MEASURED AND PREDICTED LOSS GENERATION IN TRANSONIC TURBINE BLADING.

For a typical transonic turbine rotor blade, designed for use with coolant ejection, the trailing edge, or base loss is three to four times the profile boundary layer loss. The base region of such a profile is dominated by viscous effects and it seems essential to attack the problem of loss prediction by solving the compressible Navier-Stokes equations. However, such an approach is inevitably compromised by both numerical accuracy and turbulence modelling constraints. This paper describes a Navier-Stokes solver written for 2D blade-blade flows and employing a simple two-layer mixing length eddy viscosity model. Then, measured and predicted losses and base pressures are presented for two transonic rotor blades and attempts are made to assess the capabilities of the Navier-Stokes solver and to outline areas for future work.

An accelerated 3D Navier-Stokes solver for flows in turbomachines

A new three-dimensional Navier-Stokes solver for flows in turbomachines has been developed. The new solver is based on the latest version of the Denton codes, but has been implemented to run on Graphics Processing Units (GPUs) instead of the traditional Central Processing Unit (CPU). The change in processor enables an order-of-magnitude reduction in run-time due to the higher performance of the GPU. Scaling results for a 16 node GPU cluster are also presented, showing almost linear scaling for typical turbomachinery cases. For validation purposes, a test case consisting of a three-stage turbine with complete hub and casing leakage paths is described. Good agreement is obtained with previously published experimental results. The simulation runs in less than 10 minutes on a cluster with four GPUs. Copyright © 2009 by ASME.

Shallow water equations: Magic and limitations

D Liang from Cambridge University explains the shallow water equations and their applications to the dam-break and other steep-fronted flow modeling. They assume that the horizontal scale of the flow is much greater than the vertical scale, which means the flow is restricted within a thin layer, thus the vertical momentum is insignificant and the pressure distribution is hydrostatic. The left hand sides of the two momentum equations represent the acceleration of the fluid particle in the horizontal plane. If the fluid acceleration is ignored, then the two momentum equations are simplified into the so-called diffusion wave equations. In contrast to the SWEs approach, it is much less convenient to model floods with the Navier-Stokes equations. In conventional computational fluid dynamics (CFD), cumbersome treatments are needed to accurately capture the shape of the free surface. The SWEs are derived using the assumptions of small vertical velocity component, smooth water surface, gradual variation and hydrostatic pressure distribution.

Classical plume theory: 1937-2010 and beyond

Developing a theoretical description of turbulent plumes, the likes of which may be seen rising above industrial chimneys, is a daunting thought. Plumes are ubiquitous on a wide range of scales in both the natural and the man-made environments. Examples that immediately come to mind are the vapour plumes above industrial smoke stacks or the ash plumes forming particle-laden clouds above an erupting volcano. However, plumes also occur where they are less visually apparent, such as the rising stream of warmair above a domestic radiator, of oil from a subsea blowout or, at a larger scale, of air above the so-called urban heat island. In many instances, not only the plume itself is of interest but also its influence on the environment as a whole through the process of entrainment. Zeldovich (1937, The asymptotic laws of freely-ascending convective flows. Zh. Eksp. Teor. Fiz., 7, 1463-1465 (in Russian)), Batchelor (1954, Heat convection and buoyancy effects in fluids. Q. J. R. Meteor. Soc., 80, 339-358) and Morton et al. (1956, Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A, 234, 1-23) laid the foundations for classical plume theory, a theoretical description that is elegant in its simplicity and yet encapsulates the complex turbulent engulfment of ambient fluid into the plume. Testament to the insight and approach developed in these early models of plumes is that the essential theory remains unchanged and is widely applied today. We describe the foundations of plume theory and link the theoretical developments with the measurements made in experiments necessary to close these models before discussing some recent developments in plume theory, including an approach which generalizes results obtained separately for the Boussinesq and the non-Boussinesq plume cases. The theory presented - despite its simplicity - has been very successful at describing and explaining the behaviour of plumes across the wide range of scales they are observed. We present solutions to the coupled set of ordinary differential equations (the plume conservation equations) that Morton et al. (1956) derived from the Navier-Stokes equations which govern fluid motion. In order to describe and contrast the bulk behaviour of rising plumes from general area sources, we present closed-form solutions to the plume conservation equations that were achieved by solving for the variation with height of Morton's non-dimensional flux parameter Γ - this single flux parameter gives a unique representation of the behaviour of steady plumes and enables a characterization of the different types of plume. We discuss advantages of solutions in this form before describing extensions to plume theory and suggesting directions for new research. © 2010 The Author. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Structural sensitivity of spiral vortex breakdown

Previous numerical simulations have shown that vortex breakdown starts with the formation of a steady axisymmetric bubble and that an unsteady spiralling mode then develops on top of this. We investigate this spiral mode with a linear global stability analysis around the steady bubble and its wake. We obtain the linear direct and adjoint global modes of the linearized Navier-Stokes equations and overlap these to obtain the structural sensitivity of the spiral mode, which identifies the wavemaker region. We also identify regions of absolute instability with a local stability analysis. At moderate swirls, we find that the m=-1 azimuthal mode is the most unstable and that the wavemaker regions of the m=-1 mode lie around the bubble, which is absolutely unstable. The mode is most sensitive to feedback involving the radial and azimuthal components of momentum in the region just upstream of the bubble. To a lesser extent, the mode is also sensitive to feedback involving the axial component of momentum in regions of high shear around the bubble. At an intermediate swirl, in which the bubble and wake have similar absolute growth rates, other researchers have found that the wavemaker of the nonlinear global mode lies in the wake. We agree with their analysis but find that the regions around the bubble are more influential than the wake in determining the growth rate and frequency of the linear global mode. The results from this paper provide the first steps towards passive control strategies for spiral vortex breakdown. © 2013 Cambridge University Press.

