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In order to quantitatively assess the influence of the retractable dome on the observational performance of the 4-m Chinese Large Telescope (CLT), an integrated analysis method based on computational fluid dynamics (CFD) and sub-harmonic phase screen is proposed in this paper. The pressure, the temperature, and the speed of air surrounding the retractable dome are attained by CFD simulations, and then the fluctuation of refractive index of air is calculated. Based on sub-harmonic phase screen algorithm, three kinds of performance evaluation parameters are presented: irradiance, phase of the target, and Full Width Half Maximum (FWHM). The wind tunnel tests (WT) with a 1:120 scaled model of the retractable dome for the CLT are conducted to verify the calculated precision of the CFD. The results show that the fluctuation of air refractive index surrounding the CLT is mainly caused by the inhomogeneous distribution of temperature and speed, and with the help of pier’s height the impact of inhomogeneous air temperature from the ground layer on the fluctuation of air refractive index can be effectively decreased. Furthermore, the lower of the air speed is, the better performance of the retractable dome will be, and when the speed of air is less than 5m/s, the dome seeing induced by the retractable dome on the observational wave front is less than 0.13 arcsec.
© 2015 Optical Society of America
1. Introduction
Ground-based optical telescopes are usually housed in confined domes to avoid harmful weather conditions [1–3]. However, due to the existence of the dome, the temperature,the pressure and the speed of air is nonuniform, leading the refractive index of air to form a nonuniform distribution. The optical path varies when transmits through the air, then dome seeing is generated [4–7], as a result, the quality of imaging is degraded.
Turbulence around the dome and the telescope, the mixing of hot and cold air from surfaces inside and outside the dome including the telescope, and the mixing of cold air from the near ground layer are regarded as the main cause of dome seeing [8–11]. While the retractable dome allows the telescope fully expose to the ambient night atmosphere, any warm air will quickly dispersed into the ambient air, and since the shell of the dome is left down, big eddy(turbulence) around the dome will not be produced, as for the mixing of cold air from the near ground layer, it can be reduced by installing the telescope above the boundary layer of the ground [12]. And it is much easier for telescope to track a fast-moving target, once the telescope selects the retractable dome. Because of these advantages, more and more ground-based optical telescopes have adopted this structure [13–15]. Thus, it is very significance for ground-based optical telescopes to investigate the influence of the retractable dome on the observational performance.
The performance analysis of domes for the past only focused on their aerodynamic property, for instance, Ando [16], Siegmund [17], and Schneermann [18], using WT tests to study the flushing time, uplift effect and the pressure inside dome. Young [19,20], Vogiatzis [21], and Chylek [22] utilized CFD simulation, avoiding a long preparation period and high expense of WT tests, to study the wind load of the telescope and search for the optimization plan of dome. There are few articles use these aerodynamic parameters to establish relations with the quality of telescopes’ observational performance. Since ΔT, the temperature difference between the air inside and outside of the dome, is small for retractable dome, even the empirical formula of dome seeing proposed by Racine [4] is no longer applicable for retractable dome.
For the limitation of these methods and the eager to analysis the influence of retractable dome on the observational performance for CLT, an optical performance evaluated method based on CFD and sub-harmonic phase screen algorithm is presented in this paper. First, the refractive index of air surrounding retractable dome is calculated by CFD under different conditions. Then, the irradiance, the phase of the target observed by telescope is attained by sub-harmonic phase screen algorithm to assess the influence of retractable dome on the observational performance.
2. Refractive index of air and its relationship with the temperature, pressure and speed of air
As we all know, the quality of optical transmission in the medium is directly related to the homogeneousness of refractive index, and the refractive index of air n can be computed from the density of the air, according to the Gladstone–Dale relation [23]
where is the Gladstone-Dale constant, depending on observing wavelength (um), can be expressed asIf the observing wavelengths of telescope are in the range of 0.4~1um, approximates to 2.23 × 10−4.Since the air surrounding the dome has a low speed (~10 m/s) [16–22], the Mach number (Ma)<0.3, the total pressure of flow field is equal to the sum of static pressure and dynamic pressure
where is the density of air at the standard condition, u, v, and w are the velocity components of the three coordinate directions.According to the state equation and Eq. (3), the density of the air can be expressed as
where is the temperature of air, and R is the gas constant.On the basis of Eq. (1)- Eq. (5), the refractive index of the air can be expressed as
Comparing with the vacuum, there has, relating to the speed, temperature and pressure of air, in the refractive index of air. Thus, the homogeneousness of the refractive index is under the control of the pressure, the temperature and the speed of air, and the variations in the refractive index(total differential of refractive index) can be expressed as
Thus, in order to evaluate the influence of retractable dome on the observational performance of CLT, the temperature, the speed and the pressure of air surrounding the dome must be calculated. Fortunately, as a fluid, the equations for conservation of energy, mass, and momentum of air are used to solve the temperature, the speed and the pressure of air [24].Taking into account the complexity of the problem and the boundary conditions, analytic solutions of the equations for conservation of mass, momentum and energy are difficult to attain. With the development of CFD and the computational power, the complex flow field around the dome can be accurately modeled by CFD both spatially and temporally, each of the steps is described below.
