Transonic Flow Problems Research Articles

Recycling Krylov subspaces for efficient partitioned solution of aerostructural adjoint systems

Robust and efficient solvers for coupled-adjoint linear systems are crucial to successful aerostructural optimization. Monolithic and partitioned strategies can be applied. The monolithic approach is expected to offer better robustness and efficiency for strong fluid-structure interactions. However, it requires a higher implementation cost and convergence may depend on appropriate scaling and initialization strategies. On the other hand, the modularity of the partitioned method enables a straightforward implementation while its convergence may require relaxation. In addition, a partitioned solver often leads to a higher number of iterations to get the same level of convergence as the monolithic one.The objective of this paper is to accelerate the fluid-structure coupled-adjoint partitioned solver by considering techniques borrowed from approximate invariant subspace recycling strategies adapted to sequences of linear systems with varying right-hand sides. Indeed, in a partitioned framework, the structural source term attached to the fluid block of equations affects the right-hand side with the nice property of quickly converging to a constant value. For the fluid block of equations, we also consider deflation of approximate eigenvectors in conjunction with advanced inner-outer Krylov solvers.To demonstrate the effectiveness of our proposed recycling strategy, we compute the coupled derivatives for two emblematic problems in transonic viscous flow. The first one is an aeroelastic configuration of the ONERA-M6 fixed wing. For this exercise, the fluid grid was coupled to a structural model specifically designed to exhibit a high flexibility. The second one is a wing-body aeroelastic configuration of the Common Research Model. All computations are performed using a fully linearized one-equation Spalart-Allmaras turbulence model. Numerical simulations show up to 39% reduction in matrix-vector products for GCRO-DR and up to 25% for the nested FGCRO-DR solver.

An accuracy-enhanced transonic flow prediction method fusing deep learning and a reduced-order model

It is difficult to accurately predict the flow field over an aircraft in the presence of shock waves due to its strong nonlinear characteristics. In this study, we developed an accuracy-enhanced flow prediction method that fuses deep learning and a reduced-order model to achieve accurate flow field prediction for various aerodynamic shapes. Herein, we establish a convolutional neural network/proper orthogonal decomposition (CNN-POD) model for mapping geometries to the overall flow field. Then, local flow regions containing nonlinear flow structures can be identified by the POD reconstruction to build the enhanced model. A CNN model is established to map geometries to the local flow field. The proposed method was applied to two cases involving the prediction of transonic flow over airfoils. The results indicate that the proposed accuracy-enhanced flow prediction method can reduce the prediction error for flow properties in regions with nonlinear flow structures by values ranging from 13% to 66.27%. Additionally, the proposed method demonstrates better efficiency and robustness in comparison to existing methods, and it can also address the prediction problem of complex transonic flow with multiple strong nonlinear structures.

The Finite Element Numerical Study of Transonic Flow of Moist Air with Nonequilibrium Condensation

The finite element numerical simulation of the transonic flow with condensation is studied in this paper. The transformed weak form is numerically discretized by using the classical Galerkin finite element spatial discretization scheme and the backward Euler difference time discretization scheme. Because the coupled Euler equations are convection equations, the results obtained by the classical Galerkin method often have nonphysical numerical oscillations, so the stabilization method is introduced. Another important feature of compressible Euler equation is the existence of shock wave. Because the thickness of shock wave is small and difficult to capture, the shock wave can be better captured by adding artificial viscosity term. Here, the streamline upwind/Petrov–Galerkin (SUPG) finite element method with good stability and the isotropic diffusion method with shock capture ability are used to solve the problem of transonic flow with condensation. In the simulation of steady problems, in order to improve the calculation accuracy of the existing area of the shock wave, the adaptive mesh refinement technology is added, which refines the mesh in the region of the shock wave to improve the resolution of the shock wave. Numerical experiments verify the feasibility and stabilization of the numerical method. Finally, the simulation of transonic flow around NACA-0012 and parallel-jet nozzle A1 has obtained the better numerical results.

On mixed problem of Chaplygin's hodograph equation in hyperbolic domain near the parabolic degenerate line

AbstractThe transonic flow problem is an important one in fluid dynamics or gas dynamics. Local solvability of the mixed problem of Chaplygin's hodograph equation in hyperbolic bounded domains near the degenerate line is considered by using abc method.

