Higher-order Panel Method Research Articles

Aerodynamic Shape Optimization of Subsonic/Supersonic Flows Integrating Variable-Fidelity Longitudinal Trim Analysis

Most existing studies on aerodynamic shape optimization have not considered longitudinal trim under control surface deflection, typically achieving self-trim through a constraint of zero pitching moment or adjusting the optimized configuration for longitudinal trim. However, adjustments to the optimized configuration might introduce additional drag, reducing overall optimization benefits. In this paper, a novel approach of incorporating control surface deflection for longitudinal trim in aerodynamic optimization is proposed. Firstly, an aerodynamic computation program based on the high-order panel method was developed, introducing velocity perturbations on specific mesh surfaces to simulate actual control surface deflections. Subsequently, a comprehensive optimization framework was established, encompassing parametric modeling, aerodynamic computation, and variable-fidelity control surface deflection analysis. Finally, aerodynamic optimization analysis was conducted under both subsonic and supersonic conditions. Thirty-one design variables were selected with the trimmed lift-to-drag ratio in cruising condition as the objective function and the control surface deflection angle as the constraint. The results indicated an 8.52% increase in the trimmed lift-to-drag ratio compared to the baseline model under subsonic conditions and an 8.1% increase under supersonic conditions.

Two-Way Coupled Aero-Structural Optimization of Stable Flying Wings

An aero-structural algorithm to optimize a flying wing in cruise conditions for preliminary design is developed using two-way interaction between the structure and aerodynamics. A particle swarm routine is employed to solve the multi-objective optimization, aiming to reduce the weight of the structure and the aerodynamic drag at the design point. Different shapes are evaluated during the optimization process until the algorithm reaches the optimal wing aspect ratio, taper ratio, angle of incidence, twist angle, swept angle, and airfoil shape, where a six-parameters method is employed to allow reflex airfoils. A main isotropic I-beam models the wing structure. An extended vortex lattice model is employed to model the aerodynamics, along with a high-order panel method with fully coupled viscous interaction. The finite element method is used to solve the flying-wing structure under static loads. An algorithm is developed to iterate between the deflection of the wing and its impact on the aerodynamics until convergence is reached. Different constraints are implemented into the objective function to fulfil the structural criteria and the longitudinal static stability. A comparison against a baseline optimization is carried out, achieving higher efficiency and promising results in elliptical lift distribution, and a high static margin, without the use of non-constant twist. The results suggest that combining both reflex airfoils and sweep with washout is the optimal solution to reduce the drag and weight, keeping the longitudinal static stability criteria for tailless aircraft in the lower end of the transonic regime.

Accuracy of Küchemann’s prediction for the locus of aerodynamic centres on swept wings compared to an inviscid panel method

AbstractThe locus of aerodynamic centres of a finite wing is the collection of all spanwise section aerodynamic centres, and depends on aspect ratio, wing sweep and planform shape. This locus is of great importance in the positioning of vortex elements in lifting-line theory. Traditionally, these vortex elements are placed along the quarter-chord of a wing, leading to inaccurate predictions of aerodynamic coefficients for swept wings due to the discontinuity in the line of vorticity at the wing root. An analytical solution was presented by Küchemann in 1956 to determine the locus of aerodynamic centres as a function of sweep. While experimental studies have been performed to visualise this locus, no large amount of data is available to fully evaluate the accuracy of Küchemann’s analytical solution. In the present study, a numerical approach is taken using a high-order panel method for inviscid, incompressible flow to calculate the locus of aerodynamic centres for elliptic wings over a wide range of sweep angles, aspect ratios and profile thicknesses. An inviscid panel method is chosen over full CFD solutions because of their ability to isolate the inviscid phenomena. Küchemann’s prediction is compared to this numerical data. The root mean square error is calculated for each wing in a broad design space to determine the accuracy of Küchemann’s theory. It is shown to be remarkably accurate over the range of cases studied, with the root mean square error staying below 4% for all wings with aft sweep and aspect ratios higher than $R_A=5$ . The actual difference between Küchemann’s prediction and numerical data is lower than that for the majority of the span for many of the wing designs considered, with the RMS error being skewed by the results at the tip. Results demonstrate that Küchemann’s analytical equations can be used as an accurate approximation for the locus of aerodynamic centres and could be used in modern numerical lifting-line algorithms*.

