Karhunen–Loéve expansion of Burgers’ model of turbulence

Abstract

Characteristics of the Karhunen–Loéve expansion of a strongly inhomogeneous random process possessing small viscous length scales and a large outer scale have been investigated in relation to the application of the expansion to turbulent flow fields. Monte Carlo simulations of a randomly forced Burgers’ equation with zero velocity boundary conditions generate the random process numerically and the Karhunen–Loéve (KL) eigenfunctions and the eigenvalue spectra are computed for different Reynolds numbers. The eigenfunctions possess thin viscous boundary layers at the walls and are independent of Reynolds number in the core, where the random process is quasihomogeneous. The eigenfunctions and eigenvalues of the outer, large scale motions obey a principle of Reynolds number similarity. Eigenvalue spectra contain much of the energy in the first few modes, but they are as broad as ordinary trigonometric power spectra. The rate at which the expansion converges to within 90% of the total energy decreases with increasing Reynolds numbers and the expansion of the mean plus the fluctuation converges more rapidly than the expansion of the fluctuation alone.

Neural Network Modeling for Near Wall Turbulent Flow

“…The randomly forced Burgers equation is known to generate solutions showing turbulentlike fluctuations [11]. It has been used as a benchmark problem for testing modeling and control techniques using neural networks [24], and for testing the POD [11].…”

“…It has been used as a benchmark problem for testing modeling and control techniques using neural networks [24], and for testing the POD [11].…”

“…However, the application of the POD to three-dimensional turbulent flows is hindered by the size of the computational problem; studies of the performance of the POD have shown that as the Reynolds number increases the convergence rate of the truncated POD expansion to a given fraction of the total flow energy decreases, requiring larger numbers of eigenfunctions to represent the flow adequately [11].…”

In situ acceleration in the Galactic Center Arc

“…In the vicinity of a strong magnetic distortion, like the Arc filaments, Burgers turbulence is expected to be excited (Chambers et al 1988). Thus, we choose a turbulence spectral index of q = 2.…”

Reduced-Order Models of Unsteady Transonic Viscous Flows in Turbomachinery

“…However, the techniques developed have been used to model both linear and nonlinear phenomena (Noor 1994;Stone & Cutler 1996). The dynamically linearized reduced-order modeling technique has been applied to a wide variety of systems, such as Burger's model of turbulence (Canuto et al 1988;Chambers et al 1988), full potential equation , Euler equations, Navier}Stokes equations (Deane et al 1988), Raleigh}BeH nard convection (Holmes et al 1996), turbulence, and boundary layer models (Liu et al 1994;Sirovich 1987a, b, c). Additionally, fully nonlinear normal modes and reduced-order models were also investigated for low-dimensional systems (Shaw & Pierre 1993.…”