A direct numerical simulation (DNS) of a channel flow with one curved surface was performed at moderate Reynolds number (Re_tau = 395 at the inlet). The adverse pressure gradient was obtained by a wall curvature through a mathematical mapping from physical coordinates to Cartesian ones. The code, using spectral spanwise and normal discretization, combines the advantage of a good accuracy with a fast integration procedure compared to standard numerical procedures for complex geometries. The turbulent flow slightly separates on the profile at the lower curved wall and is at the onset of separation at the opposite flat wall. The thin separation bubble is characterized with a reversal flow fraction. Intense vortices are generated near the separation line on the lower wall but also at the upper wall. Turbulent normal stresses and kinetic energy budget are investigated along the channel.
In the last two decades, the control of flow around an airfoil is of strong interest. The aim is to delay or to suppress the detachment of the turbulent boundary layer in several flight phases of an airplane. The turbulent boundary layer submitted to an adverse pressure gradient is thus one of the main research effort in aeronautics and more generally in transport. It is imperative to improve our knowledge on near wall turbulence submitted to pressure gradient in order to develop new strategies for the control acting on the turbulent coherent structures.
Most of the knowledge on this type of flow have been obtained experimentally using hot wire anemometry. There is several ways to generate a pressure gradient experimentally, which lead to very different pressure gradient configurations, constant or varying in space, with or without separation. It is not the aim, here, to make an extended review of the subject but just to present relevant contributions related to the present study. Skare & Krogstad [19,9] did a detailed study of a boundary-layer in a strong adverse pressure gradient with a constant skin friction coefficient, they provided a thorough analysis of the turbulence statistics of the flow, in particular the budget of the kinetic energy. Webster et al [23] also provide a detailed analysis of the turbulence statistics of their experiment of a boundary-layer flow, but their adverse pressure gradient is created by a surface bump (quite similar to the one used in this paper) with concave and convex regions. In their experimental study of a boundary layer flow with an adverse pressure gradient, Dengel & Fernholz [4] compares different cases of pressure distribution, with and without reverse flow (transitory detachment in the classification of Simpson [18]). They show that the presence of recirculation leads to a significant change in the near-wall properties of the flow, even upstream of the reverse flow region. These different studies provided a useful description of the velocity fluctuation near the wall but the use of numerical simulations, which give all the information of the flow field, should allow to bring a new insight on wall turbulence with pressure gradient.
One of the easiest way to introduce an adverse pressure gradient is to use suction at one wall of a plane channel flow or to directly prescribe the adverse pressure gradient. Na and Moin [15,16] performed a DNS of separated boundary layer flow on a flat plate using suction-blowing velocity distribution at the upper wall and second order finite difference scheme for spatial derivative. The inflow condition was taken from Spalart’s temporal zero pressure gradient (ZPG) simulation. Chong et al [3] used the data from the DNS of Na & Moin and from the Spalart’s DNS of ZPG flow [21] to analyze the topology of near wall coherent structures using invariant of the velocity gradient tensor. Spalart [22] compared experimental and DNS results of a boundary layer flow with pressure gradient. The DNS of Spalart was performed using a spectral code with a fringe region in order to deal with periodic conditions in the non-homogeneous streamwise direction and a friction velocity at the edge of the boundary layer was prescribed to reproduce the pressure gradient of the experiment. More recently, Skote and Henningson [20] performed DNS of separated boundary layer flow with two different adverse pressure gradients at Reynolds numbers Re * δ = 400 at the inlet (where δ * is the momentum thickness).The advantage of those methods, with a prescription of the pressure gradient, is the possibility to use a numerical code adapted to plane boundary layers or plane channel flows. However the use of curved walls is more challenging. Some recent simulations of such flows have been conducted with finite differences or finite volume codes which are usually less efficient than spectral codes to perform a highly resolved simulation. Neumann et al [17]) have investigated the effect of flow control on the flow separation which requires a fine enough spatial discretization to capture the smallest structures of the flow. It means that only DNS or highly resolved LES can be used. Wu et al [24] have performed a LES of a boundary layer over a smooth bump and have compared the results to an experiment conducted earlier by Wester et al [23]. However, the use of coarse resolution with an eddy viscosity model did not allow an accurate investigation of very small coherent structures close to the wall.
The numerical code used for the present study combines the advantages of the good accuracy of a spectral resolution and a fast integration procedure for simulations over a smooth profile. This code was originally developed to study the 2D and 3D instabilities of a boundary layers over a bump [11,12]. It was here adapted to investigate the effect of a pressure gradient on the turbulent structures at moderate Reynolds number: Re = hU max /ν = 7900 where h is half the channel width and U
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