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Engineering Comprehensive Model of Complex Wind Fields for Flight Simulation

Engineering Comprehensive Model of Complex Wind Fields for Flight Simulation aerospace Article Engineering Comprehensive Model of Complex Wind Fields for Flight Simulation Jianwei Chen *, Liangming Wang *, Jian Fu and Zhiwei Yang School of Energy and Power Engineering, Nanjing University of Science and Technology, Nanjing 210094, China; fujian@njust.edu.cn (J.F.); patch1205b@163.com (Z.Y.) * Correspondence: cjwrko@njust.edu.cn (J.C.); lmwang802@163.com (L.W.); Tel.: +86-(025)-8431-4923 (L.W.) Abstract: A complex wind field refers to the typical atmospheric disturbance phenomena existing in nature that have a great influence on the flight of aircrafts. Aimed at the issues involving large volume of data, complex computations and a single model in the current wind field simulation approaches for flight environments, based on the essential principles of fluid mechanics, in this paper, wind field models for two kinds of wind shear such as micro-downburst and low-level jet plus three-dimensional atmospheric turbulence are established. The validity of the models is verified by comparing the simulation results from existing wind field models and the measured data. Based on the principle of vector superposition, three wind field models are combined in the ground coordinate system, and a comprehensive model of complex wind fields is established with spatial location as the input and wind velocity as the output. The model is applied to the simulated flight of a rocket projectile, and the change in the rocket projectile’s flight attitude and flight trajectory under different wind fields is analyzed. The results indicate that the comprehensive model established herein can reasonably and efficiently reflect the influence of various complex wind field environments on the flight process of aircrafts, and that the model is simple, extensible, and convenient to use. Keywords: flight simulation; wind field model; comprehensive model Citation: Chen, J.; Wang, L.; Fu, J.; Yang, Z. Engineering Comprehensive Model of Complex Wind Fields for Flight Simulation. Aerospace 2021, 8, 145. https://doi.org/10.3390/ 1. Introduction aerospace8060145 A complex wind field refers to the typical atmospheric disturbance phenomena in nature, for instance gusts, typhoons, wind shear, turbulence and so on, which are closely Academic Editor: Carlo E.D. Riboldi related to factors such as weather and terrain. In flying aircrafts, rockets, missiles, and air- ships, etc., the whole flight process is affected by complex wind fields. For example, Received: 28 April 2021 the flight stability and safety of aircrafts will reduce, and the firing accuracy of rockets and Accepted: 21 May 2021 missiles will become worse. Therefore, establishing a computational model of complex Published: 24 May 2021 wind fields suitable for flight simulation is of great significance to the design of aircraft and the study of flight control. Publisher’s Note: MDPI stays neutral At present, there are three types of modeling method for complex wind fields. The first with regard to jurisdictional claims in method is the wind field data measurement method. The data obtained from this method published maps and institutional affil- are authentic and reliable, but the volume of data is large and the computation cost is iations. high. [1,2], respectively, substituted the exploratory data of 14 Atlantic tropical storms from 1982 to 1989 collected by the Hurricane Research Division (HRD) and the wind vector data from the HY-2 satellite into the wind field data model of the National Center for Environmental Prediction (NCEP) for calculation, analysis and verification. [3] used particle Copyright: © 2021 by the authors. filter and Monte Carlo methods to process the measured data and set up the wind velocity Licensee MDPI, Basel, Switzerland. model. [4] reanalyzed the atmospheric circulation data taken from the Black Sea region This article is an open access article during the period of 1958 to 2001 and established the distribution of circulation wind field distributed under the terms and affected by sea temperature. The second method is the numerical method of atmospheric conditions of the Creative Commons dynamics, which needs to solve the computation-heavy nonlinear differential equations, Attribution (CC BY) license (https:// due to which the methodology is too complicated. [5] used the mesoscale model (MM5) of creativecommons.org/licenses/by/ the National Center for Atmospheric Research of Pennsylvania State University to calculate 4.0/). Aerospace 2021, 8, 145. https://doi.org/10.3390/aerospace8060145 https://www.mdpi.com/journal/aerospace Aerospace 2021, 8, 145 2 of 18 the distribution of low-level winds located over Antarctica. [6] used the RNG turbulence closure model along with the SIMPLEC pressure correction algorithm to establish the nature of wind field distribution around different buildings. The third method is the engineering simulation method. This method begins with the flow characteristics of airflow in various wind fields and describes the law of airflow movement with simple fluid dynamics equations. This method is simple and intuitive, and can highlight the influence of primary physical parameters. For example, refs. [7–9] built the engineering models of micro-downburst using the vortex ring method and wind profile model, respectively. Ref. [10] established a wind distribution model over a large-scale ridge with temperature as the vertical coordinate, and [11,12] developed the atmospheric turbulence models for flight simulation. In the actual flight simulation applications, it is frequently necessary to quickly adjust the parameters according to the changing flight environments to complete the simulation. As a result, it is difficult for both the measured wind field data method and the atmospheric dynamics numerical method to meet such real-time and fast-paced requirements. The engineering simulation method has been widely used because of its simplicity and low computation requirement [13–16]. It can be seen from the above discussion about the available literature that there are two major deficiencies in the current research on wind field modeling in flight environment. Firstly, the model is relatively simple, because it considers only one form of wind field, and neglects the case that the natural wind appears simultaneously in the form of wind shear, turbulence, and other forms under the influence of terrain and weather. Secondly, owing to the large volume of model data and the complex calculations, special fluid dynam- ics analysis software is needed. Additionally, it is difficult to meet the requirements of fast simulation or real-time simulation in some practical engineering applications. To address these problems, in this paper, the wind field models of two typical types of wind shear, namely micro-downburst and low-level jet, and atmospheric turbulence are established by the using engineering simulation method. Then, taking the ground coordinate system as the unified reference coordinate system, the three wind field models are combined together to construct a comprehensive wind field model with spatial location coordinates as the input and the wind velocity as the output. Finally, the ballistic simulation of a rocket projectile is used as an example to verify the practicability and efficiency of the comprehensive wind field models. 2. Typical Wind Field Models 2.1. Micro-Downburst A micro-downburst (abbreviated as MD in this section) is a type of low-level wind shear associated with convective weather [17,18]. Figure 1 shows a schematic diagram of airflow distribution in the MD wind field. It can be noticed that the MD manifests itself as a local vertically downward airflow in the strong convective cloud cluster. After the airflow sinks and touches the ground, it diverges in all directions and curls up to form an area of vortex ring above the ground. In view of such flow characteristics, a bunch of ground-symmetric vortex rings were constructed in the vertical direction of the horizontal plane to simulate the vertical airflow generation [19]. The coordinate system of the model is illustrated in Figure 2. Considering the ground coordinate system oxyz as the datum and a point O above the plane xoz as the center, a closed vortex ring of radius R is set, named the main vortex ring, and its curve equation is: 2 2 (x x ) + (y y ) = R p p (1) z = z p Aerospace 2021, 8, 145 3 of 18 Aer Aerosp ospace ace 20 2021 21, , 8 8, , x FO x FOR P R PEE EER R RE REVIEW VIEW 3 3 of of 21 21 clouds clouds Clouds Clouds downdraght downdraght D D o ow wn nd dr ra ag gh ht t Horizontal vortex Horizontal vortex Flow Flow f fr ro on nt t external flow external flow E xternal E xternal Ground Ground flow flow Figure Figure 1. 1. Airflow Airflow distribution distribution of of micr micro o-downburst. -downburst. Figure 1. Airflow distribution of micro-downburst.  Main vortex ring Main vortex ring O Main vortex ring Main vortex ring P P min min zz rr M max max Z r Z Aircraft P Aircraft Ground symmetric point Ground symmetric point G Gr ro ou un nd d c co oo or rd diin na at te e system system Y Mirror vortex ring  Mirror vortex ring  II II Mirror vortex ring Mirror vortex ring Aircraft Aircraft Figure 2. Coordinate system of micro-downburst model. Figure 2. Coordinate system of micro-downburst model. Figure 2. Coordinate system of micro-downburst model. The Consider circulation ing thline e ground equation coordin of theate main system vortex ox is:yz as the datum and a point Considering the ground coordinate system oxyz as the datum and a point O xxoz oz R above above th the e pl plan ane e as as th the e ce cent nter, er, a a clo closed sed vo vortex rtex ring ring o of f r radius adius R i is s s set, et, named named th the e y = (r + r )F(k) (2) P max min main vortex ring, and its curve equation is: main vortex ring, and its curve equation is: 2p where G denotes the intensity of the (x− vortex x ) ring, + (y− determined y ) = R by vertical velocity V (0) of (x− x ) + (y− y ) = R z  pp  pp (1) the vortex ring center and the vortex ring radius R: (1)  zz = zz =  p  p  G = 2RV (0) (3) The circulation line equation of the main vortex is: The circulation line equation of the main vortex is: r and r are the maximum and the minimum distances of the main vortex ring max min from an arbitrary point O (x , y , z=+ ) in(the r ground r )F coor (k )dinate system, and F(k) is the =+ (r r )F (k ) ( (2 2) ) M M M M P max min P max min 2 2 elliptic integral function, where: where  denotes the intensity of the vo rtex ring, d etermined by vertical velocity where denotes the intensity of the vortex ring, determined by vertical velocity V (0) r r V (0) max min z k = (4) r + r of the vortex ring center and the vortex ring maxradius R : of the vortex ring center and the vortex ring radius min R : Based on the theories of higher functions [7], when 0  k  1, F(k) is approximated as: = 2RV (0) = 2RV (0) ( (3 3) ) 0.788k rr r F(k)  p (5) max min max min and are the maximum and the minimum distances of the main vortex ring and are the maximum and the minimum distances of the main vortex ring 0.25 + 0.75 1 k O (x , y , z ) Fk () O (x , y , z ) Fk () M M M M M M M M from an arbitrary point in the ground coordinate system, and is from an arbitrary point in the ground coordinate system, and is Based on Equation (2), the induced radial and axial velocities of the main vortex ring th the el e ell liptic iptic integra integral functio l function, where n, where:: can be calculated by partial derivatives: rr − rr − max min max min ¶y k = 1 k = P P (4) (4) v = r ¶z rr P + R rr + max min (6) max min ¶y P P v = r ¶r P P Aerospace 2021, 8, 145 4 of 18 2 2 where r is distance of the point O from the axis of vortex ring (r = (x x ) + (y y ) ). P M P M P M P By decomposing the induced velocities calculated by Equation (6) along the ox and oz axes of the ground system, the velocity components can be written as: x x P M P P v = v x r (7) y y P M P P v = v y r In the actual circumstances, the vertical velocity component on the ground should be zero after the vertical airflow of the vortex ring center reaches the ground. For that reason, by setting the method of mirror vortex ring for the symmetry of main vortex ring about the plane xoy, the vertical induced velocities on the ground are reversed with equal value and thus cancel each other out. The signs of the streamline equations for the two vortex rings are opposite y = y . Given the mirrored vortex ring center O (x , y ,z ), its connection I P I p p p line O O with the main vortex ring center O is vertical to the ground. According to P I P the streamline equation of the mirrored vortex ring, analogous to the derivation of the induced velocity of the main vortex ring, the induced velocities at the spatial points of the mirrored vortex ring can be calculated. Velocity superposition is performed by combining Equations (6) and (7), and the resultant velocity at point O can be obtained as: 0 1 0 1 0 1 P I w v v MDx x x P I @ A @ A @ A w = v + v (8) MDy y y P I w v v MDz z z Then, the streamline equation of point O is: " # 0 0 2 0 0.788k (r + r ) G 0.788k (r + r ) max min max min y = y + y = p p (9) P I 2 0 2 2p 0.25 + 0.75 1 k 0.25 + 0.75 1 k 0 0 where r and r denote the maximum and the minimum distances of the mirrored max min 0 0 r r max 0 min vortex ring from the spatial point O , and similarly k = . M 0 0 r +r max min The main parameters [20] of the vortex ring model are shown in Table 1. Table 1. Parameters of micro-downburst model. Parameter Value O (x , y , z ) (1000 m, 0, 800 m) p p p R 1100 m V (0) 10 m/s The simulation results of the model are shown in Figure 3. It can be noted that the simulated wind field distribution of the vortex ring model is consistent with the measured data. Aerospace 2021, 8, x FOR PEER REVIEW 5 of 21 ' ' r r where and denote the maximum and the minimum distances of the max min '' rr − max min mirrored vortex ring from the spatial point , and similarly . k = '' rr + max min The main parameters [20] of the vortex ring model are shown in Table 1. Table 1. Parameters of micro-downburst model. Parameter Value O (x , y , z ) P p p p (1000 m, 0, 800 m) 1100 m V (0) −10 m/s Aerospace 2021, 8, 145 5 of 18 The simulation results of the model are shown in Figure 3. It can be noted that the simulated wind field distribution of the vortex ring model is consistent with the measured data. 1.6 15.0m/s 1.4 1.2 1.0 0.8 0.6 0.4 0.2 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 X(km) (a) (b) 2.0 15.0m/s 1.5 1.0 0.5 -0.5 -1.0 -1.5 -2.0 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 Z(km) (c) (d) Figure 3. Comparisons between simulation results and measured data of micro-downburst. (a) Simulation wind vector Figure 3. Comparisons between simulation results and measured data of micro-downburst. (a) Simulation wind vector diagram of horizontal section. (b) Flow pattern of 7 July 1990 Orlando downburst from NTRS-NASA Technical Reports diagram of horizontal section. (b) Flow pattern of 7 July 1990 Orlando downburst from NTRS-NASA Technical Reports Server [21]. (c) Simulation wind vector diagram of vertical section. (d) Flow pattern of 1988 microburst event of DEN from NTRS-NASA Technical Reports Server [22]. 