Compressible Flow Model Research Articles

A comparison of the application of RSM and LES turbulence models in the numerical simulation of thermal and flow patterns in a double-circuit Ranque-Hilsch vortex tube

The present paper reports an investigation of a double-circuit Ranque-Hilsch vortex tube using a fully three-dimensional numerical model of the turbulent compressible air flow. The main purpose of the paper is to perform a detailed analysis of the thermal and flow patterns obtained with RSM-LRR and LES turbulence models in order to understand which one is able to represent the Ranque-Hilsch energy separation effect observed in a double-circuit vortex tube in the most effective way.The LES turbulence model is found to represent the qualitative turbulence of the flow better than the RSM-LRR model. The detailed analysis of turbulence characteristics estimated by LES model indicates that turbulence has a significant impact on the energy separation phenomenon.Both LES and RSM-LRR turbulence models are shown to have a weak accuracy in the prediction of the integral characteristics of a double-circuit vortex tube.An attempt was made to improve the results achieved with LES turbulence model by means of a grid refinement and an increase of Smagorinsky constant value, but they were found to have a limited impact. This leads to the conclusion that more advanced kinds of LES turbulence model should be investigated in order to increase the accuracy of the simulation of a double-circuit vortex tube.

A HLL-type Riemann solver for two-phase flow with surface forces and phase transitions

We present a new HLL-type approximate Riemann solver for a compressible two-phase flow model with phase transition and surface forces such as surface tension or electric forces. The solver is obtained following the main ideas of the HLL-approach. Due to the nonlinearity in the kinetic relation driving the phase transition, this solver involves the solution of one single nonlinear equation, contrary to a single-phase HLL-type solver, where no nonlinear equations have to be solved. We present some illustrative numerical examples to show the performance and accuracy of the new solver, comparing it to the relaxation Riemann solver from the literature as well as to the two-shock approximation, which is the most accurate interface Riemann solver currently available for the present situation, but suffers from high computational costs.

Entropic lattice Boltzmann model for gas dynamics: Theory, boundary conditions, and implementation.

We present in detail the recently introduced entropic lattice Boltzmann model for compressible flows [N. Frapolli et al., Phys. Rev. E 92, 061301(R) (2015)PLEEE81539-375510.1103/PhysRevE.92.061301]. The model is capable of simulating a wide range of laminar and turbulent flows, from thermal and weakly compressible flows to transonic and supersonic flows. The theory behind the construction of the model is laid out and its thermohydrodynamic limit is discussed. Based on this theory and the hydrodynamic limit thereof, we also construct the boundary conditions necessary for the simulation of solid walls. We present the inlet and outlet boundary conditions as well as no-slip and free-slip boundary conditions. Details necessary for the implementation of the compressible lattice Boltzmann model are also reported. Finally, simulations of compressible flows are presented, including two-dimensional supersonic and transonic flows around a diamond and a NACA airfoil, the simulation of the Schardin problem, and the three-dimensional simulation of the supersonic flow around a conical geometry.

A Roe-type scheme with low Mach number preconditioning for a two-phase compressible flow model with pressure relaxation

We describe two-phase compressible flows by a hyperbolic six-equation single-velocity two-phase flow model with stiff mechanical relaxation. In particular, we are interested in the simulation of liquid-gas mixtures such as cavitating flows. The model equations are numerically approximated via a fractional step algorithm, which alternates between the solution of the homogeneous hyperbolic portion of the system through Godunov-type finite volume schemes, and the solution of a system of ordinary differential equations that takes into account the pressure relaxation terms. When used in this algorithm, classical schemes such as Roe’s or HLLC prove to be very efficient to simulate the dynamics of transonic and supersonic flows. Unfortunately, these methods suffer from the well known difficulties of loss of accuracy and efficiency for low Mach number regimes encountered by upwind finite volume discretizations. This issue is particularly critical for liquid-gasmixtures due to the large and rapid variation in the flow of the acoustic impedance. To cure the problem of loss of accuracy at low Mach number, in this work we apply to our original Roe-type scheme for the two-phase flow model the Turkel’s preconditioning technique studied by Guillard–Viozat [Computers & Fluids, 28, 1999] for the Roe’s scheme for the classical Euler equations.We present numerical results for a two-dimensional liquid-gas channel flow test that show the effectiveness of the resulting Roe-Turkel method for the two-phase system.

