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A method for optimizing the aerodynamic layout of a helicopter that reduces the effects of aerodynamic interaction
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A method for optimizing the aerodynamic layout of a helicopter that reduces the effects of aerodynamic interaction
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National Key Laboratory of Rotorcraft Aeromechanics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
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Yang Lu ∗ , Taoyong Su, Renliang Chen, Pan Li, Yu Wang
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Article history: Received 9 August 2018 Received in revised form 30 December 2018 Accepted 3 March 2019 Available online xxxx Keywords: Helicopter Aerodynamic interaction Aerodynamic layout optimization Vortex particle method Class function/shape function transformation (CST) method
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The aerodynamic environment during the flight of a helicopter is complex because of the severe aerodynamic interaction between various components. To fully consider the effect of aerodynamic interaction in the initial stages of helicopter design and to eliminate or reduce its adverse effects, a comprehensive design optimization method for the aerodynamic layout of a helicopter that is capable of reducing the adverse effects of aerodynamic interaction is developed in this paper. To satisfy the requirements for precision and efficiency in the calculation model, an aerodynamic interaction analysis model of various helicopter components was established based on a viscous vortex particle and the unsteady panel hybrid method. To simultaneously consider the influences of the position and shape of the aerodynamic components on the aerodynamic interaction during the optimization process, parameter modeling of the helicopter’s shape was performed based on the class function/shape function transformation (CST) method. A Kriging surrogate model of the objective function was further developed and combined with a hybrid sequential quadratic algorithm and genetic algorithm optimization strategy to establish a comprehensive optimization flow for the aerodynamic layout of a helicopter that reduces the adverse effects of aerodynamic interaction. Verification was carried out based on a fuselage shape derived from UH-60 helicopter. The optimization results showed that the use of the comprehensive optimization method for the aerodynamic layout of a helicopter can effectively reduce the adverse effects of aerodynamic interaction. Based on the optimization objectives, the efficiency of hovering increased by 4.7%, the hovering ceiling increased by 3.48%, the speed stability derivative increased by 264.7%, and the angle of attack stability derivative decreased by 26.4%. © 2019 Elsevier Masson SAS. All rights reserved.
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1. Introduction
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When a helicopter is hovering and during low-speed forward flight, strong mutual interaction occurs between the rotor wake and the other aerodynamic components of the helicopter. The aerodynamic interaction between various components of a helicopter will directly affect its flight characteristics. Under a same flight condition, the aerodynamic interaction between various components of a helicopter is significantly determined by the shape and relative position of each aerodynamic component; that is, the aerodynamic layout of the helicopter. As modern helicopter design progresses toward a more compact layout, greater disk loading, and better maneuverability, aerodynamic interaction between various components of a helicopter will become more se-
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*
Corresponding author. E-mail address: [email protected] (Y. Lu).
https://doi.org/10.1016/j.ast.2019.03.005 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.
vere, and the effect on the flight characteristics of the helicopter will be more serious. To meet the higher flight characteristics requirements of modern helicopters, the aerodynamic interaction between various components should be fully considered during the design of the aerodynamic layout of a helicopter to reduce its adverse effects as much as possible and to comprehensively enhance the flight characteristics of the helicopter. For a long time, during the process of helicopter design, several aerodynamic layout combinations for a helicopter have been proposed mainly based on the experience of engineering designers. The aerodynamic interaction between various components is analyzed using wind tunnel tests. However, this method is only able to consider a limited number of schemes for the aerodynamic layout of a helicopter; it is unable to carry out a multi-objective, multiparameter optimization for the aerodynamic layout that addresses the issue of aerodynamic interaction between various components of the helicopter. Therefore, this method can no longer meet the increasing requirements of modern helicopter design.
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Researchers have performed extensive studies on issues related to aerodynamic interaction between various components of a rotorcraft, including rotor/fuselage [1–4], rotor/tail rotor [5,6], rotor/tail plane [7,8] and proprotor/wing [9] aerodynamic interactions. The research methods mainly include experimental methods [10–13], prescribed vortex wake method combined with low-order fuselage panel modeling [14,15], free wake method combined with fuselage panel modeling [16,17], and the computational fluid dynamics (CFD) method [18,19]. Current studies mainly focus on the calculation and analysis of aerodynamic interaction, but almost no studies have focused on the optimization of the aerodynamic layout of a helicopter to reduce the adverse effects of aerodynamic interaction. One of the key issues in performing a comprehensive design optimization for the aerodynamic layout to reduce the adverse effects of aerodynamic interaction is the development of a model for calculating helicopter aerodynamic interaction that is suited to the optimization algorithm. This type of model requires both a high degree of precision to accurately calculate the aerodynamic disturbance force and high computational efficiency to meet the optimization requirements. Currently, it is generally necessary to use a CFD method to accurately calculate the aerodynamic interaction [20]. However, the CFD method has very high requirements for computational resources and consumes an excessively long amount of time. For example, it takes 50-80 hours to calculate the aerodynamic interaction in a model with approximately 12 million grid cells on a single workstation equipped with an Intel Xeon X5570 central processing unit (CPU), and it can easily take several months or even years to perform a comprehensive optimization using CFD as the calculation kernel, which is difficult to accept in engineering applications. On the other hand, the main consideration in existing aerodynamic layout optimizations of helicopters is the effects of the relative position parameters of the aerodynamic components on the flight characteristics of the helicopter; the effect of the shape parameters of the aerodynamic components is considered less [21–23]. However, the shapes of various aerodynamic components have a major effect on the flight characteristics of a helicopter. To comprehensively consider aerodynamic interaction during the optimization, one should also consider the effects of changes in the shapes of components. To this end, it is necessary to carry out parameter modeling of the shapes of the aerodynamic components on a helicopter. To solve these problems, the effects of aerodynamic interaction between various components on a helicopter are considered, and a comprehensive optimization method for the aerodynamic layout of a helicopter that can reduce the adverse effects of aerodynamic interaction and improve the flight characteristics is developed in this paper. To address the problem of calculating the flight characteristics of a helicopter, the viscous vortex particle method (VVPM) is introduced in this paper [24,25]. By solving the vorticity dynamics equation under the Lagrangian description system, this method depends little on the empirical parameters and is able to accurately predict the distribution of the rotor wake vorticity. Compared with the CFD method, the VVPM is able to simultaneously guarantee the computational efficiency and computational precision of the underlying analytical module for the optimization problem. To address the issue of geometric parameter modeling in the optimization, the class function/shape function transformation (CST) method is introduced to perform the shape parameter modeling of the aerodynamic components of the helicopter [26]. The shape control parameters of the CST method have explicit geometric meanings. In addition, it is able to effectively control the optimization space with smaller quantity of parameters and higher fitting precision.
A combined global optimization and local optimization strategy based on a progressive surrogate model is used in this paper, which guarantees the optimization speed and precision while fully utilizing the ability of global optimization to traverse the overall design space. The main contents of this paper are as follows. First, the comprehensive optimization problem for the aerodynamic layout of a helicopter to reduce the adverse effects of aerodynamic interaction is defined, and the optimization flow is determined in connection with the characteristics of comprehensive optimization. The CST method is then introduced, and the shape parameter modeling process of the method is described in detail using the nose of a helicopter as an example. Then, a calculation model for the flight characteristics of a helicopter is established based on the VVPM and the vortex panel method and the aerodynamic interaction effect is considered. Finally, a fuselage shape derived from UH-60 helicopter is used as an example to perform the verification of the optimization method, in which the shape and position parameters of the main aerodynamic components are selected as the design variables, and indicators of the flight characteristics, such as the efficiency in hovering and the angle of attack stability derivative, are used as the optimization objectives.
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2. Definition of the comprehensive optimization of the aerodynamic layout of a helicopter that reduces the adverse effects of aerodynamic interaction The definition of the comprehensive optimization includes three aspects: the optimization objectives, design variables, and constraint conditions.
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2.1. Optimization objectives
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As stated above, the purpose of the optimization research in this paper is to reduce the adverse effects of aerodynamic interaction between various components through the optimization of the aerodynamic layout of the helicopter and then to improve the relevant flight characteristics of the helicopter. The flight characteristics of a helicopter include numerous performance indices. When selecting the optimization objectives, one can choose indices that are more significantly affected by aerodynamic interaction than others. For example, to reduce the adverse effects of aerodynamic interaction during hovering, the helicopter efficiency in hovering and the hovering ceiling can be chosen. To reduce the adverse effects of aerodynamic interference during flight at a transition speed, the speed stability derivative and the damping ratio of the Dutch roll mode can be chosen. Finally, during the aerodynamic layout optimization process, corresponding indices are selected as optimization objectives based on the actual need. In this paper, the flight characteristics of a helicopter as the aforementioned optimization objectives were obtained by calculations using the VVPM. Because each optimization parameter has a different physical meaning, the physical units and magnitudes also differ; therefore, it was necessary to nondimensionalize the dimensions of each parameter before the optimization was carried out. Afterwards, the directions of the optimization for the various optimization objectives were further clarified. For example, the direction of optimization for the hovering efficiency was maximization, and the direction of optimization for collective pitch was minimization. Obviously, the aerodynamic layout optimization of a helicopter to reduce aerodynamic interaction is a multi-objective optimization problem, and the weighted stacking method is used in this paper to solve this problem.
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The final mathematical expression for the optimization objectives in this paper can be represented by:
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N min
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Minimize:
Obj =
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wi
i =1
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yi si
−
N max j =1
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wj
yj sj
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(1)
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where Obj is the total optimization objective, y is each optimization sub-objective, s is the reference value of each optimization objective, w is the weight value, N min is the number of optimization objectives with minimization as the direction of optimization, and N max is the number of optimization objectives with maximization as the direction of optimization. The weight value w can be determined based on the magnitude of the effect that aerodynamic interaction has on the parameters of the optimization objectives.
