Airfoil Problem Research Articles

Implementation of a PAFV turbulence model for airfoil flow simulation on OpenFOAM

To further develop a more effective turbulence model and to improve the calculation accuracy of the flow around airfoil, a new PAFV turbulence model has here been constructed by using a deformation rate tensor and the grouping of an average fluctuation velocity. To evaluate the applicability of the PAFV turbulence model, the numerical calculations of flow around the airfoil have here been implemented, which was based on the OpenFOAM calculation platform. On the basis of grid independence research, the model was used to calculate the low-speed flow-around problem for the plano-convex airfoil NACA4412 and the symmetric airfoil NACA0012. It was also compared with the S-A (Spalart-Allmaras) and SST (Shear Stress Transport) k-ω turbulence models. Firstly, the maximum lift angle-of-attack case of the NACA4412 airfoil was calculated. Thereafter, numerical calculations were performed for the flow around the airfoil in the multi-angle-of-attack case of the NACA0012 airfoil. The results showed that the NACA4412 airfoil had an obviously separated vortex at the trailing edge of the airfoil at the maximum lift angle of attack. Also, there was a certain velocity loss downstream of the trailing edge, which was calculated by all three models. However, the results of the PAFV turbulence model were found to be better than those of the S-A and SST turbulence models. The three turbulence models showed comparable accuracies for the calculations of the surface pressure coefficients of the NACA0012 airfoil. However, the S-A and SST k-ω turbulence models were slightly better for the calculations of the mean velocity profiles of the NACA0012 airfoil. Also, the PAFV turbulence model was more accurate for the calculations of the lift and drag coefficients. In conclusion, the PAFV model can make effective predictions of the airfoil low-speed flow around the problem at hand, which in turn preliminarily verifies the applicability of this turbulence model for the low-speed flow around the airfoil problems.

Denoising autoencoder genetic programming: strategies to control exploration and exploitation in search

Denoising autoencoder genetic programming (DAE-GP) is a novel neural network-based estimation of distribution genetic programming approach that uses denoising autoencoder long short-term memory networks as a probabilistic model to replace the standard mutation and recombination operators of genetic programming. At each generation, the idea is to capture promising properties of the parent population in a probabilistic model and to use corruption to transfer variations of these properties to the offspring. This work studies the influence of corruption and sampling steps on search. Corruption partially mutates candidate solutions that are used as input to the model, whereas the number of sampling steps defines how often we re-use the output during model sampling as input to the model. We study the generalization of the royal tree problem, the Airfoil problem, and the Pagie-1 problem, and find that both corruption strength and the number of sampling steps influence exploration and exploitation in search and affect performance: exploration increases with stronger corruption and lower number of sampling steps. The results indicate that both corruption and sampling steps are key to the success of the DAE-GP: it permits us to balance the exploration and exploitation behavior in search, resulting in an improved search quality. However, also selection is important for exploration and exploitation and should be chosen wisely.

Multifidelity Prediction Framework with Convolutional Neural Networks Using High-Dimensional Data

This paper proposes two novel multifidelity neural network architectures developed for high-dimensional inputs such as computational flowfields. We employed a two-dimensional flow-varying transonic supercritical airfoil problem while exploring and comparing our methods with the former “multifidelity deep neural networks” from the literature. We call these novel methods “modified multifidelity deep neural networks” and “multifidelity convolutional neural networks.” The main objective of this study is to establish an advanced multifidelity prediction framework that can be applied to any computational data; however, here, we applied our methods to the prediction of aerodynamic coefficients using pressure coefficient fields around the airfoil. To generate the dataset, first, a coarse grid is employed using the SU2 Euler solver for low-fidelity data; then, a relatively finer grid is used to obtain the viscous solutions by using the Spalart–Allmaras turbulence model for the high-fidelity data. The performance metrics to compare the methods are determined as the test accuracy, the physical training time, and the size of the high-fidelity samples. The results demonstrate that the proposed multifidelity neural network architectures outperform the multifidelity deep neural networks in predictive modeling using high-dimensional inputs by improving the multifidelity prediction accuracy significantly.

