Analysis of Upstream Turbulence Impact on Wall Heat Transfer in an Acoustic Liner with Large-Eddy Simulations

Abstract

Acoustic liners of aircraft fan ducts generate synthetic jets that interact with the boundary layer of the incident grazing flow. Such an interaction leads to complex wall heat transfer, which has been scarcely studied. The objective of the present work is to evaluate the flow dynamics and the heat transfer mechanisms that occur in acoustic liners by the use of Large-Eddy Simulations. To do so, two configurations, linked by a principle of similarity, are considered: a lab-scale one, for which dimensions have been multiplied by a factor of 6.25 to ease measurements, and an engine-scale configuration. The lab-scale configuration is used to validate the numerical methodology and, although some limitations are pointed out, the similitude is validated. As a main outcome of these detailed simulations, synthetic jets are found to completely drive the flow dynamics in the jet wakes. Upstream turbulence is also shown to impact the development of the first jet rows as well as the wall heat transfer between jets.

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, Duchaine [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. Méry 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, introduced recently [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, Holman 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 Holman 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 $\frac{T}{2}=\frac{1}{2f}$:

$${\displaystyle \overline{W}=2f\frac{1}{A}{\int }_A{\int }_0^{1/2f}w(t,x)\mathrm{d}t\mathrm{d}A,}$$

(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:

$$Re=\frac{\overline{W}D}{\nu },L_0=\frac{\overline{W}}{2f},M=\frac{\overline{W}}{U_0},$$

(2)

with $\nu$ being the air kinematic viscosity. Although wall heat transfer has not been much studied within acoustic liners, such issues have been considered for other applications of synthetic jets. Most of the applications treating heat transfer with synthetic jets target impinging jet applications [11] while, in acoustic liners, it is the perforated plate crossed by the jets that is of interest. Such a configuration has been studied by Manning et al. [12], who evidenced an enhanced wall heat transfer with synthetic jets compared to natural convection only. They also pointed out that a multi-perforated configuration is more efficient in increasing wall heat transfer than a single perforation. However, their configuration considered only four perforations in a quiescent environment. Another field of research provided information of interest: flow control, more specifically flow separation delay. In this context, the jets are submitted to a grazing flow. Zhong et al. [13] experimentally identified different flow structures. Depending on the synthetic jet Reynolds number, stroke length and velocity ratio, hairpins, tilted vortices or vortex rings can be observed. Jabbal and Zhong [14] determined the flow structure impacts on the wall through thermo-sensitive methods: the tilted vortices are evacuated very early out of the boundary layer, while the two other categories have a greater impact on the wall. Zhou and Zhong [15] numerically reproduced their experiment and observed similar flow structures. They distinguished two kinds of structures, depending on their footprint on the wall heat flux and wall shear stress that can present one (distorted vortex rings) or two (hairpins and stretched vortex rings) streamwise streaks. More recently, Wen and Tang [16] showed the impact of turbulence on hairpin formation: hairpins are more bent-shaped, asymmetric, and are characterised by a higher velocity and dissipation rate in the turbulent case, leading to a wider but weaker impact on the turbulent boundary layer (visible on velocity and gradient profiles) and wall shear stress. These configurations bring information about wall heat transfer around a synthetic jet but are limited to a single jet and their results cannot be directly extended to liners, where a large number of jets interact with each other. Wen et al. [17] studied two in-line synthetic jets and highlighted the importance of the jet phase difference $\Delta \varphi$ in the jet development: the hairpins can merge into combined vortices ($\Delta \varphi =90°$), partially interact ($\Delta \varphi =0°$ or 180°) or remain separated ($\Delta \varphi =270°$).

