Steady Transonic Flow Research Articles

Turbulent wedge modeling in intermittency-based transition models at high Reynolds numbers

This article presents an approach to set turbulent wedges and transition trippings in local correlation-based intermittency transport transition models. Turbulent wedges are created by increasing the intermittency at the wedge apex or the tripping location. Downstream, no further interference with the transition model is required. The method is demonstrated for the NASA CRM-NLF configuration in a transonic high Reynolds number flow. Prior research on this and other configurations has already shown the important influence of turbulent contamination and boundary layer trippings on the overall aerodynamics. Turbulent wedges and a tripping along a polyline are successfully created for two different intermittency transport models. It is observed that the wedge angles are too large compared to the experimental data. Grid spacing, initial disturbance size, and scaling of the diffusion term only have a minor effect on the wedge angles. This indicates the need to improve the overall transport behavior of the intermittency transport equation for three-dimensional flows. Reynolds number effects are investigated to demonstrate limits of the approach as the transition model gets less sensitive to local disturbances for decreasing Reynolds numbers. In addition to steady transonic flows, the method is also shown for an unsteady transonic flow.

Invariant Vortex-Force Theory Extending Classical Aerodynamic Theories to Transonic Flows

Recent studies about the Lamb vector have led to the development of the vortex-force theory: a formulation able to predict the aerodynamic force in compressible flows and decompose it into lift, lift-induced drag, and profile drag. Here, a revised formulation of the vortex-force theory developed at ONERA in collaboration with the University of Naples is presented and applied to steady transonic flows. In the mathematical developments, special care is given to the presence of shock wave discontinuities within the flowfield. The equivalence between the new definition, the Kutta–Joukowski lift theorem, Maskell’s lift-induced drag formula (“Progress Towards a Method for the Measurement of the Components of the Drag of a Wing of Finite Span,” Procurement Executive, Ministry of Defence, Royal Aircraft Establishment TR 72232, Farnborough, England, U.K., 1972) and Betz’s profile drag formula (“A Method for the Direct Determination of Wing-Section Drag,” NACA TM 337, 1925) are emphasized. The revised formulation also presents several practical advantages: the decomposition is naturally invariant to the reference point chosen for the calculation of moment transformations, and the computation can be performed in a fine part of the grid, close to the body skin. The formulation is finally tested on the NASA Common Research Model wing–fuselage configuration in cruise flight conditions.

Multidimensional transonic shock waves and free boundary problems

We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic–hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection–diffraction, and the Prandtl–Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs.

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.

Steady transonic dense gas flow past two‐dimensional compression/expansion ramps revisited

AbstractSupersonic flow past compression/expansion ramps represents a canonical problem in fluid dynamics which has been studied extensively in the past assuming perfect gas behaviour. More recently real gas effects on such flows have received increasing attention as part of efforts, among others, to increase the efficiency of Organic Rankine Cycles for decentralized power plants. Of special interest in this connection are dense gases of Bethe–Zel'dovich–Thompson fluids which have the distinguishing property that the so called fundamental derivative of gasdynamics Γ takes on negative values in the general neighbourhood of the thermodynamic critical point. Asymptotic analysis of the transonic flow regime has so far concentrated on cases where the unperturbed state with pressure p0, density ρ0 and entropy s0 is close to the transition curve Γ = 0 in the pressure specific volume plane such that |Γ0| ≪ 1, , Kluwick and Cox [1], [2]. These studies have revealed a number of new phenomena and a non‐uniqueness of solutions exceeding that known from the theory of perfect gases.Here we extend the analysis to cover flows where the unperturbed state in the p, 1/ρ plane is in the vicinity of the point on the transition line where Γ and Λ vanish simultaneously. Such flows have |Γ0| ≪ 1, |Λ0| ≪ 1, and are of considerable practical interest as isentropes that can be realized experimentally at present just barely enter the negative Γ region. Again an increased wealth of possible solutions is observed.

Computational study of flow around a NACA 0012 by using Roe FVM scheme and davis-yee TVD scheme

<p>Currently CFD had been considered as an important tool for solving engineering problems. The application of CFD had been used intensively in aircraft industries in design a new aircraft or in the effort of improvement on the exiting aircraft. In term of CFD computer code, the CFD code differs with any others may due to the difference in the numerical scheme have been used. Therefore, the present work presents the comparison result between two developed computer codes with ANSYS-FLUENT software to the case of transonic steady flow past through airfoil NACA 0012. The first computer code used a finite difference method with numerical scheme according to Davis-Yee TVD scheme. Meanwhile, the second computer code used a Roe’s cell centre finite volume scheme. The flow analysis is carried out at two Mach number, M (0.65 & 0.8). Each Mach number applied to two different angles of attacks (0° & 5°). The flow domain discretized by use of C-topology with 193x63 grid points. The comparison in term of the pressure coefficient, along the airfoil surface are presented. From the result, indicated that developed computer code is able to capture the presence of shock wave in the flow field.</p>

Weak shock reflection in steady transonic dense gas flow

AbstractFlows of dense gases may exhibit a number of phenomena which are impossible in the case of perfect gases and thus are of considerable theoretical interest. These include, among others, the formation of rarefaction shock waves which accelerate rather than decelerate the gas and a substantial increase of the lower critical Mach number. As a consequence there is no danger of flow separation if rarefaction shocks interact with boundary layers in high Reynolds number flow. Experimental work is needed to clarify whether these phenomena predicted by theory can be realised in a technical environment and be used to increase the efficiency of turbomachines in Organic Rankine Cycles for decentralised power plants in low popularised areas. Special interest here is for steady transonic flow. The present work is intended to support such experimental efforts. As a first step the most simple but canonical case of slightly supersonic flow past compression/compression ramps was considered in Kluwick & Cox [1]. The present study is intended to complete the understanding of such flows by investigating the reflection problem arising from the presence of an opposing solid wall in confined geometries.

Steady small-disturbance transonic dense gas flow past two-dimensional compression/expansion ramps

The behaviour of steady transonic dense gas flow is essentially governed by two non-dimensional parameters characterising the magnitude and sign of the fundamental derivative of gas dynamics ($\unicode[STIX]{x1D6E4}$) and its derivative with respect to the density at constant entropy ($\unicode[STIX]{x1D6EC}$) in the small-disturbance limit. The resulting response to external forcing is surprisingly rich and studied in detail for the canonical problem of two-dimensional flow past compression/expansion ramps.

High-efficiency solving method for steady transonic flow field

High-efficiency solving method for steady transonic flow field

Stability and asymptotic behavior of transonic flows past wedges for the full Euler equations

The existence, uniqueness, and asymptotic behavior of steady transonic flows past a curved wedge, involving transonic shocks, governed by the two-dimensional full Euler equations are established. The stability of both weak and strong transonic shocks under the perturbation of the upstream supersonic flow and the wedge boundary is proved. The problem is formulated as a one-phase free boundary problem, in which the transonic shock is treated as a free boundary. The full Euler equations are decomposed into two algebraic equations and a first-order elliptic system of two equations in Lagrangian coordinates. With careful elliptic estimates by using appropriate weighted Hölder norms, the iteration map is defined and analyzed, and the existence of its fixed point is established by performing the Schauder fixed point argument. The careful analysis of the asymptotic behavior of the solutions reveals particular characters of the full Euler equations.

Steady transonic dense gas flow past two‐dimensional compression/expansion ramps

AbstractThe dynamic behaviour of compressible fluids depends crucially on the curvature of isentropes in the pressure/specific volume diagram. Most conveniently this curvature is expressed in form of a non‐dimensional quantity Γ now commonly referred to as the fundamental derivative of gasdynamics, Thompson [5]. Bethe‐Zel'dovich‐Thompson (BZT) fluids have the distinguishing property that they exhibit embedded regions in the general neighbourhood of the thermodynamic critical point where Γ is negative in contrast to classical gases of low molecular complexity including perfect gases where Γ is strictly positive.The behaviour of steady transonic flows of such fluids is essentially governed by two non‐dimensional parameters: (Γ) and its derivative with respect to the density at constant entropy (Λ), Cramer and Fry [2], Kluwick [4]. The resulting response to external forcing is surprisingly rich in nonclassical phenomena such as rarefaction shocks, sonic shocks, split shocks, etc. and is studied in detail for the canonical problem of two‐dimensional flow past compression/expansion ramps. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Aircraft Lift and Drag Decomposition in Transonic Flows

