Boundary integral equation for wave equation with moving boundary and applications to compressible potential aerodynamics of airplanes and helicopters

This work presents a general boundary-integral-equation methodology for the solution of the wave equation around objects moving in arbitrary motion, with applications to compressible potential aerodynamics of airplanes and helicopters. The paper includes the derivation of the boundary integral equation for the wave equation, in a frame of reference moving in arbitrary motion (in particular, in translation and in rotation). The formulation is then applied to study unsteady potential compressible aerodynamic flows around streamlined bodies, such as airplanes and helicopters. The formulation is given in terms of the velocity potential, for which an explicit treatment of the wake is required; a discussion of the formulation for the wake transport is included. The advantages of the velocity-potential formulation over the acceleration-potential formulation are discussed. The boundary-element algorithm used for the computational implementation is briefly outlined. Validation of the formulation is presented for airplane wings and helicopter rotors in hover. The test cases fall into two categories. prescribed-wake and free-wake analyses. The validation of the prescribed-wake analysis is presented for compressible flows, subsonic for helicopter rotors, transonic for airplanes. The numerical validation of the free-wake analysis of helicopter rotors is presented for incompressible flows.     

Boundary integral equations and conservative dissipation schemes for full potential transonic flows

A boundary integral formulation for the nonlinear aerodynamic analysis of three-dimensional full-potential transonic flows is presented. The emphasis here is on the analysis of the effects on the solution of artificial dissipation schemes, which are necessary in order to capture properly the physics of the phenomenon. The main novelty is the use of conservative schemes, never previously used in boundary integral formulations where all the existing approaches are based on non-conservative ones. The conservative scheme presented here is an adaptation of concepts used in the CFD community. Specifically, a linear dissipation term is added directly to the continuity equation: hence the name artificial mass-generation scheme. Both conservative and non-conservative full-potential expressions for the nonlinear terms are discussed. The corresponding TSP (transonic small perturbation) formulation are also analyzed. Numerical results, for two-dimensional steady flows are presented in order to assess the different schemes. Good agreement is obtained with existing finite-difference and finite-volume results.     

Computation of transonic viscous flow over airfoils with control by an elastic membrane and comparison with control by passive ventilation

Summary The performance of transonic wings can be influenced by control of the shock/boundary layer interaction (SBLI) using an adaptive surface in the shock region. This is achieved by inserting a cavity into the airfoil and covering it with an elastic membrane. The theoretical methods for the computation of transonic viscous flow around airfoils with control are presented. The zonal solution method consists of numerical and analytical parts. The influence of the adaptive wall on the flow field over a modern transonic airfoil has been investigated. The flow parameters have been varied up to off-design conditions to understand the physical effects of the control. Previous investigations of a second way of control, the passive ventilation, allow a comparison of these two methods.     

High-order BEM for potential transonic flows

A high-order boundary element method (BEM) for the analysis of the steady two-dimensional full-potential transonic equation is presented. The use of a high-order (piecewise cubic on the boundary and piecewise bi-cubic in the field) numerical formulation, the main novelty of the present work, is important in two respects: first, the convergence of the solution as h vanishes is faster than the zeroth-order (the piecewise constant) one, yielding more accurate results with coarser grid resolutions. In addition, in supercritical flows, the derivation of the velocity field from the high-order representation for potential gives, in the vicinity of the shock, a sharper discontinuity, and allows for an in-depth analysis of the shock properties. Both aspects are analyzed in the present paper through applications to steady, two-dimensional, full-potential flows. All the results obtained using the present method are validated through comparison to other computational fluid dynamics (CFD) solutions of the full-potential flows and, when applicable, Euler equations. A comparison to a zeroth-order BEM, based on the same integral formulation is also included.     

The continuous adjoint approach to the k–ω SST turbulence model with applications in shape optimization

In this article, the gradient of aerodynamic objective functions with respect to design variables, in problems governed by the incompressible Navier–Stokes equations coupled with the k–ω SST turbulence model, is computed using the continuous adjoint method, for the first time. Shape optimization problems for minimizing drag, in external aerodynamics (flows around isolated airfoils), or viscous losses in internal aerodynamics (duct flows) are considered. Sensitivity derivatives computed with the proposed adjoint method are compared to those computed with finite differences or a continuous adjoint variant based on the frequently used assumption of frozen turbulence; the latter proves the need for differentiating the turbulence model. Geometries produced by optimization runs performed with sensitivities computed by the proposed method and the ‘frozen turbulence’ assumption are also compared to quantify the gain from formulating and solving the adjoint to the turbulence model equations.     

