). .. , xvi 2 Thermal efficiency for the ideal and real processes at different inlet combustion chamber temperature for compressor and turbine efficiency set to 0.9 (a) and propulsive efficiency for the jet engine based on the ratio between outlet and inlet velocity (b), Ideal and real (denoted ') Brayton cycle in T-S (a) and P-V diagrams

, Flow behaviour of a fluid initially at rest put in motion by a rotating disc (a) and a rotating fluid over a stationary disc (b). From Schlichting [8]

, Top: axial velocity, middle: azimuthal velocity, bottom: radial velocity. The analytical derivation and computation of the velocity profiles is proposed in Bridel-Bertomeu [12], Auto-similar velocity profiles of the laminar flow between a rotor and a stator for different inter-disc Reynolds numbers Re h

, Categories of flow encountered in enclosed rotor-stator cavity

, Sketch of turbine cavity and main annulus with main central core and pressure increase with radius

, Observed flow patterns in the wheel space for simple rim seal at various non-dimensionalized purge flow rate C w supplied in the cavity. From Phadke and Owen [19]

, The different rim seal geometries (a) and the corresponding variation of C w,min with Re ? with G c = 0.01. Adapted from Phadke and Owen, vol.19

, The effect of Re ? on the variation of C w,min with mainflow Reynolds Re u for axial rim seal at two gap ratio Gc = 0.01 (a) and Gc = 0.02 (b)

, Sealing effectiveness for simple (a) and axial overlapping (b) rim seal s/r 0 = 0.0048, at various mainstream swirl flow fraction. Adapted from Phadke and Owen

, Streamlines of mainstream gas (red) and rim seal gas (blue) indicating a toroidal recirculation in the rim seal. From Savov et al. [28] (b)

, The effect of C w on the observed flow patterns in the wheel space for Gc = 0.01

, 70 to 50 m.s ?1 ) and pressure (right: -15 to 15 kPa), Re ? = 6x10 6 . From Cao et al, Instantaneous radial velocities, p.16

. .. Turbomachine, 30 2.3 Control volume located around a turbine stage. Example of viscous and thermal irreversibility creation from exergy approach

, The split of exergy ? in the flow mechanical work potential ? and flow thermal work potential, marked on a h-s diagram (a) and a control volume located around a turbine stage exhibiting the thermal creation from the flow mechanical work potential approach (b)

, Adapted From Zlatinov [54] . 46 2.7 Unsteady flow structures with corresponding characteristic length scale and frequency. From Lagraff et al. [66]

, Re ?start )/end (Re ? end ) of transition based on momentum thickness Reynolds number over a flat plate without pressure gradient depending on the free stream turbulence (b). From Abu-Ghannam and Shaw [71]

, Sketch of possible transition process on aft part of low pressure turbine suction side depending on external disturbances (FST: Free Stream Turbulence, WNJ: Wake Negative Jet), p.57

, 60 2.12 Mixing of two streams in a constant area duct corresponding to a simplified configuration of mixing process downstream a blade trailing edge. From Greitzer et al. [65]

, Secondary flow structures in the passage proposed by, vol.93, p.65

, Side (a) and downstream view (b) of simplified secondary flow structures in the passage proposed by

, Evolution of an upstream wake in the passage of a downstream row, contours of entropy (a) and negative jet phenomenon (b)

, Entropy contour at exit of the stator (left) and rotor (right)

, Generation of entropy due to shear layer in an axisymmetric (1)/3D configuration (2) (a)

, Radial velocity across the blade due to crossflow (1), blockage (2) contributions (b)

F. Zlatinov, Decomposition of entropy production according to the velocity gradients (see eq.(3.12) p.146 for more details), vol.54, p.79

, View of the experimental set up, p.83

. .. Rim-seal-geometry,

, Inlet profiles at one axial chord length upstream of the blade leading edge: (a): azimuthal angle, (b): radial angle, (c): inlet total pressure (d): inlet total temperature. The height has been normalized by the height of the main annulus. (P ref , T ref )=(101 325 Pa, 300 K), p.84

, Turbulence intensity decay in the axial direction downstream the turbulence grid (a) and five-holes probe measurement positions (b), p.85

, Structured multi-block 0-6H tolopogy

, hub and shroud (bottom) for the RANS simulation. LE and TE stand respectively for leading and trailing edge, y + distribution around the wetted surfaces: unwrapped blade (top)

, hub and shroud (bottom) for the LES elsA simulation (temporal average). LE and TE stand respectively for leading and trailing edge, y + distribution around the wetted surfaces: unwrapped blade (top), p.91

, Averaged grid dimension at the wall for the LES elsA simulation, s + : streamwise, n + : normal, r + : spanwise

