A boundary element method for the aerodynamic analysis of aircraft in arbitrary motions

A new boundary integral formulation for the aerodynamic analysis of an aircraft (in particular, a tiltrotor) in arbitrary motion is presented. The formulation is based on the velocity potential for compressible flows, and as such is an extension of past work of the authors. The distinguishing feature is that the boundary integral representation is written for a surface in arbitrary motion with respect to a frame of reference which in turn moves in arbitrary motion with respect to the undisturbed air. Thus, the integrals are evaluated on the emission surface, which is the locus of the emitting points at the locations (in the moving frame) that they had when the signal influencing a given point at a given time was emitted. The differences with respect to related formulations (e.g., Ffowcs Williams and Hawkings) are outlined. Also, the advantages of the present formulation with respect to the preceding ones by the authors are discussed. Numerical validation results are presented for the limited case of helicopter rotors in hover.     

Aerodynamics and aeroacoustics of wings and rotors via BEM - unsteady, transonic, and viscous effects

Recent developments on a general boundary integral formulation for the aerodynamic and aeroacoustic analyses of lifting bodies (e.g., wings and rotors) are reviewed. The emphasis is on recent numerical results, specifically on the effects of the unsteadiness, of the viscosity, and of the transonic nonlinearities. The boundary-element full-potential formulation for bodies in arbitrary motion is outlined along with the its extension to viscous/inviscid interaction. The effects of viscosity are taken into account through a coupled boundary-layer/full-potential technique similar to that used in the CFD community. Numerical results obtained with the present formulation are compared against numerical and experimental results available in literature. They include (i) aerodynamics and aeroacoustics of helicopter rotors in hover and forward flight in subsonic flows; (ii) transonic aerodynamics/acoustics results for steady potential flows around airfoils and hovering rotors, and (iii) viscous flows (subsonic and transonic) around airfoils.     

Numerical analysis of aerodynamic performance of rotors with leading edge slats

A three dimensional compressible Navier-Stokes analysis capable of computing flows around wings and rotors with partial-span flaps or slats is described. This analysis is cast in a moving body-fitted coordinate system permitting arbitrary motion of the solid surfaces to be directly modeled. This methodology is validated through several single and multi-element rotor and wing configurations. The analysis is subsequently applied to two helicopter rotors. The effects of leading edge slats on the aerodynamic performance of these rotors in hover is studied. It is demonstrated that leading edge slats can significantly improve the hover performance at high pitch settings, with an increase in thrust and a reduction in torque. At low pitch settings, the slats were found to be detrimental to the rotor performance. Many practical issues such as surface imperfections, proper sealing of the slat during retraction, control loads and other implementation aspects should be carefully evaluated in the event of a practical design of a slat. However, in the present study only the aerodynamic effects of slats are addressed and no practical issues are considered.This work was supported by the U.S. Army Research Office under the Center of Excellence in Rotorcraft Technology (CERT) program. Dr. Thomas Doligalski was the technical monitor.     

On the vorticity-generated sound for moving surfaces

 A formulation for analyzing the mechanism of vorticity-generated sound is presented, which is based upon an extension of a boundary integral equation method for unsteady quasi-potential compressible flows. The extension is obtained through an exact decomposition of the velocity field into potential and rotational contributions, with the distinguishing feature that the vortical contribution vanishes in much of the field. The novelty is that the construction of the vortical contribution of the velocity is such that discontinuities across the wake of the two (potential and rotational) contributions are eliminated, thereby facilitating possible implementations. Received 6 November 2000     

Wake capturing in potential flow calculations

Summary In this paper, we present a formulation for irrotational, isentropic flows in terms of velocity components which is equivalent to the classical potential equation, except the wake is not fixed and it is captured as part of the solution with no special treatment. Preliminary numerical results are presented to validate this new approach.     

