Dr Olaf Marxen

Code_Saturne, an open-source computational fluid dynamics (CFD) code, has been applied to a range of problems related to turbomachinery internal air systems. These include a closed rotor-stator disc cavity, a co-rotating disc cavity with radial outflow and a co-rotating disc cavity with axial throughflow. Unsteady Reynolds-averaged Navier–Stokes (RANS) simulations and large eddy simulations (LES) are compared with experimental data and previous direct numerical simulation and LES results. The results demonstrate Code_Saturne's capabilities for predicting flow and heat transfer inside rotating disc cavities. The Boussinesq approximation was implemented for modelling centrifugally buoyant flow and heat transfer in the rotating cavity with axial throughflow. This is validated using recent experimental data and CFD results. Good agreement is found between LES and RANS modelling in some cases, but for the axial throughflow cases, advantages of LES compared to URANS are significant for a high Reynolds number condition. The wall-modelled large eddy simulation (WMLES) method is recommended for balancing computational accuracy and cost in engineering applications.

Successful design of aerospace missions requires accurate modelling of the physical phenomena in a hypersonic boundary layer. The behaviour of hypersonic boundary layers is strongly influenced by finite-rate thermochemical effects. These effects can be captured by including finite-rate thermochemistry models in computational tools. Such models are typically highly parametrised, introducing non-equilibrium features into the flow that generate significant uncertainty, which reflects on the output quantities of interest. The way such phenomena interact with solid boundaries and roughness is fundamentally unknown and the additional uncertainty is unquantifiable. In the present work, we propose the investigation of the linear response of n-periodic roughness arrays using an efficient mathematical framework. We extend an existing computational tool to investigate the effect of roughness, including real-gas and finite-rate chemistry effects by coupling with the Mutation++ library. The proposed framework allows to study efficiently and in parallel the linear flow response and wake synchronisation without the restricting idealised periodicity constraint. The results are extracted from reduced-order geometries using automatic linearisation tools. This framework can be potentially combined with sensitivity analysis tools to identify critical roughness configurations. In this paper, we provide the necessary background and present preliminary results for canonical flat plate hypersonic boundary layers.

Buoyancy-induced flows occur in the rotating cavities of gas turbine internal air systems, and are particularly challenging to model due to the inherently unsteadiness of these flows. While the global features of such flows are well documented, detailed analyses of the unsteady structure and turbulent quantities have not been reported. In this work we use a high-order numerical method to perform large-eddy simulation (LES) of buoyancyinduced flow in a sealed rotating cavity with either adiabatic or heated disks. New insight is given into long-standing questions regarding the flow characteristics and nature of the boundary layers. The analyses focus on showing time-averaged quantities, including temperature and velocity fluctuations, as well as on the effect of the centrifugal Rayleigh number on the flow structure. Using velocity and temperature data collected over several revolutions of the system, the shroud and disk boundary layers are analysed in detail. The instantaneous flow structure contains pairs of large, counter-rotating convection rolls, and it is shown that unsteady laminar Ekman boundary layers near the disks are driven by the interior flow structure. The shroud thermal boundary layer scales as approximately Ra−1/3 , in agreement with observations for natural convection under gravity.

The evolution of two- and three-dimensional small-amplitude disturbances in the laminar part of a laminar separation bubble is investigated in detail. We apply a combination of local linear stability theory, results from different experimental measurement campaigns and direct numerical simulations to identify two different discrete eigenmodes in the laminar part of the bubble. A stable eigenmode, the outer mode, governs unsteady oscillations in the upstream part of the bubble. However, this perturbation is quickly overtaken by an unstable eigenmode, the inner mode, which eventually leads to transition of the detached shear layer. Such a behaviour is observed due to an acceleration region with a favourable pressure gradient preceding the adverse-pressure-gradient region. The flow is stable in the acceleration region, in which the outer mode is only moderately damped, while the inner mode is strongly damped. At the onset of instability for the unstable eigenmode upstream of separation, both viscous Tollmien-Schlichting and inviscid Kelvin-Helmholtz instability mechanisms contribute to amplification, while deeper inside the bubble only the inviscid mechanism is active. If the explicit forcing is moved to a region downstream of the favourable pressure gradient, only the unstable eigenmode appears. The same behaviour is found for two-dimensional and weakly oblique waves. © 2012 Cambridge University Press.