A vorticity-based exact theory for the analysis of the aerodynamic force is here applied to three-dimensional aircraft configurations in steady transonic flow by postprocessing numerical solutions. A rigorous and unambiguous definition of lift-induced drag in compressible flows and its distinction from the profile component have been obtained. The equation is based on field integrals of the Lamb vector field highlighting the generation of the aerodynamic force in the rotational part of the flow only. Encountered numerical difficulties, arisen in high-transonic flow conditions, are first described. They have been overcome by a proper treatment of the numerical integration in the shock region where the accuracy of the numerical flow solution is poorer. Thus, a new exact formula is obtained for an accurate aerodynamic force computation in high-transonic and supersonic flows. In addition, a proper identification of the shock wave wake in the numerical solution allows for the decomposition of the profile drag in viscous and wave contributions. Applications are shown in the case of airfoil, elliptic wing and for the NASA Common Research Model wing–body configuration. Comparisons with classical drag breakdown methods are presented and the improvements obtained in the lift-induced drag analysis are discussed.

Effect of nose shape on the shock standoff distance at nearsonic flows

This paper describes a numerical solution of the bow shock shape ahead of some blunt and sharp axisymmetric noses containing sphere, blunt cone, and sharp cone at steady transonic flow in the Mach number range of 1.01 to 1.2. For validating the results, one sphere and three blunt cones are modeled, and their shock standoff distance is compared with other experimental and numerical studies. The flow over other noses with similar geometric parameters is then solved and compared with each other. In this study, the Reynolds-averaged Navier—Stokes equations are solved using the Spalart—Allmaras turbulence model. The purpose of this study is to determine the shock standoff distance for some blunt and sharp noses at low supersonic free flight speed. The shock standoff distance is determined from the Mach number curve on the symmetry line. The present numerical simulations reach down to M8=1.01 a range where it is almost very difficult to set in experimental studies. The shock wave locations were found to agree well with previous numerical and experimental studies. Our results are closer to the experimental results compared to other numerical studies. In addition, the results for shock standoff distances over paraboloids in these speed ranges have not been previously published as far as we know.

A hybrid time stepping scheme combining explicit Runge–Kutta with implicit LU‐SGS for Navier–Stokes equations

SummaryA hybrid time stepping scheme is developed and implemented by a combination of explicit Runge–Kutta with implicit LU‐SGS scheme at the level of system matrix. In this method, the explicit scheme is applied to those grid cells of blocks that have large local time steps; meanwhile, the implicit scheme is applied to other grid cells of blocks that have smaller allowable local time steps in the same flow field. As a result, the discretized governing equations can be expressed as a compound of explicit and implicit matrix operator. The proposed method has been used to compute the steady transonic turbulent flow over the RAE 2822 airfoil. The numerical results are found to be in excellent agreement with the experimental data. In the validation case, the present scheme saved at least 50% of the memory resources compared with the fully implicit LU‐SGS. Copyright © 2015 John Wiley & Sons, Ltd.

Similarity solutions for the generalized equation of steady transonic gas flow with a singular source

In this brief paper, we consider the generalized equation of steady transonic gas flow with the addition of a singular source term. While the addition of a source term often destroys self-similarity of such flows, we demonstrate that a self-similar solution can still exist in the case of a singular source. We first reduce the governing nonlinear partial differential equation into an ordinary differential equation for a class of similarity solutions. Then, we study the existence of solutions for this similarity equation. After that, several explicit solution forms are given. In constructing exact solutions analytically, we demonstrate that dual solution branches may exist for some parameter regimes. For those parameter regimes where exact or analytical solutions are not possible, we obtain numerical solutions. The results demonstrate interesting properties of the solutions which warrant further study.

An adaptive time-stepping strategy for the implicit solution of steady transonic flow

This study addresses a novel adaptive time stepping procedure, which leads to selection of larger time steps allowed by the physics of the problem under consideration. Information about the gradients of the flow variables can be regarded as an indicator for determining proper amount of time step, in which the system evolved. In this study, the signals from the pressure sensors, which act according to the pressure gradients, are chosen as a measure to determine the magnitude of the local CFL number. Thus, the aimed methodology for the selection of the local time step with the use of 'pressure sensor' introduces optimal time steps to the implicit solution method by accounting for the pressure gradient in the solution domain, such that sharp pressure gradients encourages small time steps and vice versa. To illustrate the effect of proposed procedure, Newton Krylov (NK), with implicit pseudo time stepping method, has been employed to solve the compressible Euler equations for steady transonic case by turning on the pressure switch. Numerical experiments show that the introduced adaptive time stepping procedure decreases the computation time and the number of iterations, effectively.

