Qiang LIU, Zhenbing LUO, Xiong DENG, Lin WANG, Yan ZHOU

College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China

KEYWORDS

Abstract In order to improve the control ability of synthetic jets in compressible boundary layer,a novel control method based on dual synthetic cold/hot jets coupled control of velocity profile and temperature profile was proposed.As fundamental investigations on the effects of synthetic jet temperature on the jet behavior and flow field characteristics were essentially necessary, preliminary numerical simulations were conducted to study the influence of temperature (200 K and 400 K)on the flow field characteristics of synthetic jets using Large Eddy Simulations (LES) model.Time-averaged flow fields showed that different temperatures led to variable behavior of two strands of jets.For dual synthetic cold jets,a potential-core arose apparently with its height ranging from 0.01 to 0.03 m,while for dual synthetic hot jets,two strands of jets emerged downstream.The modal decomposition of instantaneous flow fields had been done using both Proper Orthogonal Decomposition(POD)and Dynamic Mode Decomposition(DMD).Various modes showed different characteristics of the flow fields. As the POD method focuses on the energy of flow while the DMD method focuses on the frequency, the first two modes had many similarities, but the third and fourth modes demonstrated completely different vortex structures.The current researches play a role of preliminary investigations for further and comprehensive exploration of novel flow control measures in global velocity field.

1. Introduction

The boundary layer transition,turbulent flow and flow separation of flight vehicle’s inner and outer flow fields are often accompanied with complex changes of aerodynamic force,aerodynamic heating and aerodynamic noise. For the outflow of the vehicle, the superiority of lower friction resistance for laminar boundary layer requires to delay the transition from laminar to turbulent flow,1,2whereas,to pursue a high starting performance of inlet flow field,engineers often adapt flow control technologies to accelerate the transition to turbulent flow because the turbulent boundary layer is more resistant to flow separation.3–5Therefore, flow control technologies can bring lift enhancement, drag reduction, heat reduction and noise reduction to flight vehicles, and have a boosting prospect in engineering application, thus having become the forefront and hot issue of fluid mechanics researches in recent decades.Flow control technologies can be divided into passive ones and active ones. Therein, simple design and high efficiency are the advantages of passive flow control methods, but the control effect would be undesirable if working beyond the designed state.6Meanwhile, active flow control technologies are widely studied because of their flexible controllability and wide applications.7–9As one type of active flow control technologies, zero-mass-flux synthetic jets actuator has been attracting much attention for its simple structure, compact design, flexible control ability and high stability.

‘‘Synthetic shear jet flow” was first proposed and experimentally verified by Savasand and Coles10in 1985. Smith and Glezer11then put forward the concept of zero-net-massflux synthetic jet in 1998. They pointed out that the essence of synthetic jets is that the reciprocating motions of vibrating membrane make the fluid generate periodic blowing and suction, and then a series of unsteady vortex rings form and flow downstream under the shear stress at the orifice.Subsequently,foreign and domestic scholars carried out series of researches on the characteristics of synthetic jets and its applications to boundary layer control. Hong et al.12–14experimentally investigated the effectiveness of synthetic jet to prevent flow separation caused by adverse pressure gradient, and found that the key of effective control was to utilize synthetic jet as the Tollmien-Schlichting (T-S) instability trigger to accelerate the viscous transition and thus to control the Kelvin-Helmholtz(K-H) instability. Zhong et al.15–18numerically investigated the interactions between synthetic jet and cross-flow or separated flow, and the formation and evolution characteristics of the vortex in laminar and turbulent boundary layer were also studied respectively. Numerical simulations were conducted by Liu et al. to investigate the influence of secondary flow in compressor cascade controlled by synthetic jet.19They found that with the excitation of synthetic jets,velocity fluctuations could be suppressed effectively,leading to a more stable flow status. Tang et al.20proposed to use Synthetic Jet Actuator (SJA) arrays for flow separation control over a low-speed wing model, making the lift coefficient and drag coefficient improve up to 27.4%and 19.6%respectively. The mechanism was that the injection of vortex structures altered boundary layer velocity profile and brought high-momentum fluids into the near-wall region, thus overcoming the flow separation.Xu and Zhou21placed synthetic jet actuators on the leading edge of a flying wing,finding that the selection of jet frequency would achieve different control effects of the longitudinal and lateral–directional aerodynamic characteristic. Emanuele and Zhong22used synthetic jets array to test the reduction of skin friction drag in turbulent boundary layer, and Particle Image Velocimetry (PIV) results showed that region of reduced skin friction drag arose.They proposed the model of vortical structures where the induced hairpin vortex legs created a downwash region and thus led to a local increase of streamwise velocity. As the failure and rupture of vibrating diaphragm might arise due to the pressure differential between the controlled flow and the environment, Luo et al.23,24innovatively placed both sides of the vibrating diaphragm in the controlled flow fields and invented a novel kind of actuators consisting of a piezoelectric vibrating diaphragm, and two cavities with slots, which are called Dual Synthetic Jets actuator (DSJ).