. .. Rans), 95 3.12 Convergence of continuum, first momentum, energy and turbulent residuals (logarithm scale) for the RANS simulation, Evolution of the mass flow rate at the inlet, cavity inlet and outlet based on the number of iterations performed, p.95

, Evolution of the mass flow rate at the inlet, outlet (a) and cavity inlet (b) in time for the LES elsA simulation

, Evolution of the mass flow rate at the inlet, outlet (a) and cavity inlet (b) in time for the LES AVBP simulation

, 4%, (b): 6%, (c): 50% and pressure coefficient downstream of the blade (d) for the A05 configuration for the standard and refined mesh grid (RANS simulation)

, Comparison of the pressure loss coefficient downstream of the blade for the A05 configuration for coarse and fine grids

, Comparison of the pressure coefficient distribution around the blade at (a): 4%, (b): 6%, (c): 50% span and pressure coefficient downstream of the blade for the A05 configuration for k-? and SpalartAllmaras turbulence model (RANS)

. .. , Suction side boundary layer thickness from k-? turbulence model used in the study (a) and Spalart-Allmaras model (b), p.101

, 4%, (b): 6%, (c): 50% span for the A05 configuration using RANS approach and experimental results at the different blade height (d)

, Pressure coefficient around the blade at 4, 6 and 50% span for the A05 configuration using the LES elsA approach, p.103

, Pressure coefficient around the blade at (a): 4%, (b): 6%, (c): 50% span for the A05 configuration using the LES AVBP approach, p.104

, Pressure coefficient around the blade at (a): 4%, (b): 6%, (c): 50% span the A05 configuration using the LES Pro-LB approach, p.105

, Pressure loss coefficient downstream the blade for the A05 configuration for the different solvers (a): RANS, (b): LES elsA, (c): LES AVBP, (d): Pro-LB

, Hub (a) and shroud (b) boundary layer thickness from the RANS simulation

, Streak lines (a) and skin friction (b) at the shroud wall exhibiting rolling process of the horse shoe vortices from the LES simulation, p.108

. .. , Streak lines on the suction (left) and pressure side (right) for the A05 configuration from the RANS simulation (unwrapped blade), vol.109

, Velocity vector close to the rim seal: face to leading edge (a)(c) and at the center of the channel (b)(d) for the A05, vol.05, p.110

, a) and horseshoe vortex interaction with the cavity flow based on a iso q-criterion q = 10 6 colored by the streamwise vorticity from an instantaneous LES solution (b

, 112 3.30 Streak lines at the hub from the temporally averaged LES (a) and sight from downstream to upstream of the secondary flows in the passage obtained using iso q-criterion q = 10 6 colored by vorticity from instantaneous LES solution (b)

, Pressure coefficient and vorticity downstream the blade from instantaneous LES solution where experimental measurements have been performed

, Spectrum and corresponding summed modes (1, 2, 3 and 4) related to the horse shoe vortex process at the shroud obtained from Dynamical Mode Decomposition (LES simulation)

, Pressure standard deviation ?(p) around the blade suction side (a), at hub and shroud (b)

, Three-dimensional (a) and two-dimensional modes related to KelvinHelmholtz instability based on fully three-dimensional and two-dimensional meridional plane face to the blade leading edge (FB) and at the center of the passage (CP) (b)

, Pressure coefficient and domain of fluctuation at 4% blade height (a) and downstream blade (b) where the characteristic time T corresponds to the time for a particle to be convected of one axial chord, vol.118

, Density frequency spectrum in x, z cuts and around the blade with the A05 configuration and corresponding to the fundamental KelvinHelmholtz density mode

, Positions in the linear cascade where the boundary layer profiles have been extracted for the RANS and LES simulations, p.120

, 120 3.39 Boundary layer profiles for the RANS (a) and LES simulation (b) upstream of the blade leading edge, see 5 in Fig. 3.37 for the corresponding extraction position

. .. , Streak lines on the suction and pressure side for the A05 configuration from the LES simulation (unwrapped blade), p.121

, Oil-painting visualization on the blade suction (a) and pressure side (b)

, Boundary layer profiles on the blade suction side in the favourable pressure gradient portion for the RANS (a) and the LES simulations (b), see position 6, 7 and 8 in Fig

, Boundary layer profiles on the blade suction side in the adverse pressure gradient portion for the RANS (a) and LES simulations (b), see 9, 10, 11 in Fig. 3.37 for the corresponding extraction positions, p.124

, by the vorticity showing the turbulence injection at the inlet of the domain in the first time steps of the AVBP simulation (a) and the turbulent kinetic energy decay from the inlet of the domain to the blade leading edge based on a theoretical HIT decay and obtained in AVBP simulation (b)

, iso-contour of viscous entropy colored by temperature (b) exhibiting the secondary vortices downstream of the blade trailing edge (trailing shed and passage vortices)