In this paper large-eddy simulation is used to study buoyancy-induced flow in a rotating cavity with an axial throughflow of cooling air. This configuration is relevant in the context of secondary air systems of modern gas turbines, where cooling air is used to extract heat from compressor disks. Although global flow features of these flows are well understood, other aspects such as flow statistics, especially in terms of the disk and shroud boundary layers, have not been studied. Here, previous work for a sealed rotating cavity is extended to investigate the effect of an axial throughflow on flow statistics and heat transfer. Time- and circumferentially-averaged results reveal that the thickness of the boundary layers forming near the upstream and downstream disks is consistent with that of a laminar Ekman layer, although it is shown that the boundary layer thickness distribution along the radial direction presents greater variations than in the sealed cavity case. Instantaneous profiles of the radial and azimuthal velocities near the disks show good qualitative agreement with an Ekman-type analytical solution, especially in terms of the boundary layer thickness. The shroud heat transfer is shown to be governed by the local centrifugal acceleration and by a core temperature, which has a weak dependence on the value of the axial Reynolds number. Spectral analyses of time signals obtained at selected locations indicate that, even though the disk boundary layers behave as unsteady laminar Ekman layers, the flow inside the cavity is turbulent and highly intermittent. In comparison with a sealed cavity, cases with an axial throughflow are characterised by a broader range of frequencies, which arise from the interaction between the laminar jet and the buoyant flow inside the cavity.

Rotating fluids are well-known to be susceptible to waves. This has received much attention from the geophysics, oceanographic and atmospheric research communities. Inertial waves, which are driven by restoring forces, for example the Coriolis force, have been detected in the research fields mentioned above. This paper investigates inertial waves in turbine rim seal flows in turbomachinery. These are associated with the large-scale unsteady ow structures having distinct frequencies, unrelated to the main annulus blading, identified in many experimental and numerical studies. These unsteady flow structures have been shown in some cases to reduce sealing effectiveness and are difficult to predict with conventional steady Reynolds-averaged Navier-Stokes (RANS) approaches. Improved understanding of the underlying ow mechanisms and how these could be controlled is needed to improve the efficiency and stability of gas turbines. This study presents large-eddy simulations for three rim seal configurations { chute, axial and radial rim seals { representative of those used in gas turbines. Evidence of inertial waves is shown in the axial and chute seals, with characteristic wave frequencies limited within the threshold for inertial waves given by classic linear theory (i.e. jf=frelj 2), and instantaneous flow fields showing helical characteristics. The radial seal, which limits the radial fluid motion with the seal geometry, restricts the Coriolis force and suppresses the inertial wave.

A numerical investigation of the disturbance amplification in a Mach 4.8 flat-plate boundary layer with a localized two-dimensional roughness element is presented. The height of the roughness is varied and reaches up to approximately 70% of the boundary-layer thickness. Simulations are based on a time-accurate integration of the compressible Navier-Stokes equations, with a small disturbance of fixed frequency being triggered via blowing and suction upstream of the roughness element. The roughness element considerably alters the instability of the boundary layer, leading to increased amplification or damping of a modal wave depending on the frequency range. The roughness is also the source of an additional perturbation. Even though this additional mode is stable, the interaction with the unstable mode in the form of constructive and destructive interference behind the roughness element leads to a beating and therefore transiently increased disturbance amplitude. Far downstream of the roughness, the amplification rate of a flat-plate boundary layer is recovered. Overall, the two-dimensional roughness element behaves as disturbance amplifier with a limited bandwidth capable of filtering a range of frequencies and strongly amplifying only a selected range. © 2010 Cambridge University Press.

A novel flow configuration devised for investigation of active control of separated airfoil flows using synthetic jets is presented. The configuration consists of a flat plate, with an elliptic leading edge and a blunt trailing edge, at zero incidence in a free stream. Flow separation is induced on the upper surface of the airfoil at the aft-chord location by applying suction and blowing on the top boundary of the computational domain. Typical separated airfoil flows are generally characterized by at least three distinct frequency scales corresponding to the shear layer instability, the unsteadiness of the separated region and the vortex shedding in the wake, and all these features are present in the current flow. Two-dimensional Navier-Stokes simulations of this flow at a chord Reynolds number of 6 104 have been carried out to examine the nonlinear dynamics in this flow and its implications for synthetic-jet-based separation control. The results show that there is a strong nonlinear coupling between the various features of the flow, and that the uncontrolled as well as the forced flow is characterized by a variety of lock-on states that result from this nonlinear coupling. The most effective separation control is found to occur at the highest forcing frequency for which both the shear layer and the separated region lock on to the forcing frequency. The effects of the Reynolds number on the scaling of the characteristic frequencies of the separated flow and its subsequent control are studied by repeating some of the simulations at a higher Reynolds number of 1 105. © 2010 Cambridge University Press.

A high-order numerical method is employed to investigate flow in a rotor/stator cavity without heat transfer and buoyant flow in a rotor/rotor cavity. The numerical tool used employs a spectral element discretisation in two dimensions and a Fourier expansion in the remaining direction, which is periodic and corresponds to the azimuthal coordinate in cylindrical coordinates. The spectral element approximation uses a Galerkin method to discretise the governing equations, similarly to a finite element method, but employs high-order polynomials within each element to obtain spectral accuracy. A second-order, semi-implicit, stiffly stable algorithm is used for the time discretisation, and no subgrid modelling is included in the governing equations. Numerical results obtained for the rotor/stator cavity compare favourably with experimental results for Reynolds numbers up to Re1 = 106 in terms of velocities and Reynolds stresses. For the buoyancy-driven flow, the energy equation is coupled to the momentum equations via the Boussinesq approximation, which has been implemented in the code considering two different formulations. Numerical predictions of the Nusselt number obtained using the traditional Boussinesq approximation are considerably higher than available experimental data. Much better agreement is obtained when the extended Boussinesq approximation is em-ployed. It is concluded that the numerical method employed has considerable potential for further investigations of rotating cavity flows.