As synthetic jet/dual synthetic jets can only provide velocity-profile control, its control ability is limited in some special environments, such as compressible boundary layer.However, temperature change of fluids can also play a certain control effect on high-speed boundary layer. Mack25numerically investigated supersonic boundary layer using linear stability theory, and found that the wall cooling could make the second mode more unstable. The cooling wall will make the compressibility of the fluids more obvious and enhance the coherent structures of the turbulence.26,27

Recently, the author proposed and designed dual synthetic cold/hot jets actuator by adding cooling or heating modules into the cavities.28Coupled with both velocity control and temperature control, the kinetic energy and thermal energy can operate on the boundary layer simultaneously.Unlike wall temperature control,cold jets or hot jets inject directly into the boundary layer,resulting in a quicker heat exchange and more intensified movements of the vortex and waves, which achieve effective control of compressible boundary layer. Compared with high power-consuming plasma jet or wall cooling, this novel control method has the potential to achieve compressible boundary layer control with a low power consumption.

But by far, researches of dual synthetic cold/hot jets have been mainly aimed at its application and no investigation has been conducted on the characteristics of dual synthetic cold/hot jets themselves. We still know little about the effects of jet temperature on the flow field characteristics of synthetic jets.It is of great importance to conduct preliminary investigation on the effects of jet temperature on the formation and development of dual synthetic jets,and complex flow phenomena involve vortex instability, deformation, fusion and breakup. This paper will give numerical simulations of dual synthetic cold/hot jets using large eddy simulation method and investigate the effects of temperature on synthetic jets’behavior and dynamic flow structures using both Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD). The rest of this paper is organized as follows: Section 2 discusses the numerical computations employed in this paper. Section 3 briefly explains the POD and DMD algorithms.Section 4 gives the main results and discussion in detail and Section 5 presents conclusions of this work.

2. Numerical method

Fig.1 gives the configuration of DSJ,29a unique feature of which is that it contains two cavities with slots. When the piezoelectric vibrating diaphragm oscillates back and forth,two strands of unsteady jets are produced with a phase difference of 180°. The jets interact with each other when moving downstream and emerge into a new synthetic jet finally. Dual synthetic cold/hot jets actuator was proposed by adding heating modules or cooling modules into the cavities.

Fig.1 Configuration of dual synthetic jets actuator.29

Fig.2 Mesh structures in computational domain.

The mesh structures in the computational domain are shown in Fig.2, with the origin of coordinate located on the center of the circular diaphragm. In this paper,xcorresponds to the breadthwise direction,ythe streamwise direction andzthe spanwise direction. The distance to the orifices plane is defined as the heighth,which is shown in Fig.1.Both the flow fields inside the cavities and the orifices were simulated to get more accurate results.Total number of grids was 1.378 million with both structured meshes and unstructured meshes. The actuator diaphragm radius was 0.023 m, with a sinusoidal velocity boundary condition,u(t)=umsin(2πft), whereumwas set as 0.4 m/s,fthe driving frequency 10 Hz andtthe time step size. The temperature of cold jets was set to 200 K and that of hot ones 400 K. Free stream boundary condition was imposed on the controlled flow fields, and its initial temperature was set at 300 K.

The flow fields were computed employing LES model that discretized the incompressible Navier-Stokes equations and the continuity equations based on finite-volume codes.Dynamic Smagorinsky model was employed to solve subgrid-scale stress proposed by Lilly.30Second-order Crank-Nicolson method was performed for time integration, and pressure term and convective terms were solved with secondorder spatial discretization and third-order MUSCL scheme respectively. Δy+at the first layer off the orifice was less than 0.5 and corresponding Δx+was less than 29 and Δz+less than 40.

