LU Daju,ZHANG Kai,DONG Hang,XU Ming,SU Hua

(1.Key Laboratory of Science and Technology on High Energy Laser，CAEP, Mianyang 621900,China; 2.Institute of Applied Electronics, CAEP, Mianyang 621900, China; 3. Institute of Applied Physics and Computational Mathematics, Beijing 100094, China)

Abstract:This study simulated the aero-optical effect of beams with different projection directions around conformal turrets with different radii(400 mm and 2 000 mm). The Mach number was fixed, and the Reynolds number remained similar in all cases. The space-time characteristic of beam quality factor and beam tilt angle caused by the aero-optical effect was comprehensively studied. The aero-optical effect was mainly determined based on the mean flow effect, and the characteristic frequency of the mean flow effect was determined based on that of the flow around the turret. The spatial characteristic of the aero-optical effect was accurately reproduced in the reduced-scale experiments; however, the temporal characteristic was poorly simulated . It is pointed out that the wavefront distortion statistics and fluctuation values caused by aero-optical effect aero-optical effect appear to be minimized at a projection direction of 40° zenith in the forward direction, and the increase of projection radius can result in fast increaseing of the mean flow effect while maintaining the turbulent effect almost constant.

Key words:radius of turret; mean flow effect; turbulent effect; beam quality; tilt

Introduction

Hemisphere-on-cylinder turret provides a convenient mechanical system to point-and-track the laser beam, and is widely used in many practical applications, such as airborne laser(ABL), advanced tactical laser(ATL), airborne aero-optics laboratory(AAOL), and so on. However, a turret bluff-body shape creates a complex turbulent flow around it, and the turbulent fluctuations create unsteady density fluctuations around the turret and impose detrimental aero-optical effects on the outgoing beam. In the last decade or so, flow features around turrets and related aero-optical distortions have being extensively studied and fairly well-understood[1-10].

As the development of wind tunnel technology, the same Mach number, Reynolds number and geometry experiments become true, but almost all the experiments are reduced-size experiments. The model size affects the characteristic frequency of the flow over turrets, and induces the difference of aero-optical effect between the experiment and the reality.

The Strouhal number represents a measure of the ratio of inertial forces due to the unsteadiness of the flow or local acceleration to the inertial forces due to changes in velocity from one point to another in the flow field. The dominant frequency of the sound generated by the cylinder can be calculated using the Strouhal number.

The current work explores the space-time characteristic of beam quality factor and beam tilt angle caused by the aero-optical effect over conformal hemisphere-on-cylinder turrets for different radii, and focus on the scale effect, the effect of projecting radius and direction, the ratio of mean flow effect to turbulent effect, and the optimization of the projection radius.

1 Computation model

1.1 Flow compution

The 400 mm diameter conformal turret at surface environment and 2 000 mm diameter conformal turret models flying at 12 km altitude were analyzed using the large fluid calculation software SAED[11-13]. The mesh strategy of SAED is structured multi-block composted classic Euler mesh, which ensures the flux conservative across the blocks. As for the discrete scheme, finite volume method is adopted to ensure the flux conservation during solving the governing equation-S equations, where the nonlinear convective term is discretized by upwind Roe scheme with 2nd order accuracy, the viscous term is done by shifted control volume method. Reynolds-averaged Navier-Stokes equations (RANS) modeling is used, where the Reynolds stress is modeled by Chien’sk-εversion,a two-equation modeling suitable for turbulent boundary layer flow. The parallel computations are realized by using message passing interface (MPI) to enhance the efficiency of the solve.

The turret model is a hemisphere-cylinder structure, and the height of the cylinder is the same as the diameter of the hemisphere. The time characteristic of aero-optical effect was analyzed by unsteady calculation. The total grid scale is more than 30 million, and local encryption is taken near the hemisphere of the turret; the calculation step takes 5 μs, and a total of about 28.000 steps is calculated.

