## 1 Introduction  
Acoustic liners are widely used in modern aircraft engines to reduce fan noise. Placed within the fan duct, they are composed by a network of cavities covered by a perforated plate. Thanks to such geometries, the energy of the acoustic excitation generated by the fan is converted into kinetic energy through complex mechanisms leading to vortex shedding: when the pressure outside the liner increases, the flow is sucked into the cavities through the perforations. Then, the outside pressure decreases leading to an ejection phase. This phenomenon is periodically repeated, leading to the formation of oscillatory jets and vortices around the perforations. Consequently, the cavities act as resonators: when triggered at their resonant frequency by the acoustic signal, they amplify the oscillatory phenomena. The response of perforations to acoustic signals has been known for several decades [1] and acoustic liners have been widely studied and optimised since then [2]. Most studies on those devices cover acoustic performances and the heat transfer that takes place along liners is still not well known. Few recent studies tackled this issue. To the authors' knowledge, [PERSON] [3] was the first to have considered heat transfer issues within fan duct liners. He numerically evidenced the significant sensitivity of the wall heat transfer to upstream perturbations, whether acoustical or turbulent, over the upstream part of the plate; after some distance, the heat transfer tended to a plateau, which was pretty similar for the different considered conditions. [PERSON] et al. [4] experimentally considered the impact of a temperature gradient across an acoustic liner on its performance. Interestingly, they also measured the liner temperature under different conditions and observed the impact of the acoustics on the thermal gradient: when a high SPL acoustic signal is switched on, a liner surface temperature decrease of about 10 K is noticed. To better understand the complex flow dynamics triggered by acoustic liners, it is of interest to consider the general case of synthetic jets. This terminology, introducedrecently [5], refers to periodic jets without mass injection generated by acoustic liners and different kinds of actuators, such as pistons [6] or piezoelectric membranes [7]. They require no complex plumbing, which is advantageous when compared to classical, continuous jets and they have been proven useful for different applications, such as flow control [8] or electronic cooling [9]. Note in particular, and this is the adopted terminology throughout this article, [PERSON] et al. [10] defined synthetic jets as \"the appearance of a time-averaged outward velocity along the jet axis\" which \"corresponds to the generation and subsequent convection or escape of a vortex ring\". With such a definition, over a complete oscillation period, the time-averaged jet velocity is null. Thus, dedicated parameters have to be defined to characterise these jets. As said, the conventions proposed by [PERSON] et al. [10] are used here: the jet velocity \(\overline{W}\) is defined as the velocity averaged over the perforation section \(A\) and over the ejection mid-period \(\frac{T}{2}=\frac{1}{2f}\):  
\[\overline{W}=2f\frac{1}{A}\int_{A}\int_{0}^{1/2f}w(t,x)\mathrm{d}t\mathrm{d}A, \tag{1}\]  
where \(f\) is the frequency of oscillation and \(w(t,x)\) the instantaneous vertical velocity. In a quiescent environment, synthetic jets are then completely characterised by two dimensionless parameters: the Reynolds number \(Re\), based on the perforation diameter \(D\), and the dimensionless stroke length \(\frac{L_{0}}{D}\). In the presence of a grazing flow, a third parameter has to be taken into account: the flow velocity ratio \(M\) defined as the ratio of the jet velocity to the grazing flow velocity \(U_{0}\). These three parameters are defined as follows:  
\[Re=\frac{\overline{W}D}{\
u},\ L_{0}=\frac{\overline{W}}{2f},\ M=\frac{ \overline{W}}{U_{0}}, \tag{2}\]  
with \(\
u\) being the air kinematic viscosity.