Inviscid Transonic Flow Research Articles

Singularity and mesh divergence of inviscid adjoint solutions at solid walls

• Comprehensive review of the mesh dependence anomaly observed in two and three dimensional numerical adjoint solutions for subsonic and transonic inviscid flows past wings, airfoils and blunt bodies. • Review of the analytic adjoint solution obtained for the 2D incompressible Euler equations. • Identification of the origin of the mesh dependence anomaly as a singularity of the analytic solution. • Cross-comparison of analytic and numerical adjoint solutions for lift and drag.

A face-based immersed boundary method for compressible flows using a uniform interpolation stencil

In this study, an immersed boundary method developed for compressible viscous flows (Ramakrishnan, R., Girdhar, A., & Ghosh, S. (2016). Immersed Boundary Methods for Compressible Laminar Flows) is modified to improve their stability and robustness. The embedded object is represented as a set of line segments in two dimensions with their outward unit normal vectors specified. A forcing method that leverages the finite volume approach is used, wherein the solution at the cell interfaces that lie near the boundaries of the embedded solid is reconstructed to implicitly satisfy boundary conditions at the immersed surface. The proposed immersed boundary method is validated for transonic inviscid flow past a bump in a channel, supersonic flow past a circular cylinder, transonic viscous flow past a NACA0012 airfoil, and supersonic viscous flow past a circular cylinder. The results are compared with simulations from the literature using contours of flow properties, surface pressure, or Mach number plots and show good agreement.

2차원 압축성 유동 해석자의 OpenACC를 이용한 GPU 활용

A two-dimensional compressible flow solver was ported to GPU using OpenACC. The performance of this program was tested in three cases: viscous flow on a flat plate, inviscid transonic flow in a channel with a bump, and inviscid transonic flow around n0012 airfoil. The GPU program running on RTX 3090 showed up to 20 times of speedup compared to the original program using a single core of Ryzen 5950X, and up to 10 times of speedup compared to the program that uses all resources on the CPU. Among the computation steps of the program, implicit residual smoothing was the worst performing step on the GPU. To address this problem, skipping residual smoothing and using the Cyclic Reduction algorithm were tried as alternatives. These approaches improved speedup in almost every case, increasing the maximum speedups to 24 times compared to the single core program and 11 times compared to the 16-core program. The results show that GPU is a suitable tool for accelerating CFD solvers due to its memory bandwidth and parallel computing capability and using OpenACC is a viable method for porting programs to GPU.

Registration-Based Model Reduction in Complex Two-Dimensional Geometries

We present a general—i.e., independent of the underlying equation—egistration procedure for parameterized model order reduction. Given the spatial domain \(\varOmega \subset \mathbb {R}^2\) and the manifold \(\mathcal {M}= \{ u_{\mu } : \mu \in \mathcal {P} \}\) associated with the parameter domain \(\mathcal {P} \subset \mathbb {R}^P\) and the parametric field \(\mu \mapsto u_{\mu } \in L^2(\varOmega )\), our approach takes as input a set of snapshots \(\{ u^k \}_{k=1}^{n_\mathrm{train}} \subset \mathcal {M}\) and returns a parameter-dependent bijective mapping \({\varPhi }: \varOmega \times \mathcal {P} \rightarrow \mathbb {R}^2\): the mapping is designed to make the mapped manifold \(\{ u_{\mu } \circ {\varPhi }_{\mu }: \, \mu \in \mathcal {P} \}\) more amenable for linear compression methods. In this work, we extend and further analyze the registration approach proposed in [Taddei, SISC, 2020]. The contributions of the present work are twofold. First, we extend the approach to deal with annular domains by introducing a suitable transformation of the coordinate system. Second, we discuss the extension to general two-dimensional geometries: towards this end, we introduce a spectral element approximation, which relies on a partition \(\{ \varOmega _{q} \}_{q=1} ^{N_\mathrm{dd}}\) of the domain \(\varOmega \) such that \(\varOmega _1,\ldots ,\varOmega _{N_\mathrm{dd}}\) are isomorphic to the unit square. We further show that our spectral element approximation can cope with parameterized geometries. We present rigorous mathematical analysis to justify our proposal; furthermore, we present numerical results for a heat-transfer problem in an annular domain, a potential flow past a rotating symmetric airfoil, and an inviscid transonic compressible flow past a non-symmetric airfoil, to demonstrate the effectiveness of our method.