A stabilized finite element formulation for the transonic small‐disturbance system

A stabilized finite element formulation for the transonic small‐disturbance system of equations is developed and used to solve a variety of problems in transonic aerodynamics. An adaptive mesh refinement technique and a common discontinuity capturing operator are used to resolve regions with large gradients in the velocity field. The scheme works well in both flow regimes, subsonic and supersonic, and captures shocks naturally. Agreement with available experimental observations and theoretical approximations is very good. Copyright © 2001 John Wiley & Sons, Ltd.     

A velocity-decomposition formulation for the incompressible Navier–Stokes equations

The principle of velocity decomposition is used to combine field discretization and boundary-element techniques to solve for steady, viscous, external flows around bodies. The decomposition modifies the Navier–Stokes boundary-value problem and produces a Laplace problem for a viscous potential, and a new Navier–Stokes sub-problem that can be solved on the portion of the domain where the total velocity has rotation. The key development in the decomposition is the formulation for the boundary condition on the viscous potential that couples the two components of velocity. An iterative numerical scheme is described to solve the decomposed problem. Results are shown for the steady laminar flow over a sectional airfoil, a circular cylinder with separation, and the turbulent flow around a slender body-of-revolution. The results show the viscous potential is obtainable even for massively separated flows, and the field discretization must only encompass the vortical region of the total velocity.     

振荡压气机叶栅叶片表面非定常响应以及气弹稳定性分析

发展并验证了一种适用于叶轮机内部非定常跨音流动诱导的叶片气弹问题的高效、准确的数值模拟方法。采用有限体积的多块结构化网格形式,多重网格方法加速收敛,隐式的双时间步时间推进,Spalart-Allmaras（S-A）湍流模型求解非定常雷诺平均Navier-Stokes方程。通过气动弹性标准算例10,叶片在高亚音和跨音流动下做弯曲振动,分析了流动状态、折合频率以及叶片间相位角对叶片表面非定常气动力响应以及叶栅气弹稳定性的影响。分析结果表明激波在此跨音振荡压气机叶栅中起失稳作用,叶片间相位角对气弹稳定性的影响在高折合频率下被加强。     

Field panel method with grid stretching technique for solving transonic potential flow around arbitrary airfoils

The Field Panel Method (FPM) with grid stretching technique, presented in this paper, was developed for solving transonic full potential flow around arbitrary airfoils at incidence. In this method, the total potential values are represented by boundary integrals together with a volume integral. The volume integral domain includes both inside and finite outside of the configuration and can be discretisized in a Cartesian grid which may penetrate into the configuration surface. Thus, we avoid the very difficult task of generating body-fitted grids around complex configurations. The boundary potential values are obtained by implementing a standard panel method (symmetrical singularity model), whereas the field potential values are estimated by solving the full potential equation (using AF3 scheme in a Cartesian grid) with approximate inner and proper outer boundary conditions.Furthermore, the grid stretching technique has been utilized that allows to capture the shock waves in a much better quality. It is also shown that both field grid and panel distribution have to be stretched at the same time.Results for transonic potential flows about NACA0012 and RAE2822 airfoils at different Mach numbers and incidences are obtained and compared with other numerical solutions. Great improvement in shock wave quality was achieved by using the present method.Supported by Alexander von Humboldt Foundation, Germany.Supported by DFG (Deutsche Forschungsgemeinschaft) (Wa 424/8).     

Computation of three-dimensional transonic viscous flow using the JUMB03D code

Summary A vertex based finite volume method for the solution of the three dimensional Reynolds averaged Navier-Stokes equations has been developed. The computations can be carried out blockwise after dividing the computational domain into smaller blocks to reduce the memory requirement for a single processor computer and also to facilitate parallel computing. A five stage Runge-Kutta scheme has been used to advance the solution in time. Enthalpy damping, implicit residual smoothing, local time stepping, and grid sequencing are used for convergence acceleration. In order to get smooth convergence for transonic, viscous flows, the artificial dissipation has been modified by using the time step for advective and diffusive equations. An algebraic turbulence model has been used to determine the turbulent eddy viscosity. The method has been used to compute transonic flow over a cropped delta wing and the ONERA M-6 wing, and subsonic flow over a launch vehicle configuration. The results obtained show good agreement with available experimental data.     

Subsonic interfacial (Stoneley) waves in anisotropic multiferroic bimaterials with a viscous interface

The problem of subsonic interfacial (Stoneley) wave propagation in anisotropic multiferroic bimaterials with a viscous interface is treated. A concise analytical method is constructed for deduction of possible subsonic interfacial wave with varying viscosity of the interface. A numerical scheme and several calculations are given based on the method, which demonstrate interesting results. For an interface constructed by a piezoelectric half-space and a piezomagnetic half-space, when assumed to be non-viscous, calculation shows that it does not permit any subsonic interfacial wave. Yet when the same interface is assumed to be viscous, at least one possible subsonic interfacial wave speed appears which varies with the viscosity of the interface. By introducing the relation between viscosity of certain adhesives and temperature, the possibility of control of interfacial wave speeds through accommodating the working temperature is put forward.     