Adjoint-based limit cycle oscillation instability sensitivity and suppression

Dynamical systems often exhibit limit cycle oscillations (LCOs), self-sustaining oscillations of limited amplitude. LCOs can be supercritical or subcritical. The supercritical response is benign, while the subcritical response can be bi-stable and exhibit a hysteretic response. Subcritical responses can be avoided in design optimization by enforcing LCO stability. However, many high-fidelity system models are computationally expensive to evaluate. Thus, there is a need for an efficient computational approach that can model instability and handle hundreds or thousands of design variables. To address this need, we propose a simple metric to determine the LCO stability using a fitted bifurcation diagram slope. We develop an adjoint-based formula to efficiently compute the stability derivative with respect to many design variables. To evaluate the stability derivative, we only need to compute the time-spectral adjoint equation three times, regardless of the number of design variables. The proposed adjoint method is verified with finite differences, achieving a five-digit agreement between the two approaches. We consider a stability-constrained LCO parameter optimization problem using an analytic model to demonstrate that the optimizer can suppress the instability. We also consider a more realistic LCO speed and stability-constrained airfoil problem that minimizes the normalized mass and stiffness. The proposed method could be extended to optimization problems with a partial differential equation (PDE)-based model, opening the door to other applications where high-fidelity models are needed.

The reduced-order method of continuous space-time finite element scheme for the non-stationary incompressible flows

In this paper, we mainly concern with the order reduction for the unknown solution coefficient vectors about the classical continuous space-time finite element (CCSTFE) method of the two-dimensional (2D) non-stationary incompressible Navier-Stokes equations. Therefore, we first build the CCSTFE scheme for the 2D non-stationary Navier-Stokes equations with respect to stream–vorticity functions, give the existence and stability together with convergence to the CCSTFE solutions, and rewrite the CCSTFE scheme as a matrix form. And then, we use a proper orthogonal decomposition (POD) technique to build a reduced-order extrapolated continuous space-time finite element (ROECSTFE) scheme in the matrix form including very few unknowns but possessing high enough accuracy, employ matrix analysis to demonstrate the existence and stability together with convergence of the ROECSTFE solutions such that the theoretical analysis becomes very concise and convenient, and summarize the flowchart for solving the ROECSTFE scheme. Finally, we take advantage of the numerical simulations for the back-step flow and the flow around airfoil problem to verify the correctness of theoretical results.

Multi-fidelity non-intrusive reduced-order modelling based on manifold alignment

This work presents the development of a multi-fidelity, parametric and non-intrusive reduced-order modelling method to tackle the problem of achieving an acceptable predictive accuracy under a limited computational budget, i.e. with expensive simulations and sparse training data. Traditional multi-fidelity surrogate models that predict scalar quantities address this issue by leveraging auxiliary data generated by a computationally cheaper lower fidelity code. However, for the prediction of field quantities, simulations of different fidelities may produce responses with inconsistent representations, rendering the direct application of common multi-fidelity techniques challenging. The proposed approach uses manifold alignment to fuse inconsistent fields from high- and low-fidelity simulations by individually projecting their solution onto a common latent space. Hence, simulations using incompatible grids or geometries can be combined into a single multi-fidelity reduced-order model without additional manipulation of the data. This method is applied to a variety of multi-fidelity scenarios using a transonic airfoil problem. In most cases, the new multi-fidelity reduced-order model achieves comparable predictive accuracy at a lower computational cost. Furthermore, it is demonstrated that the proposed method can combine disparate fields without any adverse effect on predictive performance.

High-Order Overset Flux Reconstruction Method for Dynamic Moving Grids

Overset meshes have a unique advantage in handling moving boundary problems as remeshing is often unnecessary. Recently, overset Cartesian and strand meshes were used successfully to compute complex flow over rotorcraft. Although it is quite straightforward to deploy a high-order finite difference method on the Cartesian mesh, the near-body solver for the strand mesh is often limited to second-order accuracy. In the present study, a high-order flux reconstruction solver, hpMusic, is developed on both the near-body and background grids, and it is extended to handle moving boundary problems. Accuracy studies are carried out, and the designed order of accuracy is obtained for both inviscid and viscous flows. The method is then tested for a benchmark dynamic airfoil problem from the 4th International Workshop on High-Order CFD Methods. Mesh and order-convergent results are obtained and compared with those from other groups. Finally flow over a hovering rotor is simulated to compare with experimental data. In this case, the present high-order solver is capable of generating and propagating tip vortices with high resolution. Good agreement is achieved with experimental data in tip vortex core size, location, and the swirl velocity at third-order accuracy.

On a global supersonic-sonic patch characterized by 2-D steady full Euler equations

Supersonic-sonic patches are ubiquitous in regions of transonic flows and they boil down to a family of degenerate hyperbolic problems in regions surrounded by a streamline, a characteristic curve and a possible sonic curve. This paper establishes the global existence of solutions in a whole supersonic-sonic patch characterized by the two-dimensional full system of steady Euler equations and studies solution behaviors near sonic curves, depending on the proper choice of boundary data extracted from the airfoil problem and related contexts. New characteristic decompositions are developed for the full system and a delicate local partial hodograph transformation is introduced for the solution estimates. It is shown that the solution is uniformly $C^{1,\frac{1}{6}}$ continuous up to the sonic curve and the sonic curve is also $C^{1,\frac{1}{6}}$ continuous.