2.2. Low-Level Jet Low-level jet (abbreviated as LLJ in this section) refers to the wind velocity zone in the lower troposphere, which is significantly affected by the mesoscale weather system. It is a surface inversion phenomenon occurring in the stable surface boundary layer [23]. According to the principle of the plane wall jet, when a plane free jet with a large width flows through a narrow slit, the velocity distribution of the jet near the wall is similar to that of the LLJ [24]. For that reason, the plane wall jet is employed to simulate the wind velocity distribution of the LLJ wind field, as shown in Figure 4. According to the principles of fluid dynamics, the relationship between the horizontal velocity component u(x, H) of a free jet and its maximum jet velocity u (x) is: u(x, H) H = 1 th (k ) (10) u (x) x Y(km) Y(km) Aerospace 2021, 8, x FOR PEER REVIEW 6 of 21 2.2. Low-Level Jet Low-level jet (abbreviated as LLJ in this section) refers to the wind velocity zone in the lower troposphere, which is significantly affected by the mesoscale weather system. It is a surface inversion phenomenon occurring in the stable surface boundary layer [23]. According to the principle of the plane wall jet, when a plane free jet with a large width flows through a narrow slit, the velocity distribution of the jet near the wall is similar to Aerospace 2021, 8, 145 6 of 18 that of the LLJ [24]. For that reason, the plane wall jet is employed to simulate the wind velocity distribution of the LLJ wind field, as shown in Figure 4. Wind profile u(x,H) H jet Figure Figure 4. 4. Diagram Diagram of of the the plane plane wall wall jet. jet. where k denotes a scale factor and th refers to hyperbolic tangent function. To simplify According to the principles of fluid dynamics, the relationship between the the model, the basic flow characteristics of LLJ are retained, while the wind velocity is horizontal velocity component of a free jet and its maximum jet velocity u(x,H ) evenly distributed in the horizontal direction x, that is, ux () is: u(x, H) H H = 1 th (C ) (11) u(x, H ) H u (x) H m 2 s =− 1 th (k ) (10) u () x x where H denotes the height of maximum velocity of the free jet with symmetrical distribu- tion, C is employed to describe the relationship between H and the width B of the jet in s s where denotes a scale factor and th refers to hyperbolic tangent function. To the vertical 1 direction; and the width of jet denoting the velocity range determined by 7% of the maximum velocity. The expression is simplify the model, the basic flow characteristics of LLJ are retained, while the wind velocity is evenly distributed in the horizontal direction , that is, B = 4 (12) u(x, H ) HH − C2 =− 1 th (C ) s (11) u () x H ms By superimposing the velocity distribution of the free jet onto the exponential model of the mean wind in the boundary layer, the vertical velocity profile of LLJ on the surface where denotes the height of maximum velocity of the free jet with symmetrical boundary layer can be obtained: C H distribution, is employed to describe the relationship between  and the width s s H H H u(x, H) = u ( ) + u 1 th (C ) (13) R s s B of the jet in the vertical direction; and the width of jet denoting the velocity range H H R s determined by 7% of the maximum velocity. The expression is In the formula above, u is the wind velocity corresponding to the height of H , where R R the distribution index m can be calculated in keeping with the relevant meteorological B= 4 (12) data [25]: h i 0.5 m = 1/ ln ( H H ) /Z 0.0403 ln(u /6) (14) p 0 0 R where Z denotes terrain roughness; analogous to Equation (13), the deviation of wind direction between height H and height H is derived as: H H H H 0 L a( H) = a + arctan[ tan(a a )] + a 1 th (C ) (15) H H H H L 0 0 L H H H T 0 L where H , H and H denote reference height, height of maximum deviation of wind 0 L T direction in the jet stream layer and height of top of the jet stream layer, respectively; a , a and a are the angles included between the wind direction and the geostrophic wind H H L T Aerospace 2021, 8, 145 7 of 18 at the three corresponding heights, respectively; w and w are wind velocities at heights 0 L H and H , respectively. 0 L Thus, the wind velocity components in the ground coordinate system are calculated as: w = u(x, H) cos(a( H)) LL Jx (16) w = u(x, H) sin(a( H)) LL Jz The model parameters [26] are configured as shown in Table 2 and the simulation results are shown in Figure 5. Table 2. Parameters of low-level jet model. Z H w H w H a a a C C 0 0 0 L L T H H H s L 0 L T Aerospace 2021, 8, x FOR PEER REVIEW 8 of 21 1 1 2.5 m 3.5 m 5 ms 180 m 10 ms 800 m 0 30 60 0.8 0.3 0 10 20 30 40 wind speed(m/s) (a) (b) u(x,H ) Figure 5. Comparisons between simulation results and measured data of low-level jet. (a) Wind profiles (u(x, H)) of Figure 5. Comparisons between simulation results and measured data of low-level jet. (a) Wind profiles ( ) of low-level jet by simulation. (b) Wind profiles derived from the NCEP-NCAR reanalyzes and from the PACS-SONET low-level jet by simulation. (b) Wind profiles derived from the NCEP-NCAR reanalyzes and from the PACS-SONET upper-air observations [27] (© American Meteorological Society. Used with permission). upper-air observations [27] (© American Meteorological Society. Used with permission). It can be seen from Figure 5a,b that the wall jet model can accurately simulate the It can be seen from Figure 5a,b that the wall jet model can accurately simulate the wind velocity distribution of the actual LLJ wind field. wind velocity distribution of the actual LLJ wind field. 2.3. Atmospheric Turbulence 2.3. Atmospheric Turbulence Atmospheric turbulence is the irregular and uneven random eddy motion of the Atmospheric turbulence is the irregular and uneven random eddy motion of the Earth’s atmosphere. The Dryden model is a classical form of atmospheric turbulence, Earth’s atmosphere. The Dryden model is a classical form of atmospheric turbulence, wherein the longitudinal and transverse correlation functions [28] are expressed as: wherein the longitudinal and transverse correlation functions [28] are expressed as: x /L − /L f (x) = e fe () = (17) x /L g(x) = e (1 ) 2L (17) − /L ge ( )=− (1 ) The spatial correlation function [29] of one-dimensional turbulence is: 2L 2 x /L The spatial correlation function [29] of one-dimensional turbulence is: R(x) = s (1 x /2L)e (18) 2/ − L R( )=−  (1  / 2L)e (18) For two (or three)-dimensional turbulence, the spatial correlation function is:  + 2/ − L Re (, ,) = (1− ) uu 1 2 3 2L   + 2/ − L Re (, ,) = (1− ) (19) vv 1 2 3 2L  + 2/ − L Re (, ,) = (1− ) ww 1 2 3 2L    where denotes the location difference between the two spatial points, , , 1 2 3 are the components of in the ground coordinate system, is the turbulence u v intensity parameter, L is the turbulence scale parameter, and the subscripts , , denote the components of turbulent velocity in the ground coordinate system. For a one-dimensional turbulent sequence, the recursion equation is: w(x)= aw(x−h)+br(x) (20) height(m) Aerospace 2021, 8, 145 8 of 18 For two (or three)-dimensional turbulence, the spatial correlation function is: 2 2 x +x > 2 x /L 2 3 R (x , x , x ) = s e (1 ) > u 1 2 3 2Lx 2 2 x +x 2 x /L 1 3 (19) R (x , x , x ) = s e (1 ) 1 2 3 v 2Lx 2 2 x +x : 2 x /L 2 1 R (x , x , x ) = s e (1 ) w 2 3 1 w 2Lx where x denotes the location difference between the two spatial points, x , x , x are the 1 2 3 components of x in the ground coordinate system, s is the turbulence intensity parameter, L is the turbulence scale parameter, and the subscripts u, v, w denote the components of turbulent velocity in the ground coordinate system. For a one-dimensional turbulent sequence, the recursion equation is: w(x) = aw(x h) + br(x) (20) where W(x) denotes the turbulent velocity along the one-dimensional x direction, r(x) is the one-dimensional white noise sequence, h is the simulation step size, and a, b are the recursion parameters to be solved. According to the correlation function definitions of the Dryden model, 2 2 R = E[w(x)w(x)] = a R + b 0 0 (21) R = E[w(x)w(x h)] = aR 1 0 Combining Equations (18) and (21), where R = R(i h), it is obtained that h/L a = (1 h/2L)e (22) b = s 1 a Thus, the one-dimensional atmospheric turbulent velocity can be calculated. Similarly to Equation (20), the recursion equation of two-dimensional turbulence can be written as: w(x, y) = a w(x h, y) + a w(x, y h) + a w(x h, y h) + br(x, y) (23) 1 2 3 The correlation function values are R = E[w(x, y)w(x, y)] R = E[w(x, y)w(x, y)] (24) R = E[w(x, y)w(x + h, y)] R = E[w(x, y)w(x + h, y + h)] The equation obtained by expanding Equation (24) is R = a R + a R + a R + b 00 1 10 2 01 3 11 R = a R + a R + a R 2 00 3 01 1 11 10 (25) > R = a R + a R + a R 10 1 00 2 11 3 01 R = a R + a R + a R 2 3 00 11 1 01 10 Substitute R , R , R , R calculated by Equation (19) into Equation (25), where 00 01 10 11 R = R (i h, j h), so that the parameters a , a , a , b can be obtained. Thus, the two- u 2 3 i j 1 dimensional atmospheric turbulent velocity can be calculated according to the recursive Equation (23). For the three-dimensional space turbulence, the construction idea is as follows: taking one-dimensional turbulence as the boundary value to figure out the two-dimensional plane turbulence sequence, and subsequently taking the two-dimensional plane turbulence sequence as the boundary value to figure out the three-dimensional space turbulence, as shown in Figure 6. Aerospace 2021, 8, 145 9 of 18 Aerospace 2021, 8, x FOR PEER REVIEW 10 of 21 w(x, y) w(x , y + jh) wx() w() x + h w() x + ih 0 0 w(x , y ) w(x + ih, y ) y y z z one-dimensional two-dimensional three-dimensional Figure 6. Construction process of three-dimensional atmospheric turbulence. Figure 6. Construction process of three-dimensional atmospheric turbulence. Analogous to Equations (20) and (23), the three-dimensional turbulence recursion Analogous to Equations (20) and (23), the three-dimensional turbulence recursion equation is formulated as equation is formulated as w(x, y, z)= a w(x− h, y− h, z− h)+ a w(x− h, y− h, z)+ a w(x, y− h, z− h) 1 2 3 w(x, y, z) = a w(x h, y h, z h) + a w(x h, y h, z) + a w(x, y h, z h) 1 2 3 +a w(x h, y, z h) + a w(x, y, z h) + a w(x h, y h, z) (26) + 4a w(x− h, y, z− h)+5a w(x, y, z− h)+ 6 a w(x− h, y− h, z) (26) 4 5 6 +a w(x h, y, z) + br(x, y, z) + a w(x− h, y, z)+ br(x, y, z) The correlation function of the three-dimensional space turbulence can be expressed as The correlation function of the three-dimensional space turbulence can be ex- pressed as R = E[w(x, y, z)w(x + ih, y + jh, z + kh)] (i, j, k = 0, 1) (27) i jk R = E[w(x, y, z)w(x+ih, y+ jh, z+kh)] (i, j,k = 0,1) ijk (27) Substituting Equation (19) into Equation (27), where R = R (i  h, j  h, k  h), i jk the following equations are obtained R = R (i *h, j *h,k *h) Substituting Equation (19) into Equation (27), where , ijk u the following equations are obtained R = a R + a R + a R + a R + a R + a R + a R + s 000 1 111 2 110 3 011 4 101 5 001 6 010 7 100 R = a R + a R + a R + a R + a R + a R + a R 001 R 1=110 a R 2+ a 111R 3+ a 010R + 4 a 100R + 5 a000 R + 6a 011 R +7a 101 R + 000 1 111 2 110 3 011 4 101 5 001 6 010 7 100 R = a R + a R + a R + a R + a R + a R + a R 010 1 101 2 100 3 001 4 111 5 011 6 000 7 110 R = a R + a R + a R + a R + a R + a R + a R 001 1 110 2 111 3 010 4 100 5 000 6 011 7 101 R = a R + a R + a R + a R + a R + a R + a R 011 1 100 2 101 3 000 4 110 5 010 6 001 7 111 (28) R = a R + a R + a R + a R + a R + a R + a R 100 1 011 2 010 3 111 4 001 5 101 6 110 7 000 R = a R + a R + a R + a R + a R + a R + a R 010 1 101 2 100 3 001 4 111 5 011 6 000 7 110 R = a R + a R + a R + a R + a R + a R + a R 101 1 010 2 011 3 110 4 000 5 100 6 111 7 001 R = a R + a R + a R + a R + a R ++ a R a R 011 1 100 2 101 3 000 4 110 5 010 6 001 7 111 R = a R + a R + a R + a R + a R + a R + a R 110 1 001 2 000 3 101 4 011 5 111 6 100 7 010 (28) R = a R + a R + a R + a R + a R + a R + a R 111 1 000 2 001 3 100 4 010 5 110 6 101 7 011 R = a R + a R + a R + a R + a R + a R + a R 100 1 011 2 010 3 111 4 001 5 101 6 110 7 000 R = a R + a R + a R + a R + a R + a R + a R By solving Equation (28), the recursive parameters a  a and b can be obtained, 101 1 010 2 011 3 110 4 000 5 1 100 6 111 7 001 and then the velocity recursion equation of three-dimensional space turbulence w (x, y, z) R = a R + a R + a R + a R + a R + a R + a R 110 1 001 2 000 3 101 4 011 5 111 6 100 7 010 is obtained. By using the method above, w (x, y, z) and w (x, y, z) can also be obtained. v w R = a R + a R + a R + a R + a R + a R + a R AT refers to the atmospheric turbulence herein. 