When analyzing the data, we need pay attention to the variation of Reynolds number and characteristic frequency of flow field for different calculation models, and Mach number is chosen as 0.7 for all conditions. The Reynolds number is

(1)

Whereρis the density of air，fis characteristic frequency，Uis the velocity of plane，Dis the diameter of turret,μis viscosity coefficient,Sris the Strouhal number. The Strouhal number for a flowed cylinder is around 0.2[14]. In this paper,Sris taken as 0.215, and the atmosphere parameter is taken from the standard atmosphere 1976[15]: at the surface,ρis 1.225 kg·m-3, pressure is 1.013 3×105Pa, temperature is 288.15 K, μ is 1.789 4×10-5kg·s2m-1; at the height of 12 km,ρis 0.311 9 kg·m-3, pressure is 1.94×104Pa, temperature is 216.65 K,μis 1.421 6×10-5kg·s2m-1. The results are shown in Table 1, and we can see that Reynolds number of the two models are similar, and the effects of turret diameter on aero-optical effect can be studied by comparison.

Aero-optical effect is the function of relative diameter. The relative diameter is defined as the ratio of the emission diameter to the turret diameter, and in this paper, the relative diameter is taken as 0.33 unless otherwise specified, which is the same as the AAOL experiment.

The characteristics of the flow field around the turret are different in different areas of the turret, such as the front, side, top, rear, etc. And the corresponding aero-optical effects are also very different. Fig.1 is the flow line around a turret ,and it shows a classical flow around the turret.

Fig.1 Illustration of flow around turret

In order to compare the effect of different areas of the flow field on aero-optical effects, we analyzed and compared the changes of aero-optical effects with different emission directions. Figure 2 shows 9 emission directions. Directions 1 to 5 are in the central axis which is the line that passes through the center of the turret along the flow field. Zenith angle and azimuth are used to define different directions. The zenith angle is the angle between the direction of the emission and the direction perpendicular to the direction of the central axis. The azimuth is centered on the central axis, is 0° along the direction of the flow, and rotated from 0° to 360° counterclockwise . On the central axis, the windward direction is 180° and the downwind direction is 0°. The angular coordinates of the 9 directions from 1 to 9 are(35°，180°)，(14°，180°)，(0°，0°)，(14°，0°)，(35°，0°)，(14°，90°)，(14°，270°)，(14°，135°)，(14°，225°). It is obvious that directions 6 & 7 and directions 8 & 9 are mutually symmetrical. Their aero-optical effects should be similar, which can also be used as an indicator to measure the reliability of the simulation results.

Fig.2 Illustration of projection directions

1.2 Aero-optical effects computation

The optical path lengthOPL, the optical path differenceOPD, and the root mean square of optical path difference over the apertureOPDrmsis computed from the density distribution as follows, and the wavefront is the negative of OPD :

(2)

By superimposing the wavefront distortion caused by the aero-optical effect on the plane wave, the far-field intensity realization is obtained to find the beam-quality and tilt. The beam quality factorβis defined as the ratio of radius of far-field spot with 83.7% of total power to ideal condition , and the beam tilt is the deviation of the centroid of far-field spot to the screen center.

2 Time characteristics of aero-optical effect

2.1 Time characteristics of aero-optical effect and spectrum analysis

In Fig.3～8, the beam tilt and beam quality factor as a function of time caused by aero-optics effects in different directions of Model 1 and Model 2 are compared. The method of FFT has been applied to analysis on the specific direction of Model 1. FFT cannot be applied to analysis on Model 2 because of the long period. Its calculation time is only equivalent to 2 to 3 periods. The characteristic frequency of Model 2 receives directly according to the period.

Fig.3 Tilt angle of 400 mm diameter turret as function of time(in directions 1，3，and 5)

Fig.4 Beam quality factor of 400 mm diameter turret as a function of time(in direction 1)

Fig.5 Beam quality factor of 400 mm diameter turret as a function of time(in directions 6 and 7)

Fig.6 FFT analysis of beam quality factor of 400 mm diameter turret in direction 5

Fig.7 Tilt angle of 2 000 mm diameter turret as a function of time(in directions 2,3, and 5)

Fig.8 Beam quality factor of 2 000 mm diameter turret as a function of time(in directions 2,8, and 9)

The results show that the beam tilt caused by aero-optical effect varying with time is very similar in the symmetry of the directions 6&7, as well as the directions 8&9, which confirms the reliability of the calculation results to a certain extent. Figure 6 shows that the characteristic frequency of Model 1 is 122 Hz whenSris 0.215. Figure 7 and 8 show that the periods of beam quality and beam tilt are both 0.044 ms in the windward direction of Model 2, when the characteristic frequency is 22.7 Hz andSris 0.22.