General Approach to Lifting-Line Theory, Applied to Wings with Sweep

Implementations of lifting-line theory predict the lift of a finite wing using a sheet of semi-infinite vortices extending from a vortex filament placed along the locus of aerodynamic centers of the wing. Prandtl’s classical implementation is restricted to straight wings in flows without side slip. In this Paper, it is shown that lifting-line theory can be extended to swept wings if, at the control points where induced velocity is calculated, the second derivative of the locus of aerodynamic centers is zero and the trailing vortices are perpendicular to the locus of aerodynamic centers. Therefore, a general implementation of lifting-line theory is presented that conditionally forces the second derivative of the locus of aerodynamic centers to zero at each control point and joints each trailing vortex such that there is a finite segment of the trailing vortex that lies perpendicular to the locus of aerodynamic centers. Consideration is given to modeling the locus of aerodynamic centers and section aerodynamic properties of swept wings. The resulting general formulation is analyzed to determine sensitivity to closure parameters, accuracy, and numerical convergence. The general implementation demonstrates approximately second-order convergence when the control points are cosine clustered and produces results that closely match those of a higher-order panel method and experimental data, though requiring only a fraction of the computational cost.

Calculation of Hinge Moments for a Folding Wing Aircraft Based on High-Order Panel Method

Calculating hinge moments during the morphing process is a critical aspect in the folding wing design. The deficiencies of the traditional flat-plate aerodynamic model in the calculation are expounded in this work, and a flight simulation platform based on a high-order panel method is established. On the basis of the platform, a typical flight-folding process of the aircraft is simulated, and the results of different aerodynamic models are compared. Results show that airfoil thickness has a great influence on the aerodynamic loading distribution of wing surfaces and thus affects the hinge moments during the folding process. The flat-plate method, which ignores the influence of the airfoil thickness, shows a great simulation error in hinge moment, whereas the high-order panel method can effectively describe the thickness effect and obtain reliable simulation results.

Advances on simulation of wave-body interactions under consideration of the nonlinear free water surface

ABSTRACT The paper refers to a boundary element method (BEM) in the time domain for the simulation of wave-body interactions considering the free water surface. The approach solves the boundary integral equations for the velocity and the acceleration potential. The model of the free water surface in this approach is based on the mixed-Eulerian-Lagrangian (MEL) method. Validation results are presented for two well-studied test cases. The first validation case is dedicated to the free water surface. Here, the nonlinear wave loads on a truncated cylinder are analysed and compared with a higher-order panel method and experimental data. Second, results of a Wigley hull moving with forward speed in head waves are shown and compared with experimental results published by Kashiwagi. The validation results in both test cases generally show good agreement with published data and demonstrate the ability of BEM to investigate different types of hydrodynamic problems related to wave-body interactions.

삼각형 패널 상에 선형적으로 분포된 다이폴 강도를 갖는 패널법의 정식화

A high-order potential-based panel method based on Green"s theorem, with piecewise-linear dipole strength on triangular panels, is formulated for the analysis of potential flow around a three-dimensional wing. Previous low-order panel methods adopt square panels with piecewise-constant dipole strength, which results in inherent errors. Square panels can not represent a high curvature lifting body, such as propellers, since the four vertices of the square panel do not locate at the same flat plane. Moreover the piecewise-constant dipole strength induces inevitable errors due to the steps in dipole strength between adjacent panels. In this paper a high-order panel method is formulated to improve accuracy by adopting a piecewise linear dipole strength on triangular panels. Firstly, the square panels are replaced by triangular panels in order to increase the geometric accuracy in representing the shape of the object with large curvature. Next, the step difference of the dipole strength between adjacent panels is removed by adopting piecewise-linear dipole strength on the triangular panels. The calculated results by the present method is compared with analytical ones for simple non-lifting geometries, such as ellipsoid. The results for an elliptic wing with zero thickness at finite angle of attack are compared with Jordan"s results. The comparison shows reasonable agreements for the both lifting and non-lifting bodies.

