A Preconditioned Multigrid Method for Efficient Simulation of Three-dimensional Compressible and Incompressible Flows

To develop an efficient and robust aerodynamic analysis method for numerical optimization designs of wing and complex configuration, a combination of matrix preconditioning and multigrid method is presented and investigated. The time derivatives of three-dimensional Navier-Stokes equations are preconditioned by Choi-Merkle preconditioning matrix that is originally designed for two-dimensional low Mach number viscous flows. An extension to three-dimensional viscous flow is implemented, and a method improving the convergence for transonic flow is proposed. The space discretizaition is performed by employing a finite-volume cell-centered scheme and using a central difference. The time marching is based on an explicit Rtmge-Kutta scheme proposed by Jameson. An efficient FAS multigrid method is used to accelerate the convergence to steady-state solutions. Viscous flows over ONERA M6 wing and M100 wing are numerically simulated with Mach numbers ranging from 0.010 to 0.839. The inviscid flow over the DLR-F4 wing-body configuration is also calculated to preliminarily examine the performance of the presented method for complex configuration. The computed results are compared with the experimental data and good agreement is achieved. It is shown that the presented method is efficient and robust for both compressible and incompressible flows and is very attractive for aerodynamic optimization designs of wing and complex configuration.

Robust design of natural laminar flow supercritical airfoil by multi-objective evolution method

Abstract A transonic, high Reynolds number natural laminar flow airfoil is designed and studied. The γ-θ transition model is combined with the shear stress transport （SST） k-w turbulence model to predict the transition region for a laminar-turbulent boundary layer. The non-uniform free-form deformation （NFFD） method based on the non-uniform rational B-spline （NURBS） basis function is introduced to the airfoil parameterization. The non-dominated sorting genetic algorithm-II （NSGA-II） is used as the search algo- rithm, and the surrogate model based on the Kriging models is introduced to improve the efficiency of the optimization system. The optimization system is set up based on the above technologies, and the robust design about the uncertainty of the Mach number is carried out for NASA0412 airfoil. The optimized airfoil is analyzed and compared with the original airfoil. The results show that natural laminar flow can be achieved on a supercritical airfoil to improve the aerodynamic characteristic of airfoils.

Investigation of vortical flow over bluff bodies with base cavities

Based on our previous research about drag reduction in term of the base cavity length using two dimensional simulations, this paper describes a numerical study of a bluff body of which the number of base cavities is successively increased and the cavity geometries are also modified to assume different shapes. Here we attempt to find an effective configuration to reduce the drag by increasing the number of base cavities. The numerical simulations examining varied number of base cavities reveal the presence of different strength of vortices in the wake zone which is the reason why the drag coefficients are distinctly different for different cases. In the case with double and triple rectangular cavities, we use the pressure contours snapshots at successive time instants to describe the wake evolution. We further investigate the effect of variable base cavity shapes for a constant cavity length at an identical time instant. A total of two different geometries of base cavities are discussed here： the rectangular and the sinusoidal cavities with sharp and rounded trailing edges, respectively. The numerical results reveal that the for- mer is an effective drag reduction configuration which can produce a significant base pressure recovery corresponding to the strength of the vortices shown in the pressure contour figures. While the latter shows no obvious reduction in drag coefficient and a similar intensity of vortex in the wake zone compared with the unmodified case. Reductions in drag are observed for all the investigated cavity configurations, and additionally it is found that the magnitude of the reduction bears a direct relationship with the number of the cavities up to a certain minimum value.

