We relate the justification of Kolmogorov’s hypotheses on the local isotropy and small-scale universality in real turbulent flows to an observed universality of basis independence for the global energy spectrum and energy flux of small-scale turbulence. To readily examine the small-scale universality, an approach is suggested that investigates the global energy spectrum in a general spectral space for which the nonlinear interscale interaction may not be Fourier-triadic. Specific verifications are performed based on direct numerical simulations of turbulence in a spherical geometry and reexaminations of several existing results for turbulent channel flows.

Summary We present a parallel fully implicit algorithm for the large eddy simulation (LES) of incompressible turbulent flows on unstructured meshes in three dimensions. The LES governing equations are discretized by a stabilized Galerkin finite element method in space and an implicit second‐order backward differentiation scheme in time. To efficiently solve the resulting large nonlinear systems, we present a highly parallel Newton‐Krylov‐Schwarz algorithm based on domain decomposition techniques. Analytic Jacobian is applied in order to obtain the best achievable performance. Two benchmark problems of lid‐driven cavity and flow passing a square cylinder are employed to validate the proposed algorithm. We then apply the algorithm to the LES of turbulent flows passing a full‐size high‐speed train with realistic geometry and operating conditions. The numerical results show that the algorithm is both accurate and efficient and exhibits a good scalability and parallel efficiency with tens of millions of degrees of freedom on a computer with up to 4096 processors. To understand the numerical behavior of the proposed fully implicit scheme, we study several important issues, including the choices of linear solvers, the overlapping size of the subdomains, and, especially, the accuracy of the Jacobian matrix. The results show that an exact Jacobian is necessary for the efficiency and the robustness of the proposed LES solver.

SUMMARYThis paper presents a global Galerkin spectral method for solving the incompressible Navier-Stokes equations in three-dimensional bounded domains. The method is based on helical-wave decomposition (HWD), which uses the vector eigenfunctions of the curl operator as orthogonal basis functions. We shall first review the general theory of HWD in an arbitrary simply connected domain, along with some new developments. We then employ the HWD to construct a Galerkin spectral method. The current method innovates the existing HWD-based spectral method by (a) adding a series of auxiliary fields to the HWD of the velocity field to fulfill the no-slip boundary condition and to settle the convergence problem of the HWD of the curl fields, and (b) providing a pseudo-spectral method that utilizes a fast spherical harmonic transform algorithm and Gaussian quadrature to calculate the nonlinear term in the Navier-Stokes equations. The auxiliary fields are uniquely determined by solving the Stokes and Stokes-like equations under adequate boundary conditions. The implementation of the method under the spherical geometry is presented in detail. Several numerical examples are provided to validate the proposed method. The method can be easily extended to other domains once the helical-wave bases, which depend only on the geometry of the domains, are available.