Exact solutions to non-classical steady nozzle flows of Bethe-Zel'dovich-Thompson fluids

Steady nozzle flows of Bethe-Zel'dovich-Thompson fluids - substances exhibiting non-classical gasdynamic behaviour in a finite vapour-phase thermodynamic region in close proximity to the liquid-vapour saturation curve - are examined. Non-classical flow features include rarefaction shock waves, shock waves with either upstream or downstream sonic states and split shocks. Exact solutions for a mono-component single-phase fluid expanding from a reservoir into a stationary atmosphere through a conventional converging-diverging nozzle are determined within the quasi-one-dimensional inviscid flow approximation. The novel analytical approach makes it possible to elucidate the connection between the adiabatic, possibly non-isentropic flow field and the underlying local isentropic-flow features, including the possible qualitative alterations in passing through shock waves. Contrary to previous predictions based on isentropic-flow inspection, shock disintegration is found to occur also from reservoir states corresponding to a single sonic point. The global layout of the flow configurations produced by a monotonic decrease in the ambient pressure, namely the functioning regime, is examined for reservoir conditions resulting in single-phase flows. Accordingly, a classification of steady nozzle flows into 10 different functioning regimes is proposed. Flow conditions determining the transition between the different classes of flow are investigated and each functioning regime is associated with the corresponding thermodynamic region of reservoir states

Effects of Molecular Complexity and Reservoir Conditions on the Discharge Coefficient of Adapted Planar Nozzles

The transonic flow at throat section of a convergent-divergent nozzle is studied in adapted conditions to assess the influence of the fluid molecular complexity and total thermodynamic state on the discharge coefficient. The standard Sauer method is applied to solve the transonic perturbation potential equation in the vicinity of the nozzle throat. An analytic expression is derived that allows one to compute the discharge coefficient in terms of the nozzle curvature at the throat section and of the value of the fundamental derivative of gasdynamics at sonic conditions, which depends on the fluid molecular complexity and on the thermodynamic state in the reservoir. A linear dependence of the discharge coefficient on the sonic value of the fundamental derivative of gasdynamics is exposed

Efficient Lagrangian particle tracking algorithms for distributed-memory architectures

This paper focuses on the solution of the dispersed phase of Eulerian–Lagrangian one-way coupled particle laden flows. An efficient two-constraint domain partitioning for 2D and 3D unstructured hybrid meshes is proposed and implemented in distributed memory architectures. A preliminary simulation, using a set of representative particles, is performed first to suitably tag the cells with a weight proportional to the probability of being crossed by a particle. In addition, an innovative parallel ray-tracing location algorithm is presented. A global identifier is assigned to each particle resulting in a significant reduction of the overall communication among processes. The proposed approaches are verified against two steady reference cases for ice accretion simulation: a NACA 0012 profile and a NACA 64A008 swept horizontal tail. Furthermore, a cloud droplet impact test case starting from an unsteady flow around a 3D cylinder is performed to evaluate the code performances on unsteady problems

A robust 3D particle tracking solver for in-flight ice accretion using arbitrary precision arithmetic

A particle tracking code is presented to compute droplet trajectories within a known Eulerian flow field for in-flight ice accretion simulations. The implementation allows for hybrid or unstructured meshes used by common CFD solvers. A known vicinity algorithm was devised to identify particles inside the mesh by computing the intersection between the particle trajectory and the faces of the mesh elements. Arbitrary precision arithmetic is used in the intersection evaluation in order to avoid errors when selecting the exit face if the intersection point is close to or coincident with a vertex or an edge. State-of- the-art wall interaction models are used to take into account droplet rebound, splash and spread at the walls. Non planar surface elements are assumed to improve the accuracy in evaluating the trajectory of secondary re-emitted particles. The software exhibits almost linear scaling when running in parallel on a distributed memory system. The particle tracking code is assessed against the experimental results regarding the impingement of Supercooled Large Droplets over a wing

Finite volume and finite element schemes for the euler equations in cylindrical and spherical coordinates

A numerical scheme is presented for the solution of the compressible Euler equations over unstructured grids in cylindrical and spherical coordinates. The proposed scheme is based on a mixed finite volume / finite element approach. Numerical simulations are presented for the explosion problem in two spatial dimensions in cylindrical and spherical coordinates, and the numerical results are compared with the one-dimensional simulation for cylindrically and spherically symmetric explosions