From the perspective of beam quality and beam tilt, the aero-optical effects of the scaled model and the actual size model are compared with time. It can be seen that they have the same calibration law, that is, the characteristic frequency of the aerodynamic effect is equal to that of the turret and the aircraft. Further analysis shows that the fluctuation spectrum of the beam quality is more complicated for different directions of emission. It has fine structure, andSralso changes slightly. The main reason is that in the aero-optical effect, the mean flow field effect is relatively large, which determines the characteristics of the aero-optical effect with time, and the characteristic frequency of the mean flow field effect is determined by the characteristic frequency of the flow field.

2.2 Statistical properties of aero-optical effect

In this section, we studied the statistical results of aero-optical effects of different models along different emission directions, including the average value of the beam quality caused by the aero-optical effect, the mean square error of the beam quality, the coefficient of variation of the beam quality, the average value of the beam tilt angle, the mean square error of the beam tilt angle and the coefficient of variation of the beam tilt angle, where the coefficient of variation is the ratio of the mean squared difference to the mean value.

The beam quality factor and the average of beam tilt angle can be considered to be mainly influenced by the mean flow field effect, while the fluctuation value is mainly affected by the turbulence effect, so the coefficient of variation can be used as a parameter to measure the turbulence effect. The mean flow field effect is derived from the entire flow field, and the characteristic frequency of the flow field is determined by the velocity of the plane, the diameter of the turret andSr. Because the absolute value of mean flow field is large, the beam quality and the beam tilt still have a period determined by the characteristic frequency of the flow field.

From Fig.9 to Fig.11, it can be found that in the emission direction 1, the average value and fluctuation value of the wavefront distortion caused by the aero-optical effect are significantly lower than other directions. We can find the best launch angle around this direction. In the windward directions (1, 2, 8, 9), the flow field is a nearly constant flow field, and the coefficients of variation of beam quality and beam tilt are relatively small. In the emission direction 3, due to the existence of a stable shock wave, the coefficient of variation of the beam quality is small, and the absolute value of the beam tilt is small, while the fluctuation value is large. It indicates that the beam tilt in this direction is mainly due to the turbulence effect. In the directions 4, 5, 6, and 7, it is a typical unsteady flow field, so the coefficient of variation is relatively larger.

Fig.9 Mean beam quality factors of 400 nn and 2 000 mm diameter turrets versus projecting directions

Fig.10 Mean tilt angles of 400 mm and 2 000 mm diameter turrets versus projecting directions

Fig.11 Coefficient of variation as a function of different projecting directions

From the comparison of the two models, it can be found that the beam tilt of the reduced-scale model is significantly larger than the actual model, and the beam quality factor is slightly smaller than the actual model. Because the beam tilt is inversely proportional to the emission aperture. For the same wavefront distortion, the smaller the emission aperture, the larger the beam tilt, so the beam tilt of the reduced-scale model is about five times that of the actual model. There is no directly relationship between the beam quality and the emission aperture. The beam quality is mainly determined by the wavefront distortion. In the case of the same geometric shape, the wavefront distortion is determined by the Mach number and the Reynolds number. The Reynolds number of the reduced-scale model is slightly small, so the beam quality is also slightly smaller than the actual size model.The coefficient of variation is similar for both models, indicating that for both models, the contribution of the turbulent effect is similar. In summary, the statistical characteristics of the aero-optical effect of the actual size model, including the mean value and the fluctuation value, can be effectively simulated by the reduced-scale experiment under the condition that the Mach number and the Reynolds number are the same.

2.3 Time related properties of aero-optical effect

In this section we analyzed the time-related properties of the aero-optical effects of the turret by using the ideally corrected method. The specific method is to calculate the beam tilt and beam quality caused by the residual wavefront , where the residual wavefront is the difference between the current wavefront and the wavefront after a certain time interval, and compare the beam tilt and beam quality caused by the current wavefront. For a certain time interval Δt, the same operation is performed for all time arrays, the average value is taken as the analysis index, and the correction frequency is the reciprocal of the time interval Δt. Obviously, a ratio greater than 1 indicates that there is no correction result. This method can roughly calculate the time-related properties of the aero-optical effect. Figure 12～15 show the results of beam tilt and beam quality factor changed with the corrected frequency in different projection directions around turrets with different radii (400 mm and 2 000 mm).

