Studies on the interference of wings and propeller slipstreams

The propeller slipstream is numerically simulated by solving three-dimensional unsteady Euler equations. The method of unstructured overset grids is applied to the simulation of relative motion between the propeller and airframe. The rotationalmotion of the propeller is simulated by establishing a rotational subzone, and the airframe is contained in the stationary subzone. During the unsteady simulation with the rotational subzone rotating, there is always an overlapping area between the rotational and stationary subzones, and the overlapping area is used to convect the flow variables of the two subzones. The governing equations are solved by using the dual-time method and standard explicit four-stage Runge-Kutta method. The unsteady flowfields of different working conditions are calculated, and the influence of propeller slipstream is analyzed. Some calculating results are compared with experimental data, and they have a good agreement with each other. It is demonstrated that the present method is efficient and robust for the simulation of an unsteady propeller slipstream flow.

Given the greater efficiency of algorithms, coupled with ever-faster computers with increased memory capacity, it is becoming more practical to calculate flow over complex configurations using direct solutions of the differential equations (Euler equations for inviscid flows, Navier-Stokes for viscous flows) representing mass, momentum and energy conservation laws. The solutions must comply with the appropriate initial and boundary conditions. The availability of large vector computers has enabled researchers to attempt the modeling of increasingly more complex geometries and the calculation of flow at higher angles of attack by solving Euler or Navier-Stokes equations. The recent development of parallel computers may facilitate further increases in the speed and complexity of such calculations. This chapter will be devoted to the discussion of the methods of solving Euler equations and their application to studies of flows at high angles of attack. The solutions of the Navier-Stokes equations and their application to computations of high angles of attack flows will be discussed in Chapter 9.

Solution of the Euler equations on Cartesian grids

A numerical procedure has been developed for the computation of the steady, inviscid supersonic flow over aircraft configurations. The technique accounts for major nonlinear effects (shock waves, blunt leading edges) at low to moderate supersonic speeds. A fully implicit marching technique for the full potential equation is utilized in a stereographically projected, conformally mapped, spherical coordinate frame. Cross-flow planes are efficiently solved by type-dependen t relaxation techniques. Results are presented for several delta wing configurations and bodies of revolution, and are compared with existing experimental data, Euler's equations solutions, and results from linearized theories (panel methods). I. Introduction T HE goal of the present investigation is the development of a highly efficient and general numerical procedure for the computation of the steady, inviscid supersonic flow over aircraft configurations. In particular, this paper will concentrate on the development and verification of the method through application to isolated delta wings and bodies of revolution. Emphasis will be given to the prediction of nonlinear effects at imbedded shock waves and blunt leading edges in the low to moderate supersonic Mach number range. Existing computational techniques in this regime may be broadly characterized as panel methods and finite-difference methods. Panel methods, which have been demonstrated to be widely versatile and quite efficient, have reached a very advanced stage of development and utility for subsonic flows. Efforts are continuing to improve their capabilities at supersonic speeds (i.e., Ehlers et al. *). These methods address the solution to the first-order linear small-disturbance equation with various approximations for the boundary conditions. Finite-difference techniques for wing planforms in the supersonic regime concentrate on solutions to Euler's equations with differing treatments of shock waves, shock capturing2 or shock fitting,3'5 and a variety of methods for grid generation (i.e., Moretti5) and geometry input. The major drawbacks of these methods are associated with their large computational times which are due primarily to the fine grids necessary to resolve thin wings and the accompanying strict stability requirements. Other problem areas for the Euler solutions in the Mach number range considered here are due to subsonic Mach numbers in the axial marching direction, initial conditions at the apex of the wing or body, and vortical singularities of the leeward surface. In a recent paper by Mason and daForno6 the opportunities for performance gains through nonlinear aerodynamic effects are discussed including the impact on nonlinear computational analysis and design. A convincing case is made for the necessity of the inclusion of nonlinear effects particularly for wing leading edges and imbedded shock waves for the aerodynamic design and analysis of wings.

A linearized Euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and blade motion is presented. Using the linearized Euler technique, one decomposes the flow into a mean (or steady) flow plus an unsteady, harmonically varying, small disturbance flow. Linear variable coefficient equations describe the small disturbance behavior of the flow, and are solved using a pseudo-time marching Lax-Wendroff scheme. For the blade motion problem, a harmonically deforming computational rid that conforms to the motion of vibrating blades eliminates large error producing mean flow gradient terms that would otherwise appear in the unsteady flow tangency boundary condition. The paper also presents a new, numerically exact, nonreflecting far-field boundary condition based on an eigenanalysis of the discretized equations. Computed flow solutions demonstrate the computational accuracy and efficiency of the present method. The solution of the linearized Euler equations requires one to two orders of magnitude less computer time than solution of the nonlinear Euler equations using traditional time-accurate time-marching techniques. In addition, the deformable grid significantly improves the accuracy of the solution.

