defined on the unit sphere. Therefore, Birkhoff–Khinchin theory is applied to calculate the time average of square harmonic and sub harmonic amplitudes. Moreover, this paper is also focused on studying the numerical solutions of both simple and complex structure of the dynamical system by using Runge–Kutta method with random initial conditions. The solution of the dynamical system is examined at different signs of the weight factors, where the bounded solutions of this system are found at both positive and negative signs. However, in another study of a dynamical system, an explosive instability is noted at a negative sign for only one of the weight factors, where all study cases are related to the choice of wave vectors. The random initial conditions are applied to both simple and complex dynamical system to study the behaviour of system solutions. The coupling different triads within the dynamical system lead to chaotic turbulence regime.