dovich-Thompson Fluids Research Articles

Ideal and non-ideal planar compressible fluid flows in radial equilibrium

Two-dimensional compressible flows in radial equilibrium are investigated in the ideal dilute-gas regime and the non-ideal single-phase regime close to the liquid–vapour saturation curve and the critical point. Radial equilibrium flows along constant-curvature streamlines are considered. All properties are therefore independent of the tangential streamwise coordinate. A differential relation for the Mach number dependency on the radius is derived for both ideal and non-ideal conditions. For ideal flows, the differential relation is integrated analytically. Assuming a constant specific heat ratio $\gamma$ , the Mach number is a monotonically decreasing function of the radius of curvature for ideal flows, with $\gamma$ being the only fluid-dependent parameter. In non-ideal conditions, the Mach number profile also depends on the total thermodynamic conditions of the fluid. For high molecular complexity fluids, such as toluene or hexamethyldisiloxane, a non-monotone Mach number profile is admissible in single-phase supersonic conditions. For Bethe–Zel'dovich–Thompson fluids, non-monotone behaviour is observed in subsonic conditions. Numerical simulations of subsonic and supersonic turning flows are carried out using the streamline curvature method and the computational fluid dynamics software SU2, respectively, both confirming the flow evolution from uniform flow conditions to the radial equilibrium profile predicted by the theory.

Nonlinear wave propagation in dense vapor of Bethe–Zel'dovich–Thompson fluids subjected to temperature gradients

Flows of fluids made of complex organic molecules exhibit unconventional fluid dynamic behavior in the vapor phase if their thermodynamic state is close to that of the vapor–liquid critical point. If the molecule is sufficiently complex, this thermodynamic domain is characterized by negative values of the fundamental derivative of gasdynamics Γ, the fluid is called Bethe–Zel'dovich–Thompson (BZT) fluid, and atypical phenomena, such as rarefaction shockwaves, are theoretically admissible. The nature of the steepening of nonlinear waves in dense vapor flows of organic fluids evolving in this thermodynamic region can be significantly affected by the presence of temperature gradients in the flow. This study investigates the evolution of finite-amplitude acoustic waves in these conditions. The steepening of the wavefront is analyzed using the wavefront expansion technique, and the deformation of the wavefront is simulated numerically by solving the Westervelt equation. The results of simulations of wave propagation in dense vapors indicate that, though Γ governs the nature of steepening waves, local gradients in sound speed and density can alter the rate of steepening and can enhance or delay shock formation in the medium, a result relevant also to the envisaged experiments aimed at proving the existence of nonclassical gasdynamics phenomena in BZT vapors.

Shock interactions in two-dimensional steady flows of Bethe–Zel’dovich–Thompson fluids

The morphology of nodes generated by the interaction of discontinuities in steady two-dimensional inviscid flows is examined. The fluids considered are Bethe–Zel’dovich–Thompson (BZT) fluids, which feature negative values of the fundamental derivative of gas dynamics in the vapour phase. The operating conditions correspond to the non-classical gas-dynamic regime where expansion shocks, compression fans and composite waves are admissible in addition to the classical compression shocks and expansion fans. Interactions caused by the crossing, overtaking and splitting of compression/expansion shocks, along with the refraction of these through a contact discontinuity, are analysed here. The well-established method of wave curves is applied to non-classical wave curves, revealing a variety of interaction patterns that are simply not admissible in classical gas dynamics. It is shown that shock waves can be reflected, transmitted and refracted as Prandtl–Meyer fans or composite waves. Based on numerical evidence, the splitting (and consequently the Mach reflection) of an expansion shock seems to be disallowed. Theoretical considerations on the admissibility of such configurations are also provided. The present analysis is relevant to applications potentially involving supersonic flows of BZT fluids, e.g. organic Rankine cycle power systems, and can also be used in front-tracking algorithms for general equations of state.