3. Aerodynamic numerical computation
3.1 Analytical model
The CLT is a state of the art optical instrument for research in atmospheric compensation and adaptive optics techniques. Many features, including active mirror support and precise mirror temperature control, have been incorporated in the telescope to permit optimal performance of the imaging system, the optical layout is shown in Fig. 1, and the primary and secondary mirrors have diameters of 4m and 0.66m. In order to minimize the degrading influence of dome seeing on imaging, a cylindrical retractable dome (see in Fig. 2) with a 15-deg sloping roof is adopted. The dome is designed to retract vertically to a position that allows the telescope an unobscured view of the horizon at all azimuthal angles. The dome outer diameter D is 24m with a wall thickness of 1m and a 12.2m diameter circular opening through which the telescope sees, and the height of pier is 14m to reduce the impact of hot air from the near ground layer on dome seeing.
For the conceptual design phase of the telescope, the site has not yet been established. Based on this uncertainty, the dome is placed on a flat surface in smooth flow. Respectively, the exterior flow field around the dome, shown in Fig. 3, is located 5D upstream, 5D downstream, 5D away from the sides of the dome, and the ceiling is located 5D above the dome. Since the Ma <0.3, the flow can be regarded as incompressible, the location of the flow field boundaries at these distances is assumed to be appropriate for these computations and the uncertainty from the domain boundaries is negligible [25]. Note that 0° azimuth orientation corresponds to the CLT opening facing the oncoming wind (see in Fig. 3), while the 0° zenith orientation corresponds to the CLT opening pointing vertically. Because of the symmetry of the dome, only three special conditions are investigated: 30° zenith and 0° azimuth, 30° zenith and 90° azimuth, and 30° zenith and 180° azimuth.
3.2 Meshing
After defining the analytical model and the exterior folw field, the calculational domain needs to be discretized. Considering the speed and the precision of calculation, the unstructured grid [24] is used, the meshing of the telescope and local flow field is shown in Fig. 4.
3.3 Boundary conditions and turbulence model
The surfaces of the telescope and dome are given a smooth and no-slip boundary, and the sides and ceiling of the calculational domain are given free-slip boundaries with a zero normal velocity component. Assuming the site of telescope located at an altitude of 3193m in Li Jiang (longitude = 100.47°,latitude = 26.83°),where the maximum average wind speed is 10m/s and the average temperature is 283.15K, the basic parameters of air are listed in Table 1, and the temperature of the ground, dome and telescope are fixed at 283.15K, but a vertical temperature gradient for the ambient air of −6 K/km is incorporated to verify the rationality of pier’s height.
Table 1. Boundary conditions for aerodynamic computation
Since the Re is 2.17 × 105>104, the flow belongs to turbulence, the effects of turbulence are incorporated by selecting the Shear Stress Transport (SST) turbulence model, which can accurately predict the beginning of flow and the separation of fluid under negative pressure gradient condition [24]. The calculations are fully three-dimensional and unsteady, and commercial software ANSYS CFX is used to solve the equations for conservation of mass, momentum, and energy.
3.3 The settings of solver
Time step is one of the most important parameters in unsteady CFD simulation. Fortunately, the time step can be set according to the courant number∈ [2,10] [26,27]. The definition of the courant number is
where Δt is the time step and elementsize is the minimum grid size. If the courant is 2, the speed of air is 10 m/s, and the minimum grid size is 20 mm, the time step calculated through Eq. (8) is 0.004 s.The simulation of the flow past the CLT model is calculated by HP Z800 Workstation, the memory (RAM) is 44GB, and the CPU is X5675@3.07GHz. In order to avoid overflow of numerical solution, the total analytical time of the solution is 5s.
