Non-ideal Fluid Research Articles

Accuracy assessment of the Non-Ideal Computational Fluid Dynamics model for siloxane MDM from the open-source SU2 suite

Abstract The first-ever accuracy assessment of a computational model for Non-Ideal Compressible-Fluid Dynamics (NICFD) flows is presented. The assessment relies on a comparison between numerical predictions, from the open-source suite SU2, and pressure and Mach number measurements of compressible fluid flows in the non-ideal regime. Namely, measurements regard supersonic flows of siloxane MDM (Octamethyltrisiloxane, C 8 H24O 2 Si 3 ) vapor expanding along isentropes in the close proximity of the liquid–vapor saturation curve. The model accuracy assessment takes advantage of an Uncertainty Quantification (UQ) analysis, to compute the variability of the numerical solution with respect the uncertainties affecting the test-rig operating conditions. This allows for an uncertainty-based assessment of the accuracy of numerical predictions. The test set is representative of typical operating conditions of Organic Rankine Cycle systems and it includes compressible flows expanding through a converging–diverging nozzle in mildly-to-highly non-ideal conditions. All the considered flows are well represented by the computational model. Therefore, the reliability of the numerical implementation and the predictiveness of the NICFD model are confirmed.

Design Method and Performance Prediction for Radial-Inflow Turbines of High-Temperature Mini-Organic Rankine Cycle Power Systems

The realization of commercial mini organic Rankine cycle (ORC) power systems (tens of kW of power output) is currently pursued by means of various research and development activities. The application driving most of the efforts is the waste heat recovery from long-haul truck engines. Obtaining an efficient mini radial inflow turbine, arguably the most suitable type of expander for this application, is particularly challenging, given the small mass flow rate, and the occurrence of nonideal compressible fluid dynamic effects in the stator. Available design methods are currently based on guidelines and loss models developed mainly for turbochargers. The preliminary geometry is subsequently adapted by means of computational fluid-dynamic calculations with codes that are not validated in case of nonideal compressible flows of organic fluids. An experimental 10 kW mini-ORC radial inflow turbine will be realized and tested in the Propulsion and Power Laboratory of the Delft University of Technology, with the aim of providing measurement datasets for the validation of computational fluid dynamics (CFD) tools and the calibration of empirical loss models. The fluid dynamic design and characterization of this machine is reported here. Notably, the turbine is designed using a meanline model in which fluid-dynamic losses are estimated using semi-empirical correlations for conventional radial turbines. The resulting impeller geometry is then optimized using steady-state three-dimensional computational fluid dynamic models and surrogate-based optimization. Finally, a loss breakdown is performed and the results are compared against those obtained by three-dimensional unsteady fluid-dynamic calculations. The outcomes of the study indicate that the optimal layout of mini-ORC turbines significantly differs from that of radial-inflow turbines (RIT) utilized in more traditional applications, confirming the need for experimental campaigns to support the conception of new design practices.

Interaction pressure tensor on high-order lattice Boltzmann models for nonideal fluids.

In this work we address the application of pseudopotentials directly on high-order lattice Boltzmann models. We derive a general expression for the pressure tensor on high-order lattices considering all nonideal interactions, including intra- and intermolecular interactions, following the discrete lattice theory introduced by X. Shan [Phys. Rev. E 77, 066702 (2008)PLEEE81539-375510.1103/PhysRevE.77.066702]. From the derived expression, a generalized continuum approximation, truncated at fourth-order isotropy, is obtained that is readily applicable to high-order lattices. With this, we demonstrate that high-order lattice models with pseudopotentials can satisfy thermodynamic consistency. The derived generalized expression and continuum approximation are validated for the case of a flat interface and compared against the standard definition available from the literature. The generalized expression is also shown to accurately reproduce the Laplace experiment for a variety of high-order lattice structures. This work sets the preliminary steps towards the application of high-order lattice models for simulating nonideal fluid mixtures.

