Total Energy Equation Research Articles

Arbitrary-rate relaxation techniques for the numerical modeling of compressible two-phase flows with heat and mass transfer

We describe compressible two-phase flows by a single-velocity six-equation flow model, which is composed of the phasic mass and total energy equations, one volume fraction equation, and the mixture momentum equation. The model contains relaxation source terms accounting for volume, heat and mass transfer. The equations are numerically solved via a fractional step algorithm, where we alternate between the solution of the homogeneous hyperbolic portion of the system via a HLLC-type wave propagation scheme, and the solution of a sequence of three systems of ordinary differential equations for the relaxation source terms driving the flow toward mechanical, thermal and chemical equilibrium. In the literature often numerical relaxation procedures are based on simplifying assumptions, namely simple equations of state, such as the stiffened gas one, and instantaneous relaxation processes. These simplifications of the flow physics might be inadequate for the description of the thermodynamical processes involved in various flow problems. In the present work we introduce new numerical relaxation techniques with two significant properties: the capability to describe heat and mass transfer processes of arbitrary relaxation time, and the applicability to a general equation of state. We show the effectiveness of the proposed methods by presenting several numerical experiments.

Restoring the conservativity of characteristic-based segregated models: Application to the hybrid lattice Boltzmann method

A general methodology is introduced to build conservative numerical models for fluid simulations based on segregated schemes, where mass, momentum, and energy equations are solved by different methods. It is especially designed here for developing new numerical discretizations of the total energy equation and adapted to a thermal coupling with the lattice Boltzmann method (LBM). The proposed methodology is based on a linear equivalence with standard discretizations of the entropy equation, which, as a characteristic variable of the Euler system, allows efficiently decoupling the energy equation with the LBM. To this extent, any LBM scheme is equivalently written under a finite-volume formulation involving fluxes, which are further included in the total energy equation as numerical corrections. The viscous heat production is implicitly considered thanks to the knowledge of the LBM momentum flux. Three models are subsequently derived: a first-order upwind, a Lax–Wendroff, and a third-order Godunov-type schemes. They are assessed on standard academic test cases: a Couette flow, entropy spot and vortex convections, a Sod shock tube, several two-dimensional Riemann problems, and a shock–vortex interaction. Three key features are then exhibited: (1) the models are conservative by construction, recovering correct jump relations across shock waves; (2) the stability and accuracy of entropy modes can be explicitly controlled; and (3) the low dissipation of the LBM for isentropic phenomena is preserved.

A reduced model for compressible viscous heat-conducting multicomponent flows

In the present paper we propose a reduced temperature non-equilibrium model for simulating multicomponent flows with inter-phase heat transfer, diffusion processes (including the viscosity and the heat conduction) and external energy sources. We derive three equivalent formulations for the proposed model. The first formulation consists of balance equations for partial densities, the mixture momentum, the mixture total energy, and phase volume fractions. The second formulation is symmetric and obtained by replacing the equations for the mixture total energy and volume fractions in the first formulation with balance equations for the phase total energy. Replacing one of the phase total energy equation of the second formulation with the mixture total energy equation gives the third formulation. All the three formulations assume velocity and pressure equilibrium across the material interface. These equivalent forms provide different physical perspectives and numerical conveniences. Temperature equilibration and continuity across the material interfaces are achieved with the instantaneous thermal relaxation. Temperature equilibrium is maintained during the heat conduction process. The proposed models are proved to respect the thermodynamical laws. For numerical solution, the model is split into a hyperbolic partial differential equation (PDE) system and parabolic PDE systems. The former is solved with the high-order Godunov finite volume method that ensures the pressure–velocity–temperature (PVT) equilibrium conduction. The parabolic PDEs are solved with both the implicit and the explicit locally iterative method (LIM) based on Chebyshev parameters. Numerical results are presented for several multicomponent flow problems with diffusion processes. Furthermore, we apply the proposed model to simulate the target ablation problem that is of significance to inertial confinement fusion. Comparisons with one-temperature models in literature demonstrate the ability to maintain the PVT property and superior convergence performance of the proposed model in solving multicomponent problems with diffusions.

