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Oblique waves in steady supersonic flows of Bethe-Zel'dovich-Thompson fluids
Full text linkSteady self-similar solutions to the supersonic flow of Bethe-Zel'dovich-Thompson fluids past compressive and rarefactive ramps are derived. Inviscid, non-heat-conducting, non-reacting and single-phase vapour flow is assumed. For convex isentropes and shock adiabats in the pressure-specific volume plane (classical gas dynamic regime), the well-known oblique shock and centred Prandtl-Meyer fan occur at a compressive and rarefactive ramp, respectively. For non-convex isentropes and shock adiabats (non-classical gas dynamic regime), four additional wave configurations may possibly occur; these are composite waves in which a Prandtl-Meyer fan is adjacent up to two oblique shock waves. The steady two-dimensional counterparts of the wave curves defined for the one-dimensional Riemann problem are constructed. In the present context, such curves consist of all the possible states connected to a given initial state (namely, the uniform state upstream of the ramp/wedge) by means of a steady self-similar solution. In addition to the classical case, as many as six non-classical wave-curve configurations are singled out. Moreover, the necessary conditions leading to each type of wave curves are analysed and a map of the upstream states leading to each configuration is determined
Exact solutions to non-classical steady nozzle flows of Bethe-Zel'dovich-Thompson fluids
Get PDFSteady 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
On the numerical simulation of non-classical quasi-1D steady nozzle flows: Capturing sonic shocks
Full text linkThe suitability of Roe-type upwind schemes for the computation of steady quasi-1D flows of non-ideal fluids using explicit integration in the pseudo time is discussed. Based on the particular Roe linearization and entropy fix technique applied, several numerical difficulties can arise in modeling shock waves that are sonic either on the upstream or downstream side of the shock. These so-called sonic shocks typically occur away from stationary points of the cross-sectional area distribution, namely, where the geometrical source term does not vanish. The problem of selecting suitable formulations for the accurate simulation of non-classical quasi-1D steady flows is therefore addressed. Numerical experiments indicate a limited influence of the chosen Roe linearization technique, provided the so-called Property U, allowing the exact representation of steady shocks, is satisfied. Nevertheless, application of standard entropy fixes may either predict an incorrect steady-state transonic expansion neighbouring the sonic shock or even fail to attain a discrete steady state. In the latter case, lack of convergence is due to numerical unbalancing of the flux difference and source term integral over the transonic expansion which occurs in the close proximity of sonic shocks approaching their steady-state position. A simple modification to the synchronous splitting technique of [28] is proposed, which is able to produce the desired steady-state balance and allows substantial improvement in the resolution of sonic shocks
A unified description of oblique waves in ideal and non-ideal steady supersonic flows around compressive and rarefactive corners
Full text linkAccording to classical gas dynamic theory, if a steady supersonic parallel flow encounters a sudden change in the wall slope, two very different phenomena may occur. If the flow expands around a sharp corner, the well-known isentropic Prandtl–Meyer fan is observed. Conversely, a shock wave occurs if the flow is compressed: for wedge angles smaller than the detachment value, which depends on the uniform upstream state, an oblique shock originates at the corner; at larger deviation angles, a detached shock is formed. A unified description of these flows is presented here to extend the validity of the common (Formula presented.)–(Formula presented.) (shock angle–deflection angle) diagram for shocked non-isentropic flows into the realm of isentropic expansions. The new graph allows for a straightforward identification of the wave angles for self-similar flow fields around compressive and rarefactive corners. Besides, it clarifies the relation between shock waves and rarefaction fans in the neighbourhood of the (Formula presented.) axis, where shock waves are weak enough to be fairly well approximated by isentropic compressions. At (Formula presented.), indeed, shock and rarefaction curves are demonstrated to be first order continuous. This result is interpreted in view of the bisector rule for oblique shock waves. Exemplary diagrams are reported for both ideal-gas flows, dilute-gas flows and non-ideal flows of dense vapours in the close proximity of the liquid–vapour saturation curve and critical point. The application of the new diagram is illustrated for the textbook case of the supersonic flow past a diamond-shaped airfoil
Non-ideal oblique shock waves
Full text linkFrom the analysis of the isentropic limit of weak compression shock waves, oblique shock waves in which the post-shock Mach number is larger than the pre-shock Mach number, named non-ideal oblique shocks, are admissible in substances characterized by moderate molecular complexity and in the close proximity to the liquid-vapour saturation curve. Non-ideal oblique shocks of finite amplitude are systematically analysed, clarifying the roles of the pre-shock thermodynamic state and Mach number. The necessary conditions for the occurrence of non-ideal oblique shocks of finite amplitude are singled out. In the parameter space of pre-shock thermodynamic states and Mach number, a new domain is defined which embeds the pre-shock states for which the Mach number increase can possibly take place. The present findings are confirmed by state-of-the-art thermodynamic models applied to selected commercially available fluids, including siloxanes and hydrocarbons currently used as working fluids in renewable energy systems
Non-ideal compressible-fluid effects in oblique shock waves
Full text linkThe non-monotone dependence of the speed of sound along adiabatic transformations is demonstrated to result in the admissibility of non-ideal increase of the flow Mach number across oblique shock waves, for pre-shock states in close proximity of the liquid-vapour saturation curve. This non-ideal behaviour is primarily associated with a less-than-unity value of the fundamental derivative of gasdynamics and, therefore, non-ideal shock waves are expected to be observed in flows of fluids with moderate molecular complexity. The simple yet qualitatively sound van der Waals model is used to confirm the present findings and to provide exemplary non-ideal shock waves
Non-ideal effects on the typical trailing edge shock pattern of ORC turbine blades
No full textAt 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, C8H24O2Si3), as an exemplary organic fluid. The present results can be arguably extended to most vapors currently employed in ORC applications
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