4. Results of CFD simulation
The mesh independence of the CFD computational results is established by comparisons of the aerodynamic computational results that are predicted on different meshes [28]. Figure 5 is an example of such a test. Variations in the number of tetrahedral elements cause small differences. Therefore, the CFD computational results are independent of the mesh size.
Fig. 5 Distribution of monitoring points(a) and the average speed of the monitoring points on the different mesh sizes (b,mesh number 1 denotes 600,142 tetrahedral elements, mesh number 2 denotes 1,350,801 tetrahedral elements, mesh number 3 denotes 2,075,502tetrahedral elements, mesh number 4 denotes 10,120,129 tetrahedral elements,and mesh number 5 denotes 49,776,455 tetrahedral elements).
To obtain the optimal efficiency, the 2,075,502 tetrahedral-element CFD computational grid is chosen in this study, because this number of tetrahedral elements is not too large and the price of the computations is not prohibitive.
In order to reduce rounding error, the pressure in CFX will be expressed as gauge pressure [29], the average total pressure of the primary mirror for the three different analytical conditions is shown in Fig. 6,the maximum absolute value of the total pressure pulsations are 34.55pa, 42.25pa and 31.91pa, which are far less than the atmospheric pressure(68Kpa). Thus, the variations in the refractive index caused by the fluctuation of pressure will be small, and the variations in the refractive index will be mainly related to the speed of air and the temperature of air.
Fig. 6 Mean pressure on the mirror:(a) 30° zenith and 0° azimuth,(b) 30° zenith and 90° azimuth, and (c) 30° zenith and 180° azimuth.
The distributions of average speed for the three different analytical conditions are presented in Fig. 7 When the speed of inlet stream is 10 m/s, the maximum speed of the flow for the three different analytical conditions are14.39m/s, 15.17m/s, and 14.58m/s, the increase of the speed is about 40-50%. Although the distribution of speed are different for different analytical situations, the maximum speed of air are all occurred at the windward edge of the dome, the reason is that the blocking effect of dome decreases the static pressure of the flow, according to the principle of Bernoulli [30], the speed of air will increase. And when the flow gets through the dome, the flow separation, shear layer, and large-scale vortex are formed, the homogenous of flow field is degraded. A prediction is made that the impact of dome on the disturbances of natural stream flow, which is very inhomogeneous, will be greater.
Fig. 7 Flow speed in the section: (a) 30° zenith and 0° azimuth,(b) 30° zenith and 90° azimuth, and (c) 30° zenith and 180° azimuth.
The distributions of average temperature for the three different analytical conditions are presented in Fig. 8. Since a vertical temperature gradient is incorporated for the ambient air, the air temperature is higher in the regions which are close to the ground. However, due to the existence of the dome and telescope, the homogenous of the air temperature is worse in these the regions, especially in the upwind region, the main reason is that the flow separation, shear layer, and large-scale vortex are formed with the help of the dome and telescope, the homogenous of the air is broken. Fortunately, we have taken into account this effect and the CLT is installed at the height of 14m, and the temperature gradient of the air around the telescope is about 0.2K.
Fig. 8 Air temperature in the section: (a) 30° zenith and 0° azimuth,(b) 30° zenith and 90° azimuth, and (c) 30° zenith and 180° azimuth.
5. WT tests
In order to verify the results of CFD simulation, WT tests are conducted. The 1:120 scaled geometry model of the CFD simulations is used in the WT test (see in Fig. 9). Since the site has not been chosen, the geomorphology around the dome is not yet considered, the model of WT test is placed upon a flat square plane. And the test is performed in an open-jet WT, which is 0.55m wide, 0.4m high, velocity uniformity is ± 3%, and turbulence intensity is 5%. In this configuration, the WT test section has no walls, only the floor. The air flows from the upstream nozzle to the downstream collector, and the boundary conditions has been listed in Table 1. Nine pressure monitoring points located on the primary mirror (see in Fig. 10) and three speed monitoring points (see in Fig. 11) are selected to set up a database to communicate CFD simulation, the speed of the monitoring point is recorded by the hot-wire anemometer with a sampling frame frequency of 10 kHz and total pressure of the monitoring point is recorded by the pressure scanner.
