Effect of Discrete Roughness Elements on Flow over Low-Pressure Turbines

In this paper, the influence of an array of discrete roughness elements on the evolution of three-dimensional instabilities of a boundary layer subject to a strong adverse pressure gradient is investigated by means of direct numerical simulation (DNS). The flow configuration is a linear T106 blade cascade with and without roughness elements installed, operating at an isentropic exit Reynolds number of 60,000 and an exit Mach number of 0.4. The height of the roughness elements considered yields the local roughness Reynolds number of 415, at which steady laminar streamwise vortices form. These steady streaks reduce the size of the suctionside flow separation by accelerating the breakdown process. The streaky vortical structures distort the separated shear layer, resulting in short-span Kelvin–Helmholtz vortices. Further downstream, the streamwise vortices lift the low-momentum fluid from the nearwall region, leading to a strong instability of the streaks and rapid breakdown to turbulence. INTRODUCTION Decreasing aerofoil drag is of great importance since it can provide a significant reduction in aircraft fuel burn. In a modern aircraft engine, the low-pressure turbine (LPT) blades generally operate at low Reynolds numbers, ranging from 50,000 to 500,000. This results in the flow on most of the suction side being laminar, except in the region close to the trailing edge where flow separation can occur due to an adverse pressure gradient and the separated shear layer typically undergoes rapid transition to turbulence. Reducing the size of the resulting separation bubble is desirable to obtain an increase in the efficiency of the LPT, which can result in a decrease in fuel consumption. For a typical adverse-pressure-gradient (APG) boundary layer, transition to turbulence occurs via an inflectional inviscid instability. The mechanisms of this instability can be explained as follows. When the flow is decelerating, the velocity profile becomes inflectional. The boundary layer then separates from the surface. This separated shear layer rolls up into two-dimensional coherent vortices, often referred to as ‘Kelvin–Helmholtz’ (KH) rollers. After the KH rollers break down to turbulence, the separated shear layer reattaches to the surface (for a recent review of various routes to transition of boundary layers in turbine flows, see Coull & Hodson, 2011). When a single or distributed roughness is placed in a zeropressure-gradient (ZPG) boundary layer, counter-rotating streamwise vortices appear behind it. This results in an upstream shift of the transition front of an undisturbed boundary layer. The roughness wake vortices can be either steady or unsteady depending on the local roughness Reynolds number Rek = ukk/ν , where uk is the velocity at the roughness height k, and ν is the kinematic viscosity. The critical Reynolds number is in the range between 600 and 900, depending on roughness shape (Klebanoff et al., 1992). Since roughness elements can accelerate the transition process, they can be employed to APG flow to promote earlier reattachment, which results in a decrease in the size of a separation bubble. Recent numerical works reveal that roughness elements can shorten the bubble if they are placed upstream of the transition point (Rao et al., 2014; Karaca & Gungor, 2016). However, since those simulations investigated the influence of the roughness elements on a flat plate with an adverse pressure gradient, some effects that may appear only in a real blade configuration (e.g. surface curvature or circulation changes) were not included. The aim of this work is to investigate the effect of spanwise periodic arrays of roughness elements on a low pressure turbine blade. The focus is on how the streaky vortical structures alter the inviscid instability of LPT flow. NUMERICAL APPROACH The compressible Navier–Stokes equations for conservative variables are solved in curvilinear coordinates via an in-house code, HiPSTAR. The flow solver employs a 4th-order optimised compact central finite-difference scheme (Kim & Sandberg, 2012) to discretise all the spatial derivatives in the axial (x) and pitchwise (y) directions, while a Fourier method is used in the spanwise (z) direction. Temporal advancement is obtained by an ultra-low-storage five-step 4th-order Runge–Kutta scheme (Kennedy et al., 2000). To enhance the stability of the code, a skew-symmetric splitting scheme of the convective terms and a pentadiagonal compact filter (Kim, 2010), applied after every full time-step with a cut-off wavenumber of 0.88, are used. The code has been validated with various benchmark cases and recently been applied to perform DNS of a linear LPT cascade (Sandberg et al., 2015; Michelassi et al., 2015). To represent discrete roughness elements, a second-order boundary data immersion method (BDIM) of Schlanderer et al. (2017) is employed. The reference velocity and length for the LPT DNS are respectively the blade exit velocity u2is and the blade chord length C. Hence, the dimensionless parameters (i.e. Reynolds number Re and Mach number M) are defined based on the blade exit conditions. The Prantdl number and the ratio of the specific heat are set to be constant at Pr = 0.72 and γ = 1.4. The molecular viscosity μ is computed using Sutherland’s law with the Sutherland constant of 0.3686. The system of equations is closed by solving the nondimensional equation of state to obtain the pressure. RESULTS In this section, the effect of roughness elements on a boundary layer is explored. A simulation of flat-plate flow with roughness elements is first conducted to validate the implementation of the BDIM. Then, results of flow over a LPT blade with and without roughness elements are presented. Roughness Elements embedded on a Flat Plate Before exploring the effect of the roughness elements on the real blade geometry, the capability of the numerical technique used to represent a three-dimensional roughness element is first investigated by computing flow over a flat plate with an array of roughness