Shock-induced energy conversion of entropy in non-ideal fluids

From shaping cosmic structures in space to producing intense sounds in aircraft engines, shock waves in fluids ineluctably convert entropy fluctuations into swirling motions and sound waves. Studies of the corresponding conversion from internal energy to kinetic energy have so far been restricted to ideal (or idealised) fluids. Yet, many substances do not obey the ideal-gas law (including those in the above two examples). The present work demonstrates that non-ideal thermodynamic properties provide a remarkable degree of control over the conversion to solenoidal and dilatational kinetic energies. Of particular interest is the ability to suppress much of the emitted acoustic field whilst promoting mixing downstream of the shock. This is made possible by exploiting the convexity (or lack thereof) of the shock adiabats. Whilst illustrated here using dense vapours near the thermodynamic critical point, this ability to design and control specific shock-induced energy transfers extends beyond near-critical-point phenomena; e.g. shocked mixtures (high-speed dusty flows on Mars, nanoparticle formation in supersonic expanders for drug manufacturing), reacting fronts (supersonic combustion, rocket propulsion), ionising shocks (reentry systems, inertial confinement fusion) or fronts in active fluids (bacterial and crowd flows). This theoretical work, which demonstrates the predictive capabilities of linear theory, lays the foundation for future experimental investigations ultimately aimed at delivering novel shock-based flow-control strategies exploiting the thermodynamic properties of the fluid.

Linear instability of Poiseuille flows with highly non-ideal fluids

The objective of this work is to investigate linear modal and algebraic instability in Poiseuille flows with fluids close to their vapour–liquid critical point. Close to this critical point, the ideal gas assumption does not hold and large non-ideal fluid behaviours occur. As a representative non-ideal fluid, we consider supercritical carbon dioxide ($\text{CO}_{2}$) at a pressure of 80 bar, which is above its critical pressure of 73.9 bar. The Poiseuille flow is characterized by the Reynolds number ($Re=\unicode[STIX]{x1D70C}_{w}^{\ast }u_{r}^{\ast }h^{\ast }/\unicode[STIX]{x1D707}_{w}^{\ast }$), the product of the Prandtl ($Pr=\unicode[STIX]{x1D707}_{w}^{\ast }C_{pw}^{\ast }/\unicode[STIX]{x1D705}_{w}^{\ast }$) and Eckert numbers ($Ec=u_{r}^{\ast 2}/C_{pw}^{\ast }T_{w}^{\ast }$) and the wall temperature that in addition to pressure determine the thermodynamic reference condition. For low Eckert numbers, the flow is essentially isothermal and no difference with the well-known stability behaviour of incompressible flows is observed. However, if the Eckert number increases, the viscous heating causes gradients of thermodynamic and transport properties, and non-ideal gas effects become significant. Three regimes of the laminar base flow can be considered: the subcritical (temperature in the channel is entirely below its pseudo-critical value), transcritical and supercritical temperature regimes. If compared to the linear stability of an ideal gas Poiseuille flow, we show that the base flow is modally more unstable in the subcritical regime, inviscid unstable in the transcritical regime and significantly more stable in the supercritical regime. Following the principle of corresponding states, we expect that qualitatively similar results will be obtained for other fluids at equivalent thermodynamic states.

Design Methodology for Supersonic Radial Vanes Operating in Nonideal Flow Conditions

The stator vanes of high-temperature organic Rankine cycle (ORC) radial-inflow turbines (RIT) operate under severe expansion ratios and the associated fluid-dynamic losses account for nearly two-thirds of the total losses generated within the blading passages. The efficiency of the machine can strongly benefit from specialized high-fidelity design methods able to provide shapes attenuating shock wave formation, consequently reducing entropy generation across the shock-wave and mitigating shock-wave boundary layer interaction. Shape optimization is certainly a viable option to deal with supersonic ORC stator design, but it is computationally expensive. In this work, a robust method to approach the problem at reduced computational cost is documented. The method consists of a procedure encompassing the method of characteristics (MoC), extended to nonideal fluid flow, for profiling the diverging part of the nozzle. The subsonic section and semibladed suction side are retrieved using a simple conformal geometrical transformation. The method is applied to design a supersonic ORC stator working with Toluene vapor, for which two blade shapes were already available. The comparison of fluid-dynamic performance clearly indicates that the MoC-Based method is able to provide the best results with the lowest computational effort, and is therefore suitable to be used in a systematic manner for drawing general design guidelines.