Fig. 10 Model of main tub:(a) main tube,(b) distribution of pressure monitoring points on the primary mirror, and(c) pressure scanner.
Fig. 11 Distribution of monitoring points for 30° zenith and 0° azimuth: (a) Point1, (b) Point2, and (c) Point3.
6. Comparison between CFD simulation and WT tests
Since the actual flow is unsteady, which means it is almost impossible to fully reproduce by unsteady numerical simulation, there are certain numerical deviations between CFD simulation and WT tests.
From Tables 2-4, we can see that the maximum deviation of the average speed of the monitoring points between the CFD simulation and WT tests are 15.53%,17.83% and13.99%, all are smaller than 20% [31]. And from the Tables 5-7, the deviations of average total pressure of the pressure monitoring points on the primary mirror all are less than 10%. It can be deduced from the comparison, the results of the CFD simulation are accurate, reliable and this CFD numerical model can be used to investigate the refractive index of air surrounding the retractable dome.
Table 2. Average speed and speed deviation of the monitoring points for 30° zenith and 0° azimuth
Table 3. Average speed and speed deviation of the monitoring points for 30° zenith and 90° azimuth
Table 4. Average speed and speed deviation of the monitoring points for 30° zenith and 180° azimuth
Table 5. Total pressure of the monitoring points on the primary mirror for 30° zenith and 0° azimuth
Table 6. Total pressure of the monitoring points on the primary mirror for 30° zenith and 90° azimuth
Table 7. Total pressure of the monitoring points on the primary mirror for 30° zenith and 180° azimuth
After validating the results of CFD simulation, according to Eq. (7) and boundary conditions listed in Table 1, the mean variations of refractive index for the three different analytical conditions can be attained(see in Fig. 12). The distribution of refractive index is inhomogenous, verified the validity of our conjecture, the inhomogeneous of refractive index become much more prominent in the windward side of the dome, which is similar to the distribution of speed(see in Fig. 7), and the average fluctuation of the refractive index in the flow field is in the range of 10−12~10−7 for the different analytical conditions.
Fig. 12 Mean fluctuation of refractive index in the section: (a) 30° zenith and 0° azimuth,(b) 30° zenith and 90° azimuth,and (c) 30° zenith and 180° azimuth.
Since the monitoring point 2 (see in Fig. 5) is located inside the main tube, the variation of refractive index for point 2 will reflect the general rule of the variation of refractive index in the light path of imaging. The fluctuation of refractive index calculated from CFD simulation for the point 2 is shown in Fig. 13, the fluctuation of refractive index of air is in the range of 1 × 10−7~5 × 10−7<<0.0003, and the power spectrum of the fluctuation of the refractive index is shown in Fig. 14, which is accord with the Von Karman PSD(~f-11/3) [2].
Fig. 13 Fluctuation of refractive index for the point 2: (a) 30° zenith and 0° azimuth, (b) 30° zenith and 90° azimuth,and (c) 30° zenith and 180° azimuth.
Fig. 14 Power spectrum of the fluctuation of refractive index for the point2: (a) 30° zenith and 0° azimuth, (b) 30° zenith and 90° azimuth,and (c) 30° zenith and 180° azimuth.
7. Sub-harmonic phase screen algorithm
To determine the influence of dome on the quality of imaging, we need to investigate the change of irradiance and phase after the light transmits through the turbulent atmosphere around the dome.