To validate the code, numerical simulation was carried out with the same condition which was experimentally investigated by Lee et al.31Fig.3 gives the centerline distributions of dimensionless mean streamwise velocity curves(dis the orifice width,his the height away from orifice andv/vmaxis dimensionless mean streamwise velocity).The simulation result using the code in this paper is in good agreement with the experiment conducted by Lee et al.,31proving that the code could be used to simulate synthetic jets effectively.Grid independence analysis was also performed employing three different grids:a small grid with 7.4×105cells, a medium grid with 1.4×106cells and a big grid with 2.8×106cells. The pressure of the monitoring point depicts that the results with medium grid and big grid are consistent with each other, higher than that of the small grid. Therefore, to reduce the computational costs,the medium grid was selected to conduct the numerical simulations.

Fig.3 Validation of code-centerline distributions of mean streamwise velocities.

Fig.4 Velocity magnitude contour maps of meanflow fields.

As the working frequency of the actuator was 10 Hz, the time step was set as 7.8125×10-5s, so that the total number of time steps was 1280 in each period. To get the developed flow fields, the simulations were run for 50 periods until the residual convergence was less than 10-4.

3. Model reduction techniques

This paper adapted the classical POD technique and novel DMD technique to extract major large-scale coherent structures. Proper orthogonal decomposition32and dynamic mode decomposition33methods have been proved to be effective to identify dominant characteristics and extra classical vortex structures based on energy or frequency. One of the advantages of these methods is that they can be employed to analyze complex flow structures with reduced portions of the spatial domain. Both of them have been applied to many fields, such as confined turbulent jet,34circular cylinder wake,35–37flow around a hemisphere-cylinder,38,39flow past a multi-stream nozzle40and zero-net-mass-flux jet.41Therefore, both of the methods are applied in this paper to detect dominant flow structures and growing instabilities temporally and spatially.The following part gives a brief introduction to the methods.

3.1. POD method

As put by Lumley et al.,32POD technique is a useful and powerful technique to analyze the complex flow phenomenon and extract large-scale coherent structures from massive and highdimensional data originated from experiments and numerical simulations.

The basic idea of POD is to find a set of optimal orthogonal basesai(t) and φi(x) to describe the target fieldu(x,t) that change in both temporal and spatial space. The target field is presented in the form of

Asai(t) and φi(x) are orthogonal, the above equation decouplesu(x,t)into two individual parts,namely time coefficients and spatial modes respectively.Meyer et al.42found that the maximum problem can be transformed into an eigenvalue problem of matrix and proposed the snapshots POD. Construct the velocity field information ofNsnapshots into a matrixU=[u1,u2, ...,uN], and the autocovariance matrixC=UTUcan be gotten.

Fig.5 Central line velocity distributions of actuator orifice.

Fig.6 Time-averaged streamwise velocities for different h.

Fig.7 Time-averaged spanwise velocities for different h.

By solving the eigenvalues of the above matrix, an optimal decomposition can be obtained. The specific steps are as follows:CAi=λiAi,where λiis the eigenvalue,with the corresponding eigenvectorAi. Arrange the eigenvalues in descending order, and the first few modes will occupy a large amount of energy. After that, we can construct POD modes:

The snapshots POD method has a smaller demand for computer memories, is suitable for the processing of data with a much higher spatial resolution than the temporal resolution,and thus has been widely adapted.

3.2. DMD method

DMD is a new method of flow field decomposition developed from Koopman mode analysis in recent years.43–45This method can obtain the characteristics of flow field in both temporal and spatial aspects, escaping the limits of flow types.Besides, frequencies and growth rates of modes can also be obtained. Like POD technique, a series of time-varying snapshots are constructed as sequence matrix:

where the real part of ωidenotes the growth rate and the imaginary part denotes the frequency. Unlike POD, the DMD modes are sorted by their amplitudes ||Φi||. Besides, eigenvalues ωican also be used to study the stability characteristics of DMD modes. Place the eigenvalues in the complex domain,and if they lie on a unit circle, it means that those modes are stable with zero growth rates. Meanwhile, if the eigenvalues lie inside and outside of the unite circle, it means that those modes are damped and undamped respectively.