Polynomial-Chaos–Kriging with Gradient Information for Surrogate Modeling in Aerodynamic Design

This paper proposes a variant of gradient-enhanced surrogate model based on polynomial-chaos–kriging (PCK) to assist aerodynamic design exploration. The main aim is to improve the accuracy of kriging when gradient information is available via the adjoint method. In contrast to the standard gradient-enhanced kriging (GEK), the gradient information is used to enrich both the trend function and the stochastic part, using the polynomial-chaos expansion (PCE) as the trend function. The optimal polynomial terms are selected by scanning through various polynomial orders in which the coefficients are calculated by means of the generalized least squares or the least angle regression algorithm. The performance of the gradient-enhanced PCK (GEPCK) was compared with GEK, gradient-enhanced PCE (GEPCE), and their nongradient counterparts on a set of analytical and a suite of aerodynamic problems consisting of aerodynamic design exploration of a blended-wing–body configuration, an inviscid transonic flow over an airfoil, and a low-fidelity aerostructural wing design. The results show notable improvements obtained by GEPCK in terms of performance and robustness compared with other gradient-enhanced surrogate models. Based on these results, it can be concluded that enhancing the trend part with gradient is highly beneficial in improving the accuracy of the kriging surrogate model.

Output-based adaptive aerodynamic simulations using convolutional neural networks

This paper presents a new method to perform output error estimation and mesh adaptation in computational fluid dynamics (CFD) using machine-learning techniques. The error of interest is the functional output error induced by the numerical discretization, including the finite computational mesh and approximation order. Given the data of adaptive flow simulations guided by adjoint-based error estimates, a surrogate model is trained to predict the output error and drive the mesh adaptation with only the low-fidelity solution as input. The goal is to generalize the error estimation and mesh adaptation knowledge from the simulation data at hand. The proposed method uses an encoder-decoder type convolutional neural network (CNN), supervised by both the adaptive error indicator field and the total output error, to capture both the local and global features related to the numerical error. To handle geometries and irregular meshes in adaptive simulations, topology mapping and local projection are introduced into traditional CNN models. The feasibility of the proposed machine-learning approach for error prediction and mesh adaptation is demonstrated in inviscid transonic flow simulations over airfoils. Both the output error and the localized adaptive indicators are well predicted by the trained CNN model, which is then used to drive the mesh adaptation as an alternative to standard adjoint-based methods. The good performance and relatively simple deployment encourage more study and development of the proposed method.

Supplemental-frequency harmonic balance: A new approach for modeling aperiodic aerodynamic response

Presented in this work is a novel and easy-to-implement supplemental-frequency harmonic balance (SF-HB) approach to efficiently compute dynamically aperiodic systems, which can be cumbersome to model using the nominal high-dimensional harmonic balance (HDHB) technique. The stability of the time-spectral operator and the SF-HB solver involving multiple excitation frequencies is ensured by introducing a group of supplemental frequencies based on the fact that aperiodic (or almost aperiodic) response would contain all possible frequencies. Together with the excitation (primary) frequencies, the frequency set forms a nearly-harmonic series leading to a small condition number for the Fourier transformation matrices directly impacting the stability of the HDHB solver. In contrast to the similar work reported in the literature where optimal unequally-spaced sub-time levels are determined through a rather complicated procedure, the current approach simply uses equally-spaced sub-time levels spanning a time period that depends on a predicted base frequency. At convergence, the original excitation frequency modes dominate the solution as desired. This new technique is verified for a forced pitching airfoil in the transonic inviscid flow regime subjected to two excitation frequencies. Results of both periodic and aperiodic cases show that with the help of Fourier interpolation, the entire time history of the dynamic response can be obtained through a single run of the SF-HB solver, for which the efficiency and robustness of the traditional HDHB method is preserved.

