Direct numerical simulation of the ﬂow around a rectangular cylinder at a moderately high Reynolds number

We report a Direct Numerical Simulation (DNS) of the ﬂow around a rectangular cylinder with a chord-to-thickness ratio B/D = 5 and Reynolds number Re = 3000. Global and single-point statistics are analysed with particular attention to those relevant for industrial applications such as the behaviour of the mean pressure coeﬃcient and of its variance. The mean and turbulent ﬂow is also assessed. Three main recirculating regions are found and their dimensions and turbulence levels are characterized. The analysis extends also to the asymptotic recovery of the equilibrium conditions for self-similarity in the fully developed wake. Finally, by means of two-point statistics, the main unsteadinesses and the strong anisotropy of the ﬂow are highlighted. The overall aim is to shed light on the main physical mechanisms driving the complex behaviour of separating and reattaching ﬂows. Furthermore, we provide well-converged statistics not aﬀected by turbulence modelling and mesh resolution issues. Hence, the present results can also be used to quantify the inﬂuence of numerical and modelling inaccuracies on relevant statistics for the applications.


Introduction
works, see e.g. Nakamura et al. [4], Ohya et al. [5], Hourigan et al. [6], and 2. Direct Numerical Simulation and statistical convergence a rectangular cylinder. The evolution of the flow is governed by the continuity 94 and momentum equations, where x = x 1 (u = u 1 ), y = x 2 (v = u 2 ), z = x 3 (w = u 3 ) are the stream- like V = v , uv and ∂ · /∂ỹ. In conclusion, the average of a generic quantity 146 β is defined as where the sum and difference of the two integrals is given by the symmetric or  i.e. the lift, C l , and drag, C d , coefficients. Obviously, for symmetry reasons, 158 the average lift coefficient is null, C l = 0 where · denotes the time average.

159
As shown in figure 3(a), instantaneously the lift coefficient is not zero, but it 160 fluctuates in time. Different time scales are recognized and can be studied by 161 considering the frequency spectrum defined as where· denotes the Fourier transform, * the complex conjugate and St the 163 dimensionless frequency, i.e. the so-called Strouhal number. As shown in figure   164 3(b), the frequency spectrum confirms the presence of different temporal scales.

165
In particular, a clear peak for St ≈ 0.14 is present and will be shown in the quantitatively agree with those observed in Kiya and Sasaki [18,19] for very 173 large Reynolds numbers. In accordance with these works, we argue that the 174 peak at St ≈ 0.14 is related with a large scale shedding of vortices from the 175 main recirculating region while the very slow peak at St ≈ 0.042 is due to 176 the presence of a low-frequency unsteadiness encompassing the entire flow field.

177
Concerning the drag coefficient, we measure that C d = 0.96.

178
Let us now consider the topology of the mean flow field. As shown in figure   179 4, the streamlines of the mean flow highlight the presence of a large scale recir-

218
This region of negative shear is the near-wall footprint of the primary vortex. 219 Actually, the primary vortex is responsible also for the previously observed (c) apparent, the wake centerline velocity approaches the self-similar decay, It is worth noting that the above self-similar behaviour implies [22] that and, hence, that the wake spreads as a power law, i.e.ỹ 1/2 ∼x 1/2 .
are the symmetric and antisymmetric part of the velocity gradient tensor. This quantity β can be written as,

395
R ββ (x, y, r z ) = β ′ (x, y, z + r z /2, t)β ′ (x, y, z − r z /2, t) β ′ β ′ (x, y) . (10) In figure 11, the spanwise correlation function of the three components of in the spanwise direction and in time, the spectrum of turbulent kinetic energy 427 q = u i u i /2 can be defined as where k z and St are the spanwise wavenumber and frequency, while (·) denotes 429 the two-dimensional Fourier transform with respect to the spanwise direction and, analogously, the one-dimensional frequency spectrum is computed by inte-435 grating with respect to k z , By using the Taylor's hypothesis of frozen turbulence we also address the puta-437 tive wavenumber spectrum in the streamwise direction defined as The main unsteadinesses of the flow are analysed in figure 12 by means of