Three-dimensional Euler Code Research Articles

Stability of Vortex Pairs over Slender Conical Bodies: Analysis and Numerical Computation

DOI: 10.2514/1.33498 Analytical studies and computational fluid dynamics simulations are presented to study the formation and stability of stationary symmetric and asymmetric vortex pairs over slender conical bodies in an inviscid incompressible flow at high angles of attack. The analytical method is based on an eigenvalue analysis on the motion of the vortices under small perturbations. A three-dimensional time-accurate Euler code is used to compute five typical flowsstudiedbythe analytical methodon extraordinarily finegrids withstrict convergence criteria. Both the theory and the computation show that the vortices over a delta wing are stable and those over a wing–body configuration at the low angle of attack are symmetric and stable, but become asymmetric and bistable at higher angles of attack; that is, the vortices shift to one of two stable mirror-imaged asymmetric configurations. The computational results agree well with the analytical predictions, demonstrating the existence of a global inviscid hydrodynamic instability mechanism responsible for the asymmetry of separation vortices over slender conical bodies.

Efficient Prediction of Helicopter BVI Noise under Different Conditions of Wake and Blade Deformation

Predictions of helicopter BVI noise using three-dimensional Euler code with a single blade grid are conducted under three different conditions: BVI noise caused by (1) interaction between rotating blades and vortex shed from fixed wing vortex generator, (2) interaction between rotating blades and tip vortices shed from preceding blades, and (3) interaction between rotating blades with elastic deformation and shed tip vortices. In the CFD calculation, the Field Velocity Approach (FVA) and Scully’s vortex model are used to import the wake information into the calculation grid and to determine the induced velocity made by tip vortices, respectively (cases 1–3). Beddoes generalized wake model is used to prescribe the tip vortices position in the wake (cases 2 and 3). Information about blade elastic deformation is imported from HART II project experimental data into the calculation (case 3). Acoustic analyses based on Ffowcs-Williams and Hawkings (FW-H) equation are conducted subsequently in each case. The results from the calculations show good agreement with experiments in all three cases, indicating that application of FVA, Scully’s model, and Beddoes generalized wake model is effective for BVI noise prediction in this study, which is intended for low calculation cost using a single blade grid. Also, use of blade elastic deformation data in the calculation shows marked improvement in calculation precision. Consequently, the method used in this study can predict BVI noise under various conditions of wake or blade deformation with acceptable precision and low calculation cost.

전산유체/전산구조 연계 방법을 사용한 항공기날개의 정적 공탄성 해석

A static aeroelastic analysis for supersonic aircraft wing equipped with external store under the wing lower surface is performed using computational fluid dynamics (CFO) and computational structural technology(CST) coupling methodology. Two mapping algorithms, which are the pressure mapping algorithm and the displacement mapping algorithm, are used for CFD/CST coupling. A three-dimensional unstructured Euler code and finite element analysis program are used to calculate the flow properties and the structural displacements, respectively. The coupling procedure is repeated in an iterative manner until a specified convergence criterion is satisfied. Static aeroelastic analysis for a typical supersonic flight wing is performed and final converged wing configuration is obtained after several iterations.

Flap-Deflection Optimization for Transonic Cruise Performance Improvement of Supersonic Transport Wing

Wing flap deflection angles of a supersonic transport are optimized to improve transonic cruise performance. For this end, a numerical optimization method is adopted using a three-dimensional unstructured Euler code and a discrete adjoint code. Deflection angles often flaps; five for leading edge and five for trailing edge, are employed as design variables. The elliptic equation method is adopted for the interior grid modification during the design process. Interior grid sensitivities are neglected for efficiency. Also tested is the validity of the approximate gradient evaluation method for the present design problem and found that it is applicable for leading edge flap design in cases of no shock waves on the wing surface. The BFGS method is used to minimize the drag with constraints on the lift and upper surface Mach numbers. Two design examples are conducted; one is leading edge flap design, and the other is simultaneous design of leading edge and trailing edge flaps. The latter gave a smaller drag than the former by about two counts.

