Total Energy Conservation Equation Research Articles

Numerical Analysis and Experimental Investigation of Cavitating Flows Considering Thermal and Compressibility Effects

This article deals with the numerical simulation of unsteady cavitating flow around hydrofoils, supported by experimental research realized in a cavitation tunnel situated in the Centre of Hydraulic Research. Two straight NACA hydrofoils (NACA0020 and NACA2412) were employed. The comprehensive unsteady CFD analysis was based on scale-resolving simulations (hereinafter SRS) with the aim of capturing correctly the interactions between the cavitation structures and re-entrant flow as well as the compressibility and thermal effects of cavitation. The static pressure fluctuations during the cavity oscillation cycles and the evaporation and condensation processes are discussed in detail. To predict correctly the high-pressure peaks during the bubble cloud collapses and the pressure pulse propagation speed, the real properties of water and the mixture total energy conservation equation were considered. In addition, the estimated content of undissolved air was taken into account. The numerical simulations were validated by means of already published experiments or compared with experiments conducted by the authors, and with good agreement.

Study on the non-equilibrium temperature behavior of normal shock waves and the bulk viscosity

The abnormality of small overshoot in the temperature profile of strong normal shock waves has been resolved, and the bulk viscosity of dilute monatomic gases cannot be ignored. By introducing the total energy conservation equation rather than the internal energy equation, an equation for the non-equilibrium temperature has been suggested. The postulated temperature equation gives a monotonically increasing profile of Monte Carlo data. The Navier–Stokes approximation derives shock equations in one differential equation and one algebraic equation and gives that the ratio of the bulk viscosity to the shear viscosity is equal to 1/15 for dilute monatomic gases.

CMMSE 2019: an explicit algorithm for the simulation of non-isothermal multiphase multicomponent flow in a porous medium

An explicit computational algorithm is developed to simulate complex flow of multiphase multicomponent slightly compressible fluid through porous media in view of heat sources. The proposed algorithm is based on the original model of porous media flow constructed by the analogy with the quasigasdynamic system of equations including the total energy conservation equation and modified to change the continuity equation type from parabolic to hyperbolic. It is approximated by a three-level explicit difference scheme having the second order of approximation in time and in space with a rather mild stability condition. The model takes into account gravitational and capillary forces, includes strongly nonlinear functions of the relative phase permeability. For the description of mass and energy transfer between the phases the model is generalized to account for multicomponent fluid structure. Conservation laws are formulated now for the components in terms of mass concentrations of components in the phases. The dependence of phase density and dynamic viscosity on pressure, temperature and multicomponent composition should be noted. Constants of the phase equilibrium close the system of equations. The created approach has been verified by test predictions of two- and three-phase fluid flows. Physically correct results were obtained, a good agreement with results by other authors was observed.

A pressure-based semi-implicit space–time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier–Stokes equations at all Mach numbers