Numerical solutions of 2-D steady compressible natural convection using high-order flux reconstruction

Low Mach number flows are common and typical in industrial applications. When simulating these flows, performance of traditional compressible flow solvers can deteriorate in terms of both efficiency and accuracy. In this paper, a new high-order numerical method for two-dimensional(2-D)state low Mach number flows is proposed by combining flux reconstruction(FR)and preconditioning. Firstly, a Couette flow problem is used to assess the efficiency and accuracy of preconditioned FR. It is found that the FR scheme with preconditioning is much more efficient than the original FR scheme. Meanwhile,this improvement still preserves the numerical accuracy. Using this new method and without the Boussinesq assumption,classic natural convection is directly simulated for cases of small and large temperature differences. For the small temperature difference, a p and h refinement study is conducted to verify the grid convergence and accuracy. Then, the influence of the Rayleigh number (Ra) is analyzed. By comparing with the reference results, the numerical results of preconditioned FR is very close to that calculated by incompressible solvers. Furthermore, a large temperature difference test case is calculated and analyzed, indicating this method is not limited by the Boussinesq assumption and is also applicable to heat convection with large temperature differences.

Improving Solution of Euler Equations by a Gas-Kinetic BGK Method

Aim.The well known JST(Jameson-Schmidt-Turkel) scheme requires the use of a dissipation term.We propose using gas-kinetic BGK(Bhatnagar-Gross-Krook) method,which is based on the more fundamental Boltzmann equation,in order to obviate the use of dissipation term and obtain,we believe,an improved solution.Section 1 deals essentially with three things:(1) as analytical solution of molecular probability density function at the cell interface has been obtained by the Boltzmann equation with BGK model,we can compute the flux term by integrating the density function in the phase space;eqs.(8) and(11) require careful attention;(2) the integrations can be expressed as the moments of Maxwellian distribution with different limits according to the analytical solution;eqs.(9) and(10) require careful attention;(3) the discrete equation by finite volume method can be solved using the time marching method.Computations are performed by the BGK method for the Sod′s shock tube problem and a two-dimensional shock reflection problem.The results are compared with those of the conventional JST scheme in Figs.1 and 2.The BGK method provides better resolution of shock waves and other features of the flow fields.

Simulation of natural convection under high magnetic field by means of the thermal lattice Boltzmann method

The thermal lattice Boltzmann method （TLBM）, which was proposed by J. G. M. Eggels and J. A. Somers previously, has been improved in this paper. The improved method has introduced a new equilibrium solution for the temperature distribution function on the assumption that flow is incompressible, and it can correct the effect of compressibility on the macroscopic temperature computed. Compared to the previous method, where the half- way bounce back boundary condition was used for non-slip velocity and temperature, a non-equilibrium extrapolation scheme has been adopted for both velocity and temperature boundary conditions in this paper. Its second-order accuracy coincides with the ensemble accuracy of lattice Boltzmann method. In order to validate the improved thermal scheme, the natural convection of air in a square cavity is simulated by using this method. The results obtained in the simulation agree very well with the data of other numerical methods and benchmark data. It is indicated that the improved TLBM is also successful for the simulations of non-isothermal flows. Moreover, this thermal scheme can be applied to simulate the natural convection in a non-uniform high magnetic field. The simulation has been completed in a square cavity filled with the aqueous solutions of KC1 （llwt%）, which is considered as a diamagnetic fluid with electrically low-conducting, with Grashof number Gr=4.64~104 and Prandtl number Pr----7.0. And three cases, with different cavity locations in the magnetic field, have been studied. In the presence of a high magnetic field, the natural convection is quenched by the body forces exerted on the electrically low-conducting fluids, such as the magnetization force and the Lorentz force. From the results obtained, it can be seen that the quenching efficiencies decrease with the variation of location from left, symmetrical line, to the right. These phenomena originate from the different distributions of the magnetic field strengths in the zones of the symmetrical central line of the magnetic fields. The results are also compared with those without a magnetic field. Finally, we can conclude that the improved TLBM will enable effective simulation of the natural convection under a high magnetic field.

