Total Energy Equation Research Articles

Symmetry-preserving discretizations in Lagrangian cell-centered hydrodynamics

This paper discusses constructing discretizations in Lagrangian cell-centered hydrodynamics (CCH) that preserve cylindrical symmetry on unequal-angle-zoned grids in two-dimensional Cartesian geometry. We achieve this by modifying the nodal solver (Corot and Mercier, 2018) and updating the total and internal energy equations. The method is a unique solution to the challenging problem of ensuring symmetry in vectors. A criterion is established for determining whether or not this symmetry correction should be applied. We prove that both nodal and zonal quantities maintain a symmetry distribution. Numerical illustrations using unequal-angle initial zoning are presented to demonstrate the efficiency of the scheme.

Forced-Vibration Characteristics of Bowtie-Shaped Honeycomb Composite Sandwich Panel with Viscoelastic Damping Layer.

The incorporation of viscoelastic layers in laminates can markedly enhance the damped dynamic characteristics. This study focuses on integrating viscoelastic layers into the composite facesheet of the bowtie-shaped honeycomb core composite sandwich panel (BHC-CSP). The homogenization of the damped BHC-CSP is performed by employing the variational asymptotic method. Based on the generalized total energy equation, the energy functional of the representative unit cell of the damped BHC-CSP is asymptotically analyzed. The warping function, derived following the principle of minimum potential energy, provides a basis for obtaining the corresponding Euler-Lagrange equation to ascertain the equivalent elastic properties of the damped BHC-CSP. Utilizing the developed two-dimensional equivalent model, the free-vibration characteristics of the damped BHC-CSP are examined across diverse boundary conditions while delving into the impact of an external viscous damping layer on the natural frequency of the damped BHC-CSP. The results reveal that intensified boundary constraints effectively diminish the effective vibration region of the damped BHC-CSP, thereby enhancing its overall stability. The introduction of a PMI foam layer proves effective in adjusting the stiffness and mass distribution of the damped BHC-CSP. Resonance characteristics are explored through frequency and time-domain analyses, highlighting the pivotal roles of the excitation position and receiver point in influencing the displacement and velocity responses. Although the stiffness is improved by incorporating a PMI foam layer, its effect on the damping performance of the damped BHC-CSP is minimal when compared to the T-SW308 foam layer.

Hybrid compressible lattice Boltzmann method for supersonic flows with strong discontinuities

Within the framework of the hybrid recursive regularized lattice Boltzmann (HRR-LB) model, we propose a novel hybrid compressible LB method to ensure the conservation of total energy in simulating compressible flows with strong discontinuities. This method integrates a LB solver to handle the mass and momentum conservation equations via collision-streaming steps on standard lattices, while a finite volume method (FVM) is employed for the conservation of the total energy equation. The flux reconstruction in the FVM is achieved through a momentum coupled method (MCM). The interface momentum, crucial for reconstructing the convective fluxes and determining the upwind extrapolation of passive scalar quantities in MCM, is derived from the LB method. The validity and accuracy of the proposed method are evaluated through six test cases: (I) isentropic vortex convection in subsonic and supersonic regimes; (II) non-isothermal acoustic pulse; (III) one-dimensional Riemann problems; (IV) two-dimensional Riemann problem; (V) double Mach reflection of a Mach 10 shock wave; and (VI) shock–vortex interaction. Numerical results demonstrate that this method surpasses the previous HRR-LB model by Guo et al. [“Improved standard thermal lattice Boltzmann model with hybrid recursive regularization for compressible laminar and turbulent flows,” Phys. Fluids 32, 126108 (2020)] in terms of accuracy and robustness when dealing with strong shock waves.

A Detailed Limited-Area Atmospheric Energy Cycle for Climate and Weather Studies

Lorenz’ seminal work on global atmospheric energetics improved our understanding of the general circulation. With the advent of Regional Climate Models (RCMs), it is important to have a limited-area energetic budget available that is applicable for both weather and climate, analogous to Lorenz’ global atmospheric energetics. A regional-scale energetic budget is obtained in this study by applying Reynolds decomposition rules to quadratic forms of the kinetic energy K and the available enthalpy A, to obtain time mean and time deviation contributions. According to the employed definition, the time mean energy contributions are decomposed in a component associated with the time-averaged atmospheric state and a component due to the time-averaged statistics of transient eddies; these contributions are suitable for the study of the climate over a region of interest. Energy fluctuations (the deviations of instantaneous energies from their climate value) that are appropriate for weather studies are split into quadratic and linear contributions. The sum of all the contributions returns exactly to the total primitive kinetic energy and available enthalpy equations.