Prediction of laminar-turbulent transition is a key factor in the design of the heat shield of vehicles (re-)entering a planetary atmosphere. To investigate the transition by means of numerical simulation, accurate and efficient computational methods are necessary. Here, the compressible Navier-Stokes equations are solved for a gas where properties such as specific heat, thermal conductivity, viscosity, and specific gas constant depend on one or two thermodynamic variables. Our approach models a mixture of perfect gases in local thermodynamic equilibrium. The gas properties are provided either by means of direct calls to a library based on statistical mechanics and kinetic theory or indirectly in the form of look-up tables. In the first part of the paper, our method of handling a high-temperature gas in thermochemical equilibrium is described and verified. In the second part, the method is applied to the investigation of linear and non-linear boundary-layer instability. We carry out numerical simulations for a laminar flat-plate boundary layer at Mach 10 with a small, convectively amplified perturbation for both Earth and Martian atmospheres. Amplification of the perturbations shows favorable agreement with results obtained from linear theory. The secondary instability of the boundary layer in the presence of a large-amplitude two-dimensional wave is investigated. We observe that the non-linear mechanism of fundamental resonance becomes active and leads to a strong increase in amplification of three-dimensional disturbance waves. © 2011 American Institute of Physics.

Steady linear three-dimensional disturbances are investigated in a two-dimensional laminar boundary layer. The boundary layer is subject to a streamwise favourable-to-adverse pressure gradient and eventually undergoes separation. The separating flow corresponds to the first part of a pressure-induced laminar-separation bubble on a flat plate. Streamwise disturbance development in such a flow is studied by means of direct numerical simulation, a water-tunnel experiment and an adjoint-based parabolic theory suited to study spatial optimal growth. A complete overview of the disturbance evolution in various areas of the favourable-to-adverse pressure gradient laminar boundary layer is given. Results from all investigation methods show overall good agreement with respect to disturbance growth and shape within the entire domain. In the favourable pressure-gradient region and, again, slightly downstream of separation, transient growth caused by the lift-up effect dominates disturbance behaviour. In the adverse pressure-gradient region, a modal instability is observed. Evidence is presented that this instability is of Grtler type. © 2009 Cambridge University Press.

© © 2014 Cambridge University Press.Nonlinear instability in a supersonic boundary layer at Mach 4.8 with two-dimensional roughness is investigated by means of spatial direct numerical simulations (DNS). It was previously found that an important effect of a two-dimensional roughness is to increase significantly the amplitude of two-dimensional waves downstream of the roughness in a certain frequency band through enhanced instability and transient growth, while waves outside this band are damped. Here, we investigate the nonlinear secondary instability induced by a large-amplitude two-dimensional wave, which has received a significant boost in amplitude from this additional roughness-induced amplification. Both subharmonic and fundamental secondary excitation of the oblique secondary waves are considered. We found that even though the growth rate of the secondary perturbations increases compared to their linear amplification, only in some of the cases was a fully resonant state attained by the streamwise end of the domain. A parametric investigation of the amplitude of the primary wave, the phase difference between the primary and the secondary waves, and the spanwise wavenumber has also been performed. The transient growth experienced by the primary wave was found to not influence the secondary instability for most parameter combinations. For unfavourable phase relations between the primary and the secondary waves, the phase speed of the secondary wave decreases significantly, and this hampers its growth. Finally, we also investigated the strongly nonlinear stage, for which both the primary and the subharmonic secondary waves had a comparable, finite amplitude. In this case, the growth of the primary waves was found to vanish downstream of the transient growth region, resulting in a lower amplitude than in the absence of the large-amplitude secondary wave. This feedback also decreases the amplification rate of the secondary wave.

A laminar boundary layer separates in a region of adverse pressure gradient on a flat plate and undergoes transition. The detached shear layer rolls up into spanwise vortices that rapidly break down into small-scale turbulence. Finally, the turbulent boundary layer reattaches, forming a laminar separation bubble. Development and role of three-dimensional disturbances for transition in such a separation bubble are studied by means of direct numerical simulation with controlled disturbance input. In the present case, the level of incoming three-dimensional perturbations is not relevant due to an absolute secondary instability of these disturbances in the region of convective two-dimensional shear layer rollup. In particular, this is true for steady perturbations up to moderate amplitudes. Following their generation by nonlinear interaction of disturbance waves in the region of favorable pressure gradient, these steady disturbances develop as streaks. Their downstream evolution can be first attributed to transient behavior, depending on initial excitation, followed by a universal state with characteristics of a modal instability. Numerical results are confirmed by a comparison with experimental data.