Modeling and Computer Simulation of Centrifugal CO2 Compressors at Supercritical Pressures

The use of supercritical carbon dioxide (SC-CO2) as a working fluid in energy conversion systems has many benefits, including high efficiency, compact turbomachinery, and the abundance of CO2. A very important issue for design optimization and performance analysis of future SC Brayton cycles is concerned with the SC-CO2 flow inside high-speed compressors and turbines. The objective of this paper is to present a novel modeling approach to, and its use in numerical simulations of, SC-CO2 flow inside a high-speed compact compressor. The proposed approach capitalizes on using three different physical and mathematical formulations of one-dimensional (1D) models, i.e., compressible and incompressible flow models using actual properties of SC-CO2 and a compressible ideal gas model, as a reference to verify the predictive capabilities of a three-dimensional (3D) incompressible flow model. The incompressible model has been used to perform simulations for a complete detailed multidimensional model of an SC-CO2 high-speed compact compressor. The advantages of the new model include numerical stability, computational efficiency, and physical accuracy. In particular, it has been shown that the model's predictions are consistent with selected published technical data.

Relative entropy based error estimates for discontinuous Galerkin schemes

These notes give an overview on how the relative entropy stability framework can be employed to derive a posteriori error estimates for semi-(spatially)-discrete discontinuous Galerkin schemes approximating systems of hyperbolic conservation laws endowed with one strictly convex entropy. We also show how these methods can be extended as to cover a related, higher order, model for compressible multiphase flows with non-convex energy.

Freshwater generation from a solar chimney power plant

A modified solar chimney power plant is presented which is not only a solar thermal system able to achieve output power but also is able to extract freshwater from the air. This solar chimney power plant has no greenhouse canopy which is replaced by a collector made of black tubes around the chimney to warm the water and air. The effectiveness of this engineering structure is analyzed in comparison to natural precipitation at nine cities in China; one-dimensional compressible flow and heat transfer mathematical model has been developed to describe the moist air which cools down along the chimney and condenses above the lifting condensation level. The recovery of water is a partial duplication of natural atmospheric convection processes because of the unstable environment lapse rate. The results showed that there is often a high-strength positive correlation between the natural precipitation and the water production by this modified solar chimney power plant in those cities, Water production by the device in the conditions of the study can produce up to 37.9×106tons of water per year in Guangzhou, and 29.7×106tons of water per year in Chengdu, the correlation coefficient is even up to 0.875. Moreover, this device is more efficient in arid regions, might increase the rainfalls and enhance the regional water cycle. The amount of potable water produced by this modified solar chimney power plant is remarkable and might be able to benefit to several millions of people.

Algebraic Reynolds stress modeling of turbulence subject to rapid homogeneous and non-homogeneous compression or expansion

A recently developed explicit algebraic Reynolds stress model (EARSM) by Grigoriev et al. [“A realizable explicit algebraic Reynolds stress model for compressible turbulent flow with significant mean dilatation,” Phys. Fluids 25(10), 105112 (2013)] and the related differential Reynolds stress model (DRSM) are used to investigate the influence of homogeneous shear and compression on the evolution of turbulence in the limit of rapid distortion theory (RDT). The DRSM predictions of the turbulence kinetic energy evolution are in reasonable agreement with RDT while the evolution of diagonal components of anisotropy correctly captures the essential features, which is not the case for standard compressible extensions of DRSMs. The EARSM is shown to give a realizable anisotropy tensor and a correct trend of the growth of turbulence kinetic energy K, which saturates at a power law growth versus compression ratio, as well as retaining a normalized strain in the RDT regime. In contrast, an eddy-viscosity model results in a rapid exponential growth of K and excludes both realizability and high magnitude of the strain rate. We illustrate the importance of using a proper algebraic treatment of EARSM in systems with high values of dilatation and vorticity but low shear. A homogeneously compressed and rotating gas cloud with cylindrical symmetry, related to astrophysical flows and swirling supercritical flows, was investigated too. We also outline the extension of DRSM and EARSM to include the effect of non-homogeneous density coupled with “local mean acceleration” which can be important for, e.g., stratified flows or flows with heat release. A fixed-point analysis of direct numerical simulation data of combustion in a wall-jet flow demonstrates that our model gives quantitatively correct predictions of both streamwise and cross-stream components of turbulent density flux as well as their influence on the anisotropies. In summary, we believe that our approach, based on a proper formulation of the rapid pressure-strain correlation and accounting for the coupling with turbulent density flux, can be an important element in CFD tools for compressible flows.

Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables.

We present a new version of conservative ADER-WENO finite volume schemes, in which both the high order spatial reconstruction as well as the time evolution of the reconstruction polynomials in the local space-time predictor stage are performed in primitive variables, rather than in conserved ones. To obtain a conservative method, the underlying finite volume scheme is still written in terms of the cell averages of the conserved quantities. Therefore, our new approach performs the spatial WENO reconstruction twice: the first WENO reconstruction is carried out on the known cell averages of the conservative variables. The WENO polynomials are then used at the cell centers to compute point values of the conserved variables, which are subsequently converted into point values of the primitive variables. This is the only place where the conversion from conservative to primitive variables is needed in the new scheme. Then, a second WENO reconstruction is performed on the point values of the primitive variables to obtain piecewise high order reconstruction polynomials of the primitive variables. The reconstruction polynomials are subsequently evolved in time with a novel space-time finite element predictor that is directly applied to the governing PDE written in primitive form. The resulting space-time polynomials of the primitive variables can then be directly used as input for the numerical fluxes at the cell boundaries in the underlying conservative finite volume scheme. Hence, the number of necessary conversions from the conserved to the primitive variables is reduced to just one single conversion at each cell center. We have verified the validity of the new approach over a wide range of hyperbolic systems, including the classical Euler equations of gas dynamics, the special relativistic hydrodynamics (RHD) and ideal magnetohydrodynamics (RMHD) equations, as well as the Baer-Nunziato model for compressible two-phase flows. In all cases we have noticed that the new ADER schemes provide less oscillatory solutions when compared to ADER finite volume schemes based on the reconstruction in conserved variables, especially for the RMHD and the Baer-Nunziato equations. For the RHD and RMHD equations, the overall accuracy is improved and the CPU time is reduced by about 25 %. Because of its increased accuracy and due to the reduced computational cost, we recommend to use this version of ADER as the standard one in the relativistic framework. At the end of the paper, the new approach has also been extended to ADER-DG schemes on space-time adaptive grids (AMR).

Implementation of k-k<sub>L</sub>-ω Turbulence Model for Compressible Flow into OpenFOAM

The article deals with results of the implementation of the k-kL-ω turbulence model for compressible transitional flow into OpenFOAM. This model was firstly proposed by Walters and Leylek [2] and utilizes the approach of the laminar kinetic energy in order to predict the transition between laminar and turbulent flows. The performance of the implemented model has been tested for the case of flow over a flat plate and the flow through VKI and SE 1050 turbine cascades. The properties of the implementation of the model for compressible flow simulations into OpenFOAM are discussed.

Computations of two-fluid models based on a simple and robust hybrid primitive variable Riemann solver with AUSMD

This paper is to continue our previous work in 2008 on solving a two-fluid model for compressible liquid-gas flows. We proposed a pressure-velocity based diffusion term original derived from AUSMD scheme of Wada and Liou in 1997 to enhance its robustness. The proposed AUSMD schemes have been applied to gas and liquid fluids universally to capture fluid discontinuities, such as the fluid interfaces and shock waves, accurately for the Ransom's faucet problem, air-water shock tube problems and 2D shock-water liquid interaction problems. However, the proposed scheme failed at computing liquid-gas interfaces in problems under large ratios of pressure, density and volume of fraction. The numerical instability has been remedied by Chang and Liou in 2007 using the exact Riemann solver to enhance the accuracy and stability of numerical flux across the liquid-gas interface. Here, instead of the exact Riemann solver, we propose a simple AUSMD type primitive variable Riemann solver (PVRS) which can successfully solve 1D stiffened water-air shock tube and 2D shock-gas interaction problems under large ratios of pressure, density and volume of fraction without the expensive cost of tedious computer time. In addition, the proposed approach is shown to deliver a good resolution of the shock-front, rarefaction and cavitation inside the evolution of high-speed droplet impact on the wall.