Simplified Closed-Form Method to Large Deflection Stability Analysis of Thin Rectangular Plate

At the moment, the analyses of postbuckling behavior of thin rectangular plates of constant thickness are generally based on von Karman’s large deflection equations which make use of stress functions and trigonometric variables. These equations are coupled, nonlinear partial differential equations of fourth order each. Getting a closed-form solution of these equations is near impossible and tedious. The present work presents a simplified closed-form general equation using a variational approach for large deflection analysis. The approach adopted here is devoid of Airy’s stress functions. A new strain-displacement equation is formulated, and the total potential energy equation is minimized. The resulting compatibility equation was solved to obtain the general governing stability equation under large deflection. This governing equation was applied to a plate simply supported all-around using polynomial displacement function and numerical results were obtained. To validate the numerical results obtained, they were compared with the values obtained by Levy whose results are generally acclaimed as exact and with two others. It was observed that the minimum and maximum percentage differences were 0% and 41.56% at stress parameters 3.66 and 21.45 respectively. Also, for deflection to thickness ratio (w/t), the present results showed a close agreement with those of Levy with a minimum and maximum percentage difference of 0.07% and 41.56% at w/t of 0 and 3.376 respectively. Importantly, the present result lies between two other research results. We, therefore, conclude that the present work is adequate and a new simple closed-form approach to understanding and predicting the postbuckling strength of thin rectangular plates.

Exact mathematical formulas for wall-heat flux in compressible turbulent channel flows

In this paper, several exact expressions for the mean heat flux at the wall ($q_w$) for the compressible turbulent channel flows are derived by using the internal energy equation or the total energy equation. Two different routes, including the FIK method and the RD method, can be applied. The direct numerical simulations data of compressible channel flows at different Reynolds and Mach numbers verify the correctness of the derived formulas. Discussions related to the different energy equations, and different routes are carried out, and we may arrive at the conclusion that most of the formulas derived in the present work are just mathematically ones which generally are lack of clear physical interpretation. They can be used to estimate $q_w$, but might not be suitable for explore the underlying physics.

Exact Three-Dimensional Stability Analysis of Plate Using an Direct Variational Energy Method

In this paper, direct variational calculus was put into practical use to analyses the three dimensional (3D) stability of rectangular thick plate which was simply supported at all the four edges (SSSS) under uniformly distributed compressive load. In the analysis, both trigonometric and polynomial displacement functions were used. This was done by formulating the equation of total potential energy for a thick plate using the 3D constitutive relations, from then on, the equation of compatibility was obtained to determine the relationship between the rotations and deflection. In the same way, governing equation was obtained through minimization of the total potential energy functional with respect to deflection. The solution of the governing equation is the function for deflection. Functions for rotations were obtained from deflection function using the solution of compatibility equations. These functions, deflection and rotations were substituted back into the energy functional, from where, through minimizations with respect to displacement coefficients, formulas for analysis were obtained. In the result, the critical buckling loads from the present study are higher than those of refined plate theories with 7.70%, signifying the coarseness of the refined plate theories. This amount of difference cannot be overlooked. However, it is shown that, all the recorded average percentage differences between trigonometric and polynomial approaches used in this work and those of 3D exact elasticity theory is lower than 1.0%, confirming the exactness of the present theory. Thus, the exact 3D plate theory obtained, provides a good solution for the stability analysis of plate and, can be recommended for analysis of any type of rectangular plates under the same loading and boundary condition. Doi: 10.28991/CEJ-2022-08-01-05 Full Text: PDF

ПРИНЦИП НАИМЕНЬШЕГО ДЕЙСТВИЯ В РАСЧЕТЕ ВОДОСЛИВА С ШИРОКИМ ПОРОГОМ

Purpose: to apply the variation principle of least action to create a theory for calculating the broad-crested weir, to compare the results obtained with the data of various domestic and foreign researchers and to draw conclusions about the accuracy of the solution obtained. Materials and methods. The minimum flow depth on a broad-crested weir was determined by minimizing the Lagrangian liquid flow. This equation is used to close the equation of specific total flow energy. The verification of the obtained method in determining the minimum depth at the weir crest was carried out by a laboratory experiment and computer simulation of the flow movement through the weir, carried out in the ANSYS 17.0 software package. The results of determining weir capacity were compared with the experimental and theoretical data of domestic and foreign authors. Results. The comparison of the results of calculations of the broad-crested weir capacity with a wide threshold with the proposed method showed good agreement with the data of various authors, the difference did not exceed 3 %. The comparison of the results of depth calculations at the weir crest showed their agreement with the data of various authors, our own laboratory experiments and computer simulation. The adequacy of the obtained dependence was confirmed on the basis of determination coefficient and the Fisher variance ratio. Conclusions. A new theoretical method for hydraulic calculation of a broad-crested weir has been obtained, which made it possible to determine the liquid flow depth at the weir crest and its capacity. An analysis of the applicability of the method showed that the depth by the principle of least action is realized only in cases where the flow has no obstacles for free discharge. The limitation of the proposed method is the mandatory presence of a section with a relatively streamline flow and hydrostatic pressure distribution on the weir crest, as well as the absence of flooding of the structure.

A productive exposition for the thermal post buckling problem of orthotropic circular plates by using relevant admissible function for lateral displacement

A suitable mathematical formulation is used to study the thermal post buckling problem of orthotropic circular plates is presented herein. The total energy equation is reduced by assuming algebraic function for lateral displacement w. The radial tension per unit length is calculated by substituting the value of orthotropic parameter under simply supported and clamped boundary conditions. The thermal post buckling load carrying capacity of the plate is evaluated using integrated average for various values of orthotropic parameter β, and the numerical results obtained from the present investigation are compared with the results obtained from the known literatures. The results obtained for both cases are concurs well with reference values and the numerical results using integrated average shows more accurate values for the post buckling load.

A test of the standard cosmological model with geometry and growth

We perform a general test of the ΛCDM and wCDM cosmological models by comparing constraints on the geometry of the expansion history to those on the growth of structure. Specifically, we split the total matter energy density, Ω M , and (for wCDM) dark energy equation of state, w, into two parameters each: one that captures the geometry, and another that captures the growth. We constrain our split models using current cosmological data, including type Ia supernovae, baryon acoustic oscillations, redshift space distortions, gravitational lensing, and cosmic microwave background (CMB) anisotropies. We focus on two tasks: (i) constraining deviations from the standard model, captured by the parameters ΔΩ M ≡ Ω M grow-Ω M geom and Δw ≡ w grow-w geom, and (ii) investigating whether the S 8 tension between the CMB and weak lensing can be translated into a tension between geometry and growth, i.e. ΔΩ M ≠ 0, Δw ≠ 0. In both the split ΛCDM and wCDM cases, our results from combining all data are consistent with ΔΩ M = 0 and Δ w = 0.If we omit BAO/RSD data and constrain the split wCDM cosmology, we find the data prefers Δw < 0 at 3.6σ significance and ΔΩ M > 0 at 4.2σ evidence.We also find that for both CMB and weak lensing, ΔΩ M and S 8 are correlated, with CMB showing a slightly stronger correlation. The general broadening of the contours in our extended model does alleviate the S 8 tension, but the allowed nonzero values of ΔΩ M do not encompass the S 8 values that would point toward a mismatch between geometry and growth as the origin of the tension.

LES of Reacting Flow in a Hydrogen Jet into Supersonic Crossflow Combustor Using a New Turbulent Combustion Model

Modelling the complete flow physics and chemical kinetics of supersonic combustion is a particularly complex and daunting task that requires significant computational resources. To foster performance evaluation tools for future hypersonic vehicles, developing accurate yet computationally efficient solution methods is of great importance. In this work, a new subgrid combustion model for large eddy simulations is derived and used in a three-dimensional in-house flow solver to provide simulations of experimental scramjet ground tests. In particular, this paper introduces a hybrid model closure with the reaction-rate approach to close the filtered chemical source terms in the governing equations for species mass fractions and total energy. The model developed here makes use of a linear bridging function, depending on the segregation rate of the mixture fraction, between a resolved contribution issued from a perfectly stirred reactor (PSR) estimation, and a subgrid-scale (SGS) contribution where a closure that approximates the Lagrangian trajectory in the composition space is retained. The new model considers the effect of fluctuations of compositions and can be extended to take into account, for example, the fluctuations of temperature. The new approach is tested using a hydrogen-fueled scramjet combustor from circular injector into a Mach 2 vitiated airflow for total pressure and temperature of 0.40 MPa and 1695 K, respectively. The selected operating conditions are representative of the LAPCAT-II dual-mode ramjet/scramjet combustion. Chemistry is described using a four-step reduced mechanism. The results obtained with the present modelling proposal are compared to those issued from numerical simulations performed with the quasi-laminar chemistry or PSR approach. These results do show that, even for a highly resolved computational mesh, the effects of composition fluctuations remain significant, especially in the vicinity of the injection where the SGS fluctuations of the scalar field are non-negligible.

Use of Higher Order Plate Theory in dynamic analysis of SSFS and CSFS thick rectangular plates in orthogonal polynomials

&#x0D; &#x0D; &#x0D; This study applied polynomial expressions as displacement and shear deformation functions in the free-vibration study of thick and moderately thick isotropic rectangular plates. Rectangular plates with two different edge conditions investigated in this work are: one with simple supports at three of its edges and with no support at the other edge denoted with the acronym (SSFS) and a rectangular plate with simple supports at opposite edges while the other opposite edges has a fixed support at one edge and no support at the other edge, this is denoted with the acronym (CSFS). The total potential energy of the plate was derived using the general theory of elasticity. The general governing equation of the plate was derived by minimizing the total potential energy equation of the plate. Edge conditions of the SSFS and CSFS plates were met and substituted into the general governing equation to obtain a linear expression which was solved to generate fundamental natural frequency function for the plates with various span-depth proportion (m/t) and planar dimensions proportion (n/m). The results obtained from this research were found to agree favourably with the results of similar problems in the literature upon comparison.&#x0D; &#x0D; &#x0D;

A priori analysis of subgrid-scale terms in compressible transcritical real gas flows

Of interest to the research community dealing with real gas flows, this study analyzes the influence of the physical complexity of real gases on the amplitude of subgrid-scale (SGS) terms present in the filtered Navier–Stokes equations to be solved in large eddy simulations. The direct numerical simulation results of three academic configurations (homogeneous isotropic turbulence, mixing layer, and channel flow) are filtered from the largest scale in the domain down to the Kolmogorov length scale. The analysis of the filtered flow variables consistently shows that the SGS turbulent stress and the SGS pressure cannot be neglected in the momentum equation. In the total energy equation, SGS pressure work and SGS internal and kinetic fluxes are found to be significant in the inertial zone of the turbulent kinetic energy spectrum. Since in the inertial zone, which corresponds to large filter sizes, specific models have not yet been designed for some of these terms, this study calls for such a modeling effort that will benefit the real gas and organic Rankine cycles research communities.

Energy-consistent finite difference schemes for compressible hydrodynamics and magnetohydrodynamics using nonlinear filtering

In this paper, an energy-consistent finite difference formulation for the compressible hydrodynamic and magnetohydrodynamic (MHD) equations is introduced. For the compressible magnetohydrodynamics, an energy-consistent finite difference formulation is derived using the product rule for the spatial difference. The conservation properties of the internal, kinetic, and magnetic energy equations can be satisfied in the discrete level without explicitly solving the total energy equation. The shock waves and discontinuities in the numerical solution are stabilized by nonlinear filtering schemes. An energy-consistent discretization of the filtering schemes is also derived by introducing the viscous and resistive heating rates. The resulting energy-consistent formulation can be implemented with the various kinds of central difference, nonlinear filtering, and time integration schemes. The second- and fifth-order schemes are implemented based on the proposed formulation. The conservation properties and the robustness of the present schemes are demonstrated via one- and two-dimensional numerical tests. The proposed schemes successfully handle the most stringent problems in extremely high Mach number and low beta conditions.

Conservativity and weak consistency of a class of staggered finite volume methods for the Euler equations

We address a class of schemes for the Euler equations with the following features: the space discretization is staggered, possible upwinding is performed with respect to the material velocity only and the internal energy balance is solved, with a correction term designed on consistency arguments. These schemes have been shown in previous works to preserve the convex of admissible states and have been extensively tested numerically. The aim of the present paper is twofold: we derive a local total energy equation satisfied by the solutions, so that the schemes are in fact conservative, and we prove that they are consistent in the Lax-Wendroff sense.

Line-finite-element implementation for driven steel H-piles in layered sands considering post-driving residual stresses

Driven Steel H-piles are commonly used in the deep foundation of being cost-effective and easy in workmanship. The post-driving residual stresses acting on the pile could be large when being driven in cohesionless sands, significantly affecting its compressive strength. Current design methods, mostly based on the empirical assumptions of using safety factors, are unable to consider the post-driving residual stresses on piles accurately and usually very conservative in practical applications. Such consideration is further complicated when the pile is embedded in layered sands. Moreover, the soil-structure interaction (SSI) between pile and ground medium is usually complicated in the analysis and should be appropriately considered in a successful design. This paper proposes a line-finite-element implementation method, based on the Euler-Bernoulli pile element formulations, for robustly and efficiently analyzing the driven steel H-piles in layered sands with explicitly modelling the SSI. The effects resulting from the post-driving residual stresses are considered in the total potential energy equation for generating the secant relations. The derivation procedure is elaborated with details. Consequentially, verification examples are given for validating the accuracy of the proposed method. This work would be helpful for improving the numerical efficiency and accuracy for designing the steel H-piles in layered sands.

A numerical derivation of internal energy equation under a staggered mesh system for 1-D thermal hydraulic system analysis codes

This paper introduces a modified internal energy equation derived for multiphase flow in various flow conditions for a staggered mesh system. The pressure drop and heat dissipation terms in the internal energy conservation equation currently being developed based on the assumption that the scalar and momentum variables, which originate from the total energy and mechanical energy equations, respectively, are all located in the cell center of the control volume were redefined such that two different pressures and velocities stored in both cell centers and faces were imposed for the internal energy conservation. To achieve this, first, a modified internal energy conservation equation for a staggered mesh was derived by subtracting the mechanical energy equation from the total energy equation. The equation was then discretized classifying each term by its origin; variables that originated from the total energy equation were defined in the cell center, whereas terms that came from the mechanical energy equation were identified at faces. Since the discretized form of the proposed equation contained face velocity and cell pressure for the heat dissipation and pressure drop terms, respectively, these two terms were calculated implicitly leading to enhanced numerical stability. The accuracy of the modified internal energy equation in predicting the system pressure, fluid temperature, and heat balance for various flow channels was assessed. The verification of the proposed equation was completed through simulations of multiple theoretical problems including saturated liquid depressurization, adiabatic expansion of hydrogen, heat transfer in a helical steam generator, and energy transfer in a converging pipe. An improved result was obtained with the modified equation as the numerical calculation results agreed well with the analytic solutions with relative deviations less than 0.1% for most cases, while the solution obtained by the conventional internal energy equation showed a significant amount of deviation from the analytic solution.

Motion paths and mechanical behavior of origami-inspired tunable structures

Origami structures have received tremendous attention for their excellent kinematic and mechanical performance. Here, the inherent relationship between kinematics and mechanics of origami-based structures is investigated. The coupling phenomena between the axial displacement and the rotational displacement are systematically analyzed. Versatile folding motions of Kresling-inspired spring truss structures (KTS) are analytically calculated to demonstrate the feature of compression bifurcation, snapback behavior, reverse rotation, multipath, and multistability. Popular spring truss models are introduced, and the detailed static analysis of the KTS is performed by total potential energy equations and constraint conditions. The result reveals the relationship between motion paths and bifurcation, snapback and reverse rotation with varied initial geometries. Multipath property significantly broadens the application of origami structures in terms of bearing capacity and energy storage capacity. Moreover, the stable configuration distribution is given with prescribed initial geometries. The principles in this work set forth a significant avenue for designing novel mechanical metamaterials and functional engineering systems.