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2.2. Design variables
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The position parameters and shape parameters that are closely related to aerodynamic interaction are selected as the optimization variables. To consider the problem from the perspective of reducing aerodynamic interaction, the position parameters usually include the vertical position of the rotor, the longitudinal position of the horizontal tail, and the longitudinal position of the tail rotor, and the shape parameters usually include the shape control parameters for the nose of the fuselage, the engine cowling, and the tail boom. As stated above, the CST method is used in this paper to perform the shape parameterization of the various components on the helicopter, and the control point coordinates and control indices of the required key positions are the shape parameters in the design variables.
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2.3. Constraint conditions
Fig. 1. The flow of establish the progressive global surrogate model.
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The constraint conditions are directly used to limit the ranges of values of the optimization variables.
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3. Comprehensive optimization flow for the aerodynamic layout of a helicopter that reduces the adverse effects of aerodynamic interaction
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Compared with the CFD method, the VVPM used in this paper was able to reduce the computational time for the performance indices of the flight characteristics of a helicopter. However, as a calculation kernel to be integrated into the optimal design, the VVPM still requires a longer overall optimization time; therefore, a surrogate modeling technique based on the Kriging method was used during the optimization to further reduce the computational time [27]. Specifically, a progressive global surrogate model was used to construct a surrogate model that satisfied the prediction precision [28]; that is, based on the initial surrogate model, new sets of sample points were gradually added globally and locally in accordance with a certain strategy to continuously improve the global fitting precision and the fitting precision of the local region that had larger than normal errors until the satisfactory precision was reached to obtain a surrogate model with high global precision. The flow chart for this process is shown in Fig. 1. In the calculation model for the flight characteristics of a helicopter considering the aerodynamic interaction effects, which is characterized by a high degree of nonlinearity and multiple extrema, a Multi-Island Genetic Algorithm (MIGA) [29] and Sequential Quadratic Programming (SQP) [30] combined optimization strategy was used to utilize the advantages of the global optimization algorithm in traversing the overall design space and the advantages of the gradient optimization algorithm for local optimization.
The optimization flow established in this paper is as follows. First, determine the relative positions and aerodynamic shapes of the aerodynamic components based on the position parameters and shape parameters. Next, calculate the flight characteristics of the helicopter based on the VVPM, and combine it with the Kriging method to establish a surrogate model. Then, use the surrogate model as the calculation kernel for integration into the global optimization and the local optimization to decrease the calculated cost, use the combined optimization strategy to perform the comprehensive optimization of the helicopter aerodynamic layout, and obtain the optimal solution. Finally, substitute the optimization results from the Kriging surrogate model into the calculation model for the flight characteristics of a helicopter established based on the VVPM to obtain the actual optimal solution. The overall optimization flow is shown in Fig. 2. The two core issues in the aforementioned optimization flow are the parameter modeling of the fuselage shape and the calculation of the flight characteristics of the helicopter considering aerodynamic interaction effects, which are described in detail below.
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4. Parameter modeling for the fuselage based on the CST method
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4.1. CST method for the shape of the helicopter
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In designing the shape of the helicopter, the fuselage, engine cowling, and other longitudinal contours are parts of the general curve. The CST method used in connection with the general curve can be expressed as:
ζ (ψ) = ψ N 1 (1 − ψ) N 2
N i =0
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A i ψ i + ψζ T
(2)
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Fig. 2. Optimization flow for the aerodynamic layout of a helicopter that reduces the adverse effects of aerodynamic interaction.
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where ψ = x/c, ζ = y /c, ζ T = Z T E /c, N 1 and N 2 are control indices, N is the number of curve control points, A i is the polynomial coefficient, (x, y) are the coordinates of a point on the general curve, c is the length of the curve in the x direction, and Z T E is the height of the curve in the y direction. The class function of the CST method is defined as:
C (ψ) = ψ
N1
(1 − ψ)
N2
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(3)
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and the shape function is defined as:
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S (ψ) =
N
Fig. 3. Longitudinal contours of the nose of the helicopter.
Ai ψ i
(4)
i =0
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Different shapes for the cross-sectional curves of the fuselage can be obtained by changing N 1 and N 2 in the class function. The longitudinal contours for the shape of the helicopter are more complex. If only one function formula is used to represent the entire contour, the polynomial order may be too high due to the excessive number of control points required, which may cause severe morbidity to appear on the curve fitted by the CST method. To solve this problem, the curve can be divided into several segments, the modeling of each segment can use the same method as for the general curve, and the segments of the curve can be stitched together to complete the modeling of this class of curves.
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4.2. Shape parameter modeling for the nose of the helicopter based on the CST method
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The CST method can perform parameter modeling in connection with the shape of every component on the helicopter. The nose of a certain helicopter is used as an example to show the specific implementation process of the CST method in the parameter modeling of the helicopter fuselage.
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As shown in Fig. 3, the shape of the nose of the helicopter is represented by the longitudinal contours and the cross-sectional surface. The longitudinal contours of the nose of the helicopter include the longitudinal contour on the XY plane and the longitudinal contour on the ZX plane. The longitudinal contour on the XY plane is defined based on Formula (2). This curve is divided into two symmetrical curve segments, and the control indices N 1 and N 2 in the class function are determined based on the slopes of the tangent lines at the endpoints of the curves, which in this example are 0.5 and 1, respectively. The coefficients A i of the shape function s (ψ) are determined by the coordinates of the curve control points. In this example, the solution process is as follows. Separately take the vertical coordinates Z T E1 , Z T E2 , Z T E3 , Z T E4 , and Z T E5 of the curve at ψ1 = 1/11l, ψ2 = 3/11l, ψ3 = 5/11l, ψ4 = 7/11l, and ψ5 = 9/11l, respectively, and substitute the nondimensionalized horizontal and vertical coordinates of these five control points into the equations to obtain a fourth-order shape function. Only five parameters are needed to describe the longitudinal contours of the shape of the nose of this helicopter. The parameter modeling method for the longitudinal contour on the ZX plane is similar to that for the longitudinal contour on
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Fig. 4. Shape of the longitudinal contour on the ZX plane.
Fig. 7. Grid calculation model for the nose of the helicopter.
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5.1. Helicopter flight dynamics model considering aerodynamic interaction effects
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Fig. 5. Shape of the first cross-section of the nose.
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Fig. 6. Shape of the second cross-section of the nose.
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the XY plane. Divide the longitudinal contour on the ZX plane into two parts. As shown in Fig. 4, the upper part is formed by the combination of two curves, and the lower part is composed of one curve. The longitudinal contour within the ZX plane can be obtained by performing parameter modeling of the three curves. In this example, the cross-sectional shapes for the nose of the helicopter are shown in Figs. 5 and 6. Divide the cross-section into two parts. Then, take a constant for the shape function to control the width to height ratio of the curve. Because the left and right sides of the cross-section are symmetrical, the values of N 1 and N 2 are equal. In this example, as shown in Fig. 5, the control index for the class function of the first cross-section is 0.4. The control index for the class function of the second cross-section, which is shown in Fig. 6, takes the control parameter for the upper part of the first cross-section of 0.25, and the lower part is 0.35. After the parameter model for the nose of the helicopter was established using the CST method, automatic generation and data export of the grid for the nose of the helicopter was performed based on the secondary development of the computer-aided threedimensional interactive application (CATIA). Fig. 7 shows the grid generated for the nose of the helicopter.
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5. VVPM-based calculation model for the flight characteristics of a helicopter considering aerodynamic interaction effects
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In this section, the modeling method for a VVPM-based calculation model for the flight characteristics of a helicopter considering aerodynamic interaction effects is presented, and a UH-60 helicopter is used as an example to verify the accuracy of the established model.
Three equations for the forces and three equations for the momentums equilibrium acting on the center of gravity of the helicopter were established in the orthogonal body axes system, including the aerodynamic forces and torques of the rotor, fuselage, tail rotor, horizontal tail, and vertical tail. The rotor model included the rotor blade model and the wake model, the blade model was developed using the vortex panel method, and the rotor wake model was developed using the VVPM. The fuselage was represented by the vortex panel method, and the tail rotor model was the same as the rotor model. Modeling of the horizontal tail and the vertical tail was similar to that of the rotor blade. To consider the effects of aerodynamic interaction on the flight characteristics of a helicopter, it is necessary to couple the interaction effects of other components in the calculation model to the aerodynamic forces of each component. For example, the aerodynamic interaction between the rotor and fuselage is mainly reflected in four aspects: (1) the pass-through effect of the bound vortex panel of the rotor blade on the fuselage, which affects the pressure distribution of the fuselage surface; (2) the effect of the induced velocity field of the rotor wake on the pressure distribution of the fuselage surface; (3) the disturbance velocity field caused by the fuselage vortex panel, which affects the strength of the bound vortex panel of the blade; and (4) the effect of the disturbance velocity field caused by the fuselage vortex panel on the movement and distortion of the rotor wake, which changes the induced velocity field of the wake and therefore affects the strength of the bound vortex of the blade. The calculation model for the aerodynamic interaction of each component on the helicopter is given below, and the rotor is used as an example to illustrate the modeling process of the VVPM and the vortex panel method under an interaction flow field.