Sonic-supersonic Solutions to a Mixed-type Boundary Value Problem for the Two-Dimensional Full Euler Equations

We are concerned with the sonic-supersonic structures extracted from the transonic flow problems in gas dynamics. A local classical supersonic solution for the two-dimensional steady full Euler equations is established in an angular region bounded by the sonic curve and the characteristic curve. In order to overcome the challenges caused by the coupling of nonlinearity and degeneracy at the corner point, we develop a new iteration pattern to show the convergence of the iterative sequence generated by the Euler equations in terms of a partial hodograph coordinate system. The pattern developed here will be useful for studying the degenerate mixed-type boundary value problems for other related nonlinear hyperbolic systems.

A fourth-order entropy condition scheme for systems of hyperbolic conservation laws

Based on the analysis of the entropy condition scheme formulation, the accuracy order comes from initial value interpolation and flux reconstruction. Following the limiters of the traditional second-order Total Variation Diminishing scheme, higher accuracy order and non-oscillatory nature are retained with a newly proposed smoothness threshold method. Then, the scheme using the solution formula method in Dong et al. [(2002). High-order discontinuities decomposition entropy condition schemes for Euler equations. CFD Journal, 10(4), 563–568] is generalized to fully-discrete fourth-order accuracy, which retains the advantages of former schemes, i.e. it is a fully-discrete, one-step scheme with no need to perform with a Runge–Kutta process in time; an entropy condition is satisfied with no need of an entropy fix with artificial numerical viscosity; and an exact solution is achieved for linear cases if CFL→1. Numerical experiments are given with a 1D scalar equation for a shock-tube problem, a blast-wave problem, and Shu–Osher problem, a 2D Riemann problem, a double Mach reflection problem and a transonic airfoil flow problem for NACA0012. All tests are compared with a fifth-order Weighted Essentially Non-Oscillatory (WENO) scheme. Numerical experiments and efficiency comparisons show that the efficiency of the new fourth-order scheme is superior to the fifth-order WENO scheme.

On a degenerate hyperbolic problem for the 3-D steady full Euler equations with axial-symmetry

AbstractThe transonic channel flow problem is one of the most important problems in mathematical fluid dynamics. The structure of solutions near the sonic curve is a key part of the whole transonic flow problem. This paper constructs a local classical hyperbolic solution for the 3-D axisymmetric steady compressible full Euler equations with boundary data given on the degenerate hyperbolic curve. By introducing a novel set of dependent and independent variables, we use the idea of characteristic decomposition to transform the axisymmetric Euler equations as a new system which has explicitly singularity-regularity structures. We first establish a local classical solution for the new system in a weighted metric space and then convert the solution in terms of the original variables.

A relative adequacy framework for multimodel management in multidisciplinary design optimization

In previous work, we presented a novel relative adequacy framework to manage the employment of a set of available computational models in (single-disciplinary) design optimization problems. In this paper, we extend our method to solve multidisciplinary design optimization problems with particular emphasis on strongly coupled fluid-structure interactions. We illustrate that these interactions can have a significant impact on multimodel management: models that may be selected in a single-disciplinary analysis context can be inadequate in a multidisciplinary analysis one. We implement our method for two multidisciplinary design optimization architectures: the monolithic multidisciplinary feasible formulation and a penalty-based distributed interdisciplinary feasible formulation. We illustrate the proposed multimodel management methodology by means of two example problems: a flexible beam fluid-structure interaction problem and a transonic fan flow problem. The obtained results demonstrate that our framework is accurate and efficient while exhibiting significant computational cost benefits, especially when disciplinary coupling is tight.

On a degenerate mixed-type boundary value problem to the 2-D steady Euler equation

This paper is concerned with the structure of sonic-supersonic solutions near sonic curves arising from the transonic channel flow problems in gas dynamics. Given two smooth curves, one of which is a sonic curve and the other is a characteristic curve, we construct a local classical supersonic solution for the two-dimensional steady Euler equations in the angular region. A novel set of dependent and independent variables are introduced to transform the Euler equations into a linear system with explicitly singularity-regularity structures. We use the iteration method to establish the existence and uniqueness of smooth solutions for the new system and then express the solution in terms of the original variables.

Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves

In this paper we are concerned with a class of double phase energy functionals arising in the theory of transonic flows. Their main feature is that the associated Euler equation is driven by the Baouendi–Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. After establishing a weighted inequality for the Baouendi–Grushin operator and a related compactness property, we establish the existence of stationary waves under arbitrary perturbations of the reaction.

