Particle image velocimetry (PIV) was used to obtain the mean axial velocity , the mean radial velocity and the Reynolds stress statistics in an isothermal flow formed by discharging a central jet in an annular stream of swirling air flow. This is a typical geometry used in swirl-stabilised burners, where the central jet is the fuel. The flow Reynolds number was 29000, based on the area-averaged velocity of 8.46 m/s at the exit and the diameter of swirling co-flow of 50.8 mm. Three swirl numbers, S, of 0.3, 0.58, and 1.07 of the annular swirling stream were considered. The effects of the swirl number on the behaviours of the Reynolds stresses, and their gradients, which appear in the Reynolds-averaged Navier-Stokes (RANS) equations are analysed. The results show pronounced asymmetry in the mean and the fluctuating quantities for all swirl numbers. The flow asymmetricity is found to be related to the radial velocity fluctuations. It was postulated that for swirling flows with constant Reynolds number the change in the flow swirl number only, does not necessary result in increase of turbulent intensity. An 'effective' turbulent viscosity was shown to be independent of the flow swirl number for constant Reynolds number.