An improved adaptive sampling and experiment design method for aerodynamic optimization

Experiment design method is a key to construct a highly reliable surrogate model for numerical optimization in large-scale project. Within the method, the experimental design criterion directly affects the accuracy of the surrogate model and the optimization efficient. According to the shortcomings of the traditional experimental design, an improved adaptive sampling method is proposed in this paper. The surrogate model is firstly constructed by basic sparse samples. Then the supplementary sampling position is detected according to the specified criteria, which introduces the energy function and curvature sampling criteria based on radial basis function （RBF） network. Sampling detection criteria considers both the uniformity of sample distribution and the description of hypersurface curvature so as to significantly improve the prediction accuracy of the surrogate model with much less samples. For the surrogate model constructed with sparse samples, the sample uniformity is an important factor to the interpolation accuracy in the initial stage of adaptive sam- pling and surrogate model training. Along with the improvement of uniformity, the curvature description of objective function surface gradually becomes more important. In consideration of these issues, crowdness enhance function and root mean square error （RMSE） feedback function are introduced in C criterion expression. Thus, a new sampling method called RMSE and crowd- ness enhance （RCE） adaptive sampling is established. The validity of RCE adaptive sampling method is studied through typical test function firstly and then the airfoil/wing aerodynamic opti- mization design problem, which has high-dimensional design space. The results show that RCE adaptive sampling method not only reduces the requirement for the number of samples, but also effectively improves the prediction accuracy of the surrogate model, which has a broad prospects for applications.

Smoothing methods based on coordinate transformation in a linear space and application in airfoil aerodynamic design optimization

In detailed aerodynamic design optimization,a large number of design variables in geometry parameterization are required to provide sufficient flexibility and obtain the potential optimum shape.However,with the increasing number of design variables,it becomes difficult to maintain the smoothness on the surface which consequently makes the optimization process progressively complex.In this paper,smoothing methods based on B-spline functions are studied to improve the smoothness and design efficiency.The wavelet smoothing method and the least square smoothing method are developed through coordinate transformation in a linear space constructed by B-spline basis functions.In these two methods,smoothing is achieved by a mapping from the linear space to itself such that the design space remains unchanged.A design example is presented where aerodynamic optimization of a supercritical airfoil is conducted with smoothing methods included in the optimization loop.Affirmative results from the design example confirm that these two smoothing methods can greatly improve quality and efficiency compared with the existing conventional non-smoothing method.

Aerodynamic design optimization of nacelle/pylon position on an aircraft

The arbitrary space-shape free form deformation （FFD） method developed in this paper is based on non-uniform rational B-splines （NURBS） basis function and used for the integral parameterization of nacelle-pylon geometry. The multi-block structured grid deformation technique is established by Delaunay graph mapping method. The optimization objects of aerodynamic characteristics are evaluated by solving NavierStokes equations on the basis of multi-block structured grid. The advanced particle swarm optimization （PSO） is utilized as search algorithm, which com-bines the Kriging model as surrogate model during optimization. The optimization system is used for optimizing the nacelle location of DLR-F6 wing-body-pylon-nacelle. The results indicate that the aerodynamic interference between the parts is significantly reduced. The optimization design system established in this paper has extensive applications and engineering value.

Aerodynamic optimization of rotor airfoil based on multi-layer hierarchical constraint method

Rotor airfoil design is investigated in this paper. There are many difficulties for this highdimensional multi-objective problem when traditional multi-objective optimization methods are used. Therefore, a multi-layer hierarchical constraint method is proposed by coupling principal component analysis(PCA) dimensionality reduction and e-constraint method to translate the original high-dimensional problem into a bi-objective problem. This paper selects the main design objectives by conducting PCA to the preliminary solution of original problem with consideration of the priority of design objectives. According to the e-constraint method, the design model is established by treating the two top-ranking design goals as objective and others as variable constraints. A series of bi-objective Pareto curves will be obtained by changing the variable constraints, and the favorable solution can be obtained by analyzing Pareto curve spectrum. This method is applied to the rotor airfoil design and makes great improvement in aerodynamic performance. It is shown that the method is convenient and efficient, beyond which, it facilitates decision-making of the highdimensional multi-objective engineering problem.