© 2015 AIP Publishing LLC.The control of flow around a canonical airfoil-like geometry with laminar separation bubble is analyzed using linear stability theory. The theoretical predictions are compared to data from Navier-Stokes simulations [Kotapati et al., "Nonlinear dynamics and synthetic-jet-based control of a canonical separated flow," J. Fluid Mech. 654, 65-97 (2010)], in which the flow was controlled through a zero-net-mass-flux actuator. Very good agreement between the two approaches is found for a range of frequencies from low to high relative to the most dominant frequency for convective instability. The uncontrolled case exhibits periodic vortex shedding from the separation bubble due to an absolute instability. Linear modes with intermediate frequencies are found to exhibit strongest convective amplification, and forcing at these frequencies is most effective in order to reduce the size and extent of the separation bubble. The corresponding physical mechanism relies on a Kelvin-Helmholtz instability of the separated shear layer in conjunction with the non-linear effect of the mean flow deformation. For low frequencies, the front part of the bubble still diminishes due to the interaction of a vortex that starts from the actuator with the wall. This vortex transiently amplifies downstream due to the Orr mechanism. Actuation at high frequencies leads to visible, amplified instability waves in the shear layer, but is not effective in reducing the size of the bubble.

The paper describes a numerical investigation of linear and nonlinear instability in high-speed boundary layers. Both a frozen gas and a finite-rate chemically reacting gas are considered. The weakly nonlinear instability in the presence of a large-amplitude two-dimensional wave is investigated for the case of fundamental resonance. Depending on the amplitude of this two-dimensional primary wave, strong growth of oblique secondary perturbations occurs for favourable relative phase differences between the two. For essentially the same primary amplitude, secondary amplification is almost identical for a reacting and a frozen gas. Therefore, chemical reactions do not directly affect the growth of secondary perturbations, but only indirectly through the change of linear instability and hence amplitude of the primary wave. When the secondary disturbances reach a sufficiently large amplitude, strongly nonlinear effects stabilize both primary and secondary perturbations. © Cambridge University Press 2014.

The mutual interaction of laminar-turbulent transition and mean flow evolution is studied in a pressure-induced laminar separation bubble on a flat plate. The flat-plate boundary layer is subjected to a sufficiently strong adverse pressure gradient that a separation bubble develops. Upstream of the bubble a small-amplitude disturbance is introduced which causes transition. Downstream of transition, the mean flow strongly changes and, due to viscous-inviscid interaction, the overall pressure distribution is changed as well. As a consequence, the mean flow also changes upstream of the transition location. The difference in the mean flow between the forced and the unforced flows is denoted the mean flow deformation. Two different effects are caused by the mean flow deformation in the upstream, laminar part: a reduction of the size of the separation region and a stabilization of the flow with respect to small, linear perturbations. By carrying out numerical simulations based on the original base flow and the time-averaged deformed base flow, we are able to distinguish between direct and indirect nonlinear effects. Direct effects are caused by the quadratic nonlinearity of the Navier-Stokes equations, are associated with the generation of higher harmonics and are predominantly local. In contrast, the stabilization of the flow is an indirect effect, because it is independent of the Reynolds stress terms in the laminar region and is solely governed by the non-local alteration of the mean flow via the pressure. © 2010 Cambridge University Press.

A laminar boundary layer separates in a region of adverse pressure gradient on a flat plate and undergoes transition. Finally the turbulent boundary layer reattaches, forming a laminar separation bubble (LSB). Laminar-turbulent transition within such a LSB is investigated by means of Laser-Doppler-Anemometry (LDA), Particle Image Velocimetry (PIV), and direct numerical simulation (DNS). The transition mechanism occurring in the flow-field under consideration is discussed in detail. Observations for the development of small disturbances are compared to predictions from viscous linear instability theory (Tollmien-Schlichting instability). Non-linear development of these disturbances and their role in final breakdown to turbulence is analyzed.

Large-eddy simulations (LES) of centrifugal convection in a closed rotating cavity with different thermal boundary conditions are presented for Rayleigh numbers (Ra) 107∼109. Previous results for radial convection with a positive radial temperature gradient and for axial convection with low Ra are confirmed. For axial convection, turbulence develops in the core flow at Ra=109, but is considerably weaker than the turbulence for the radial and mixed convection. In mixed convection, the mean core temperature shows little radial variation as in radial convection, but has a significant axial temperature gradient as in axial convection. Compared to the radial and axial convection cases, both shroud and disc heat transfer are increased in mixed convection. Changes in disc heat transfer from the axial convection case are particularly strong. In all the conditions, laminar kinematic Ekman layers occur on the discs, but the disc thermal boundary layers for axial and mixed convection are considerably thicker. •Open-source CFD code is applied to the pure centrifugal buoyant driven flow in a rotating disc cavity, with results comparing well to the published data. The calculations are then extended to axial convection and mixed convection with Rayleigh numbers up to 109.•Under axial convection and mixed convection, temperature gradient and mean flow circulation along the axial direction are developed. In the mixed convection, turbulence level in the cavity and heat transfer on the shroud are both increased.

