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

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

Non-ideal effects on the typical trailing edge shock pattern of ORC turbine blades

Abstract At the trailing edge of supersonic high-pressure turbine vanes, a typical shock pattern, the so-called fish-tail shocks, originates due to the flow rotation imposed by its finite thickness. In addition, shock and shock/fan systems can arise in case of a post-expanded channel design or at off-design conditions. ORC turbine stator blades are particularly prone to this phenomena since they are designed to provide a high outlet Mach number, especially at the first stage. In the close proximity of the saturation curve, near the critical point, molecularly complex organic fluids for ORC applications may exhibit a number of non-ideal gasdynamic effects, including a remarkable dependency of the shock properties on the upstream thermodynamic state of the fluid, at a fixed upstream Mach number. The influence of thermodynamic conditions on the shock pattern is assessed as a function of the flow deviation and compared against the ideal gas case, for which the shock properties depends on the upstream Mach number only. Non-ideal effects are investigated here using siloxane vapor MDM (Octamethyltrisiloxane, C 8 H 24 O 2 Si 3 ), as an exemplary organic fluid. The present results can be arguably extended to most vapors currently employed in ORC applications

Non-Classical Gas Dynamics of Vapour Mixtures

The non-classical gas dynamics of binary mixtures of organic fluids in the vapour phase is investigated for the first time. A predictive thermodynamic model is used to compute the relevant mixture properties, including its critical point coordinates and the local value of the fundamental derivative of gas dynamics Gamma. The considered model is the improved Peng-Robinson Stryjek-Vera cubic equation of state, complemented by the Wong-Sandler mixing rules. A finite thermodynamic region is found where the nonlinearity parameter Gamma is negative and therefore non-classical gas dynamics phenomena are admissible. A non-monotone dependence of Gamma on the mixture composition is observed in the case of binary mixtures of siloxane and perfluorocarbon fluids, with the minimum value of Gamma in the mixture being always larger than that of its more complex component. The observed dependence indicates that non-ideal mixing has a strong influence on the gas dynamics behaviour, either classical or non-classical, of the mixture. Numerical experiments of the supersonic expansion of a mixture flow around a sharp corner show the transition from the classical configuration, exhibiting an isentropic rarefaction fan centred at the expansion corner, to non-classical ones, including mixed expansion waves and rarefaction shock waves, if the mixture composition is changed

Development of a Background-Oriented Schlieren Technique with Telecentric Lenses for Supersonic Flow

Background oriented schlieren (BOS) is a quantitative optical technique which exploits light deflection occurring in non-homogeneous transparent media. It allows to indirectly measure the density gradients by analysing the apparent displacement of features of a background pattern when imaged through the investigated flow. Thanks to its simple set-up and to the consolidated data reduction technique based on cross-correlation algorithms the BOS technique has progressively attracted the interest of the researchers. In this work a BOS system using a telecentric lens system has been set up in order to improve measurement accuracy and to avoid 3D effects arising from using conventional entocentric lenses. The design of the telecentric lens system is reported along with an analysis of its performance in term of spatial resolution. Some preliminary tests on a supersonic flows are also reported

Critical Point Anomalies Include Expansion Shock Waves

From first-principle fluid dynamics, complemented by a rigorous state equation accounting for critical anomalies, we discovered that expansion shock waves may occur in the vicinity of the liquid-vapor critical point in the two-phase region. Due to universality of near-critical thermodynamics, the result is valid for any common pure fluid in which molecular interactions are only short-range, namely, for so-called 3-dimensional Ising-like systems, and under the assumption of thermodynamic equilibrium. In addition to rarefaction shock waves, diverse non-classical effects are admissible, including composite compressive shock-fan-shock waves, due to the change of sign of the fundamental derivative of gasdynamics

The admissibility domain of rarefaction shock waves in the near-critical vapour-liquid equilibrium region of pure typical fluids

Application of the scaled fundamental equation of state of Balfour et al. (Phys. Lett. A, vol. 65, 1978, pp. 223-225) based upon universal critical exponents, demonstrates that there exists a bounded thermodynamic domain, located within the vapour-liquid equilibrium region and close to the critical point, featuring so-called negative nonlinearity. As a consequence, rarefaction shock waves with phase transition are physically admissible in a limited two-phase region in the close proximity of the liquid-vapour critical point. The boundaries of the admissibility region of rarefaction shock waves are identified from first-principle conservation laws governing compressible flows, complemented with the scaled fundamental equations. The exemplary substances considered here are methane, ethylene and carbon dioxide. Nonetheless, the results are arguably valid in the near-critical state of any common fluid, namely any fluid whose molecular interactions are governed by short-range forces conforming to three-dimensional Ising-like systems, including, e.g. water. Computed results yield experimentally feasible admissible rarefaction shock waves generating a drop in pressure from 1 to 6 bar and pre-shock Mach numbers exceeding 1.5

Three-Dimensional Simulation of a Complete Vertical Axis Wind Turbine Using Overlapping Grids

Three-dimensional simulations of the aerodynamic field around a three-blade straight-axis Vertical Axis Wind Turbine (VAWT) are presented for two values of the Tip Speed Ratio λ (TSR), namely λ=1.52 and λ=2.5. Numerical simulations were carried out using the over-set grid solver ROSITA (ROtorcraft Software ITAly). The Reynolds-Averaged Navier-Stokes equations are completed by the Spalart-Allmaras turbulence model. A strong interaction between the blade and the blade wakes is evidenced. Dynamic stall is observed in the case λ=2.5. The computed flow-field presents diverse three-dimensional effects, including the interaction between the blades and the tip vortices and the aerodynamic disturbances from the turbine shaft and the support arms. Three-dimensional effects are more relevant for λ=2.5. The comparison to experimental data confirms the general features of the flow

SU2: The Open-Source Software for Non-ideal Compressible Flows

The capabilities of the open-source SU2 software suite for the numerical simulation of viscous flows over unstructured grid are extended to non-ideal compressible-fluid dynamics (NICFD). A built-in thermodynamic library is incorporated to account for the non-ideal thermodynamic characteristics of fluid flows evolving in the close proximity of the liquid-vapour saturation curve and critical point. The numerical methods, namely the Approximate Riemann Solvers (ARS), viscous fluxes and boundary conditions are generalised to non-ideal fluid properties. Quantities of interest for turbomachinery cascades, as loss coefficients and flow angles, can be automatically determined and used for design optimization. A variety of test cases are carried out to assess the performance of the solver. At first, numerical methods are verified against analytical solution of reference NICFD test cases, including steady shock reflection and unsteady shock tube. Then, non-ideal gas effects in planar nozzles and past turbine cascades, typically encountered in Organic Rankine Cycle applications, are investigated and debated. The obtained results demonstrate that SU2 is highly suited for the analysis and the automatic design of internal flow devices operating in the non-ideal compressible-fluid regime