Low Mach number limit of multidimensional steady flows on the airfoil problem

In this paper, we justify the low Mach number limit of the steady irrotational Euler flows for the airfoil problem, which is the first result for the low Mach number limit of the steady Euler flows in an exterior domain. The uniform estimates on the compressibility parameter $\varepsilon$, which is singular for the flows, are established via a variational approach based on the compressible-incompressible difference functions. The limit is on the H\"{o}lder space and is unique. Moreover, the convergence rate is of order $\varepsilon^2$. It is noticeable that, due to the feature of the airfoil problem, the extra force dominates the asymptotic decay rate of the compressible flow to the infinity. And the effect of extra force vanishes in the limiting process from compressible flows to the incompressible ones, as the Mach number goes to zero.

Comparison of Uncertainty Quantification Approaches in a Supersonic Biplane Airfoil Problem

Comparison of Uncertainty Quantification Approaches in a Supersonic Biplane Airfoil Problem

Free surface effects on 2-D airfoils and 3-D wings moving over water

The iterative boundary element method (IBEM) developed originally before for cavitating two-dimensional (2-D) and three-dimensional (3-D) hydrofoils moving under free surface is modified and applied to the case of 2-D (two-dimensional) airfoils and 3-D (three-dimensional) wings over water. The calculation of the steady-state flow characteristics of an inviscid, incompressible fluid past 2-D airfoils and 3-D wings above free water surface is of practical importance for air-assisted marine vehicles such as some racing boats including catamarans with hydrofoils and WIG (Wing-In-Ground) effect crafts. In the present paper, the effects of free surface both on 2-D airfoils and 3-D wings moving steadily over free water surface are investigated in detail. The iterative numerical method (IBEM) based on the Green\'s theorem allows separating the airfoil or wing problems and the free surface problem. Both the 2-D airfoil surface (or 3-D wing surface) and the free surface are modeled with constant strength dipole and constant strength source panels. While the kinematic boundary condition is applied on the airfoil surface or on the wing surface, the linearized kinematic-dynamic combined condition is applied on the free surface. The source strengths on the free surface are expressed in terms of perturbation potential by applying the linearized free surface conditions. No radiation condition is enforced for downstream boundary in 2-D airfoil and 3-D wing cases and transverse boundaries in only 3-D wing case. The method is first applied to 2-D NACA0004 airfoil with angle of attack of four degrees to validate the method. The effects of height of 2-D airfoil from free surface and Froude number on lift and drag coefficients are investigated. The method is also applied to NACA0015 airfoil for another validation with experiments in case of ground effect. The lift coefficient with different clearance values are compared with those of experiments. The numerical method is then applied to NACA0012 airfoil with the angle of attack of five degrees and the effects of Froude number and clearance on the lift and drag coefficients are discussed. The method is lastly applied to a rectangular 3-D wing and the effects of Froude number on wing performance have been investigated. The numerical results for wing moving under free surface have also been compared with those of the same wing moving above free surface. It has been found that the free surface can affect the wing performance significantly.

Dimensionality Reduction Using Principal Component Analysis Applied to the Gradient

The cost of dimensionality reduction in aerodynamic design applications involving high-dimensional design spaces and computational fluid dynamics is often prohibitive. In an attempt to overcome this challenge, a new method for dimensionality reduction is presented that scales as , where is the number of design variables. It works by taking advantage of adjoint design methods to compute the covariance matrix of the gradient. This information is then used with principal component analysis to develop a linear transformation that allows an aerodynamic optimization problem to be reformulated in an equivalent coordinate system of lower dimensionality. To demonstrate its feasibility, the method is tested on a two-dimensional staggered airfoil problem, intentionally chosen as an abstraction of a more realistic overwing nacelle integration problem, and shown to exhibit similarities with the latter. Results show that the method rivals typical screening methods used in aerospace engineering in terms of effectiveness but outperforms them either in terms of cost or the ability to capture nonlinear effects. Furthermore, the method is found to show good agreement with optimization results obtained without using the method. Overall, results offer strong evidence in support of the proposed approach, setting the stage for larger analytical efforts such as design space exploration, where dimensionality reduction is unavoidable to cope with the necessity of gradient-free approaches.