, Hub and shroud boundary layer viscous entropy production with wallnormal contributions (du s /dr and du c /dr)

, Blade boundary layer entropy production with wall-normal contribution (du s /dc)

, Entropy production at the rim seal interface between the left corner at x/C x = -0.26 and right corner at x/C x = -0.04 including the terms with high contribution to entropy: shear layer for axial and tangential velocity gap (du c /dr and du s /dr) (a) and variation of radial velocity u r close to rim seal right corner (b)

, Axial cuts in the passage colored by viscous entropy production related to du r /dc gradients

, 9 to the end of the domain x/C x = 3 including the terms with high contribution to entropy: trailing shed vortex process (du c /ds) (a) and radial velocity gradients (b)

. .. , Thermal entropy production along the domain with the large contribution of the radial variation of temperature (dT /dr), p.151

, and (e)-(f): 50% span for the axial rim seal at the different purge flow rates based on the RANS approach (left figures) and experiments (right figures), p.4

, Pressure loss coefficient downstream of the blade for the axial rim seal at the different purge flow rates obtained from RANS simulations (a) and experiments (b)

, Influence of the purge flow rate on entropy production for the total (a), blade (b), hub (c), shroud (d) and remaining term (e), p.155

, Pressure coefficient around blade at 4, 6 and 50% span for the different rim seal geometries at high purge flow rate (1%) based on the RANS approach and experiments

, Pressure loss coefficient downstream of the blade at high purge flow rate (1%) for the different rim seal geometries obtained from RANS simulations (a) and experiments (b)

, Influence of the rim seal geometry (axial and simple overlapping) on entropy production for total (a), blade (b), hub (c) contributions, p.158

, Radial cut close to the hub colored by the radial velocity for axial (a) and single overlapping geometry (b)

, Radial cut close to the hub colored by temperature for axial (a) and single overlapping geometry (b)

, Resolved and subgrid contributions to the viscous entropy production based on the LES simulation without turbulence injection (bottom)

, Total entropy production for the different subdomains in the LES simulation without turbulence injection

, Total entropy production for the different subdomains in the LES simulation with turbulence injection

, Low pressure turbine experimental test rig

, Simulation domain representing 1/24 of the full test rig, p.170

, Grid refinement for rotor 1 including wake refinement of stator 1 and refinement in the cavities

, R2 (top) and at the hub surface (bottom), y + resolution around the different blade suction sides S1, R1, vol.2

, Temporal evolution of the mass flow rate at the inlet, outlet (a) and at the inlet of the cavities (b)

. .. , 50% span of the stator 2 row and experimental measurements at the different nozzle guide vane height (d), p.20

, Pressure loss coefficient downstream of stator 1 (a), rotor 1 (b) and stator 2 (c) obtained numerically compared to experimental data, p.178

, Evolution of the dimensionless volume-averaged kinetic energy for the entire cavity 1 (a) and evolution of the dimensionless kinetic energy of the rotor boundary layer (b) and stator boundary layer at the cavity mid-height (c)

, Radial and tangential velocities in the cavity 1 in a meridional plane, vol.181

, Radial and tangential velocities into the cavities 2 and 3 in a meridional plane

, Three-dimensional streamlines in the cavities 1, 2 and 3, p.182

, Azimuthally-averaged radial (a) and azimuthal (b) velocity profiles at cavity 1 mid-height

, Azimuthally-averaged radial (a) and azimuthal (b) velocity profiles at cavity 2 mid-height

, Azimuthally-averaged radial (a) and azimuthal (b) velocity profiles at cavity 3 mid-height

, Axial velocity in the labyrinth (a) and pressure evolution normalized by pressure cavity 2 (b)

, Repartition of cooling flow in cavity 1 and 2 on the three rim seal, p.185

, Wall friction on the surfaces of stator 1 and on an axial plane downstream of the blade with pressure coefficient. The shroud and cavities are omitted

, Boundary layer profile on the blade suction side at mid-span for stator row 1 (a) and 2 (b)

, Suction side boundary layer thickness for stator 1 (a) and 2 (b), p.187

, Total pressure, whirl and radial angle downstream of the first rotor row (compared to the nominal turning of the blade 64.6 ? and no radial angle) from simulation (top) and experiments (bottom, p.188

, Wall friction on the surfaces of stator 2 and axial plane downstream of the blade with pressure coefficient. The casing, rotor shrouding and cavities are omitted

, Wall friction on the surfaces of stator 1, rotor 1 and axial plane downstream of the blade with total pressure showing wakes and secondary flow influence on the pressure coefficient downstream of the blade. The casing, rotor shrouding and cavities are omitted, p.190