Smooth Transonic Flows of Meyer Type in De Laval Nozzles

A smooth transonic flow problem is formulated as follows: for a de Laval nozzle, one looks for a smooth transonic flow of Meyer type whose sonic points are all exceptional and whose flow angle at the inlet is prescribed. If such a flow exists, its sonic curve must be located at the throat of the nozzle and the nozzle should be suitably flat at its throat. The flow is governed by a quasilinear elliptic–hyperbolic mixed type equation and it is strongly degenerate at the sonic curve in the sense that all characteristics from the sonic points coincide with the sonic curve and never approach the supersonic region. For a suitably flat de Laval nozzle, the existence of a local subsonic–sonic flow in the convergent part and a local sonic–supersonic flow in the divergent part is proved by some elaborate elliptic and hyperbolic estimates. The precise asymptotic behavior of these two flows near the sonic state is shown and they can be connected to a smooth transonic flow whose acceleration is Lipschitz continuous. The flow is also shown to be unique by an elaborate energy estimate. Moreover, we give a set of infinitely long de Laval nozzles, such that each nozzle admits uniquely a global smooth transonic flow of Meyer type whose sonic points are all exceptional, while the same result does not hold for smooth transonic flows of Meyer type with nonexceptional points.

Semi-hyperbolic patches of solutions to the two-dimensional compressible magnetohydrodynamic equations

We construct semi-hyperbolic patches of solutions, in which one family out of two families of wave characteristics start on sonic curves and end on transonic shock waves, to the two-dimensional (2D) compressible magnetohydrodynamic (MHD) equations. This type of flow patches appear frequently in transonic flow problems. In order to use the method of characteristic decomposition to construct such a flow patch, we also derive a group of characteristic decompositions for 2D self-similar MHD equations.

Sensitivity assessment of optimal solution in aerodynamic design optimisation using SU2

Computational fluid dynamics has become the method of choice for aerodynamic shape optimisation of complex engineering problems. However, the sensitivity of the final aerodynamic shape to numerical parameters has been largely underestimated to date. The purpose of this work is to investigate the influence that numerical parameters have on the optimisation results for two aerofoil problems (NACA 0012 and RAE 2822) in transonic flow, and to provide compact guidelines for best practice. Numerical parameters include: a) two parameterisation methods, Hicks–Henne bump functions and free-form deformation; b) numerical settings related to the tuning of each parameterisation method; and c) closure coefficients of Spalart–Allmaras (SA) turbulence model. All optimisations were performed using the open-source software tool SU2, and gradients were computed using the continuous adjoint method. It was found that: a) the optimisation result of NACA 0012 aerofoil exhibits strong dependence on all numerical parameters investigated, whereas the optimal design of RAE 2822 aerofoil is insensitive to those numerical settings; b) the degree of sensitivity reflects the difference in the design space, particularly of the local curvature on the optimised shape; c) the closure coefficients of SA model affect the final optimisation performance, raising the need for a good calibration of the turbulence model.

Transonic flow over localised heating elements in boundary layers

The problem of transonic flow past an array of micro-electro-mechanical-type (MEMS-type) heating elements placed on a flat surface is investigated using the triple-deck theory. The compressible Navier–Stokes equations supplemented by the energy equation are considered for large Reynolds numbers. The triple-deck problem is formulated with the aid of the method of matched expansions. The resulting nonlinear viscous lower deck problem, coupled with the upper deck problem governed by the nonlinear Kármán–Guderley equation, is solved using a numerical method based on Chebyshev collocation and finite differences. Our results show the differences in subsonic and supersonic flow behaviour over heated elements. The results indicate the possibility of using the elements to favourably control the transonic flow field.

A Switch Function-Based Gas-Kinetic Scheme for Simulation of Inviscid and Viscous Compressible Flows

AbstractIn this paper, a switch function-based gas-kinetic scheme (SF-GKS) is presented for the simulation of inviscid and viscous compressible flows. With the finite volume discretization, Euler and Navier-Stokes equations are solved and the SF-GKS is applied to evaluate the inviscid flux at cell interface. The viscous flux is obtained by the conventional smooth function approximation. Unlike the traditional gas-kinetic scheme in the calculation of inviscid flux such as Kinetic Flux Vector Splitting (KFVS), the numerical dissipation is controlled with a switch function in the present scheme. That is, the numerical dissipation is only introduced in the region around strong shock waves. As a consequence, the present SF-GKS can well capture strong shock waves and thin boundary layers simultaneously. The present SF-GKS is firstly validated by its application to the inviscid flow problems, including 1-D Euler shock tube, regular shock reflection and double Mach reflection. Then, SF-GKS is extended to solve viscous transonic and hypersonic flow problems. Good agreement between the present results and those in the literature verifies the accuracy and robustness of SF-GKS.