111 1 000 2 001 3 100 4 010 5 110 6 101 7 011 w = w (x, y, z) ATx u aa By solving Equation (28), the recursive parameters and b can be ob- w = w (x, y, z) (29) ATy tained, and then the velocity recursion equation of three-dimensional space turbulence w = w (x, y, z) ATz w (x, y, z) w (x, y, z) w (x, y, z) is obtained. By using the method above, and can u v w According to the parameters in Table 3, the three-dimensional atmospheric turbulence also be obtained. AT refers to the atmospheric turbulence herein. was simulated in space. The simulation results are shown in Figures 7 and 8. From Figure 8, Aerospace 2021, 8, x FOR PEER REVIEW 11 of 21 w = w (x, y, z) AT−x u w = w (x, y, z) (29) AT−y v w = w (x, y, z) AT−z w Aerospace 2021, 8, 145 10 of 18 According to the parameters in Table 3, the three-dimensional atmospheric turbu- lence was simulated in space. The simulation results are shown in Figures 7 and 8. From Figure 8, the random variation of the atmospheric turbulent wind velocity in space is the random variation of the atmospheric turbulent wind velocity in space is observed. observed. As presented in Figure 8, the theoretical value is calculated according to cor- As presented in Figure 8, the theoretical value is calculated according to correlation func- relation function (Equation (19)) of the Dryden model. It can be seen that the variation tion (Equation (19)) of the Dryden model. It can be seen that the variation trend in the trend in the correlation value of turbulent velocity simulated by simulation is actually correlation value of turbulent velocity simulated by simulation is actually the same as that the same as that obtained from the Dryden model, which proves the rationality and ef- obtained from the Dryden model, which proves the rationality and effectiveness of the fectiveness of the established model. established model. Table 3. Parameters of atmospheric turbulence model [30]. Table 3. Parameters of atmospheric turbulence model [30]. Parameter Value Parameter Value L== L L 150 m uvw L = L = L 150 m u v w −1  ==   1 1.5 ms uvw s = s = s 1.5 ms u v w h 50 m h 50 m -4 -8 -5 z/h x/h Aerospace 2021, 8, x FOR PEER REVIEW 12 of 21 Figure 7. Atmospheric turbulent wind velocity in space (H = 600 m). Figure 7. Atmospheric turbulent wind velocity in space (H = 600 m). 1.2 Si mul aton resul ts Theoreti cal resul ts 0.8 0.6 0.4 0.2 0 200 400 600 800 1000 1200  (m) Figure 8. Comparisons of correlation values between simulation results and theoretical calcula- Figure 8. Comparisons of correlation values between simulation results and theoretical calculation. tion. 3. Comprehensive Wind Field Model 3. Comprehensive Wind Field Model Since wind velocity is a vector including magnitude and direction, it meets the princi- ple of vector superposition. For any spatial point P, assuming that wind field A induces Since wind velocity is a vector including magnitude and direction, it meets the principle of vector superposition. For any spatial point P , assuming that wind field A induces the wind velocity vector at point P and wind field B induces the wind velocity vector at point P , the total wind velocity vector at point P induced by the wind fields A and B can be obtained by the following equation: W=+ W W (30) P P P AB Based on Equation (30), in this section, a superposition method is adopted to syn- thesize the established typical wind field models. The input and output parameters of the three typical wind field models are listed in Table 4. Table 4. Input and output parameters of the three typical wind field models established. Wind Field Input Output (x, y, z) Micro-downburst w w w , , MD−x MD−z MD−y H() H = y Low-level jet w w LLJ−x LLJ−z (x, y, z) w w w Atmospheric turbulence , , AT−x AT−z AT−y It can be seen that the inputs of the three models are the spatial positions, and the outputs are the wind velocity components. Therefore, the inputs and outputs are com- bined, and the ground coordinate system is used as the frame of reference to establish a comprehensive model. The application structure of this model is shown in Figure 9. w(m/s) Aerospace 2021, 8, 145 11 of 18 the wind velocity vector W at point P and wind field B induces the wind velocity vector ! ! W at point P, the total wind velocity vector W at point P induced by the wind fields A P P and B can be obtained by the following equation: ! ! ! W = W + W (30) P P P A B Based on Equation (30), in this section, a superposition method is adopted to synthe- size the established typical wind field models. The input and output parameters of the three typical wind field models are listed in Table 4. Table 4. Input and output parameters of the three typical wind field models established. Wind Field Input Output Micro-downburst (x, y, z) w , w , w MDx MDy MDz Low-level jet H( H = y) w , w LL Jx LL Jz Atmospheric turbulence (x, y, z) w , w , w ATx ATy ATz It can be seen that the inputs of the three models are the spatial positions, and the outputs are the wind velocity components. Therefore, the inputs and outputs are com- Aerospace 2021, 8, x FOR PEER REVIEW 13 of 21 bined, and the ground coordinate system is used as the frame of reference to establish a comprehensive model. The application structure of this model is shown in Figure 9. Application settings Aircraft flight Terrain & climate model environment Model selection Input (x,y,z) Parameter setting Typical Complex wind models Micro-downburst Low-level jet Extended Atmospheric model model Turbulence model models Wind speed superposition Output (w , w , w ) x y z Comprehensive model Figure 9. Application structure of the comprehensive wind field model. Figure 9. Application structure of the comprehensive wind field model. As can be seen from Figure 9, the calculation process of the comprehensive wind field As can be seen from Figure 9, the calculation process of the comprehensive wind model consists of four key steps: field model consists of four key steps: Input spatial location parameters; • Input spatial location parameters; Select the wind field models according to simulation requirements; • Select the wind field models according to simulation requirements; Calculate the total wind velocity value; • Calculate the total wind velocity value; Substitute the wind velocity into the flight simulation. • Substitute the wind velocity into the flight simulation. To ensure the scalability of the model, an expansion module (yellow box in the chart) To ensure the scalability of the model, an expansion module (yellow box in the is set up to add other wind field models such as gust and mountain flow. chart) is set up to add other wind field models such as gust and mountain flow. 4. Model Application 4.1. Flight Simulations under Different Wind Field Conditions In order to verify the calculation effect on the whole model, a rocket projectile is taken as an example and the six degrees-of-freedom rigid body trajectory equation of the rocket projectile is considered as the flight simulation model. The basic parameters [31] of the rocket projectile are listed in Table 5. Table 5. Basic parameters of the rocket projectile. Parameter Value Diameter of rocket 0.122 m Length of rocket 2.9 m Specific impulse 250 s Working time of the engine 3 s −1 Initial velocity 40 ms Firing angle 50 deg Firing direction 0 deg For the flight state of a rocket projectile, the main concern is its trajectory and flight attitude. The flight trajectory can be directly analyzed by calculating the wind Aerospace 2021, 8, 145 12 of 18 4. Model Application 4.1. Flight Simulations under Different Wind Field Conditions In order to verify the calculation effect on the whole model, a rocket projectile is taken as an example and the six degrees-of-freedom rigid body trajectory equation of the rocket projectile is considered as the flight simulation model. The basic parameters [31] of the rocket projectile are listed in Table 5. Table 5. Basic parameters of the rocket projectile. Parameter Value Diameter of rocket 0.122 m Length of rocket 2.9 m Specific impulse 250 s Working time of the engine 3 s Initial velocity 40 ms Firing angle 50 deg Aerospace 2021, 8, x FOR PEER REVIEW 14 of 21 Firing direction 0 deg For the flight state of a rocket projectile, the main concern is its trajectory and flight three-dimensional trajectory curve of the rocket projectile, while the flight attitude needs attitude. The flight trajectory can be directly analyzed by calculating the three-dimensional to be reflected by the flight attack angle. trajectory curve of the rocket projectile, while the flight attitude needs to be reflected by the Figure 10 shows the conceptual schematic diagram of the attack angle of rocket, flight attack angle. where the coordinate system is taken as the reference coordinate system, O is O Figure 10 shows the conceptual schematic diagram of the attack angle of rocket, the center of rocket mass, the axis coincides with the axis of the rocket, the O O where the coordinate system Oxhz is taken as the reference coordinate system, O is the axis points vertically upward, and the O axis is determined according to the right center of rocket mass, the Ox axis coincides with the axis of the rocket, the Oh axis points hand rule. The red vector denotes the velocity v of the rocket’s centroid, the blue vector vertically upward, and the Oz axis is determined according to the right hand rule. The red v v v v represents the three components , and of in the coordinate system vector denotes the velocity v of the rocket’s  centr oid, the blue vector represents the three components ,  repre v , sents v and the v ang of le inc v in luded the bet coor wee dinate n the rock system et axO is xhz, d and repr th esents e veloci the - angle O h O x z included between the rocket axis Ox and the velocity v, and is called the total attack angle, v   ty , and is called the total attack angle, is called the pitch attack angle, and is 1 2 d is called the pitch attack angle, and d is called the direction attack angle. The attack 1 2   angles d, d and d of the rocket describe the positional  relationship between the projectile called the direction attack angle. The attack angles , and of the rocket de- 1 2 1 2 axis and the velocity direction during the flight of the rocket. Through the curve of the scribe the positional relationship between the projectile axis and the velocity direction attack during angle, the flig the ht attitude of the rocket changes . Throu and gh the the stability curve of of th the e attac rocket k ang during le, the the attitude flight can be seen. change The s and specific the stability flight simulation of the rocket calculation during the steps flight ar can e shown be seen. in Th Figur e speci e 11 fic . flight simulation calculation steps are shown in Figure 11. Figure 10. Attack angle diagram of the rocket projectile. Figure 10. Attack angle diagram of the rocket projectile. Comprehensive wind model Start Position Trajectory End Stop trajectory coordinates model of trajectory condition calculation (x , y , z ) rocket calculation i i i (x , y , z ) i+1 i+1 i+1 N update Figure 11. Flight simulation process. wind Aerospace 2021, 8, x FOR PEER REVIEW 14 of 21 three-dimensional trajectory curve of the rocket projectile, while the flight attitude needs to be reflected by the flight attack angle. Figure 10 shows the conceptual schematic diagram of the attack angle of rocket, where the coordinate system is taken as the reference coordinate system, O is O the center of rocket mass, the O axis coincides with the axis of the rocket, the O axis points vertically upward, and the O axis is determined according to the right hand rule. The red vector denotes the velocity of the rocket’s centroid, the blue vector v v v v represents the three components , and of in the coordinate system    ,  represents the angle included between the rocket axis and the veloci- O O v   ty , and is called the total attack angle, is called the pitch attack angle, and is 1 2   called the direction attack angle. The attack angles  , and of the rocket de- 1 2 scribe the positional relationship between the projectile axis and the velocity direction during the flight of the rocket. Through the curve of the attack angle, the attitude changes and the stability of the rocket during the flight can be seen. The specific flight simulation calculation steps are shown in Figure 11. Aerospace 2021, 8, 145 13 of 18 Figure 10. Attack angle diagram of the rocket projectile. Comprehensive wind model Start Position Trajectory End Stop trajectory coordinates model of trajectory condition calculation (x , y , z ) rocket calculation i i i (x , y , z ) i+1 i+1 i+1 update Figure 11. Flight simulation process. Figure 11. Flight simulation process. While using the integral method to solve the trajectory, the coordinates of the rocket’s position within each calculation step are substituted into the wind field model to obtain the wind velocity, which is then substituted into the trajectory equations to calculate the next trajectory parameters. These trajectory parameters include the position coordinates (x, y, z) in addition to attack angles d , d of the rocket projectile. The detailed solution 1 2 of the trajectory equations can be found in the literature [31] (pp. 141–143). Table 6 lists two wind field conditions employed for flight simulation of a rocket projectile. The flight processes under the two wind field conditions are simulated, and are compared with the flight state of the rocket projectile under windless condition. Table 6. Simulation conditions of complex wind field. Serial Number Climatic Condition Wind Field Condition 1 Clear sky Low-level jet Micro-downburst and 2 Thunderstorm atmospheric turbulence Figure 12 shows the change curves of the attack angles of the rocket projectile with flight time under the influence of the two wind fields listed in Table 6. The one on the left- hand side of the figure is the attack angle of the whole trajectory, while the small one on the right-hand side is the local attack angle of the first 5 s of the trajectory. It can be noticed that in a windless environment, the amplitude of attack angle of rocket projectile persistently decreases with the increase in the flight time and converges to nearly 0 degrees, indicating that the flight attitude of the rocket projectile gradually tends to become stable. Under the influence of the low-level jet in condition 1, the rocket projectile has a low velocity and weak anti-interference capability in the initial stages of launch. So, the amplitude of attack angle increases suddenly under the action of transient but strong airflow. As the rocket projectile continues to fly, the attack angle converges in a similar way as it would in a windless condition. Under the influence of the MD in condition 2, the attack angle of the rocket projectile also increases suddenly, similar to that observed in condition 1. However, due to the influence of the atmospheric turbulence at the same time, the amplitude of attack angle cannot converge further, fluctuates continuously, and the flight attitude is not stable anymore. Figure 13 shows the three-dimensional curve of flight trajectory of rocket projectile under wind field condition 1, wind field condition 2 and the windless environment. As can be seen from Figure 13, under the influence of conditions 1 and 2, the maximum trajectory height of the rocket projectile decreases, the range decreases, and the whole flight trajectory deviates significantly. In particular, the lateral deviation of the landing point under the influence of condition 1 increases much more than that under condition 2, which reflects the difference in the influence of the two wind conditions. Aerospace 2021, 8, 145 14 of 18 Aerospace 2021, 8, x FOR PEER REVIEW 16 of 21 Aerospace 2021, 8, x FOR PEER REVIEW 16 of 21 No wind condition 1 condition 2 No wind condition 1 condition 2 6 6 6 6 0 20 40 60 80 0 0 1 2 3 4 5 0 20 40 60 80 0 1 2 3 4 5 t(s) t(s) t(s) t(s) - 2 - 2 - 2 - 2 - 4 - 4 - 4 - 4 - 6 - 6 - 6 - 6 0 20 40 60 80 0 1 2 3 4 5 0 20 40 60 80 0 1 2 3 4 5 t(s) t(s) t(s) 3 t(s) - 3 - 3 - 3 0 20 40 60 80 - 3 0 1 2 3 4 5 0 20 40 60 80 0 1 2 3 4 5 t(s) t(s) t(s) t(s) Figure 12. Attack angle curves of the rocket projectile. Figure 12. Attack angle curves of the rocket projectile. Figure 12. Attack angle curves of the rocket projectile. 