A recent experimental study proposed by Giachetti et al. [18] at Pprime Institute (France) investigated the wall heat transfer on a similitude configuration that matches the different aspects of acoustic liners through a scaling: it is multi-perforated (50 perforations distributed in 10 rows of 5) and submitted to a grazing flow. They evidenced the interest of a multi-perforation configuration and obtained Nusselt increases up to 175% when comparing synthetic jets’ results with no jet cases. Moreover, they pointed out the interest of a high $M=\overline{W}/U_0$ value: the higher it is, the higher the heat transfer enhancement that is obtained. The goal of the present study based on the experiment of Giachetti is twofold. The first objective is to validate the numerical methodology used to simulate heat transfer in acoustic liners. To do so, Large-Eddy Simulation (LES) results are compared to experimental data obtained with the test rig. A previous study [19] has shown promising results and provided a preliminar validation of the numerical set-up. However, the upstream turbulence was not considered in that case while a previous study [3] evidenced that inlet turbulence played a key role in the development of the first rows of perforation. To conclude, the similitude is validated comparing numerical results on the lab-scale configuration with the engine-scale predictions.

The paper is organised as follows. The configurations and similitude methodology are first described, then results from the lab scale are used to briefly describe the flow behaviour. The engine scale results are then compared to the lab scale, along with experimental data validating the numerical approach, in order to discuss the validity of the similitude hypothesis. The impact of upstream turbulence on synthetic jet development and wall heat transfer is finally investigated.

2. Configuration Descriptions

2.1. Lab-Scale Configuration

The experimental lab-scale configuration is composed of a perforated plate with 50 apertures of diameter $D=6.25$ mm arranged in 10 lines of 5 rows (Figure 1b), which is placed within a wind tunnel test section (Figure 1a). Each perforation is extended by a duct that ends into a piston, which is used to generate synthetic jets. All of the pistons are connected to the same engine, meaning that the jets are synchronised. In the numerical study, the piston motion is approximated by a sinusoidal signal:

$$Z_{pistons}\left(t\right)=Z_{mean}+Ksin\left(2\pi ft\right),$$

(3)

where $Z_{mean}$ is the piston vertical mean position while K and f are the imposed amplitude and frequency, respectively.

To study wall heat transfer, the perforated plate is heated. Giachetti et al. [18] used a regression method to find out the heat transfer coefficient h, based on Newton’s cooling law: $\mathsf{\Phi }=h(T_w-T_{ref})$. The methodology relies on the imposition of several values of wall heat flux $\mathsf{\Phi }$ and the measurement of the wall temperature $T_w$: h is then given by the slope of the linear relation between $T_w$ and $\mathsf{\Phi }$ and the reference temperature $T_{ref}$ as the temperature for null heat flux.

Different operating points were considered during the experimental study, defined by the frequency f, the amplitude K of the pistons and the grazing flow velocity $U_0$. The present study focuses on a specific point defined in

Table 1. Numerical operating points.

${\mathit{U}}_0$ (m.s${}^{-1}$)	f (Hz)	K (mm)	$\mathbf{Re}$	${\mathit{L}}_0/\mathit{D}$	M
12.8	12.8	11	829.2	12.94	0.16

. The inlet velocity profile is turbulent (Figure 2), and the experimental data provide a maximum of turbulent intensity equal to 10.5% (Figure 2). Since one of the objectives is to determine the impact of upstream turbulence on heat transfer, two different kinds of simulations are performed: the first one reproduces the experimental turbulence by injecting a fluctuating velocity at the inlet, while the second kind ignores it.

2.2. Similitude and Engine-Scale Configuration

Synthetic jets are completely characterised by the Reynolds number $Re$, the dimensionless stroke length $L_0/D$ and the velocity ratio M. The lab-scale configuration is based on a similitude in order to be representative of a classical acoustic liner, with perforation diameters equal to $D=1$ mm. To do so, the dimensions of the rig are obtained by multiplying that of the liner perforated plate by $1/\chi =6.25$. The different relations linking the geometrical and flow parameters of both scales, through the parameter $\chi$, are summarised in

Table 2. Similitude ratios between the lab and engine scales.

Similitude Ratios		$\mathit{\chi }=1/6.25$
Parameter$\mathit{X}$	${\mathit{X}}_{\mathbf{lab}}/{\mathit{X}}_{\mathbf{engine}}$	${\mathit{X}}_{\mathbf{lab}}$	${\mathit{X}}_{\mathbf{engine}}$
Geometrical parameters
D (mm)	$1/\chi$	6.25	1
L (mm)	$1/\chi$	300	48
Flow parameters
$U_0$ (m.s${}^{-1})$	$\chi$	12.8	80
$\overline{W}$ (m.s${}^{-1})$	$\chi$	2.07	12.94
f (Hz)	${\chi }^2$	12.8	500
Heat transfer parameter
h (W.m${}^{-2}$.K${}^{-1}$)	$\chi$	-	-

.