Reliability and Sensitivity Analysis of Transonic Flutter Using Improved Line Sampling Technique

The improved line sampling （LS） technique, an effective numerical simulation method, is employed to analyze the probabilistic characteristics and reliability sensitivity of flutter with random structural parameter in transonic flow. The improved LS technique is a novel methodology for reliability and sensitivity analysis of high dimensionality and low probability problem with implicit limit state function, and it does not require any approximating surrogate of the implicit limit state equation. The improved LS is used to estimate the flutter reliability and the sensitivity of a two-dimensional wing, in which some structural properties, such as frequency, parameters of gravity center and mass ratio, are considered as random variables. Computational fluid dynamics （CFD） based unsteady aerodynamic reduced order model （ROM） method is used to construct the aerodynamic state equations. Coupling structural state equations with aerodynamic state equations, the safety margin of flutter is founded by using the critical velocity of flutter. The results show that the improved LS technique can effectively decrease the computational cost in the random uncertainty analysis of flutter. The reliability sensitivity, defined by the partial derivative of the failure probability with respect to the distribution parameter of random variable, can help to identify the important parameters and guide the structural optimization design.

Influence of leading edge with real manufacturing error on aerodynamic performance of high subsonic compressor cascades

To investigate the influence of real leading-edge manufacturing error on aerodynamic performance of high subsonic compressor blades,a family of leading-edge manufacturing error data were obtained from measured compressor cascades.Considering the limited samples,the leadingedge angle and leading-edge radius distribution forms were evaluated by Shapiro-Wilk test and quantile–quantile plot.Their statistical characteristics provided can be introduced to later related researches.The parameterization design method B-spline and Bezier are adopted to create geometry models with manufacturing error based on leading-edge angle and leading-edge radius.The influence of real manufacturing error is quantified and analyzed by self-developed non-intrusive polynomial chaos and Sobol’indices.The mechanism of leading-edge manufacturing error on aerodynamic performance is discussed.The results show that the total pressure loss coefficient is sensitive to the leading-edge manufacturing error compared with the static pressure ratio,especially at high incidence.Specifically,manufacturing error of the leading edge will influence the local flow acceleration and subsequently cause fluctuation of the downstream flow.The aerodynamic performance is sensitive to the manufacturing error of leading-edge radius at the design and negative incidences,while it is sensitive to the manufacturing error of leading-edge angle under the operation conditions with high incidences.

Efficient aero-structural design optimization: Coupling based on reverse iteration of structural model

Traditional coupled multi-disciplinary design optimization based on computational fluid dynamics/computational structure dynamics(CFD/CSD)aims to optimize the jig shape of aircraft,and general multi-disciplinary design optimization methodology is adopted.No special consideration is given to the aircraft itself during the optimization.The main drawback of these methodologies is the huge expanse and the low efficiency.To solve this problem,we put forward to optimize the cruise shape directly based on the fact that the cruise shape can be transformed into jig shape,and a methodology named reverse iteration of structural model(RISM)is proposed to get the aero-structural performance of cruise shape.The main advantage of RISM is that the efficiency can be improved by at least four times compared with loosely-coupled aeroelastic analysis and it maintains almost the same fidelity of loosely-coupled aeroelastic analysis.An optimization framework based on RISM is proposed.The aerodynamic and structural performances can be optimized simultaneously in this framework,so it may lead to the true optimal solution.The aerodynamic performance was predicted by N-S solver in this paper.Test shows that RISM predicts the aerodynamic and structural performances very well.A wing-body configuration was optimized by the proposed optimization framework.The drag and weight of the aircraft are decreased after optimization,which shows the effectiveness of the proposed framework.