Steady transonic dense gas flow past two‐dimensional compression/expansion ramps revisited

AbstractSupersonic flow past compression/expansion ramps represents a canonical problem in fluid dynamics which has been studied extensively in the past assuming perfect gas behaviour. More recently real gas effects on such flows have received increasing attention as part of efforts, among others, to increase the efficiency of Organic Rankine Cycles for decentralized power plants. Of special interest in this connection are dense gases of Bethe–Zel'dovich–Thompson fluids which have the distinguishing property that the so called fundamental derivative of gasdynamics Γ takes on negative values in the general neighbourhood of the thermodynamic critical point. Asymptotic analysis of the transonic flow regime has so far concentrated on cases where the unperturbed state with pressure p0, density ρ0 and entropy s0 is close to the transition curve Γ = 0 in the pressure specific volume plane such that |Γ0| ≪ 1, , Kluwick and Cox [1], [2]. These studies have revealed a number of new phenomena and a non‐uniqueness of solutions exceeding that known from the theory of perfect gases.Here we extend the analysis to cover flows where the unperturbed state in the p, 1/ρ plane is in the vicinity of the point on the transition line where Γ and Λ vanish simultaneously. Such flows have |Γ0| ≪ 1, |Λ0| ≪ 1, and are of considerable practical interest as isentropes that can be realized experimentally at present just barely enter the negative Γ region. Again an increased wealth of possible solutions is observed.

Thermodynamic Properties of Dodecamethylpentasiloxane, Tetradecamethylhexasiloxane, and Decamethylcyclopentasiloxane

Siloxanes are widely used in the chemical industry and in process and power engineering. For example, they are used as working fluids of organic Rankine cycle power plants since 20 years ago. For the process design and optimization, thermodynamic properties, such as enthalpy, entropy, speed of sound, density, and vapor–liquid equilibrium, are required. While the properties of short-chained siloxanes, such as hexamethyldisiloxane (MM) or octamethylcyclotetrasiloxane (D₄), have already been investigated comprehensively, information on thermophysical properties of higher order siloxanes is limited. Therefore, measurements on density and speed of sound in the liquid state of dodecamethylpentasiloxane (MD₃M), tetradecamethylhexasiloxane (MD₄M), and decamethylcyclopentasiloxane (D₅) are presented here. On the basis of these measurements and other experimental data from the literature, new fundamental equations of state were developed for these three fluids. The equations are based on the Helmholtz energy and, thus, allow for the calculation of any thermodynamic state property by means of derivatives with respect to the natural variables, namely temperature and density. The obtained models also feature a correct extrapolation behavior in regions where no data are available in order to ensure the applicability of the equations to mixture models in the future. On the basis of the present equations of state and recently published equations for other siloxanes, the possibility of the siloxanes belonging to the group of Bethe–Zel’dovich–Thompson fluids is also investigated.

Oblique waves in steady supersonic flows of Bethe–Zel’dovich–Thompson fluids

Steady 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

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.

Shape optimization of an airfoil in a BZT flow with multiple-source uncertainties

Bethe–Zel’dovich–Thompson fluids (BZT) are characterized by negative values of the fundamental derivative of gasdynamics for a range of temperatures and pressures in the vapor phase, which leads to non-classical gasdynamic behaviors such as the disintegration of compression shocks. These non-classical phenomena can be exploited, when using these fluids in Organic Rankine Cycles (ORCs), to increase isentropic efficiency. A predictive numerical simulation of these flows must account for two main sources of physical uncertainties: the BZT fluid properties often difficult to measure accurately and the usually fluctuating turbine inlet conditions. For taking full advantage of the BZT properties, the turbine geometry must also be specifically designed, keeping in mind the geometry achieved in practice after machining always slightly differs from the theoretical shape. This paper investigates some efficient procedures to perform shape optimization in a 2D BZT flow with multiple-source uncertainties (thermodynamic model, operating conditions and geometry). To demonstrate the feasibility of the proposed efficient strategies for shape optimization in the presence of multiple-source uncertainties, a zero incidence symmetric airfoil wave-drag minimization problem is retained as a case-study. This simplified configuration encompasses most of the features associated with a turbine design problem, as far the uncertainty quantification is concerned. A preliminary analysis of the contributions to the variance of the wave-drag allows to select the most significant sources of uncertainties using a reduced number of flow computations. The resulting mean value and variance of the objective are next turned into metamodels. The optimal Pareto sets corresponding to the minimization of various substitute functions are obtained using a genetic algorithm as optimizer and their differences are discussed.

Siloxanes: A new class of candidate Bethe-Zel’dovich-Thompson fluids

This paper presents a new class of Bethe-Zel’dovich-Thompson fluids, which are expected to exhibit nonclassical gasdynamic behavior in the single-phase vapor region. These are the linear and cyclic siloxanes, light silicon oils currently employed as working fluids in organic Rankine cycle turbines. State-of-the-art multiparameter equations of state are used to describe the thermodynamic properties of siloxanes and to compute the value of the fundamental derivative of gasdynamics Γ, whose negative sign is the herald of nonclassical gasdynamics. Siloxane fluids starting from D6 and cyclic siloxanes of greater complexity, and MD3M and linear siloxanes of greater complexity are predicted to exhibit a thermodynamic region in which Γ is negative and hence nonclassical wavefields are admissible. As an exemplary case, a nonclassical rarefaction shock wave propagating in fluid D6 is studied to demonstrate the possibility of using siloxane fluids in nonclassical gasdynamic applications and to experimentally verify the existence of nonclassical wavefields in the vapor phase. The sensitivity of the present results to the considered thermodynamic model of the fluid is also briefly discussed.