The test section and perforated plate of the engine-scale configuration are similar to that of the lab scale, except for their dimensions, which are $1/\chi =6.25$ smaller than that of the lab-scale configuration. The ducts and pistons below each of the perforations are replaced by a single cavity, the dimensions of which are chosen to get a quarter-wave resonator, corresponding to a resonance frequency $f_{1/4}=500$ Hz. The choice of a quarter-wave resonator leads to a cavity length around 190 mm, which is too long for an acoustic liner. In real applications, a Helmholtz resonator would be privileged to target such a resonant frequency while maintaining the minimum footprint required in an aeronautical context. However, the quarter-wave resonator is much simpler to design and the choice of resonator does not impact the synthetic jet generation. Figure 3 shows the geometrical differences between both configurations. An acoustic signal is superimposed on the grazing flow at the inlet of the domain, characterised by its frequency $f=500$ Hz and Sound Pressure Level $SPL=148.5$ dB.

Two main limitations of the similitude can be pointed out at this stage. First of all, it has been shown that the actuator geometry, especially that of the neck, can impact the formation of vortices [20]. Here, the lab-scale geometry does not exactly correspond to the classical cavity and neck configuration of a liner with a long hole geometry (Figure 3). The other important point is the jet triggering mechanism. In the lab scale, the jets are generated by the imposed hydrodynamics while in the engine scale they are due to the acoustics. This point raises another issue: the propagation of an acoustic wave along the liner takes some time, which implies that the wave does not impact all of the perforations and resonators at the same time. A direct consequence is that the different synthetic jets generated by the wave are not fully in phase. On the contrary, the experimental rig is based on the use of pistons that are all moved by the same engine, and are thus all perfectly in phase. Differences could therefore be expected if comparing detailed data obtained from the experimental rig and a classic acoustic liner. In that respect, Wen et al. [17] showed on an in-line twin synthetic jet configuration that hairpins have a completely different behaviour regarding the phase difference between the two jets. Note, however, that the phase difference between two successive jets is around 1.5° on the engine-scale configuration, which is very close to the null phase difference of the lab scale.

3. Numerical Modelling

All simulations discussed hereafter are performed with the AVBP solver, which solves the three-dimensional compressible Navier–Stokes equations. Note that the buoyancy effect has been neglected as forced convection is predominant in this configuration. Wall-resolved LESs are targeted for all cases and a second order in time and space numerical scheme, the Lax–Wendroff (LW) scheme [21], is chosen. Concerning the sub-grid-scale model, the $\sigma$ model [22] is used. The correct behaviour of close-to-wall turbulent viscosity is correctly reproduced by this model, a requirement for wall-resolved simulations.

The numerical domain corresponds to a section of the wind tunnel test section beginning at the perforated plate, as can be seen in Figure 1. At this location, the experimental data for the flow velocity are available. The coordinate origin is as indicated in Figure 1. The Navier–Stokes Characteristic Boundary Conditions (NSCBCs) formalism [23] is used for the inlet and outlet. For all of the simulations, a velocity profile provided by the experiment is imposed at the inlet (Figure 2) and the similitude relations are used to adapt the profile to the engine scale. Note that the engine scale inlet requires an additional treatment since an acoustic signal is superposed to the velocity profile. To investigate the effet of inlet turbulence, two different simulations are run on the similitude case: one with no turbulence injection at the inlet and one with an injection of fluctuating velocities. When turbulence is injected, the fluctuating velocity $u_{rms}$ profile, adapted from the experimental data (Figure 2), is imposed at the inlet using the Kraichnan/Celik method [24]. Regarding the outlet, a constant pressure $P_0$ = 101,325 Pa is imposed and the three-dimensional extension of the classical LODI (Local One-Dimensional Inviscid) formulation with space-averaged values as detailed in Granet et al. [25] is used. The ceiling and side walls of the test section are far enough away from the perforated plate not to interact with heat transfer; thus, they are treated as slip walls. On the contrary, all other walls, i.e., the plate, the ducts, the lab scale pistons and the engine-scale cavity walls, are no-slip walls. For the lab-scale configuration, the ALE (Arbitrary Lagrangian–Eulerian) formalism [26] is used to simulate the piston motion. Finally, a constant heat flux is imposed on the perforated plate. The same method as Giachetti et al., based on Newton’s cooling law, is used to determine the heat transfer coefficient h. Thus, for each of the configurations, two simulations are run with two different flux values.