Experimental and Numerical Investigation of the Flow in a Centrifugal Compressor Volute

This paper presents the experimental and numerical investigation of an outward volute of rectangular cross section. The investigation is carried out at the level of stage performance, volute performance, and detailed flow field study at selected peripheral positions for various operating points. The objective of the investigation was to gain further knowledge about the flow structure and loss mechanism in the volute. Simultaneously with the experimental investigation, a numerical simulation of the flow in the volute was carried out. A three-dimensional Euler code was used in which a wall friction term and a tuned artificial dissipation term account for viscous effects. A reasonable agreement between the experimental and numerical results is observed. As a result a good and detailed knowledge about the pressure recovery and loss mechanism in the volute is obtained. [S0889-504X(00)00301-9]

Wing-Tip Vortex Calculations Using a High-Accuracy Scheme

The objective of the research described in this paper is to investigate the use of computational aeroacoustics methodology and parallel computers to simulate the development of a tip vortex. A threedimensional Euler code, implemented using the message passing library and Fortran 90 on the IBM SP2, is used to perform calculations of the steady e ow. Results are presented for low-speed e ow over a e nite span NACA 0012 wing. Comparisons with experimental data show that the tip vortex is predicted to diffuse too rapidly, but the high-accuracy scheme does reduce numerical errors. It is argued that Navier ‐ Stokes calculations on a much e ner mesh would improve the solutions, but they would be prohibitively expensive for unsteady calculations.

Time-Accurate Euler Simulation of Interaction of Nozzle Wake and Secondary Flow With Rotor Blade in an Axial Turbine Stage Using Nonreflecting Boundary Conditions

The objective of this paper is to investigate the three-dimensional unsteady flow interactions in a turbomachine stage. A three-dimensional time-accurate Euler code has been developed using an explicit four-stage Runge–Kutta scheme. Three-dimensional unsteady nonreflecting boundary conditions are formulated at the inlet and the outlet of the computational domain to remove the spurious numerical reflections. The three-dimensional code is first validated for two-dimensional and three-dimensional cascades with harmonic vortical inlet distortions. The effectiveness of the nonreflecting boundary conditions is demonstrated. The unsteady Euler solver is then used to simulate the propagation of nozzle wake and secondary flow through the rotor and the resulting unsteady pressure field in an axial turbine stage. The three-dimensional and time-dependent propagation of nozzle wakes in the rotor blade row and the effects of nozzle secondary flow on the rotor unsteady surface pressure and passage flow field are studied. It was found that the unsteady flow field in the rotor is highly three dimensional and the nozzle secondary flow has significant contribution to the unsteady pressure on the blade surfaces. Even though the steady flow at the midspan is nearly two dimensional, the unsteady flow is three dimensional and the unsteady pressure distribution cannot be predicted by a two-dimensional analysis.

Quantitative comparison between interferometric measurements and Euler computations for supersonic cone flows

Dual-reference-beam, plane-wave digital holographic interferometry has been applied to obtain quantitative interferometric data in the three-dimensional supersonic flow over circular cones. The interferometric data are compared quantitatively on a two-dimensional grid with the postprocessed results of an Euler code that simulates three-dimensional inviscid compressible flows. The comparison involves two different combinations of cone angle and angle of incidence. The maximum deviations between the interferometric data and the numerical data are found to lie in the error interval [-2.0%, +2.0%]. O validate numerical codes for the simulation of threedimensional compressible flows, extensive experimental flowfield data are required. The nonintrusiveness and the high resolution of the data provided by a single measurement (512 x 512 data points) make digital holographic interferometry (DHI) in some respects a good alternative for conventional measuring techniques like pressure probe measurements. DHI is the optical measuring technique that combines holographic interferometry to visualize the flowfield in an interferogram, phase stepping of the interferogram and digital image processing to compute the phase map from the digitized interferograms. The phase map reflects the deformation of the wave fronts of a laser light beam that has passed through the flowfield inside a test section. The phase is proportional to the mean density over the width of the test section along the direction of light propagation. At the same time, this explains the inherent disadvantage of DHI. In three-dimensional flows it provides only quantitative information about the average density along light rays through the flowfield. In this paper the experimental data provided by DHI will be compared with the numerical data obtained by a computer code for inviscid compressible flow (three-dimensional Euler code). The supersonic flow around circular cones will serve as the test flowfield in the comparison. Several good reasons exist for choosing this flowfield as the test flowfield. First, if the bow shock is attached, it represents a relatively simple flowfield because of its conical symmetry. In this case no dominant viscous interaction effects are present and the assumption of an inviscid fluid holds. Furthermore, extensive results of inviscid flow computations are available for several Mach numbers M^, angles of incidence a, and cone half-angles Oc.l~3 Finally, the applicability of holographic interferometry in the investigation of the supersonic flow around circular cones has been successfully demonstrated before.4-5

Three-Dimensional Euler Simulations of Flow in Hydraulic Turbine Runners. Study on Simulation Results for Blade Surface Pressures of Francis Runner.