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5.2. VVPM-based calculation of the aerodynamic forces of the rotor under an interaction flow field
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The rotor blade is represented by several bound vortex panels span-wise and chord-wise, and Kutta panels are set up in the airflow direction at the trailing edge of the blade to satisfy the Kutta condition for the trailing edge of the blade. The strength of each bound vortex panel on the blade was determined by the impenetrable condition, and the strength of the Kutta vortex panel was determined assuming that the vortex strength of the lifting body at the trailing edge is zero. The rotor wake model was established based on the VVPM. During the process of solving for the strength of the bound vortex panels of the blade, the Kutta panel was used to calculate the induced velocity in the form of vortex panels. The Kutta vortex panel was then replaced by an equivalent vortex particle, and the rotor
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+
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p wb f ∇ utrb + ∇ ui + ∇ ui + ∇ u∞ · αi i i N + N tr + N p w
3
+ 2ν
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wake was represented by the equivalent vortex particles generated by the trailing edge of the blade, as shown in Fig. 8. In transforming the vortex panel into equivalent vortex particles, it is necessary to satisfy the equal vortex strength and conservation of the first-order moment of the vorticity, in which the equal vortex strength can be represented by
α j (t ) = −τ NC c NC j sj (t ) − sj (t − t ) + u T . E t sj −1 (t ) − sj +1 (t ) /2
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and the conservation of the first-order moment of the vorticity can be represented by
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α j × xj =
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where a j and x j are the vector-valued vorticity and position, respectively, of particle j, c NC j and sj are the length of the leading edge and vortex strength, respectively, of the Kutta vortex panel j, t is the current calculation time, u T . E is the vector-valued local velocity of the trailing edge, τ NC is the unit tangent vector of the leading edge of the Kutta vortex panel, t is the iteration time step, S K utaj is the Kutta vortex panel that was made equivalent, γ K utaj is the vortex strength vector of the Kutta vortex panel that was made equivalent, and S (x) is the area of the Kutta vortex panel. For the equivalent vortex particles generated in the rotor wake, the vorticity transport and diffusion phenomena in the wake were simulated by solving for the Navier–Stokes(N-S) equations expressed in vorticity-velocity form in the Lagrangian description system. The changes in the positions of the vortex particles in the rotor wake and the change in vorticity satisfy the discrete vortex dynamics equation. Considering the aerodynamic interaction effect of each component, the movement equation and the vorticity transport equation of the rotor vortex particles are:
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dxi
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dt
=−
N 1
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−
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K (ρ )(xi − x j ) × α j 3
N tr 1 j =1
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Npw
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−
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+
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dα i dt
K (ρ )(xi − xtrj ) × αtrj 3
σi j
1 j =1
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spectively, of the tailing rotor wake particle j, x j and a j are the position and vector-valued vorticity, respectively, of the horizontal tail wake particle j, ubin is the velocity induced by the rotor
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Nb
σi3j
K (ρ
u bin +
n =1
pw )(xi − x j ) ×
N trb n =1
tr
= [∇ u i ] + ∇ u i
p wb
utrb in + u in
utrb in
pw
is the velocity induced by the tail rotor blade n, f u in
p wb ui
is the velocity induced by the horizontal tail, is the velocity inν is the kinematic viscosity, V is the volume duced by the fuselage,
of particle, ρ = xi − x j /σi j is the dimensionless distance parame-
ter,
σi j = σi2 + σ j2 /2 is the symmetric smoothing parameter, the
vorticity distribution function ξσi j (xi − x j ) uses the Gaussian distri2 bution function, K (ρ ) = [G (ρ ) − ξ (ρ )] /ρ is the kernel function, √ρ
1 and G (ρ ) = 4πρ Erf
2 √2
π
is Green’s function after smoothing cor-
s 0
e −x dx is the error function. 2
For the non surface boundary, the vorticity of the symmetric particle strength exchange (PSE) in Formula (8) is conserved, but in the vicinity of the boundary of the surface, a partial vorticity which through the boundary is lost. In this paper, the conservation of the vorticity is guaranteed by mirror mapping [31]. The movement equation and the vorticity transport equation of the vortex particles in the tail rotor wake and those of the vortex particles in the horizontal tail wake are similar to those for the rotor, and the relevant equations are not presented in this paper. Based on the established equation for the movement equation and the vorticity transport equation of the wake vortex particles, the second-order precision Adams-Bashforth time integration algorithm is used to obtain the vorticity position vector and strength vector of the wakes of the rotor, tail rotor, and horizontal tail over time. Combined with the impenetrable condition for the vortex panel, the equation for the strength of the bound vortex panel of the rotor blade that considers the aerodynamic interaction effects is:
bj ai j
−
w iw
pw
+ wi
+ us + ω p wb
+ wi
b s × xi f
+
w tr i
+
w trb i
α
f
+ u in − u ∞
p w b + ∇ ui + ∇ ui · αi
(7)
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f
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panel i; w iw , w tr , w trb , wi , wi and w i are the velocity of i i the modeling objects control points induced by main rotor wake, tail rotor wake, tail rotor vortex panel, horizontal tail wake, horizontal tail vortex panel and fuselage vortex panel, respectively; u s is the linear velocity of the main rotor blade; ω s is the angular velocity of the main rotor blade; nbi is the normal unit vector at the
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control point of the main rotor vortex panel i, is the position vector of the control point of the main rotor vortex panel i. Similar to Formula (9), the equation for the strength of the bound panel of the fuselage that considers the aerodynamic interaction effects of various components is:
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f f j bi j + w bi + w iw − u f − ω f × xi + w tr i pw p wb f trb • ni = 0 + wi + wi + wi
pw j
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(9)
where bj is the vortex strength of the main rotor panel j; ai j is influence coefficient of the rotor vortex panel j on the vortex p wb
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+ w i • nbi = 0
pw
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xbi
σ j =1 i j
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(6)
S K utaj
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γ K utaj × xdS (x)
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rection, and Erf (s) =
(5)
69
(8)
where xtrj and atrj are the position and vector-valued vorticity, re-
blade n, Fig. 8. VVPM-based rotor wake model.
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( V i α j − V j α i )ξσi j (xi − x j )
pw
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σi j
j =1
5
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f
114 115 116 117 118 120 121 122 123 124 125 126
(10)
where j is the strength of the fuselage panel j; b i j is influence coefficient of the fuselage panel j on the panel i; u f is the linear velocity of the fuselage; w bi is the velocity of the modeling objects control points induced by the main rotor bound vortex; ω f is the
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Fig. 9. Rotor wake interacting with rotor/fuselage,
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angular velocity of the fuselage; ni is the normal unit vector at f
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p wb j ci j
+
pw wi
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+ − u p ws − ω f pw trb + w tr =0 i + w i + w i • ni
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pw p ws × xi
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(11)
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p wb
where j is the vortex strength of the horizontal tail vortex panel j; c i j is influence coefficient of the horizontal tail vortex panel j on the vortex panel i; u p ws is the linear velocity of the horizontal tail; ω p ws is the angular velocity of the horizontal tail; pw ni is the normal unit vector at the control point of the horizontal
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tail vortex panel i; xi is the position vector of the control point of the horizontal tail vortex panel i. The tail rotor blade and the rotor blade were treated in the same manner. The equation for the strength of the bound vortex panel of the tail rotor blade in an aerodynamic interaction environment is:
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pw
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Fig. 11. Variation of the collective pitch of the helicopter with changing flight speed.
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f
tr b trb trb j d i j + w i + w i + w i − ωtrs × xi − u trs pw p wb • ntrb + w iw + w i + w i i =0
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(12)
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trb j
where is the vortex strength of the tail rotor vortex panel j; di j is influence coefficient of the tail rotor vortex panel j on the vortex panel i; utrs is the linear velocity of the tail rotor blade; ω p ws is the angular velocity of the tail rotor blade; ntrb is the nori mal unit vector at the control point of the tail rotor vortex panel i; xtrb is the position vector of the control point of the tail rotor vori tex panel i. By solving for the aforementioned control equations, one can obtain the aerodynamic loads of the rotor, fuselage, tail rotor, and horizontal tail under an interaction flow field as well as the flow field velocity distribution and vorticity distribution and then calculate the flight characteristics of the helicopter. To verify the VVPM-based model of a helicopter considering aerodynamic interaction, the rotor-fuselage interaction (ROBIN) configuration [32] was used as an example. Fig. 9 show the vorticity isosurfaces of the rotor wake interacting with fuselage. The calculation results of time averaged surface pressure distribution were compared with the experimental data [33] in Fig. 10. The CFD calculation results are also shown in Fig. 10. From the comparison, the calculated results of VVPM-based model are consistent with the experimental data, which are in the same order of magnitude as those calculated by CFD. 5.3. Example verification of the calculation model for the flight characteristics of a helicopter considering aerodynamic interaction effects
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μ = 0.23.
the control point of the fuselage vortex panel i, xi is the position vector of the control point of the fuselage vortex panel i. The equation for the strength of the bound vortex panel of the horizontal tail in an aerodynamic interaction environment is:
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Fig. 10. Pressure coefficient on the fuselage upper surface,
To verify the developed calculation model for the flight characteristics of a helicopter, an UH-60 helicopter was used as an
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Fig. 12. Variation of the longitudinal cyclic pitch of the helicopter with changing flight speed.
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example. See [34] and [35] in the literature for the parameters of the UH-60 helicopter. The calculation results were compared with the flight test data [35] for verification. In addition, to demonstrate that the model could accurately consider the aerodynamic interaction effects during the flight of a helicopter, the trim results using a semi-empirical method to calculate the aerodynamic interaction [36] were compared. Figs. 11 to 14 give the results of the comparisons of the collective pitch of the rotor, the pitch angle of the helicopter, the longitudinal cyclic pitch, and the required power with changes in the flight speed, respectively. The figures show that the results of the VVPM-based calculations for the flight characteristics of the helicopter considering aerodynamic interaction effects have a higher degree of consistency with the
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Table 1 Weight of each parameter of the optimization objectives.