Absolute and convective instability in gas turbine fuel injectors

Hydrodynamic instabilities in gas turbine fuel injectors help to mix the fuel and air but can sometimes lock into acoustic oscillations and contribute to thermoacoustic instability. This paper describes a linear stability analysis that predicts the frequencies and strengths of hydrodynamic instabilities and identifies the regions of the flow that cause them. It distinguishes between convective instabilities, which grow in time but are convected away by the flow, and absolute instabilities, which grow in time without being convected away. Convectively unstable flows amplify external perturbations, while absolutely unstable flows also oscillate at intrinsic frequencies. As an input, this analysis requires velocity and density fields, either from a steady but unstable solution to the Navier-Stokes equations, or from time-averaged numerical simulations. In the former case, the analysis is a predictive tool. In the latter case, it is a diagnostic tool. This technique is applied to three flows: a swirling wake at Re = 400, a single stream swirling fuel injector at Re - 106, and a lean premixed gas turbine injector with five swirling streams at Re - 106. Its application to the swirling wake demonstrates that this technique can correctly predict the frequency, growth rate and dominant wavemaker region of the flow. It also shows that the zone of absolute instability found from the spatio-temporal analysis is a good approximation to the wavemaker region, which is found by overlapping the direct and adjoint global modes. This approximation is used in the other two flows because it is difficult to calculate their adjoint global modes. Its application to the single stream fuel injector demonstrates that it can identify the regions of the flow that are responsible for generating the hydrodynamic oscillations seen in LES and experimental data. The frequencies predicted by this technique are within a few percent of the measured frequencies. The technique also explains why these oscillations become weaker when a central jet is injected along the centreline. This is because the absolutely unstable region that causes the oscillations becomes convectively unstable. Its application to the lean premixed gas turbine injector reveals that several regions of the flow are hydrodynamically unstable, each with a different frequency and a different strength. For example, it reveals that the central region of confined swirling flow is strongly absolutely unstable and sets up a precessing vortex core, which is likely to aid mixing throughout the injector. It also reveals that the region between the second and third streams is slightly absolutely unstable at a frequency that is likely to coincide with acoustic modes within the combustion chamber. This technique, coupled with knowledge of the acoustic modes in a combustion chamber, is likely to be a useful design tool for the passive control of mixing and combustion instability. Copyright © 2012 by ASME.

The acoustic analogy in an annular duct with swirling mean flow

This paper is concerned with modelling the effects of swirling flow on turbomachinery noise. We develop an acoustic analogy to predict sound generation in a swirling and sheared base flow in an annular duct, including the presence of moving solid surfaces to account for blade rows. In so doing we have extended a number of classical earlier results, including Ffowcs Williams & Hawkings' equation in a medium at rest with moving surfaces, and Lilley's equation for a sheared but non-swirling jet. By rearranging the Navier-Stokes equations we find a single equation, in the form of a sixth-order differential operator acting on the fluctuating pressure field on the left-hand side and a series of volume and surface source terms on the right-hand side; the form of these source terms depends strongly on the presence of swirl and radial shear. The integral form of this equation is then derived, using the Green's function tailored to the base flow in the (rigid) duct. As is often the case in duct acoustics, it is then convenient to move into temporal, axial and azimuthal Fourier space, where the Green's function is computed numerically. This formulation can then be applied to a number of turbomachinery noise sources. For definiteness here we consider the noise produced downstream when a steady distortion flow is incident on the fan from upstream, and compare our results with those obtained using a simplistic but commonly used Doppler correction method. We show that in all but the simplest case the full inclusion of swirl within an acoustic analogy, as described in this paper, is required. © 2013 Cambridge University Press.

Modal stability theory

This article contains a review of modal stability theory. It covers local stability analysis of parallel flows including temporal stability, spatial stability, phase velocity, group velocity, spatio-temporal stability, the linearized Navier-Stokes equations, the Orr-Sommerfeld equation, the Rayleigh equation, the Briggs-Bers criterion, Poiseuille flow, free shear flows, and secondary modal instability. It also covers the parabolized stability equation (PSE), temporal and spatial biglobal theory, 2D eigenvalue problems, 3D eigenvalue problems, spectral collocation methods, and other numerical solution methods. Computer codes are provided for tutorials described in the article. These tutorials cover the main topics of the article and can be adapted to form the basis of research codes. Copyright © 2014 by ASME.