Fig.12 Relative residual errors of 2 000 mm diameter turret versus correction frequency

Fig.13 Relative residual errors of 400 mm diameter turret versus correction frequency

Fig.14 Relative residual error of beam quality factor of 400 mm diameter turret versus correction frequency

Fig.15 Relative residual error of beam quality factor of 400 mm diameter turret versus correction frequency

It shows that the correction effect varies greatly for different directions of emission. The flow field in the windward direction is relatively stable, especially in the direction of emission 1, and the correction frequency is higher than the characteristic frequency of the flow field, which has a very good effect. The zenith direction and the leeward direction as well as the lateral direction require a high frequency to have a good correction effect due to the turbulent effect. At the same time, the larger turret diameter is, the lower the correction frequency required to achieve the same correction effect, and the reason is that the larger the turret diameter corresponds to the lower characteristic frequency of the flow field.

It can be seen that the time related properties of model of 400 mm turret is worse than model of 2 m turret in most directions, which indicates that the time related properties depends on turret diameter. The characteristic frequency of the reduced-scale model is usually higher than the actual model, meaning that it is often impossible to accurately simulate the time related properties of the aero-optical effects of actual flight conditions with reduced-scale experiment.

3 Space characteristics of aero-optical effect

3.1 Aero-optical effect effects versus projection direction

The above results show that in addition to the windward direction, the aero-optical effects in other directions can cause severe distortion of the emitted beam. Therefore, rectification technology is usually used to improve the characteristics of the flow field[16]. But the aero-optical effect of forward emission is relatively weak, and is suitable as emission condition. We investigated the aero-optical effect for different emission directions under the forward emission condition. Fig.16～18 show the aero-optical effect for different emission directions with the zenith angle from 20° to 70 °. Here the projection radius is 0.67 m and the emission direction is the central axis. We investigated the mean value and root mean square values of the beam quality and beam tilt caused by aero-optical effects. Roughly speaking, the average value reflects the average flow field effect, and the root mean square value of the fluctuation reflects the turbulent effect.

Fig.16 and 17 show the absolute values of beam quality and tilt angle. It can be seen that the fluctuation values are relatively small for different emission directions, the reason is that the flow field in the windward direction of the turret is constant and stable. At the zenith angle of about 40°, the average value of the beam quality and the fluctuation value are relatively small; the beam tilt is relatively serious, but the beam tilt is easy to overcome, so this direction is chosen as the candidate for the favorite emission direction. In Fig.18, the variation coefficient is shown as the variation of the coefficient of variation with the angle of emission. The coefficient of variation is defined as the ratio of the mean square error to the mean value, which can be used as a parameter to measure the relative magnitude of the turbulent effect. It can be seen that the turbulence effect accounts for 10% to 15% near the zenith angle of 40 °.

Fig.16 Beam quality factor versus projection direction

Fig.17 Tilt angle versus projection direction

Fig.18 Coefficients of variation versus projection direction

In the direction deviating from the central axis, the asymmetry of the expected flow field causes large distortion of the wavefront, resulting in large beam quality factor. Fig.19 shows the beam quality versus azimuth when zenith angle is 40°. Here the 0° of the azimuth is defined as the windward direction of the central axis. It shows that the beam quality increases by 3 to 4 times as the emission direction deviates from the central axis, which is significantly deteriorated. Based on the above analysis, we can choose 40° of zenith angle and 0° of the azimuth along the central axis as the optimal emission direction considering the aero-optical effect.

Fig.19 Beam quality factor versus azimuth

3.2 Aero-optical effect versus projection radii

The far-field spot is reduced due to the diffraction effect when the projection radius is increased. However，as the range of the flow field increases, the wavefront distortion caused by the aero-optical effect increases, and the far-field spot increases. The average of the beam quality and the beam tilt angle can be considered to be mainly contributed by the average flow field effect, while the fluctuation value is mainly contributed by the turbulence effect, so the coefficient of variation can be used as a parameter to measure the turbulent effect. Table 2 shows aero-optical effect versus projection radii for 40° of zenith angle in the central axis. It can be seen that as the projection radius increases, the average beam quality increases, but the coefficient of variation decreases. At the same time, the tilt angle decreases, and the coefficient of variation of the tilt angle increases.