10.0 10.0 8.0 No wind 8.0 No wind condition1 condition1 condition2 6.0 condition2 6.0 4.0 4.0 2.0 2.0 - 2.0 - 2.0 - 1.0 - 1.0 Z/km 35.0 0 30.0 Z/km 25.0 35.0 20.0 30.0 15.0 25.0 10.0 20.0 1.0 5.0 15.0 10.0 1.0 5.0 0 X/km X/km Figure Figure113. 3. Trajecto Trajectories ries of of the r the r oc ocket ket projectile projectile un under der d dif ifffer erent wi ent wind nd ffield ield co conditions. nditions. Figure 13. Trajectories of the rocket projectile under different wind field conditions. Figure 14 shows the dispersion of the rocket projectile under the influence of condition 2 using the Monte-Carlo method. According to the flight simulation results shown in Figures 12–14, it can be concluded that the comprehensive model of complex wind field established herein, can reflect the law of influence of different wind fields on the aircraft when applied in flight simulation, which reflects the practicability and rationality of the model. Y/km δ (°) Y/km δ (°) δ(°) 1 δ(°) δ (°) δ (°) δ (°) δ (°) δ (°) 1 δ(°) 1 δ(°) δ (°) 2 Aerospace 2021, 8, 145 15 of 18 Aerospace 2021, 8, x FOR PEER REVIEW 17 of 21 0.5 No wind Condition 2 - 0.5 - 1.0 - 1.5 34.55 34.60 34.65 34.70 34.75 34.80 X/km Figure 14. Impact dispersion of the rocket projectile under the influence of condition 2 (simulation Figure 14. Impact dispersion of the rocket projectile under the influence of condition 2 (simulation time = 100). time = 100). 4.2. Analysis on Influence of Different Model Parameters on Flight Process According to the flight simulation results shown in Figures 12–14, it can be con- cluded that the comprehensive model of complex wind field established herein, can re- For a certain kind of typical wind field, it is necessary to study the effect of differ- flect the law of influence of different wind fields on the aircraft when applied in flight ent model parameters on the flight process of aircrafts. Taking the low-level jet wind simulation, which reflects the practicability and rationality of the model. field(condition 1 in Section 4.1) as an example, the simulations are made with different values of w (strength parameter) and H (scale parameter). The main ballistic parameters L T 4.2. Analysis on Influence of Different Model Parameters on Flight Process of simulations are listed in Tables 7 and 8. The attack angle curves (within the first 5 s of the flight trajectory) of d and d are shown in Figure 15. For a certain kind of 1 typica 2 l wind field, it is necessary to study the effect of different model parameters on the flight process of aircrafts. Taking the low-level jet wind Table 7. Main ballistic parameters with different w . field(condition 1 in Section 4.1) as an example, the simulations are made with different values Wof (stren Flight gth Tparam ime eter) Down and Range (scale Cross param Range eter). The Tm erminal ain balli Velocity stic pa- L T (m/s) (s) (km) (km) (m/s) rameters of simulations are listed in Tables 7 and 8. The attack angle curves (within the 0 (No wind) 104.7 34.38 0.009 366   first 5 s of the flight trajectory) of and are shown in Figure 15. 1 2 6 91.7 32.18 2.501 347 10 89.8 31.81 2.738 345 14 88.9 31.42 3.067 343 Table 7.Main ballistic parameters with different w . 18 86.1 31.04 3.182 340 w Flight Time Down Range Cross Range Terminal Velocity From Tables 7 and 8, it can be seen that the ballistic characteristics changed under (s) (km) (km) (m/s) (m/s) the influence of low-level jet. The changes are specifically manifested in the reduction in 0 (No wind) 104.7 34.38 −0.009 366 flight time and terminal velocity, the decrease in down range and the increase in cross 91.7 32.18 3 −2.501 6 347 range (the order of magnitude from 10 to 10 ). The simulation results also show that the 89.8 31.81 −2.738 influence 10 degree of low-level jet on the flight process is positively correlated to 345 the strength parameter w and the scale parameter H . L 88.9 31.42 T −3.067 14 343 From the attack angle curves in Figure 15, the attitude changes of the rocket projectile 86.1 31.04 −3.182 18 340 under different wind field model parameters can be directly observed. It is shown that the low-level jet causes a sudden change and a convergence process of the attack angle of the attack angle d , which result in the large change in the cross range (in Tables 7 and 8) of the rocket. Z/km Aerospace 2021, 8, x FOR PEER REVIEW 18 of 21 Table 8.Main ballistic parameters with different . H Flight Time Down Range Cross Range Terminal Velocity (s) (km) (km) (m/s) (m) 104.7 34.38 −0.009 0 (No wind) 366 Aerospace 2021, 8, 145 16 of 18 90.6 31.95 −3.005 400 347 89.8 31.81 −2.738 500 345 89.1 31.72 −2.692 600 345 Table 8. Main ballistic parameters with different H . 88.6 31.61 −2.461 700 T 343 W Flight Time Down Range Cross Range Terminal Velocity From Tables 7 and 8, it can be seen that the ballistic characteristics changed under (m/s) (s) (km) (km) (m/s) the influence of low-level jet. The changes are specifically manifested in the reduction in 0 (No wind) 104.7 34.38 0.009 366 flight time and terminal velocity, the decrease in down range and the increase in cross 400 90.6 31.95 3.005 347 range (the order of magnitude from 10 to 10 ). The simulation results also show that the 500 89.8 31.81 2.738 345 influence degree of low-level jet on the flight process is positively correlated to the 600 89.1 31.72 2.692 345 700 w88.6 31.61 2.461 343 strength parameter and the scale parameter H . - 2 No wind w =6m/s - 4 -1 No wind w =10m/s w =6m/s -2 L - 6 w =14m/s w =10m/s L L -3 w =14m/s w =18m/s - 8 w =18m/s -4 - 10 -5 0 1 2 3 4 5 t(s) 0 1 2 3 4 5 t(s) w w (a)  − t curve with different values of (b)  − t curve with different values of L L 1 2 2 4 - 2 No wind No wind H =6m/s - 4 H =400m T - 2 H =10m/s H =500m T H =14m/s H =600m T - 6 - 4 H =18m/s H =700m - 8 - 6 0 1 2 3 4 5 t(s) 0 1 2 t(s) 3 4 5 H H (c)  − t curve with different values of (d)  − t curve with different values of T T 1 2 Figure 15. Attack angle curves with different wind field model parameters. 4.3. Discussions of Use Conditions of the Comprehensive Model In fact, the real wind field in nature changes not only in space but also in time, which can significantly influence the accuracy of flight simulation. It is necessary to give some conditions of use for the established comprehensive model. According to the atmospheric dynamics [25], the wind field disturbance can be expressed as: W = w + Dw (31) δ (°) δ (°) δ (°) δ (°) 2 Aerospace 2021, 8, x FOR PEER REVIEW 19 of 21 Figure 15. Attack angle curves with different wind field model parameters. From the attack angle curves in Figure 15, the attitude changes of the rocket projec- tile under different wind field model parameters can be directly observed. It is shown that the low-level jet causes a sudden change and a convergence process of the attack angle of the attack angle , which result in the large change in the cross range (in Ta- bles 7 and 8) of the rocket. 4.3. Discussions of Use Conditions of the Comprehensive Model In fact, the real wind field in nature changes not only in space but also in time, which can significantly influence the accuracy of flight simulation. It is necessary to give some conditions of use for the established comprehensive model. According to the at- mospheric dynamics [25], the wind field disturbance can be expressed as: Aerospace 2021, 8, 145 17 of 18 (31) W = w+w Equation (31) indicates that the total disturbance W is composed of the mean wind w and the stochastic term w . In the conditions of small time and spatial Equation (31) indicates that the total disturbance W is composed of the mean wind scales, can be described as a certain wind field, such as micro-downburst, and the w and thewstochastic term Dw. In the conditions of small time and spatial scales, w can be described as a w certain wind field, such as micro-downburst, and the wind profile, Dw, wind profile, , can be described as the atmospheric turbulence. The diagram is can be described as the atmospheric turbulence. The diagram is shown in Figure 16. shown in Figure 16. Total wind disturbance Mean wind Stochastic term Certain wind field Wind profiles Atmospheric turbulence Figure 16. Composition of the wind disturbance in small time and spatial scales. Figure 16. Composition of the wind disturbance in small time and spatial scales. Based Based on onFigur Figure e 16 16, , the theconditions conditionsof of use use for forthe the established establishedcompr compreh ehensive ensivemodel model are given as: are given as: • When When the the flight flight si simulation mulation iis s in ina asm small all time time scscale ale (in (in con condition dition 1 o1 f Section of Section 4.1, 4.1 th,e the rocket went through the low-level jet area within 2 s) or in a small spatial scale (the rocket went through the low-level jet area within 2 s) or in a small spatial scale (the low-level jet area has a height of 800 m and the rocket has a flight altitude of 10 km), low-level jet area has a height of 800 m and the rocket has a flight altitude of 10 km), the established comprehensive model can be used to obtain some reasonable results. the established comprehensive model can be used to obtain some reasonable When the flight simulation is in a large time scale (or in a large spatial scale), such as the results. total flight process of a long-range missile, an airplane or an airship, the comprehensive • When the flight simulation is in a large time scale (or in a large spatial scale), such model might cause significant errors. as the total flight process of a long-range missile, an airplane or an airship, the comprehensive model might cause significant errors. 5. Conclusions Built on the basic principles of fluid mechanics, the engineering wind velocity cal- 5. Conclusions culation models of three typical wind fields, namely micro-downburst, low-level jet and Built on the basic principles of fluid mechanics, the engineering wind velocity atmospheric turbulence, are herein established. By combining the three models in the calculation models of three typical wind fields, namely micro-downburst, low-level jet ground coordinate system and unifying the input and output parameters, a comprehensive and atmospheric turbulence, are herein established. By combining the three models in the model of complex wind field is established. The wind field model simulation and appli- ground coordinate system and unifying the input and output parameters, a cation simulation indicate that the comprehensive model can reasonably and effectively comprehensive model of complex wind field is established. The wind field model describe the flow characteristics of the relevant wind field, and has the characteristics of simple calculation and model scalability. Additionally, as shown in Section 4.2, the com- prehensive model can be used not only to analyze the influence of different wind fields on the flight process, but also to study the influence of different model parameters on the flight simulations. As an important factor of the simulation accuracy of the wind field model, the wind field data collection methods in different time and space scales of the simulated flights are the main points of our further research. Meanwhile, developing a reasonable wind field model for large time and space scales with the least amount of collection data is also an extension of the work in this paper. Author Contributions: Conceptualization, J.C.; methodology, J.C.; software, J.C., Z.Y.; validation, J.C.; formal analysis, J.C. and J.F..; investigation, J.C. and J.F.; resources, J.C. and Z.Y.; writing—original draft preparation, J.C.; writing—review and editing, J.C.; supervision, L.W. and J.F.; project adminis- tration, L.W.; funding acquisition, L.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Science Foundation of China (No. 61603191). Data Availability Statement: Not applicable. Aerospace 2021, 8, 145 18 of 18 Conflicts of Interest: The authors declare no conflict of interest. References 1. Tuleya, R.E.; Lord, S.J. The Impact of Dropwindsonde Data on GFDL Hurricane Model Forecasts Using Global Analyses. Weather Forecast. 2010, 12, 307–323. [CrossRef] 2. Bo, M.; Lin, M.; Peng, H. WindSim Computational Flow Dynamics Model Testing Using Databases from Two Wind Measurement Stations. 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Flight Principle in Atmospheric Turbulence, 1st ed.; National Defense Industry Press: Beijing, China, 1993; pp. 27–47. 26. Chen, J.; Wang, L.; Li, Z. Influence of two typical kinds of low-level wind shear on ballistic performance of rockets. J. Beijing Univ. Aeronaut. Astronaut. 2018, 44, 1008–1017. 27. Marengo, J.A.; Soares, W.R.; Saulo, C.; Nicolini, M. Climatology of the Low-Level Jet East of the Andes as Derived from the NCEP–NCAR Reanalyses: Characteristics and Temporal Variability. J. Clim. 2004, 17, 12–19. [CrossRef] 28. Beal, T.R. Digital Simulation of Atmospheric Turbulence for Dryden and von Karman Models. J. Guid. Control. Dynam. 1993, 16, 132–138. [CrossRef] 29. Jing, G.; Guanxin, H.; Zaoqing, L. Theory and method of numerical simulation for 3D atmospheric turbulence field based on Von Karman model. J. Beijing Univ. Aeronaut. Astronaut. 2012, 38, 736–740. 30. Halsig, S.; Artz, T.; Iddink, A.; Nothnagel, A. Using an atmospheric turbulence model for the stochastic model of geodetic VLBI data analysis. Earth Planets Space 2016, 68, 106. [CrossRef] 31. Han, Z.P. Exterior Ballistics of Shells and Rockets, 2nd ed.; Beijing University of Technology Press: Beijing, China, 2008; pp. 141–143. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Aerospace Multidisciplinary Digital Publishing Institute