On an Exact Step Length in Gradient-Based Aerodynamic Shape Optimization—Part II: Viscous Flows

This study presents an extension of a previous study (On an Exact Step Length in Gradient-Based Aerodynamic Shape Optimization) to viscous transonic flows. In this work, we showed that the same procedure to derive an explicit expression for an exact step length βexact in a gradient-based optimization method for inviscid transonic flows can be employed for viscous transonic flows. The extended numerical method was evaluated for the viscous flows over the transonic RAE 2822 airfoil at two common flow conditions in the transonic regime. To do so, the RAE 2822 airfoil was reconstructed by a Bezier curve of degree 16. The numerical solution of the transonic turbulent flow over the airfoil was performed using the solver ANSYS Fluent (using the Spalart–Allmaras turbulence model). Using the proposed step length, a gradient-based optimization method was employed to minimize the drag-to-lift ratio of the airfoil. The gradient of the objective function with respect to design variables was calculated by the finite-difference method. Efficiency and accuracy of the proposed method were investigated through two test cases.

On an Exact Step Length in Gradient-Based Aerodynamic Shape Optimization

This study proposeda novel exact expression for step length (size) in gradient-based aerodynamic shape optimization for an airfoil in steady inviscid transonic flows. The airfoil surfaces were parameterized using Bezier curves. The Bezier curve control points were considered as design variables and the finite-difference method was used to compute the gradient of the objective function (drag-to-lift ratio) with respect to the design variables. An exact explicit expression was derived for the step length in gradient-based shape optimization problems. It was shown that the derived step length was independent of the method used for calculating the gradient (adjoint method, finite-difference method, etc.). The obtained results reveal the accuracy of the derived step length.

Machine learning for nonintrusive model order reduction of the parametric inviscid transonic flow past an airfoil

Fluid flow in the transonic regime finds relevance in aerospace engineering, particularly in the design of commercial air transportation vehicles. Computational fluid dynamics models of transonic flow for aerospace applications are computationally expensive to solve because of the high degrees of freedom as well as the coupled nature of the conservation laws. While these issues pose a bottleneck for the use of such models in aerospace design, computational costs can be significantly minimized by constructing special, structure-preserving surrogate models called reduced-order models. In this work, we propose a machine learning method to construct reduced-order models via deep neural networks and we demonstrate its ability to preserve accuracy with a significantly lower computational cost. In addition, our machine learning methodology is physics-informed and constrained through the utilization of an interpretable encoding by way of proper orthogonal decomposition. Application to the inviscid transonic flow past the RAE2822 airfoil under varying freestream Mach numbers and angles of attack, as well as airfoil shape parameters with a deforming mesh, shows that the proposed approach adapts to high-dimensional parameter variation well. Notably, the proposed framework precludes the knowledge of numerical operators utilized in the data generation phase, thereby demonstrating its potential utility in the fast exploration of design space for diverse engineering applications. Comparison against a projection-based nonintrusive model order reduction method demonstrates that the proposed approach produces comparable accuracy and yet is orders of magnitude computationally cheap to evaluate, despite being agnostic to the physics of the problem.