Table 2 Comparison of aero-optical effect under different projection radii

Fig.20 and 21 show the beam quality and beam tilt versus time under different projection radii. It can be seen that the temporal characteristics of aero-optical effects are basically the same under different projection radii.

Fig.20 Time dependence of beam quality factor under different projection radii

Fig.21 Time dependence of tilt angle under different projection radii

The time-related properties of the aero-optical effects of the turret were further analyzed by an ideal correction method, and the results are shown in Fig.22 and Fig.23 , where the relative residual error is defined as the residual wavefront error divided by the initial wavefront. It is obvious that the relative correction result is better when the projection radii increase, the reason is that, when the projection radii increase, the turbulence effects remain unchanged and the mean flow field effect increases. As the characteristic frequency of the mean flow field is relatively low, the aero-optical effect caused by the mean flow field can be well corrected. The absolute value of the ideal correction shows that the beam quality after correction under different projection radii is not much different, especially for the high-frequency correction, which indicates that for the low-frequency correction, the mean flow field effect is basically corrected, but the turbulence effect still exists. The beam quality caused by the turbulence effect does not vary with the projection radii, it is basically fixed. This result shows that increasing the projection radius can effectively increase the power density of target under the condition that the mean flow field effect can be corrected.

Fig.22 Beam quality factor versus correction frequency under different projection radii

Fig.23 Relative residual errors of beam quality factor versus correction frequency under different projection radii

3.3 Wavefront correlation characteristics

Different from atmosphere propagation, the wavefront induced by aero-optical effect is anisotropic. To analyze the wavefront correlation, the correlation function is chosen as:

(3)

whereφis the wavefront,ris the position of wavefront, Δris the length between wavefronts, andF(Δr) is the correlation coefficient.

As an example, for the condition of 2 000 mm diameter turret, with projection direction 1(zenith 35°，azimuth 180°) and projection radius 400 mm, the wavefront correlation is analyzed. Fig.24 shows the wavefront correlation coefficients of streamwise and spanwise directions. It is obvious that , the wavefront correlation coefficients of spanwise is remarkably larger than that of streamwise, and this may be important for the application of adaptive optics.

Fig.24 Correlation coefficients of spanwise and streamwise direction

4 Conclusions

The space-time characteristics of the beam quality and beam tilt angle caused by the aero-optical effect around conformal turrets are studied by using the unsteady calculation with fixed Mach number and similar Reynolds number. The results indicate that the statistical characteristics of aero-optical effects can be effectively simulated with reduced-scale experiment while the time-related properties of aero-optical effects are poorly simulated. For the unsteady calculation of the actual size model, the flow field with short time step, high spatial resolution and long enough duration is given. The changes of aero-optical effects in different emission directions are analyzed in detail, and the aero-optical effect around turrets is studied comprehensively. The result indicates that the aero-optical effect has a characteristic frequency determined by the frequency of the flow field, and the characteristic frequency of the flow field is determined by the velocity of plane, the turret diameter andSr, the reason is that the mean flow field effect is derived from the entire flow field. Due to the large absolute value of the mean flow field effect, the characteristic frequency of the aero-optical effect is determined by the characteristic frequency of the flow field under the combined action of the mean flow field and turbulence.

In the windward direction, the absolute value and fluctuation value of the wavefront distortion caused by the aero-optical effect are relatively small. The wavefront distortion is mainly affected by the mean flow field of the stable low frequency, and the turbulence effect is less. The simulation results give an optimal selection of the emission direction and the effect of the projection radii. It is pointed out that with the increase of the projection radii, the high-frequency turbulence effect is basically unchanged, and the low-frequency mean flow field effect will increase accordingly. The increase of the mean flow field effect will significantly deteriorate the beam quality. The result shows that increasing the projection radius can effectively increase the power density of target under the condition that the mean flow field effect can be corrected.