Direct numerical simulation of 2D transonic flows around airfoils Tapan K. Sengupta ⇑ , Ashish Bhole, N.A. Sreejith Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India article info Article history: Received 4 May 2013 Received in revised form 7 August 2013 Accepted 20 August 2013 Available online 28 August 2013 Keywords: Navier–Stokes equation Transonic flow Airfoil aerodynamics Shock capturing Shock-boundary layer interaction abstract High-accuracy, time-accurate compressible Navier–Stokes solvers have been developed for transonic flows. These solvers use optimized upwind compact schemes (OUCS) and four-stage, fourth order explicit Runge–Kutta (RK4) time integration scheme, details of which can be obtained in Sengupta [Sengupta TK. High Accuracy Computing Methods, fluid flows and wave phenomena. UK: Cambridge Univ. Press; 2013]. Although these compact schemes have been developed originally for direct simulation of incompressible flows, it is shown here that the same can be used for compressible flows, with shock-boundary layer interactions clearly captured for flow past NACA 0012 and NLF airfoils. Numerical higher order diffusion terms which are used for incompressible flows, have been replaced here by the pressure-based artificial diffusions proposed by Jameson et al. [Jameson A, Schmidt W, Turkel E. Numerical solution of the Euler equations by finite volume methods using Ruge–Kutta time stepping schemes. AIAA Paper 1981-1259. AIAA 14th fluid and plasma dynamics conference. Palo Alto, CA; 1981]. Such second and fourth order dif- fusion terms are used adaptively at selective points, located by the pressure switch. Developed compu- tational methods used here are validated for cases with and without shocks, for which experimental results are available. Apart from surface pressure coefficient, contours of physical quantities are pre- sented to explain the time-accurate results. Presented methods are robust and the results can be gainfully used to study shock formation, drag divergence and buffet onset of flow over airfoils. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Transonic flows represent many complex phenomena, such as creation of shock, contact discontinuities, shock-boundary-layer interaction and buffet onset. Such flows are strongly time-depen- dent and cannot be fully understood without reference to viscous nature of the fluid, as viscous terms are important in their interac- tion with the convective process [1]. In real viscous flows, changes in flow field occur in non-monotonic fashion and produce surface fluctuations, even in the absence of unsteady boundary conditions [2]. Therefore, transonic flows must be described by the time- dependent, compressible Navier–Stokes equations. Numerical methods for solving Navier–Stokes equations must be capable of predicting all the above flow features accurately. In [3], authors summarize the developments in CFD ranging from solution of po- tential equation to Reynolds Averaged Navier–Stokes equation (RANS) and use of software packages to model transonic flows. Use of numerical diffusion and turbulence models are common in the simulation of transonic flows. In [4,5] turbulence models have been used, whereas direct numerical simulation (DNS) of transonic flows can be found in [6]. The authors report DNS results to inves- tigate upstream moving pressure waves over a supercritical airfoil. In the literature, various numerical schemes have been proposed for calculation of compressible flows. For example, Allaneu and Jame- son [7] proposed kinetic energy preserving scheme; Chiu et al. [8] have used a conservative meshless scheme. Other notable refer- ences are with implicit factored scheme [9], essentially non-oscilla- tory (ENO) shock capturing scheme [10], weighted essentially non- oscillatory (WENO) scheme [11]. In the present work, we use high accuracy, dispersion relation preserving (DRP), optimized upwind compact schemes (OUCS), which were originally developed for incompressible flows [12–14]. DNS of two-dimensional wall bounded turbulent flow is also reported in [15], where OUCS3 scheme for spatial discretization have been used. In [16], symme- trized version of the OUCS4 scheme have been used in parallel com- puting framework, to solve three-dimensional unsteady compressible Navier–Stokes equations for supersonic flow past a cone–cylinder for M 1 = 4 and Re 1 = 1.12  10 6 . Parallel computing technique have been developed in this reference by solving the con- vection of a shielded vortex in subsonic flow. By solving and vali- dating two-dimensional transonic flows around NACA 0012 and SHM-1 airfoils using two OUCS schemes, we show here that the same compact schemes can be used for transonic flows as well. Since the validity of computation is based on available experimental results, knowledge of experimental conditions and possible errors associated with the data should be known quantitatively. In [5,17], problems related to transonic wind tunnel 0045-7930/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compfluid.2013.08.007 ⇑ Corresponding author. Tel.: +91 512 2597945. E-mail address: tksen@iitk.ac.in (T.K. Sengupta). Computers & Fluids 88 (2013) 19–37 Contents lists available at ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid