Turbulence model and numerical scheme assessment for buffet computations

The prediction of shock‐induced oscillations over transonic rigid airfoils is important for a better understanding of the buffeting phenomenon. The unsteady resolution of the Navier–Stokes equations is performed with various transport‐equation turbulence models in which corrections are added for non‐equilibrium flows. The lack of numerical efficiency due to the CFL stability condition is circumvented by the use of a wall law approach and a dual time stepping method. Moreover, various numerical schemes are used to try and be independent of the numerical discretization. Comparisons are made with the experimental results obtained for the supercritical RA16SC1 airfoil. They show the interest in using the SST correction or realizability conditions to get correct predictions of the frequency, amplitude and pressure fluctuations over the airfoil. Copyright © 2004 John Wiley & Sons, Ltd.     

Adaptive strategy of the supersonic turbulent flow over a backward‐facing step

An adaptive strategy incorporating mesh remeshing and refining is developed to study the supersonic turbulent flow over a backward‐facing step on a mixed quadrilateral–triangular mesh. In the Cartesian co‐ordinate system, the unsteady Favre‐averaged Navier–Stokes equations with a low‐Reynolds‐number k–εturbulence model are solved using a locally implicit scheme with an anisotropic dissipation model. In the present adaptive strategy, two error indicators for both mesh remeshing and refining, respectively, are presented. The remeshing error indicator incorporates unified magnitude of substantial derivative of pressure and that of vorticity magnitude, whereas the refining error indicator incorporates unified magnitude of substantial derivative of pressure and that of weighted vorticity magnitude. To assess the present approach, the transonic turbulent flow around an NACA 0012 airfoil is performed. Based on the comparison with the experimental data, the accuracy of the present approach is confirmed. According to the high‐resolutional result on the adaptive mesh, the structure of backstep corner vortex, expansion wave and oblique shock wave is distinctly captured. Copyright © 2004 John Wiley & Sons, Ltd.     

Implicit application of non‐reflective boundary conditions for Navier–Stokes equations in generalized coordinates

The non‐reflective boundary conditions (NRBC) for Navier–Stokes equations originally suggested by Poinsot and Lele (J. Comput. Phys. 1992; 101 :104–129) in Cartesian coordinates are extended to generalized coordinates. The characteristic form Navier–Stokes equations in conservative variables are given. In this characteristic‐based method, the NRBC is implicitly coupled with the Navier–Stokes flow solver and are solved simultaneously with the flow solver. The calculations are conducted for a subsonic vortex propagating flow and the steady and unsteady transonic inlet‐diffuser flows. The results indicate that the present method is accurate and robust, and the NRBC are essential for unsteady flow calculations. Copyright © 2005 John Wiley & Sons, Ltd.     

Kriging‐based optimization applied to flow control

The automatic optimization of flow control devices is a delicate issue because of the drastic computational time related to unsteady high‐fidelity flow analyses and the possible multimodality of the objective function. Thus, we experiment in this article the use of kriging‐based algorithms to optimize flow control parameters because these methods have shown their efficiency for global optimization at moderate cost. Navier–Stokes simulations, carried out for different control parameters, are used to build iteratively a kriging model. At each step, a statistical analysis is performed to enrich the model with new simulation results by exploring the most promising areas, until optimal flow control parameters are found. This approach is validated and demonstrated on two problems, including comparisons with similar studies: the control of the flow around an oscillatory rotating cylinder and the reduction of the intensity of a shock wave for a transonic airfoil by adding a bump to the airfoil profile. Copyright © 2011 John Wiley & Sons, Ltd.     

Separated Flow and Buffeting Control

In transonic flow conditions, the shock wave/turbulent boundary layer interaction and the flow separations on the upper wing surfaces of civil aircraft induce flow instabilities, ‘buffet’ and then structural vibrations, ‘buffeting’. Buffeting can greatly affect aerodynamic behavior. The buffeting phenomenon appears when the aircraft's Machnumber or angle of attack increases. This phenomenon limits the aircraft's flight envelope. The objectives of this study are to cancel out or decrease the aerodynamic instabilities (unsteady separation, movement of the shock position) due to this type of flow by using control systems. The following actuators can be used: ‘Vortex Generators’ situated upstream of the shock position, a ‘Bump’ located at the shock position, and a new moving part designed by ONERA, situated on the trailing edge of the wing, the ‘Trailing Edge Deflector’ or TED. It looks like an adjustable ‘Divergent Trailing Edge’. It is an active actuator and can take different deflections or be driven by dynamic movements up to 250 Hz. Tests were performed in transonic 2D flow with models well equipped with unsteady pressure transducers. For high lift coefficients, a selected static position of the ‘Trailing Edge Deflector’ increases the wing's aerodynamic performances and delays the onset of buffet. Furthermore, in 2D flow buffet conditions, the ‘Trailing Edge Deflector’, driven by a closed-loop active control using the measurements of the unsteady wall static pressures, can greatly reduce buffet. The aerodynamic performances are not improved to the same extent by the bump actuator. From our experience, there is no effect on buffet or separated flow because of the incorrect positioning of the bump. All that can be observed is a local improvement on the intensity of the shock wave when the bump is very precisely situated at the shock position. Vortex generators have a great impact on the separated flow. The separated flow instabilities are greatly reduced and buffet is totally controlled even for strong instabilities. The aerodynamic performances of the airfoil are also greatly improved.     