This study investigates turbine rim seal geometry effects within the rotationally-driven ingestion regime. Computations were performed with a wall-resolved unsteady Reynolds-averaged Navier-Stokes (URANS) model and a large-eddy simulation (LES) model including near-wall boundary layer modelling, that is wall-modelled LES (WMLES). Use of simplified rim sealing models is proposed as an efficient method of ranking seal designs and investigating sensitivity to seal geometry. Four rim seal configurations, two chute seals, an axial seal and a radial seal which are representative of those used in gas turbines and in previous research were investigated. Furthermore, hybrid seals combining geometric characteristics from both the chute and radial seal were considered. Significant sensitivities of sealing performance to turbulence modelling are identified, but URANS and WMLES show similar trends in ranking of seal performance, and these are consistent with previous experimental work. The addition of an outer radial clearance section to a chute seal is effective in reducing ingestion levels.

Code_Saturne, an open-source computational fluid dynamics (CFD) code, has been applied to a range of problems related to turbomachinery internal air systems. These include a closed rotor-stator disc cavity, a co-rotating disc cavity with radial outflow and a co-rotating disc cavity with axial throughflow. Unsteady Reynolds-averaged Navier-Stokes (RANS) simulations and large eddy simulations (LES) are compared with experimental data and previous direct numerical simulation (DNS) and LES results. The results demonstrate Code_Saturne's capabilities for predicting flow and heat transfer inside rotating disc cavities. The Boussinesq approximation was implemented into the code for modelling centrifugally buoyant flow and heat transfer in the rotating cavity with axial throughflow. This is validated using recent experimental data and CFD results. Good agreement is found between LES and RANS modelling in some cases, but for the axial throughflow cases, advantages of LES compared to URANS are significant for a high Reynolds number condition. The wall-modelled large eddy simulation (WMLES) method is recommended for balancing computational accuracy and cost in engineering applications. NOMENCLATURE Roman letters í µí±Ž Inner radius of the cavity [m] í µí± Outer radius of the cavity [m] Bo Buoyancy number, Ro/ √︁ í µí»½Δí µí±‡ [-] í µí° ¶ í µí± Specific heat capacity at constant pressure [J·kg −1 ·K −1 ] í µí±‘ Axial distance between discs [m] í µí±‘ 1 Axial length between the outlet slot and the upstream disc [m] í µí±‘ 2 Axial length between upstream and downstream cobs [m] í µí±‘ ℎ Axial throughflow hydraulic diameter, 2(í µí±Ž − í µí±Ÿ í µí±) [m] Ec Eckert number, Ω 2 í µí±Ÿ í µí±š 2 /í µí° ¶ í µí± Δí µí±‡ [-] í µí±€ Angular moment [kg·m 2 ·s −2 ] í µí±› Mesh node number [-] Pr Prandtl number, í µí¼ˆ/í µí»¼ [-] Pr í µí±¡ Turbulent Prandtl number [-] í µí±ž Wall heat flux [W·m −2 ] í µí±Ÿ Local radius [m] í µí±Ÿ í µí±š Cavity mid-radius, (a+b)/2 [m] í µí±Ÿ í µí± Radius of the shaft [m] í µí±Ÿ * Non-dimensional radius, (í µí±Ÿ − í µí±Ž)/(í µí± − í µí±Ž) [-] Ra Rotational Rayleigh number, Pr í µí»½Δí µí±‡Re í µí¼™ 2 [-] Re í µí¼™ Rotational Reynolds number, Ωí µí± 2 /í µí¼ˆ [-] Ro Rossby number, í µí±Š/Ωí µí±Ž [-] í µí± Axial width of the outlet slot [m] í µí±‡ Static temperature [K] í µí±‡ + Non-dimensional temperature, (í µí±‡ í µí±¤ − í µí±‡) í µí¼Œí µí° ¶ í µí± í µí±¢ í µí¼ /í µí±ž [-] í µí±¢ Resultant velocity [m·s −1 ] í µí±ˆ + Non-dimensional velocity, í µí±¢/í µí±¢ í µí¼ [-] í µí±¢ í µí¼ Friction velocity, √︁ í µí¼/í µí¼Œ [m·s −1 ] í µí±£ Velocity component [m·s −1 ] í µí±Š Bulk velocity at inlet [m·s −1 ] í µí±¦ + Non-dimensional wall distance, í µí±¢ í µí¼ Δí µí±¦/í µí¼ˆ [-] Greek letters í µí»¼ Thermal diffusivity, í µí¼†/í µí¼Œí µí° ¶ í µí± [m 2 ·s −1 ] í µí»½ Thermal expansion coefficient, 1/í µí±‡ 0 [-] Δí µí±‡ Temperature difference, í µí±‡ í µí± − í µí±‡ 0 for configuration 3, í µí±‡ í µí±ší µí±Ží µí±¥ − í µí±‡ 0 for others [K] Δí µí±¦ First cell spacing [m] í µí¼… von Karman constant [-] í µí¼† Thermal conductivity [W·m −1 ·K −1 ] í µí¼ˆ Kinematic viscosity [m 2 ·s −1 ] Ω Rotational speed [rad·s −1 ] í µí¼Œ Density [kg·m −3 ] í µí¼ Wall shear stress [kg·m −1 ·s −2 ] 1