Interpolation-based reduced-order modelling for steady transonic flows via manifold learning

This paper presents a parametric reduced-order model (ROM) based on manifold learning (ML) for use in steady transonic aerodynamic applications. The main objective of this work is to derive an efficient ROM that exploits the low-dimensional nonlinear solution manifold to ensure an improved treatment of the nonlinearities involved in varying the inflow conditions to obtain an accurate prediction of shocks. The reduced-order representation of the data is derived using the Isomap ML method, which is applied to a set of sampled computational fluid dynamics (CFD) data. In order to develop a ROM that has the ability to predict approximate CFD solutions at untried parameter combinations, Isomap is coupled with an interpolation method to capture the variations in parameters like the angle of attack or the Mach number. Furthermore, an approximate local inverse mapping from the reduced-order representation to the full CFD solution space is introduced. The proposed ROM, called Isomap+I, is applied to the two-dimensional NACA 64A010 airfoil and to the 3D LANN wing. The results are compared to those obtained by proper orthogonal decomposition plus interpolation (POD+I) and to the full-order CFD model.

Development of the meshless finite volume particle method with exact and efficient calculation of interparticle area

The Finite Volume Particle Method (FVPM) is a meshless method based on a definition of interparticle area which is closely analogous to cell face area in the classical finite volume method. In previous work, the interparticle area has been computed by numerical integration, which is a source of error and is extremely expensive. We show that if the particle weight or kernel function is defined as a discontinuous top-hat function, the particle interaction vectors may be evaluated exactly and efficiently. The new formulation reduces overall computational time by a factor between 6.4 and 8.2. In numerical experiments on a viscous flow with an analytical solution, the method converges under all conditions. Significantly, in contrast with standard FVPM and SPH, error depends on particle size but not on particle overlap (as long as the computational domain is completely covered by particles). The new method is shown to be superior to standard FVPM for shock tube flow and inviscid steady transonic flow. In benchmarking on a viscous multiphase flow application, FVPM with exact interparticle area is shown to be competitive with a mesh-based volume-of-fluid solver in terms of computational time required to resolve the structure of an interface.

Numerical Simulation of Compressible Vortical Flows Using a Conservative Unstructured-Grid Adaptive Scheme

AbstractA two-dimensional numerical scheme for the compressible Euler equations is presented and applied here to the simulation of exemplary compressible vortical flows. The proposed approach allows to perform computations on unstructured moving grids with adaptation, which is required to capture complex features of the flow-field. Grid adaptation is driven by suitable error indicators based on the Mach number and by element-quality constraints as well. At the new time level, the computational grid is obtained by a suitable combination of grid smoothing, edge-swapping, grid refinement and de-refinement. The grid modifications—including topology modification due to edge-swapping or the insertion/deletion of a new grid node—are interpreted at the flow solver level as continuous (in time) deformations of suitably-defined node-centered finite volumes. The solution over the new grid is obtained without explicitly resorting to interpolation techniques, since the definition of suitable interface velocities allows one to determine the new solution by simple integration of the Arbitrary Lagrangian-Eulerian formulation of the flow equations. Numerical simulations of the steady oblique-shock problem, of the steady transonic flow and of the start-up unsteady flow around the NACA 0012 airfoil are presented to assess the scheme capabilities to describe these flows accurately.

Implicit gas-kinetic BGK scheme with multigrid for 3D stationary transonic high-Reynolds number flows

Instead of solving the Euler and Navier–Stokes equations directly, the gas-kinetic BGK scheme based on the Boltzmann equation has been developed and attracted more and more attentions since the early 1990s. It shows high accuracy and robustness for a wide range of flow regimes. But an obvious disadvantage of the BGK scheme is the low computational efficiency, in particular for multidimensional problems. Till now it has not been widely used as a practical tool for science and engineering applications. To overcome this drawback, in this paper some acceleration techniques, including local time stepping, implicit LU-SGS method, and multigrid strategy, are implemented into the original BGK approach and the new scheme is applied to study 3D steady transonic viscous flows. Numerical results show the significant speed up of the scheme to capture the steady state solution.