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.     

Detailed study of complex flow fields of aerodynamical configurations by using numerical methods

The mathematical physics of fluid flow in a compressible medium, leads to nonlinear partial differential equations or their equivalent integral versions. For the solution of these equations one has generally to resort to numerical methods using mostly finite difference or finite volume schemes, which are well established now. These field methods are very suitable for studying the physical features of complex flows. The present paper gives at first a short sketch of the numerical procedure and thereafter goes into the detailed analysis of the flow fields of delta wings, double-delta wings, delta shaped wing-canard combinations and space vehicles. Further examples include long span wings and wing-bodies at supercritical onflows, flows around propellers and rotors and finally some unsteady flows. The examples cited are selected topics from the extensive studies undertaken in the department of numerical aerodynamics of thedlr in Braunschweig in the course of the last few years.     

A hybrid variational Allen-Cahn/ALE scheme for the coupled analysis of two-phase fluid-structure interaction

We present a partitioned iterative formulation for the modeling of fluid-structure interaction (FSI) in two-phase flows. The variational formulation consists of a stable and robust integration of three blocks of differential equations, viz, an incompressible viscous fluid, a rigid or flexible structure, and a two-phase indicator field. The fluid-fluid interface between the two phases, which may have high density and viscosity ratios, is evolved by solving the conservative phase-field Allen-Cahn equation in the arbitrary Lagrangian-Eulerian coordinates. While the Navier-Stokes equations are solved by a stabilized Petrov-Galerkin method, the conservative Allen-Cahn phase-field equation is discretized by the positivity preserving variational scheme. Fully decoupled implicit solvers for the two-phase fluid and the structure are integrated by the nonlinear iterative force correction in a staggered partitioned manner and the generalized-α method is employed for the time marching. We assess the accuracy and stability of the phase-field/ALE variational formulation for two- and three-dimensional problems involving the dynamical interaction of rigid bodies with free surface. We consider the decay test problems of increasing complexity, namely, free translational heave decay of a circular cylinder and free rotation of a rectangular barge. Through numerical experiments, we show that the proposed formulation is stable and robust for high density ratios across fluid-fluid interface and for low structure-to-fluid mass ratio with strong added-mass effects. Overall, the proposed variational formulation produces results with high accuracy and compares well with available measurements and reference numerical data. Using unstructured meshes, we demonstrate the second-order temporal accuracy of the coupled phase-field/ALE method via decay test of a circular cylinder interacting with the free surface. Finally, we demonstrate the three-dimensional phase-field FSI formulation for a practical problem of internal two-phase flow in a flexible circular pipe subjected to vortex-induced vibrations due to external fluid flow.     

Wind turbine aerodynamics using ALE–VMS: validation and the role of weakly enforced boundary conditions

In this article we present a validation study involving the full-scale NREL Phase VI two-bladed wind turbine rotor. The ALE–VMS formulation of aerodynamics, based on the Navier–Stokes equations of incompressible flows, is employed in conjunction with weakly enforced essential boundary conditions. We find that the ALE–VMS formulation using linear tetrahedral finite elements is able to reproduce experimental data for the aerodynamic (low-speed shaft) torque and cross-section pressure distribution of the NREL Phase VI rotor. We also find that weak enforcement of essential boundary conditions is critical for obtaining accurate aerodynamics results on relatively coarse boundary layer meshes. The proposed numerical formulation is also successfully applied to the aerodynamics simulation of the NREL 5MW offshore baseline wind turbine rotor.     

Interaction of a transonic dislocation with subsonic dislocation and point defect clusters

The moving speeds of all observed dislocations in crystals are subsonic. There has been a view in the literature that the speed of subsonic dislocations can not be accelerated above the speed of sound because the energy required would be infinitely large. Recent molecular dynamics (MD) simulation had shown that it is possible to generate dislocations with an initial moving speed higher than the velocity of sound in solids. This raises a question: what will happen when a supersonic dislocation meets other defects along its moving path? This work reports the results of MD simulation on the interaction of a transonic dislocation with other subsonic dislocations as well as with point defect clusters. The results show that a vacancy cluster such as a void has an insignificant slow-down effect on the transonic dislocation, while a subsonic dislocation slows down the transonic dislocation to subsonic one. In some cases, the subsonic dislocation (or a subsonic part of a transonic dislocation) can overcome the traditional sound barrier.     