The CDF and its sensitivity analysis of stochastic structure with stochastic excitation by advanced stratified line sampling

For the stochastic structure with stochastic excitation, an advanced stratified line sampling (SLS) method is presented to obtain the cumulative distribution function (CDF) of the structural response and its sensitivity. The advanced stratified line sampling method introduces a set of middle failure subsets firstly. And for each subset, the conventional line sampling can be used to obtain the corresponding value of the response's CDF. At the same time, the sensitivity estimations of each failure subset can also be computed by modifying the important direction and corresponding reliability coefficients. The properties of CDF sensitivity are proved while the performance function is linear with normal random variables. After two simple examples are used to demonstrate the properties of CDF sensitivity and the feasibility of the presented method, the method employed to analyze the CDF and corresponding sensitivity of root bending moment (RBM) responses for the stochastic BAH is wing with gust excitation to a square-edged gust and to a Dryden gust. The results show that the parameters of the second and the fifth order modals exert more influence on the CDF of response than the other ones, and the presented SLS method can more significantly reduce the computational cost compared with Monte Carlo simulation (MCS).

A Robust Blade Design Method based on Non-Intrusive Polynomial Chaos Considering Profile Error

To weaken the influence of profile error on compressor aerodynamic performance, especially on pressure ratio and efficiency, a robust design method considering profile error is built to improve the robustness of aerodynamic performance of the blade. The characteristics of profile error are random and small-scaled, which means that to evaluate the influence of profile error on blade aerodynamic performance is a time-intensive and high-precision work. For this reason, non-intrusive polynomial chaos(NIPC) and Kriging surrogate model are introduced in this robust design method to improve the efficiency of uncertainty quantification(UQ) and ensure the evaluate accuracy. The profile error satisfies the Gaussian distribution, and NIPC is carried out to do uncertainty quantification since it has advantages in prediction accuracy and efficiency to get statistical behavior of random profile error. In the integrand points of NIPC, several surrogate models are established based on Latin hypercube sampling(LHS)+ Kriging, which further reduces the costs of optimization design by replacing calling computational fluid dynamic(CFD) repeatedly. The results show that this robust design method can significantly improve the performance robustness in shorter time(40 times faster) without losing accuracy, which is meaningful in engineering application to reduce manufacturing cost in the premise of ensuring the aerodynamic performance. Mechanism analysis of the robustness improvement samples carried out in current work can help find out the key parameter dominating the robustness under the disturbance of profile error, which is meaningful to further improvement of compressor robustness.

Buffeting numerical simulation coupled with aerodynamics and structure based on MDDES

Unsteady effect of seriously separated flow is the main factor of modern aircraft buffeting. So accurate simulation of this complex flow becomes the basis associated with the research of aircraft buffeting. This paper constructs an unsteady numerical simulation method for separation flow based on modified delayed detached eddy simulation (MDDES) method by considering both modern computer resources and the credibility of simulating separation flow. The proposed method is also verified through the simulation of the separated flow by a typical fighter at high angle of attack. And then a robust and efficient technology for deforming mesh is established using radial basis function (RBF) and infinite interpolation method. Moreover, the platform for numerical simulation of buffeting is set up in combination with the structural dynamics equations in the modal space, by which the research of vertical tail buffeting caused by edge vortex is carried out on a fighter at large angle of attack. Through spectrum analysis of time-domain response of pressure pulsation on the location of vortex rupture, the results show that the pulsation frequency of vortex structure with different scales covers the inherent modal frequency of vertical tail structure. Compared to the Reynolds-averaged Navier-Stokes equations, the MDDES method can distinguish the more detailed and higher frequency small-scale vortex structure. Unlike flutter, displacement acceleration response of each mode in buffeting is dominated by its own mode. There exists strong coupling between the first bending mode and first torsion mode, and it leads to acceleration and large inertia impact of structure, which is the main factor causing structural fatigue. In sum, the obtained results verify the validity of the numerical means and the corresponding methods in the paper.