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Direction of optimization
Weight
ηhover
Maximization Maximization Maximization Minimization
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Optimization sub-objective
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Fig. 13. Variations of the pitch and attitude angles of the helicopter with changing flight speed.
Fig. 15. Grid calculation model of the UH-60 helicopter fuselage.
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Table 2 Design optimization variables and constraint conditions.
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Design optimization variable
Initial value
Constraint condition
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hC 1 lC 1 nC 1 nC 2 hmain_rotor hhorizontal_tail htail_rotor
500 mm 800 mm 0.25 0.35 775 cm 1735 cm 1825 cm
400 mm∼600 mm 600 mm∼1000 mm 0.15∼0.6 0.15∼0.6 760 cm∼820 cm 1730 cm∼1830 cm 1800 cm∼1890 cm
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Fig. 14. Variation of the required power of the rotor with changing flight speed.
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flight test results than the semi-empirical method, particularly in the low-speed and transition-speed flight phases. Compared with the results from using a semi-empirical method to calculate the aerodynamic interaction, the new model can more accurately take into account the effects of the aerodynamic interaction from various components of the helicopter on its flight characteristics and the maximum error was reduced from 25% to about 5% in low speed forward flight. These results demonstrate the accuracy and effectiveness of the developed method, which can provide a reliable foundation and basis for calculating the optimized design for the aerodynamic layout of a helicopter.
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6. Example verification of comprehensive optimization of the aerodynamic layout of a helicopter that reduces the adverse effects of aerodynamic interaction
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To verify the effectiveness of the developed optimization method, a fuselage shape derived from UH-60 helicopter is used to carry out an optimization of the aerodynamic layout of a helicopter that reduces the adverse effects of aerodynamic interaction, and the efficiency of the optimization calculation is assessed.
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6.1. Example verification
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Because the UH-60 helicopter is a transport helicopter, several indicators that are important for transport helicopters were selected as the optimization objectives for the aerodynamic layout, including the efficiency in hovering ηhover , the hovering ceiling
hhover , the speed stability derivative S velocit y , and the angle of attack stability derivative S angle . To focus on the effect of aerodynamic interaction, the speed stability derivative and the angle of attack stability derivative were based on the amount of trim at four forward flight speeds (10 km/h, 20 km/h, 30 km/h, and 40 km/h) and were obtained by taking the mean value after stacking. The weight of each optimization objective and the directions of optimization are shown in Table 1. During hovering and low-speed forward flight, the shape of the nose has a significant impact on the aerodynamic interaction effect of the helicopter. In this example, the shape parameters for controlling the nose of the fuselage were selected as the shape design variables, which are the height of the first control point h C 1 , the length of the upper curve l C 1 in x direction, and the upper curve control index nC 1 and the lower curve control index nC 2 on the cross-section; this results in four shape design variables. Three position parameters were selected: the vertical position of the rotor hmain_rotor , the longitudinal position of the horizontal tail hhorizontal_tail , and the longitudinal position of the tail rotor htail_rotor . The values of the positions of the aerodynamic components are related to the coordinate system. In this paper, the coordinate system is the same as the one of the Ref. [34]. According to this coordinate system, the vertical and longitudinal positions of the center of gravity of the helicopter are 5.87 m and 8.912 m, respectively. After the shape of the nose was determined in accordance with the control parameters, a geometric model for the entire helicopter was developed by bridging with the mid-section of the fuselage. The entire helicopter was then divided into grids based on the geometric model, and the vortex panel method was used to develop the fuselage calculation model. The fuselage grid model corresponding to the initial values of the shape parameters, as well as the coordinate system are shown in Fig. 15. The initial values of the position variables are shown in Table 2. In this example, the ranges of variation of the design variables were selected as the constraint conditions, which are also shown in Table 2.
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Table 3 Fitting precisions of various optimization objectives in the surrogate model. Optimization sub-objective
S angle
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Fitting precision
ηhover hhover S velocit y
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9
10 km/h 95.8% 10 km/h 93.6%
90.7% 92.5% 20 km/h 30 km/h 93.1% 99.9% 20 km/h 30 km/h 91.4% 99.9%
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40 km/h 99.9% 40 km/h 99.9%
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After establishing the optimization problem, it was combined with the calculation model for the flight characteristics of a helicopter to establish the Kriging surrogate model. The samples were acquired using the Latin hypercube method, and the final number of sample points was 96. Due to the speed stability derivative and the angle of attack stability derivative were based on the value at four forward flight speeds, the fitting precision of these two optimization objects should be analyzed at four speeds. The precision of the surrogate model is shown in Table 3; the results show that the surrogate model can substitute for the calculation model based on VVPM and can be used for the comprehensive optimization. In this example, the Kriging surrogate model used a combined multi-island genetic algorithm and sequential quadratic programming optimization strategy to perform the comprehensive optimization of the design. The multi-island genetic algorithm had a population of 15, the number of islands was 10, and the hereditary algebra was 20. The optimal solution acquired from the multiisland genetic algorithm was used as the initial value of the sequential quadratic programming to complete the aerodynamic layout optimization of the helicopter. The process of changing the various optimization objectives during optimization is shown in Fig. 16. The grid calculation model for the shape of the fuselage after the optimization is shown in Fig. 17, and the optimization results for the position parameters of the components are shown in Table 4. After completing the optimization based on the surrogate model, the optimal solution was substituted into the calculation model for the flight characteristics of a helicopter considering aerodynamic interaction effects to obtain the actual optimal solution. The actual optimization results of the various optimization objectives are shown in Table 5. The result of the overall optimization objective was a reduction by 157.5%, and the four optimization objectives all changed in the directions of optimization. In the actual optimization results, the efficiency in hovering increased by 4.7%, the hovering ceiling increased by 3.48%, the speed stability derivative increased by 264.7%, and the angle of attack stability derivative decreased by 26.4%. In this sample, the results for optimal values of the vertical position of the rotor and the longitudinal position of the tail rotor are absolutely predictable. The reason of the changes in the longitudinal position of the horizontal tail is further investigate here. This example is focused on the hover and low speed forward flight. In these conditions, the angle of attack of the horizontal tail is positive, which can be obtained from Ref. [35]. So the horizontal tail will generate a positive lift in these forward conditions, which will provide a pitch down moment on the helicopter. The speed stability of the helicopter could be deteriorated by increasing the longitudinal position of the horizontal tail. So the optimal result of the longitudinal position of the horizontal tail may be caused by the compromise choice of the layout optimization. In this example, the calculation model used in the optimization considering the aerodynamic interaction effects of various components on a helicopter, and the flight characteristics that are affected more by the aerodynamic interaction were selected as optimization objectives. The optimization results showed that the
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Fig. 16. Changes in various optimization objectives.
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Fig. 17. Grid calculation model for the shape of the fuselage after the optimization.
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Table 4 Optimization results for the position parameters of the components. Position parameter
Optimal solution
hmain_rotor hhorizontal_tail htail_rotor hC 1 lC 1 nC 1 nC 2
820 cm 1771.77 cm 1890 cm 435.86 mm 955.15 mm 0.2187 0.5738
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Table 5 Results of the aerodynamic layout optimization of a helicopter. Optimization objective
Initial value
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5
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−0.4
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0.67 3190 m 0.00017 0.0007
0.6835 3331 m 0.00068 0.000557
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the aerodynamic layout of a helicopter that reduces the adverse effects of aerodynamic interaction and is able to effectively improve the flight characteristics of the helicopter.
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6.2. Assessment of the computational cost
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In this example, the development of the surrogate model was the most time-consuming component; it required approximately 192 hours on a single workstation equipped with an Intel Xeon X5570 CPU. The combined optimization process required less than 2 minutes, and verification of the calculation results required approximately 2 hours. The total amount of time was approximately 194 hours, which satisfies the engineering design requirements for computational efficiency.
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7. Conclusion
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None declared.
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References
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strategy established in this paper for the optimization of the aerodynamic layout of a helicopter can effectively improve the flight characteristics of the helicopter and reduce the adverse effects of aerodynamic interaction.