The different meshes used for both scales are composed of tetrahedra. An automatic mesh adaptation strategy, based on a double criterion inspired from Odier et al. [27], is used. The idea is to run a simulation on a coarse, user-defined mesh to get preliminary results. Then, the mesh is refined in the zones of interest that are defined by a high velocity gradient (corresponding to the boundary layer and the synthetic jets) and high normalised wall distance $Y^{+}$ (to target wall-resolved simulations). More information about this strategy can be found in [19] and the original and adapted meshes are illustrated in Figure 4. The adapted meshes contain around 66 million cells for the lab scale and 55 million cells for the engine scale. The adaptation was particularly efficient for the lab scale and led to a $Y^{+}$ value averaged over the perforated plate equal to 3.7, while it is equal to 9.7 for the engine scale. The CPU costs for simulations along a period T on these adapted meshes with the IDRIS Jean Zay computer (Intel Cascade Lake 6248 processors, with 20 cores at 2.5 GHz) are approximately 57,000 hCPU and 9000 hCPU for the engine and lab scales, respectively. To ensure the convergence of the statistics, the different quantities presented have been time-averaged on five periods of the piston motion for the lab scale and inlet acoustic pulsation for the engine scale.

4. General Flow Description

Figure 5 illustrates the flow evolution over a period T of the piston motion, for the lab-scale configuration with no turbulence injection. Upstream of the first row of perforations, the grazing flow boundary layer as well as the wall shear stress present similar evolutions for the four considered instants: the boundary layer thickens, leading to a decreasing wall shear stress. This is characteristic of a classical flat plate. Around the first row of perforations, the flow is then completely disturbed by the synthetic jets and its evolution is no longer comparable to that of a flat plate. Two time phases have to be distinguished: the ejection and suction mid-periods. The ejection mid-period is characterised by the interaction of the jets and the grazing flow boundary layer, which is destabilised by unstable shear layers. The turbulence levels rise and the formation of several coherent flow structures can be observed (cf. Figure 6). Horseshoe vortices are formed upstream of the first perforations, due to a blockage effect, and counter-rotating vortex pairs (CRVP) can be observed downstream the perforations. The greatest observable structures are hairpins, which have been found to interact with the boundary layer and to affect the wall heat transfer [13,14,15,16]. The upstream structures are more coherent than those that are formed downstream, where all the structures interact and the flow transitions to turbulence. The impact of flow structures on the plate can be noticed through the wall shear stress. This impact is similar all along the perforated zone, except for the first row of perforations where the flow is laminar. Contrarily, the aspiration phase is characterised by a heterogenous wall shear stress, and two regions can be identified: the jet axis and the in-between-jets axis. In the jet axis, the boundary layer is sucked through the perforations. A boundary layer is then established after each perforation. This leads to significant wall shear stress. Between jet lines, the flow behaviour is totally different. The decreasing evolution of the wall shear stress corresponds to a thickening boundary layer. Overall, the boundary layer is deeply impacted by the synthetic jets, which, through the wall shear stress modifications, leads to a wall temperature decrease [19].

Concerning wall heat transfer, since the vortical structures impact the whole perforated surface from the third perforation row in a similar way during the ejection mid-period, the temperature is more homogeneous on the last portion of the plate compared to the upstream part where the flow is mainly laminar (Figure 7). During the suction phase, two different behaviours are observed. In the jet axis, the heat transfer is particularly efficient due to the formation of new, thin boundary layers. However, the thickening of these boundary layers in between the jets decreases the heat transfer. The wall temperature is hence highly heterogenous: the plate is efficiently cooled down in the jet axis but not in between them.

