Statistical behavior of supersonic turbulent boundary layers with heat transfer at M 1 ¼ 2 M.S. Shadloo a,b , A. Hadjadj b,⇑ , F. Hussain c a LHEEA Lab., Ecole Centrale de Nantes/CNRS (UMR 6598), 44321 Nantes cedex 3, France b CORIA-UMR 6614, Normandie University, CNRS-University & INSA of Rouen, 76800 Saint Etienne du Rouvray, France c Mechanical Engineering Department, Texas Tech. University, Lubbock, TX 79409-1021, USA article info Article history: Received 13 August 2014 Received in revised form 19 January 2015 Accepted 23 February 2015 Keywords: Supersonic turbulent boundary layer Compressible flow Isothermal wall Wall heat transfer Direct numerical simulation (DNS) Turbulent kinetic energy (TKE) budget abstract Direct numerical simulations (DNS) of supersonic turbulent boundary layers (STBL) over adiabatic and isothermal walls are performed to investigate the effects of wall heat transfer on turbulent statistics and near wall behaviors. Four different cases of adiabatic, quasi-adiabatic, and uniform hot and cold wall temperatures are considered. Based on the analysis of the current database, it is observed that even though the turbulent Mach number is below 0.3, the wall heat transfer modifies the behavior of near-wall turbulence. These modifications are investigated and identified using both instantaneous fields (i.e. scat- ter plots) and mean quantities. Morkovin’s hypothesis for compressible turbulent flows is found to be valid for neither heated nor cooled case. It is further uncovered that although some near-wall asymptotic behaviors change upon using weak or strong adiabatic walls, respectively denote the isothermal and iso- flux walls, basic turbulent statistics are not affected by the thermal boundary condition itself. We also show that among different definition of Reynolds number used in STBL, the Reynolds number based on the friction velocity has some advantages data comparison regarding the first and second order sta- tistical moments. More in depth analyses are also performed using the balance equation for turbulent kinetic energy (TKE) budget, as well as the dissipation rate. It is found that the dilatational to solenoidal dissipation ratio increases/decreases when heating/cooling the walls. The DNS of the current STBLs are available online for the community. Ó 2015 Elsevier Inc. All rights reserved. 1. Introduction Turbulence modeling of high-speed flows is often based on the compressibility assumptions. Morkovin suggested that the turbu- lence is weakly affected by compressibility if the turbulent Mach number is smaller than unity (i.e. M t ¼ ffiffiffiffiffiffiffiffi u 0 i u 0 i q = c  1). This condi- tion is satisfied for flows at moderate free-stream Mach numbers (M 1 < 5). The Morkovin’s hypothesis is described as follows (Bradshaw, 1977): In non-hypersonic wall-bounded shear flows where the pressure and the total temperature fluctuations are much smaller than their corresponding mean quantities (i.e. respectively, the acoustic and entropic modes are negligible), the coupling between sound and thermal fields primarily arises due to the temporal and spatial variations of density, viscosity and heat conductivity. One consequence of this coupling is that as long as variations of the mean flow properties are taken into account, the high-speed (compressible) bounded shear flow resembles that of the low-speed one (incompressible). This is the main reason why the density-weighted velocity scaling of Van Driest for adiabatic walls was successful (Van Driest, 1951; Dussauge and Smits, 1996). The full analogy between the temperature and velocity fields is the other consequence of Morkovin’s hypothesis. The analogy appears at three different levels between: (i) temperature variation and wall friction (Bradshaw, 1977; Schlichting et al., 1968; Lele, 1994), (ii) mean and local velocity and temperature fields (the so called Corocco–Busemann relation(Busemann, 1931; Crocco, 1932)), and (iii) turbulent stresses and heat fluxes. The last is the most important consequence of Morkovin’s hypothesis, known as strong Reynolds analogy (SRA) (Morkovin, 1962). Several scientists attempted to solve bounded shear layer flow problems numerically using direct numerical simulation (DNS) and large eddy simulation (LES) for the wide range of problem parameters (Coleman et al., 1995; Sarkar, 1995; Foysi et al., 2004; Zhang et al., 2012, 2014; Morinishi et al., 2004; Brun et al., 2008). However, concerning heat transfer in a compressible turbu- lent boundary layer over an isothermal flat-plate, only a few http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.02.004 0142-727X/Ó 2015 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail address: hadjadj@coria.fr (A. Hadjadj). International Journal of Heat and Fluid Flow 53 (2015) 113–134 Contents lists available at ScienceDirect International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff