Three-dimensional Inverse Method Research Articles

A quantitative three-dimensional inverse design rule based on cross-flow control of profiled end wall

Profiled end wall provides a novel and effective solution for end wall flow control in turbines and compressors. However, the application of profiled end walls in compressors still lacks quantitative design rules. This paper presents a quantitative design rule for end wall profiling based on a three-dimensional inverse method and numerically validates it on a highly loaded compressor cascade. The current inverse method can solve the corresponding end wall shape from a given end wall pressure distribution, but the determination of the end wall pressure distribution heavily relies on empirical knowledge. This study establishes a model between streamline curvature and cross-passage pressure gradient (CPG) through the circumferential equilibrium equation in the S1 stream surface, thereby providing a quantitative basis for determining the end wall pressure distribution. The quantitative design rule proposed in this paper is expressed as follows: at the axial position where the separation begins on the suction surface (SS), within the range of 0.1-0.2 pitch away from the SS, the end wall boundary layer fluid with a higher velocity than the corner region average velocity should possess the same streamline curvature as the fluid within the viscous sublayer. The inverse-designed profiled end wall using the quantitative design rule enhances the local cross-flow near the SS by imposing a stronger CPG, thus encouraging the end wall boundary layer fluid with relatively higher momentum to arrive at the SS earlier and enhancing the radial migration on the SS. Consequently, the intensified cross-flow entrains relatively higher momentum into the corner region, while the enhanced radial migration drives the low-momentum fluid away from the corner region towards the midspan. Finally, the inverse-designed profiled end wall reduces the half-span mass-flow-weighted average loss by 4.3%.

A three-dimensional inverse method based on moving no-slip boundary for profiled end wall design

How to utilize existing flow control mechanisms to make profiled end wall design more flexible, efficient, and physical is a meaningful challenge. This study presents a three-dimensional inverse method for profiled end wall design to achieve the application of flow control mechanisms. The predetermined pressure distribution on the end wall is reached by modifying the end wall geometry during flow field calculation. A motion velocity model is derived from the normal momentum equation of the moving no-slip boundary to modify the end wall geometry. A Reynolds-Averaged Navier-Stokes (RANS) solver based on the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm is adopted to simulate the flow field. Based on the mechanism understanding obtained through numerical optimization results, this study adopts the inverse method to redesign an optimized end wall in a compressor cascade. The results indicate that the redesigned end wall exhibits better loss reduction, reducing the overall total pressure loss by 5.5%, whereas the optimized end wall reduces it by 3%. The inverse method allows the imposition of desired influences on the end wall flow without constructing a database, making it highly flexible, efficient, and physical.

A Novel Three-Dimensional Inverse Method for Axial Compressor Blade Surface Design

A novel three-dimensional inverse method based on the time-accurate solution of Navier–Stokes equations for axial compressor design is proposed in this work. The main novelty lies in the derivation of an inverse design boundary condition established on the conservation of Riemann invariant in order to directly design the blade surface. Specifically, a dynamic mesh technique is employed to update the grids and reduce the computational costs. In addition, some restrictions are imposed on the blade surface movement in order to avoid unrealistic airfoil profiles and guarantee computational robustness. Two redesign procedures are presented, including shock wave strength restrain for the NASA Rotor 37 stage and integrated controlled diffusion airfoil concept design for the Stage 35. Results indicate that this novel inverse method is effective for detailed axial compressor design.

A practical study of joints in three-dimensional Inverse Receptance Coupling Substructure Analysis method in a horizontal milling machine

Abstract Machine tool components are connected by joints that provide relative motion between contact interfaces. These joints add flexibility to the load path from the tool tip to the foundation and, therefore, the correct determination of joint properties becomes a crucial step if the whole machine performance is to be analyzed. A well suited experimental method to characterize these properties is the Inverse Receptance Coupling Substructure Analysis (IRCSA). In the existing literature, this technique has been widely applied to characterize single two-dimensional connections between two parts, where few measured FRFs are necessary in the implementation. However, a more complex multi-body structure with several joints demands a three-dimensional formulation, where the large number of required FRFs limits the application of IRCSA due to the inherent ill-conditioning of the inverse formulation. The work presented in this paper deals with the mentioned limitation, developing a novel IRCSA approach to characterize completely the linear guideways of an industrial machine tool. To avoid the use of excessive FRFs in the formulation, a procedure to study each joint independently is developed, where an appropriate number of connection points in the contact areas between the rails and the carriages is determined. This way, well-conditioning of the formulation is guaranteed while the required precision is preserved. Furthermore, the presented procedure requires only the measurement of translational degrees of freedom (TDOF), which avoids the need to measure rotational degrees of freedom (RDOF). Comparison of numerically calculated and experimentally measured results has demonstrated the validity of the proposed procedure.

Design method of internal waverider inlet under non-uniform upstream for inlet/forebody integration

A novel Bump-integrated three-dimensional internal waverider inlet (IWI) design method is presented for high-speed inlet/forebody integration. The low-kinetic-energy (boundary layer) flow generated by a blunted leading-edge and forebody boundary layer represents an extreme challenge in the integration of aircraft forebody and inlet. In this method, such an inlet's flowpath is divided into the entrance shockwave segment, the isentropic compression segment and the isolator. First, a three-dimensional inverse method of characteristics (3D-IMOC) is developed to obtain a compression surface that can generate a requested entrance shock wave in non-uniform upstream flow. This configuration realizes the integration of IWI and aircraft fuselage by incorporating a Bump to remove most of the boundary layer flow. This is followed by a three-dimensional, isentropic compression flow-path with cross sectional areas conforming to the specified Mach number distribution. Finally, a new three-dimensional Bump-integrated IWI was tested in M = 6 wind tunnel, under a rather thick boundary layer upstream flow (37% height of inlet entrance). Both of the experimental data and numerical simulation results show that, the new method of IWI and Bump can overcome serious boundary layer flow problems and improve the inlet performance.

Three-dimensional inverse method for aerodynamic optimization in compressor

Design experience plays an important role in compressor design. Accumulated design experience is used to reduce the number of simulations and to make time for the whole optimization process to be compatible with industrial standards. However, the major drawbacks of this design strategy are that the design result depends on talented designers with rich design experience and this method does not easily produce better configurations than existing designs. These drawbacks are related to the parametric description of the blade, which is conventionally performed using only geometric parameters. A good solution to this problem is to use a blade parametrization based on an inverse design method. Inverse design methods have been widely used for the design of various kinds of turbomachines, proving that it is a valuable alternative to the iterative use of direct methods. One main design parameter in the inverse design approach is the blade loading on both the hub and the shroud along the meridional direction. The blade loading distributions have a more direct relationship to the aerodynamic performance because they influence the flow field in a more straight-forward way. Fewer design parameters are then required to describe the blade shape than a purely geometric expression of the blade. Therefore, an optimization design method using the inverse method to parameterize the blade geometry can reduce the overall optimization time. The optimization design process then gives the optimal blade loading distributions, instead of the optimal combination of the geometric parameters. This is a more general result which can be applied to similar design problems without repeating the optimization process.

Temperature and heat flux fast estimation during rolling process

Monitoring and controlling flatness during the rolling process becomes critical for ensuring the product quality. Flatness defects are due to highly three-dimensional phenomena. Indeed, strips with different widths are rolled during the same campaign and cooling systems are heterogeneous along the axial direction to modify the thermal expansion of the roll. Therefore this paper presents a fully three-dimensional inverse analytical method to determine the temperature field and heat fluxes (especially at the surface of the roll) by interpreting measurements of temperature done with several thermocouples fully embedded in the roll body and aligned along the axial direction. Since the method is dedicated to on-line interpretation and designed as a tool for adapting the rolling parameters during the rolling process, iterative methods are not studied to avoid long computation times, which justifies the development of an analytical solution of the problem. The computation time displayed by Scilab 5.3 with a quadcore 2.8 GHz is around 0.5 s by cycle for accurate computation and 0.07 s by cycle for rough computation. This paper improves a previous work (2D and relying on four assumptions designed for the prediction of wear). In the present contribution the 3D unsteady heat equation of the rotating roll is solved analytically with only one assumption in order to deal with the restriction of the measurement system (i.e., measurement according to successive times). Therefore not only radial and tangential heat fluxes are taken into account but also axial heat flux. The solution is validated by comparing the outputs of the method and some prescribed analytical temperature fields. Good agreement is obtained. Noise sensitivity is estimated by adding artificial random numbers to the inputs, and good accuracy is observed. Moreover sensitivity to sensor depth is estimated and demonstrated to be not compromising.