As an electromagnetic phenomenon, optical transmission is governed by Maxwell's equations. Generally, the atmosphere is considered a source-free, nonmagnetic, and isotropic medium, so the wave equation for the field variable can be written as [32]
where U is the field component vector, k is the vacuum optical wave number, and n is the refractive index of air.According to the theory of Kolmogorov [33], in the inertial region(l0<l<L0) Eq. (9) can be simplified as
where l0 is the inner scale of turbulence, typically a few millimeters to a few centimeters, and L0 is the outer scale of turbulence, typically tens to hundreds of meters.If the l0 >>λ, there only exists the small-angle forward scattering, the optical transmission can be simplified according to the paraxial approximation. And if the field variable U can expressed as
Then, Eq. (10) can be written aswhere if the fluctuation of refractive index(dn) is small, Eq. (12) can be simplified asSince the operator is the square root of another operator, different approximate methods can attain different equations, the most simple approximation is Taylor expanded form, which can be expressed asAccording to Eq. (6), Eq. (14) can be further expressed asIn this case, Eq. (13) can be expressed asIf the optical transmission conducts in vacuum, theat the right side of Eq. (16) is zero, then the distribution of field component at can be solved by Green function, if there exists a point source at [34]
And if we only consider the influence of turbulence on the refractive index of air, there only has at the right side of Eq. (16), the solution of the equation at this time iswhere S is the phase. In fact, Eq. (18) is the phase modulation in the transmission path of light wave. If is very small, S is also small, then the optical transmission in vacuum and turbulence atmosphere can be regarded as two independent and simultaneous processes, the turbulent atmosphere can be simplified as a series of parallel plate(phase screen, see in Fig. 15), which thickness is Δz. The optical field from the front to the back of the flat plate can be calculated through Eq. (17), and the final optical field is obtained by the phase modulation of the plate (expressed in Eq. (18)). Repeat this process, the optical field of a target transports in turbulence atmosphere will be able to obtain.Through the above analysis of optical transmission in turbulent atmosphere, in the direction of transmission, the optical field from the plane to the planecan be obtained through the distance of Δz transmission in vacuum, and the phase modulation of phase screen, according to Eq. (16), the field component can be expressed as
The difficultness of solving Eq. (19) with analytic method is hard to imagine because of the random nature of ,so the numerical method is the only solution, Fourier transform is the most widely used among the numerical methods [35,36].The Fourier transform of Eq. (19) is
where and are the spatial wave number, and the inverse Fourier transform of Eq. (21) isIt can be deduced from Eq. (22), the most important thing for the numerical solution of optical transmission in turbulence medium is the construction of phase screen, which can correctly describe the fluctuation of refractive index. By far, the easiest way to construct phase screen is spectral inversion put forward by McGlamery [37], the phase in the frequency domain can be expressed as
whereand are the random numbers subjected to the standard normal distribution, andis the spectral density function of expressed aswhere is the spectral density of the refractive index. The Fourier transform of Eq. (23) is the specific expression of the phase can be expressed asFrom the results of CFD simulation, the spectral density of refractive index accords with the function of Von Karman, namely,
where z is the direction of transmission, and is the refractive index structure constant expressed as [32]Obviously, Eq. (25) is the form of continuous transformation, as for the numerical solution, the discrete form is needed, if the length of phase screen is equal to the width of phase screen, phase screen is divided into N × N squares grid, and the width of each grid is Δx, the schematic of phase screen shown in Fig. 16, according to the sampling theory, the wave number intervals in phase space is ΔK = 2π/(NΔx), and the corresponding phase space of wave numbers is Kx = 0, ± ΔK, ± 2ΔK,… ± (N/2)ΔK.So the discrete expression for phase is
And the discrete phase distribution in real space isAccording to Eq. (28), the Eq. (29) can be expressed aswhere is the random number subjected to the standard normal distribution.Since the power spectral density (see in Fig. 14) of the fluctuations of atmospheric refractive index is higher at low frequency region, the energy of phase is more concentrated at these region. In order to accurately describe the fluctuations of the low-order wave front, very high sampling frequency is needed at low frequency region, however, this is almost impossible for spectral inversion.In order to compensate for the undersampling in the low frequency region of spectral inversion, sub-harmonic method is adopted [38], the expression can be written as
where is order of the sub-harmonic, andEquation (31)and Eq. (32) are the low and high frequency components of the phase screen respectively. Thus, the sum of Eq. (31) and Eq. (32) is the actual phase, and optical field is calculated by Eq. (22).Under the same conditions, the phase screen constructed by spectral inversion and sub-harmonic is shown in Fig. 17, through adding sub-harmonics, the phase of the low-frequency part has been improved significantly.