Experimental Investigation of a Linear Cascade with Large Solidity Using Pressure Sensitive Paint and Dual-Camera System

Pressure Sensitive Paint(PSP)technique has been increasingly applied to the experimental research of aerodynamics and thermodynamics due to its strengths of non-contact,high resolution results and large coverage area,etc.However,rarely has this technique been successfully used to the study of internal flow such as compressor cascade,since narrow flow passages would heavily restrict the acquisition of PSP images.In this paper,PSP technique was used to study the pressure distribution on a linear compressor cascade with large solidity of 2.3,where the view of recording camera can be heavily blocked due to adjacent blade surfaces.To help get integrated PSP images of the internal flow passage,dual camera system along with image processing tools like 3D reconstruction and image integration were adopted.The results showed that with the aid of such assistance,image results with good quality and readability could be obtained.Meanwhile,pressure data given by PSP were compared with data from traditional way of pressure taps and showed good consistency.Massive results of the entire cascade passage surface were given with different inlet Mach numbers and incidence angles.The results showed that PSP technique can integrally measure cascade tunnel of large solidity with the help of dual-camera system.

Research on multi-fidelity aerodynamic optimization methods

Constructing high approximation accuracy surrogate model with lower computational cost has great engineering significance. In this paper, using co-Kriging method, an efficient multi-fidelity surrogate model is constructed based on two independent high and low fidelity samples. Co-Kriging method can use a greater quantity of low-fidelity information to enhance the accuracy of a surrogate of the high-fidelity model by modeling the correlation between high and low fidelity model, thus computational cost of building surrogate model can be greatly reduced. A wing-body problem is taken as an example to compare characteristics of co-Kriging multi-fidelity （CKMF） model with traditional Kriging based multi-fidelity （KMF） model. A sampling convergence of the CKMF model and the KMF model is conducted, and an appropriate sampling design is selected through the sampling convergence analysis. The results indicate that CKMF model has higher approximation accuracy with the same high-fidelity samples, and converges at less high-fidelity samples. A wing-body drag reduction optimization design using genetic algorithm is implemented. Satisfying design results are obtained, which validate the feasibility of CKMF model in engineering design.

Robust design of NLF airfoils

A robust optimization design approach of natural laminar airfoils is developed in this paper. First, the non-uniform rational B-splines （NURBS） free form deformation method based on NURBS basis function is introduced to the airfoil parameterization. Second, aerodynamic characteristics are evaluated by solving Navier-Stokes equations, and theγ-Reθt transition model coupling with shear-stress transport （SST） turbulent model is introduced to simulate boundary layer transition. A numerical simulation of transition flow around NLF0416 airfoil is conducted to test the code. The comparison between numerical simulation results and wind tunnel test data approves the validity and applicability of the present transition model. Third, the optimization system is set up, which uses the separated particle swarm optimization （SPSO） as search algorithm and combines the Kriging models as surrogate model during optimization. The system is applied to carry out robust design about the uncertainty of lift coefficient and Mach number for NASA NLF-0115 airfoil. The data of optimized airfoil aerodynamic characteristics indicates that the optimized airfoil can maintain laminar flow stably in an uncertain range and has a wider range of low drag.

Side force control on slender body by self-excited oscillation flag

Strong asymmetrical vortices appear on the leeward of slender body at high angles of attack, which has very unfavorable effect on the stability and control of the aircraft. A method is developed to control the side force of slender body at high angles of attack, and is verified in wind tunnel. A thin-film triangular self-excited oscillation flag is fixed at the tip of the slender body model whose semi-apex angle is 10°. Side force is approximately linearly proportional to roll-setting angle of self-excited oscillation flag at high angles of attack, and the slop of fitting straight line obtained by the least square method is -0.158. The linear relationship between side force and roU-setting angle provides convenience for developing side force control law of slender body at high angles of attack. Experimental data shows that the side force coefficients vary linearly with roll-setting angles when a specific plastic self-excited oscillation flag is used as the control flag. The range of side force coefficient and roll-setting angle are, respectively, -3.2 to 3.0 and -20° to 20°. The device is simple, effective, and is of great potential in engineering application.