Optimal airfoil shapes for viscous transonic flows of Bethe–Zel’dovich–Thompson fluids

High-performance airfoils for transonic viscous flows of dense gases are constructed using an efficient high-order accurate flow solver coupled with a multi-objective genetic algorithm. Dense gases are theoretically characterized by reversed behavior of the speed of sound in isentropic perturbations for a range of temperatures and pressures in the vapor phase. A class of dense gases, namely the so-called Bethe–Zel’dovich–Thompson (BZT) fluids, might exhibit nonclassical gasdynamic behaviors in the transonic and supersonic regimes, such as the disintegration of compression shocks. Utilizing BZT gases as working fluids may result in low drag exerted on airfoils operating at high transonic speeds thanks to an increase in the airfoil critical Mach number. This advantage can be further improved by a proper design of the airfoil shape, also leading to the enlargement of the airfoil operation range within which BZT effects are significant. Such a result is of particular interest in view of the exploitation of BZT fluids for the development of high-efficiency turbomachinery.

Airfoil Shape Optimization for Transonic Flows Bethe-Zel'dovich-Thompson Fluids

High-performance airfoils for transonic flows of Bethe-Zel'dovich-Thompson fluids are constructed using a robust and efficient Euler flow solver coupled with a multi-objective genetic algorithm. Bethe-Zel'dovich- Thompson fluids are characterized by negative values of the fundamental derivative of gasdynamics for a range of temperatures and pressures in the vapor phase, which leads to nonclassical gasdynamic behaviors such as the disintegration of compression shocks. Using Bethe-Zel'dovich-Thompson gases as working fluids may result in low drag exerted on airfoils operating at high transonic speeds, due to a substantial increase in the airfoil critical Mach number. This advantage can be further improved by a proper design of the airfoil shape, also leading to the enlargement of the airfoil operation range within which Bethe-Zel'dovich-Thompson effects are significant. Such a result is of particular interest in view of the exploitation of Bethe-Zel'dovich-Thompson fluids for the development of high-efficiency turbomachinery.

Roe-type schemes for dense gas flow computations

Dense gas dynamics studies the dynamic behavior of gases in the thermodynamic region close to the liquid–vapor critical point, where the perfect gas law is no longer valid, and has to be replaced by more complex equations of state. In such a region, some fluids, known as the Bethe–Zel’dovich–Thompson fluids, can exhibit non-classical nonlinearities, such as expansion shocks, and mixed shock-fan waves. In the present work, the problem of choosing a suitable numerical scheme for dense gas flow computations is addressed. In particular, some extensions of classical Roe’s scheme to real gas flows are reviewed and their performances are evaluated for flow problems involving non-classical nonlinearities. A simplification to Roe’s linearization procedure is proposed, which does not satisfy the U-property exactly, but significantly reduces complexity and computational costs. Such simplification introduces an additional error O(δ x 2), with δ x the mesh size, with respect to the first-order accurate Roe’s scheme, and O(δ x 6) with respect to its higher-order MUSCL extensions. Numerical experiments, concerning a one-dimensional dense gas shock tube, supersonic flow of a BZT gas past a forward-facing step, and transonic dense gas flow through a turbine cascade, show a negligible influence of the adopted linearization procedure on the solution accuracy, whereas it significantly affects computational efficiency.

Construction and Validation of a Dense Gas Shock Tube

Results from initial tests of a shock tube designed to verify the nonclassical dynamics of Bethe-Zel'dovich-Thompson (BZT) fluids are presented. These tests employed nitrogen at pressures and temperatures required for shock-tube initial conditions that will produce nonclassical phenomena in a BZT fluid. A single-diaphragm, stainless steel shock tube is enclosed in a resistance-heated tube furnace. Each of the 16 segments of the furnace is controlled by a proportional integral differential loop to sustain a uniform temperature along the tube. Water-cooled static and dynamic pressure transducers and thermocouples are used to monitor the static initial conditions and the dynamic wave field. The nitrogen test cases are compared to the ideal gas Riemann problem. The pressure differential of the incident expansion wave matches the theoretical magnitude to within 2%, whereas the average wave speed agrees within 7%