We propose a new arbitrary high order accurate semi-implicit space–time discontinuous Galerkin (DG) method for the solution of the two and three dimensional compressible Euler and Navier–Stokes equations on staggered unstructured curved meshes. The method is pressure-based and semi-implicit and is able to deal with all Mach number flows.The new DG scheme extends the seminal ideas outlined in [1], where a second order semi-implicit finite volume method for the solution of the compressible Navier–Stokes equations with a general equation of state was introduced on staggered Cartesian grids. Regarding the high order extension we follow [2], where a staggered space–time DG scheme for the incompressible Navier–Stokes equations was presented. In our scheme, the discrete pressure is defined on the primal grid, while the discrete velocity field and the density are defined on a face-based staggered dual grid. Then, the mass conservation equation, as well as the nonlinear convective terms in the momentum equation and the transport of kinetic energy in the energy equation are discretized explicitly, while the pressure terms appearing in the momentum and energy equation are discretized implicitly. Formal substitution of the discrete momentum equation into the total energy conservation equation yields a linear system for only one unknown, namely the scalar pressure. Here the equation of state is assumed linear with respect to the pressure. The enthalpy and the kinetic energy are taken explicitly and are then updated using a simple Picard procedure. Thanks to the use of a staggered grid, the final pressure system is a very sparse block five-point system for three dimensional problems and it is a block four-point system in the two dimensional case. Furthermore, for high order in space and piecewise constant polynomials in time, the system is observed to be symmetric and positive definite. This allows to use fast linear solvers such as the conjugate gradient (CG) method. In addition, all the volume and surface integrals needed by the scheme depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessing stage. This leads to significant savings in terms of computational effort for the time evolution part. In this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step. The viscous terms and the heat flux are also discretized making use of the staggered grid by defining the viscous stress tensor and the heat flux vector on the dual grid, which corresponds to the use of a lifting operator, but on the dual grid. The time step of our new numerical method is limited by a CFL condition based only on the fluid velocity and not on the sound speed. This makes the method particularly interesting for low Mach number flows. Finally, a very simple combination of artificial viscosity and the a posteriori MOOD technique allows to deal with shock waves and thus permits also to simulate high Mach number flows. We show computational results for a large set of two and three-dimensional benchmark problems, including both low and high Mach number flows and using polynomial approximation degrees up to p=4.

Dynamic modeling and simulation of reaction, slag behavior, and heat transfer to water-cooling wall of shell entrained-flow gasifier

A mathematical model is developed to simulate a pilot Shell entrained-flow coal gasifier. Submodels of specific structures of the gasifier are established to simulate the complicated gasification process. The model includes the total energy conservation equation and mass conservation equations for the gas components, solid flow, and gas flow. It simulates the influence of the gasifier structure and dimensions and can calculate the effects of changing almost every important operation parameter, e.g., the syngas composition, gasification temperature, carbon conversion ratio, walllayer temperature, and slag mass flow rate. The model can predict the syngas composition under a limited residence time condition. Furthermore, it considers the heat transfer coefficient of each layer of the water wall to calculate its heat loss and temperature. Thus, the model also reflects the influence of performance parameters of the gasifier’s water wall. The slag mass flow rate on the wall is calculated using a slag submodel.

Computing supersonic non-premixed turbulent combustion by an SMLD flamelet progress variable model

This paper presents a statistically more likely distribution (SMLD) approach for the evaluation of the presumed probability density function (PDF) in flamelet progress variable (FPV) models for non-premixed supersonic combustion. The numerical simulation of the NASA Langley Research Center supersonic H2–Air combustion chamber is performed using two approaches: the first one is a standard FPV model, built presuming the functional shape of the PDFs of the mixture fraction, Z, and of the progress parameter, Λ; the second approach employs the SMLD technique to presume the joint PDF of Z and Λ. The standard and FPV-SMLD models have been developed using the low Mach number assumption. In both cases, the temperature is evaluated by solving the total-energy conservation equation, providing a more suitable approach for the simulation of supersonic combustion. By comparison with experimental data, the proposed SMLD model is shown to provide a clear improvement with respect to the standard FPV model, especially in the auto-ignition and stabilization regions of the flame.

Molecular simulation of mass transport in phosphoric acid doped poly(2,5-benzimidazole) polymer electrolyte membranes

In this paper, a molecular model based on quantum mechanics and molecular dynamics (MD) is developed to study the transfer mechanisms of proton, phosphoric acid (PA) molecule and dihydrogen phosphate anion (H2PO4−) in PA-doped poly(2,5-benzimidazole) (ab-PBI) membrane. This model couples with intermolecular potential energy, interaction force between two particles, linear momentum conservation equation, angular momentum conservation equation, total energy conservation equation, the radial distribution function and Einstein's law of diffusion. The simulated molecular system consisting of ab-PBI macromolecules doped with PA is optimized to obtain an optimal structure of the amorphous cell unit system. The effects of PA doping level, temperature and electric field intensity on the mass transfer in the ab-PBI membrane are studied. A comparison between this simulation results and experimental data gotten from literatures shows this model and its simulation are reliable. Two kinds of proton transfer mechanisms are evaluated, and the first transfer mechanism, in which H2PO4− takes as a carrier and [−C=N−] takes as a stronghold, is considered as the main proton transfer mechanism in an ab-PBI membrane at the studied doping levels. The diffusion coefficient of PA is about 2.5 times of that of H2PO4−, which indicates the diffusion of H2PO4− controls the mass transfer in the ab-PBI membrane.