Evaluation of contact stress during rolling process, by three dimensional analytical inverse method

A fully analytical three-dimensional inverse method has been developed in order to evaluate contact stress in the roll bite during rolling process. Stress measurements can be done by inserting fibre optics inside the roll body, (however no real measurements were available and simulations have been used instead). The inverse method takes as inputs three independent measured (or simulated) components of the stress tensor under the surface of the roll, and evaluates surface tractions of the roll especially in the contact in the roll gap. Stress, deformation and displacements can be obtained in the whole roll as well. This approach uses the theory of 3D isotropic elasticity and relies on displacement harmonic and bi-harmonic potentials expanded into a double Fourier series along the circumferential and axial directions. The identification of the solution involves matrices of size (3×3) computed off-line. This simple solution involves mainly the on-line computation of Fast Fourier Transform (fft) of three inputs, which takes 0.5s for each (processor 2.8GHz, time displayed by Scilab 5.3). Good accuracy is obtained, and the number of sensors along the axial direction is studied. Thus, this work encourages the development of an on-line industrial tool.

A three-dimensional inverse finite-element method applied to experimental eddy-current imaging data

Eddy-current techniques can be used to create electrical conductivity mapping of an object. The eddy-current imaging system in this paper is a magnetic induction tomography (MIT) system. MIT images the electrical conductivity of the target based on impedance measurements from pairs of excitation and detection coils. The inverse problem here is ill-posed and nonlinear. Current state-of-the-art image reconstruction methods in MIT are generally based on linear algorithms. In this paper, a regularized Gauss-Newton scheme has been implemented based on an edge finite-element forward solver and an efficient formula for the Jacobian matrix. Applications of Tikhonov and total variation regularization have been studied. Results are presented from experimental data collected from a newly developed MIT system. The paper also presents further progress in using an MIT system for molten metal flow visualization in continuous casting by applying the proposed algorithm in a real experiment in a continuous casting pilot plant of Corus RD&T, Teesside Technology Centre.

A three-dimensional inverse method using navier—stokes equations for turbomachinery blading

A new numerical method for solving fully three-dimensional inverse shape design problem of turbomachinery blading has been developed. The general inverse problem refers to the problem in which the pressure distributions on suction and pressure surfaces of blade are given, but the corresponding blade profile is unknown. In this paper, the calculations are based on the 3D Navier-Stokes equations expressed in terms of nonorthogonal curvilinear coordinates and corresponding nonorthogonal velocity components, and the explicit time marching algorithm and Baldwin-Lomax turbulence model are adopted. A special treatment for boundary conditions on blade surfaces is employed to satisfy the given pressure distribution. In computational process, an initial blade profile is supposed at starting, and then the blade surfaces will move regularly with time steps in the time marching process until the convergence is reached. The movement velocities at every point of blade surfaces are obtained from the solution of the Navier-Stokes equations. After each revision of the blade profile, the grid is reconstructed, and the aerodynamic parameters need to be transferred between the old and new grid points by an accurate interpolation method. Thus the viscous inverse problem is solved in a new process. The computational results for two test cases indicate that the method presented in this paper is very effective.

A new compressor and turbine blade design method based on three-dimensional euler computations with moving boundaries

A new three-dimensional inverse method for the design of compressor and turbine blades is presented. The method solves the time dependent Euler equations in a numerical domain of which some boundaries (the blade walls) are iteratively modified until a prescribed pressure distribution is reached. Each iteration of the procedure starts with a blade modification, based on the transpiration model and the permeable wall concept. After generating a new mesh the flow field is updated by performing one finite volume time iteration, taking into account the mesh points movement during the time stepping. The blade modifications and time marching computation converge simultaneously to the required geometry and steady-state flow solution. The method makes use of a high resolution three-dimensional finite volume Euler solver, with an upwind-biased evaluation of the advective fluxes for sharp shock wave capturing and low numerical entropy generation. The wall boundary conditions respect the hyperbolic character of the time dependent Euler equations. Non-reflecting boundary conditions are applied along the inlet and outlet boundaries, to favor convergence and limit errors due to shock wave reflection. Applications to turbine and compressor blade design illustrate the capabilities of the method and demonstrate its industrial applicability.

Practical Use of Three-Dimensional Inverse Method for Compressor Blade Design

The practical utility of a three-dimensional inverse viscous method is demonstrated by carrying out a design modification of a first-stage rotor in an industrial compressor. In this design modification study, the goal is to improve the efficiency of the original blade while retaining its overall aerodynamic, structural, and manufacturing characteristics. By employing a simple modification to the blade pressure loading distribution (which is the prescribed flow quantity in this inverse method), the modified blade geometry is predicted to perform better than the original design over a wide range of operating points, including an improvement in choke margin.