Flows induced by centrifugal buoyancy occur in rotating systems in which the centrifugal force is large when compared to other body forces and are of interest for geophysicists and also in engineering problems involving rapid rotation and unstable temperature gradients. In this numerical study we analyse the onset of centrifugal buoyancy in a rotating cylindrical cavity bounded by two plane, insulated disks, adopting a geometrical configuration relevant to fundamental studies of buoyancy-induced flows occurring in gas turbine’s internal air systems. Using linear stability analysis, we obtain critical values of the centrifugal Rayleigh number and corresponding critical azimuthal wavenumbers for the onset of convection for different radius ratios. Using direct numerical simulation, we integrate the solutions starting from a motionless state to which small sinusoidal perturbations are added, and show that nonlinear triadic interactions occur before energy saturation takes place. At the lowest Rayleigh number considered, the final state is a limit-cycle oscillation affected by the presence of the disks, having a spectrum dominated by a certain mode and its harmonics. We show that, for this case, the limit-cycle oscillations only develop when no-slip end walls are present. For the largest considered chaotic motion occurs, but the critical wavenumber obtained from the linear analysis eventually becomes the most energetic even in the turbulent regime.

The convective primary amplification of a forced two-dimensional perturbation initiates the formation of essentially two-dimensional large-scale vortices in a laminar separation bubble. These vortices are then shed from the bubble with the forcing frequency. Immediately downstream of their formation, the vortices get distorted in the spanwise direction and quickly disintegrate into small-scale turbulence. The laminar-turbulent transition in a forced laminar separation bubble is dominated by this vortex formation and breakup process. Using numerical and experimental data, we give an in-depth characterization of this process in physical space as well as in Fourier space, exploiting the largely periodic character of the flow in time as well as in the spanwise direction. We present evidence that a combination of more than one secondary instability mechanism is active during this process. The first instability mechanism is the elliptic instability of vortex cores, leading to a spanwise deformation of the cores with a spanwise wavelength of the order of the size of the vortex. Another mechanism, potentially an instability of flow in between two consecutive vortices, is responsible for three-dimensionality in the braid region. The corresponding disturbances possess a much smaller spanwise wavelength as compared to those amplified through elliptic instability. The secondary instability mechanisms occur for both fundamental and subharmonic frequency, respectively, even in the absence of continuous forcing, indicative of temporal amplification in the region of vortex formation. © 2013 Cambridge University Press.

Fast and accurate prediction of the nonlinear evolution of instability waves in high-speed boundary layers requires specialized numerical algorithms, and augmenting limited observation in this extreme flow regime is challenging. The deep operator networks (DeepONet) has been shown to be an effective tool for providing accurate and fast physics-informed predictions. DeepONet is trained to map an incoming perturbation to the associated downstream flow field within the nonlinear flow regime. The training is performed using high-fidelity data from direct numerical simulations of the compressible Navier–Stokes equations, when the gas can be approximated as calorically perfect and when chemical non-equilibrium effects must be computed. The performance and requirements of training the DeepONet in each case are evaluated. In addition, we show that informing the training of the DeepONet with the continuity equation improves the accuracy of the results, especially in absence of sufficient training data. Success of the physics-informed DeepONet to predict missing fields depends on the observables. Specifically, prediction of a unique solution depends on the available measurements. These results are a promising step towards applications of neural operator networks to more complex high-speed flow configurations and to data assimilation. •DeepONets learn Navier-Stokes operator for high-Mach boundary-layer instabilities.•DeepONets accurately estimate instabilities and species with finite-rate chemistry.•Partial and missing data effectively mitigated using physics-informed DeepONets.

For measuring three components of velocity in unknown flow fields, multi-hole pressure probes possess a significant advantage. Unlike methods such as hot-wire anemometry, laser-Doppler velocimetry and particle-image velocimetry, multi-hole pressure probes can provide not only the three components of local velocity, but also static and stagnation pressures. However, multi-hole probes do require exhaustive calibration. The traditional technique for calibrating these probes is based on either look-up tables or polynomial curve fitting, but with the low cost and easy availability of powerful computing resources, neural networks are increasingly being used. Here, we explore the possibility to further reduce measurement uncertainty by implementing neural-network-based methods that have not been previously used for probe calibration, including supervised and unsupervised learning neural networks, regression models and elastic-map methods. We demonstrate that calibrating probes in this way can reduce the uncertainty in flow angularity by as much as 50% compared to conventional techniques.