Online Projective Integral with Proper Orthogonal Decomposition for Incompressible Flows Past NACA0012 Airfoil

The projective integration method based on the Galerkin-free framework with the assistance of proper orthogonal decomposition (POD) is presented in this paper. The present method is applied to simulate two-dimensional incompressible fluid flows past the NACA0012 airfoil problem. The approach consists of using high-accuracy direct numerical simulations over short time intervals, from which POD modes are extracted for approximating the dynamics of the primary variables. The solution is then projected with larger time steps using any standard time integrator, without the need to recompute it from the governing equations. This is called the online projective integration method. The results by the projective integration method are in good agreement with the full scale simulation with less computational needs. We also study the individual function of each POD mode used in the projective integration method. It is found that the first POD mode can capture basic flow behaviors but the overall dynamic is rather inaccurate. The second and the third POD modes assist the first mode by correcting magnitudes and phases of vorticity fields. However, adding the fifth POD mode in the model leads to some incorrect results in phase-shift forms for both drag and lift coefficients. This suggests the optimal number of POD modes to use in the projective integration method.

Dimension Reduction for Aerodynamic Design Optimization

The search for an optimal design in a high-dimensional design space of a multivariate problem requires a sample size proportional or even exponential to the number of variables of the problem. This ‘curse of dimensionality’ places a computational burden on the cost of optimization, especially when the problem uses expensive high fidelity simulations and may force one to try to reduce the dimensions of a problem. Traditional variable screening techniques reduce the dimensionality of the problem by removing variables that seem irrelevant to the design problem. This practice fails when all the variables are equally relevant in the problem or when some variables are relevant only in some parts of the design space. The present work describes a dimension reduction method called generative topographic mapping based on non-linear latent models which transform a high-dimensional data set into a low-dimensional latent space, without removing any variables. It is first illustrated on a two dimensional Branin function and then applied to a thirty-dimensional airfoil problem. The method is then compared with a global optimizer (a genetic algorithm), other dimension reduction methods (principle component analysis and Gaussian process latent variable models) and with Kriging surrogate models. The method improves when the initial sample used for dimension reduction is filtered to contain only good designs.

Time Spectral Method for Periodic and Quasi-Periodic Unsteady Computations on Unstructured Meshes

For flows with strong periodic content, time-spectral methods can be used to obtain time-accurate solutions at substantially reduced cost compared to traditional time-implicit meth- ods which operate directly in the time domain. However, these methods are only applicable in the presence of fully periodic flows, which represents a severe restriction for many aerospace en- gineering problems. This paper presents an extension of the time-spectral approach for problems that include a slow transient in addition to strong periodic behavior, suitable for applications such as transient turbofan simulation or maneuvering rotorcraft calculations. The formulation is based on a collocation method which makes use of a combination of spectral and polynomial basis func- tions and results in the requirement of solving coupled time instances within a period, similar to the time spectral approach, although multiple successive periods must be solved to capture the transient behavior. The implementation allows for two levels of parallelism, one in the spatial dimension, and an- other in the time-spectral dimension, and is implemented in a modular fashion which minimizes the modifications required to an existing steady-state solver. For dynamically deforming mesh cases, a formulation which preserves discrete conservation as determined by the Geometric Con- servation Law is derived and implemented. A fully implicit approach which takes into account the coupling between the various time instances is implemented and shown to preserve the baseline steady-state multigrid convergence rate as the number of time instances is increased. Accuracy and efficiency are demonstrated for periodic and non-periodic problems by comparing the performance of the method with a traditional time-stepping approach using a simple two-dimensional pitching airfoil problem, a three-dimensional pitching wing problem, and a more realistic transitioning rotor problem.

A robust high-order residual distribution type scheme for steady Euler equations on unstructured grids

A robust high-order algorithm is proposed to solve steady Euler equations on unstructured grids. The main ingredients of the algorithm include a standard Newton method as the outer iterative scheme and a linear multigrid method as the inner iterative scheme with the block lower-upper symmetric Gauss–Seidel (LU-SGS) iteration as its smoother. The Jacobian matrix of the Newton-iteration is regularized by the local residual, instead of using the commonly adopted time-stepping relaxation technique based on the local CFL number. The local Jacobian matrix of the numerical fluxes are computed using the numerical differentiation, which can significantly simplify the implementations by comparing with the manually derived approximate derivatives. The approximate polynomial of solution on each cell is reconstructed by using the values on centroid of the cell, and limited by the WENO hierarchical limiting strategy proposed by Xu et al. [Z.L. Xu, Y.J. Liu, C.W. Shu, Hierarchical reconstruction for discontinuous galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells, Journal of Computational Physics 228 (2009) 2194–2212]. It is found that the proposed algorithm is insensitive to the parameters used. More precisely, in our computations, only one set of the parameters (namely, the proportional constant α for the local residual, the relaxation parameter τ in the Newton-iteration, the weight μ in the WENO scheme and the number of smoothing steps in the multigrid solver) is employed for various geometrical configurations and free-stream configurations. The high-order and robustness of our algorithm are illustrated by considering two-dimensional airfoil problems with different geometrical configurations and different free-stream configurations.

