Recent experimental and direct numerical simulation data of two-dimensional, isothermal wall-bounded incompressible turbulent flows indicate that Reynolds-number effects are not only present in the outer layer but are also quite noticeable in the inner layer. The effects are most apparent when the turbulence statistics are plotted in terms of inner variables. With recent advances made in Reynolds-stress and near-wall modeling, a near-wall Reynolds-stress closure based on a recently proposed quasi-linear model for the pressure strain tensor is used to analyse wall-bounded flows over a wide range of Reynolds numbers. The Reynolds number varies from a low of 180, based on the friction velocity and pipe radius/channel half-width, to 15406, based on momentum thickness and free stream velocity. In all the flow cases examined, the model replicates the turbulence statistics, including the Reynolds-number effects observed in the inner and outer layers, quite well. Furthermore, the model reproduces the correlation proposed for the location of the peak shear stress and an appropriately defined Reynolds number, and the variations of the near-wall asymptotes with Reynolds numbers. It is conjectured that the ability of the model to replicate the asymptotic behavior of the near-wall flow is most responsible for the correct prediction of the Reynolds-number effects.

Topological refractions created by valley sonic crystals (VSCs) have attracted great attentions in the communities of physics and engineering owing to the advantage of zero reflection of sound and the potential for designing advanced acoustic devices. In previous works, topological refractions of valley edge states are demonstrated to be determined by the projections of the valleys K and K ′ , and two types of topological refractions generally exist at opposite terminals or different frequency bands. However, the realization of tunable topological refractions at the fixed frequency band and terminal still poses great challenge. To overcome this, we report the realization of tunable topological refractions by VSCs with triple valley Hall phase transitions. By simply rotating rods, we realize 3 types of topological waveguides (T1, T2, and T3) composed of two VSCs, in which the projections of the observed valley edge states can be modulated between K and K ′ . Additionally, based on the measured transmittance spectra, we experimentally demonstrate that these valley edge states are almost immune to backscattering against sharp bends. More importantly, we realize tunable topological refractions at the fixed frequency band and terminal, and experimentally observe the coexistence of positive and negative refractions for T1 and T3, and negative refractions for T2. The proposed tunable topological refractions have potential applications in designing multi-functional sound antennas and advanced communication devices.

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The stress field in a rotating turbulent internal flow is highly anisotropic. This is true irrespective of whether the axis of rotation is aligned with or normal to the mean flow plane. Consequently, turbulent rotating flow is very difficult to model. This paper attempts to assess the relative merits of three different ways to account for stress anisotropies in a rotating flow. One is to assume an anisotropic stress tensor, another is to model the anisotropy of the dissipation rate tensor, while a third is to solve the stress transport equations directly. Two different near-wall two-equation models and one Reynolds stress closure are considered. All the models tested are asymptotically consistent near the wall. The predictions are compared with measurements and direct numerical simulation data. Calculations of turbulent flows with inlet swirl numbers up to 1.3, with and without a central recirculation, reveal that none of the anisotropic two-equation models tested is capable of replicating the mean velocity field at these swirl numbers. This investigation, therefore, indicates that neither the assumption of anisotropic stress tensor nor that of an anisotropic dissipation rate tensor is sufficient to model flows with medium to high rotation correctly. It is further found that, at very high rotation rates, even the Reynolds stress closure fails to predict accurately the extent of the central recirculation zone. NOTATION b ijanisotropy tensor (u i u j À 2kä ij =3)=(2k) C D dimensionless constant with an assigned value of 1.68 C s model constant with an assigned value of 0.11 C 1 , C Ã 1 , C 2 , C Ã 3 , C 5 model constants = 3.4, 1.8, 4.2, 1.3 and 0.4 respectively C å , C å1 , C å2 , C å3 , C å5 model constants = 0.12, 1.5, 1.83, (1.9), 2.9556 and 5.8 respectively C ì model constant with an assigned value of 0.096 (0.09) D pipe diameter D ij production tensor D T ij turbulent diffusion tensor f w1 damping function exp[À(Re t =200) 2 ] f w2 damping function exp[À(Re t =40) 2 ] f å damping function 1 À 2 9 exp[À(Re t =6) 2 ] f ì damping function h channel half-width k turbulent kinetic energy L, M, N model constants 2:25, 0:5 and 0.57 respectively n i wall unit normal vector (0, 1, 0) p fluctuating pressure P mean pressure P ij production tensor P production of turbulent kinetic energy Àu i u j (@U i =@x j ) r radial coordinate R pipe radius Re Reynolds number U m D=í Re t turbulent Reynolds number k 2 =åí Re å turbulent Reynolds number (åí) 1=4 y=í Ro rotation number W 0 =U m or 2jÙjh=U m S swirl number [2 R 0 U W r 2 dr]=(R 3 U 2 m ) S ij mean strain rate tensor (@U i =@x j @U j =@x i )=2 S Ã ij dimensionless strain rate tensor S ij (k=å) 193 The MS was

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