A closed form for low-order panel methods

A BEM-isogeometric method for the ship wave-resistance problem

SummaryThree low-order panel methods developed for the analysis of supersonic flows have been used to make calculations for a number of test configurations, with the aim of establishing the range of problems for which this class of computational method yields adequate solutions. The programs investigated were the Woodward USSAERO/C program, the related NLRAERO program, and the British Aerospace Warton supersonic panel program. Results obtained using these programs have been evaluated against theoretical and experimental data for a number of test cases covering wing-alone, body-alone and wing-body geometries. It is concluded that low-order panel methods can provide adequate solutions for supersonic flows about wings, bodies and wing-body combinations, provided the assumptions implicit in the linearised potential flow model are not violated. Examples show that the prediction of lift and pitching moment curve slopes for quite complex configurations may be acceptable, but that the detailed pressure distributions are not always predicted satisfactorily. In particular, serious problems are encountered in calculating the flow about wings with rounded supersonic leading edges due to the linearised flow model which is used.

A zonal method based on the iterative coupling between an in viscid panel method and a Navier-Stokes (NS) method has been developed to analyze the complex flowfield associated with the V/STOL flow environment in order to improve the prediction of aerodynamic characteristics. The low-order panel method, VSAERO, developed by Analytical Methods, Inc. is used to solve for the entire flow region, which includes the V/STOL configuration and jet boundary model. In two separate investigations presented here, ARC3D, a well-known compressible thin layer NS code developed at NASA Ames Research Center, and APPL, a parabolized NS code developed at Amtec Engineering, are used to solve for the details of the jet region subject to the boundary condition provided by VSAERO. Overlapping boundaries are used for the purpose of coupling. Comparisons with the available experimental data for a jet-in-crossflow and a V/STOL configuration with lift-jet in a subsonic freestream show qualitative agreement with experiment and improved correlations over those by inviscid computations alone.

As the interest in arctic shipping and arctic exploration of oil and gas is increasing in recent years, the number of ice class vessels is increasing rapidly. Also the choices for propulsion devices are getting wider and these include podded propulsion systems. This study is a framework for the numerical prediction of the ice interaction loads acting on a podded propeller blade. The results of this study will help us to understand the propeller-ice interaction problem more comprehensively. Several studies for propeller-ice interaction have been carried out in the past few decades. Propeller-ice interaction, however, is a complicated process with a high level of uncertainties due to ice properties, ship operating conditions, and environmental conditions. Full-scale measurements involve high costs. In order to overcome these difficulties, model tests were carried out with model ice in an ice tank. The model tests provide well-controlled ice properties and interaction conditions to reduce the uncertainties. The tests were carried out in the ice tank with scaled down model ice at the National Research Council of Canada’s Institute for Ocean Technology. The ice loads acting on the propeller blade were measured with a six-component force and moment load cell fitted to the shaft and one of the propeller blades. Based on the experimental results, a numerical prediction model was developed to estimate the ice loads on the propeller blade. The numerical prediction is composed of three parts: the hydrodynamic calculations including separable and inseparable hydrodynamic loads, and the ice milling loads calculation. The separable and inseparable hydrodynamic loads can be obtained from clear water and blocked flow respectively. The hydrodynamic calculations were done by a low order panel method. The subroutines for calculating the ice milling loads are implemented into the panel method. The numerical prediction model for ice milling loads is described and the results are compared with those of experiments.

Flow around circular and noncircular cylinders by a low-order panel method

The added mass coefficients for two and three-dimensional submerged bodies were calculated using a low-order panel code. The source and dipole strengths, and the panel surface area for each panel, were used to compute the integrals needed for added mass in all six degrees of motions. Several applications of this method were used in comparing the results with the theoretical, when available, experimental or other numerical results. The method was found to be successful in predicting the added mass coefficients using relatively low numbers of panels.

A B-spline panel method is developed to solve a boundary integral equation for three-dimensional potential flow problems. In particular, a nonuniform rational B-spline (NURBS) surface is introduced to accommodate a precise geometric description of a given body. For the unknown (potential) description over the surface, uniform B-spline basis functions are used. A collocation approach is adopted for numerical computations, and influence coefficients are evaluated in a robust manner over the surface defined by NURBS. Convergence characteristics of the present method have been investigated in the numerical experiments on the unbounded problem of a sphere, the radiation problem of a floating hemisphere, and the diffraction problem of a submerged spheroid. It is shown that the numerical results are in excellent agreement with analytical solutions in all cases considered. It is concluded that the NURBS panel method is superior to existing panel methods in geometric description of body surface, and convergent solutions can be achieved rapidly with a small number of panels.

A suitable low order panel method is introduced, that allows the calculation of free vortex sheets emanating from sharp-edged wings in subsonic flow. The wing surfaces as well as the separated shear layers are discretized into panels, each carrying a constant doublet distribution. Bodies — e.g. fuselages — can be modeled by a network of source panels with elementwise constant strength.

Fourier–Chebyshev approximation of Kochin functions in the Fourier–Kochin approach to panel methods

Results from the application of a second generation low-order panel method to a number of powerplant installations are presented. The method, which retains the advantages of ease of use and low operating cost of earlier panel methods while retaining the aerodynamic modelling rigor of higher-order methods was used to predict the flow in and around typical nacelles and aircraft for a wide range of conditions. The ability of the second generation low-order analysis to calculate internal flows without the leakage problems associated with earlier programs is demonstrated and correlation between calculated and measured surface pressures and flow behavior, including predicting streamlines and locating regions of separated flow, is presented.

