Inviscid Transonic Flow Research Articles

An hp‐adaptive unstructured finite volume solver for compressible flows

SummaryThe idea of hp‐adaptation, which has originally been developed for compact schemes (such as finite element methods), suggests an adaptation scheme using a mixture of mesh refinement and order enrichment based on the smoothness of the solution to obtain an accurate solution efficiently. In this paper, we develop an hp‐adaptation framework for unstructured finite volume methods using residual‐based and adjoint‐based error indicators. For the residual‐based error indicator, we use a higher‐order discrete operator to estimate the truncation error, whereas this estimate is weighted by the solution of the discrete adjoint problem for an output of interest to form the adaptation indicator for adjoint‐based adaptations. We perform our adaptation by local subdivision of cells with nonconforming interfaces allowed and local reconstruction of higher‐order polynomials for solution approximations. We present our results for two‐dimensional compressible flow problems including subsonic inviscid, transonic inviscid, and subsonic laminar flow around the NACA 0012 airfoil and also turbulent flow over a flat plate. Our numerical results suggest the efficiency and accuracy advantages of adjoint‐based hp‐adaptations over uniform refinement and also over residual‐based adaptation for flows with and without singularities.

Global optimization of benchmark aerodynamic cases using physics-based surrogate models

The paper proposes the combination of physics-based surrogate models, adaptive sampling of the design space and evolutionary optimization towards the solution of aerodynamic design problems. The Proper Orthogonal Decomposition is used to extract the main features of the flow field and Radial Basis Function networks allow the surrogate model to predict the target response over the entire design space. In order to train accurate and usable surrogates, ad hoc in-fill criteria are provided which smartly rank and select new samples to enrich the model database. The solution of two aerodynamic benchmark problems is proposed within the framework of the AIAA Aerodynamic Design Optimization Discussion Group. The two benchmark problems consist respectively in the drag minimization of the RAE 2822 airfoil in transonic viscous flow and of the NACA 0012 airfoil in transonic inviscid flow. The shape parameterization approach is based on the Class-Shape Transformation (CST) method with a sufficient degree of Bernstein polynomials to cover a wide range of shapes. Mesh convergence is demonstrated on single-block C-grid structured meshes. The in-house ZEN flow solver is used for Euler/RANS aerodynamic solution. Results show that, thanks to the combined usage of surrogate models and adaptive training in an evolutionary optimization framework, optimal candidates may be located even with limited computational resources with respect to plain evolutionary approaches and similar standard methodologies.

Numerical calculations of 2D transonic flow in GAMM channel and over the profile

The aim of this work is to make a 2D numerical model of the solution of the transonic inviscid and viscous compressible flow around the profile. In a case of viscous flow several turbulent models are used. For the verification of the calculation Baldwin-Lomax model is compared with Wilcox k-omega model and SST turbulent model. Calculations are done in GAMM channel computational domain with 10% DCA profile and in the turbine cascade computational domain with 8% DCA profile. Numerical methods are based on a finite volume solution. Comparisons are done with the experimental data for the 8% DCA profile.

Enabling off-design linearised aerodynamics analysis using Krylov subspace recycling technique

The major computational challenge, when using frequency domain linearised computational fluid dynamics in the analysis of aeroelastic problems such as aircraft flutter, gust response or shock buffet, are the excessive memory and CPU time requirements to solve the large sparse linear systems of equations. To address these issues found with the generalised minimal residual linear equation solver, the generalised conjugate residual solver with deflated restarting is adopted here in which an invariant Krylov subspace is recycled both between restarts when solving a single linear frequency domain problem and for a sequence of equations when varying the system matrix and forcing terms. The proposed method is implemented in an industrial code and applied to three test cases including the forced excitation and buffet onset of a pitch-plunge aerofoil, a realistic passenger aircraft in inviscid transonic flow and a generic half wing-body model at a pre-buffet condition. The memory requirements for the problems investigated are reduced by up to an order of magnitude, while the CPU times are reduced by up to a factor of three.

Virtual boundary method for inviscid transonic flow over supercritical airfoils

A direct virtual boundary method is presented for flow modeling over complex geometries. The main characteristic of the method is that the internal boundary condition at the surface of the embedded body is generated using a function that determinate the suitable body force that ensures the no slip condition directly from the momentum equation. The body force is spread from the Lagrangian to the Eulerian mesh using a Dirac delta function combined with a smooth distribution function. The solver is based on the finite difference approach; integration is carried out with a predictor corrector method, which has second order of precision in both space and time. The flow model is based on the equations of Euler for inviscid compressible flow in two dimensions. The method is employed to simulate the flow at transonic regime over the supercritical airfoils RAE 2822 and NLR 7301. Results show good agreement between numerical and experimental data.

