International Journal of Rotating Machinery

Volume 2015 (2015), Article ID 650783, 15 pages

http://dx.doi.org/10.1155/2015/650783

## Turbulent Kinetic Energy Production in the Vane of a Low-Pressure Linear Turbine Cascade with Incoming Wakes

Institut für Hydromechanik, Universität Karlsruhe, Kaiserstrasse 12, 76128 Karlsruhe, Germany

Received 5 November 2014; Accepted 4 January 2015

Academic Editor: Funazaki Ken-ichi

Copyright © 2015 V. Michelassi and J. G. Wissink. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Incompressible large eddy simulation and direct numerical simulation of a low-pressure turbine at and with discrete incoming wakes are analyzed to identify the turbulent kinetic energy generation mechanism outside of the blade boundary layer. The results highlight the growth of turbulent kinetic energy at the bow apex of the wake and correlate it to the stress-strain tensors relative orientation. The production rate is analytically split according to the principal axes, and then terms are computed by using the simulation results. The analysis of the turbulent kinetic energy is followed both along the discrete incoming wakes and in the stationary frame of reference. Both direct numerical and large eddy simulation concur in identifying the same production mechanism that is driven by both a growth of strain rate in the wake, first, followed by the growth of turbulent shear stress after. The peak of turbulent kinetic energy diffuses and can eventually reach the suction side boundary layer for the largest Reynolds number investigated here with higher incidence angle. As a consequence, the local turbulence intensity outside the boundary layer can grow significantly above the free-stream level with a potential impact on the suction side boundary layer transition mechanism.

#### 1. Introduction

Typically, the time-averaged flow field of undisturbed plane wakes consists of two equally strong, slowly diverging vortex sheets of opposite orientation. The rate at which the sheets diverge is proportional to , where is the distance to the origin of the wake (see, e.g., Schlichting [1]), indicating that for large the two vortex sheets will be roughly parallel. Each vortex sheet corresponds to a plane shear layer not unlike a flat plate boundary layer. In a turbulent plane wake, each shear layer is associated with a peak in the turbulent kinetic energy, , indicating that the production of is concentrated near the regions of high shear.

Castro and Bradshaw [2] and Gibson and Rodi [3] analysed the flow and turbulence structure of a highly curved mixing layer. The mixing layer under investigation bounds a normally impinging plane irrotational jet. In the experiment, turbulence was discovered to be first attenuated. Then, approximately after the first 50–60% of the curve, the normal and shear stresses were first amplified and exceeded the plane-layer values reached in relaxation region further downstream when the flow curvature vanishes. The analysis indicated that the overshoot of turbulent quantities was mostly due to shear-stress production.

In the absence of strain, Moser et al. [4] observed a self-similar evolution of plane wakes. The effect of the presence of uniform mean strain was studied by Rogers [5], who performed direct numerical simulations of turbulent, time-evolving strained wakes using a pseudo-spectral method. In all his simulations, the strain was applied to the same self-similar wake flow field. He found that though the main flow reacts quickly on the applied strain, the response of turbulence to strain is slower; changes in the turbulence intensity could not keep pace with changes in the mean wake velocity. Turbulence is produced by two competing mechanisms: shear and strain. Rogers found that, when the direction of compression is parallel to the centre-line of the wake (case C), the wake-width grows exponentially as the wake velocity deficit increases. The normal Reynolds stresses, , , , and also are all found to increase in time while the typical structure of the time-averaged wake, that is, the two more or less parallel shear layers, remains intact. When the direction of expansion is parallel to the centre-line of the wake (case D), the wake-width, the wake velocity deficit, and the Reynolds stresses all decrease in time, eventually degrading the structure of the wake.

In summary, the measurements by Castro and Bradshaw [2] allow studying the evolution of shear layers (i.e. half portion of a wake) in presence of strong flow-core turning, whereas the DNS by Rogers [5] focus on the effect of strain on planar straight wakes. Both are relevant to the flow in a turbomachine in which the wakes produced by the preceding blade row are periodically ingested into a blade vane. In particular, while a plane wake travels through the passage between two turbine blades it is severely strained and distorted by the main flow. In contrast to the study of Rogers, the actual direction of the mean strain relative to the centre-line of the wake varies with the actual location. Moreover, the direction of shear, individuated by the wakes, differs from the flow direction because of the relative motion between blades and wakes. Hence, differences arise with respect to the flow geometry by Castro and Bradshaw too, in which the direction of shear is aligned with the core flow.

Wu and Durbin [6] performed the DNS of the flow in a low-pressure (LP) linear turbine rotor blade with periodic incoming wakes. The wakes are subject to both flow curvature (as in Castro and Bradshaw [2]) and strain (as in Rogers [5]). The simulations revealed a peak in the turbulent kinetic energy located near the bow-apex of the wake, where the direction of compression is aligned with the centre-line of the wake, corresponding to case C of Rogers. Near the pressure side Wu and Durbin observed that the direction of expansion was almost aligned with the centre-line of the wake, corresponding to case D of Rogers. In the present paper we aim to further analyse the production of turbulence in the plane turbulent wake as it travels through the free-stream region of a turbine cascade.

#### 2. Description of the Test Cases

The geometry under investigation is that of the low-pressure aft-loaded turbine blade T106. The flow around this blade assembled in a linear cascade was measured by Stadtmüller [7] and by Stadtmüller and Fottner [8]. In the experiments the blade aspect ratio (h/c) is 1.76, and the blade chord is 100 mm. Therefore the flow at mid-span can be considered nearly two-dimensional and the three-dimensional computer simulations can be performed under the assumption of a homogeneous flow in the span-wise direction. The measurements were carried out at Re = 5.18 × 10^{4} and 2 × 10^{5} (based on inlet conditions and axial chord) with the effect of an upstream row of blades simulated by a moving bar wake generator. The bar diameter to pitch ratio is . Unfortunately, for the larger Re only, the bar-to-blade pitch ratio is not an integer number (0.4/0.525). To yield an exactly periodic flow, the simulations should hence be carried out by using eight blade vanes, which would require an excessive computational effort. Therefore we will focus on the DNS data by Wissink [9] and the LES data by Michelassi et al. [10], which refer both to the flow at the lower Reynolds number with a wake-to-blade pitch ratio of 0.5 (see test L in Table 1). The same geometry was also selected by Wu and Durbin [6] for a DNS at the significantly larger Re of 1.48 × 10^{5} with a wake-to-blade pitch ratio of one. This DNS was used as a reference data set by Michelassi et al. [10] who performed the LES of the same flow. The latter data set will also be used for further analysis under different operating conditions with respect to [9, 10] (see test H in Table 1).