Parallel simulation of unsteady hovering rotor wakes

Numerical simulation using low diffusion schemes, for example free‐vortex or vorticity transport methods, and theoretical stability analyses have shown the wakes of rotors in hover to be unsteady. This has also been observed in experiments, although the instabilities are not always repeatable. Hovering rotor wake stability is considered here using a finite‐volume compressible CFD code. An implicit unsteady, multiblock, multigrid, upwind solver, and structured multiblock grid generator are presented, and applied to lifting rotors in hover. To allow the use of very fine meshes and, hence, better representation of the flow physics, a parallel version of the code has been developed, and parallel performance using upto 1024 CPUs is presented. A four‐bladed rotor is considered, and it is demonstrated that once the grid density is sufficient to capture enough turns of the tip vortices, hover exhibits oscillatory behaviour of the wake, even using a steady formulation. An unsteady simulation is then performed, and also shows an unsteady wake. Detailed analysis of the time‐accurate wake history shows that three dominant unsteady modes are captured, for this four‐bladed case, with frequencies of one, four, and eight times the rotational frequency. A comparison with theoretical stability analysis is also presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Unstructured mesh methods for the solution of the unsteady compressible flow equations with moving boundary components

A review of a procedure for the simulation of time-dependent, inviscid and turbulent viscous, compressible flows involving geometries that change in time is presented. The adopted discretization technique employs unstructured meshes and both explicit and implicit time-stepping schemes. A dual time-stepping procedure and an ALE formulation enable flows involving moving boundary components to be included. Techniques that have been developed to maintain the validity of the unstructured mesh and to allow for the capture of moving flow features are also reviewed. Using the in-house developed techniques, some examples are included to demonstrate the use of the approach for the simulation of a number of flows of practical industrial interest.     

A general higher-order shell theory for compressible isotropic hyperelastic materials using orthonormal moving frame

The primary objective of this study is threefold: (1) to present a general higher-order shell theory to analyze large deformations of thin or thick shell structures made of general compressible hyperelastic materials; (2) to formulate an efficient shell theory using the orthonormal moving frame, and (3) to develop and apply the nonlinear weak-form Galerkin finite element model for the proposed shell theory. The displacement field of the line normal to the shell reference surface is approximated by the Taylor series/Legendre polynomials in the thickness coordinate of the shell. The use of an orthonormal moving frame makes it possible to represent kinematic quantities (e.g., the determinant of the deformation gradient) in a far more efficient manner compared with the nonorthogonal covariant bases. Kinematic quantities for the shell deformation are obtained in a novel way in the surface coordinate described in the appendix of this study with the help of exterior calculus. Furthermore, the governing equation of the shell deformation has been derived in the general surface coordinates. To obtain the nonlinear solution in the quasi-static cases, we develop the weak-form finite element model in which the reference surface of the shell is modeled exactly. The general invariant based compressible hyperelastic material model is considered. The formulation presented herein can be specialized for various other nonlinear compressible hyperelastic constitutive models, for example, in biomechanics and other soft-material problems (e.g., compressible neo-Hookean material, compressible Mooney–Rivlin material, Saint Venant–Kirchhoff model, and others). A number of numerical examples are presented to verify and validate the formulation presented in this study. The scope of potential extensions are outlined in the final section of this study.     

A direct boundary integral equation formulation for the Oseen flow past a two-dimensional cylinder of arbitrary cross-section

Summary A novel boundary integral formulation is presented for the direct solution of the classical problem of slow flow past a two-dimensional cylinder of arbitrary cross section in an unbounded viscous medium, the equations of motion having first been linearised by the Oseen approximation. It is shown how the governing partial differential equations of motion, together with the no-slip boundary conditions on the cylinder, may be reformulated as a pair of coupled integral equations of the second kind, which may be manipulated further to yield the lift and drag coefficients explicitly, as well as flow characteristics anywhere in the flowfield.The present formulation requires a non-iterative numerical solution procedure which is applicable to low Reynolds number flows. The method is not restricted in its ability to deal with complicated cylinder geometries, as the discretisation of only the cylinder surface is required.Results of the present method are shown to be in good agreement with those of previous analytical and numerical investigations.With 2 Figures     

Transonic wing flow computations using the field-panel method with a shock-fitting technique

An integral equation field-panel scheme for solving the full-potential equation for compressible flows with and without shocks is presented. The full-potential equation is written in the form of the Poisson's equation. Compressibility is treated as non-homogeneity. The integral equation solution in terms of velocity field is obtained by Green's theorem. The solution consists of wing (or a general body) surface integral term(s) of vorticity/source distribution(s), wake surface integral term(s) of free-vortex sheet(s), a volume integral term of compressibility over a small limited domain around the source of disturbance, and a shock surface integral term of source distributions for the shock-fitting purpose. Solutions are obtained through an iterative procedure. Instead of using a grid (field-panel) refinement procedure, a shock-fitting technique is used to fit the shock. The present scheme is applied to non-lifting flows around both sharp and round leading edge rectangular wings at high-subsonic and transonic flow conditions.     