Note that the above method presented here is valid only for weak fluctuation of refractive index, which is often evaluated by log-amplitude. The definition of log-amplitude variance can be expressed as
For plane-wave, the log-amplitude variance evaluates toweak fluctuations are associated with< 0.25, and strong fluctuations with >>0.25.The flow chart of sub-harmonic phase screen algorithm is shown in Fig. 18. First, geometric and atmospheric parameters must be determined, for example, the observing wavelength (λ), the aperture of telescope(D), the distance of transmission(Δz), the inner scale (l0), the outer scale(L0) of turbulence atmosphere and the refractive index structure constant(C2n). Next, the calculation of log-amplitude variance have to be done, if is not much greater than 0.25, the fluctuation of refractive index is weak, sub-harmonic phase screen algorithm can be applied. Then, the objective source needs to be modeled, for example, the function of sinc can be used to simulate a point source, and the number of phase screen (Nz), the number of grid of phase screen (N), the grid spacing (Δx) must be determined, as for the change of phase in the phase screen can be generated by the random function (randn). Since the transmission of optical field from a point source to a observation screen is a Fresnel process, the angular spectrum algorithm is adopted [39]. Finally, the statistic of irradiance and phase in observation screen is done to obtain the final optical field.
8. The performance analysis of CLT retractable dome
For the performance analysis of CLT retractable dome, assuming the incident light comes from a point source 20km from the telescope, observation wavelength of the CLT is 0.5um, and the size of observation screen located at the bottom of the primary mirror is 2 m × 2 m, the refractive index structure constant in optical path of transmission is calculated from the average fluctuation of refractive index of the 12 points located in the main tube(see in Fig. 19), since the optical design of CLT adopts catadioptric structure, the value of z in the Eq. (27) is 10m, twice as much as the length of the main tube, and in order to describe the impact of retractable dome on the observational wave front more accurately, the atmospheric seeing of site and the error of optical system will not be considered.
To simplify the calculation, the air is divided into 11 layers, in other words, the number of phase screen Nz is 11, the minimum required number of grid points N is 29, so the size of the phase screen is 512 × 512, gird spacing Δx is 1cm, the inner scale l0 of turbulence atmosphere surrounding the dome is 4cm, and the outer scale L0 of turbulence atmosphere is 10m.
The average fluctuations of refractive index of the 12 monitoring points for the three different analytical conditions calculated by CFD is shown in Table 8, and the refractive index structure constant are 7.373e-16m-2/3, 6.5885e-16m-2/3, and 8.2816e-16m-2/3. According to the known geometric and atmospheric parameters, log-amplitude variance of the optical field for the three different analytical conditions are shown in Table 9, the log-amplitude variance are all less than 0.25, so the fluctuation of refractive index in the optical path is weak, sub-harmonic phase screen algorithm can be used to simulate the optical transmission in the air surrounding the retractable dome.
Table 8. The average fluctuations of refractive index of the 12 monitoring points in the main tube of CLT.
Table 9. The log-amplitude variance of the optical field.
The first step is to perform a vacuum simulation. The purpose of performing a vacuum simulation is for comparison to the turbulent simulations. Often, we want to know how much the performance of an optical system is degraded by turbulence, so we need to know how the system performs in a vacuum for comparison. Images of the resulting irradiance and phase of target is shown in Fig. 20. Clearly, the irradiance in plot (a) is nearly uniform over the region of interest, and the phase in plot (b) is flat (after collimation).
Through the turbulence atmosphere surrounding the dome, images of the resulting irradiance and phase of target for the three different analytical conditions are shown in Fig. 21. Compared with the vacuum transmission, the distributions of irradiance are relatively diffuse, and there are an obvious decrease of irradiance, despite existed some local highlights, spatial scales between the strong and weak region is about 1.4~2cm, which is close to l0 (4cm). The phase increase their components, and the change of phase consistent with the changes of irradiance. It is concluded from the uniformity of irradiance, when the degree of azimuth is 90, the impact of the refractive index on imaging quality is smallest.
Fig. 21 Irradiance(a1-a3) and phase (b1-b3)resulting from a turbulence propagation of the model point source, (a1,b1,30° zenith, and 0° azimuth, a2,b2,30° zenith, and 90° azimuth, a3,b3,30° zenith, and 180° azimuth).
To confirm the turbulent simulation program, we compute the coherence factor in the observation plane. The modulus of the coherence factor μ in the observational plane can be computed as [39]
where (rad/m) is angular spatial frequency, Δz(m) is propagation distance, and is the power spectral density of refractive-index fluctuations.If the von Kolmogorov PSD is used, the coherence factor for the observation plane evaluates to
Coherence factor in the observation plane for the simulation is shown in Fig. 22, and the deviation between the simulated and theoretical values of coherence factor is shown in Table 10, we can see that there is a good match between theory and the simulation result, which means the turbulence model and the optical transmission method expressed in this paper accurately describe the statistical regularity of turbulent atmosphere around the dome and the irradiance, phase distribution.Fig. 22 Coherence factor in the observation plane: (a) zenith angle of 30 and azimuth angle of 0,(b) zenith angle of30 and azimuth angle of 90,and (c) zenith angle of30 and azimuth angle of 180.