Buoyancy-induced flows occur in the rotating cavities of gas turbine internal air systems, and are particularly challenging to model due to the inherently unsteadiness of these flows. While the global features of such flows are well documented, detailed analyses of the unsteady structure and turbulent quantities have not been reported. In this work we use a high-order numerical method to perform large-eddy simulation (LES) of buoyancyinduced flow in a sealed rotating cavity with either adiabatic or heated disks. New insight is given into long-standing questions regarding the flow characteristics and nature of the boundary layers. The analyses focus on showing time-averaged quantities, including temperature and velocity fluctuations, as well as on the effect of the centrifugal Rayleigh number on the flow structure. Using velocity and temperature data collected over several revolutions of the system, the shroud and disk boundary layers are analysed in detail. The instantaneous flow structure contains pairs of large, counter-rotating convection rolls, and it is shown that unsteady laminar Ekman boundary layers near the disks are driven by the interior flow structure. The shroud thermal boundary layer scales as approximately Ra−1/3, in agreement with observations for natural convection under gravity.

This study investigates turbine rim seal geometry effects within the rotationally-driven ingestion regime. Computations were performed with a wall-resolved unsteady Reynolds-averaged Navier-Stokes (URANS) model and a large-eddy simulation (LES) model including near-wall boundary layer modelling , that is wall-modelled LES (WMLES). Use of simplified rim sealing models is proposed as an efficient method of ranking seal designs and investigating sensitivity to seal geometry. Four rim seal configurations, two chute seals, an axial seal and a radial seal which are representative of those used in gas turbines and in previous research were investigated. Furthermore, hybrid seals combining geometric characteristics from both the chute and radial seal were considered. Significant sensitivities of sealing performance to turbulence modelling are identified, but URANS and WMLES show similar trends in ranking of seal performance, and these are consistent with previous experimental work. The addition of an outer radial clearance section to a chute seal is effective in reducing ingestion levels.

in a rotor/stator cavity without heat transfer and buoyant flow in a rotor/rotor cavity. The numerical tool used employs a spectral element discretisation in two dimensions and a Fourier expansion in the remaining direction, which is periodic and corresponds to the azimuthal coordinate in cylindrical coordinates. The spectral element approximation uses a Galerkin method to discretise the governing equations, but employs high-order polynomials within each element to obtain spectral accuracy. A second-order, semi-implicit, stiffly stable algorithm is used for the time discretisation. Numerical results obtained for the rotor/ stator cavity compare favourably with experimental results for Reynolds numbers up to Re1 = 106 in terms of velocities and Reynolds stresses. The buoyancy-driven flow is simulated using the Boussinesq approximation. Predictions are compared with previous computational and experimental results. Analysis of the present results shows close correspondence to natural convection in a gravitational field and consistency with experimentally observed flow structures in a water-filled rotating annulus. Predicted mean heat transfer levels are higher than the available measurements for an air-filled rotating annulus, but in agreement with correlations for natural convection under gravity.

A new method, enabling the computation of steady solutions of the Navier-Stokes equations in globally unstable configurations, is presented. We show that it is possible to reach a steady state by damping the unstable (temporal) frequencies. This is achieved by adding a dissipative relaxation term proportional to the high-frequency content of the velocity fluctuations. Results are presented for cavity-driven boundary-layer separation and a separation bubble induced by an external pressure gradient. © 2006 American Institute of Physics.

Short laminar separation bubbles can develop on a flat plate due to an externally imposed pressure gradient. Here, these bubbles are computed by means of direct numerical simulations. Laminar-turbulent transition occurs in the bubble, triggered by small disturbance input with fixed frequency, but varying amplitude, to keep the bubbles short. The forcing amplitudes span a range of two orders of magnitude. All resulting bubbles differ with respect to their mean flow, linear-stability characteristics and distance between transition and mean reattachment locations. Mechanisms responsible for these differences are analysed in detail. Switching off the disturbance input or reducing it below a certain, very small threshold causes the short bubble to grow continuously. Eventually, it no longer exhibits typical characteristics of a short laminar separation bubble. Instead, it is argued that bursting has occurred and the bubble displays characteristics of a long-bubble state, even though this state was not a statistically steady state. This hypothesis is backed by a comparison of numerical results with measurements. For long bubbles, the transition to turbulence is not able to reattach the flow immediately. This effect can lead to the bursting of a short bubble, which remains short only when sufficiently large disturbances are convected into the bubble. Large-scale spanwise-oriented vortices at transition are observed for short but not for long bubbles. The failure of the transition process to reattach the flow in the long-bubble case is ascribed to this difference in transitional vortical structures. © 2011 Cambridge University Press.

Predicting the degree of hot gas ingestion into turbine disc cavities is a challenge for computational fluid dynamics due to the complex unsteady flow dynamics in turbine rim seals and sensitivity to operating condition and seal geometry. This paper reports research aimed at clarifying the effect of operating conditions on seal performance and turbulence modelling requirements. A systematic study of sealing performance for an axial rim seal is reported, comparing an unsteady Reynolds-averaged Navier-Stokes (URANS) model and wall-modelled large-eddy simulation (WMLES). The conditions considered are classed as rotation-driven ingestion, pressure-driven ingestion, and combined mechanism ingestion. WMLES and URANS results showed similar ingestion levels and seal flows within the pressure-driven regime. For the rotationally-driven condition URANS displays larger, more coherent vortical flow structures than the WMLES. The larger vortices in the URANS drive ingress into the wheel-space resulting in higher levels of ingestion than indicated by the WMLES. For the combined ingestion condition, WMLES shows higher levels of ingestion and flow unsteadiness than URANS. The present results give some explanation for the mixed results reported for the performance of URANS models in previous studies.