Calculation of transonic flows around profiles with blunt and angled leading edges

Transonic flow is a mixed flow of subsonic and supersonic regions. Because of this mixture, the solution of transonic flow problems is obtained only when solving the differential equations of motion with special treatments for the transition from subsonic region to supersonic region and vice versa. We built codes solving the full potential equation and Euler equations by applying the finite difference method and finite volume method, and also associated with software Fluent to consider the viscous effects. The analysis of results calculated for cases of transonic flow over profiles with blunt and angled leading edges shows more clearly the physical nature of the gas - solid interaction at leading edges in the mixed flow and the optimal application of each profile in transonic flows.

Free boundary problems in shock reflection/diffraction and related transonic flow problems.

Shock waves are steep wavefronts that are fundamental in nature, especially in high-speed fluid flows. When a shock hits an obstacle, or a flying body meets a shock, shock reflection/diffraction phenomena occur. In this paper, we show how several long-standing shock reflection/diffraction problems can be formulated as free boundary problems, discuss some recent progress in developing mathematical ideas, approaches and techniques for solving these problems, and present some further open problems in this direction. In particular, these shock problems include von Neumann's problem for shock reflection-diffraction by two-dimensional wedges with concave corner, Lighthill's problem for shock diffraction by two-dimensional wedges with convex corner, and Prandtl-Meyer's problem for supersonic flow impinging onto solid wedges, which are also fundamental in the mathematical theory of multidimensional conservation laws.

Comparison Between Roe's Scheme and Cell-centered scheme For Transonic Flow Pass Through a Turbine Blades Section

The finite volume method (FVM) is a discretization technique for partial differential equations, especially those that arise from physical conservation laws. FVM uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations [1]. The present work employs the Cell- centred and Roe's scheme to investigate the inviscid compressible flow pass through a two dimensional blade on two types of blade shapes: Blazek and SE1050 blade models. The governing equation of fluid motion of the flow problem in hand is the Euler Equation. The behavior of this equation will depend on the local Mach number if the governing equation stated in the form as the governing equation of steady flow problem. 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. Elimination of the presence of those two types governing equation for the case of transonic flow problem can be achieved by representing the Euler equation in unsteady form. So the equation becomes hyperbolic with respect to time. There are various Finite Volume Methods that can used for solving hyperbolic type of equation, such as Cell-centered scheme [2], Roe Upwind Scheme [3] and TVD Scheme [1]. Those computer codes apply in the case of inviscid two dimensional compressible flow pass through blades of turbine. The present work focuses on two computer codes, first based on Cell Centre scheme and the second one based on Roe's scheme.

An algebraic multigrid solver for transonic flow problems

This article presents the latest developments of an algebraic multigrid (AMG) based on full potential equation (FPE) solver for transonic flow problems with emphasis on advanced applications. The mathematical difficulties of the problem are associated with the fact that the governing equation changes its type from elliptic (subsonic flow) to hyperbolic (supersonic flow). The flow solver is capable of dealing with flows from subsonic to transonic and supersonic conditions and is based on structured body-fitted grids approach for treating complex geometries. The computational method was demonstrated on a variety of problems to be capable of predicting the shock formation and achieving residual reduction of roughly an order of magnitude per cycle both for elliptic and hyperbolic problems, through the entire range of flow regimes, independent of the problem size (resolution).

Toward Applying Algebraic Multigrid to Transonic Flow Problem

This article presents a new approach to solve the transonic flow problem by the algebraic multigrid (AMG) method. The proposed algorithm is demonstrated to attain excellent convergence rates, independent of the problem size, through the entire range of flow regimes. The mathematical difficulties of the problem are associated with the fact that the governing equation changes its type from elliptic (subsonic flow) to hyperbolic (supersonic flow). A pointwise relaxation method when applied directly to the upwind discrete operator, in the supersonic flow regime, is unstable. Resolving this difficulty is the main achievement of this work. A variety of issues regarding the AMG coarsening and construction of transfer operators is addressed in order to achieve the required efficiency for the problems under consideration. We demonstrate the AMG performance on a variety of model problems involving the quasi-linear full potential equation in two dimensions for subsonic, sonic, and supersonic flow and various flow directions (with respect to the grid). In addition, we present a preliminary result verifying the capabilities of the constructed algorithm to deal with a nonlinear problem—transonic small-disturbance equation.