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Conflict of interest statement
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A design optimization method for the aerodynamic layout of a helicopter that is capable of reducing the adverse effects of aerodynamic interaction was proposed in this paper and, for the first time, the aerodynamic interaction effects between various components were considered using numerical calculations in the optimization. To take into account the aerodynamic interaction effects and to comprehensively consider the computational precision and efficiency, a calculation model for the flight characteristics of a helicopter was developed based on the VVPM. Based on this foundation, a combined genetic algorithm/sequential quadratic programming optimization algorithm based on the progressive Kriging surrogate model was used and combined with the CST method to consider the effects of the shapes of various components and optimize the aerodynamic layout of a helicopter. The feasibility and effectiveness of the optimization method was verified using a fuselage shape derived from UH-60 helicopter as an example. The following conclusions were obtained from this study: (1) The calculation model for the flight characteristics of a helicopter considering aerodynamic interaction effects performed based on the VVPM has not only higher precision but also a higher computational efficiency than other models. In addition, it can act as the underlying analytical module for integration into the comprehensive design optimization of the aerodynamic layout of a helicopter to reflect the aerodynamic interaction effects of various components. (2) The CST parameter method can efficiently perform parameter modeling of the aerodynamic shape of a helicopter, which makes it possible to consider the effects of changes in the shape of aerodynamic components on the flight characteristics of the helicopter during the aerodynamic optimization. (3) The optimization results of the fuselage shape derived from UH-60 helicopter showed that after the design optimization, the helicopter’s efficiency in hovering increased by 4.7%, the hovering ceiling increased by 3.48%, the speed stability derivative increased by 264.7%, and the angle of attack stability derivative decreased by 26.4%. These results demonstrate that the optimization strategy established in this paper can be used in the design optimization of
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A method for optimizing the aerodynamic layout of a helicopter that reduces the effects of aerodynamic interaction
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National Key Laboratory of Rotorcraft Aeromechanics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
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Article history: Received 9 August 2018 Received in revised form 30 December 2018 Accepted 3 March 2019 Available online xxxx Keywords: Helicopter Aerodynamic interaction Aerodynamic layout optimization Vortex particle method Class function/shape function transformation (CST) method
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a b s t r a c t
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The aerodynamic environment during the flight of a helicopter is complex because of the severe aerodynamic interaction between various components. To fully consider the effect of aerodynamic interaction in the initial stages of helicopter design and to eliminate or reduce its adverse effects, a comprehensive design optimization method for the aerodynamic layout of a helicopter that is capable of reducing the adverse effects of aerodynamic interaction is developed in this paper. To satisfy the requirements for precision and efficiency in the calculation model, an aerodynamic interaction analysis model of various helicopter components was established based on a viscous vortex particle and the unsteady panel hybrid method. To simultaneously consider the influences of the position and shape of the aerodynamic components on the aerodynamic interaction during the optimization process, parameter modeling of the helicopter’s shape was performed based on the class function/shape function transformation (CST) method. A Kriging surrogate model of the objective function was further developed and combined with a hybrid sequential quadratic algorithm and genetic algorithm optimization strategy to establish a comprehensive optimization flow for the aerodynamic layout of a helicopter that reduces the adverse effects of aerodynamic interaction. Verification was carried out based on a fuselage shape derived from UH-60 helicopter. The optimization results showed that the use of the comprehensive optimization method for the aerodynamic layout of a helicopter can effectively reduce the adverse effects of aerodynamic interaction. Based on the optimization objectives, the efficiency of hovering increased by 4.7%, the hovering ceiling increased by 3.48%, the speed stability derivative increased by 264.7%, and the angle of attack stability derivative decreased by 26.4%. © 2019 Elsevier Masson SAS. All rights reserved.
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1. Introduction
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When a helicopter is hovering and during low-speed forward flight, strong mutual interaction occurs between the rotor wake and the other aerodynamic components of the helicopter. The aerodynamic interaction between various components of a helicopter will directly affect its flight characteristics. Under a same flight condition, the aerodynamic interaction between various components of a helicopter is significantly determined by the shape and relative position of each aerodynamic component; that is, the aerodynamic layout of the helicopter. As modern helicopter design progresses toward a more compact layout, greater disk loading, and better maneuverability, aerodynamic interaction between various components of a helicopter will become more se-
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*
Corresponding author. E-mail address: [email protected] (Y. Lu).
https://doi.org/10.1016/j.ast.2019.03.005 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.
vere, and the effect on the flight characteristics of the helicopter will be more serious. To meet the higher flight characteristics requirements of modern helicopters, the aerodynamic interaction between various components should be fully considered during the design of the aerodynamic layout of a helicopter to reduce its adverse effects as much as possible and to comprehensively enhance the flight characteristics of the helicopter. For a long time, during the process of helicopter design, several aerodynamic layout combinations for a helicopter have been proposed mainly based on the experience of engineering designers. The aerodynamic interaction between various components is analyzed using wind tunnel tests. However, this method is only able to consider a limited number of schemes for the aerodynamic layout of a helicopter; it is unable to carry out a multi-objective, multiparameter optimization for the aerodynamic layout that addresses the issue of aerodynamic interaction between various components of the helicopter. Therefore, this method can no longer meet the increasing requirements of modern helicopter design.
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Researchers have performed extensive studies on issues related to aerodynamic interaction between various components of a rotorcraft, including rotor/fuselage [1–4], rotor/tail rotor [5,6], rotor/tail plane [7,8] and proprotor/wing [9] aerodynamic interactions. The research methods mainly include experimental methods [10–13], prescribed vortex wake method combined with low-order fuselage panel modeling [14,15], free wake method combined with fuselage panel modeling [16,17], and the computational fluid dynamics (CFD) method [18,19]. Current studies mainly focus on the calculation and analysis of aerodynamic interaction, but almost no studies have focused on the optimization of the aerodynamic layout of a helicopter to reduce the adverse effects of aerodynamic interaction. One of the key issues in performing a comprehensive design optimization for the aerodynamic layout to reduce the adverse effects of aerodynamic interaction is the development of a model for calculating helicopter aerodynamic interaction that is suited to the optimization algorithm. This type of model requires both a high degree of precision to accurately calculate the aerodynamic disturbance force and high computational efficiency to meet the optimization requirements. Currently, it is generally necessary to use a CFD method to accurately calculate the aerodynamic interaction [20]. However, the CFD method has very high requirements for computational resources and consumes an excessively long amount of time. For example, it takes 50-80 hours to calculate the aerodynamic interaction in a model with approximately 12 million grid cells on a single workstation equipped with an Intel Xeon X5570 central processing unit (CPU), and it can easily take several months or even years to perform a comprehensive optimization using CFD as the calculation kernel, which is difficult to accept in engineering applications. On the other hand, the main consideration in existing aerodynamic layout optimizations of helicopters is the effects of the relative position parameters of the aerodynamic components on the flight characteristics of the helicopter; the effect of the shape parameters of the aerodynamic components is considered less [21–23]. However, the shapes of various aerodynamic components have a major effect on the flight characteristics of a helicopter. To comprehensively consider aerodynamic interaction during the optimization, one should also consider the effects of changes in the shapes of components. To this end, it is necessary to carry out parameter modeling of the shapes of the aerodynamic components on a helicopter. To solve these problems, the effects of aerodynamic interaction between various components on a helicopter are considered, and a comprehensive optimization method for the aerodynamic layout of a helicopter that can reduce the adverse effects of aerodynamic interaction and improve the flight characteristics is developed in this paper. To address the problem of calculating the flight characteristics of a helicopter, the viscous vortex particle method (VVPM) is introduced in this paper [24,25]. By solving the vorticity dynamics equation under the Lagrangian description system, this method depends little on the empirical parameters and is able to accurately predict the distribution of the rotor wake vorticity. Compared with the CFD method, the VVPM is able to simultaneously guarantee the computational efficiency and computational precision of the underlying analytical module for the optimization problem. To address the issue of geometric parameter modeling in the optimization, the class function/shape function transformation (CST) method is introduced to perform the shape parameter modeling of the aerodynamic components of the helicopter [26]. The shape control parameters of the CST method have explicit geometric meanings. In addition, it is able to effectively control the optimization space with smaller quantity of parameters and higher fitting precision.
A combined global optimization and local optimization strategy based on a progressive surrogate model is used in this paper, which guarantees the optimization speed and precision while fully utilizing the ability of global optimization to traverse the overall design space. The main contents of this paper are as follows. First, the comprehensive optimization problem for the aerodynamic layout of a helicopter to reduce the adverse effects of aerodynamic interaction is defined, and the optimization flow is determined in connection with the characteristics of comprehensive optimization. The CST method is then introduced, and the shape parameter modeling process of the method is described in detail using the nose of a helicopter as an example. Then, a calculation model for the flight characteristics of a helicopter is established based on the VVPM and the vortex panel method and the aerodynamic interaction effect is considered. Finally, a fuselage shape derived from UH-60 helicopter is used as an example to perform the verification of the optimization method, in which the shape and position parameters of the main aerodynamic components are selected as the design variables, and indicators of the flight characteristics, such as the efficiency in hovering and the angle of attack stability derivative, are used as the optimization objectives.
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2. Definition of the comprehensive optimization of the aerodynamic layout of a helicopter that reduces the adverse effects of aerodynamic interaction The definition of the comprehensive optimization includes three aspects: the optimization objectives, design variables, and constraint conditions.
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2.1. Optimization objectives
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As stated above, the purpose of the optimization research in this paper is to reduce the adverse effects of aerodynamic interaction between various components through the optimization of the aerodynamic layout of the helicopter and then to improve the relevant flight characteristics of the helicopter. The flight characteristics of a helicopter include numerous performance indices. When selecting the optimization objectives, one can choose indices that are more significantly affected by aerodynamic interaction than others. For example, to reduce the adverse effects of aerodynamic interaction during hovering, the helicopter efficiency in hovering and the hovering ceiling can be chosen. To reduce the adverse effects of aerodynamic interference during flight at a transition speed, the speed stability derivative and the damping ratio of the Dutch roll mode can be chosen. Finally, during the aerodynamic layout optimization process, corresponding indices are selected as optimization objectives based on the actual need. In this paper, the flight characteristics of a helicopter as the aforementioned optimization objectives were obtained by calculations using the VVPM. Because each optimization parameter has a different physical meaning, the physical units and magnitudes also differ; therefore, it was necessary to nondimensionalize the dimensions of each parameter before the optimization was carried out. Afterwards, the directions of the optimization for the various optimization objectives were further clarified. For example, the direction of optimization for the hovering efficiency was maximization, and the direction of optimization for collective pitch was minimization. Obviously, the aerodynamic layout optimization of a helicopter to reduce aerodynamic interaction is a multi-objective optimization problem, and the weighted stacking method is used in this paper to solve this problem.
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The final mathematical expression for the optimization objectives in this paper can be represented by:
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N min
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Minimize:
Obj =
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wi
i =1
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3
yi si
−
N max j =1
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wj
yj sj
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(1)
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where Obj is the total optimization objective, y is each optimization sub-objective, s is the reference value of each optimization objective, w is the weight value, N min is the number of optimization objectives with minimization as the direction of optimization, and N max is the number of optimization objectives with maximization as the direction of optimization. The weight value w can be determined based on the magnitude of the effect that aerodynamic interaction has on the parameters of the optimization objectives.