Engineering Comprehensive Model of Complex Wind Fields for Flight Simulation

Aerospace , Volume 8 (6) – May 24, 2021

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aerospace Article Engineering Comprehensive Model of Complex Wind Fields for Flight Simulation Jianwei Chen *, Liangming Wang *, Jian Fu and Zhiwei Yang School of Energy and Power Engineering, Nanjing University of Science and Technology, Nanjing 210094, China; fujian@njust.edu.cn (J.F.); patch1205b@163.com (Z.Y.) * Correspondence: cjwrko@njust.edu.cn (J.C.); lmwang802@163.com (L.W.); Tel.: +86-(025)-8431-4923 (L.W.) Abstract: A complex wind field refers to the typical atmospheric disturbance phenomena existing in nature that have a great influence on the flight of aircrafts. Aimed at the issues involving large volume of data, complex computations and a single model in the current wind field simulation approaches for flight environments, based on the essential principles of fluid mechanics, in this paper, wind field models for two kinds of wind shear such as micro-downburst and low-level jet plus three-dimensional atmospheric turbulence are established. The validity of the models is verified by comparing the simulation results from existing wind field models and the measured data. Based on the principle of vector superposition, three wind field models are combined in the ground coordinate system, and a comprehensive model of complex wind fields is established with spatial location as the input and wind velocity as the output. The model is applied to the simulated flight of a rocket projectile, and the change in the rocket projectile’s flight attitude and flight trajectory under different wind fields is analyzed. The results indicate that the comprehensive model established herein can reasonably and efficiently reflect the influence of various complex wind field environments on the flight process of aircrafts, and that the model is simple, extensible, and convenient to use. Keywords: flight simulation; wind field model; comprehensive model Citation: Chen, J.; Wang, L.; Fu, J.; Yang, Z. Engineering Comprehensive Model of Complex Wind Fields for Flight Simulation. Aerospace 2021, 8, 145. https://doi.org/10.3390/ 1. Introduction aerospace8060145 A complex wind field refers to the typical atmospheric disturbance phenomena in nature, for instance gusts, typhoons, wind shear, turbulence and so on, which are closely Academic Editor: Carlo E.D. Riboldi related to factors such as weather and terrain. In flying aircrafts, rockets, missiles, and air- ships, etc., the whole flight process is affected by complex wind fields. For example, Received: 28 April 2021 the flight stability and safety of aircrafts will reduce, and the firing accuracy of rockets and Accepted: 21 May 2021 missiles will become worse. Therefore, establishing a computational model of complex Published: 24 May 2021 wind fields suitable for flight simulation is of great significance to the design of aircraft and the study of flight control. Publisher’s Note: MDPI stays neutral At present, there are three types of modeling method for complex wind fields. The first with regard to jurisdictional claims in method is the wind field data measurement method. The data obtained from this method published maps and institutional affil- are authentic and reliable, but the volume of data is large and the computation cost is iations. high. [1,2], respectively, substituted the exploratory data of 14 Atlantic tropical storms from 1982 to 1989 collected by the Hurricane Research Division (HRD) and the wind vector data from the HY-2 satellite into the wind field data model of the National Center for Environmental Prediction (NCEP) for calculation, analysis and verification. [3] used particle Copyright: © 2021 by the authors. filter and Monte Carlo methods to process the measured data and set up the wind velocity Licensee MDPI, Basel, Switzerland. model. [4] reanalyzed the atmospheric circulation data taken from the Black Sea region This article is an open access article during the period of 1958 to 2001 and established the distribution of circulation wind field distributed under the terms and affected by sea temperature. The second method is the numerical method of atmospheric conditions of the Creative Commons dynamics, which needs to solve the computation-heavy nonlinear differential equations, Attribution (CC BY) license (https:// due to which the methodology is too complicated. [5] used the mesoscale model (MM5) of creativecommons.org/licenses/by/ the National Center for Atmospheric Research of Pennsylvania State University to calculate 4.0/). Aerospace 2021, 8, 145. https://doi.org/10.3390/aerospace8060145 https://www.mdpi.com/journal/aerospace Aerospace 2021, 8, 145 2 of 18 the distribution of low-level winds located over Antarctica. [6] used the RNG turbulence closure model along with the SIMPLEC pressure correction algorithm to establish the nature of wind field distribution around different buildings. The third method is the engineering simulation method. This method begins with the flow characteristics of airflow in various wind fields and describes the law of airflow movement with simple fluid dynamics equations. This method is simple and intuitive, and can highlight the influence of primary physical parameters. For example, refs. [7–9] built the engineering models of micro-downburst using the vortex ring method and wind profile model, respectively. Ref. [10] established a wind distribution model over a large-scale ridge with temperature as the vertical coordinate, and [11,12] developed the atmospheric turbulence models for flight simulation. In the actual flight simulation applications, it is frequently necessary to quickly adjust the parameters according to the changing flight environments to complete the simulation. As a result, it is difficult for both the measured wind field data method and the atmospheric dynamics numerical method to meet such real-time and fast-paced requirements. The engineering simulation method has been widely used because of its simplicity and low computation requirement [13–16]. It can be seen from the above discussion about the available literature that there are two major deficiencies in the current research on wind field modeling in flight environment. Firstly, the model is relatively simple, because it considers only one form of wind field, and neglects the case that the natural wind appears simultaneously in the form of wind shear, turbulence, and other forms under the influence of terrain and weather. Secondly, owing to the large volume of model data and the complex calculations, special fluid dynam- ics analysis software is needed. Additionally, it is difficult to meet the requirements of fast simulation or real-time simulation in some practical engineering applications. To address these problems, in this paper, the wind field models of two typical types of wind shear, namely micro-downburst and low-level jet, and atmospheric turbulence are established by the using engineering simulation method. Then, taking the ground coordinate system as the unified reference coordinate system, the three wind field models are combined together to construct a comprehensive wind field model with spatial location coordinates as the input and the wind velocity as the output. Finally, the ballistic simulation of a rocket projectile is used as an example to verify the practicability and efficiency of the comprehensive wind field models. 2. Typical Wind Field Models 2.1. Micro-Downburst A micro-downburst (abbreviated as MD in this section) is a type of low-level wind shear associated with convective weather [17,18]. Figure 1 shows a schematic diagram of airflow distribution in the MD wind field. It can be noticed that the MD manifests itself as a local vertically downward airflow in the strong convective cloud cluster. After the airflow sinks and touches the ground, it diverges in all directions and curls up to form an area of vortex ring above the ground. In view of such flow characteristics, a bunch of ground-symmetric vortex rings were constructed in the vertical direction of the horizontal plane to simulate the vertical airflow generation [19]. The coordinate system of the model is illustrated in Figure 2. Considering the ground coordinate system oxyz as the datum and a point O above the plane xoz as the center, a closed vortex ring of radius R is set, named the main vortex ring, and its curve equation is: 2 2 (x x ) + (y y ) = R p p (1) z = z p Aerospace 2021, 8, 145 3 of 18 Aer Aerosp ospace ace 20 2021 21, , 8 8, , x FO x FOR P R PEE EER R RE REVIEW VIEW 3 3 of of 21 21 clouds clouds Clouds Clouds downdraght downdraght D D o ow wn nd dr ra ag gh ht t Horizontal vortex Horizontal vortex Flow Flow f fr ro on nt t external flow external flow E xternal E xternal Ground Ground flow flow Figure Figure 1. 1. Airflow Airflow distribution distribution of of micr micro o-downburst. -downburst. Figure 1. Airflow distribution of micro-downburst.  Main vortex ring Main vortex ring O Main vortex ring Main vortex ring P P min min zz rr M max max Z r Z Aircraft P Aircraft Ground symmetric point Ground symmetric point G Gr ro ou un nd d c co oo or rd diin na at te e system system Y Mirror vortex ring  Mirror vortex ring  II II Mirror vortex ring Mirror vortex ring Aircraft Aircraft Figure 2. Coordinate system of micro-downburst model. Figure 2. Coordinate system of micro-downburst model. Figure 2. Coordinate system of micro-downburst model. The Consider circulation ing thline e ground equation coordin of theate main system vortex ox is:yz as the datum and a point Considering the ground coordinate system oxyz as the datum and a point O xxoz oz R above above th the e pl plan ane e as as th the e ce cent nter, er, a a clo closed sed vo vortex rtex ring ring o of f r radius adius R i is s s set, et, named named th the e y = (r + r )F(k) (2) P max min main vortex ring, and its curve equation is: main vortex ring, and its curve equation is: 2p where G denotes the intensity of the (x− vortex x ) ring, + (y− determined y ) = R by vertical velocity V (0) of (x− x ) + (y− y ) = R z  pp  pp (1) the vortex ring center and the vortex ring radius R: (1)  zz = zz =  p  p  G = 2RV (0) (3) The circulation line equation of the main vortex is: The circulation line equation of the main vortex is: r and r are the maximum and the minimum distances of the main vortex ring max min from an arbitrary point O (x , y , z=+ ) in(the r ground r )F coor (k )dinate system, and F(k) is the =+ (r r )F (k ) ( (2 2) ) M M M M P max min P max min 2 2 elliptic integral function, where: where  denotes the intensity of the vo rtex ring, d etermined by vertical velocity where denotes the intensity of the vortex ring, determined by vertical velocity V (0) r r V (0) max min z k = (4) r + r of the vortex ring center and the vortex ring maxradius R : of the vortex ring center and the vortex ring radius min R : Based on the theories of higher functions [7], when 0  k  1, F(k) is approximated as: = 2RV (0) = 2RV (0) ( (3 3) ) 0.788k rr r F(k)  p (5) max min max min and are the maximum and the minimum distances of the main vortex ring and are the maximum and the minimum distances of the main vortex ring 0.25 + 0.75 1 k O (x , y , z ) Fk () O (x , y , z ) Fk () M M M M M M M M from an arbitrary point in the ground coordinate system, and is from an arbitrary point in the ground coordinate system, and is Based on Equation (2), the induced radial and axial velocities of the main vortex ring th the el e ell liptic iptic integra integral functio l function, where n, where:: can be calculated by partial derivatives: rr − rr − max min max min ¶y k = 1 k = P P (4) (4) v = r ¶z rr P + R rr + max min (6) max min ¶y P P v = r ¶r P P Aerospace 2021, 8, 145 4 of 18 2 2 where r is distance of the point O from the axis of vortex ring (r = (x x ) + (y y ) ). P M P M P M P By decomposing the induced velocities calculated by Equation (6) along the ox and oz axes of the ground system, the velocity components can be written as: x x P M P P v = v x r (7) y y P M P P v = v y r In the actual circumstances, the vertical velocity component on the ground should be zero after the vertical airflow of the vortex ring center reaches the ground. For that reason, by setting the method of mirror vortex ring for the symmetry of main vortex ring about the plane xoy, the vertical induced velocities on the ground are reversed with equal value and thus cancel each other out. The signs of the streamline equations for the two vortex rings are opposite y = y . Given the mirrored vortex ring center O (x , y ,z ), its connection I P I p p p line O O with the main vortex ring center O is vertical to the ground. According to P I P the streamline equation of the mirrored vortex ring, analogous to the derivation of the induced velocity of the main vortex ring, the induced velocities at the spatial points of the mirrored vortex ring can be calculated. Velocity superposition is performed by combining Equations (6) and (7), and the resultant velocity at point O can be obtained as: 0 1 0 1 0 1 P I w v v MDx x x P I @ A @ A @ A w = v + v (8) MDy y y P I w v v MDz z z Then, the streamline equation of point O is: " # 0 0 2 0 0.788k (r + r ) G 0.788k (r + r ) max min max min y = y + y = p p (9) P I 2 0 2 2p 0.25 + 0.75 1 k 0.25 + 0.75 1 k 0 0 where r and r denote the maximum and the minimum distances of the mirrored max min 0 0 r r max 0 min vortex ring from the spatial point O , and similarly k = . M 0 0 r +r max min The main parameters [20] of the vortex ring model are shown in Table 1. Table 1. Parameters of micro-downburst model. Parameter Value O (x , y , z ) (1000 m, 0, 800 m) p p p R 1100 m V (0) 10 m/s The simulation results of the model are shown in Figure 3. It can be noted that the simulated wind field distribution of the vortex ring model is consistent with the measured data. Aerospace 2021, 8, x FOR PEER REVIEW 5 of 21 ' ' r r where and denote the maximum and the minimum distances of the max min '' rr − max min mirrored vortex ring from the spatial point , and similarly . k = '' rr + max min The main parameters [20] of the vortex ring model are shown in Table 1. Table 1. Parameters of micro-downburst model. Parameter Value O (x , y , z ) P p p p (1000 m, 0, 800 m) 1100 m V (0) −10 m/s Aerospace 2021, 8, 145 5 of 18 The simulation results of the model are shown in Figure 3. It can be noted that the simulated wind field distribution of the vortex ring model is consistent with the measured data. 1.6 15.0m/s 1.4 1.2 1.0 0.8 0.6 0.4 0.2 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 X(km) (a) (b) 2.0 15.0m/s 1.5 1.0 0.5 -0.5 -1.0 -1.5 -2.0 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 Z(km) (c) (d) Figure 3. Comparisons between simulation results and measured data of micro-downburst. (a) Simulation wind vector Figure 3. Comparisons between simulation results and measured data of micro-downburst. (a) Simulation wind vector diagram of horizontal section. (b) Flow pattern of 7 July 1990 Orlando downburst from NTRS-NASA Technical Reports diagram of horizontal section. (b) Flow pattern of 7 July 1990 Orlando downburst from NTRS-NASA Technical Reports Server [21]. (c) Simulation wind vector diagram of vertical section. (d) Flow pattern of 1988 microburst event of DEN from NTRS-NASA Technical Reports Server [22]. 