To illustrate the unsteadiness of the the flow, Figure 8 presents the different values that can be taken by the temperature through several periods of the piston motion along two axes: the jet axis and the in-between-jets axis. The time-averaged values are provided for the whole period (“global mean”) as well as the ejection mid-period (“ejection mean”) and the aspiration mid-period (“suction mean”). To complement this statistical view of the profiles, PDF distributions are obtained by recording, for each position $x/D$, all of the temperature values that can be reached over time. The sampling chosen here is 40 savings per period T. Envelopes are coloured relatively to the PDF for the symmetry plane and between the jets: the darker the colour, the higher the probability to reach this temperature value. For both axes, the distribution is characterised by a very narrow range of possible temperature values upstream of the perforated zone, which is coherent with the steady development of a thermal boundary layer along a flat plate. Along the jet axis (Figure 8a), the range of temperature values is much wider. After a peak around 370 K around the first perforation, the maximum temperature decreases and tends to plateau around 340 K. After the third perforation, a pattern appears between each of the perforations. Further information provided by this figure is the difference between the aspiration and the ejection mid-periods. Indeed, the lower part of the temperature envelope, under the global mean value, can be attributed to the suction phase while the upper part is caused by the ejection. It appears that the difference between the upper part of the temperature enveloppe and the ejection mean value is larger than the difference between the lower part of the temperature envelope and the suction mean value. This can be compared with the previous analysis: in the aspiration, the flow is completely driven by the suction while the boundary layer is stabilised. The flow is cooled down efficiently and the temperature values remain close to the mean value; a narrow range of values can be observed. This explains why the suction mean value is so close to the minimal value. On the contrary, during the ejection mid-period, the jets generate turbulence and several flow structures that impact the plate. It is thus expected to get a large range of temperatures. Regarding the PDF, two main “paths” can be observed. Each of them matches one of the mid-period means, supporting the hypothesis that these two paths can be attributed to each of the mid-periods. The one at the bottom of the envelope, which corresponds to the suction mid-period, is narrow. If the ejection phase was excluded, this would correspond to a PDF with a small standard deviation. On the contrary, the ejection PDF is broader, corresponding to a larger standard deviation. The flow is stabilised during the aspiration; the thin, new boundary layer after each of the perforations contributes to an efficient cooling of the wall. The heat transfer, as well as the flow dynamics, is totally driven by the suction. On the contrary, during the ejection phase the jets increase the turbulence levels and the different flow structures impact the plate in a more chaotic way. Finally, between the jets (Figure 8b), the temperature range is wider due to the effect of turbulent structures emitted from the synthetic jets. It should be noted that the two mid-period mean temperature profiles seem to be in contradiction with the previous analysis: the aspiration profile reveals higher temperature values than those obtained for the ejection. It was shown that, during the aspiration, the boundary layer between the perforations evolves along the flat plate, leading to an increasing temperature. During the ejection, the plate is impacted by the flow structures, which bring cold flow near the wall, leading to lower temperatures. Therefore, the opposite trend should be expected, with a lower ejection mean temperature than the aspiration one. One explanation for the observed counter-intuitive result is that, due to the latency between the jet formation and their impact on this zone, a delay in heat transfer response can appear. As can be seen in Figure 7 with the temperature field evolution along the whole plate through a period, it takes the whole ejection mid-period for the jets to flow in the zone between the perforations. At the middle of the ejection phase ($t=0$), the vortices are still not spread enough to reach the axis between the jets and the temperatures are therefore still high. In the same way, it takes the whole suction phase to evacuate all of the flow structures: at the middle of the aspiration ($t=T/2$), their impact can still be observed from the fifth perforation row to the end of the plate. The ejection mid-period is hence efficient to decrease the plate temperature, with wall temperatures equal to that observed in the jet axis. However, during the aspiration the thermal boundary layer thickens in a way similar to that of a flat plate, leading to high temperatures at the end of the perforated area. The maximum temperatures observed for this axis are close to that of the flat plate case. Around the first perforations, it is even higher. With the PDF, two similar “paths” can be observed along the first part of the plate, from $x/D=7$ to $x/D=20$. For $x/D\ge 20$, no specific path is seen and the standard deviations of both ejection and aspiration are much larger.