4. Results and discussion

4.1. Flow field characteristics

Stroke is an important parameter to characterize the formation and evolution of synthetic jet actuator. Based on the‘‘slug” modal proposed by Smith and Glezer,11dimensionless stroke is defined as

Fig.8 Power spectrum analysis of streamwise velocity.

Fig.9Contour maps of streamwise turbulent fluctuation intensities 〈v ′v′〉.

Fig.5 plots streamwise velocity and spanwise velocity distribution along the central line of one orifice. In the vicinity of the actuator orifice,streamwise velocity and spanwise velocity of dual synthetic cold jet were higher than that of the hot ones. However, when the height exceeded 0.04 m, streamwise velocity of the former fell below that of the latter, whereas the spanwise velocities of the both were approximately equivalent and decreased in a slow trend.

Fig.10 Time-averaged temperature distributions at different h.

The velocity distributions at different heights are specifically demonstrated in Figs. 6 and 7. As apparently shown in Fig.6, at almost every height, the peak values of streamwise velocity of dual synthetic cold jets located on thez-axis. Nevertheless, when the height was lower than 0.04 m, there were two peaks in the case of dual synthetic hot jets.One interesting phenomenon was that all of curves plotted in Fig.6 were symmetric around thez-axis whereas Fig.7 depicted that the curves of spanwise velocities demonstrated an antisymmetric distribution.This was because the essence of dual synthetic jets was that two cavities compressed mutually to form two strands of synthetic jet with a phase difference of 180°. Thus, for the streamwise velocities of time-averaged flow fields, they were symmetrical due to the average of a large number of data.But considering that the self-induced velocities of vortices on both sides were opposite, the direction of the spanwise velocities was also different and the curves demonstrated an antisymmetric shape eventually. As the height increased, the vortex structures dissipated and the fluctuations of spanwise velocity decreased gradually, with distribution lines tending to have lower slopes. Overall, the spanwise velocity of synthetic cold jets was larger than that of the hot ones when the height exceeded 0.01 m, which meant that the diffusion rate of the former was faster than the latter.

To detect the dominant frequencies of the flow fields, spectrum analysis was performed using the velocity database obtained by a number of monitoring points. Fig.8 shows the power spectrum results of instantaneous velocity at different height of the central line. When the heighth=0 m, the frequencies corresponding to peak value of the power spectra were in accordance with actuator’s driving frequency, with high Power Spectral Density (PSD) in both cases. Nevertheless, as the height of the monitoring points increased, a harmonic frequency 20 Hz was observed in each case. This phenomenon was due to the periodic blowing and suction of the jets. And the power spectral densities decreased significantly at first and then decreased slowly, predicating that the energy was mainly concentrated in the vicinity of orifice and dissipated gradually with the increase of height. Besides, the power spectrum density of the cold jets was higher than that of the hot ones, indicating that the flow field energy of synthetic cold jets was more concentrated around the orifice.

Time-averaged temperature fields in Fig.10 demonstrated that in quiescent condition, dual synthetic cold/hot jets could effectively change the temperature of surrounding flow fields.Even if the height was up to 0.09 m,the temperature remained a discrepancy by about 5 K as well.

Fig.11 Iso-surfaces of instantaneous Q=1×106 at maximum blowing stage.46

Fig.11 gives the instantaneous coherent structures at maximum blowing stage identified byQ-criterion method.46There were apparently differences of iso-surfaces of instantaneousQ=1×106between cold jets and hot jets.For dual synthetic cold jets, the vortices were mainly concentrated in the near field and tilted each other to middle area of the two orifices,forming the potential-core eventually. By contrast, dual synthetic hot jets seemed to have a larger stroke and produce more stable vortex structures which flowed downstream straightforward.

4.2. POD analysis

In Section 4.1, the analysis was mainly based on the timeaveraged flow fields. To get a better understanding of vortex structures contained in the instantaneous flow fields, POD and DMD methods were applied. Both POD and DMD decompositions were based on unsteady database from the last five periods, a total of 3200 snapshots. The time interval Δtbetween each snapshot was 1.5625×10-4s.