Using the perturbation method to solve the problem of separated incompressible flow past thin airfoils

A model for separated incompressible flow past thin airfoils in the neighborhood of the “shockless entrance” condition is constructed based on the averaging of the vortex shedding flow past the airfoil edges. By approximation of the vortex shedding by two vortex curves, determination of the average hydrodynamic parameters is reduced to a twofold solution of an integral singular equation equivalent to the equation describing steady-state nonseparated airfoil flow. In this case, the calculation time is two orders of magnitude smaller than the time required for the solution of the corresponding evolution problem. The results of a test calculation using the proposed method are in fair agreement with available results of calculations and experiments.

내재적 이중시간 전진기법과 DADI 기법을 이용한 비정상 Navier-Stokes 코드개발

본 연구에서는 비정상 유동해석을 위한 CFD 코드의 개발을 위해 대각화 ADI 기법을 적용한 정상 해석기법과 내재적 이중시간 전진기법을 도입하였다. 정상상태 Navier-Stokes 방정식의 Jacobian 행렬은 비점성항에 대해서만 적용하였고 여기에 내재적 인공점성 연산자를 첨가하여 블록 5대각 행렬을 유도하였다. 시간단축을 위해 스칼라 5대각 행렬로 대체하였다. 가상시간에 대한 정상상태기법에 실시간에 대한 미분항이 포함된 새로운 잔류항을 정의하였다. 가상시간에 대해 수렴된 해로부터 실시간 해를 구하고 시간에 대해 적분을 수행하는 내재적 이중시간 전진기법을 이용한 비정상 Navier-Stokes 코드를 개발하였다. 이에 대한 검증으로 정지한 유체속에 진동하는 평판문제, 원기둥 후방의 주기적인 Karman 와류생성, 이중원호 익형주위의 충격파 진동문제등을 수치해석하여 이론치, 실험치, 타연구자의 계산결과와 비교, 분석하였다. In present study, a two dimensional unsteady Navier-Stokes solver has been developed using the Diagonalized ADI (DADI) method and implicit dual time stepping method. The jacobian matrices in steady state Navier-Stokes equations are introduced from inviscid flux terms. The implicit treatment of artificial dissipation terms results in a block penta-diagonal matrix system and it becomes a scalar penta-diagonal matrix by diagonalization. In steady state equations about fictitious time, a new residual including a real time derivative term is introduced. From a converged solution about fictitious time, a real time unsteady solution can be obtained, which is called 'implicit dual time stepping method'. For code validation, an oscillating flat plate, a regular Karman vortices past a circular cylinder and shock buffeting around a bicircular airfoil problems are numerically solved. And they are compared with a theoretical solution, experiments and other researcher's computations.

Multipoint inverse shape design of airfoils based on variable‐domain variational principle

As a kind of free (unknown) boundary problem, inverse shape design of airfoils has attracted extensive attention in recent years. By variable‐domain variational theory, free boundary condition can be coupled with the governing equations for flow field, which makes it possible to calculate the flow field with the free boundary simultaneously. In this paper, the variational principle (VP) of 2D airfoil shape design problem is obtained from the basic dimensionless velocity potential equations for 2D compressible flow by using a systematic approach and variable‐domain variational formula. The deformable finite element method based on the VP is used to segmentally design airfoil at two design points (angles of attack) and four design points, respectively. The results show that the present method is highly effective and accurate for solving multipoint inverse problem of airfoils.

Adaptive grid generation by minimizing residuals

AbstractWe describe recent developments in the method of fluctuation splitting, in particular the minimization of residuals with respect to both the nodal values of the solution and also the nodal co‐ordinates. For elliptic problems, including external (lifting) flows, obtaining good results requires that the residuals be evaluated to third order. Solutions to typical airfoil problems can then be obtained with very economical grids. For hyperbolic problems, only a second‐order evaluation of the residual is needed. Shocks can be incorporated with arbitrarily high resolution by allowing elements to shrink to zero area, but to exclude the possibility of rarefaction shocks an adaptive quadrature must be employed. Copyright © 2002 John Wiley & Sons, Ltd.