To conclude this thermal analysis, the plate temperature averaged over five periods of the piston motion is given in Figure 9. The observations made in the analysis of the instantaneous field are retrieved with the development of a stationary boundary layer from the inlet to the perforations, the competition of the blockage effect during the ejection and aspiration phases of the first row of synthetic jets, the predominance of the aspiration phase in the jet wakes and the mixing in between the jets.

5. Validation of the Similitude

A preliminary validation of the lab-scale configuration with no turbulence injection was presented in a previous study [28]. It confirmed the key role of the mesh adaptation strategy and the validity of the numerical set-up. In the present study, the goal is now to assess the relevance of the similitude methodology that allows the use of the lab-scale configuration in place of a real geometry and operating condition to perform experiments. To do so, velocity profiles corresponding to both lab and engine scales are compared to the experimental data (Figure 10). Recall that these cases correspond to the simulations without turbulence injection at the inlet. The experimental uncertainty, shown by the error bars in the velocity profiles, is equal to 4% for mean velocities and 8% for rms velocities [18]. The provided profiles coincide with the following positions: $x/D=0$, $x/D=8$, $x/D=16.4$, $x/D=28.4$, $x/D=33.2$ and $x/D=40$, shown in Figure 1b. To clarify the figures, the dimensionless velocity profiles are shifted with the addition of $x/X_0$, where $X_0=0.1$ m for axial velocities and $X_0=0.05$ m for vertical velocities.

The first observation is the destabilisation and thickening of the boundary layer due to the synthetic jets. The numerical results are similar for the two scales and the boundary layer thickness is particularly well retrieved, as seen with the axial velocity $u/U_0$ profiles in Figure 10a. Note, nonetheless, that both numerical simulations overpredict the jet penetration within the boundary layer compared to experimental results, as seen in the $w/\overline{W}$ profiles (Figure 10b). This leads to some overprediction of the boundary layer displacement thickness close to the wall, and particularly for the first three perforation rows, observed for $u/U_0$ (Figure 10a) and $u_{rms}/U_0$ (Figure 10c). As the flow evolves along the plate, the experimental axial and vertical velocity profiles are then better reproduced for both the lab- and the engine-scale simulations. The first profiles present some differences while downstream of the seventh row of jets numerical results reproduce the experimental measurements very well.

The normalised wall heat transfer coefficent h obtained by LES is compared to the experimental results in Figure 11. Here, h is normalised by $H_0$, which equals 1 W.m${}^{-2}$.K${}^{-1}$ for the lab scale and 6.25 W.m${}^{-2}$.K${}^{-1}$ for the engine scale. The considered axes correspond to those shown in Figure 1b. First, the simulations retrieve the right order of magnitude and both the lab scale and the engine scale present similar results. Some differences from the experimental data are nonetheless present. To begin with, upstream of the perforations, the heat coefficient h profiles present a similar shape for the three data sets, corresponding to a developing boundary layer along a flat plate. The slope is, however, larger for the numerical results than for the experimental data. The perforated area can be divided into two main areas: the jet wakes and the zone between the jets. Along the jet wakes, numerical predictions underestimate the heat transfer coefficient h close to the first perforation whereas it is well reproduced from the second jet row. This observation is linked to the absence of inlet freestream turbulence in the simulation, which seems to be critical for the first row of jets. On the contrary, between the jets, predicted h evolutions differ from the experimental data. These differences are of two kinds and are explained by different factors: (i) The wavy aspect obtained for the experimental data between the perforations is directly due to the experimental set-up. Indeed, the heating system is not totally homogeneous. Since the boundary conditions imposed for the simulations are homogeneous wall heat fluxes, these patterns can not be retrieved by the numerical results and the profiles are smooth. (ii) The numerical results between the perforations are quite far from the mean value of the experimental data wavy part and this difference is still present downstream of the perforations. There, the experimental heating system is way more homogeneous and thus does not explain this difference. Further analyses pointed out the importance of the upstream turbulence, as detailed in the following section.