Dynamics of gravitational instability excitation in viscoelastic polytropic fluids

The evolutionary excitation dynamics of the gravitational instability in a self-gravitating viscoelastic non-thermal polytropic complex fluid is semi-analytically explored on the astro-scales of space and time. The polytropic equation of state is well validated for the hydrostatic equilibrium established by a perfect heating-cooling balancing in the uni-component complex fluid. We apply a generalized gravitating hydrodynamic model in the concurrent presence of buoyancy, thermal fluctuations, volumetric expansion, and so forth. A normal mode (local) analysis yields a quadratic linear dispersion relation with a unique set of multi-parametric coefficients. The analytical reliability is checked by comparing with the existing reports on purely ideal inviscid nebular fluids and non-ideal viscoelastic fluids in isolation. It is seen that, unlike the normal instability mechanisms, the instability here remains unaffected due to the thermo-mechanical diffusion processes. The stabilizing (destabilizing) and accelerating (decelerating) factors of the instability are illustratively explored. The instability features are judged in the light of both impure non-ideal viscoelastic fluid and pure ideal inviscid nebular fluid scenarios. The relevancy of our exploration in superdense compact viscoelastic astro-objects and their surrounding atmospheres is summarily outlined.

Rate-Controlled Constrained-Equilibrium Application in Shock Tube Ignition Delay Time Simulation

The rate-controlled constrained-equilibrium (RCCE), a model order reduction method, assumes that the nonequilibrium states of a system can be described by a sequence of constrained-equilibrium kinetically controlled by relatively a small number of constraints within acceptable accuracies. The full chemical composition at each constrained-equilibrium state is obtained by maximizing (or minimizing) the appropriate thermodynamic quantities, e.g., entropy (or Gibbs functions) subject to the instantaneous values of the constraints. Regardless of the nature of the kinetic constraints, RCCE always guarantees correct final equilibrium state. Ignition delay times measured in shock tube experiments with low initial temperatures are significantly shorter than the values obtained by constant volume models. Low initial temperatures and thus longer shock tube test times cause nonideal heat transfer and fluid flow effects such as boundary layer growth and shock wave attenuation to gradually increase the pressure (and simultaneously increase the temperature) before ignition. To account for these effects, in this paper, the RCCE prescribed enthalpy and pressure (prescribed h/p) model has been further developed and has been applied to methane shock tube ignition delay time simulation using GRI-Mech 3.0. Excellent agreement between RCCE predictions and shock tube experimental data was achieved.

Methodology to develop off-design models of heat exchangers with non-ideal fluids

Supercritical CO2 (sCO2) closed Brayton cycles are promising heat engines for next-generation thermal power plants since they are efficient, highly scalable, and compatible with a variety of heat sources. A potential application for these cycles is load-following concentrating solar power plants with thermal storage, which will frequently operate at off-design conditions. Accurate and computationally efficient models of the cycle’s heat exchangers and turbomachinery are required to assess and optimise its off-design performance. The printed circuit heat exchangers (PCHEs) used in the sCO2 closed Brayton cycle are challenging to model since they exhibit non-ideal-gas effects and typically use zigzag channels, for which flow patterns and heat transfer mechanisms are not completely understood. Moreover, heat transfer correlations that capture all effects relevant to a given geometry and flow conditions are often unavailable.We present a methodology to develop accurate and computationally efficient on- and off-design models of heat exchangers that exhibit complex nonlinear behaviours. This methodology involves fitting a 1D discretised heat exchanger model to experimental data using nonlinear least-squares optimisation. Unknown internal heat exchanger geometric parameters are used as fitting parameters.We demonstrate the proposed methodology by developing numerical models for two PCHEs: (1) an sCO2–sCO2 PCHE operating far from CO2’s critical point and (2) an oil–sCO2 PCHE operating close to CO2’s critical point, where non-ideal fluid property variations are significant. Test data spans heat loads from 6–48% and 15–39% of name-plate duty for heat exchangers (1) and (2) respectively. Across these operating ranges, the maximum and mean heat transfer prediction residuals are 0.91% and 0.36% for heat exchanger (1) and 3.04% and 1.24% for heat exchanger (2). Additionally, we show that good accuracy can be obtained when using only the channel hydraulic diameter as a fitting parameter. Due to their low computational cost and high accuracy, models developed using the proposed methodology are eminently suitable for off-design modelling and optimisation of the sCO2 closed Brayton cycle and other power cycles or industrial processes where heat exchangers exhibit complex nonlinear behaviour.