A Lagrange Multiplier-based Technique within the Nonlinear Finite Element Method in Cracked Columns

In this research, two energy-based techniques, called Lagrange multiplier and conversion matrix, are applied to involve crack parameters into the non-linear finite element relations of Euler-Bernoulli beams made of functionally graded materials. The two techniques, which divide a cracked element into three parts, are implemented to enrich the secant and tangent stiffness matrices. The Lagrange multiplier technique is originally proposed according to the establishment of a modified total potential energy equation by adding continuity conditions equations of the crack point. The limitation of the conversion matrix in involving the relevant non-linear equations is the main motivation in representing the Lagrange multiplier. The presented Lagrange multiplier is a problem-solving technique in the cracked structures, where both geometrical nonlinearity and material inhomogeneity areas are considered in the analysis like the post-buckling problem of cracked functionally graded material columns. Accordingly, some case-studies regarding the post-buckling analysis of cracked functionally graded material columns under mechanical and thermal loads are used to evaluate the results.

The formulation of the RANS equations for supersonic and hypersonic turbulent flows

AbstractAccurate prediction of supersonic and hypersonic turbulent flows is essential to the design of high-speed aerospace vehicles. Such flows are mainly predicted using the Reynolds-Averaged Navier–Stokes (RANS) approach in general, and in particular turbulence models using the effective viscosity approximation. Several terms involving the turbulent kinetic energy (k) appear explicitly in the RANS equations through the modelling of the Reynolds stresses in such approach, and similar terms appear in the mean total energy equation. Some of these terms are often ignored in low, or even supersonic, speed simulations with zero-equation models, as well as some one- or two-equation models. The omission of these terms may not be appropriate under hypersonic conditions. Nevertheless, this is a widespread practice, even for very high-speed turbulent flow simulations, because many software packages still make such approximations. To quantify the impact of ignoring these terms in the RANS equations, two linear two-equation models and one non-linear two-equation model are applied to the computation of five supersonic and hypersonic benchmark cases, one 2D zero-pressure gradient hypersonic flat plate case and four shock wave boundary layer interaction (SWBLI) cases. The surface friction coefficients and velocity profiles predicted with different combinations of the turbulent kinetic energy terms present in the transport equations show little sensitivity to the presence of these terms in the zero-pressure gradient case. In the SWBLI cases, however, comparisons show that inclusion of k in the mean flow equations can have a strong effect on the prediction of flow separation. Therefore, it is highly recommended to include all the turbulent kinetic energy terms in the mean flow equations when dealing with simulations of supersonic and hypersonic turbulent flows, especially for flows with SWBLIs. As a further consequence, since k may not be obtained explicitly in zero-equation, or certain one-equation, models, it is debatable whether these models are suitable for simulations of supersonic and hypersonic turbulent flows with SWBLIs.

Non-Linear Response of Torsional Buckling Piezoelectric Cylindrical Shell Reinforced with DWBNNTs Under Combination of Electro-Thermo-Mechanical Loadings in Elastic Foundation

Nanocomposites provide new properties and exploit unique synergism between materials. Polyvinylidene fluoride (PVDF) is an ideal piezoelectric matrix applicable in nanocomposites in a broad range of industries from oil and gas to electronics and automotive. And boron nitride nanotubes (BNNTs) show high mechanical, electrical and chemical properties. In this paper, the critical torsional load of a composite tube made of PVDF reinforced with double-walled BNNTs is investigated, under a combination of electro-thermo-mechanical loading. First, a nanocomposite smart tube is modeled as an isotropic cylindrical shell in an elastic foundation. Next, employing the classical shell theory, strain-displacement equations are derived so loads and moments are obtained. Then, the total energy equation is determined, consisting of strain energy of shell, energy due to external work, and energy due to elastic foundation. Additionally, equilibrium equations are derived in cylindrical coordinates as triply orthogonal, utilizing Euler equations; subsequently, stability equations are developed through the equivalent method in adjacent points. The developed equations are solved using the wave technique to achieve critical torsional torque. Results indicated that critical torsional buckling load occurred in axial half-wave number m = 24 and circumferential wave number n = 1, for the investigated cylindrical shell. The results also showed that with the increase in the length-to-radius ratio and in the radius-to-shell thickness ratio, the critical torsional buckling load increased and decreased, respectively. Lastly, results are compared in various states through a numerical method. Moreover, stability equations are validated via comparison with the shell and sheet equations in the literature.