Forecasting the Time Evolution of the Oil Price During Months of Negative Inflation Rate

Mass conservation equation is employed to study the time evolution of the mass of oil in a reservoir, according to the actual mass flow rate of extraction. It is also possible to define the critical mass flow rate of extraction, which is the value exhausting the reservoir in an infinite time. The evolution with time of the price of the resources extracted and sold to the market is investigated in case of no-accumulation and no-depletion of the resources, i.e., when the resources are extracted and sold to the market at the same mass flow rate. The total energy conservation equation is transformed into a money or capital per time conservation equation, which allows to study the price evolution with time, which is dependent on the following parameters. The price evolution with time of the extracted resource is dependent on the parameter PIFE, “Price-Increase Factor of Extracted resource,” which is the difference between the basic interest rate of the capital, e.g., the inflation rate, and the mass flow rate of extraction. The price evolution with time of the sold resource is dependent on the parameter PIFS, “Price-Increase Factor of Sold resource,” which is the difference between the interest rate of the capital, e.g., discount or prime rate, and the mass flow rate of extraction. The parameter CIPS, “Critical Initial Price of Sold resource,” depends on the initial price of the extracted resource, on the interest rate of nonextracted resource, and on the difference between PIFS and PIFE. The parameter CIPES, “Critical Initial Price Extreme of Sold resource,” depends on the initial price of the extracted resource, on the interest rate of nonextracted resource, and PIFS. The time evolution of the oil price during the 8 months of 2009, when the inflation rate was negative, and following the economic crisis of 2008, is investigated introducing a new category of cases, i.e., the negative inflation rate one. The paper presents and discusses the results of the forecasting for different values of the interest rate of the capital, i.e., prime and discount rate, with the conclusion that the present theory can forecast the evolution of the oil price with a reasonable confidence using the prime and the discount rates as extreme limits.

Effects of Nonequilibrium Chemistry and Darcy—Forchheimer Pyrolysis Flow for Charring Ablator

The fully implicit ablation and thermal response code simulates pyrolysis and ablation of thermal protection materials and systems. The governing equations, which include energy conservation, a three-component decomposition model, and a surface energy balance, are solved with a moving grid. This work describes new modeling capabilities that are added to a special version of code. These capabilities include a time-dependent pyrolysis gas flow momentum equation with Darcy–Forchheimer terms and pyrolysis gas species conservation equations with finite-rate homogeneous chemical reactions. The total energy conservation equation is also enhanced for consistency with these new additions. Two groups of parametric studies of the phenolic impregnated carbon ablator are performed. In the first group, an Orion flight environment for a proposed lunar-return trajectory is considered. In the second group, various test conditions for arcjet models are examined. The central focus of these parametric studies is to understand the effect of pyrolysis gas momentum transfer on material in-depth thermal responses with finite-rate, equilibrium, or frozen homogeneous gas chemistry. Results indicate that the presence of chemical nonequilibrium pyrolysis gas flow does not significantly alter the in-depth thermal response performance predicted using the chemical equilibrium gas model.