Fig.12 shows the comparison of the relative mode energy and cumulative mode energy of the first 100 POD modes for streamwise velocity between dual synthetic cold/hot jets. In both cases, the first few modes almost occupied most of the energy and the remaining energy was more distributed, especially for the case of the synthetic hot jets. This was because the POD method focused on the energy of flow and the first few modes represented energetic large-scale vortex structures while the higher modes demonstrated small-scale vortex structures. It was worthy to note that for synthetic hot jets, the energy of the first mode was up to about 80%,far higher than that of the cold jets,which was at around 60%,indicating that the first POD mode of dual synthetic hot jets took up most of energy.

Fig.12 Relative mode energy and percentage of cumulative POD modes energy versus the first 100 modes.

Fig.13 POD analysis of streamwise velocity for Tj=200 K.

POD analysis results for dual synthetic cold/hot jets are shown in Fig.13 and Fig.14 respectively, including the first four modes with corresponding time history of POD coefficients and power spectra. Fig.13(a) depicts that the first two POD modes of streamwise velocity were symmetric aboutzaxis, representing the most energetic vortex structures around the jet orifice, while the latter two modes represent the trifling energy in the downstream of jets. This phenomenon was also confirmed from the results of power spectrum analysis in Fig.13(c), in which the first two modes occupied most of the energy while the energy of the latter two was lower by two orders of magnitude.Fig.13(b)reports the first four energy-sorted POD time coefficients along five jet periods. It showed that temporal coefficienta1（t） had a regular sinusoidal shape, with a period corresponding to the jet period. However, time coefficients for Mode 2,Mode 3 and Mode 4 seemed not to be so regularly sinusoidal, and their amplitudes were lower as well.

For synthetic hot jets, the first four POD modes were all symmetric aboutz-axis. Mode 1 mainly demonstrated vortex structures near the orifice and the other three modes mainly showed the evolution of jets in the downstream, despite occupying a lower energy. As to power spectrum of time coefficients, the one difference between Fig.13(c) and Fig.14(c)was that Mode 2 PSD of synthetic hot jets was lower than that of the cold ones, signifying a lower energetic mode.

Fig.14 POD analysis of streamwise velocity for Tj=400 K.

More significantly, however, it was obvious that in both cases,the PSD of Mode 4 time coefficients had two peaks.This phenomenon was explained by the contour maps of Mode 4,which represented the small-scale vortex structures near the orifice and downstream at the same time.

4.3. DMD analysis

The drawback of POD method is that it obtains the flow fields by statistical means, and thus loses the phase information of the system. As POD modes are still doped with vortex structures at different frequency,it is difficult to analyze the original flow field from the point of view of dynamics. Coincidentally,the decomposition and extraction in DMD techniques are based on the dynamics of flow fields and the modes obtained are time-irrelevant.

As the first mode of DMD results is usually the meanflow field, it is labeled as the 0th order and the other modes are sorted descendingly according to the energy level in order to compare with POD results conveniently. The first four modes are depicted in Fig.15(a). For brevity, only the real part of each mode was shown. As depicted in the figures, the decomposition results of Mode 1 and Mode 2 were similar to that of POD,demonstrating vortex structures near the orifices of actuator. The spatial scale of vortex structures shown in Mode 3 and Mode 4 decreased with the increase of frequency.Fig.15(b)gives the amplitudes of DMD modes.The amplitude of Mode 1 was the highest and its frequency was in accordance with the driving frequency of actuator. Harmonic frequency,20 Hz, appeared in Mode 2. But Mode 3 and Mode 4 had higher frequencies, denoting small-scale vortex structures produced after the full development of synthetic jets, which was consistent with contour maps in Fig.15(a).

Fig.15 DMD analysis of streamwise velocity for Tj=200 K.

To observe the temporal variation of the first four DMD modes, Fig.15(c) plots the DMD coefficients. An intuitive view of time-varying coefficients of Mode 1 and Mode 2 presented perfect periodic shapes. Jonathan believes that amplitudes of DMD coefficients reflect the trend of corresponding modes,a form of growing or decaying.47The amplitude of Mode 1 showed a slowly decay trend, and Mode 2 maintained a constant amplitude, indicating that these modes were neutral-stable. The frequencies of Mode 3 and Mode 4 were very high,but their amplitudes dropped swiftly to 0,indicating that the eigenvalues of corresponding modes fell within the unit circle.33

Fig.16 shows the DMD analysis results of dual synthetic hot jets. Like POD results, Mode 1 and Mode 2 mainly demonstrated large-scale vortex structures near the orifice and downstream, and their mode coefficients also approximately maintained constant. However, the difference was that Mode 3 and Mode 4 presented fragmentized vortex structures,with their mode coefficient presenting a decay trend in a short time.Compared to the results of dual synthetic cold jets,coefficient of Mode 1 had higher amplitude,indicating that it took up more energy. Furthermore, it was obvious that modes in high amplitudes of dual synthetic hot jets tended to lower frequencies compared with cold ones.