In order to study the applicability of flow simulations by using the three-dimensional Euler code with pseudo-compressibility, flow simulations of a Francis runner with a high specific speed were carried out and results were compared with those of experiments on the pressure distributions on blade surfaces. As a result, it is concluded that the simulation results obtained using this code indicate a good agreement with the experimental ones, and this code is useful for predicting the flow in a Francis runner.

Numerical solution of Euler equations for aerofoils in arbitrary unsteady motion

AbstractThis paper is part of a DLR research programme to develop a three-dimensional Euler code for the calculation of unsteady flow fields around helicopter rotors in forward flight. The present research provides a code for the solution of Euler equations around aerofoils in arbitrary unsteady motion. The aerofoil is considered rigid in motion, and an O-grid system fixed to the moving aerofoil is generated once for all flow cases. Jameson's finite volume method using Runge-Kutta time stepping schemes to solve Euler equations for steady flow is extended to unsteady flow. The essential steps of this paper are the determination of inviscid governing equations in integral form for the control volume varying with time in general, and its application to the case in which the control volume is rigid with motion. The implementation of an implicit residual averaging with variable coefficients allows the CFL number to be increased to about 60. The general description of the code, which includes the discussions of grid system, grid fineness, farfield distance, artificial dissipation, and CFL number, is given. Code validation is investigated by comparing results with those of other numerical methods, as well as with experimental results of an Onera two-bladed rotor in non-lifting flight. Some numerical examples other than periodic motion, such as angle-of-attack variation, Mach number variation, and development of pitching oscillation from steady state, are given in this paper.

Sensitivity derivatives for three-dimensional supersonic Euler code using incremental iterative strategy

In a recent work, an incremental strategy was proposed to iteratively solve the very large systems of linear equations that are required to obtain quasianalytical sensitivity derivatives from advanced computational fluid dynamics (CFD) codes. The technique was sucessfully demonstrated for two large two-dimensional problems: a subsonic and a transonic airfoil. The principal feature of this incremental iterative stategy is that it allows the use of the identical approximate coefficient matrix operator and algorithm to solve the nonlinear flow and the linear sensitivity equations; at convergence, the accuracy of the sensitivity derivatives is not compromised. This feature allows a comparatively straightforward extension of the methodology to three-dimensional problems; this extension is successfully demonstrated in the present study for a space-marching solution of the three-dimensional Euler equations over a Mach 2.4 blended wing-body configuration.

Multigrid Euler calculations over complete aircraft

A three-dimensional Euler code is applied to a series of configurations of increasing complexity. Comparisons are made to experiments, or to other computations when the former is not available. The method uses the multigrid approach on sets of equally spaced Cartesian grid cells. A unique and robust implementation of the body surface boundary condition on grid cells not aligned with the surface provides accurate results on relatively coarse grids. All computational results are compared to experimental data with good agreement exhibited over a wide range of flight conditions. The solution accuracy is assessed for the Onera M-6 wing, with errors due to grid resolution of less than 1% being achieved for the lift and pitching moment coefficients.

Numerical and experimental investigation of airframe-integrated inlet for high velocities

Airframe-integrated, three-dimensional fixed geometry inlets for hypersonic Mach numbers are under consideration. The numerical investigation is conducted using the three-dimensional Euler's code. The program calculates local and integral flowfield performances in duct taking into account displacement thickness along the wetted surface on the condition that the flow is nonseparated and the gas is in thermodynamic equilibrium. Sweep sidewall compression inlets with and without struts and transformed types of three-dimensi onal inlets have been examined. The results of flowfield calculations at the exit section and along the duct as a function of Mach number M are given. Two models of three-dimensi onal inlets with and without struts have been tested at A/0, = 2-6. The results of numerical and experimental investigations are compared. Nomenclature Ac = inlet frontal area AN = inlet passage area normal to axis Ox AT = area of captured stream tube at the inlet throat A T = relative throat area, = A T/A c

A three-dimensional Euler code for calculating flow fields in centrifugal compressor diffusers

In order to avoid large computer running times and storage requirements three-dimensional flow fields in turbomachines have been approximated by applying a general theory of Wu [1] (1952) up to the present day. Nowadays considerably more computer capacity is available. Therefore, the handling of more complex flow problems has become possible with a reasonable effort. In this paper a full 3-D code will be discussed which replaces the earlier quasi-3-D approach given in Ref. [2]. The code has been implemented on a conventional CYBER 990 and a CRAY-XMP. The running times of both computers will be compared to measure the effect of vectorization of the present FORTRAN code. Here the results obtained with the new 3-D time marching procedure are compared with those obtained with the earlier method. The present code proves to be a very efficient tool for solving 3-D transonic flow fields in high-pressure-ratio centrifugal compressor diffusers.