A look-up table method based on unstructured grids and its application to non-ideal compressible fluid dynamic simulations

Fast and accurate computation of thermo-physical properties is essential in computationally expensive simulations involving fluid flows that significantly depart from the ideal gas or ideal liquid behavior. A look-up table algorithm based on unstructured grids is proposed and applied to non-ideal compressible fluid dynamics simulations. The algorithm grants the possibility of a fully automated generation of the tabulated thermodynamic region for any boundary and to use mesh refinement. Results show that the proposed algorithm leads to a computational cost reduction up to one order of magnitude, while retaining the same accuracy level compared to simulations based on more complex equation of state. Furthermore, a comparison of the LuT algorithm with a uniformly spaced quadrilateral tabulation method resulted in similar performance and accuracy.

Effects of thermal fluctuations in the fragmentation of a nanoligament

We study the effects of thermally induced capillary waves in the fragmentation of a liquid ligament into multiple nanodroplets. Our numerical implementation is based on a fluctuating lattice Boltzmann (LB) model for nonideal multicomponent fluids, including nonequilibrium stochastic fluxes mimicking the effects of molecular forces at the nanoscales. We quantitatively analyze the statistical distribution of the breakup times and the droplet volumes after the fragmentation process at changing the two relevant length scales of the problem, i.e., the thermal length scale and the ligament size. The robustness of the observed findings is also corroborated by quantitative comparisons with the predictions of sharp interface hydrodynamics. Beyond the practical importance of our findings for nanofluidic engineering devices, our study also explores a novel application of LB in the realm of nanofluidic phenomena.

An adaptive ALE scheme for non-ideal compressible fluid dynamics over dynamic unstructured meshes

This paper investigates the application of mesh adaptation techniques in the Non-Ideal Compressible Fluid Dynamic (NICFD) regime, a region near the vapor-liquid saturation curve where the flow behavior significantly departs from the ideal gas model, as indicated by a value of the fundamental derivative of gasdynamics less than one. A recent interpolation-free finite-volume adaptive scheme is exploited to modify the grid connectivity in a conservative way, and the governing equations for compressible inviscid flows are solved within the Arbitrary Lagrangian Eulerian framework by including special fictitious fluxes representing volume modifications due to mesh adaptation.The absence of interpolation of the solution to the new grid prevents spurious oscillations that may make the solution of the flow field in the NICFD regime more difficult and less robust.Non-ideal gas effects are taken into account by adopting the polytropic Peng-Robinson thermodynamic model. The numerical results focus on the problem of a piston moving in a tube filled with siloxane $\mathrm{MD_4M}$, a simple configuration which can be the core of experimental research activities aiming at investigating the thermodynamic behavior of NICFD flows. Several numerical tests involving different piston movements and initial states in 2D and 3D assess the capability of the proposed adaption technique to correctly capture compression and expansion waves, as well as the generation and propagation of shock waves, in the NICFD and in the non-classical regime.