We investigate the hydrodynamic stability of compressible boundary layers over adiabatic walls with fluids at supercritical pressure in the proximity of the Widom line (also known as the pseudo-critical line). Depending on the free-stream temperature and the Eckert number that determines the viscous heating, the boundary-layer temperature profile can be either sub-, trans- or supercritical with respect to the pseudo-critical temperature, Tpc. When transitioning from sub- to supercritical temperatures, a seemingly continuous phase change from a compressible liquid to a dense vapour occurs, accompanied by highly non-ideal changes in thermophysical properties. Using linear stability theory (LST) and direct numerical simulations (DNS), several key features are observed. In the sub- and supercritical temperature regimes, the boundary layer is substantially stabilized the closer the free-stream temperature is to Tpc and the higher the Eckert number. In the transcritical case, when the temperature profile crosses Tpc, the flow is significantly destabilized and a co-existence of dual unstable modes (Mode II in addition to Mode I) is found. For high Eckert numbers, the growth rate of Mode II is one order of magnitude larger than Mode I. An inviscid analysis shows that the newly observed Mode II cannot be attributed to Mack’s second mode (trapped acoustic waves), which is characteristic in high-speed boundary-layer flows with ideal gases. Furthermore, the generalized Rayleigh criterion (also applicable for non-ideal gases) unveils that, in contrast to the trans- and supercritical regimes, the subcritical regime does not contain an inviscid instability mechanism.

•Efficient DeepONets are developed for the non-equilibrium chemistry at high Mach numbers.•We obtain good prediction when the Mach number is out of the input.•We develop a new inference tool, DeepMMnet for complex multiphysics and multiscale models.•DeepMMnets assimilate sparse data using pre-trained DeepONets in a “plug-and-play” mode.•The codes are compact and easy to adopt in diverse domains as well as maintain or enhance further. In high-speed flow past a normal shock, the fluid temperature rises rapidly triggering downstream chemical dissociation reactions. The chemical changes lead to appreciable changes in fluid properties, and these coupled multiphysics and the resulting multiscale dynamics are challenging to resolve numerically. Using conventional computational fluid dynamics (CFD) requires excessive computing cost. Here, we propose a totally new efficient approach, assuming that some sparse measurements of the state variables are available that can be seamlessly integrated in the simulation algorithm. We employ a special neural network for approximating nonlinear operators, the DeepONet [23], which is used to predict separately each individual field, given inputs from the rest of the fields of the coupled multiphysics system. We demonstrate the effectiveness of DeepONet for a benchmark hypersonic flow involving seven field variables. Specifically we predict five species in the non-equilibrium chemistry downstream of a normal shock at high Mach numbers as well as the velocity and temperature fields. We show that upon training, DeepONets can be over five orders of magnitude faster than the CFD solver employed to generate the training data and yield good accuracy for unseen Mach numbers within the range of training. Outside this range, DeepONet can still predict accurately and fast if a few sparse measurements are available. We then propose a composite supervised neural network, DeepM&Mnet, that uses multiple pre-trained DeepONets as building blocks and scattered measurements to infer the set of all seven fields in the entire domain of interest. Two DeepM&Mnet architectures are tested, and we demonstrate the accuracy and capacity for efficient data assimilation. DeepM&Mnet is simple and general: it can be employed to construct complex multiphysics and multiscale models and assimilate sparse measurements using pre-trained DeepONets in a “plug-and-play” mode.

Direct numerical simulation data of bypass transition in flat-plate boundary layers are analysed to examine the characteristics of turbulence in the transitional regime. When intermittency is 50 % or less, the flow features a juxtaposition of turbulence spots surrounded by streaky laminar regions. Conditionally averaged turbulence statistics are evaluated within the spots, and are compared to standard time averaging in both the transition region and in fully turbulent boundary layers. The turbulent-conditioned root-mean-square levels of the streamwise velocity perturbations are notably elevated in the early transitional boundary layer, while the wall-normal and spanwise components are closer to the levels typical for fully turbulent flow. The analysis is also extended to include ensemble averaging of the spots. When the patches of turbulence are sufficiently large, they develop a core region with similar statistics to fully turbulent boundary layers. Within the tip and the wings of the spots, however, the Reynolds stresses and terms in the turbulence kinetic energy budget are elevated. The enhanced turbulence production in the transition zone, which exceeds the levels from fully turbulent boundary layers, contributes to the higher skin-friction coefficient in that region. Qualitatively, the same observations hold for different spot sizes and levels of free-stream turbulence, except for young spots which do not yet have a core region of developed turbulence.