, Wall friction on the surfaces of rotor 1 and axial plane downstream of the blade with total pressure showing wakes, secondary and bypassed flow influence on the pressure coefficient downstream of the blade. The casing and cavities are omitted

, Streamlines showing the recirculation zone at rim seal interface S1-R1. A: wake region, B: passage vortex. Rotor blade 1 and 2 have been omitted

, Pressure distribution at the rim seal interface 1 (S1-R1), vol.3

, Radial evolution at the rim seal interface at the beginning (x/C x = 6.3), middle (x/C x = 6.4) and end of the cavity ((x/C x = 6.5) for axial (a), azimuthal (b), total pressure (c) and static temperature (d), p.193

, Extracted power on first rotor along blade (a) and cumulated power (b), the first rotor blade extends from x/C x = 6, vol.9, p.195

, Extracted power on second rotor along blade (a) and cumulated power (b), the second rotor blade extends from x/C x = 10

, Evolution of total viscous anergy production (resolved+subgrid) along the domain including subgrid scale model contribution, p.197

, Evolution of total anergy production (resolved+subgrid) along the domain including the artificial contribution

, Evolution of total thermal anergy production (resolved+subgrid) along the domain including subgrid scale model contribution, p.199

. .. , Temperature at the rim seal interface, along the hub and rotor 1 blade with migrating cavity flow on blade suction side

, Total entropy production at the inlet (a) and outlet (b) of the domain one axial chord downstream of rotor 2 blade trailing edge including wall normal contributions

, Evolution of the total viscous entropy production along the simulation and restricted to the hub/shroud wall normal contribution (du s /dr), p.201

, Iso-contour of entropy production along first stator row for the contribution du s /dr (left) and du c /dr (right)

, Wall friction on the surfaces of stator one rotor one and axial plane downstream blade with total pressure showing wakes and secondary flow influence on pressure coefficient downstream blade. The shroud is omitted

, Evolution of the total viscous entropy production along the simulation and restricted to the hub/shroud wall normal contribution (du c /dr), p.203

, Total entropy production along the domain and blade wall normal contribution (du s /dc)

, Axial cuts at the trailing edge colored by viscous entropy production related to du s /dc gradient for stator 1 and

, Total entropy production along the domain and entropy production related to du r /dc gradients

, Axial cuts along the passage domain colored by viscous entropy production related to du r /dc gradient for stator one and two, p.205

, Iso-surface of q-criterion colored by streamwise vorticity showing upstream secondary vortices from stator 1 impacting rotor 1 in an upstream (a) and side view (b)

. .. , Pressure coefficient downstream of the rotor 1, p.206

, Contribution of viscous and thermal anergy to the decrease of exergy or the flow

, Contribution of the work extracted on the shaft to the decrease of exergy of the flow

, Illustration of different scales to describe a flow. From left to right, macroscopic, mesoscopic and microscopic scales, p.223

, Velocity distribution in three dimension (D3Q19) to recover the weakly compressible isothermal Navier-Stokes equations (b)

A. , Link between velocity, space and time discretization in a Cartesian grid229 A.4 Schematic view of steps 2 to 4 required to perform one iteration in LBM approach from a D1Q3 lattice, Courtesy of Coreixas

R. Les and . Gravemeier, A.5 Turbulent spectrum with degrees of modeling: DNS
URL : https://hal.archives-ouvertes.fr/hal-01714740

A. , Reynolds number variation in a medium-sized gas turbine engine (a) and engine flight envelope Reynolds number for a medium-sized low pressure turbine (b), p.234

, Sketch of the scale splitting principle in LES due to the mesh

A. , Computational domain with incoming and outgoing waves at the inlet and outlet of the domain

, Hybrid (a) and structured meshes (b) around a low pressure turbine blade at mid-span

, Sketch of the influence of the grid resolution on the theoretical turbulence cascade (b), A.10 Grid requirements for LES and hybrid RANS-LES following Piomelli and Balaras [195] and adapted from Tucker, vol.158

A. , Decomposition of the sources of pressure loss downstream of a blade in a simplified configuration

B. , Normalized entropy distribution across a shock

, M = 1.5, upstream entropy taken as 0, downstream value =1.0

, Mach and isentropic Mach number to estimate boundary layer thickness280 C.2 Boundary layer thickness around blade suction side for linear cascade configuration without purge flow obtained from elsA (a) and boundary layer module (b)

, Characteristics of the cascade rig

, Numerical setup for the Navier-Stokes numerical approaches, p.94

, Numerical setup for the LBM numerical approach, p.94

, Exergy balance at two different reference conditions, p.133

, Characteristics of the stator and rotor rows

, Purge mass flow rate (kg.s ?1 ) for the different cases studied, p.169

, A.1 Velocity particle and weight of Gauss-Hermite quadrature of the D3Q19

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