Compressible closure models for turbulent multifluid mixing

This paper studies governing equations describing the turbulent fluid mixing behavior effectively. The goal is to propose a closure for compressible multiphase flow models with transport and surface tension, which satisfy the boundary conditions at the mixing zone edges, the conservation requirements, and an entropy inequality constraint. Implicitness of positivity for the entropy of averaging requires entropy inequality as opposed to conservation of entropy for microphysically adiabatic processes.

Measurements of the Stack Metal Temperature During a Natural Gas Blowdown Event Through a Full-Bore Blowdown Valve

Experiments were used in conjunction with a compressible flow model to investigate the temperature recovery phenomenon along a blowdown stack during a high-pressure natural gas pipeline blowdown. The test rig involved instrumented 2 in. blowdown stacks mounted on a full-bore valve. Stacks with two wall thicknesses and stagnation pressures of approximately 3000 kPa-a and 5600 kPa-g were tested, giving a total of four test cases. Using the compressible flow model, which was calibrated using static pressure measurements, the stack-gas temperature was calculated to range from −38 °C to −18 °C for the four test cases. The respective stack wall temperatures were measured to range between −13 °C and 0 °C; thus, the temperature recovery ranged between 18 °C and 26 °C. Empirical correlations available in the literature, which were developed for aeronautical applications, were tested against the experimental results. Poor agreement was found between the measured temperature recovery factor and that predicted by five empirical correlations: the coefficient of determination (R2) between the measured and correlation-calculated recovery factor was found to be negative for all five correlations.

On the reduction of Blackstock׳s model of thermoviscous compressible flow via Becker׳s assumption

Issues regarding the derivation and applicability of the weakly non-linear model presented by Brunnhuber and Kaltenbacher in their article [Well-posedness and asymptotic behavior of solutions for the Blackstock–Crighton–Westervelt equation, Discret. Contin. Dyn. Syst. Ser. A 34 (2014) 4515–4535] are revisited and clarified; in particular, its links to both Blackstock׳s thermoviscous compressible flow model and Becker׳s assumption are established.

Compositional flow modeling using a multi-point flux mixed finite element method

We present a general compositional formulation using multi-point flux mixed finite element (MFMFE) method on general hexahedral grids. The mixed finite element framework allows for local mass conservation, accurate flux approximation, and a more general treatment of boundary conditions. The multi-point flux inherent in MFMFE scheme allows the usage of a full permeability tensor. The proposed formulation is an extension of single and two-phase flow formulations presented by Wheeler and Yotov, SIAM J. Numer. Anal. 44(5), 2082–2106 (35) with similar convergence properties. Furthermore, the formulation allows for black oil, single-phase and multi-phase incompressible, slightly and fully compressible flow models utilizing the same design for different fluid systems. An accurate treatment of diffusive/dispersive fluxes owing to additional velocity degrees of freedom is also presented. The applications areas of interest include gas flooding, CO 2 sequestration, contaminant removal, and groundwater remediation.

An implicit numerical model for multicomponent compressible two-phase flow in porous media

We introduce a new implicit approach to model multicomponent compressible two-phase flow in porous media with species transfer between the phases. In the implicit discretization of the species transport equation in our formulation we calculate for the first time the derivative of the molar concentration of component i in phase α (cα, i) with respect to the total molar concentration (ci) under the conditions of a constant volume V and temperature T. The species transport equation is discretized by the finite volume (FV) method. The fluxes are calculated based on powerful features of the mixed finite element (MFE) method which provides the pressure at grid-cell interfaces in addition to the pressure at the grid-cell center. The efficiency of the proposed model is demonstrated by comparing our results with three existing implicit compositional models. Our algorithm has low numerical dispersion despite the fact it is based on first-order space discretization. The proposed algorithm is very robust.