Higher‐order and adaptive discontinuous Galerkin methods with shock‐capturing applied to transonic turbulent delta wing flow

Discontinuous Galerkin (DG) methods allow high‐order flow solutions on unstructured or locally refined meshes by increasing the polynomial degree and using curved instead of straight‐sided elements. DG discretizations with higher polynomial degrees must, however, be stabilized in the vicinity of discontinuities of flow solutions such as shocks. In this article, we device a consistent shock‐capturing method for the Reynolds‐averaged Navier–Stokes and k‐ ω turbulence model equations based on an artificial viscosity term that depends on element residual terms. Furthermore, the DG method is combined with a residual‐based adaptation algorithm that targets at resolving all flow features. The higher‐order and adaptive DG method is applied to a fully turbulent transonic flow around the second Vortex Flow Experiment (VFE‐2) configuration with a good resolution of the vortex system.Copyright © 2013 John Wiley & Sons, Ltd.     

An exact non‐linear Navier–Stokes compressible‐flow solution for CFD code verification

An exact similarity solution of the compressible‐flow Navier–Stokes equations is presented, which embeds supersonic, transonic, and subsonic regions. Describing the viscous and heat‐conducting high‐gradient flow in a shock wave, the solution accommodates non‐linear temperature‐dependent viscosity as well as heat‐conduction coefficients and provides the variation of all the flow variables and their derivatives. Also presented are methods to obtain time‐dependent and/or multi‐dimensional solutions as well as verification benchmarks of increasing severity. Comparisons between the developed analytical solution and CFD solutions of the Navier–Stokes equations, with determination of convergence rates and orders of accuracy of these solutions, illustrate the utility of the developed exact solution for verification purposes. Copyright © 2012 John Wiley & Sons, Ltd.     

Shock-unsteadiness model applied to oblique shock wave/turbulent boundary-layer interaction

Reynolds-averaged Navier–Stokes prediction of shock wave/turbulent boundary layer interactions can yield significant error in terms of the size of the separation bubble. In many applications, this can alter the shock structure and the resulting surface properties. Shock-unsteadiness modification of Sinha et al. (Physics of Fluids, Vol.15, No.8, 2003) has shown potential in improving separation bubble prediction in compression corner flows. In this article, the modification is applied to oblique shock wave interacting with a turbulent boundary layer. The challenges involved in the implementation of the shock-unsteadiness correction in the presence of multiple shock waves and expansion fans are addressed in detail. The results show that a robust implementation of the model yields appreciable improvement over standard k–ω turbulence model predictions.     

On coupling the Reynolds‐averaged Navier–Stokes equations with two‐equation turbulence model equations

Two methods for coupling the Reynolds‐averaged Navier–Stokes equations with the q–ω turbulence model equations on structured grid systems have been studied; namely a loosely coupled method and a strongly coupled method. The loosely coupled method first solves the Navier–Stokes equations with the turbulent viscosity fixed. In a subsequent step, the turbulence model equations are solved with all flow quantities fixed. On the other hand, the strongly coupled method solves the Reynolds‐averaged Navier–Stokes equations and the turbulence model equations simultaneously. In this paper, numerical stabilities of both methods in conjunction with the approximated factorization‐alternative direction implicit method are analysed. The effect of the turbulent kinetic energy terms in the governing equations on the convergence characteristics is also studied. The performance of the two methods is compared for several two‐ and three‐dimensional problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Frequency lock-in phenomenon for oscillating airfoils in buffeting flows