Maximum Intensity of Rarefaction Shock Waves for Dense Gases

Modern thermodynamic models indicate that fluids consisting of complex molecules may display non-classical gasdynamic phenomena such as rarefaction shock waves (RSWs) in the vapour phase. Since the thermodynamic region in which non-classical phenomena are physically admissible is finite in terms of pressure, density and temperature intervals, the intensity of RSWs is expected to exhibit a maximum for any given fluid. The identification of the operating conditions leading to the RSW with maximum intensity is of paramount importance for the experimental verification of the existence of non-classical phenomena in the vapour phase and for technical applications taking advantage of the peculiarities of the non-classical regime. This study investigates the conditions resulting in an RSW with maximum intensity in terms of pressure jump, wave Mach number and shock strength. The upstream state of the RSW with maximum pressure drop is found to be located along the double-sonic locus formed by the thermodynamic states associated with an RSW having both pre- and post-shock sonic conditions. Correspondingly, the maximum-Mach thermodynamic and maximum-strength loci locate the pre-shock states from which the RSW with the maximum wave Mach number and shock strength can originate. The qualitative results obtained with the simple van der Waals model are confirmed with the more complex Stryjek–Vera–Peng–Robinson, Martin–Hou and Span–Wagner equations of state for selected siloxane and perfluorocarbon fluids. Among siloxanes, which are arguably the best fluids for experiments aimed at the generation and measurement of an RSW, the state-of-the-art Span–Wagner multi-parameter equation of state predicts a maximum wave Mach number close to 1.026 for D6 (dodecamethylcyclohexasiloxane, [O-Si-(CH3)2]6). Such value is well within the capacity of the measurement system of a newly built experimental set-up aimed at the first-ever demonstration of the existence of RSWs in dense vapours

Optimisation of a Nacelle Electro-Thermal Ice Protection System for Icing Wind Tunnel Testing

Abstract Aircraft are equipped with ice protection systems (IPS), to avoid, delay or remove ice accretion. Two widely used technologies are the thermo-pneumatic IPS and the electro-thermal IPS (ETIPS). Thermo-pneumatic IPS requires air extraction from the engine negatively affecting its performances. Moreover, in the context of green aviation, aircraft manufacturers are moving towards hybrid or fully electric aircraft requiring all electric on-board systems. In this work, an ETIPS has been designed and optimised to replace the nacelle pneumatic-thermal system. The aim is to minimise the power consumption while assuring limited or null ice formation and that the surface temperature remains between acceptable bounds to avoid material degradation. The design parameters were the length and heat flux of each heater. Runback ice formations and surface temperature were assessed by means of the in-house developed PoliMIce framework. The optimisation was performed using a genetic algorithm, and the constraints were handled through a linear penalty method. The optimal configuration required 33% less power with respect to the previously installed thermo-pneumatic IPS. Furthermore, engine performance is not affected in the case of the ETIPS. This energy saving resulted in an estimated reduction of specific fuel consumption of 3%, when operating the IPS in anti-icing mode

An explicit primitive conservative solver for the Euler equations with arbitrary equation of state

This work presents a procedure to solve the Euler equations by explicitly updating, in a conservative manner, a generic thermodynamic variable such as temperature, pressure or entropy instead of the total energy. The presented procedure is valid for any equation of state and spatial discretization. When using complex equations of state such as Span–Wagner, choosing the temperature as the generic thermodynamic variable yields great reductions in the computational costs associated to thermodynamic evaluations. Results computed with a state of the art thermodynamic model are presented, and computational times are analyzed. Particular attention is dedicated to the conservation of total energy, the propagation speed of shock waves and jump conditions. The procedure is thoroughly tested using the Span–Wagner equation of state through the CoolProp thermodynamic library and the Van der Waals equation of state, both in the ideal and non-ideal compressible fluid-dynamics regimes, by comparing it to the standard total energy update and analytical solutions where available

Numerical simulation of aileron buzz using an adaptive-grid compressible flow solver for dynamic meshes

The paper presents numerical results from a novel scheme for the solution of the flow equations in two dimensional domains by an Arbitrary Lagrangian Eulerian formulation able to cope with deforming and adaptive two dimensional grids without recurring to any explicit interpolation scheme. The method is applied to the investigation of a classical transonic aeroelastic instability phenomenon: the aileron buzz. By resorting to deforming and adaptive grids, the method allows to highlight the dependency of the aeroelastic stability boundaries on the mesh spacing