Koopman-Based Approach to Nonintrusive Reduced Order Modeling: Application to Aerodynamic Shape Optimization and Uncertainty Propagation

A methodology for nonintrusive projection-based nonlinear model reduction originally presented by Renganathan et al. (“Koopman-Based Approach to Nonintrusive Projection-Based Reduced-Order Modeling with Black-Box High-Fidelity Models,” AIAA Journal, Vol. 56, No. 10, 2018, pp. 4087–4111) is further extended toward parametric systems with a focus on application to aerospace design. Specifically, the method is extended to address static systems with parametric geometry (that deforms the mesh) in addition to parametric freestream boundary conditions. The main idea is to first perform a transformation on the governing equations such that it is lifted to a higher-dimensional but linear underdetermined system. This enables one to extract the system matrices easily as compared to that of the original nonlinear system. The underdetermined system is closed with a set of model-dependent nonlinear constraints upon which the model reduction is finally performed. The methodology is validated on the subsonic and transonic inviscid flows past the NACA0012 and the RAE2822 airfoils with parametrized shapes. The utility of the approach is further demonstrated by applying it to two common problems in aerospace design, namely, derivative-free global optimization and parametric uncertainty quantification with Monte Carlo sampling. Overall, the methodology is shown to achieve accuracy up to 5% and computational speedup of two to three orders of magnitude relative to the full order model. Comparison against another nonintrusive model reduction method revealed that the proposed approach is more robust, accurate, and retains the consistency between the state variables.

Arbitrary Lagrangian-Eulerian unstructured finite-volume lattice-Boltzmann method for computing two-dimensional compressible inviscid flows over moving bodies.

The objective of this study is to develop and apply an arbitrary Lagrangian-Eulerian unstructured finite-volume lattice-Boltzmann method (ALE-FVLBM) for solving two-dimensional compressible inviscid flows around moving bodies. The two-dimensional compressible form of the LB equation is considered and the resulting LB equation is formulated in the ALE framework on an unstructured body-fitted mesh to correctly model the body shape and properly incorporate the mesh movement due to the body motion. The spatial discretization of the resulting system of equations is performed by a second-order cell-centered finite-volume method on arbitrary quadrilateral meshes and an implicit dual-time stepping method is utilized for the time integration. To stabilize the numerical solution, appropriate numerical dissipation terms are added to the formulation. At first, the shock tube problem is computed to examine the accuracy of the solution obtained by applying the proposed FVLBM for this unsteady test case which includes shock, expansion wave, and contact discontinuity in the flow domain. Then, the stationary isentropic vortex is simulated on both the stationary and moving meshes to assess the implementation of the geometric conservation law in enhancing the solution accuracy of the ALE-FVLBM. The compressible inviscid flow in the transonic regime is then computed around the stationary NACA0012 airfoil in order to further study the sensitivity of the solution method to the user defined parameters. Now, the transonic inviscid flow is simulated over the pitching or plunging NACA0012 airfoil to investigate the accuracy and capability of the proposed solution method (ALE-FVLBM) for the computation of the compressible flows over moving bodies. Finally, the pitching or plunging NACA0012 airfoil near the ground in the transonic inviscid flow is simulated as a practical and challenging problem to study the ground effect on the aerodynamic characteristics of the airfoil. It is indicated that the solution methodology proposed based on the finite-volume LBM formulated in the arbitrary Lagrangian-Eulerian framework (ALE-FVLBM) is capable of accurately computing the compressible inviscid flows around the moving bodies with and without the ground effect.

Solid wall and open boundary conditions in hybrid recursive regularized lattice Boltzmann method for compressible flows

Complex geometries and open boundaries have been intensively studied in the nearly incompressible lattice Boltzmann method (LBM) framework. Therefore, only few boundary conditions for the high speed fully compressible LBM have been proposed. This paper deals with the definition of efficient boundary conditions for the compressible LBM methods, with the emphasis put on the newly proposed hybrid recursive regularized D3Q19 LBM (HRR-LBM) with applications to compressible aerodynamics. The straightforward simple extrapolation-based far-field boundary conditions, the characteristic boundary conditions, and the absorbing sponge layer approach are extended and estimated in the HRR-LBM for the choice of open boundaries. Moreover, a cut-cell type approach to handle the immersed solid is proposed to model both slip and no-slip wall boundary conditions with either isothermal or adiabatic behavior. The proposed implementations are assessed considering the simulation of (i) isentropic vortex convection with subsonic to supersonic inflow and outflow conditions, (ii) two-dimensional (2D) compressible mixing layer, (iii) steady inviscid transonic flow over a National Advisory Committee for Aeronautics (NACA) 0012 airfoil, (iv) unsteady viscous transonic flow over a NACA 0012 airfoil, and (v) three-dimensional (3D) transonic flows over a German Aerospace Center (DLR) F6 full aircraft configuration.