New Three-Dimensional Inverse Method for High-Speed Vehicle Design

An inverse design method for fully three-dimensional supersonic and hypersonic bodies was developed for computercodesbasedontheEulerequations.Themethodisdesignedtobeeasilyincorporatedintoexistinganalysis codes.Theaim ofthemethodistoprovidetheaerodynamicdesignerwitha powerfultoolfordesign ofaerodynamic shapes of arbitrary cross section. Convergence acceleration techniques were developed and applied successfully to three-dimensional test cases. A three-dimensional body geometry modie cation method was developed with a new concept for the three-dimensional body slope angle. Examples are presented applying the method to a nonaxisymmetricfuselage-typepressuredistributionandacamberedwing-typeapplication.Themethodperforms equally well for both nonlifting and lifting cases. Nomenclature A ML = coefe cient of surface pressure-body shape rule; Eqs. (10) and (12) a = major axis length b = minor axis length Cp = pressure coefe cient C T p = target pressure coefe cient at the body surface c = coefe cient dee ned by Eq. (13) i = marching plane index jdim = number of grid points along the circumferential direction

Design Method for Turbomachine Blades With Finite Thickness by the Circulation Method

This paper presents a procedure to extend a recently developed three-dimensional inverse method for infinitely thin blades to handle blades with finite thickness. In this inverse method, the prescribed quantities are the blade pressure loading and the blade thickness distributions, and the calculated quantity is the blade mean camber line. The method is formulated in the fully inverse mode whereby the blade shape is determined iteratively using the flow-tangency condition along the blade surfaces. Design calculations are presented for an inlet guide vane, an impulse turbine blade, and a compressor blade in the two-dimensional inviscid- and incompressible-flow limit. Consistency checks are carried out for these design calculations using a panel analysis method and the analytical solution for the Gostelow profile.

A Fully Three-Dimensional Inverse Method for Turbomachinery Blading in Transonic Flows

This paper presents a procedure to extend a recently developed fully three-dimensional inverse method for highly loaded turbomachine blades into the transonic-flow regime. In this inverse method, the required three-dimensional blade profile to produce a prescribed swirl schedule is determined iteratively using the blade boundary conditions. In the present implementation, the flow is assumed to be inviscid and the blades are assumed to be infinitely thin. The relevant equations are solved in the conservative forms and are discretized in all three directions using a finite-volume technique. Calculations are carried out for the design of high-pressure axial- and centrifugal-compressor rotors. These examples include prescribed swirl schedules corresponding to blade designs that are shock-free and blade designs that have rapid compression regions in the blade passage.

A Three-Dimensional Inverse Method for Turbomachinery: Part II—Experimental Verification

The performance of an impeller of a low-speed radial-inflow turbine, designed using a three-dimensional inverse technique, was evaluated experimentally. This performance was compared with that achieved by a rotor typical of the present technology. Besides measuring overall quantities, in special efficiency, some traverses of flow velocity were carried out. The results of the tests showed that the new design had a peak total-to-static efficiency 1.4 points better than the conventional build. The traverses indicated that the level of swirl at exhaust of the new impeller was only half as big as that for the conventional rotor, in spite of the fact that both impellers were designed to have zero swirl at outlet. It is also shown that the rotor loss for the new impeller is considerably lower than for the conventional wheel. This research points to the desirability of using a three-dimensional inverse method for the design of turbomachines with significant three-dimensional flows.

A three-dimensional inverse calculation of an axial-flow rotor by the finite-pitch actuator duct method.

A fully three-dimensional inverse solution to the axial-flow rotor of a constant hub to tip ratio by means of the finite-pitch actuator duct method for inviscid and incompressible flow is presented. The blade is expressed by a vortex sheet and a source sheet superposed on the mean surface, and the blade shape is determined to satisfy the specified thickness and loading distributions for free-vortex type flow. Two-dimensional numerical calculations are performed by modifying the present method in order to show the difference between the blade shape obtained by the two-dimensional inverse method and that obtained by the three-dimensional inverse method. Three-dimensional numerical calculations show the influences of the stacking position and the blade spanwise pressure distributions. The direct solutions for the designed two-dimensional and three-dimensional rotors are calculated in order to examine the numerical accuracy of the present inverse calculation.

三次元遷音速逆解法を用いた主翼空力設計について

三次元遷音速逆解法を用いた主翼空力設計について