An official website of the United States government
The turbulent channel flow database is produced from a direct numerical simulation (DNS) of wall bounded flow with periodic boundary conditions in the longitudinal and transverse directions, and no-slip conditions at the top and bottom walls. In the simulation, the Navier-Stokes equations are solved using a wall {normal, velocity {vorticity formulation. Solutions to the governing equations are provided using a Fourier-Galerkin pseudo-spectral method for the longitudinal and transverse directions and seventh-order Basis-splines (B-splines) collocation method in the wall normal direction. De-aliasing is performed using the 3/2-rule [3]. Temporal integration is performed using a low-storage, third-order Runge-Kutta method. Initially, the flow is driven using a constant volume flux control (imposing a bulk channel mean velocity of U = 1) until stationary conditions are reached. Then the control is changed to a constant applied mean pressure gradient forcing term equivalent to the shear stress resulting from the prior steps. Additional iterations are then performed to further achieve statistical stationarity before outputting fields.
more »
« less
Wavelet‐Based Wavenumber Spectral Estimate of Eddy Kinetic Energy: Idealized Quasi‐Geostrophic Flow
Abstract A wavelet‐based method is re‐introduced in an oceanographic and spectral context to estimate wavenumber spectrum and spectral flux of kinetic energy and enstrophy. We apply this to a numerical simulation of idealized, doubly periodic quasi‐geostrophic flows, that is, the flow is constrained by the Coriolis force and vertical stratification. The double periodicity allows for a straightforward Fourier analysis as the baseline method. Our wavelet spectra agree well with the canonical Fourier approach but with the additional strengths of negating the necessity for the data to be periodic and being able to extract local anisotropies in the flow. Caution is warranted, however, when computing higher‐order quantities, such as spectral flux.
more »
« less
- PAR ID:
- 10402572
- Publisher / Repository:
- DOI PREFIX: 10.1029
- Date Published:
- Journal Name:
- Journal of Advances in Modeling Earth Systems
- Volume:
- 15
- Issue:
- 3
- ISSN:
- 1942-2466
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
A new type of ensemble Kalman filter is developed, which is based on replacing the sample covariance in the analysis step by its diagonal in a spectral basis. It is proved that this technique improves the approximation of the covariance when the covariance itself is diagonal in the spectral basis, as is the case, e.g., for a second-order stationary random field and the Fourier basis. The method is extended by wavelets to the case when the state variables are random fields which are not spatially homogeneous. Efficient implementations by the fast Fourier transform (FFT) and discrete wavelet transform (DWT) are presented for several types of observations, including high-dimensional data given on a part of the domain, such as radar and satellite images. Computational experiments confirm that the method performs well on the Lorenz 96 problem and the shallow water equations with very small ensembles and over multiple analysis cycles.more » « less
-
Abstract Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $$\mathcal L u = f$$. The framework posits the existence of a left-Fredholm regulator for $$\mathcal L$$ and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to Fourier finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann–Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients.more » « less
-
Nguyen, Dinh-Liem; Nguyen, Loc; Nguyen, Thi-Phong (Ed.)This paper is concerned with the numerical solution to the direct and inverse electromagnetic scattering problem for bi-anisotropic periodic structures. The direct problem can be reformulated as an integro-di erential equation. We study the existence and uniqueness of solution to the latter equation and analyze a spectral Galerkin method to solve it. This spectral method is based on a periodization technique which allows us to avoid the evaluation of the quasiperiodic Green's tensor and to use the fast Fourier transform in the numerical implementation of the method. For the inverse problem, we study the orthogonality sampling method to reconstruct the periodic structures from scattering data generated by only two incident fields. The sampling method is fast, simple to implement, regularization free, and very robust against noise in the data. Numerical examples for both direct and inverse problems are presented to examine the efficiency of the numerical solvers.more » « less
-
Abstract We formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of time-stepping the initial value problem are proposed, an explicit Runge–Kutta (ERK) method and an exponential time-differencing (ETD) scheme. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.more » « less