A study of wall boundary conditions in pseudopotential lattice Boltzmann models

The effect of fluid–solid interactions on the hydrodynamics of non-ideal fluids and wettability of surfaces is investigated. We integrate the interaction forces, simulated by pseudopotentials, into two on-site boundary conditions: standard bounce-back (SBB) and Zou and He (ZH) [12] to determine the distribution functions of the boundary nodes. Three different interaction forces are tested: pseudopotential-based interaction (ψ), modified pseudopotential-based interaction (mψ), and a ZH-based interaction, which is proposed by this study based on the ZH method. Therefore, the schemes are ψ-SBB, mψ-SBB, mψ-ZH, and ZH-ZH. The first criterion is the achievement of macroscopic Poiseuille flow. The second criterion is the achievement of a wide range of contact angles. The main method of simulation is multipseudopotential interaction [30]. It is found that the scheme of ψ-SBB creates a relatively large fluctuation of density across the channel. Whilst, the schemes of mψ-SBB, mψ-ZH, and ZH-ZH generate much less density variation across the channel. Among them, ZH-ZH treatment is superior based on density fluctuation and the error associated with the resolution, relaxation time, and compressibility. We found that all four boundary conditions can form a wide of range of contact angles. The ψ-SBB scheme creates largest density fluctuation inside a drop on wettable surfaces. The schemes of mψ-SBB and mψ-ZH create almost the same density fluctuation which is larger than ZH-ZH. Moreover, mψ interaction generates spurious velocities as high as six times a free drop with SBB and eight times with ZH while spurious velocities in ψ-SBB and ZH-ZH are very close to the free drop. Therefore, ZH-ZH performs best, also, in wettability tests.

Corner-transport-upwind lattice Boltzmann model for bubble cavitation.

Aiming to study the bubble cavitation problem in quiescent and sheared liquids, a third-order isothermal lattice Boltzmann model that describes a two-dimensional (2D) fluid obeying the van der Waals equation of state, is introduced. The evolution equations for the distribution functions in this off-lattice model with 16 velocities are solved using the corner-transport-upwind (CTU) numerical scheme on large square lattices (up to 6144×6144 nodes). The numerical viscosity and the regularization of the model are discussed for first- and second-order CTU schemes finding that the latter choice allows to obtain a very accurate phase diagram of a nonideal fluid. In a quiescent liquid, the present model allows us to recover the solution of the 2D Rayleigh-Plesset equation for a growing vapor bubble. In a sheared liquid, we investigated the evolution of the total bubble area, the bubble deformation, and the bubble tilt angle, for various values of the shear rate. A linear relation between the dimensionless deformation coefficient D and the capillary number Ca is found at small Ca but with a different factor than in equilibrium liquids. A nonlinear regime is observed for Ca≳0.2.

Discharge of Non-Reactive Fluids from Vessels

Abstract This paper presents simulations of discharges from pressure vessels that consistently account for non-ideal fluid behavior in all the required thermodynamic properties and individually considers all the chemical components present. The underlying assumption is that phase equilibrium occurs instantaneously inside the vessel and, thus, the dynamics of the fluid in the vessel comprises a sequence of equilibrium states. The formulation leads to a system of differential-algebraic equations in which the component mass balances and the energy balance are ordinary differential equations. The algebraic equations account for the phase equilibrium conditions inside the vessel and at the discharge point, and for sound speed calculations. The simulator allows detailed predictions of the condition inside the vessel and at the discharge point as a function of time, including the flow rate and composition of the discharge. The paper presents conceptual applications of the simulator to predict the effect of leaks from vessels containing mixtures of light gases and/or hydrocarbons and comparisons to experimental data.

Active subspaces for the optimal meanline design of unconventional turbomachinery

The preliminary fluid dynamic design of turbomachinery operating with non-standard working fluids and unusual operating conditions and specifications can be very challenging because of the lack of know-how and guidelines. Examples are the design of turbomachinery for small-capacity organic Rankine cycle and supercritical CO2 cycle power plants, whereby the efficiency of turbomachinery components has also a strong influence on the net conversion efficiency of the system. These machines operate with the fluid in thermodynamic states which, for part of the process, largely deviate from those obeying to the ideal gas law. This in turn implies the presence of so-called non-ideal compressible fluid dynamics effects.Active subspaces, a model reduction technique, is at the basis of the methodology presented here, which is aimed at the optimal meanline design of unconventional turbomachinery. The resulting surrogate model depends on a very small set of non-physical variables, called active variables. The procedure integrates into a single constrained optimization framework the selection of the working fluid, the thermodynamic cycle calculation and the preliminary sizing of the turbomachinery component.As a demonstration of the advantages of the proposed approach, the design of a 10kW mini organic Rankine cycle turbine with a turbine inlet temperature of 240°C is illustrated. In this case, approximately the same maximum efficiency is estimated for three dissimilar turbines operating with different working fluids and rather different thermodynamic cycles. The use of active subspaces allows the seamless evaluation of the sensitivity of results to input parameters, both those related to the machine and the working fluid. The novel design procedure is compared in terms of computational efficiency to a conventional approach based on the coupling of a genetic algorithm directly with a meanline code. Results show that the calculation based on the use of surrogate models is more than two orders of magnitude faster. The surrogate can be used to solve any design problem within the specified boundaries of the design envelope. Results are affected by uncertainty on the estimation of losses and of non-ideal compressible fluid dynamics effects, which, in turn, do not affect the applicability of the method, which will become quantitatively accurate once this information will be available. Work to this end is underway in various laboratories.