Numerical investigation of transonic flow over deformable airfoil with plunging motion

In this article, the transonic inviscid flow over a deformable airfoil with plunging motion is studied numerically. A finite volume method based on the Roe scheme developed in a generalized coordinate is used along with an arbitrary Lagrangian-Eulerian method and a dynamic mesh algorithm to track the instantaneous position of the airfoil. The effects of different governing parameters such as the phase angle, the deformation amplitude, the initial angle of attack, the flapping frequency, and the Mach number on the unsteady flow field and aerodynamic coefficients are investigated in detail. The results show that maneuverability of the airfoil under various flow conditions is improved by the deformation. In addition, as the oscillation frequency of the airfoil increases, its aerodynamic performance is significantly improved.

Surrogate-based Airfoil Design with Space Mapping and Adjoint Sensitivity

Abstract This paper presents a space mapping algorithm for airfoil shape optimization enhanced with adjoint sensitivities. The surrogate-based algorithm utilizes low-cost derivative information obtained through adjoint sensitivities to improve the space mapping matching between a high-fidelity airfoil model, evaluated through expensive CFD simulations, and its fast surrogate. Here, the airfoil surrogate model is constructed though low-fidelity CFD simulations. As a result, the design process can be performed at a low computational cost in terms of the number of high-fidelity CFD simulations. The adjoint sensitivities are also exploited to speed up the surrogate optimization process. Our method is applied to a constrained drag minimization problem in two-dimensional inviscid transonic flow. The problem is solved for several low-fidelity model termination criteria. The results show that when compared with direct gradient-based optimization with adjoint sensitivities, the proposed approach requires 49-78% less computational cost while still obtaining a comparable airfoil design.

Predictor-Corrector LU-SGS Discontinuous Galerkin Finite Element Method for Conservation Laws

Efficient implicit predictor-corrector LU-SGS discontinuous Galerkin (DG) approach for compressible Euler equations on unstructured grids is investigated by adding the error compensation of high-order term. The original LU-SGS and GMRES schemes for DG method are discussed. Van Albada limiter is employed to make the scheme monotone. The numerical experiments performed for the transonic inviscid flows around NACA0012 airfoil, RAE2822 airfoil, and ONERA M6 wing indicate that the present algorithm has the advantages of low storage requirements and high convergence acceleration. The computational efficiency is close to that of GMRES scheme, nearly 2.1 times greater than that of LU-SGS scheme on unstructured grids for 2D cases, and almost 5.5 times greater than that of RK4 on unstructured grids for 3D cases.

Adjoint‐based airfoil optimization with discretization error control

SUMMARYIn this article, we develop a new airfoil shape optimization algorithm based on higher‐order adaptive DG methods with control of the discretization error. Each flow solution in the optimization loop is computed on a sequence of goal‐oriented h‐refined or hp‐refined meshes until the error estimation of the discretization error in a flow‐related target quantity (including the drag and lift coefficients) is below a prescribed tolerance. Discrete adjoint solutions are computed and employed for the multi‐target error estimation and adaptive mesh refinement. Furthermore, discrete adjoint solutions are employed for evaluating the gradients of the objective function used in the CGs optimization algorithm. Furthermore, an extension of the adjoint‐based gradient evaluation to the case of target lift flow computations is employed. The proposed algorithm is demonstrated on an inviscid transonic flow around the RAE2822, where the shape is optimized to minimize the drag at a given constant lift and airfoil thickness. The effect of the accuracy of the underlying flow solutions on the quality of the optimized airfoil shapes is investigated. Copyright © 2014 John Wiley & Sons, Ltd.

Method for solving the equations describing the triple-deck interaction of a three-dimensional boundary layer with an outer inviscid transonic flow

An approach is developed for solving boundary value problems describing three-dimensional steady flows in the region where a laminar boundary layer interacts with an outer inviscid transonic flow. By applying the method, the flow over a roughness element is computed within the classical triple-deck theory, the asymptotic height of the roughness element corresponding to nonseparated flow is determined, and separated flow patterns are constructed.

Influence of divergent section on flow fields and discharge coefficient of ISO 9300 toroidal-throat sonic nozzle

The effect of divergent section of ISO 9300 toroidal-throat nozzle on discharge coefficient was analyzed based on the inviscid transonic flow model and laminar boundary layer theory. A series of numerical simulations were conducted to verify the results of theory, and investigate the effect of divergent section length L and diffuser angle θ operated at different Reynolds numbers. Combined with the numerical results in this study and the experimental data reported by Nakao, it showed the discharge coefficient increases with the rise of diffuser angle θ or the drop of divergent section length L. A lot of new results about the effect of divergent section were obtained. It indicated that the effect of divergent section on discharge coefficient of ISO 9300 toroidal-throat nozzle should be considered when Re<1.1×104. At last, a concept of effective critical flow was proposed to discuss the effect of divergent section on discharge coefficient.