Aerodynamic characteristics of an airfoil, NACA 6409, flying over a wavy wall is investigated numerically. An Euler code based on the LU-factored algo rithm and higher-order upwind scheme is constructed and its accuracy is tested on two benchmark problems : NACA 4412 airfoil moving over a level ground and NACA 0012 airfoil in free-flight pitching oscillation. The calculated flow about NACA 6409 airfoil over the wavy ground represented by a moving sine function indicates that the aerodynamic property of the airfoil becomes sensitive if the wave number or amplitude of the wavy ground is increased, and/or if the proximity of the airfoil to the ground is smaller. Introduction When an aircraft flies near the ground, the aerodynamic performance of the wing is much changed from that of the free flight. In particular, a wing in proximity to the ground manifests reduced upwash, downwash, and tip vortices which cause enhancement of lift and pitching moment, and reduction of induced drag. These effects, in overall, are called ‘the ground effect’ whereas the wing taking advantage of these effects is called ‘wing in the ground effect’ or WIG, in short. There has been recently considerable interest in development of WIG crafts in such countries as Russia[1,2], Japan[3,4], Germany[5,6], and China[7] due to its energy saving feature as a means of passenger and cargo transportation. Preliminary design of a WIG airplane is under progress in South Korea[8] and the present research is motivated by this particular program. The gain in the lift-todrag ratio might be achieved by a WIG craft at the cost of stability due to the increased pitching moment. The flight properties of a WIG hence need to be investigated thoroughly during the aircraft development process. Recently, a few reports have appeared on the aerodynamic performance of WIG wings and airfoils[4-7,9] flying over a level *Doctoral candidate, Dept. of Aerospace Engineering. tprofessor, Dept. of Aerospace Engineering, Member AIAA. Copyright @ 2000 by YEHOO” Im. Kcun-Shik Chmg. Publidld by the Americ-n Institute of Aeronautica md Aatronautica. Inc. with permhion. ground. NACA Cdigit airfoils are, in general, known to have positive ground effect. However, a symmetric airfoil with large thickness, NACA 0012, for example, can exhibit negative ground effect at a small angle of attack[4,7]. The WIG craft operated over the sea is quite often expected to face with rough wavy surfaces. This rather periodic terrain will cause ground effect different from that of the level ground. In the literature, the unsteady flow past a twodimensional airfoil moving over a wavy ground has been investigated with the lifting surface theory by Ando et al.[lO], and with the unsteady panel method by Morishita and Ashihara[ll]. Mizutani and Suzuki[l2] used Rankine source and boundary element method to compute the wing aerodynamics over the coastal free surface. Their results have certainly offered some useful data for fundamental study of the lifting airfoils. However, the solutions of Euler and Navier-Stokes equations are still wanted in order to study the possible compressibility and viscosity effect of the WIG craft. In this paper we have numerically solved the Euler equations with the LU-factored algorithm[l3] and high-resolution upwind scheme for the unsteady WIG airfoil moving over a wavy wall. Fortunately, Mizutani and Suzuki[l2] have found that deformation of the free surface caused by the proximate flight of a WIG craft is only negligible. We verified accuracy of the present computer code by reproducing two earlier results : NACA 4412 airfoil over a level ground[l4], and the NACA 0012 airfoil in pitching oscillation[l5]. Various parameter effects on NACA 6409 airfoil flying over a wavy wall with a subsonic flight Mach number, M, = 0.3, is elaborated in this paper. Governing Equations We consider unsteady twodimensional Euler equations in the computational domain (t, 5, n), 1 OF 8 aQ ai aF z+-+-=O ac 877 with the flux vectors (1) AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2000-0657