Navier-Stokes based computer simulations are conducted to determine the aerodynamic flow field response that is observed for a NACA0012 airfoil that undergoes prescribed harmonic oscillation in transonic buffeting flows, and also in pre-buffet flow conditions. Shock buffet is the term for the self-sustained shock oscillations that are observed for certain combinations of Mach number and steady mean flow angle of attack even in the absence of structural motion. The shock buffet frequencies are typically on the order of the elastic structural frequencies, and therefore may be a contributor to transonic aeroelastic response phenomena, including limit-cycle oscillations. Numerical simulations indicate that the pre-shock-buffet flow natural frequency increases with mean angle of attack, while the flow damping decreases and approaches zero at the onset of buffet. Airfoil harmonic heave motions are prescribed to study the interaction between the flow fields induced by the shock buffet and airfoil motion, respectively. At pre-shock-buffet conditions the flow response is predominantly at the airfoil motion frequency, with some smaller response at multiplies of this frequency. At shock buffet conditions, a key effect of prescribed airfoil motions on the buffeting flow is to create the possibility of a lock-in phenomenon, in which the shock buffet frequency is synchronized to the prescribed airfoil motion frequency for certain combinations of airfoil motion frequencies and amplitudes. Aerodynamic gain-phase models for the lock-in region, as well as for the pre-shock-buffet conditions are suggested, and also a possible relationship between the lock-in mechanism and limit-cycle oscillation is discussed.     

Implementation of physical boundary conditions into computational domain in modelling of oscillatory bottom boundary layers

This paper discusses the importance of realistic implementation of the physical boundary conditions into computational domain for the simulation of the oscillatory turbulent boundary layer flow over smooth and rough flat beds. A mathematical model composed of the Reynolds averaged Navier–Stokes equation, turbulent kinetic energy (k) and dissipation rate of the turbulent kinetic energy (ε) has been developed. Control‐volume approach is used to discretize the governing equations to facilitate the numerical solution. Non‐slip condition is imposed on the bottom surface, and irrotational main flow properties are applied to the upper boundary. The turbulent kinetic energy is zero at the bottom, whereas the dissipation rate is approaching to a constant value, which is proportional to the kinematic viscosity times the second derivative of the turbulent kinetic energy. The output of the model is compared with the available experimental studies conducted in oscillatory tunnels and wave flume. It is observed that the irrotational flow assumption at the upper boundary is not realistic in case of water tunnels. Therefore, new upper boundary conditions are proposed for oscillatory tunnels. The data of wave flume show good agreement with the proposed numerical model. Additionally, several factors such as grid aspect ratio, staggered grid arrangement, time‐marching scheme and convergence criteria that are important to obtain a robust, realistic and stable code are discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

A SOLVER USING NEWTON‘S METHOD FOR UNSTEADY VISCOUS FLOWS

A solver is developed for time-accurate computations of viscous flows based on the conception of Newton‘s method. A set of pseudo-time derivatives are added into governing equations and the discretized system is solved using GMRES algorithm. Due to some special properties of GMRES algorithm, the solution procedure for unsteady flows could be regarded as a kind of Newton iteration. The physical-time derivatives of governing equations are discretized using two different approaches, I.e., 3-point Euler backward, and Crank-Nicolson formulas, both with 2nd-order accuracy in time but with different truncation errors. The turbulent eddy viscosity is calculated by using a version of Spalart~Allmaras one-equation model modified by authors for turbulent flows. Two cases of unsteady viscous flow are investigated to validate and assess the solver, I.e., low Reynolds number flow around a row of cylinders and transonic bi-circular-arc airfoil flow featuring the vortex shedding and shock buffeting problems, respectively. Meanwhile, comparisons between the two schemes of timederivative discretizations are carefully made. It is illustrated that the developed unsteady flow solver shows a considerable efficiency and the Crank-Nicolson scheme gives better results compared with Euler method.     

Numerical modeling of transonic buffeting and its control by means of tangential jet blowing

Flow past the airfoil of a civil aircraft wing is numerically modeled in the transonic buffeting regimes. The characteristics of unsteady self-oscillations of the shock on the upper surface of the airfoil are studied. To suppress them a technique of controlling the flow is proposed. It consists in blowing a jet tangent to the upper surface of the airfoil from a small-size slot nozzle. The calculations are carried out within the framework of the unsteady Reynolds equations.     

Unsteady CFD computations using vertex‐centered finite volumes for unstructured grids on Graphics Processing Units

This paper presents a Navier–Stokes solver for steady and unsteady turbulent flows on unstructured/hybrid grids, with triangular and quadrilateral elements, which was implemented to run on Graphics Processing Units (GPUs). The paper focuses on programming issues for efficiently porting the CPU code to the GPU, using the CUDA language. Compared with cell‐centered schemes, the use of a vertex‐centered finite volume scheme on unstructured grids increases the programming complexity since the number of nodes connected by edge to any other node might vary a lot. Thus, delicate GPU memory handling is absolutely necessary in order to maximize the speed‐up of the GPU implementation with respect to the Fortran code running on a single CPU core. The developed GPU‐enabled code is used to numerically study steady and unsteady flows around the supercritical airfoil OAT15A, by laying emphasis on the transonic buffet phenomenon. The computations were carried out on NVIDIA's Ge‐Force GTX 285 graphics cards and speed‐ups up to ～46 × (on a single GPU, with double precision arithmetic) are reported. Copyright © 2010 John Wiley & Sons, Ltd.     