Inverse airfoil design using variable-resolution models and shape-preserving response prediction

The paper presents a computationally efficient surrogate-based optimization algorithm for the inverse design of transonic airfoils. Our approach replaces the direct optimization of an accurate, but computationally expensive, high-fidelity airfoil model by an iterative re-optimization of two different surrogate models. Initially, for a few design iterations, a corrected physics-based low-fidelity model is employed, which is subsequently replaced by a response surface approximation model. The low-fidelity model is based on the same governing fluid flow equations as the high-fidelity one, but uses coarser mesh resolution and relaxed convergence criteria. The shape-preserving response prediction (SPRP) technique is utilized to predict the high-fidelity model response, here, the airfoil pressure distribution. In this prediction process, SPRP employs the actual changes of the low-fidelity model response (or the surface approximation model) due to the design variable adjustments. The SPRP algorithm is embedded into the trust region framework to ensure good convergence properties of the optimization procedure. Our algorithm is applied to constrained inverse airfoil design in inviscid transonic flow. A comparison with the basic version of the optimization algorithm, exploiting only a physics-based low-fidelity model, is carried out. While the performance of both versions is similar with respect to their ability to match the target pressure distribution, the improved algorithm offers substantial design cost savings, from 25 to 72 percent, depending on the test case.

Comparative accuracy and performance assessment of the finite point method in compressible flow problems

A comparative assessment of the Finite Point Method (FPM) is presented. Using a wing-fuselage configuration under transonic inviscid flow conditions as reference test case, the performance of the FPM flow solver is compared with an equivalent edge-based Finite Element (FEM) implementation. Efficiency issues have discouraged practical application of meshless methods in the past. Thus, a simplification of the basic FPM technique is proposed in order to reduce the performance gap with respect to classical grid-based algorithms. A comparative evaluation of the accuracy, computational cost and parallel performance of the meshless implementation is carried out with the objective to assess the level of maturity of the technique and identify improvements still required to tackle practical applications. The results obtained show accuracy and performance of the core algorithm comparable to a conventional FEM implementation, thus removing a major obstacle for further developments of the FPM.

Comparison between Cell-Centered Schemes Computer Code and Fluent Software for a Transonic Flow Pass through an Array of Turbine Stator Blades

The Finite Volume Method (FVM) is a discretization method which is well suited for the numerical simulation of various types (elliptic, parabolic or hyperbolic, for instance) of conservation laws; it has been extensively used in several engineering fields. The Finite volume method uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations [. the developed computer code based Cell-centered scheme and Fluent software had been used to investigate the inviscid Transonic Flow Pass Through an array of Turbine Stator Blades. The governing equation of fluid motion of the flow problem in hand is assumed governed by the compressible Euler Equation. Basically this equation behave as a mixed type of partial differential equation elliptic and hyperbolic type of partial differential equation. If the local Mach number is less than one, the governing equation will behave as elliptic type of differential equation while if the Mach number is greater than one it will behave as hyperbolic type of differential equation. To eliminate the presence a mixed type behavior, the governing equation of fluid motion are treated as the governing equation of unsteady flow although the problem in hand is steady flow problems. Presenting the Euler equation in their unsteady form makes the equation becomes hyperbolic with respect to time. There are various Finite Volume Methods can used for solving hyperbolic type of equation, such as Cell-centered scheme [, Roe Upwind Scheme [ and TVD Scheme [. The present work use a cell centered scheme applied to the case of flow pass through an array of turbine stator blades. The comparison carried out with the result provided by Fluent Software for three different value of back pressure. The developed computer code shows the result close to the Fluent software although the Fluent software use a Time Averaged Navier stokes equation as its governing equation of fluid motion.

Transonic Flow Simulation Around the Pitching Airfoil with Accurate Pressure-Based Algorithm

A new pressure based implicit procedure to solve the Euler and Navier-Stokes equations is developed to predict transonic viscous and inviscid flowsaround the pitching airfoil with high resolution scheme. In this process, nonorthogonal and non moving mesh with collocated finite volume formulation areused. In order to simulate pitching airfoil, oscillation of flow boundary condition is applied. The boundedness criteria for this procedure are determined from Normalized Variable Diagram (NVD) scheme. The procedure incorporates the k - e eddy-viscosity turbulence model. In the new algorithm, the computation time is considerably reduced. This process is tested for inviscid and turbulent transonic aerodynamic flows around pitching airfoil.The results are compared with other existing numerical solutions and with experiment data. The comparisons show that the resolution quality of the developed algorithm is considerable.