Table 10. The average deviation of coherence factor between theory and simulation
Since random variations in the air refractive index alter the phase of the wave front, the phase can be regarded as a random variable, as we known the root mean square(rms) of the random variable can be used to express the fluctuation of the random variable, the rms values of the phase for different analytical conditions are listed in Table 11, it is clearly that the rms of the phase are all small, which means the fluctuation of phase is small and indirectly reflects the fluctuation of the air refractive index is small too.
Table 11. The rms of the phase variation
In order to evaluate the influence of the retractable dome on imaging quantitatively, the the FWHM, which is the angular diameter at half height of the point spread function (PSF),is calculated, the expression of FWHM is [40]
whereis the atmospheric coherence diameter. For a plane wave source, the atmospheric coherence diameter is mathematically computed asIf the zenith angle of telescope is γ, Eq. (39) becomes [41]Since the zenith angle of CLT for the analytical situation is 30 degrees, the atmospheric coherence diameters for the three different analytical conditions are shown in Table 12.Table 12. The atmospheric coherence diameter.
According to Table 12 and Eq. (40), the seeing-limited FWHMs of the PSF for CLT under three different analytical conditions are shown in Table 13.
Table 13. The seeing-limited FWHMs of the PSF.
The maximum value of the FWHMs for the three different analytical conditions is 0.4275 arcsec, and the star images for different analytical conditions are shown in Fig. 23, compared with vacuum propagation, there exists degradation in the star images, but it is quite weak and the maximum value of FWHM at different wind speed is shown in Fig. 24, it is clearly that the lower of the air speed is, the better performance of the retractable dome will be, and the dome seeing induced by the retractable dome on the observational wave front is less than 0.13 arcsec The conclusion can be drawn from Tables 12 and 13, and Figs. 23 and 24 that the impact of retractable dome for CLT on imaging is much small and performance of the retractable dome is excellent.
Fig. 23 Star images for different analytical conditions, (a, vacuum propagation, b,30° zenith, and 0° azimuth, c,30° zenith, and 90° azimuth, d,30° zenith, and 180° azimuth).
9. Conclusion
In order to assess the performance of the retractable dome for the CLT, complementary studies, involving unsteady CFD simulations, WT tests, and optical transmission through the turbulence atmosphere around the dome are carried out. Results of the analysis are summarized as follows:
- 1. The unsteady CFD simulation model based on finite volume methods can be used with great accuracy in the flow through an optical telescope and dome.
- 2. If the atmospheric seeing of site and the error of optical system are not considered, when the inlet stream speed is 10 m/s and a vertical temperature gradient for the ambient air of −6 K/km, at three different azimuth angles(0°, 90°,and 180°), the average fluctuations of refractive index surrounding the retractable dome is about 10−12~10−7<<0.0003.
- 3. The optical transmission program based on the sub-harmonic phase screen algorithm can be used with great accuracy in the optical transmission through the retractable dome.
- 4.The maximum value of the dome seeing is 0.4275 arcsec. Furthermore, the lower of the air speed is, the better performance of the retractable dome will be, and when the speed of air less than 5m/s, the dome seeing induced by the retractable dome on the observational wave front is less than 0.13 arcsec, and performance of the retractable dome is excellent, the method expressed in this paper can provide reference for the analysis performance of the next generation of large ground-based telescope domes.
The boundary conditions of the CFD simulation and WT are simplified for ease of analysis. A number of actual engineering conditions are not considered in the simulation. Therefore, the mathematical model of the refractive index through the dome can be further improved by taking actual engineering conditions into account. Experimental validation of this model will be pursued in future research.
Acknowledgments
This research is supported by National Natural Science Foundation (NSFC) of China under project No. 60978050, and Graduate student innovational foundation of Institute of Optics and Electronics, University of Chinese Academy of Sciences under project No. C12K011. We also acknowledge the reviewers and editors whose comments are very valuable and helpful for revising and improving this paper.
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