Results are reported from linear and nonlinear stability analyses of pulsatile flow in a straight channel with a smooth constriction with 50 % occlusion. Physical mechanisms based on the energy variation of the two-dimensional modes are proposed to elucidate the equilibrium states and changes in the flow-field behaviour. When investigating how the energy of the two-dimensional modes varied with Reynolds number, an energy minimum was observed before the primary instability, so that it marked the Reynolds number value at which the flow field started to exhibit one pulse front per cycle downstream of the constriction. In addition, this flow is characterized by a bistable system in two dimensions, whose equilibrium states are related to the energy level of the two-dimensional mode. The energy difference between the equilibrium states surprisingly decreases from the lower to the higher Reynolds number, which indicates that the system should converge to a unique solution at high Reynolds number. Regarding the flow’s nature, the equilibrium state with higher energy presented a vortex pair soon after the constriction, with vorticity opposite to that of the base flow separating shear layers.

Stability and nonlinear analyses were employed to study symmetric and asymmetric steady flows through a straight channel with a smooth constriction with 50 % occlusion. Linear stability analysis was carried out to determine the unstable regions and the critical Reynolds numbers for the primary and secondary global instabilities. The primary bifurcation demonstrated an intricate aspect: the three-dimensional modes transfer their energy to the two-dimensional mode, which causes a symmetry breaking of the flow. This behaviour could be observed for Reynolds number lower than the critical, showing that this primary bifurcation is hysteretical. The secondary bifurcation also presented subcritical behaviour, exhibiting a pitchfork diagram with a large hysteretic curve. Given the subcritical character of both bifurcations, the relevance of non-normal growth in these flows were assessed, so that their convective mechanisms were exhaustively investigated. In addition, we could show that, for the secondary instability, optimal initial disturbances with large enough initial energy were able to promote a subcritical nonlinear saturation for a Reynolds number lower than the critical. The physical mechanism behind the transition process occurred by nonlinear interaction between the two- and three-dimensional modes, which established oscillatory behaviour, moreover, this energy exchange between the modes led the flow to the nonlinear saturated state. Therefore, the two-dimensional modes play a key role in the primary and secondary bifurcations of this system.

High fidelity numerical simulations at high Reylnolds number (106), although extremely expensive, are nowadays possible due to the efficiency of new algorithms on highly parallel systems. Among the computational fluid dynamics methodologies employed, high-order methods are becoming increasingly popular since the industrial sector strives to increase fidelity and accuracy for design purposes while retaining computational efficiency. To this end, the well-known highly scalable spectral/hp element framework Nektar++ is used. This approach therefore brings the possibility to have access to a wide range of spatial and temporal scales in turbulent flows and also study instability mechanisms of shear flows with a lower computational cost. In this work, we investigated the impact of compressibility in an engine intake for a range of high Reynolds numbers in absence of transonic effects and under strong adverse pressure gradients by means of computational simulations. The case consists of a flow over a flat plate which is subjected to a strong adverse pressure gradient imposed by employing an inviscid liner set as an upper boundary. Additional computational resource saving was, in this study, achieved by reducing the computational domain, so that, although only a small part of the domain was computed, it was possible to reproduce the physical behaviour expected in the whole computational domain. This methodology combined with the high-order methods demonstrated to be highly efficient and drastically reduced the computational time to perform the three-dimensional numerical simulations. In order to assess the compressibility effects in the vicinity and over the flat plate surface, compressible and incompressible high-order Implicit Large Eddy Simulations (iLES) of the Navier-Stokes (NS) equations were performed. With this methodology, we are able to assert what are the physical mechanisms behind the turbulent transition processes which the flows go through. In addition, and most important, what is the role that compressibility is playing in this flow configuration, in order to promote the design of improved geometries using high-order methods. The energetic flow conditions due to the high Reynolds numbers combined with the strong adverse pressure gradient imposed by the liner make the flow field separate. The separated flow gives rise to a separation bubble, which leads to a turbulent flow after going through a transition region. Furthermore, the compressible and incompressible instabilities were triggered by the same physical mechanisms. The Tollmien-Schlichting (T-S) instabilities trigger the Kelvin-Helmholtz (K-H) behaviour, which in turn causes the turbulent transition. Therefore, a consistent investigation of the compressibility effects and boundary layer behaviours of the compressible and incompressible solutions provided the necessary understanding of the viscous driven separation that governs the flow on an engine intake at moderate fan speed.

Direct numerical simulations (DNSs) of the T106A low-pressure turbine were conducted for various turbulence intensities and length scales to investigate their effects on flow behaviour and transition. A source-term formulation of the synthetic eddy method (SEM) was implemented in the Nektar++ spectral/hp element framework to introduce anisotropic turbulence into the flow field. A single sponge layer was imposed, which covers the inflow and outflow regions just downstream and upstream of the inflow and outflow boundaries, respectively, to avoid acoustic wave reflections on the boundary conditions. Additionally, in the T106A model, mixed polynomial orders were utilized, as Nektar++ allows different polynomial orders for adjacent elements. A lower polynomial order was employed in the outflow region to further assist the sponge layer by coarsening the mesh and diffusing the turbulence near the outflow boundary. Thus, this study contributes to the development of a more robust and efficient model for high-fidelity simulations of turbine blades by enhancing stability and producing a more accurate flow field. The main findings are compared with experimental and DNS data, showing good agreement and providing new insights into the influence of turbulence length scales on flow separation, transition, wake behaviour, and loss profiles.

Within the turbomachinery industry, components are currently assessed deploying standard second-order steady solvers. These are unable to capture complicated unsteady phenomena that have a critical impact on component performance. In this work, the high-order spectral h/p solver Nektar++ will be applied to a compressor blade to study the turbulent transition mechanisms and assess the effect of incoming disturbances with quasi-DNS resolution. The case will be modelled at an angle of incidence of $53.5°$ to match the original experimental loading at $52.8°$. At clean inflow conditions, Kelvin–Helmholtz instabilities appear on both sides of the blade due to a double separation, with the pressure side one not being reported in the experiments. The separation is gradually removed by the incoming turbulent structures but at different rates on the two sides of the blade. It will be shown that there is an optimal amount of turbulence intensity that minimises momentum thickness, which is strongly related to losses. Moreover, a discussion on the spanwise extrusion will be included, this being a major player in the modelling costs. Finally, the wall-clock time and the exact expenditure to run this case will be outlined, providing quantitative evidence of the feasibility of considering a quasi-DNS resolution in an industrial setting.

A three-dimensional direct stability analysis of the flow in a straight channel with a smooth constriction with 50% occlusion is performed. Two-dimensional steady and long-period pulsatile flows are investigated with respect to instabilities and transition to turbulence. The steady base flow is asymmetric for the Reynolds number range considered in this work, with critical value for linear instability of Re = 558.7 to three-dimensional perturbations. Although the steady flow is linearly stable for two-dimensional perturbations at Re = 700, it presents a maximum transient energy growth of the order of 107. The pulsatile flow becomes linearly unstable at Re = 249.7 to three-dimensional perturbations. The optimal initial condition that triggers the maximum growth of a two-dimensional perturbation occurs for a disturbance applied shortly after the peak velocity of the pulse for Re = 200 with a growth of the order of 103, the same order found for Re = 400, but applying the perturbation in the beginning of the pulse.