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2.2. Design variables
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The position parameters and shape parameters that are closely related to aerodynamic interaction are selected as the optimization variables. To consider the problem from the perspective of reducing aerodynamic interaction, the position parameters usually include the vertical position of the rotor, the longitudinal position of the horizontal tail, and the longitudinal position of the tail rotor, and the shape parameters usually include the shape control parameters for the nose of the fuselage, the engine cowling, and the tail boom. As stated above, the CST method is used in this paper to perform the shape parameterization of the various components on the helicopter, and the control point coordinates and control indices of the required key positions are the shape parameters in the design variables.
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2.3. Constraint conditions
Fig. 1. The flow of establish the progressive global surrogate model.
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The constraint conditions are directly used to limit the ranges of values of the optimization variables.
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3. Comprehensive optimization flow for the aerodynamic layout of a helicopter that reduces the adverse effects of aerodynamic interaction
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Compared with the CFD method, the VVPM used in this paper was able to reduce the computational time for the performance indices of the flight characteristics of a helicopter. However, as a calculation kernel to be integrated into the optimal design, the VVPM still requires a longer overall optimization time; therefore, a surrogate modeling technique based on the Kriging method was used during the optimization to further reduce the computational time [27]. Specifically, a progressive global surrogate model was used to construct a surrogate model that satisfied the prediction precision [28]; that is, based on the initial surrogate model, new sets of sample points were gradually added globally and locally in accordance with a certain strategy to continuously improve the global fitting precision and the fitting precision of the local region that had larger than normal errors until the satisfactory precision was reached to obtain a surrogate model with high global precision. The flow chart for this process is shown in Fig. 1. In the calculation model for the flight characteristics of a helicopter considering the aerodynamic interaction effects, which is characterized by a high degree of nonlinearity and multiple extrema, a Multi-Island Genetic Algorithm (MIGA) [29] and Sequential Quadratic Programming (SQP) [30] combined optimization strategy was used to utilize the advantages of the global optimization algorithm in traversing the overall design space and the advantages of the gradient optimization algorithm for local optimization.
The optimization flow established in this paper is as follows. First, determine the relative positions and aerodynamic shapes of the aerodynamic components based on the position parameters and shape parameters. Next, calculate the flight characteristics of the helicopter based on the VVPM, and combine it with the Kriging method to establish a surrogate model. Then, use the surrogate model as the calculation kernel for integration into the global optimization and the local optimization to decrease the calculated cost, use the combined optimization strategy to perform the comprehensive optimization of the helicopter aerodynamic layout, and obtain the optimal solution. Finally, substitute the optimization results from the Kriging surrogate model into the calculation model for the flight characteristics of a helicopter established based on the VVPM to obtain the actual optimal solution. The overall optimization flow is shown in Fig. 2. The two core issues in the aforementioned optimization flow are the parameter modeling of the fuselage shape and the calculation of the flight characteristics of the helicopter considering aerodynamic interaction effects, which are described in detail below.
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4. Parameter modeling for the fuselage based on the CST method
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4.1. CST method for the shape of the helicopter
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In designing the shape of the helicopter, the fuselage, engine cowling, and other longitudinal contours are parts of the general curve. The CST method used in connection with the general curve can be expressed as:
ζ (ψ) = ψ N 1 (1 − ψ) N 2
N i =0
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A i ψ i + ψζ T
(2)
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Fig. 2. Optimization flow for the aerodynamic layout of a helicopter that reduces the adverse effects of aerodynamic interaction.
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where ψ = x/c, ζ = y /c, ζ T = Z T E /c, N 1 and N 2 are control indices, N is the number of curve control points, A i is the polynomial coefficient, (x, y) are the coordinates of a point on the general curve, c is the length of the curve in the x direction, and Z T E is the height of the curve in the y direction. The class function of the CST method is defined as:
C (ψ) = ψ
N1
(1 − ψ)
N2
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(3)
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and the shape function is defined as:
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S (ψ) =
N
Fig. 3. Longitudinal contours of the nose of the helicopter.
Ai ψ i
(4)
i =0
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Different shapes for the cross-sectional curves of the fuselage can be obtained by changing N 1 and N 2 in the class function. The longitudinal contours for the shape of the helicopter are more complex. If only one function formula is used to represent the entire contour, the polynomial order may be too high due to the excessive number of control points required, which may cause severe morbidity to appear on the curve fitted by the CST method. To solve this problem, the curve can be divided into several segments, the modeling of each segment can use the same method as for the general curve, and the segments of the curve can be stitched together to complete the modeling of this class of curves.
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4.2. Shape parameter modeling for the nose of the helicopter based on the CST method
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The CST method can perform parameter modeling in connection with the shape of every component on the helicopter. The nose of a certain helicopter is used as an example to show the specific implementation process of the CST method in the parameter modeling of the helicopter fuselage.
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As shown in Fig. 3, the shape of the nose of the helicopter is represented by the longitudinal contours and the cross-sectional surface. The longitudinal contours of the nose of the helicopter include the longitudinal contour on the XY plane and the longitudinal contour on the ZX plane. The longitudinal contour on the XY plane is defined based on Formula (2). This curve is divided into two symmetrical curve segments, and the control indices N 1 and N 2 in the class function are determined based on the slopes of the tangent lines at the endpoints of the curves, which in this example are 0.5 and 1, respectively. The coefficients A i of the shape function s (ψ) are determined by the coordinates of the curve control points. In this example, the solution process is as follows. Separately take the vertical coordinates Z T E1 , Z T E2 , Z T E3 , Z T E4 , and Z T E5 of the curve at ψ1 = 1/11l, ψ2 = 3/11l, ψ3 = 5/11l, ψ4 = 7/11l, and ψ5 = 9/11l, respectively, and substitute the nondimensionalized horizontal and vertical coordinates of these five control points into the equations to obtain a fourth-order shape function. Only five parameters are needed to describe the longitudinal contours of the shape of the nose of this helicopter. The parameter modeling method for the longitudinal contour on the ZX plane is similar to that for the longitudinal contour on
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Fig. 4. Shape of the longitudinal contour on the ZX plane.
Fig. 7. Grid calculation model for the nose of the helicopter.
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5.1. Helicopter flight dynamics model considering aerodynamic interaction effects
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Fig. 5. Shape of the first cross-section of the nose.
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Fig. 6. Shape of the second cross-section of the nose.
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the XY plane. Divide the longitudinal contour on the ZX plane into two parts. As shown in Fig. 4, the upper part is formed by the combination of two curves, and the lower part is composed of one curve. The longitudinal contour within the ZX plane can be obtained by performing parameter modeling of the three curves. In this example, the cross-sectional shapes for the nose of the helicopter are shown in Figs. 5 and 6. Divide the cross-section into two parts. Then, take a constant for the shape function to control the width to height ratio of the curve. Because the left and right sides of the cross-section are symmetrical, the values of N 1 and N 2 are equal. In this example, as shown in Fig. 5, the control index for the class function of the first cross-section is 0.4. The control index for the class function of the second cross-section, which is shown in Fig. 6, takes the control parameter for the upper part of the first cross-section of 0.25, and the lower part is 0.35. After the parameter model for the nose of the helicopter was established using the CST method, automatic generation and data export of the grid for the nose of the helicopter was performed based on the secondary development of the computer-aided threedimensional interactive application (CATIA). Fig. 7 shows the grid generated for the nose of the helicopter.
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5. VVPM-based calculation model for the flight characteristics of a helicopter considering aerodynamic interaction effects
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In this section, the modeling method for a VVPM-based calculation model for the flight characteristics of a helicopter considering aerodynamic interaction effects is presented, and a UH-60 helicopter is used as an example to verify the accuracy of the established model.
Three equations for the forces and three equations for the momentums equilibrium acting on the center of gravity of the helicopter were established in the orthogonal body axes system, including the aerodynamic forces and torques of the rotor, fuselage, tail rotor, horizontal tail, and vertical tail. The rotor model included the rotor blade model and the wake model, the blade model was developed using the vortex panel method, and the rotor wake model was developed using the VVPM. The fuselage was represented by the vortex panel method, and the tail rotor model was the same as the rotor model. Modeling of the horizontal tail and the vertical tail was similar to that of the rotor blade. To consider the effects of aerodynamic interaction on the flight characteristics of a helicopter, it is necessary to couple the interaction effects of other components in the calculation model to the aerodynamic forces of each component. For example, the aerodynamic interaction between the rotor and fuselage is mainly reflected in four aspects: (1) the pass-through effect of the bound vortex panel of the rotor blade on the fuselage, which affects the pressure distribution of the fuselage surface; (2) the effect of the induced velocity field of the rotor wake on the pressure distribution of the fuselage surface; (3) the disturbance velocity field caused by the fuselage vortex panel, which affects the strength of the bound vortex panel of the blade; and (4) the effect of the disturbance velocity field caused by the fuselage vortex panel on the movement and distortion of the rotor wake, which changes the induced velocity field of the wake and therefore affects the strength of the bound vortex of the blade. The calculation model for the aerodynamic interaction of each component on the helicopter is given below, and the rotor is used as an example to illustrate the modeling process of the VVPM and the vortex panel method under an interaction flow field.