A panel method computation for oscillating aerofoil in compressible flow

A panel method using source and doublet singularity has been proposed to solve for subcritical aerodynamics of a two dimensional steady and unsteady aerofoil. The source singularities are placed on the aerofoil surface. The doublet singularity is distributed by a function along the chordline of the aerofoil; this distribution is further projected downstream into infinity. The aerodynamics of an oscillating aerofoil is investigated. The governing unsteady linearized potential equation has a Hankel function as its fundamental solution, which is a source type function. A combination of source and doublet singularity is therefore used for solving the unsteady compressible problem by means of the panel method, this methodology being an extension of a steady aerofoil formulation. Incremental effects of profile change in aerofoil and wake geometry are accounted for. A surface boundary condition is applied on the stationary mean aerofoil surface with time dependent geometrical changes accounted for. An unsteady Kutta condition of equal pressure across the trailing edge is assumed. Results are presented on the aerodynamic influence of Mach number, oscillating frequency parameter, angle of incidence and change of pivoting point. Results are also compared with linear theory, a subsonic experimental result and a subcritical solution of a transonic model.     

Studies in multigrid acceleration of some equations of interest in fluid dynamics

The paper describes the multigrid acceleration technique to compute numerical solutions of three equations of common fluid mechanical interest; Laplace equation, transonic full potential equation and Reynolds averaged Navier-Stokes equations. Starting with the simple and illustrative multigrid studies on the Laplace equation, the paper discusses its application to the cases of full potential equation and the Navier-Stokes equations. The paper also discusses some elements of multigrid strategies like V- and W-cycles, their relative efficiencies, the effect of number of grid levels on the convergence rate and the large CPU time saving obtained from the multigrid acceleration. A few computed cases of transonic flows past airfoils using the full potential equations and the Navier-Stokes equations are presented. A comparison of these results with the experimental data shows good agreement of pressure distribution and skin friction. With the greatly accelerated multigrid convergence, the full potential code typically takes about 10 seconds and the Navier-Stokes code for turbulent flows takes about 5 to 15 min of CPU time on the Convex 3820 computer on a mesh which resolves the flow quantities to good levels of accuracy. This low CPU time demand, made possible due to multigrid acceleration, on one hand, and the robustness and accuracy on the other, offers these codes as designer’s tools for evaluating the characteristics of the airfoils. Various parts of this paper have been presented at the following conferences; (i) 5th Asian Cong. on Fluid Mech., Taejon, Korea, 1992, (ii) Int. Conf. on Methods of Aerophysical Research, Novosibirsk, 1992, (iii) Fluid Dyn. Symp. in honour of Prof. R Narasimha on his 60th birthday, 1993.     

On boundary-layer transition in transonic flow

Boundary-layer transition in transonic external flow is addressed theoretically. The transonic area is rich in different flow structures, and transition paths, and the work has wide potential application in transonic aerodynamics, including special reference to the example of flow transition over an engine nacelle. The investigation is intended partly to aid, compare with, and detect any limitations of, a quasi-parallel empirical methodology for design use in the area, especially with respect to the transonic range, and partly to develop an understanding and possible control of the nonlinear natural or by-pass properties of the compressible transition present. The mechanisms behind three major factors, (a) substantial external-flow deceleration, (b) rapid boundary-layer thickening, (c) three-dimensional nonlinear interactions, are identified; these three are involved in the specific application above and in more general configurations, depending on the disturbance background present. It is found also that some similarities exist with the phenomenon of buffeting on transonic airfoils, and the relevant physics and governing equations throughout are identified. Sensitive nonlinear effects are important in all the factors (a)-(c), especially a resonance linkage between shock buffeting and boundary-layer thickening, and nonlinearly enhanced three-dimensional growth triggered by slight three-dimensional warping for instance, peculiar to the transonic range. The latter enhanced growth is perhaps the most significant finding. The implications, in the general setting as well as for the nacelle-flow context in particular, are also presented.     

Interplay of quantum size effect,anisotropy and surface stress shapes the instability of thin metal films

We show how the conformal mapping technique can be applied to analyse specific problems in the context of viscous gravity current theory. We examine the edge of steady thin planar viscous gravity currents in the presence of complex external low Reynolds flows. In addition to the uniform ambient flow we look at the case of viscous gravity currents spreading in positively strained flows and around cylindrical bodies. These external flows exert shear stress on the gravity current, which drives it in the streamwise direction. The idealised conditions are re-created in the laboratory using a Hele–Shaw cell with a point source on the bottom plate where the saline is introduced into the flow. The mapped laboratory results are compared to a known similarity solution and the agreement is good. We conclude by identifying a broad class of viscous gravity current problems where this technique may be applied.