On thin film flow in hydrodynamic bearings with a radial step at finite Reynolds number

A classical engineering approach to thin film flow problems with localised geometric step features is to use the Reynolds equation. For applications to new generation hydrodynamic bearings with very small gap clearances and lift-generating features, any abrupt changes in the thickness of a film will break the validity of the Reynolds equation, which is based on lubrication theory. In this work, formal asymptotic expansions are used to match a numerical solution of a local formulation of the full Navier–Stokes equations near a step feature to a Reynolds equation model which is valid sufficiently far from the step, i.e. with smooth film thickness variation. The approach is used to model a pressurised bearing with an axisymmetric Rayleigh step feature. An efficient and accurate mathematical model is presented using matched asymptotic expansions for both incompressible and compressible fluid flows. This work quantifies the effect of inertia at the step and considers the validation of the classical approach of patching lubrication solutions across the step with specified compatibility conditions. A parametric study is undertaken to evaluate cases where the classical engineering approach is justified.     

Finite element analysis of the transient motion of stratified viscous fluids under gravity

The present paper deals with the finite element analysis of two-dimensional two-layer density flows in a gravitational field. A fluid in each layer is replaced with a large number of discrete particles, and the motion and deformation of each layer is represented by moving those particles in a Lagrangian manner. The velocity distribution in the whole fluid region is given as the finite element solution of the Navier-Stokes equations and the equation of continuity. In the finite element calculation, free-slip conditions are used on solid wall boundaries because no-slip conditions may cause sticking of some particles to walls. Then, a new technique for the implementation of free-slip conditions on arbitrary curved boundaries is presented. As numerical examples, density flows in a rectangular closed container and Rayleigh-Taylor instability in the container with a circular cross-section have been computed.     

Acoustic calculations with the hybrid boundary element method in time domain

The boundary element method (BEM) is an efficient tool for the calculation of acoustic wave propagation in fluids. Transient waves can be solved by either using a formulation in frequency domain along with an inverse Fourier transformation or a time domain formulation. To increase the efficiency for the solver and allow for an efficient coupling with finite element domains the symmetry of the system matrices is advantageous. If Hamilton's principle is used, a symmetric variational formulation can be established with the velocity potential as field variable. The single field principle is generalized as multifield principle as basis of a hybrid BEM for the calculation of acoustic fields in compressible fluids in time domain. The state variables are separated into boundary variables, which are approximated by piecewise polynomials and domain variables, which are approximated by a superposition of weighted fundamental solutions. In both approximations the time and space dependency is separated. This is why static fundamental solution can be used for the field approximation. The domain integrals are eliminated, respectively, transformed into boundary integrals and an equation of motion with symmetric mass and stiffness matrix is obtained, which can be solved by a direct time integration scheme or by mode superposition. The time derivative of the equation of motion leads to a formulation with pressure and acoustic flux on the boundary for an easier interpretation of the variables.     

Parametric study of the dynamic response of thin rectangular plates traversed by a moving mass

The governing differential equation of motion of a thin rectangular plate excited by a moving mass is considered. The moving mass is traversing on the plate’s surface at arbitrary trajectories. Eigenfunction expansion method is employed to solve the constitutive equation of motion for various boundary conditions. Approximate and exact expressions of the inertial effects are adopted for the problem formulation. In the approximate formulation, only the vertical acceleration component of the moving mass is considered while in the exact formulation all the convective acceleration components are included in the problem formulation as well. Parametric studies are carried out to investigate the effects of moving mass weight and velocity as well as its trajectory on the dynamic response of a simply supported plate. Rectilinear and orbiting paths are considered in the parametric studies as the two limiting cases for any possible moving mass trajectories. The obtained results demonstrate the importance of the moving mass inertia with respect to the moving load in most of the cases considered. In case of the rectilinear path, the approximate formulation underestimates the plate’s maximum response for mass velocities above certain limits. Furthermore, increasing the plate’s aspect ratio or the moving mass weight further reduces the range of velocities in which the approximate formulation can be used instead of the exact formulation. For the case of an orbiting path, the approximate formulation can capture the resonance excitation frequencies of the load reasonably well, even for large mass weight and radius of the orbiting mass. Considering small orbiting mass radii, the approximate formulation would provide an upper bound for the true response of the system for all orbiting frequencies as well as the mass weights. However, for larger radii, the maximum response values resulting from the approximate formulation are considerably lower than that of the exact one, especially for frequencies near to the resonance frequencies.     