The characteristic boundary condition in pressure-based methods

This article deals with the implementation of the characteristic boundary condition, also known as the pressure far field boundary condition, in segregated pressure-based flow solvers. This boundary condition applies the Riemann invariants to determine the flow variables at domain inlets and outlets. The newly developed method is implemented in a cell-centered unstructured grid code following a finite volume discretization. Testing is performed by solving the following problems: turbulent flow over a flat plate; inviscid transonic flow over a circular are bump; inviscid supersonic flow over three equidistant circular arc bumps; turbulent flow over an axisymmetric transonic bump; turbulent flow over a NACA 0012 airfoil at 10° angle of attack; and turbulent flow over the three-dimensional ONERA M6 wing at 3.06° angle of attack. Predictions with the current pressure-based solver are compared with similar ones generated using a density-based method and with experimental data. Results are in excellent agreement.

A general use adjoint formulation for compressible and incompressible inviscid fluid dynamic optimization

A discrete adjoint formulation with an ad hoc flow solver has been recently developed and tested on transonic inviscid flow optimization problems. In the present paper the formulation is extended to various compressible flow solvers as well as to a solver for incompressible flows. An approximate, dissipative flow solver is used to develop the discrete quasi-time-dependent adjoint equations. The design problem employs a progressive optimization, i.e., a sequence of operations, containing a partially converged flow solution, followed by an adjoint solution followed by an optimization step. The procedure is performed using a sequence of progressively finer grids. The sensitivity derivative variations are limited to preserve the smoothness of the progressive procedure. First, the adjoint equations are coupled with an accurate in-house flow solver to test the approach on some inverse design problems involving two- and three-dimensional supersonic, transonic, subsonic and incompressible flows. Then, the previous design test cases are re-computed coupling the extended adjoint formulation with commercial and open source flow solvers, without noticing any relevant difference in the optimization convergence histories. Finally, incompressible design test cases are successfully computed coupling the extended adjoint formulation with a commercial solver for incompressible flows. The extended compressible adjoint formulation appears to have a wide application, insofar as it allows to perform accurate and efficient design optimization using different flow conditions, different flow solvers and even a solver for incompressible flows.

Adjoint-based airfoil optimization with adaptive isogeometric discontinuous Galerkin method

In this work, an adjoint-based airfoil shape optimization algorithm is developed based on the adaptive isogeometric discontinuous Galerkin method for compressible Euler equations to investigate the significance of each design variable of airfoil B-spline parameterization. We first parameterize the airfoil by B-spline curve approximation with some control points viewed as design variables, and build the B-spline representation of the flow field with the curve to apply the goal-oriented h-adaptive isogeometric DG method for flow solution. Then we compute and employ the discrete adjoint solutions for both multi-target error estimation in adaptive mesh refinement. With the isogeometric nature, not only all the geometrical cells but also the numerical basis functions can be analytically expressed by the design variables, indicating that the numerical solutions and objective could be differentiable with respect to those variables. Consequently, the gradient is totally computed in an accurate approach, and the sensitivity analysis is thus improved, by reducing the spatial discretization error and introducing the analytical expression of derivative, to reveal the key parameters for optimization in an intuitive and efficient manner. Although the SQP optimization algorithm is adopted in the paper, the given accurate gradient can be applied to any gradient-based optimization algorithm. The proposed algorithm is demonstrated on RAE2822 airfoil with inviscid transonic flow, where the shape is optimized to minimize the drag coefficient at a constrained lift and airfoil area. The numerical results show that the drag is much more sensitive to the design variables near the tailing edge at the beginning but sensitivity is reduced when optimal.