Fig.16 DMD analysis of streamwise velocity for Tj=400 K.

From the analysis and comparison above, the merits and drawbacks of POD and DMD methods were well demonstrated. Focuses on the energy of flow, the POD technique could capture large-scale vortex structures or large-scale behavior whereas the DMD method focuses on the frequency,and it represented the perturbation dynamics. The higher the mode was, the higher the frequency was. DMD seemed to be more suitable for identifying and tracking the vortex structures of synthetic jets.

Therefore, only the DMD method was adapted to reconstruct the flow fields. Figs. 17 and 18 give the evolution of streamwise velocity during one working circle, att0,t0+T/4,t0+T/2,t0+3T/2 of dual synthetic cold/hot jets respectively. Reconstructed fluctuating streamwise velocity contours preserved the overall characteristics of the original velocity fields and were in good accordance with the instantaneous ones.At the same time,some small-scale structures were eliminated, making the spatial characteristics of large-scale coherent structures more prominent. For dual synthetic cold jets, the fluctuating streamwise velocity was mainly concentrated near the orifices, exhibiting small vortex structures,whereas dual synthetic hot jets seemed to have a larger stroke and the vortex structures gradually dissipated downstream.

Fig.17 Fluctuating streamwise velocity evolution for Tj=200 K at t0, t0+T/4, t0+T/2, t0+3 T/2.

Fig.18 Fluctuating streamwise velocity evolution for Tj=400 K at t0, t0+T/4, t0+T/2, t0+3 T/2.

5. Conclusions

This paper proposed a novel flow control method based on dual synthetic cold/hot jets and carried out numerical simulations of jet temperature on vortex structures and flow field characteristics using POD and DMD methods.

Numerical simulations were conducted to investigate the influences of jet temperature on the flow field characteristics of synthetic jets. Time-averaged flow fields showed that at low jet temperature, two strands of jets emerged quickly in space after ejecting from orifices and formed a potential-core area. This potential-core area was responsible for higher streamwise and spanwise velocity at the height of 0.01–0.03 m. By contrast, as the temperature of dual synthetic hot jets was higher than that of the surroundings, the behavior of jets was completely different and two strands of jets separated distinctly from each other and then merged downstream.For dual synthetic cold jets, the dimensionless stroke was 2.8,whereas for hot jets, the value was 4.25.

POD analysis of the instantaneous velocity snapshots revealed that the first four POD modes of dual synthetic hot jets occupied about 90% of the energy while those of the cold ones only occupied 76%.However,power spectrum analysis of time coefficients demonstrated that POD modes sorted by occupied energy could not be distinguished upon frequency characteristics, and thus the vortex structures at different frequencies mingled together. As a result, without time independence, the extracted POD modes were often the superposition of multiple frequencies’ structures and had inconceivable applications in understanding the dynamic characteristics of flow field. DMD method was applied to remedy this defect.The first two DMD modes,selected based on mode amplitudes, showed many similarities with POD modes.Distinctively, the third and fourth modes exhibited quantity of small-scale vortex structures at high frequencies, indicating that DMD was more suitable for extracting high frequency characteristics of flow fields and tracking possible eigenstructures. Finally, the gained experience of using POD and DMD in this paper illustrated that DMD had more advantages in extracting characteristic turbulence structures from complex flow fields.

Future work is to investigate the interaction among external flow,condition inside both cavities and power limits of various actuators as well as the angle between external flow direction and normal direction of slots in detail.Based on these preliminary investigations, further and comprehensive explorations will be taken to apply this novel control method to global velocity field.

Acknowledgements

This study was supported by the National Natural Science Foundation of China (Nos. 11602299, 11502295, 11572349,11872374 and 51809271).