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5.2. VVPM-based calculation of the aerodynamic forces of the rotor under an interaction flow field
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The rotor blade is represented by several bound vortex panels span-wise and chord-wise, and Kutta panels are set up in the airflow direction at the trailing edge of the blade to satisfy the Kutta condition for the trailing edge of the blade. The strength of each bound vortex panel on the blade was determined by the impenetrable condition, and the strength of the Kutta vortex panel was determined assuming that the vortex strength of the lifting body at the trailing edge is zero. The rotor wake model was established based on the VVPM. During the process of solving for the strength of the bound vortex panels of the blade, the Kutta panel was used to calculate the induced velocity in the form of vortex panels. The Kutta vortex panel was then replaced by an equivalent vortex particle, and the rotor
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+
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p wb f ∇ utrb + ∇ ui + ∇ ui + ∇ u∞ · αi i i N + N tr + N p w
3
+ 2ν
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wake was represented by the equivalent vortex particles generated by the trailing edge of the blade, as shown in Fig. 8. In transforming the vortex panel into equivalent vortex particles, it is necessary to satisfy the equal vortex strength and conservation of the first-order moment of the vorticity, in which the equal vortex strength can be represented by
α j (t ) = −τ NC c NC j sj (t ) − sj (t − t ) + u T . E t sj −1 (t ) − sj +1 (t ) /2
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and the conservation of the first-order moment of the vorticity can be represented by
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α j × xj =
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where a j and x j are the vector-valued vorticity and position, respectively, of particle j, c NC j and sj are the length of the leading edge and vortex strength, respectively, of the Kutta vortex panel j, t is the current calculation time, u T . E is the vector-valued local velocity of the trailing edge, τ NC is the unit tangent vector of the leading edge of the Kutta vortex panel, t is the iteration time step, S K utaj is the Kutta vortex panel that was made equivalent, γ K utaj is the vortex strength vector of the Kutta vortex panel that was made equivalent, and S (x) is the area of the Kutta vortex panel. For the equivalent vortex particles generated in the rotor wake, the vorticity transport and diffusion phenomena in the wake were simulated by solving for the Navier–Stokes(N-S) equations expressed in vorticity-velocity form in the Lagrangian description system. The changes in the positions of the vortex particles in the rotor wake and the change in vorticity satisfy the discrete vortex dynamics equation. Considering the aerodynamic interaction effect of each component, the movement equation and the vorticity transport equation of the rotor vortex particles are:
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dxi
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dt
=−
N 1
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−
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K (ρ )(xi − x j ) × α j 3
N tr 1 j =1
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Npw
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−
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+
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dα i dt
K (ρ )(xi − xtrj ) × αtrj 3
σi j
1 j =1
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spectively, of the tailing rotor wake particle j, x j and a j are the position and vector-valued vorticity, respectively, of the horizontal tail wake particle j, ubin is the velocity induced by the rotor
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Nb
σi3j
K (ρ
u bin +
n =1
pw )(xi − x j ) ×
N trb n =1
tr
= [∇ u i ] + ∇ u i
p wb
utrb in + u in
utrb in
pw
is the velocity induced by the tail rotor blade n, f u in
p wb ui
is the velocity induced by the horizontal tail, is the velocity inν is the kinematic viscosity, V is the volume duced by the fuselage,
of particle, ρ = xi − x j /σi j is the dimensionless distance parame-
ter,
σi j = σi2 + σ j2 /2 is the symmetric smoothing parameter, the
vorticity distribution function ξσi j (xi − x j ) uses the Gaussian distri2 bution function, K (ρ ) = [G (ρ ) − ξ (ρ )] /ρ is the kernel function, √ρ
1 and G (ρ ) = 4πρ Erf
2 √2
π
is Green’s function after smoothing cor-
s 0
e −x dx is the error function. 2
For the non surface boundary, the vorticity of the symmetric particle strength exchange (PSE) in Formula (8) is conserved, but in the vicinity of the boundary of the surface, a partial vorticity which through the boundary is lost. In this paper, the conservation of the vorticity is guaranteed by mirror mapping [31]. The movement equation and the vorticity transport equation of the vortex particles in the tail rotor wake and those of the vortex particles in the horizontal tail wake are similar to those for the rotor, and the relevant equations are not presented in this paper. Based on the established equation for the movement equation and the vorticity transport equation of the wake vortex particles, the second-order precision Adams-Bashforth time integration algorithm is used to obtain the vorticity position vector and strength vector of the wakes of the rotor, tail rotor, and horizontal tail over time. Combined with the impenetrable condition for the vortex panel, the equation for the strength of the bound vortex panel of the rotor blade that considers the aerodynamic interaction effects is:
bj ai j
−
w iw
pw
+ wi
+ us + ω p wb
+ wi
b s × xi f
+
w tr i
+
w trb i
α
f
+ u in − u ∞
p w b + ∇ ui + ∇ ui · αi
(7)
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f
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panel i; w iw , w tr , w trb , wi , wi and w i are the velocity of i i the modeling objects control points induced by main rotor wake, tail rotor wake, tail rotor vortex panel, horizontal tail wake, horizontal tail vortex panel and fuselage vortex panel, respectively; u s is the linear velocity of the main rotor blade; ω s is the angular velocity of the main rotor blade; nbi is the normal unit vector at the
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control point of the main rotor vortex panel i, is the position vector of the control point of the main rotor vortex panel i. Similar to Formula (9), the equation for the strength of the bound panel of the fuselage that considers the aerodynamic interaction effects of various components is:
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f f j bi j + w bi + w iw − u f − ω f × xi + w tr i pw p wb f trb • ni = 0 + wi + wi + wi
pw j
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(9)
where bj is the vortex strength of the main rotor panel j; ai j is influence coefficient of the rotor vortex panel j on the vortex p wb
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+ w i • nbi = 0
pw
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xbi
σ j =1 i j
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(6)
S K utaj
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γ K utaj × xdS (x)
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rection, and Erf (s) =
(5)
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(8)
where xtrj and atrj are the position and vector-valued vorticity, re-
blade n, Fig. 8. VVPM-based rotor wake model.
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( V i α j − V j α i )ξσi j (xi − x j )
pw
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j =1
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f
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(10)
where j is the strength of the fuselage panel j; b i j is influence coefficient of the fuselage panel j on the panel i; u f is the linear velocity of the fuselage; w bi is the velocity of the modeling objects control points induced by the main rotor bound vortex; ω f is the
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Fig. 9. Rotor wake interacting with rotor/fuselage,
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angular velocity of the fuselage; ni is the normal unit vector at f
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p wb j ci j
+
pw wi
+
w bi
w iw
+ − u p ws − ω f pw trb + w tr =0 i + w i + w i • ni
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pw p ws × xi
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(11)
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p wb
where j is the vortex strength of the horizontal tail vortex panel j; c i j is influence coefficient of the horizontal tail vortex panel j on the vortex panel i; u p ws is the linear velocity of the horizontal tail; ω p ws is the angular velocity of the horizontal tail; pw ni is the normal unit vector at the control point of the horizontal
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tail vortex panel i; xi is the position vector of the control point of the horizontal tail vortex panel i. The tail rotor blade and the rotor blade were treated in the same manner. The equation for the strength of the bound vortex panel of the tail rotor blade in an aerodynamic interaction environment is:
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Fig. 11. Variation of the collective pitch of the helicopter with changing flight speed.
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f
tr b trb trb j d i j + w i + w i + w i − ωtrs × xi − u trs pw p wb • ntrb + w iw + w i + w i i =0
101 102
(12)
103 104
trb j
where is the vortex strength of the tail rotor vortex panel j; di j is influence coefficient of the tail rotor vortex panel j on the vortex panel i; utrs is the linear velocity of the tail rotor blade; ω p ws is the angular velocity of the tail rotor blade; ntrb is the nori mal unit vector at the control point of the tail rotor vortex panel i; xtrb is the position vector of the control point of the tail rotor vori tex panel i. By solving for the aforementioned control equations, one can obtain the aerodynamic loads of the rotor, fuselage, tail rotor, and horizontal tail under an interaction flow field as well as the flow field velocity distribution and vorticity distribution and then calculate the flight characteristics of the helicopter. To verify the VVPM-based model of a helicopter considering aerodynamic interaction, the rotor-fuselage interaction (ROBIN) configuration [32] was used as an example. Fig. 9 show the vorticity isosurfaces of the rotor wake interacting with fuselage. The calculation results of time averaged surface pressure distribution were compared with the experimental data [33] in Fig. 10. The CFD calculation results are also shown in Fig. 10. From the comparison, the calculated results of VVPM-based model are consistent with the experimental data, which are in the same order of magnitude as those calculated by CFD. 5.3. Example verification of the calculation model for the flight characteristics of a helicopter considering aerodynamic interaction effects
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μ = 0.23.
the control point of the fuselage vortex panel i, xi is the position vector of the control point of the fuselage vortex panel i. The equation for the strength of the bound vortex panel of the horizontal tail in an aerodynamic interaction environment is:
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Fig. 10. Pressure coefficient on the fuselage upper surface,
To verify the developed calculation model for the flight characteristics of a helicopter, an UH-60 helicopter was used as an
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Fig. 12. Variation of the longitudinal cyclic pitch of the helicopter with changing flight speed.
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example. See [34] and [35] in the literature for the parameters of the UH-60 helicopter. The calculation results were compared with the flight test data [35] for verification. In addition, to demonstrate that the model could accurately consider the aerodynamic interaction effects during the flight of a helicopter, the trim results using a semi-empirical method to calculate the aerodynamic interaction [36] were compared. Figs. 11 to 14 give the results of the comparisons of the collective pitch of the rotor, the pitch angle of the helicopter, the longitudinal cyclic pitch, and the required power with changes in the flight speed, respectively. The figures show that the results of the VVPM-based calculations for the flight characteristics of the helicopter considering aerodynamic interaction effects have a higher degree of consistency with the
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Table 1 Weight of each parameter of the optimization objectives.
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Direction of optimization
Weight
ηhover
Maximization Maximization Maximization Minimization
0.32 0.18 0.2 0.3
hhover S velocit y S angle
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Optimization sub-objective
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Fig. 13. Variations of the pitch and attitude angles of the helicopter with changing flight speed.
Fig. 15. Grid calculation model of the UH-60 helicopter fuselage.
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Table 2 Design optimization variables and constraint conditions.