Methods of determination of aerodynamic loads at flight in continuous turbulence

This paper investigates the loads which acting on the elastic high aspect ratio wing of a cargo aircraft flying in continuous turbulence. According to the requirements of CS-25, gusts that are normal to the flight path and uniform at spanwise are considered. Random gusts at continuous turbulence are described by the spectral density (PSD) of gusts suggested by Karman.The simulation of the flow around the calculated model aircraft are performed by the methods of computational aerodynamics: dipole lattice method (DLM), high order panel method (Panel), dipole lattice and constant pressure method (DLM/CPM), vortex frame method (VFM). Flow is accepted as quasi-stationary or non-stationary. The elastic mass model of the aircraft is described by beam. The discrepancies between the results of the load calculations from the continuous turbulence on the aircraft elastic wing are determined.The results show that taking into account the effects of non-stationary aerodynamics results in somewhat lower values of loads on the aircraft structure: up to 2% in the root sections of the wing and up to 5% in the end sections. The largest difference is observed using the dipole lattice method and the smallest one is in the panel method. The magnitude of the load increase from turbulence is almost the same for all the considered methods. Its main difference is observed when determining loads in horizontal flight, where the lowest values are obtained when using DLM / CPM.The main results are presented in graphical and tabular form.

Analysis of Numerical Errors of the Hess Smith Panel Method With Asymmetric Meshes

AbstractAlthough less accurate in comparison with most of the existing high-order panel methods, the Class Hess Smith panel method is still popular for potential flow calculations due to its simplicity, robustness, and especially efficiency. In the application where the method is adopted for the problems of concern, the method is usually first validated and analyzed for numerical accuracy prior to the application. This is a common practice as also seen in almost all the literature. However, the objects chosen for validation and accuracy analysis are often geometrically symmetric (most typically, spheres) and discretized into symmetric meshes. However, such analysis with symmetric meshes does not actually reveal the numerical errors as the body involved in the problem in concern or the flow around the body may not be symmetrical. In this paper, the numerical errors of the Hess Smith method are systematically analyzed with asymmetrical meshes. The values of various quantities associated with isolated panels and also the entire body are obtained and compared with the Hess Smith panel method does exhibit a very different error pattern.

Numerical study on hydrodynamic interaction between two tankers in shallow water based on high-order panel method

Abstract A three-dimensional high-order panel method based on Non-Uniform Rational B-Spline (NURBS) for predicting the ship–ship hydrodynamic interaction during meeting and overtaking in shallow water is developed. The NURBS surface is used to precisely represent the hull geometry. The velocity potential on the body surface is described by B-spline after the source density distribution on the boundary surface is determined. A collocation approach is applied to the boundary integral equation discretization, and the velocity potential is being solved at each time step. Under the low-speed assumption, the effect of free surface elevation is neglected in the numerical calculation, and the infinite image method is used to deal with the finite water depth effect. Two tankers in model tests are investigated, and the predicted hydrodynamic interaction forces and moments are compared with experimental measurements to clarify the validity of the proposed numerical method. Calculations are then conducted for different water depths, lateral ship–ship distances and ship speeds, and the detailed results are discussed to demonstrate the effects of these factors.

Multidisciplinary analysis of subsonic stealth unmanned combat aerial vehicles

In this paper, the GENUS multidisciplinary aircraft design and analysis environment is presented in its application to the conceptual design of tailless, low-observable unmanned combat aerial vehicles (UCAVs). Analysis disciplines comprise a variety of low to medium fidelity, physics-based and empirical methodologies, as well as higher order panel method aerodynamic analysis. Stealth considerations have been included in terms of a radar cross section analysis through a physical optics approximation method, with results verified against a well-known radar cross section prediction code. Preliminary results show good agreement for gross and empty masses when compared to several existing UCAV demonstrators and conceptual designs. A further validation of the presented methodologies is evaluated through the design, analysis, and optimisation of an unmanned strike fighter concept.