6. Upstream Turbulence Impact

A Q-criterion isosurface at the instant corresponding to the ejection mid-period is provided in Figure 12, for the lab-scale configuration without and with upstream turbulence. Similar flow structures are observed in both cases. However, in the presence of upstream turbulence, the hairpins are asymmetric and more bent-shaped than without upstream turbulence. This result is in agreement with the observations of Wen and Tang [16]. With a turbulent inlet, the hairpins lose their coherence faster and seem to remain lower in the boundary layer, implying that jets are more folded by the upstream flow.

Time-averaged vertical velocity profiles are compared for the lab-scale configuration for both cases with and without upstream turbulence in Figure 13 in the jet axis (in-between jet and axial velocities are not shown for the sake of brevity). These are also compared to the experimental data. The vertical velocity (Figure 13a), which characterises the jets, is better predicted in the turbulent case; without turbulence at the inlet, the jet velocities are overpredicted. Consistent with the Q-criterion analysis, the synthetic jets are more folded in the presence of upstream turbulence. Some differences are observed for the fluctuating velocities (Figure 13b) between the numerical results and experimental data in the upstream part of the plate, before getting closer and even superposed downstream of the jets, where the simulations correctly predict the velocity profiles.

Figure 14 compares the wall shear stress evolution along a period T with and without upstream turbulence for three axial positions on the plate. In order to detect general trends, the profiles are phase-averaged over five periods. Note that an increased number of periods would be necessary to totally erase the residual turbulent fluctuations, which are still seen and are indicative of the lack of statistical convergence. Note that, in the following, the edge of the plate is assumed to be far enough from the perforations not to be impacted by the jets and is thus representative of a flat plate without disturbance. It can be noticed that, for both mid-periods, the case with turbulent inlet is characterised by a higher wall shear stress than the case without upstream turbulence. Some differences are observed between the two mid-periods. During the ejection mid-period ($t=T/4$, Figure 14), apart from the first perforation rows, the wall shear stress in the jet wakes and between the jets are similar and characterised by important fluctuations. The jets impose the flow dynamics. On the contrary, during the suction mid-period ($t=3T/4$, Figure 14), specific trends can be observed. Along the jet axis, the wall shear stress is not affected by the upstream turbulence and presents a similar evolution in both simulations. Between the jets, the wall shear stress evolution, for both cases, follows that of the corresponding flat plate. This implies that the flow is not driven by the synthetic jets but rather by the upstream conditions. Wall shear stress comparisons with and without upstream turbulence show that the synthetic jets drive the flow most of the time. However, upstream turbulence can become preponderant in some specific locations and at specific instants.

The analysis of the temperature PDF in Figure 15 provides further understanding on how the turbulence role is important for the wall heat transfer. Since both reference and turbulent operating points are based on the same jet parameters, the same trends are found in the temperature PDF. Two main differences can, however, be noted. First, the PDFs for both the jet axis and between the jets are narrower in the turbulent case than in the laminar one, indicating a lower temperature standard deviation. The second point is that higher temperatures can be reached in the reference case than in the turbulent case. This is particularly visible in the jet axis plane between the first and second perforations, where a temperature peak around 360 K is found for the reference case while it reaches only 350 K in the turbulent one. This lower peak is linked to the fact that the upstream turbulence contributes to lower synthetic jets at the first perforation rows and lower hairpins. Such a difference in temperature can also be observed between the jets. For $x/D>25$, the maximum temperature reaches around 380 K in the reference case while it does not get higher than 360 K with upstream turbulence. This can be attributed to the development of the hairpins, which were shown to widen in the presence of turbulence, leading to lower temperatures farther from the jets. Moreover, the higher turbulence levels also lead to an increased mixing and therefore lower wall temperatures.