An adjoint-based grid adaptive discontinuous Galerkin method

A grid adaptation method has been developed for discontinuous Galerkin discretizations of two-dimensional Euler equations. The element contribution to the global error in interested output variable is used as an indicator to drive mesh adaptation strategies. In the error indicator construction process, the adjoint variables are obtained by solving the adjoint equations with the GMRES algorithm. The solutions of the primal and adjoint problems in the enriched space of order p +1 are approximated by taking a small number of block-diagonal Jacobi iterations on the projected solutions of order p . In order to enforce the adjoint consistency, the numerical flux and output functions defined on the wall are constructed to depend only on the exterior states. The drag coefficient of NACA0012 airfoil with different orders of accuracy is used to evaluate the reliability of the output error estimation method. The subsonic and transonic inviscid flows around NACA 0012 airfoil, and the Mach 6 hypersonic flow over a half cylinder are simulated using the developed grid adaptation method. The computation results demonstrate that the adjoint-based adaptive method is very effective in reducing discretiztion errors of the output functions such as drag coefficient. The results for the subsonic flow of NACA 0012 shows that only 17% of degrees of freedom employed by the uniform refinement is required to achieve equivalent output accuracy by the adjoint-based adaptive approach.

Numerical Investigation of the Droplet Behavior in Cascades Using a Finite Volume Method

This paper describes an Eulerian/Lagrangian two-phase model for wet steam. Two-dimensional inviscid transonic cascade flow is simulated using a cell-vertex finite volume space discretization method on structured triangular mesh. A pseudo time scheme is used to march the solution to steady state. The model provides an approach for including the interaction between the liquid and gas phases for a pure fluid. The analysis consists in the application of the discrete phase model, for modeling the liquid particle flow, and the Eulerian conservation equations to the continuous phase. The investigation permits us to know the influence of parameters such as: particle diameter, flow angle and particle velocity on deposition of drops onto stator blade surfaces. These parameters are analyzed to different inlet flow angles and drop sizes in the turbine. Deposition rate was found to be strongly dependent on increasing inlet flow angle and drop size in the example computed. The purpose of this paper is to present the results of some calculations of deposition which are based on more recent theories or less simplified flow models than previously published work

A Discrete Adjoint Formulation for Inviscid Flow Nozzle Optimization

A discrete adjoint formulation with an ad hoc flow solver has been recently developed and tested on transonic inviscid flow optimization problems. In the present paper the formulation is extended to compressible as well as incompressible flow solvers. First, the adjoint equations are coupled with an accurate in-house flow solver to test the approach on some inverse design problems involving two- and three-dimensional transonic and subsonic flows. Then, the previous design test cases are re-computed coupling the extended adjoint formulation with commercial and open source flow solvers, without noticing any relevant difference in the optimization convergence histories. Finally, incompressible design test cases are successfully computed by means of commercial solver for incompressible flows. The extended compressible adjoint formulation appears to have a wide application, insofar as it allows to perform accurate and efficient design optimization using different flow conditions, different flow solvers and even a solver for incompressible flows.

Hybrid central–WENO scheme for the large eddy simulation of turbulent flows with shocks

ABSTRACTTo provide an effective numerical method for the large eddy simulation (LES) of turbulent flows with shocks, a hybrid scheme is developed in a finite volume framework based on the fourth-order central scheme and the third-order weighted essentially non-oscillatory (WENO) scheme. A total of six easy-to-implement and promising switch functions (SFs) are examined in the hybrid central–WENO scheme for the LES of compressible turbulent flows. Both the dissipation and dispersion of the developed hybrid central–WENO scheme are theoretically confirmed using the Fourier technique. Then, the effectiveness and accuracy of this scheme and the SFs are numerically tested by three problems: decaying compressible isotropic turbulence, inviscid, and turbulent transonic flow over a bump. The numerical results show the developed hybrid scheme, coupled with the SF based on local velocity divergence and pressure gradient, has excellent capabilities of capturing shocks and resolving turbulence.