Nonlinear analysis of sloshing and floating body coupled motion in the time-domain

The aim of this paper is to develop a coupled nonlinear time-domain simulation scheme for nonlinear interactions among sloshing flows and floating body motion for both regular and irregular wave excitation. The contributions of a variety of nonlinear factors, outside waves and inside sloshing induced forces, as well as their influences on body coupled sway and roll motion were investigated. The induced forces are due to the changes in the transient wet surface of the floating body and full nonlinear sloshing. The effects of tank fill ratio and excitation wave height on the nonlinear coupled motion, as well as the relationship between sloshing and floating body nonlinear coupled motion under large wave amplitudes and severe sea conditions were also investigated and the results are presented. Finally, the numerical solutions are compared with existing experimental result. The fully nonlinear sloshing and floating body coupled motion are simulated based on the potential flow theory, with the transient position hydrodynamic assumption. The boundary value problem is solved by the B-spline higher-order panel method. The ISITIMFB (iterative semi-implicit time integration method for floating bodies) is applied to solve for the body's velocity and displacements. The sloshing energy dissipation is modeled by changing the boundary condition on the tank's solid boundaries. An extended principle to determine the energy dissipation coefficient for both regular and irregular cases is extracted. Then, the sloshing and floating body nonlinear coupled motion under large wave amplitudes and severe sea conditions are investigated, and the numerical solutions are compared with existing experimental results. The effects of tank fill ratio and excitation wave height on the nonlinear coupled motion is also investigated.

A general high order two-dimensional panel method

We develop an efficient and high order panel method with applications in airfoil design. Through the use of analytic work and careful considerations near singularities our approach is quadrature-free. The resulting method is examined with respect to accuracy and efficiency and we discuss the different trade-offs in approximation order and computational complexity. A reference implementation within a package for a two-dimensional fast multipole method is distributed freely.

Continuous-Time State-Space Unsteady Aerodynamic Modeling for Efficient Loads Analysis

The present paper proposes a continuous-time state-space formulation of the unsteady vortex lattice method, which is derived through a discretization of the governing advection equation for transport of vorticity in the wake. A continuous-time system is obtained by only discretizing the advection equation in space, while retaining the derivative with respect to time. The discretization in space is based on the discontinuous Galerkin method. The present method can be applied to any arbitrary nonuniform wake discretization and can be extended to higher-order panel methods or a nonflat wake shape. The method is extended to compressible flows by applying the Prandtl–Glauert transformation. The time-dependent terms in the small disturbance potential equation are neglected. Thus, incompressible flow solution procedures are applied with minimum modifications to unsteady compressible problems. The benefits are demonstrated by applying the model to the gust analysis of a general aircraft wing, varying the time step, and introducing a nonuniform wake discretization, resulting in a reduced model size for a given accuracy. The resulting continuous-time state-space model can be used for efficient loads analysis of general aircraft wings including the effects of compressibility and allows for easy integration with structural or flight dynamic models for efficient aero(servo)elastic analyses.

Bank effects on ship–ship hydrodynamic interaction in shallow water based on high-order panel method

ABSTRACTA non-uniform rational B-spline (NURBS) based high-order panel method is developed for predicting the bank effects on ship–ship hydrodynamic interaction for ship–ship meeting and overtaking in shallow water. NURBS is used to represent the body geometry. Velocity potential on the body surface is described by B-spline. Under the low speed assumption, the effect of free-surface elevation is neglected and the infinite image method is used to deal with the shallow water effect. The velocity potential and the position of each ship are updated at each time step. The numerical results of hydrodynamic interaction are compared with those by slender-body theory and computational fluid dynamics (CFD) method to validate the present numerical method. Vertical and sloping banks are considered, calculations are conducted for different ship–bank distance, sloping angle of bank, water depth to draft ratio and ship speed. Numerical results are presented to demonstrate the effects of these factors.

Unsteady hydrodynamic interaction between two cylindroids in shallow water based on high-order panel method

A NURBS-based high-order panel method is developed for predicting the hydrodynamic interaction of two bodies moving along parallel courses in shallow water. Source and dipole are distributed on the body surfaces, while dipole is distributed on the wake surfaces; the singularity strengths are determined by satisfying the boundary condition on the body surfaces and the Kutta condition at the trailing edges. The wave-making effect is neglected under the low speed assumption; the image method is used to deal with the effects of finite water depth and undisturbed free surface. The time-stepping method is used to update the velocity potential and the position of each body. Firstly, calculations are carried out for a cylindroid passing a stationary cylindroid and two cylindroids moving in the same direction with the same speed. Detailed convergence study with respect to panel size, time step and truncated mirrored images is carried out. On the basis of convergence study, numerical results are compared with experimental measurements to validate the numerical method. Then, calculations are conducted for two cylindroids during passing, meeting and overtaking under different lateral distances between cylindroids, different water depths and different cylindroid geometry. Numerical results are presented to demonstrate the effects of these factors.