Numerical analysis of two-dimensional compressible turbulent flows in a turbine stage

A numerical analysis based on the compressible Reynolds-averaged Navier-Stokes equation has been developed for the analysis of two-dimensional compressible turbulent flows in a turbine stage (nozzle and bucket). In the present flow analysis, governing equations are solved by the use of a time dependent explicit method and a two-equation model of turbulence is employed to estimate turbulence effects. To calculate nozzle and bucket flow fields simultaneously, a steady interaction between these flows is assumed. For spatial discretization of the governing equations, a control volume method combined with a body-fitted curvilinear coordinate system is developed to calculate the flows in arbitrarily shaped cascades. In order to assure the effectiveness of the present method, computations are carried out for a two-dimensional section at a blade midspan in a turbine stage. The method gives satisfactory results about boundary layers on blade surfaces, nozzle wake profiles and pitchwise averaged turbine design parameters at each blade exit.     

Space–time SUPG finite element computation of shallow-water flows with moving shorelines

We show that combination of the Deforming-Spatial-Domain/Stabilized Space–Time and the Streamline-Upwind/Petrov–Galerkin formulations can be used quite effectively for computation of shallow-water flows with moving shorelines. The combined formulation is supplemented with a stabilization parameter that was originally introduced for compressible flows, a compressible-flow shock-capturing parameter adapted for shallow-water flows, and remeshing based on using a background mesh. We present a number of test computations and provide comparisons to theoretical results, experimental data and results computed with nonmoving meshes.     

Diffusionlike equation for superconducting binary alloys

A diffusionlike equation is derived within the coherent potential approximation (CPA) for type II (or inhomogeneous), superconducting binary alloys. Diagrammatic formulation of the CPA is employed to extend the derivations of Eilenberger and Usadel to nondilute random substitutional alloys A _1?s B _s having arbitrary concentrations and scattering strength. The generalized diffusion coefficient satisfies a simple algebraic equation, the coefficients of which are connected to measurable physical quantities. The diffusionlike equation is valid for arbitrary values of the order parameter at arbitrary temperatures belowT _c . When the order parameter Δ is small, simple expansions in terms of Δ would lead immediately to the generalized Landau-Ginzburg equations. The upper critical magnetic field and magnetization are expressed as functions of concentrations of the binary alloy.     

Self-similar evolution of a twin boundary in anti-plane shear

This paper studies the transient motion of a twin boundary in two dimensions. The twinning deformation is described as an anti-plane shear deformation with discontinuous strains. The material is assumed to be compressible and hyperelastic with a stored energy function consisting of multiple potential wells. The quasi-steady-state evolution of a twinning step is studied. The model includes an anisotropic kinetic relation that governs the twin boundary motion in two dimensions under applied stress. A self-similar solution for the motion of the twinning step is found with a specific initial shape. General solutions to the linearized evolution equation are established in the form of an infinite series for arbitrary initial shapes. Stability of the self-similar solution is discussed.     

Time-domain analysis of electromagnetic wave fields by boundary integral equation method

Time-domain analysis of electromagnetic wave fields is popularly performed by the Finite Difference Time-Domain method. Then the Boundary Integral Equation Method (BEM) still has advantage comparing with FDM or FEM type scheme in open boundary problems, moving boundary problems and coupled problems of charge particle and electromagnetic fields. However, the time-domain boundary integral equation method still do not well developed, numerical instability in long time range calculations frequently appear except for special cases. In this paper, a stable scheme of the time-domain boundary integral equation method is presented and numerical example of particle accelerator wake fields is shown.