Energy separation in shear layers

A numerical analysis is performed to study the mechanism of energy separation in a viscous heat-conducting shear layer. Two-dimensional time-dependent Navier–Stokes equations and total energy conservation equation are solved simultaneously for four different Reynolds numbers: 100, 200, 500, and 1000. The results show that the roll-up and transport of vortices induce pressure fluctuations in the shear layer. Fluid which flows through the disturbed pressure field exchanges pressure work with the surroundings and separates into higher and lower total temperature regions. Even though some energy separation is caused by the imbalance between shear work and heat conduction, that due to the pressure fluctuation is much stronger. As the Reynolds number is increased, turbulence mixing between higher and lower total temperature regions tends to lessen the energy separation. However, the more rapid growth of pressure fluctuations overcomes the weakening of energy separation by random mixing.

On cyclotron wave heating and acceleration of solar wind ions in the outer corona

The preferential heating and acceleration of O+5 ions, as observed by Ultraviolet Coronagraph Spectrometer (UVCS) on Solar and Heliospheric Observatory (SOHO) [Kohl et al., 1998] in the solar coronal holes have been interpreted and modeled by invoking wave‐particle cyclotron resonance [Cranmer et al., 1999a, 1999b]. However, in the model of Cranmer et al. [1999a, 1999b] and in other subsequent models the assumption of a rigid slope of the wave spectrum was made in calculating the wave energy absortion by the different ion species. In the present paper it is shown that a self‐consistent treatment of the wave damping and absorption is necessary and leads to substantially different results. On the basis of quasi‐linear theory, the interaction between the ions and the ion‐cyclotron waves [Marsch et al., 1982a; Marsch, 1998] is studied. The total energy conservation equation, including the kinetic energy of the resonant particles and the wave energy, is derived and discussed in detail. The spectral evolution equation for cyclotron waves, when being controlled by the wave growth/damping rate and WKB effects, is solved self‐consistently together with the full set of anisotropic multifluid equations for the ions including the cyclotron‐resonance wave heating and acceleration rates. From the numerical results we reach the following conclusions: (1) It is physically questionable to use a spectrum with a fixed spectral slope near the cyclotron resonance when one calculates the partition of wave energy among the different ionic species and the kinetic degrees of freedom parallel and perpendicular to the magnetic field. This assumption neglects the important effects of wave absorption and the concurrent reshaping of the wave spectrum, and thus leads in the dissipation domain to extremely low amplitudes of the waves and to difficulties in supplying enough energy to balance the wave absorption at the cyclotron resonances. (2) If the spectrum is allowed to evolve self‐consistently and concurrently with the particles' heating and acceleration through wave absorption, such a high perpendicular temperature and corresponding large temperature anisotropy as observed by UVCS do not occur or cannot be maintained. We conclude that the UVCS oxygen ion observations have not yet been explained satisfactorily by the cyclotron‐resonance theory.

The foundations of informational statistical thermodynamics revisited

The aim of this paper is to review the fundamental ideas of the underlying method behind informational statistical thermodynamics. This method is set forth to deal with phenomena that occur in nonequilibrium systems. The most significant aspects of this analysis are: 1. (i) To show that Abel's theorem guarantees that in the asymptotic limit, the nonequilibrium statistical operator (NSO) obtained by MAXENT generates a stationary solution of the Liouville equation. This fact is consistent with the experimental behavior of an equilibrium system. 2. (ii) Further, it is also shown how a Liouville equation with sources can be obtained by the NSO determined by MAXENT, whose formal solution proves that the general interpretation of Abel's theorem leading to memory effects is incorrect. Rather, this theorem introduces a time smoothing function in a time interval: t 0 = −∞ < t′ < t 1 ( t 1: initial time of an observation), which is to be understood as one that connects an adiabatic perturbation for t′ < t 1. In fact, the memory effects appear in the evolution equations for the average values of the dynamical variables obtained by the NSO when these evolution equations are calculated up to second order in the perturbation Hamiltonian. 3. (iii) Also, some criticisms that have been presented against MAXENT formalism are discussed and it is shown that they are inapplicable. Such criticisms are related to the memory effects and with the inconsistency of evolution equations for macrovariables with respect to the total energy conservation equation.