A B-spline based higher order panel method for analysis of steady flow around marine propellers

The aim of this work is to provide a solver for viscoelastic multi-phase flows within the Bounded Support Spectral Solver (BoSSS) currently under development at the Chair of Fluid Dynamics at the Technical University of Darmstadt. The discretisation in BoSSS consists of a high-order Discontinuous Galerkin (DG) method for single-phase flow and a high-order eXtended Discontinuous Galerkin (XDG) method for the multi-phase purpose. The solver shall be used to investigate numerically the behaviour of viscoelastic droplets. The macroscopic Oldroyd B model which is used in a wide range of applications is chosen as the constitutive model. A detailed derivation of the system of equations including the modeling principles for the Oldroyd B model is presented. A DG discretisation of the system of equations including the Local Discontinuous Galerkin (LDG) method is presented after introducing the field of the DG method. The derivation of appropriate flux functions for the constitutive equations and the extra stress tensor are one of the key derivations of this scientific work. Difficulties arising in the numerical solution of viscoelastic flow problems for higher Weissenberg numbers for different discretisation methods are due to the convection dominated, mixed hyperbolic-elliptic-parabolic nature of the system of equations. Several strategies are presented which overcome these problems and are known from the literature. A key achievement of this scientific work is the application of the LDG method, originally developed for a hyperbolic system of equations for a Newtonian fluid, on the viscoelastic system of equations which renders methods for preserving ellipticity unnecessary. Furthermore, various strategiesn to enhance and to support convergence of the solution of the DG discretised system are presented. These are the Newton method with different approaches determining the Jacobian of the system, a homotopy continuation method based on the Weissenberg number for a better initial guess for the Newton method, and a troubled cell indicator combined with an artificial diffusion approach or an adaptive mesh refinement strategy, respectively. For the completeness of this work the XDG method is presented using a sharp interface approach with a signed distance level-set function as it is implemented in BoSSS. The single-phase solver is combined with these methods and appropriate flux functions for the interface are implemented to enable multi-phase applications for viscoelastic fluid. Several numerical experiments are conducted to verify and to validate the viscoelastic singlephase solver and to show the capability of the viscoelastic multi-phase solver to simulate viscoelastic droplets. Advantages and disadvantages of the implementation and an outlook for future research can be found in the conclusion.

This paper presents the development of a three-dimensional hydrodynamic model for wind-driven circulation using the moving element method in the vertical discretization. As most three-dimensional models use sigma coordinate transformation in the vertical discretization, the use of the moving element method is an alternative to the latter, with the advantage that there is no need for fixed subdivision of the water column. In the proposed model, the coupling of two- and three-dimensional hydrodynamic model is considered. The shallow-water equations are integrated in the vertical direction, finite elements are employed in the spatial discretization, and finite differences in the time discretization, to resolve the position of the free surface (ζ), and the vertically integrated velocities components. The three-dimensional module is used to compute the velocity profiles. In the three-dimensional module is employed the moving element method in the vertical discretization. The efficiency of the model is demonstrated through the comparison of its results with laboratory experimental results and with another model that uses sigma coordinate transformation in the vertical discretization.

We study the numerical behaviour and the relationships between some numerical methods derived from the Mixed Finite Element (MFE) formulation and from the Control Volume formulation using the Multi-Point Flux Approximation (MPFA) of Aavatsmark et al. (1996). All methods are locally mass conservative and are well suited to solve the flow problem for both steady state (elliptic case) an transient (parabolic case) on a general irregular grid with anisotropic and heterogeneous discontinuous coefficients. Hybridization of the standard MFE method allows to obtain a symmetric positive definite system which is not, in general, an M-matrix. Moreover, the mixed hybrid finite element method uses more unknowns (one per edge or face) than the finite volume formulation (one per element). On the other hand, for a general element grid, the MPFA method does not give a symmetric matrix of coefficients and monotonicity issues are known to arise for high aspect ratios combined with skewed of computational grids. In this work, we propose two variants of the MFE method: (i) the lumped-MFE method, based on the equivalence between MHFE method and the P1-nonconforming Galerkin method for the Laplace equation and (ii) the multipoint-MFE method based on Multipoint finite volume discretization using the framework of the modified mixed finite element space. We discuss connections between the different approaches and perform numerical experiments for both steady state and transient cases to compare (number of unknowns, matrix properties, condition number) and study the numerical behaviour of these methods (unphysical oscillations, CPU time).

This study is concerned with the use of integrated radial-basis-function networks (IRBFNs) for the discretisation of the velocity-pressure formulation in two dimensions on Cartesian grids. In the approximation of the field variables (i.e. velocity components and pressure), instead of using low-order polynomial interpolants, we employ global IRBFNs along grid lines (i.e. one-dimensional IRBFNs). In the imposition of boundary condition for the pressure, we propose two treatments, namely Treatment A and Treatment B. For both treatments, Neumann boundary conditions are transformed into Dirichlet ones. The former is based on values of the pressure at interior nodes along a grid line and first derivative values of the pressure at two extreme nodes of that grid line; while the latter relies on values of the pressure at interior nodes along a grid line together with both first and second derivative values of the pressure at two extreme nodes of that grid line. The proposed method is verified successfully through the simulation of a benchmark test, namely the isothermal lid-driven cavity flow problem.