Surrogate-Based Aerodynamic Shape Optimization by Variable-Resolution Models

A surrogate-based optimization algorithm for transonic airfoil design is presented. The approach replaces the direct optimization of an accurate, but computationally expensive, high-fidelity computational fluid dynamics model by an iterative reoptimization of a physics-based surrogate model. The surrogate model is constructed, during each design iteration, using the low-fidelity model and the data obtained from one high-fidelity model evaluation. The low-fidelity model is based on the same governing fluid flow equations as the high-fidelity one, but uses coarser mesh resolution and relaxed convergence criteria. The shape-preserving response prediction technique is utilized to predict the high-fidelity model response, here, the airfoil pressure distribution. In this prediction process, the shape-preserving response prediction employs the actual changes of the low-fidelity model response due to the design variable adjustments. The shape-preserving response prediction algorithm is embedded into the trust region framework that ensures good convergence properties of the optimization procedure. This method is applied to constrained airfoil lift maximization and drag minimization in two-dimensional inviscid transonic flow. The optimized designs are obtained at lower computational cost than that of two comparators. The robustness and scaling properties of the proposed algorithm are investigated.

The origin of the most probable 3:2 twin-peak quasi-periodic oscillations (QPOs) frequency ratio in the micro-quasars

In the present study, we notice that within a few gravitational radii, where the X-ray emission originates, the perturbations in the accretion disk may excite high frequency quasi-periodic oscillations (QPOs). The vertical and the radial disk oscillations frequencies due to the perturbation exhibit a 3:2 twin-peak ratio which is most commonly detected in the X-ray fluxes in many Galactic micro-quasars. The high frequency QPOs happen most likely if the oscillations occur very close to the center. We further notice that the perturbations originate in the neighborhood of the shock transition or from a discontinuity in the disk, and could be the possible mechanism to excite the high frequency QPOs. The shocks are more probable in the X-ray emission region and may form very close to the horizon particularly, when the black hole spin is very high. Studying the shock locations as a function of the black hole spin in a fully general relativistic inviscid transonic accretion flows, we calculate the disk oscillation frequencies and their ratios in that perturbed orbit. Further, from the spin dependence, we estimate the ‘most probable spin interval’ in the micro-quasars by considering the 1/M scaling hypothesis in high frequency QPOs. Finally, we find the spin (a) interval which lies for XTE 1550-564, at a∼(0.89–0.99), for the GRO 1655-40, at a∼(0.96–0.99) and for the GRS 1915+105, at a∼(0.74–0.999). The uncertainty in the mass analysis corresponds to the above spin intervals.

Fold points and singularity induced bifurcation in inviscid transonic flow

Transonic inviscid flow equation of elliptic–hyperbolic type when written in terms of the velocity components and similarity variable results in a second order nonlinear ODE having several features typical of differential–algebraic equations rather than ODEs. These features include the fold singularities (e.g. folded nodes and saddles, forward and backward impasse points), singularity induced bifurcation behavior and singularity crossing phenomenon. We investigate the above properties and conclude that the quasilinear DAEs of transonic flow have interesting properties that do not occur in other known quasilinear DAEs, for example, in MHD. Several numerical examples are included.

Stochastic response surfaces based on non-intrusive polynomial chaos for uncertainty quantification

This paper gives an overview of computationally efficient stochastic response surface techniques based on non-intrusive polynomial chaos (NIPC) for uncertainty quantification in numerical models. The results of uncertainty analysis can be used in robust and reliability-based design and optimisation studies as well as the assessment of the accuracy of the simulation results. In particular, the uncertainty quantification with NIPC methods which require no modification to the existing deterministic models are demonstrated on computational fluid dynamics (CFD) simulations in this paper. The NIPC methods have been increasingly used for uncertainty propagation in high-fidelity CFD simulations due to their non-intrusive nature and strong potential for addressing the computational efficiency and accuracy requirements associated with large-scale complex stochastic simulations. The theory and description of various NIPC methods used for non-deterministic CFD simulations are presented, which can be applied to any other computational models used in analysis and optimisation problems. Several stochastic fluid dynamics examples are given to demonstrate the application and effectiveness of NIPC methods for uncertainty quantification in fluid dynamics. These examples include stochastic computational analysis of a laminar boundary layer flow over a flat plate, supersonic expansion wave problem, and inviscid transonic flow over a three-dimensional wing with rigid and aeroelastic assumptions.

Numerical Solution of Inviscid Transonic Flow Using Hybrid Finite Volume-Finite Difference Solution Technique on Unstructured Grid

An explicit cell centered finite volume CFD solver called Pyramid2D has been developed for solving Euler’s equations in two dimensions. We modify the flow equations at the boundary faces, according to the physical boundary conditions, and solve them in finite difference discrete form suitable for unstructured grids. The flow properties at the cell faces are constructed using new reconstruction method which allows small oscillations at the shock wave region and in the same time contains enough diffusion, so it doesn’t require the use of limiters for transonic flows. The developed solver is to be used as an external aerodynamic solver; all tests showed that the proposed scheme is robust and its numerical results compares well with the published data.