Effects of vortex-induced velocity on the wake of a synthetic jet issuing into a turbulent boundary layer

A synthetic jet issued in a cross flow creates a momentum deficit in the cross flow downstream of the jet. In the literature, this deficit is ascribed to viscous blockage by the jet and the up wash of low-momentum fluid caused by the vortical structures of the jet. This paper proposes and quantifies a third effect contributing to the momentum deficit: a velocity induced by the vortical structures in the direction opposite to the cross flow. A reconstruction technique – quantifying the vortex-induced velocity – is developed to determine the momentum deficit caused by the proposed effect. This is applied to a test case of a rectangular synthetic jet (AR = 13, St = 0.5, r = 0.88) issuing into a turbulent boundary layer (Reτ = 1220, U∞ = 10 m/s, δ = 45 mm). The shape of the created vortical structures is reconstructed using a combination of planar(two-dimensional two-component) PIV in the streamwise– wall-normal plane and stereo(two-dimensional three-component) PIV in the spanwise–wall-normal plane. The reconstructed shape consists of overlapping clockwiseand counterclockwise hairpins. With this (constant) shape known, the distribution of hairpins can be determined using the spanwise-vorticity field only. From this distribution of vortical structures the induced velocity is calculated using Biot-Savart’s law. Qualitatively the induced velocity components are very similar to the equivalent measured velocity components. The streamwise momentum flux deficit per unit width at the centerline is calculated for both the induced and the measured case. After some start-up behaviour the momentum deficit for both cases becomes relatively constant. In this constant regime (x/δ > 1) the momentum deficit induced by the vortical structures accounts for 90% of the measured momentum deficit. It is reasoned that the other 10% is most likely to be caused by an increase in skin friction resulting from the up wash of low-momentum fluid (and consequential down wash of high-momentum fluid). INTRODUCTION Synthetic jets in cross flow are widely used in applications such as mixing enhancement (M’Closkey et al., 2002; Sau & Mahesh, 2008), control of turbulence (Rathnasingham & Breuer, 2003) or separation control (Dandois et al., 2007). The interaction of a synthetic jet with a cross flow leads to a momentum deficit in the cross flow downstream of the jet, causing an increase in drag. For most applications minimization of this momentum deficit is of importance for the efficiency of the pursued goal. In order to minimize it, the origin of the momentum deficit needs to be understood. In the literature, the momentum deficit is often referred to as blockage (see for example Lardeau & Leschziner (2011)), or as caused by vortex induced up wash of low-momentum fluid near the wall (see for example Rathnasingham & Breuer (2003)). However, these were qualitative descriptions and do not quantify the momentum deficit in any detail. This paper proposes a third origin and quantifies its momentum deficit: a velocity induced by the created vortical structures in the direction opposite to the cross flow. Furthermore, it will be reasoned that the effects of blockage and up wash on the momentum deficit are limited. The total momentum deficit will be a combination of these three (and possibly other) factors, i.e. viscous blockage of the cross flow, vortex induced up wash of low-momentum fluid and a vortex induced velocity in the direction opposite to the cross flow. The ratio of contributions of these effects will vary with downstream distance and depends on the flow parameters and the type of vortical structures created by the synthetic jet. A synthetic jet is formed from the working fluid by alternating blowing and suction, creating a vortex ring at the jet exit each blowing cycle. The sinusoidal velocity cycle can be characterized by a frequency, f , and a velocity magnitude, ū. The velocity magnitude used here is the mean blowing velocity, calculated as ū = 1 T ∫ T/2 0 u(t)dt. The relevant parameters of the interaction between the synthetic jet and the turbulent boundary layer are described by the velocity ratio and the Strouhal number, defined as