We present a 2D high‐order and easily accessible immersed‐boundary adaptive harmonic polynomial cell (IB‐AHPC) method to solve fully nonlinear wave‐structure interaction problems in marine hydrodynamics using potential‐flow theory. To reduce the total number of cells without losing accuracy, adaptive quad‐tree cell refinements are employed close to the free‐surface and structure boundaries. The present method is simpler to implement than the existing IB‐HPC alternatives, in that it uses standard square cells both in the fluid domain and at the boundaries, thus without having to use the more complex and expensive overlapping grids or irregular cells. The spurious force oscillations on moving structures, which is a known issue for immersed boundary methods (IBMs), are eliminated in this study by solving a separate boundary value problem (BVP) for a Lagrangian acceleration potential. We also demonstrate that solving a similar BVP for the corresponding Eulerian acceleration potential is far less satisfactory due to the involved second derivatives of the velocity potential in the body‐boundary condition, which are very difficult to calculate accurately in an IBM‐based approach. In addition, we present, perhaps for the first time since the HPC method was developed, a linear matrix‐based stability analysis for the time‐domain IB‐AHPC method. The stability analysis is also used in this study as a general guide to design robust and stable numerical algorithms, in particular related to the treatment of boundary conditions close to the intersection between a Dirichlet and a Neumann boundary, which is essential in time‐domain wave‐structure interaction analyses using IBMs. We confirm theoretically through the stability analysis that square cells have the best stability properties. The present method has been verified and validated satisfactorily by various cases in marine hydrodynamics, including a moving structure in an infinite fluid, fully nonlinear wave generation and propagation, and fully nonlinear diffraction and radiation of a ship cross section.

The unsuitability of replacing the solutions of the Navier-Stokes equations by the numerical solution of Euler's equations

The problem of unsteady wake-airfoil interaction and wake transport is of importance in nearly all fluid mechanical devices involving rotating machinery. In this study, an inviscid airfoil-to-airfoil potential flow analysis is used to predict the transport of the wakes of upstream stators through both rotating compressor and turbine airfoil passages. In the analysis, the wakes are assumed to have an infinitesimal velocity defect and are represented by inviscid flow streamlines. The dependence of wake rotation on airfoil lift and the dependence of wake distortion on airfoil geometry are both shown. Flow visualization results obtained in both a large-scale compressor and a large-scale turbine show good comparison with the predictions, of the inviscid model.

We prove uniqueness and existence of the weak solutions of Euler equations with helical symmetry, with initial vorticity in $L^{\infty}$ under “no vorticity stretching” geometric constraint. Our article follows the argument of the seminal work of Yudovich. We adjust the argument to resolve the difficulties which are specific to the helical symmetry.

ABSTRACTPhase transition in compressible flows involves capillarity effects, described by the Euler–Korteweg (EK) equations with nonconvex equation of state. Far from phase transition, that is, in the two convex parts of the equation of state, the dispersion terms vanish and one should recover the hyperbolic Euler equations of fluid dynamics. However, the solution of EK equations does not converge toward the solution of Euler equations when dispersion tends toward zero while being nonnull: it is a singular limit problem. To avoid this issue in the case of convex equation of state, a Navier–Stokes–Korteweg (NSK) model is considered, whose viscosity is chosen to counterbalance exactly the dispersive terms. In the limit of small viscosity and small dispersion, the Euler model is recovered. Numerically, an extended Lagrangian method is used to integrate the NSK equations so obtained. Doing so allows to use classical numerical schemes of Godunov type with source term. Numerical results for a Riemann problem illustrate the convergence properties with vanishing dispersion.

The 3D incompressible Euler equations with initial data characterized by uniformly large vorticity are investigated. We prove existence on long time intervals of regular solutions to the 3D incompressible Euler equations for a class of large initial data in bounded cylindrical domains. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach is based on fast singular oscillating limits, nonlinear averaging and cancellation of oscillations in the nonlinear interactions for the vorticity field. With nonlinear averaging methods in the context of almost periodic functions, resonance conditions and a nonstandard small divisor problem, we obtain fully 3D limit resonant Euler equations. We establish the global regularity of the latter without any restriction on the size of 3D initial data and bootstrap this into the regularity on arbitrary large time intervals of the solutions of 3D Euler equations with weakly aligned uniformly large vorticity at t = 0.

Thermodynamic and Gasdynamic Aspects of a Boiling Liquid Expanding Vapour Explosion

A number of papers[1–9] have appeared in recent years reporting numerical solutions of the Euler equations for flows past sharp edge delta wings with leading edge vortices. A schematic diagram of the problem under consideration is shown in Figure 1. Solutions have been compared with experiment [1,8], with incompressible panel method calculations [6], with other Euler equation solutions [8,5], and with Navier-Stokes calculations [9]. Although one cannot conclude that these comparisons show complete agreement, it is reasonable to say that they establish that finite volume calculations of the Euler equations appear to give reasonable predictions of the primary vortex generated by this configuration. A point of some concern in these papers [1,3,4] relates to the mechanism by which the Euler equations cause the flow to separate from the body and form a strong vortex which has associated with it a significant total pressure loss.

A stable moving-particle semi-implicit method for free surface flows