Studies of non-classical shock wave phenomena

Most of our physical intuition regarding compressible flows and shock wave dynamics is based on experience with perfect gases in the context of an ideal fluid description. While this model is often a good approximation, there are some hydrodynamic systems in which shocks can exhibit behavior that lies outside of the classical phenomenology. Here, we present numerical simulations of a variety of such exotic shock wave effects, including “anomalous” behavior in so-called Bethe-Zel'dovich-Thompson fluids, shock “splitting,” the D'yakov-Kontorovich instability, and the influence of nonlinear heat transfer on shock structure in strong point explosions.

Application of Bethe-Zel'dovich-Thompson Fluids in Organic Rankine Cycle Engines

Thenovel concept ofimproving organicRankinecycleengines (ORCEs)by exploiting the unusual e uid dynamic behaviorofaspecialclassofworkinge uidsknownasBethe ‐Zel’dovich‐Thompson(BZT)e uidsisexplored.ORCEs are currently manufactured and used for numerous remote terrestrial applications requiring reliable, unattended power.OneofthemajorlossmechanismsinORCEsisshockwavesgenerated intheturbinestage.Operatingunder theproperthermodynamicconditions, a BZT working e uid can potentially weaken oreliminate shock waves. This would reduce losses due to both the wave drag from shocks as well as losses from boundary-layer separation due to shock ree ections on the turbine blades. Two-dimensional e owe elds through realistic symmetric impulse turbinecascades arecompared using a e ux-limited, e nite differencenumerical schemeto assess differences in e ow efe ciencies. Resultsshowsignie cantimprovementsin turbineefe cienciesforBZTworkinge uidsoverconventional ORCE working e uids.

On the suppression of shock-induced separation in Bethe–Zel'dovich–Thompson fluids

We consider the reflection of oblique compression waves from a two-dimensional, steady, laminar boundary layer on a flat, adiabatic plate at free-stream pressures such that dense-gas effects are non-negligible. The full Navier–Stokes equations are solved through use of a dense-gas version of the Beam–Warming implicit scheme. The main fluids studied are Bethe–Zel'dovich–Thompson (BZT) fluids. These are ordinary gases which have specific heats large enough to cause the fundamental derivative of gasdynamics to be negative for a range of pressures and temperatures in the single-phase vapour regime. It is demonstrated that the unique dynamics of BZT fluids can result in a suppression of shock-induced separation. Numerical tests performed reveal that the physical mechanism leading to this suppression is directly related to the disintegration of any compression discontinuities originating in the flow. We also demonstrate numerically that the interaction of expansion shocks with the boundary layer produces no adverse effects.

Nozzle flows of dense gases

Numerical solutions for steady inviscid flows in conventional converging–diverging nozzles are obtained. The fluids considered are Bethe–Zel’dovich–Thompson fluids, i.e., those having specific heats so large that the fundamental derivative of gas dynamics is negative over a finite range of pressures and temperatures. Three general classes of flow are delineated which include two nonclassical types in addition to the usual classical flows; the latter are qualitatively similar to those of perfect gases. The nonclassical flows are characterized by isentropes containing as many as three sonic points. Numerical solutions depicting finite-strength expansion shocks, steady flows with shock waves standing upstream of the nozzle throat, and steady flows containing as many as three shock waves are presented. Flows having arbitrarily large-exit Mach numbers are found to be possible only if a sonic expansion shock is formed in the nozzle. This observation contrasts with prediction based on the perfect gas theory which states that the occurrence of shock waves always results in a subsonic exit flow.

Transonic flows of Bethe—Zel'dovich—Thompson fluids

Bethe–Zel'dovich–Thompson fluids are ordinary, single-phase fluids in which the fundamental derivative of gasdynamics is negative over a finite range of temperatures and pressures. We examine the steady transonic flow of these fluids over two-dimensional thin wings and turbine blades. The free-stream state is taken to be in the neighbourhood of one of the zeros of the fundamental derivative. It is shown that a modified form of the classical transonic small-disturbance equation governs such flows. Critical Mach number estimates are provided which take into account the non-monotone variation of the Mach number with pressure predicted in previous investigations. Critical Mach numbers well over 0.95 are predicted for conventional airfoil sections. Numerical solutions reveal substantial reductions in the strength of the compression shocks occurring in supercritical flows. Further new results include the prediction of expansion and compression shocks in the same flow and compression bow shocks in flows with subsonic free streams.