Secondary Flows in a Transonic Cascade: Comparison Between Experimental and Numerical Results

The paper presents a comparison between numerical results and experimental data about the secondary flow development in a linear transonic turbine cascade. Computations are carried out by using a three-dimensional inviscid Euler code, based on a Runge-Kutta explicit finite volume method. The experimental inlet total pressure distribution is imposed as inlet boundary condition to simulate the incoming endwall boundary layer. The comparison is made in four planes downstream of the cascade where detailed experimental data obtained in a transonic wind tunnel are available. For each of these planes secondary velocities and streamwise vorticity contour plots are presented and discussed. Moreover pitchwise mass averaged flow angle distributions showing overturning and underturning regions are shown. The comparison shows that an Euler code can predict the essential features of secondary flow phenomena like passage vortex location and intensity but a certain disagreement is found in the overturning and underturning angles evaluation. Numerical results also allow for the investigation of the development of secondary flows inside the blade channel. The investigation is carried out for three different Mach numbers: M2is = 0.5, 1.02, 1.38, in order to show the influence of compressibility on the flow vortex structure.

Application of Advanced Computational Codes in the Design of an Experiment for a Supersonic Throughflow Fan Rotor

Increased emphasis on sustained supersonic or hypersonic cruise has revived interest in the supersonic throughflow fan as a possible component in advanced propulsion systems. Use of a fan that can operate with a supersonic inlet axial Mach number is attractive from the standpoint of reducing the inlet losses incurred in diffusing the flow from a supersonic flight Mach number to a subsonic one at the fan face. The data base for components of this type is practically nonexistent; therefore, in order to furnish the required information for assessment of this type fan, a program has been initiated at the NASA Lewis Research Center to design, build, and test a fan rotor that operates with supersonic axial velocities from inlet to exit. This paper describes the design of the experiment using advanced computational codes to calculate the unique components required. The fan rotor has constant hub and tip radii and was designed for a pressure ratio of 2.7 with a tip speed of 457 m/s. The rotor was designed using existing turbomachinery design and analysis codes modified to handle fully supersonic axial flow through the rotor. A two-dimensional axisymmetric throughflow design code plus a blade element code were used to generate fan rotor velocity diagrams and blade shapes. A quasi-three-dimensional, thin shear layer Navier–Stokes code was used to assess the performance of the fan rotor blade shapes. The final design was stacked and checked for three-dimensional effects using a three-dimensional Euler code interactively coupled with a two-dimensional boundary layer code. A translating nozzle was designed to produce a uniform flow parallel to the fan up to the design axial Mach number of 2.0. The nozzle was designed with the three-dimensional Euler/interactive boundary layer code. The nozzle design in the expansion region was analyzed with a three-dimensional parabolized viscous code, which corroborated the results from the Euler code. A translating supersonic diffuser was designed using these same codes.

High-speed propeller noise predictions - Effects of boundary conditions used in blade loading calculations

The acoustics of an advanced single rotation SR-3 propeller at cruise conditions are studied employing a time-domain approach. The study evaluates the acoustic significance of the differences in blade pressures computed using nonreflecting rather than hard wall boundary conditions in the three-dimensional Euler code solution. The directivities of the harmonics of the blade passing frequency tone and the effects of chordwise loading on tone directivity are examined. The results show that the maximum difference in the computed sound pressure levels due to the use of blade pressure distributions obtained with the nonreflecting rather than the hard wall boundary conditions is about 1.5 dB. The blade passing frequency tone directivity obtained in the present study shows good agreement with jetstar flight data.

Euler analysis of transonic propeller flows

A three-dimensional Euler code, previously developed for high-speed propellers, was rewritten and vectorized for the Cray computer. First-order accurate, implicit integration was combined with approximate factorization, central differences, and constant coefficient dissipation to solve the unsteady, compressible Euler equations in general curvilinear coordinates. Calculations were compared with previous results and measurements. The new code required less memory and was more than two times faster than its predecessor. New boundary conditions produced more accurate results in terms of integrated performance characteristics. The code is a useful tool for examining the fluid dynamics and performance of propellers.

Three-Dimensional Inviscid Computation of an Impeller Flow

The flow in a centrifugal impeller is analyzed by a quasi-three-dimensional streamline curvature method, by a three-dimensional Euler code, and by a three-dimensional finite element potential flow method. Comparison with Eckardt’s published data for a backswept impeller shows that full three-dimensional methods better predict the loading at the hub and shroud.