Validating the URANS shear stress transport γ −Reθ model for low‐Reynolds‐number external aerodynamics

A numerical investigation on low‐Reynolds‐number external aerodynamics was conducted using the transitional unsteady Reynolds‐averaged Navier–Stokes shear stress transport γ ?Re_θ model and the ANSYS‐CFX computational fluid dynamics suite. The NACA 0012 airfoil was exposed to chord‐based Reynolds numbers of 5.0 ×10⁴, 1.0 ×10⁵ and 2.5 ×10⁵ at 0°, 4°and 8°angles of attack. Time‐averaged and instantaneous flow features were extracted and compared with fully turbulent shear stress transport results, XFLR5 panel e ^N method results, and published higher order numerical and experimental studies. The current model was shown to reproduce the complex flow phenomena, including the laminar separation bubble dynamics and aerodynamic performance, with a very good degree of accuracy. The sensitivity of the model to domain size, grid resolution and quality, timestepping scheme, and free‐stream turbulence intensity was also presented. In view of the results obtained, the proposed model is deemed appropriate for modelling low‐Reynolds‐number external aerodynamics and provides a framework for future studies for the better understanding of this complex flow regime. Copyright © 2011 John Wiley & Sons, Ltd.     

A domain decomposition approach to compute wave breaking (wave‐breaking flows)

A heterogeneous domain decomposition approach is followed to simulate the unsteady wavy flow generated by a body moving beneath a free surface. Attention being focused on complex free surface configurations, including wave‐breaking phenomena, a two‐fluid viscous flow model is used in the free surface region to capture the air–water interface (via a level‐set technique), while a potential flow approximation is adopted to describe the flow far from the interface. Two coupling strategies are investigated, differing in the transmission conditions. Both the adopted approaches make use of the inviscid velocity field as boundary condition in the Navier–Stokes solution. For validation purposes, two different two‐dimensional non‐breaking flows are simulated. Domain decomposition results are compared with both fully viscous and fully inviscid results, obtained by solving the corresponding equations in the whole fluid domain, and with available experimental data. Finally, the unsteady evolution of a steep breaking wave is followed and some of the physical phenomena, experimentally observed, are reproduced. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical simulation of turbulent free surface flow with two‐equation k–ε eddy‐viscosity models

This paper presents a finite difference technique for solving incompressible turbulent free surface fluid flow problems. The closure of the time‐averaged Navier–Stokes equations is achieved by using the two‐equation eddy‐viscosity model: the high‐Reynolds k–ε (standard) model, with a time scale proposed by Durbin; and a low‐Reynolds number form of the standard k–ε model, similar to that proposed by Yang and Shih. In order to achieve an accurate discretization of the non‐linear terms, a second/third‐order upwinding technique is adopted. The computational method is validated by applying it to the flat plate boundary layer problem and to impinging jet flows. The method is then applied to a turbulent planar jet flow beneath and parallel to a free surface. Computations show that the high‐Reynolds k–ε model yields favourable predictions both of the zero‐pressure‐gradient turbulent boundary layer on a flat plate and jet impingement flows. However, the results using the low‐Reynolds number form of the k–ε model are somewhat unsatisfactory. Copyright © 2004 John Wiley & Sons, Ltd.     

High-order strand grid methods for shock turbulence interaction

ABSTRACT

In this work, we examine the flux correction method for three-dimensional transonic turbulent flows on strand grids. Building upon previous work, we treat flux derivatives along strands with high-order summation-by-parts operators and penalty-based boundary conditions. A finite-volume like limiting strategy is implemented in the flux correction algorithm in order to sharply capture shocks. To achieve turbulence closure in the Reynolds-Averaged Navier–Stokes equations, a robust version of the Spalart–Allmaras turbulence model is employed that accommodates negative values of the turbulence working variable. Validation studies are considered which demonstrate the flux correction method achieves a high degree of accuracy for turbulent shock interaction flows.     

Coupled fluid structure analysis for wing 445.6 flutter using a fast dynamic mesh technology

The flow around wing 445.6 was modelled using Navier–Stokes equations and S-A model. The wing vibration and flow mesh deformation were computed using a fast dynamic mesh technology proposed by our own group. Wing 445.6 flutter was analysed through a strong coupling between the wing vibration and flow. The reduced flutter velocity was predicted and results are in good agreement with the experimental data. It is found that the subsonic flutter is mainly induced by the flow separation and the transonic and supersonic flutter are mainly caused by the oscillating shock wave and its induced flow separation. The positive aerodynamic work increases due to the oscillating shock wave when the subsonic flow becomes transonic reducing the flutter velocity. While the positive aerodynamic work induced by the oscillating shock wave decreases when the transonic flow becomes supersonic increasing the flutter velocity. That is why the transonic dip exists.