A three dimensional lattice model for thermal compressible flow on standard lattices

A three-dimensional double distribution function thermal lattice Boltzmann model has been developed for simulation of thermal compressible flows in the low Mach number limit. Both the flow field and energy conservation equation are solved by LB approach. A higher order density distribution function on standard lattices is used to solve the flow field, while an energy distribution function is employed to compute the temperature field. The equation of state of thermal perfect gas is recovered by higher order Hermite polynomial expansions in Navier–Stokes–Fourier equations. The equilibrium distribution functions of D3Q15, D3Q19 and D3Q27 lattices are obtained from the Hermite expansion. They exhibit slight differences originating in differences in the discrete lattice symmetries. The correction terms in LB models for third order derivation are added using an external force in orthogonal polynomials form. Present models are successfully assessed considering several test cases, namely the thermal Couette flow, Rayleigh–Bénard convection, natural convection in square cavity and a spherical explosion in a 3D enclosed box. The numerical results are in good agreement with both analytical solution and results given by previous authors.

Experimental and numerical analysis of compressible two-phase flows in a shock tube

The comparative study on seven equation models with two different six equations model for compressible two-phase flow analysis is proposed. The seven equations model is derived for compressible two-phase flow that is in the nonconservation form. In the present work, two different six equations model are derived for two pressures, two velocities and single temperature with the derivation of the equation of state. The closing equation for one of the six equations model is energy conservation equation while another one is closed by entropy balance equation. The partial differential form of governing equations is hyperbolic and written in the conservative form. At this point, the set of governing equations are derived based on the principle of extended thermodynamics. The method of solving single temperature from both six equation models are simple and direct solution can be obtained. Numerical simulation has been tried using one of the six equation models for air–water shock tube problems. Explicit fourth order Runge–Kutta scheme is used with Finite Volume Shock Capturing method for solving the governing equations numerically. The pressure, velocity and volume fraction variations are captured along the shock tube length through flow solver. Experimental work is carried out to magnify the initial stage of liquid injection into a gas. The outcome of six equations model for compressible two-phase flow has revealed the multi-phase flow characteristics that are similar to the actual conditions.

Supersonic Flows in Channels and Jets when Applied to the Problems of Designing Aerodynamic Windows of Gas Lasers

Consideration is given to issues associated with the development of aids for mathematical modeling of steadystate supersonic perfect compressible gas flows in aerodynamic windows of gas lasers. A comparison is made of flow patterns in supersonic jets, which appear when a uniform velocity distribution and a velocity distribution corresponding to the free vortex are realized on the nozzle section. Aids for designing channels are developed on the basis of solution of a sequence of primal problems. The calculation results show the high efficiency of the developed software as applied to solution of problems associated with the contouring of channels and nozzles and with the control over supersonic jet flows.

Microphysics of Aerodynamic Contrail Formation Processes

Abstract Aerodynamic condensation is a result of intense adiabatic cooling in the airflow over aircraft wings and behind propeller blades. Out of cloud, condensation appears as a burstlike fog (jet aircraft during takeoff and landing, propellers) or as an iridescent trail visible from the ground behind the trailing edge of the wing (jet aircraft in subsonic cruise flight) consisting of a monodisperse population of ice particles that grow to sizes comparable to the wavelength of light in ambient humidities above ice saturation. In this paper, the authors focus on aerodynamic contrail ice particle formation processes over jet aircraft wings. A 2D compressible flow model is used to evaluate two likely processes considered for the initial ice particle formation: homogeneous droplet nucleation (HDN) followed by homogeneous ice nucleation (HIN) and condensational growth of ambient condensation nuclei followed by their homogenous freezing. The model shows that the more numerous HDN particles outcompete frozen solution droplets for water vapor in a 0.5–1-m layer directly above the wing surface and are the only ice particles that become visible. Experimentally verified temperature and relative humidity–dependent parameterizations of rates of homogeneous droplet nucleation, growth, and freezing indicate that visible aerodynamic contrails form between T = −20° and −50°C and RH ≥ 80%. By contrast, combustion contrails require temperatures below −38°C and ice-saturated conditions to persist. Therefore, aerodynamic and combustion contrails can be observed simultaneously.