Prediction of hydrodynamic forces on a moored ship induced by a passing ship in shallow water using a high-order panel method

A three-dimensional high-order panel method based on non-uniform rational B-spline (NURBS) is developed for predicting the hydrodynamic interaction forces on a moored ship induced by a passing ship in shallow water. An NURBS surface is used to precisely represent the hull geometry. Velocity potential on the hull surface is described by B-spline after the source density distribution on the boundary surface is determined. A collocation approach is applied to the boundary integral equation discretization. Under the assumption of low passing speed, the effect of free surface elevation is neglected in the numerical calculation, and infinite image method is used to deal with the finite water depth effect. The time stepping method is used to solve the velocity potential at each time step. Detailed convergence study with respect to time step, panel size and Green function is undertaken. The present results of hydrodynamic forces are compared with those obtained by slender-body theory to show the validity of the proposed numerical method. Calculations are conducted for different water depths and lateral distances between ships, and the detail results are presented to demonstrate the effects of these factors.

Wash waves generated by high speed displacement ships

It is difficult to compute far-field waves in a relative large area by using one wave generation model when a large calculation domain is needed because of large dimensions of the waterway and long distance of the required computing points. Variation of waterway bathymetry and nonlinearity in the far field cannot be included in a ship fixed process either. A coupled method combining a wave generation model and wave propagation model is then used in this paper to simulate the wash waves generated by the passing ship. A NURBS-based higher order panel method is adopted as the stationary wave generation model; a wave spectrum method and Boussinesq-type equation wave model are used as the wave propagation model for the constant water depth condition and variable water depth condition, respectively. The waves calculated by the NURBS-based higher order panel method in the near field are used as the input for the wave spectrum method and the Boussinesq-type equation wave model to obtain the far-field waves. With this approach it is possible to simulate the ship wash waves including the effects of water depth and waterway bathymetry. Parts of the calculated results are validated experimentally, and the agreement is demonstrated. The effects of ship wash waves on the moored ship are discussed by using a diffraction theory method. The results indicate that the prediction of the ship induced waves by coupling models is feasible.

Hydrodynamic analysis of oscillating water column wave energy devices

A 40-chamber I-Beam attenuator-type, oscillating water column, wave energy converter is analyzed numerically based on linearized potential flow theory, and experimentally via model test experiments. The high-order panel method WAMIT by Newman and Lee (WAMIT; a radiation–diffraction panel program for wave-body interactions, 2014, http://www.wamit.com ) is used for the basic wave-structure interaction analysis. The damping applied to each chamber by the power take off is modeled in the experiment by forcing the air through a hole with an area of about 1 % of the chamber water surface area. In the numerical model, this damping is modeled by an equivalent linearized damping coefficient which extracts the same amount of energy over one cycle as the experimentally measured quadratic damping coefficient. The pressure in each chamber in regular waves of three different height-to-length ratios is measured in the experiments and compared to calculations. The model is considered in both fixed and freely floating, slack-moored conditions. Comparisons are also made to experimental measurements on a single fixed chamber. The capture width ratio in each case is predicted based on the pressures in the chambers. Good agreement is found between the calculations and the experiments.

Improved hydrodynamic analysis of marine propellers using a B-spline-based higher-order panel method

An improved hydrodynamic analysis of marine propellers based on a B-spline higher-order panel method is presented. The existing potential-based panel methods inevitably result in the degradation of the accuracy, especially near the trailing edge and the tip of the lift-generating surface. However, in the present method, the order of the B-splines used to represent body geometries and potentials can be increased without limit and, hence, solutions of any order can be obtained. In addition, the wake roll-up phenomenon of the propeller is adopted and its effects are investigated. The alignment procedure which satisfies the kinematic and dynamic boundary conditions on the shedding wake surface has been investigated by taking a close look at the case of marine propellers. In this work, we show that wake roll-up modeling gives improved results for the flow of extreme tip such as r/R = 0.99. The present method is validated by comparing it to the existing numerical and experimental results.