On the numerical simulation of non-classical quasi-1D steady nozzle flows: Capturing sonic shocks

The 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.

Chemical-potential-based lattice Boltzmann method for nonideal fluids.

Chemical potential, as an important thermodynamic quantity, has been popularly used in thermodynamic modeling for complex systems, especially for those involving the phase transitions and chemical reactions. Here we present a chemical-potential-based multiphase lattice Boltzmann model, in which the nonideal force is directly evaluated by a chemical potential. The numerical computation is more efficient than the pressure-tensor-based model [Wen et al. Europhys. Lett. 112, 44002 (2015)10.1209/0295-5075/112/44002] because the calculations of the pressure tensor and its divergence are avoided. We have derived several chemical potentials of the popular equations of state from the free-energy density function. The theoretical analyses and numerical results support that the present model satisfies thermodynamics and Galilean invariance. An effective chemical-potential boundary condition is also implemented to investigate the wettability of a solid surface, and the contact angle can be linearly tuned by the surface chemical potential.

Effect of initial densities in the lattice Boltzmann model for non-ideal fluid with curved interface

The effect of initial densities in a free energy based two-phase-flow lattice Boltzmann method for non-ideal fluids with a curved interface was investigated in the present work. To investigate this effect, the initial densities in the liquid and gas phases coming from the saturation points and the equilibrium state were adopted in the simulation of a static droplet in an open and a closed system. For the purpose of simplicity and easier comparison, the closed system is fabricated by the implementation of the periodic boundary condition at the inlet and outlet of a gas channel, and the open system is fabricated by the implementation of a constant flux boundary condition at the inlet and a free-out boundary condition at the outlet of the same gas channel. By comparing the simulation results from the two types of initial densities in the open and closed systems, it is proven that the commonly used saturation initial densities setting is the reason for droplet mass and volume variation which occurred in the simulation, particularly in the open system with a constant flux boundary condition. Such problems are believed to come from the curvature effect of the surface tension and can be greatly reduced by adopting the initial densities in the two phases from equilibrium state.

Assessment of grid adaptation criteria for steady, two-dimensional, inviscid flows in non-ideal compressible fluids

Two-dimensional simulations are carried out to assess standard grid adaptation criteria, widely used for ideal flows, for steady inviscid flows in the proximity of the liquid-vapor saturation curve, where non-ideal compressible-fluid behavior is expected. A van der Waals fluid description of the thermodynamic properties is assumed to account for non-ideal effects at least qualitatively. Nitrogen under-expanded nozzle jets are chosen as the reference flow to assess different adaptation criteria. Isotropic and anisotropic error estimators based on the derivatives of flow variables prove to be suitable to capture the rarefaction, the reflected shock and the constant-pressure jet boundary. Both density and Mach-based estimators are found to be very suitable to drive grid adaptation in the non-ideal compressible-fluid regime, which is characterized by large fluid compressibility. Then, similar adaptation criteria are used to simulate under-expanded nozzle jets of the siloxane MDM, a high molecular complexity fluid for which the van der Waals model predicts the existence of a thermodynamic region where the fundamental derivative of gasdynamics has values less than one. In this region, Mach number estimators prove to be more effective because of the non-ideal dependence of the speed of sound on the density and the temperature.