2.2. Low-Level Jet Low-level jet (abbreviated as LLJ in this section) refers to the wind velocity zone in the lower troposphere, which is significantly affected by the mesoscale weather system. It is a surface inversion phenomenon occurring in the stable surface boundary layer [23]. According to the principle of the plane wall jet, when a plane free jet with a large width flows through a narrow slit, the velocity distribution of the jet near the wall is similar to that of the LLJ [24]. For that reason, the plane wall jet is employed to simulate the wind velocity distribution of the LLJ wind field, as shown in Figure 4. According to the principles of fluid dynamics, the relationship between the horizontal velocity component u(x, H) of a free jet and its maximum jet velocity u (x) is: u(x, H) H = 1 th (k ) (10) u (x) x Y(km) Y(km) Aerospace 2021, 8, x FOR PEER REVIEW 6 of 21 2.2. Low-Level Jet Low-level jet (abbreviated as LLJ in this section) refers to the wind velocity zone in the lower troposphere, which is significantly affected by the mesoscale weather system. It is a surface inversion phenomenon occurring in the stable surface boundary layer [23]. According to the principle of the plane wall jet, when a plane free jet with a large width flows through a narrow slit, the velocity distribution of the jet near the wall is similar to Aerospace 2021, 8, 145 6 of 18 that of the LLJ [24]. For that reason, the plane wall jet is employed to simulate the wind velocity distribution of the LLJ wind field, as shown in Figure 4. Wind profile u(x,H) H jet Figure Figure 4. 4. Diagram Diagram of of the the plane plane wall wall jet. jet. where k denotes a scale factor and th refers to hyperbolic tangent function. To simplify According to the principles of fluid dynamics, the relationship between the the model, the basic flow characteristics of LLJ are retained, while the wind velocity is horizontal velocity component of a free jet and its maximum jet velocity u(x,H ) evenly distributed in the horizontal direction x, that is, ux () is: u(x, H) H H = 1 th (C ) (11) u(x, H ) H u (x) H m 2 s =− 1 th (k ) (10) u () x x where H denotes the height of maximum velocity of the free jet with symmetrical distribu- tion, C is employed to describe the relationship between H and the width B of the jet in s s where denotes a scale factor and th refers to hyperbolic tangent function. To the vertical 1 direction; and the width of jet denoting the velocity range determined by 7% of the maximum velocity. The expression is simplify the model, the basic flow characteristics of LLJ are retained, while the wind velocity is evenly distributed in the horizontal direction , that is, B = 4 (12) u(x, H ) HH − C2 =− 1 th (C ) s (11) u () x H ms By superimposing the velocity distribution of the free jet onto the exponential model of the mean wind in the boundary layer, the vertical velocity profile of LLJ on the surface where denotes the height of maximum velocity of the free jet with symmetrical boundary layer can be obtained: C H distribution, is employed to describe the relationship between  and the width s s H H H u(x, H) = u ( ) + u 1 th (C ) (13) R s s B of the jet in the vertical direction; and the width of jet denoting the velocity range H H R s determined by 7% of the maximum velocity. The expression is In the formula above, u is the wind velocity corresponding to the height of H , where R R the distribution index m can be calculated in keeping with the relevant meteorological B= 4 (12) data [25]: h i 0.5 m = 1/ ln ( H H ) /Z 0.0403 ln(u /6) (14) p 0 0 R where Z denotes terrain roughness; analogous to Equation (13), the deviation of wind direction between height H and height H is derived as: H H H H 0 L a( H) = a + arctan[ tan(a a )] + a 1 th (C ) (15) H H H H L 0 0 L H H H T 0 L where H , H and H denote reference height, height of maximum deviation of wind 0 L T direction in the jet stream layer and height of top of the jet stream layer, respectively; a , a and a are the angles included between the wind direction and the geostrophic wind H H L T Aerospace 2021, 8, 145 7 of 18 at the three corresponding heights, respectively; w and w are wind velocities at heights 0 L H and H , respectively. 0 L Thus, the wind velocity components in the ground coordinate system are calculated as: w = u(x, H) cos(a( H)) LL Jx (16) w = u(x, H) sin(a( H)) LL Jz The model parameters [26] are configured as shown in Table 2 and the simulation results are shown in Figure 5. Table 2. Parameters of low-level jet model. Z H w H w H a a a C C 0 0 0 L L T H H H s L 0 L T Aerospace 2021, 8, x FOR PEER REVIEW 8 of 21 1 1 2.5 m 3.5 m 5 ms 180 m 10 ms 800 m 0 30 60 0.8 0.3 0 10 20 30 40 wind speed(m/s) (a) (b) u(x,H ) Figure 5. Comparisons between simulation results and measured data of low-level jet. (a) Wind profiles (u(x, H)) of Figure 5. Comparisons between simulation results and measured data of low-level jet. (a) Wind profiles ( ) of low-level jet by simulation. (b) Wind profiles derived from the NCEP-NCAR reanalyzes and from the PACS-SONET low-level jet by simulation. (b) Wind profiles derived from the NCEP-NCAR reanalyzes and from the PACS-SONET upper-air observations [27] (© American Meteorological Society. Used with permission). upper-air observations [27] (© American Meteorological Society. Used with permission). It can be seen from Figure 5a,b that the wall jet model can accurately simulate the It can be seen from Figure 5a,b that the wall jet model can accurately simulate the wind velocity distribution of the actual LLJ wind field. wind velocity distribution of the actual LLJ wind field. 2.3. Atmospheric Turbulence 2.3. Atmospheric Turbulence Atmospheric turbulence is the irregular and uneven random eddy motion of the Atmospheric turbulence is the irregular and uneven random eddy motion of the Earth’s atmosphere. The Dryden model is a classical form of atmospheric turbulence, Earth’s atmosphere. The Dryden model is a classical form of atmospheric turbulence, wherein the longitudinal and transverse correlation functions [28] are expressed as: wherein the longitudinal and transverse correlation functions [28] are expressed as: x /L − /L f (x) = e fe () = (17) x /L g(x) = e (1 ) 2L (17) − /L ge ( )=− (1 ) The spatial correlation function [29] of one-dimensional turbulence is: 2L 2 x /L The spatial correlation function [29] of one-dimensional turbulence is: R(x) = s (1 x /2L)e (18) 2/ − L R( )=−  (1  / 2L)e (18) For two (or three)-dimensional turbulence, the spatial correlation function is:  + 2/ − L Re (, ,) = (1− ) uu 1 2 3 2L   + 2/ − L Re (, ,) = (1− ) (19) vv 1 2 3 2L  + 2/ − L Re (, ,) = (1− ) ww 1 2 3 2L    where denotes the location difference between the two spatial points, , , 1 2 3 are the components of in the ground coordinate system, is the turbulence u v intensity parameter, L is the turbulence scale parameter, and the subscripts , , denote the components of turbulent velocity in the ground coordinate system. For a one-dimensional turbulent sequence, the recursion equation is: w(x)= aw(x−h)+br(x) (20) height(m) Aerospace 2021, 8, 145 8 of 18 For two (or three)-dimensional turbulence, the spatial correlation function is: 2 2 x +x > 2 x /L 2 3 R (x , x , x ) = s e (1 ) > u 1 2 3 2Lx 2 2 x +x 2 x /L 1 3 (19) R (x , x , x ) = s e (1 ) 1 2 3 v 2Lx 2 2 x +x : 2 x /L 2 1 R (x , x , x ) = s e (1 ) w 2 3 1 w 2Lx where x denotes the location difference between the two spatial points, x , x , x are the 1 2 3 components of x in the ground coordinate system, s is the turbulence intensity parameter, L is the turbulence scale parameter, and the subscripts u, v, w denote the components of turbulent velocity in the ground coordinate system. For a one-dimensional turbulent sequence, the recursion equation is: w(x) = aw(x h) + br(x) (20) where W(x) denotes the turbulent velocity along the one-dimensional x direction, r(x) is the one-dimensional white noise sequence, h is the simulation step size, and a, b are the recursion parameters to be solved. According to the correlation function definitions of the Dryden model, 2 2 R = E[w(x)w(x)] = a R + b 0 0 (21) R = E[w(x)w(x h)] = aR 1 0 Combining Equations (18) and (21), where R = R(i h), it is obtained that h/L a = (1 h/2L)e (22) b = s 1 a Thus, the one-dimensional atmospheric turbulent velocity can be calculated. Similarly to Equation (20), the recursion equation of two-dimensional turbulence can be written as: w(x, y) = a w(x h, y) + a w(x, y h) + a w(x h, y h) + br(x, y) (23) 1 2 3 The correlation function values are R = E[w(x, y)w(x, y)] R = E[w(x, y)w(x, y)] (24) R = E[w(x, y)w(x + h, y)] R = E[w(x, y)w(x + h, y + h)] The equation obtained by expanding Equation (24) is R = a R + a R + a R + b 00 1 10 2 01 3 11 R = a R + a R + a R 2 00 3 01 1 11 10 (25) > R = a R + a R + a R 10 1 00 2 11 3 01 R = a R + a R + a R 2 3 00 11 1 01 10 Substitute R , R , R , R calculated by Equation (19) into Equation (25), where 00 01 10 11 R = R (i h, j h), so that the parameters a , a , a , b can be obtained. Thus, the two- u 2 3 i j 1 dimensional atmospheric turbulent velocity can be calculated according to the recursive Equation (23). For the three-dimensional space turbulence, the construction idea is as follows: taking one-dimensional turbulence as the boundary value to figure out the two-dimensional plane turbulence sequence, and subsequently taking the two-dimensional plane turbulence sequence as the boundary value to figure out the three-dimensional space turbulence, as shown in Figure 6. Aerospace 2021, 8, 145 9 of 18 Aerospace 2021, 8, x FOR PEER REVIEW 10 of 21 w(x, y) w(x , y + jh) wx() w() x + h w() x + ih 0 0 w(x , y ) w(x + ih, y ) y y z z one-dimensional two-dimensional three-dimensional Figure 6. Construction process of three-dimensional atmospheric turbulence. Figure 6. Construction process of three-dimensional atmospheric turbulence. Analogous to Equations (20) and (23), the three-dimensional turbulence recursion Analogous to Equations (20) and (23), the three-dimensional turbulence recursion equation is formulated as equation is formulated as w(x, y, z)= a w(x− h, y− h, z− h)+ a w(x− h, y− h, z)+ a w(x, y− h, z− h) 1 2 3 w(x, y, z) = a w(x h, y h, z h) + a w(x h, y h, z) + a w(x, y h, z h) 1 2 3 +a w(x h, y, z h) + a w(x, y, z h) + a w(x h, y h, z) (26) + 4a w(x− h, y, z− h)+5a w(x, y, z− h)+ 6 a w(x− h, y− h, z) (26) 4 5 6 +a w(x h, y, z) + br(x, y, z) + a w(x− h, y, z)+ br(x, y, z) The correlation function of the three-dimensional space turbulence can be expressed as The correlation function of the three-dimensional space turbulence can be ex- pressed as R = E[w(x, y, z)w(x + ih, y + jh, z + kh)] (i, j, k = 0, 1) (27) i jk R = E[w(x, y, z)w(x+ih, y+ jh, z+kh)] (i, j,k = 0,1) ijk (27) Substituting Equation (19) into Equation (27), where R = R (i  h, j  h, k  h), i jk the following equations are obtained R = R (i *h, j *h,k *h) Substituting Equation (19) into Equation (27), where , ijk u the following equations are obtained R = a R + a R + a R + a R + a R + a R + a R + s 000 1 111 2 110 3 011 4 101 5 001 6 010 7 100 R = a R + a R + a R + a R + a R + a R + a R 001 R 1=110 a R 2+ a 111R 3+ a 010R + 4 a 100R + 5 a000 R + 6a 011 R +7a 101 R + 000 1 111 2 110 3 011 4 101 5 001 6 010 7 100 R = a R + a R + a R + a R + a R + a R + a R 010 1 101 2 100 3 001 4 111 5 011 6 000 7 110 R = a R + a R + a R + a R + a R + a R + a R 001 1 110 2 111 3 010 4 100 5 000 6 011 7 101 R = a R + a R + a R + a R + a R + a R + a R 011 1 100 2 101 3 000 4 110 5 010 6 001 7 111 (28) R = a R + a R + a R + a R + a R + a R + a R 100 1 011 2 010 3 111 4 001 5 101 6 110 7 000 R = a R + a R + a R + a R + a R + a R + a R 010 1 101 2 100 3 001 4 111 5 011 6 000 7 110 R = a R + a R + a R + a R + a R + a R + a R 101 1 010 2 011 3 110 4 000 5 100 6 111 7 001 R = a R + a R + a R + a R + a R ++ a R a R 011 1 100 2 101 3 000 4 110 5 010 6 001 7 111 R = a R + a R + a R + a R + a R + a R + a R 110 1 001 2 000 3 101 4 011 5 111 6 100 7 010 (28) R = a R + a R + a R + a R + a R + a R + a R 111 1 000 2 001 3 100 4 010 5 110 6 101 7 011 R = a R + a R + a R + a R + a R + a R + a R 100 1 011 2 010 3 111 4 001 5 101 6 110 7 000 R = a R + a R + a R + a R + a R + a R + a R By solving Equation (28), the recursive parameters a  a and b can be obtained, 101 1 010 2 011 3 110 4 000 5 1 100 6 111 7 001 and then the velocity recursion equation of three-dimensional space turbulence w (x, y, z) R = a R + a R + a R + a R + a R + a R + a R 110 1 001 2 000 3 101 4 011 5 111 6 100 7 010 is obtained. By using the method above, w (x, y, z) and w (x, y, z) can also be obtained. v w R = a R + a R + a R + a R + a R + a R + a R AT refers to the atmospheric turbulence herein. 