The previous results on wall shear stress and temperature with and without upstream turbulence show that the synthetic jets drive the flow most of the time but upstream turbulence can become preponderant in some specific locations and at specific instants. As a result, the wall heat transfer coefficient h differs with the two inlet flow conditions. Figure 16 compares the heat transfer coefficient along the jet axis (Figure 16a) and between the jets (Figure 16b), for lab-scale simulations with and without upstream turbulence, with the experimental data. On the jet axis, the simulation without upstream turbulence is already really close to the experimental data. This is consistent with the conclusions obtained from the wall shear stress analysis: the synthetic jets drive the flow at this location. However, there is still one difference along the first two perforation rows, where the results are even closer to the experimental data for the turbulent case. The upstream jets are directly bent by the grazing flow, and thus by the upstream turbulence, while the downstream jets are impacted by the upstream jets, which impose their dynamics to the flow. Between the jets, the effect of the upstream turbulence is much more important than for the jet axis. In particular, the heat transfer coefficient gets higher in the turbulent case from the third perforation and remains higher until the end of the plate. This higher value matches that of the experimental data downstream of the perforations, while the simulation without upstream turbulence is characterised by an underpredicted heat transfer coefficient. Here again, this observation is consistent with the wall shear stress analysis: between the jets, the flow behaviour is no longer driven by the synthetic jets but rather by the upstream conditions. It is thus logical to better predict heat transfer at this location with the appropriate numerical upstream conditions.

To conclude this analysis, the flow is mainly driven by two different mechanisms: the synthetic jets and the upstream conditions. In the specific operating point considered for the present study, the synthetic jets are the dominant mechanism most of the time, especially along the jet axis. Neglecting the upstream turbulence does not strongly affect the numerical results. However, between the jets the upstream turbulence does have an impact and should not be neglected. The importance of upstream conditions regarding the jets is characterised by the velocity ratio $M=\overline{W}/U_0$, which is here equal to 0.16. Other behaviours should be expected at different M values: for higher M values, the upstream conditions should actually become negligible, even between the jets, while low M values would lead to the preponderance of upstream conditions over the synthetic jet flow dynamics. Indeed, as the velocity ratio M increases, stronger interactions between the jets and the grazing boundary layer occur inducing a more rapid transition to turbulence [29].

7. Conclusions

The flow dynamics generated by an acoustic liner are characterised by a phenomenon called the synthetic jet: the ambient flow is successively sucked into and ejected from cavities. This mechanism leads to a complex boundary layer development and thus a complex wall heat transfer. In order to better understand wall heat transfer within acoustic liners, numerical simulations were run based on an experimental set-up developed by Institut Pprime (France). The synthetic jets in this configuration are not generated by acoustic resonance, as for acoustic liners, but by pistons. The dimensions and operating points are representative of a classic liner through a similitude methodology. Two sets of numerical simulations were run: one called ”lab scale”, based on the test rig, and one referred to as ”engine scale” with dimensions and flow conditions directly corresponding to engine conditions. These simulations enabled the validation of the similitude methodology while pointing out its limitations. Although the mechanism is not driven in a similar way (leading notably to a phase lag that appears in the engine scale but not in the lab scale), the synthetic jets are similar for both configurations. In particular, the wall heat transfer coefficients obtained for time-averaged fields are close. The results are in part compared to the ”flat plate” case by considering the flow dynamics along the plate far from the perforations. Simulations with and without inlet turbulence are compared and analysed for lab scale, and an important element is found to strongly impact the numerical results: the upstream flow turbulence modifies the boundary layer development and thus the wall heat transfer mechanisms. In fact, two competing phenomena are responsible for the flow dynamics: the synthetic jets and the grazing flow. Depending on the velocity ratio $M=\overline{W}/U_0$, one of these two phenomena can become preponderant. As was observed here, the synthetic jets completely drive the flow dynamics in the jet axis, where the upstream conditions do not affect the boundary layer development. However, between the jets, the upstream turbulence is no longer negligible: neglecting it leads to significant errors in wall heat transfer coefficient values. It is assumed that for higher M values the upstream conditions would tend to become completely negligible, while low M synthetic jets would be much more driven by the upstream conditions.