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83
86 87
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Design optimization variable
Initial value
Constraint condition
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23
hC 1 lC 1 nC 1 nC 2 hmain_rotor hhorizontal_tail htail_rotor
500 mm 800 mm 0.25 0.35 775 cm 1735 cm 1825 cm
400 mm∼600 mm 600 mm∼1000 mm 0.15∼0.6 0.15∼0.6 760 cm∼820 cm 1730 cm∼1830 cm 1800 cm∼1890 cm
89
24 25 26 27 28
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Fig. 14. Variation of the required power of the rotor with changing flight speed.
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flight test results than the semi-empirical method, particularly in the low-speed and transition-speed flight phases. Compared with the results from using a semi-empirical method to calculate the aerodynamic interaction, the new model can more accurately take into account the effects of the aerodynamic interaction from various components of the helicopter on its flight characteristics and the maximum error was reduced from 25% to about 5% in low speed forward flight. These results demonstrate the accuracy and effectiveness of the developed method, which can provide a reliable foundation and basis for calculating the optimized design for the aerodynamic layout of a helicopter.
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6. Example verification of comprehensive optimization of the aerodynamic layout of a helicopter that reduces the adverse effects of aerodynamic interaction
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To verify the effectiveness of the developed optimization method, a fuselage shape derived from UH-60 helicopter is used to carry out an optimization of the aerodynamic layout of a helicopter that reduces the adverse effects of aerodynamic interaction, and the efficiency of the optimization calculation is assessed.
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6.1. Example verification
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Because the UH-60 helicopter is a transport helicopter, several indicators that are important for transport helicopters were selected as the optimization objectives for the aerodynamic layout, including the efficiency in hovering ηhover , the hovering ceiling
hhover , the speed stability derivative S velocit y , and the angle of attack stability derivative S angle . To focus on the effect of aerodynamic interaction, the speed stability derivative and the angle of attack stability derivative were based on the amount of trim at four forward flight speeds (10 km/h, 20 km/h, 30 km/h, and 40 km/h) and were obtained by taking the mean value after stacking. The weight of each optimization objective and the directions of optimization are shown in Table 1. During hovering and low-speed forward flight, the shape of the nose has a significant impact on the aerodynamic interaction effect of the helicopter. In this example, the shape parameters for controlling the nose of the fuselage were selected as the shape design variables, which are the height of the first control point h C 1 , the length of the upper curve l C 1 in x direction, and the upper curve control index nC 1 and the lower curve control index nC 2 on the cross-section; this results in four shape design variables. Three position parameters were selected: the vertical position of the rotor hmain_rotor , the longitudinal position of the horizontal tail hhorizontal_tail , and the longitudinal position of the tail rotor htail_rotor . The values of the positions of the aerodynamic components are related to the coordinate system. In this paper, the coordinate system is the same as the one of the Ref. [34]. According to this coordinate system, the vertical and longitudinal positions of the center of gravity of the helicopter are 5.87 m and 8.912 m, respectively. After the shape of the nose was determined in accordance with the control parameters, a geometric model for the entire helicopter was developed by bridging with the mid-section of the fuselage. The entire helicopter was then divided into grids based on the geometric model, and the vortex panel method was used to develop the fuselage calculation model. The fuselage grid model corresponding to the initial values of the shape parameters, as well as the coordinate system are shown in Fig. 15. The initial values of the position variables are shown in Table 2. In this example, the ranges of variation of the design variables were selected as the constraint conditions, which are also shown in Table 2.
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Table 3 Fitting precisions of various optimization objectives in the surrogate model. Optimization sub-objective
S angle
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Fitting precision
ηhover hhover S velocit y
7 8
9
10 km/h 95.8% 10 km/h 93.6%
90.7% 92.5% 20 km/h 30 km/h 93.1% 99.9% 20 km/h 30 km/h 91.4% 99.9%
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40 km/h 99.9% 40 km/h 99.9%
72 73 74 75 76
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After establishing the optimization problem, it was combined with the calculation model for the flight characteristics of a helicopter to establish the Kriging surrogate model. The samples were acquired using the Latin hypercube method, and the final number of sample points was 96. Due to the speed stability derivative and the angle of attack stability derivative were based on the value at four forward flight speeds, the fitting precision of these two optimization objects should be analyzed at four speeds. The precision of the surrogate model is shown in Table 3; the results show that the surrogate model can substitute for the calculation model based on VVPM and can be used for the comprehensive optimization. In this example, the Kriging surrogate model used a combined multi-island genetic algorithm and sequential quadratic programming optimization strategy to perform the comprehensive optimization of the design. The multi-island genetic algorithm had a population of 15, the number of islands was 10, and the hereditary algebra was 20. The optimal solution acquired from the multiisland genetic algorithm was used as the initial value of the sequential quadratic programming to complete the aerodynamic layout optimization of the helicopter. The process of changing the various optimization objectives during optimization is shown in Fig. 16. The grid calculation model for the shape of the fuselage after the optimization is shown in Fig. 17, and the optimization results for the position parameters of the components are shown in Table 4. After completing the optimization based on the surrogate model, the optimal solution was substituted into the calculation model for the flight characteristics of a helicopter considering aerodynamic interaction effects to obtain the actual optimal solution. The actual optimization results of the various optimization objectives are shown in Table 5. The result of the overall optimization objective was a reduction by 157.5%, and the four optimization objectives all changed in the directions of optimization. In the actual optimization results, the efficiency in hovering increased by 4.7%, the hovering ceiling increased by 3.48%, the speed stability derivative increased by 264.7%, and the angle of attack stability derivative decreased by 26.4%. In this sample, the results for optimal values of the vertical position of the rotor and the longitudinal position of the tail rotor are absolutely predictable. The reason of the changes in the longitudinal position of the horizontal tail is further investigate here. This example is focused on the hover and low speed forward flight. In these conditions, the angle of attack of the horizontal tail is positive, which can be obtained from Ref. [35]. So the horizontal tail will generate a positive lift in these forward conditions, which will provide a pitch down moment on the helicopter. The speed stability of the helicopter could be deteriorated by increasing the longitudinal position of the horizontal tail. So the optimal result of the longitudinal position of the horizontal tail may be caused by the compromise choice of the layout optimization. In this example, the calculation model used in the optimization considering the aerodynamic interaction effects of various components on a helicopter, and the flight characteristics that are affected more by the aerodynamic interaction were selected as optimization objectives. The optimization results showed that the
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Fig. 16. Changes in various optimization objectives.
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Fig. 17. Grid calculation model for the shape of the fuselage after the optimization.
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Table 4 Optimization results for the position parameters of the components. Position parameter
Optimal solution
hmain_rotor hhorizontal_tail htail_rotor hC 1 lC 1 nC 1 nC 2
820 cm 1771.77 cm 1890 cm 435.86 mm 955.15 mm 0.2187 0.5738
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Table 5 Results of the aerodynamic layout optimization of a helicopter. Optimization objective
Initial value
Optimal solution of the surrogate model
Actual optimal solution
5
Overall objective
−0.4
−1.0697
−1.03
6
ηhover
7
hhover S velocit y S angle
0.67 3190 m 0.00017 0.0007
0.6835 3331 m 0.00068 0.000557
0.7015 3301 m 0.00062 0.000515
4
8 9
the aerodynamic layout of a helicopter that reduces the adverse effects of aerodynamic interaction and is able to effectively improve the flight characteristics of the helicopter.
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6.2. Assessment of the computational cost
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In this example, the development of the surrogate model was the most time-consuming component; it required approximately 192 hours on a single workstation equipped with an Intel Xeon X5570 CPU. The combined optimization process required less than 2 minutes, and verification of the calculation results required approximately 2 hours. The total amount of time was approximately 194 hours, which satisfies the engineering design requirements for computational efficiency.
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7. Conclusion
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None declared.
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References
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strategy established in this paper for the optimization of the aerodynamic layout of a helicopter can effectively improve the flight characteristics of the helicopter and reduce the adverse effects of aerodynamic interaction.
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Conflict of interest statement
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A design optimization method for the aerodynamic layout of a helicopter that is capable of reducing the adverse effects of aerodynamic interaction was proposed in this paper and, for the first time, the aerodynamic interaction effects between various components were considered using numerical calculations in the optimization. To take into account the aerodynamic interaction effects and to comprehensively consider the computational precision and efficiency, a calculation model for the flight characteristics of a helicopter was developed based on the VVPM. Based on this foundation, a combined genetic algorithm/sequential quadratic programming optimization algorithm based on the progressive Kriging surrogate model was used and combined with the CST method to consider the effects of the shapes of various components and optimize the aerodynamic layout of a helicopter. The feasibility and effectiveness of the optimization method was verified using a fuselage shape derived from UH-60 helicopter as an example. The following conclusions were obtained from this study: (1) The calculation model for the flight characteristics of a helicopter considering aerodynamic interaction effects performed based on the VVPM has not only higher precision but also a higher computational efficiency than other models. In addition, it can act as the underlying analytical module for integration into the comprehensive design optimization of the aerodynamic layout of a helicopter to reflect the aerodynamic interaction effects of various components. (2) The CST parameter method can efficiently perform parameter modeling of the aerodynamic shape of a helicopter, which makes it possible to consider the effects of changes in the shape of aerodynamic components on the flight characteristics of the helicopter during the aerodynamic optimization. (3) The optimization results of the fuselage shape derived from UH-60 helicopter showed that after the design optimization, the helicopter’s efficiency in hovering increased by 4.7%, the hovering ceiling increased by 3.48%, the speed stability derivative increased by 264.7%, and the angle of attack stability derivative decreased by 26.4%. These results demonstrate that the optimization strategy established in this paper can be used in the design optimization of
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