111 1 000 2 001 3 100 4 010 5 110 6 101 7 011 w = w (x, y, z) ATx u aa By solving Equation (28), the recursive parameters and b can be ob- w = w (x, y, z) (29) ATy tained, and then the velocity recursion equation of three-dimensional space turbulence w = w (x, y, z) ATz w (x, y, z) w (x, y, z) w (x, y, z) is obtained. By using the method above, and can u v w According to the parameters in Table 3, the three-dimensional atmospheric turbulence also be obtained. AT refers to the atmospheric turbulence herein. was simulated in space. The simulation results are shown in Figures 7 and 8. From Figure 8, Aerospace 2021, 8, x FOR PEER REVIEW 11 of 21 w = w (x, y, z) AT−x u w = w (x, y, z) (29) AT−y v w = w (x, y, z) AT−z w Aerospace 2021, 8, 145 10 of 18 According to the parameters in Table 3, the three-dimensional atmospheric turbu- lence was simulated in space. The simulation results are shown in Figures 7 and 8. From Figure 8, the random variation of the atmospheric turbulent wind velocity in space is the random variation of the atmospheric turbulent wind velocity in space is observed. observed. As presented in Figure 8, the theoretical value is calculated according to cor- As presented in Figure 8, the theoretical value is calculated according to correlation func- relation function (Equation (19)) of the Dryden model. It can be seen that the variation tion (Equation (19)) of the Dryden model. It can be seen that the variation trend in the trend in the correlation value of turbulent velocity simulated by simulation is actually correlation value of turbulent velocity simulated by simulation is actually the same as that the same as that obtained from the Dryden model, which proves the rationality and ef- obtained from the Dryden model, which proves the rationality and effectiveness of the fectiveness of the established model. established model. Table 3. Parameters of atmospheric turbulence model [30]. Table 3. Parameters of atmospheric turbulence model [30]. Parameter Value Parameter Value L== L L 150 m uvw L = L = L 150 m u v w −1  ==   1 1.5 ms uvw s = s = s 1.5 ms u v w h 50 m h 50 m -4 -8 -5 z/h x/h Aerospace 2021, 8, x FOR PEER REVIEW 12 of 21 Figure 7. Atmospheric turbulent wind velocity in space (H = 600 m). Figure 7. Atmospheric turbulent wind velocity in space (H = 600 m). 1.2 Si mul aton resul ts Theoreti cal resul ts 0.8 0.6 0.4 0.2 0 200 400 600 800 1000 1200  (m) Figure 8. Comparisons of correlation values between simulation results and theoretical calcula- Figure 8. Comparisons of correlation values between simulation results and theoretical calculation. tion. 3. Comprehensive Wind Field Model 3. Comprehensive Wind Field Model Since wind velocity is a vector including magnitude and direction, it meets the princi- ple of vector superposition. For any spatial point P, assuming that wind field A induces Since wind velocity is a vector including magnitude and direction, it meets the principle of vector superposition. For any spatial point P , assuming that wind field A induces the wind velocity vector at point P and wind field B induces the wind velocity vector at point P , the total wind velocity vector at point P induced by the wind fields A and B can be obtained by the following equation: W=+ W W (30) P P P AB Based on Equation (30), in this section, a superposition method is adopted to syn- thesize the established typical wind field models. The input and output parameters of the three typical wind field models are listed in Table 4. Table 4. Input and output parameters of the three typical wind field models established. Wind Field Input Output (x, y, z) Micro-downburst w w w , , MD−x MD−z MD−y H() H = y Low-level jet w w LLJ−x LLJ−z (x, y, z) w w w Atmospheric turbulence , , AT−x AT−z AT−y It can be seen that the inputs of the three models are the spatial positions, and the outputs are the wind velocity components. Therefore, the inputs and outputs are com- bined, and the ground coordinate system is used as the frame of reference to establish a comprehensive model. The application structure of this model is shown in Figure 9. w(m/s) Aerospace 2021, 8, 145 11 of 18 the wind velocity vector W at point P and wind field B induces the wind velocity vector ! ! W at point P, the total wind velocity vector W at point P induced by the wind fields A P P and B can be obtained by the following equation: ! ! ! W = W + W (30) P P P A B Based on Equation (30), in this section, a superposition method is adopted to synthe- size the established typical wind field models. The input and output parameters of the three typical wind field models are listed in Table 4. Table 4. Input and output parameters of the three typical wind field models established. Wind Field Input Output Micro-downburst (x, y, z) w , w , w MDx MDy MDz Low-level jet H( H = y) w , w LL Jx LL Jz Atmospheric turbulence (x, y, z) w , w , w ATx ATy ATz It can be seen that the inputs of the three models are the spatial positions, and the outputs are the wind velocity components. Therefore, the inputs and outputs are com- Aerospace 2021, 8, x FOR PEER REVIEW 13 of 21 bined, and the ground coordinate system is used as the frame of reference to establish a comprehensive model. The application structure of this model is shown in Figure 9. Application settings Aircraft flight Terrain & climate model environment Model selection Input (x,y,z) Parameter setting Typical Complex wind models Micro-downburst Low-level jet Extended Atmospheric model model Turbulence model models Wind speed superposition Output (w , w , w ) x y z Comprehensive model Figure 9. Application structure of the comprehensive wind field model. Figure 9. Application structure of the comprehensive wind field model. As can be seen from Figure 9, the calculation process of the comprehensive wind field As can be seen from Figure 9, the calculation process of the comprehensive wind model consists of four key steps: field model consists of four key steps: Input spatial location parameters; • Input spatial location parameters; Select the wind field models according to simulation requirements; • Select the wind field models according to simulation requirements; Calculate the total wind velocity value; • Calculate the total wind velocity value; Substitute the wind velocity into the flight simulation. • Substitute the wind velocity into the flight simulation. To ensure the scalability of the model, an expansion module (yellow box in the chart) To ensure the scalability of the model, an expansion module (yellow box in the is set up to add other wind field models such as gust and mountain flow. chart) is set up to add other wind field models such as gust and mountain flow. 4. Model Application 4.1. Flight Simulations under Different Wind Field Conditions In order to verify the calculation effect on the whole model, a rocket projectile is taken as an example and the six degrees-of-freedom rigid body trajectory equation of the rocket projectile is considered as the flight simulation model. The basic parameters [31] of the rocket projectile are listed in Table 5. Table 5. Basic parameters of the rocket projectile. Parameter Value Diameter of rocket 0.122 m Length of rocket 2.9 m Specific impulse 250 s Working time of the engine 3 s −1 Initial velocity 40 ms Firing angle 50 deg Firing direction 0 deg For the flight state of a rocket projectile, the main concern is its trajectory and flight attitude. The flight trajectory can be directly analyzed by calculating the wind Aerospace 2021, 8, 145 12 of 18 4. Model Application 4.1. Flight Simulations under Different Wind Field Conditions In order to verify the calculation effect on the whole model, a rocket projectile is taken as an example and the six degrees-of-freedom rigid body trajectory equation of the rocket projectile is considered as the flight simulation model. The basic parameters [31] of the rocket projectile are listed in Table 5. Table 5. Basic parameters of the rocket projectile. Parameter Value Diameter of rocket 0.122 m Length of rocket 2.9 m Specific impulse 250 s Working time of the engine 3 s Initial velocity 40 ms Firing angle 50 deg Aerospace 2021, 8, x FOR PEER REVIEW 14 of 21 Firing direction 0 deg For the flight state of a rocket projectile, the main concern is its trajectory and flight three-dimensional trajectory curve of the rocket projectile, while the flight attitude needs attitude. The flight trajectory can be directly analyzed by calculating the three-dimensional to be reflected by the flight attack angle. trajectory curve of the rocket projectile, while the flight attitude needs to be reflected by the Figure 10 shows the conceptual schematic diagram of the attack angle of rocket, flight attack angle. where the coordinate system is taken as the reference coordinate system, O is O Figure 10 shows the conceptual schematic diagram of the attack angle of rocket, the center of rocket mass, the axis coincides with the axis of the rocket, the O O where the coordinate system Oxhz is taken as the reference coordinate system, O is the axis points vertically upward, and the O axis is determined according to the right center of rocket mass, the Ox axis coincides with the axis of the rocket, the Oh axis points hand rule. The red vector denotes the velocity v of the rocket’s centroid, the blue vector vertically upward, and the Oz axis is determined according to the right hand rule. The red v v v v represents the three components , and of in the coordinate system vector denotes the velocity v of the rocket’s  centr oid, the blue vector represents the three components ,  repre v , sents v and the v ang of le inc v in luded the bet coor wee dinate n the rock system et axO is xhz, d and repr th esents e veloci the - angle O h O x z included between the rocket axis Ox and the velocity v, and is called the total attack angle, v   ty , and is called the total attack angle, is called the pitch attack angle, and is 1 2 d is called the pitch attack angle, and d is called the direction attack angle. The attack 1 2   angles d, d and d of the rocket describe the positional  relationship between the projectile called the direction attack angle. The attack angles , and of the rocket de- 1 2 1 2 axis and the velocity direction during the flight of the rocket. Through the curve of the scribe the positional relationship between the projectile axis and the velocity direction attack during angle, the flig the ht attitude of the rocket changes . Throu and gh the the stability curve of of th the e attac rocket k ang during le, the the attitude flight can be seen. change The s and specific the stability flight simulation of the rocket calculation during the steps flight ar can e shown be seen. in Th Figur e speci e 11 fic . flight simulation calculation steps are shown in Figure 11. Figure 10. Attack angle diagram of the rocket projectile. Figure 10. Attack angle diagram of the rocket projectile. Comprehensive wind model Start Position Trajectory End Stop trajectory coordinates model of trajectory condition calculation (x , y , z ) rocket calculation i i i (x , y , z ) i+1 i+1 i+1 N update Figure 11. Flight simulation process. wind Aerospace 2021, 8, x FOR PEER REVIEW 14 of 21 three-dimensional trajectory curve of the rocket projectile, while the flight attitude needs to be reflected by the flight attack angle. Figure 10 shows the conceptual schematic diagram of the attack angle of rocket, where the coordinate system is taken as the reference coordinate system, O is O the center of rocket mass, the O axis coincides with the axis of the rocket, the O axis points vertically upward, and the O axis is determined according to the right hand rule. The red vector denotes the velocity of the rocket’s centroid, the blue vector v v v v represents the three components , and of in the coordinate system    ,  represents the angle included between the rocket axis and the veloci- O O v   ty , and is called the total attack angle, is called the pitch attack angle, and is 1 2   called the direction attack angle. The attack angles  , and of the rocket de- 1 2 scribe the positional relationship between the projectile axis and the velocity direction during the flight of the rocket. Through the curve of the attack angle, the attitude changes and the stability of the rocket during the flight can be seen. The specific flight simulation calculation steps are shown in Figure 11. Aerospace 2021, 8, 145 13 of 18 Figure 10. Attack angle diagram of the rocket projectile. Comprehensive wind model Start Position Trajectory End Stop trajectory coordinates model of trajectory condition calculation (x , y , z ) rocket calculation i i i (x , y , z ) i+1 i+1 i+1 update Figure 11. Flight simulation process. Figure 11. Flight simulation process. While using the integral method to solve the trajectory, the coordinates of the rocket’s position within each calculation step are substituted into the wind field model to obtain the wind velocity, which is then substituted into the trajectory equations to calculate the next trajectory parameters. These trajectory parameters include the position coordinates (x, y, z) in addition to attack angles d , d of the rocket projectile. The detailed solution 1 2 of the trajectory equations can be found in the literature [31] (pp. 141–143). Table 6 lists two wind field conditions employed for flight simulation of a rocket projectile. The flight processes under the two wind field conditions are simulated, and are compared with the flight state of the rocket projectile under windless condition. Table 6. Simulation conditions of complex wind field. Serial Number Climatic Condition Wind Field Condition 1 Clear sky Low-level jet Micro-downburst and 2 Thunderstorm atmospheric turbulence Figure 12 shows the change curves of the attack angles of the rocket projectile with flight time under the influence of the two wind fields listed in Table 6. The one on the left- hand side of the figure is the attack angle of the whole trajectory, while the small one on the right-hand side is the local attack angle of the first 5 s of the trajectory. It can be noticed that in a windless environment, the amplitude of attack angle of rocket projectile persistently decreases with the increase in the flight time and converges to nearly 0 degrees, indicating that the flight attitude of the rocket projectile gradually tends to become stable. Under the influence of the low-level jet in condition 1, the rocket projectile has a low velocity and weak anti-interference capability in the initial stages of launch. So, the amplitude of attack angle increases suddenly under the action of transient but strong airflow. As the rocket projectile continues to fly, the attack angle converges in a similar way as it would in a windless condition. Under the influence of the MD in condition 2, the attack angle of the rocket projectile also increases suddenly, similar to that observed in condition 1. However, due to the influence of the atmospheric turbulence at the same time, the amplitude of attack angle cannot converge further, fluctuates continuously, and the flight attitude is not stable anymore. Figure 13 shows the three-dimensional curve of flight trajectory of rocket projectile under wind field condition 1, wind field condition 2 and the windless environment. As can be seen from Figure 13, under the influence of conditions 1 and 2, the maximum trajectory height of the rocket projectile decreases, the range decreases, and the whole flight trajectory deviates significantly. In particular, the lateral deviation of the landing point under the influence of condition 1 increases much more than that under condition 2, which reflects the difference in the influence of the two wind conditions. Aerospace 2021, 8, 145 14 of 18 Aerospace 2021, 8, x FOR PEER REVIEW 16 of 21 Aerospace 2021, 8, x FOR PEER REVIEW 16 of 21 No wind condition 1 condition 2 No wind condition 1 condition 2 6 6 6 6 0 20 40 60 80 0 0 1 2 3 4 5 0 20 40 60 80 0 1 2 3 4 5 t(s) t(s) t(s) t(s) - 2 - 2 - 2 - 2 - 4 - 4 - 4 - 4 - 6 - 6 - 6 - 6 0 20 40 60 80 0 1 2 3 4 5 0 20 40 60 80 0 1 2 3 4 5 t(s) t(s) t(s) 3 t(s) - 3 - 3 - 3 0 20 40 60 80 - 3 0 1 2 3 4 5 0 20 40 60 80 0 1 2 3 4 5 t(s) t(s) t(s) t(s) Figure 12. Attack angle curves of the rocket projectile. Figure 12. Attack angle curves of the rocket projectile. Figure 12. Attack angle curves of the rocket projectile. 