Direct numerical simulation of statistically stationary and homogeneous shear turbulence and its relation to other shear flows

Abstract

Statistically stationary and homogeneous shear turbulence (SS-HST) is investigated by means of a new direct numerical simulation code, spectral in the two horizontal directions and compact-finite-differences in the direction of the shear. No remeshing is used to impose the shear-periodic boundary condition. The influence of the geometry of the computational box is explored. Since HST has no characteristic outer length scale and tends to fill the computational domain, long-term simulations of HST are 'minimal' in the sense of containing on average only a few large-scale structures. It is found that the main limit is the spanwise box width, L z , which sets the length and velocity scales of the turbulence, and that the two other box dimensions should be sufficiently large (L x 2L z , L y L z ) to prevent other directions to be constrained as well. It is also found that very long boxes, L x 2L y , couple with the passing period of the shear-periodic boundary condition, and develop strong unphysical linearized bursts. Within those limits, the flow shows interesting similarities and differences with other shear flows, and in particular with the logarithmic layer of wall-bounded turbulence. They are explored in some detail. They include a self-sustaining process for large-scale streaks and quasi-periodic bursting. The bursting time scale is approximately universal, ∼ 20S −1 , and the availability of two different bursting systems allows the growth of the bursts to be related with some confidence to the shearing of initially isotropic turbulence. It is concluded that SS-HST, conducted within the proper computational parameters, is a very promising system to study shear turbulence in general.

Vertically localised equilibrium solutions in large-eddy simulations of homogeneous shear flow

“…Before proceeding to seek equilibrium solutions, the effect of the symmetry restriction was tested in several LESes of symmetric SS-HST, which are summarised in table 1. The table also includes two reference unconstrained DNSes from Sekimoto et al (2016). Table 1 shows that the length and velocity scales found in DNS also work well in the symmetric LESes.…”

“…The discretisation uses 2/3-dealiased Fourier expansions in (x, z), and sixth-order spectral-like compact finite differences in y, with the shear-periodic boundary conditions embedded in the compact finite-difference matrices for each Fourier mode. As explained in Sekimoto et al (2016), this avoids recurrent remeshing and the resulting secular loss of enstrophy over long integration times.…”

“…It was shown in Sekimoto et al (2016) that the correct scales for the large-scale length and velocity of SS-HST are based on the spanwise box dimension, L z and SL z , so that the Reynolds number is Re z ≡ SL 2 z /ν in DNS. We will extend this definition to LES using the effective mean eddy viscosity, ν t , replacing the molecular viscosity ν.…”

“…The mean flow for turbulence statistics is Sy plus the vertically periodic correction, which is an approximately linear profile as evaluated later in this section. The numerical code for DNS is described in detail in Sekimoto et al (2016). The equations (2.1)-(2.2) for LES are formulated in terms of the vertical vorticity and of the Laplacian of the vertical velocity (Kim et al 1987;Hughes et al 2001).…”

“…Ideal unbounded homogeneous shear flow has no intrinsic length scale and is believed to grow without bound from any initial condition (Champagne et al 1970;Rogers & Moin 1987;Tavoularis & Karnik 1989;Cambon & Scott 1999) but, when simulated in a finite computational box, it reaches a statistically stationary and homogeneous state (SS-HST) characterised by repeated bursting reminiscent of near-wall and logarithmic-layer turbulence (Pumir 1996;Gualtieri et al 2007). The problem was recently revisited by Sekimoto et al (2016) using direct numerical simulations (DNSes), from where the simulation code and parameters in this paper are derived. They determined which computational boxes best mimic wall-bounded turbulence, and showed that the relevant limiting dimension is the spanwise box width.…”