10.0 10.0 8.0 No wind 8.0 No wind condition1 condition1 condition2 6.0 condition2 6.0 4.0 4.0 2.0 2.0 - 2.0 - 2.0 - 1.0 - 1.0 Z/km 35.0 0 30.0 Z/km 25.0 35.0 20.0 30.0 15.0 25.0 10.0 20.0 1.0 5.0 15.0 10.0 1.0 5.0 0 X/km X/km Figure Figure113. 3. Trajecto Trajectories ries of of the r the r oc ocket ket projectile projectile un under der d dif ifffer erent wi ent wind nd ffield ield co conditions. nditions. Figure 13. Trajectories of the rocket projectile under different wind field conditions. Figure 14 shows the dispersion of the rocket projectile under the influence of condition 2 using the Monte-Carlo method. According to the flight simulation results shown in Figures 12–14, it can be concluded that the comprehensive model of complex wind field established herein, can reflect the law of influence of different wind fields on the aircraft when applied in flight simulation, which reflects the practicability and rationality of the model. Y/km δ (°) Y/km δ (°) δ(°) 1 δ(°) δ (°) δ (°) δ (°) δ (°) δ (°) 1 δ(°) 1 δ(°) δ (°) 2 Aerospace 2021, 8, 145 15 of 18 Aerospace 2021, 8, x FOR PEER REVIEW 17 of 21 0.5 No wind Condition 2 - 0.5 - 1.0 - 1.5 34.55 34.60 34.65 34.70 34.75 34.80 X/km Figure 14. Impact dispersion of the rocket projectile under the influence of condition 2 (simulation Figure 14. Impact dispersion of the rocket projectile under the influence of condition 2 (simulation time = 100). time = 100). 4.2. Analysis on Influence of Different Model Parameters on Flight Process According to the flight simulation results shown in Figures 12–14, it can be con- cluded that the comprehensive model of complex wind field established herein, can re- For a certain kind of typical wind field, it is necessary to study the effect of differ- flect the law of influence of different wind fields on the aircraft when applied in flight ent model parameters on the flight process of aircrafts. Taking the low-level jet wind simulation, which reflects the practicability and rationality of the model. field(condition 1 in Section 4.1) as an example, the simulations are made with different values of w (strength parameter) and H (scale parameter). The main ballistic parameters L T 4.2. Analysis on Influence of Different Model Parameters on Flight Process of simulations are listed in Tables 7 and 8. The attack angle curves (within the first 5 s of the flight trajectory) of d and d are shown in Figure 15. For a certain kind of 1 typica 2 l wind field, it is necessary to study the effect of different model parameters on the flight process of aircrafts. Taking the low-level jet wind Table 7. Main ballistic parameters with different w . field(condition 1 in Section 4.1) as an example, the simulations are made with different values Wof (stren Flight gth Tparam ime eter) Down and Range (scale Cross param Range eter). The Tm erminal ain balli Velocity stic pa- L T (m/s) (s) (km) (km) (m/s) rameters of simulations are listed in Tables 7 and 8. The attack angle curves (within the 0 (No wind) 104.7 34.38 0.009 366   first 5 s of the flight trajectory) of and are shown in Figure 15. 1 2 6 91.7 32.18 2.501 347 10 89.8 31.81 2.738 345 14 88.9 31.42 3.067 343 Table 7.Main ballistic parameters with different w . 18 86.1 31.04 3.182 340 w Flight Time Down Range Cross Range Terminal Velocity From Tables 7 and 8, it can be seen that the ballistic characteristics changed under (s) (km) (km) (m/s) (m/s) the influence of low-level jet. The changes are specifically manifested in the reduction in 0 (No wind) 104.7 34.38 −0.009 366 flight time and terminal velocity, the decrease in down range and the increase in cross 91.7 32.18 3 −2.501 6 347 range (the order of magnitude from 10 to 10 ). The simulation results also show that the 89.8 31.81 −2.738 influence 10 degree of low-level jet on the flight process is positively correlated to 345 the strength parameter w and the scale parameter H . L 88.9 31.42 T −3.067 14 343 From the attack angle curves in Figure 15, the attitude changes of the rocket projectile 86.1 31.04 −3.182 18 340 under different wind field model parameters can be directly observed. It is shown that the low-level jet causes a sudden change and a convergence process of the attack angle of the attack angle d , which result in the large change in the cross range (in Tables 7 and 8) of the rocket. Z/km Aerospace 2021, 8, x FOR PEER REVIEW 18 of 21 Table 8.Main ballistic parameters with different . H Flight Time Down Range Cross Range Terminal Velocity (s) (km) (km) (m/s) (m) 104.7 34.38 −0.009 0 (No wind) 366 Aerospace 2021, 8, 145 16 of 18 90.6 31.95 −3.005 400 347 89.8 31.81 −2.738 500 345 89.1 31.72 −2.692 600 345 Table 8. Main ballistic parameters with different H . 88.6 31.61 −2.461 700 T 343 W Flight Time Down Range Cross Range Terminal Velocity From Tables 7 and 8, it can be seen that the ballistic characteristics changed under (m/s) (s) (km) (km) (m/s) the influence of low-level jet. The changes are specifically manifested in the reduction in 0 (No wind) 104.7 34.38 0.009 366 flight time and terminal velocity, the decrease in down range and the increase in cross 400 90.6 31.95 3.005 347 range (the order of magnitude from 10 to 10 ). The simulation results also show that the 500 89.8 31.81 2.738 345 influence degree of low-level jet on the flight process is positively correlated to the 600 89.1 31.72 2.692 345 700 w88.6 31.61 2.461 343 strength parameter and the scale parameter H . - 2 No wind w =6m/s - 4 -1 No wind w =10m/s w =6m/s -2 L - 6 w =14m/s w =10m/s L L -3 w =14m/s w =18m/s - 8 w =18m/s -4 - 10 -5 0 1 2 3 4 5 t(s) 0 1 2 3 4 5 t(s) w w (a)  − t curve with different values of (b)  − t curve with different values of L L 1 2 2 4 - 2 No wind No wind H =6m/s - 4 H =400m T - 2 H =10m/s H =500m T H =14m/s H =600m T - 6 - 4 H =18m/s H =700m - 8 - 6 0 1 2 3 4 5 t(s) 0 1 2 t(s) 3 4 5 H H (c)  − t curve with different values of (d)  − t curve with different values of T T 1 2 Figure 15. Attack angle curves with different wind field model parameters. 4.3. Discussions of Use Conditions of the Comprehensive Model In fact, the real wind field in nature changes not only in space but also in time, which can significantly influence the accuracy of flight simulation. It is necessary to give some conditions of use for the established comprehensive model. According to the atmospheric dynamics [25], the wind field disturbance can be expressed as: W = w + Dw (31) δ (°) δ (°) δ (°) δ (°) 2 Aerospace 2021, 8, x FOR PEER REVIEW 19 of 21 Figure 15. Attack angle curves with different wind field model parameters. From the attack angle curves in Figure 15, the attitude changes of the rocket projec- tile under different wind field model parameters can be directly observed. It is shown that the low-level jet causes a sudden change and a convergence process of the attack angle of the attack angle , which result in the large change in the cross range (in Ta- bles 7 and 8) of the rocket. 4.3. Discussions of Use Conditions of the Comprehensive Model In fact, the real wind field in nature changes not only in space but also in time, which can significantly influence the accuracy of flight simulation. It is necessary to give some conditions of use for the established comprehensive model. According to the at- mospheric dynamics [25], the wind field disturbance can be expressed as: Aerospace 2021, 8, 145 17 of 18 (31) W = w+w Equation (31) indicates that the total disturbance W is composed of the mean wind w and the stochastic term w . In the conditions of small time and spatial Equation (31) indicates that the total disturbance W is composed of the mean wind scales, can be described as a certain wind field, such as micro-downburst, and the w and thewstochastic term Dw. In the conditions of small time and spatial scales, w can be described as a w certain wind field, such as micro-downburst, and the wind profile, Dw, wind profile, , can be described as the atmospheric turbulence. The diagram is can be described as the atmospheric turbulence. The diagram is shown in Figure 16. shown in Figure 16. Total wind disturbance Mean wind Stochastic term Certain wind field Wind profiles Atmospheric turbulence Figure 16. Composition of the wind disturbance in small time and spatial scales. Figure 16. Composition of the wind disturbance in small time and spatial scales. Based Based on onFigur Figure e 16 16, , the theconditions conditionsof of use use for forthe the established establishedcompr compreh ehensive ensivemodel model are given as: are given as: • When When the the flight flight si simulation mulation iis s in ina asm small all time time scscale ale (in (in con condition dition 1 o1 f Section of Section 4.1, 4.1 th,e the rocket went through the low-level jet area within 2 s) or in a small spatial scale (the rocket went through the low-level jet area within 2 s) or in a small spatial scale (the low-level jet area has a height of 800 m and the rocket has a flight altitude of 10 km), low-level jet area has a height of 800 m and the rocket has a flight altitude of 10 km), the established comprehensive model can be used to obtain some reasonable results. the established comprehensive model can be used to obtain some reasonable When the flight simulation is in a large time scale (or in a large spatial scale), such as the results. total flight process of a long-range missile, an airplane or an airship, the comprehensive • When the flight simulation is in a large time scale (or in a large spatial scale), such model might cause significant errors. as the total flight process of a long-range missile, an airplane or an airship, the comprehensive model might cause significant errors. 5. Conclusions Built on the basic principles of fluid mechanics, the engineering wind velocity cal- 5. Conclusions culation models of three typical wind fields, namely micro-downburst, low-level jet and Built on the basic principles of fluid mechanics, the engineering wind velocity atmospheric turbulence, are herein established. By combining the three models in the calculation models of three typical wind fields, namely micro-downburst, low-level jet ground coordinate system and unifying the input and output parameters, a comprehensive and atmospheric turbulence, are herein established. By combining the three models in the model of complex wind field is established. The wind field model simulation and appli- ground coordinate system and unifying the input and output parameters, a cation simulation indicate that the comprehensive model can reasonably and effectively comprehensive model of complex wind field is established. The wind field model describe the flow characteristics of the relevant wind field, and has the characteristics of simple calculation and model scalability. Additionally, as shown in Section 4.2, the com- prehensive model can be used not only to analyze the influence of different wind fields on the flight process, but also to study the influence of different model parameters on the flight simulations. As an important factor of the simulation accuracy of the wind field model, the wind field data collection methods in different time and space scales of the simulated flights are the main points of our further research. Meanwhile, developing a reasonable wind field model for large time and space scales with the least amount of collection data is also an extension of the work in this paper. Author Contributions: Conceptualization, J.C.; methodology, J.C.; software, J.C., Z.Y.; validation, J.C.; formal analysis, J.C. and J.F..; investigation, J.C. and J.F.; resources, J.C. and Z.Y.; writing—original draft preparation, J.C.; writing—review and editing, J.C.; supervision, L.W. and J.F.; project adminis- tration, L.W.; funding acquisition, L.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Science Foundation of China (No. 61603191). Data Availability Statement: Not applicable. Aerospace 2021, 8, 145 18 of 18 Conflicts of Interest: The authors declare no conflict of interest. References 1. Tuleya, R.E.; Lord, S.J. The Impact of Dropwindsonde Data on GFDL Hurricane Model Forecasts Using Global Analyses. Weather Forecast. 2010, 12, 307–323. [CrossRef] 2. Bo, M.; Lin, M.; Peng, H. WindSim Computational Flow Dynamics Model Testing Using Databases from Two Wind Measurement Stations. 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Journal

AerospaceMultidisciplinary Digital Publishing Institute

Published: May 24, 2021

Keywords: flight simulation; wind field model; comprehensive model

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