Turbulence effect on total mechanical energy budget and energy loss of turbulent flows with different hydraulic regimes in open-channel transitions

ABSTRACT In this study, a modified form of the total mechanical energy budget equation is developed for 3-D turbulent flows with different hydraulic regimes in open-channel transitions, from which the effect of all turbulence parameters can be evaluated by applying some equalizations and RANS numerical simulations. The total energy equation is also used to extract an explicit relationship for energy loss based on turbulence parameters. The findings are presented in the form of the application of the relationship to calculate the energy loss based on simulation results for subcritical turbulent flow and hydraulic jump in open-channel transitions, which leads to identifying the effect of non-homogeneity and anisotropy of turbulence on flow energy balance. The results show that the work done by viscous shear stresses, the viscous diffusion of turbulence, and the turbulent diffusion have little to no effect on the total mechanical energy budget of open-channel turbulent flows; instead, the effect of the variations of turbulent kinetic energy and the work done by turbulence stresses is more significant. The benefit of using the modified energy equation is demonstrated by the potential for evaluating the details that were not previously considered.

Numerical simulations of underwater explosions using a compressible multi-fluid model

We present a novel solver for simulating compressible multi-fluid multiphase flow in underwater explosions (UNDEXs). The developed solver uses a modified version of Saurel's six-equation model, which includes an additional total mixture energy equation to resolve discrepancies in the thermodynamic states predicted under shock conditions. Additionally, we integrate a more precise stiffened gas equation of state (SG-EOS) that is determined using a novel method to enhance the accuracy of predicting experimental data based on a shock Hugoniot curve. We also propose a solution procedure using the modified Saurel's six-equation model on a three-dimensional (3D) structured Cartesian grid system. This involves discretizing the equation system using a Godunov scheme with a two-fluid Harten-Lax-van Leer-Contact approximate Riemann solver and a MUSCL-Hancock primitive scheme with total-variation-diminishing limiters, achieving a second-order extension. Both the dimensional splitting and fractional-step methods are utilized to model one-dimensional (1D) operators, splitting them into sequential operators. The modified model is validated for 1D and 3D problems, including the water–air shock tube, cavitation, shock–bubble interaction, and UNDEX problems in a free field, near a free surface, and near a rigid dam. Our simulations accurately predict the shockwave propagation, shock and free-surface interactions, cavitation evolution, and water jetting impact characteristics, exhibiting satisfactory agreement with those of previous studies. The proposed solver provides insight into the effects of UNDEXs on rigid structures, with potential applications in engineering and defense. The proposed method for determining the SG-EOS parameters can be applied to other areas of research involving high-pressure multi-phase flows.

A new analytical approach for nonlinear buckling and postbuckling of torsion-loaded FG-CNTRC sandwich toroidal shell segments with corrugated core in thermal environments

The design of the torsion-loaded toroidal shell segment with two functionally graded carbon nanotube-reinforced composite (FG-CNTRC) coatings and the corrugated core is presented. Round and trapezoidal shapes of the corrugated core are considered. A homogenization model for the corrugated core is applied to predict the stiffnesses of the corrugated core. The total potential energy equation, according to the Donnell shell theory, is established. The Ritz energy method is applied to determine the expressions of the buckling and postbuckling responses. The numerical investigations present the significantly beneficial effects of FG-CNTRC coatings and corrugated core on the buckling responses of the shells.

Boundary Layer Energetics of Rapid Wind and Wave Forced Mixing Events

Abstract The observed development of deep mixed layers and the dependence of intense, deep-mixing events on wind and wave conditions are studied using an ocean LES model with and without an imposed Stokes-drift wave forcing. Model results are compared to glider measurements of the ocean vertical temperature, salinity, and turbulence kinetic energy (TKE) dissipation rate structure collected in the Icelandic Basin. Observed wind stress reached 0.8 N m−2 with significant wave height of 4–6 m, while boundary layer depths reached 180 m. We find that wave forcing, via the commonly used Stokes drift vortex force parameterization, is crucial for accurate prediction of boundary layer depth as characterized by measured and predicted TKE dissipation rate profiles. Analysis of the boundary layer kinetic energy (KE) budget using a modified total Lagrangian-mean energy equation, derived for the wave-averaged Boussinesq equations by requiring that the rotational inertial terms vanish identically as in the standard energy budget without Stokes forcing, suggests that wind work should be calculated using both the surface current and surface Stokes drift. A large percentage of total wind energy is transferred to model TKE via regular and Stokes drift shear production and dissipated. However, resonance by clockwise rotation of the winds can greatly enhance the generation of inertial current mean KE (MKE). Without resonance, TKE production is about 5 times greater than MKE generation, whereas with resonance this ratio decreases to roughly 2. The results have implications for the problem of estimating the global kinetic energy budget of the ocean.

Developlment of a Thermo-Chemical Non-Equilibrium Solver for Weakly Ionized Hypersonic Flowfields

A three dimensional solver for numerical simulation of weakly ionized hypersonic reactive air flows past high speed flight vehicles had been indigenously developed. The two-temperature, seven species, eighteen reactions, thermo-chemical non equilibrium, ionizing, air-chemistry model of Park is implemented in a compressible viscous code CERANS and solved in the finite volume framework. The energy relaxation is addressed by a conservation equation for the vibrational-electron-electronic energy of the gas mixture resulting in the evaluation of its vibrational temperature. The contribution of vibrational-electron-electronic energy of the reactive mixture is accounted in total energy equation as well as in the source terms of the relaxation model. The AUSM-PW+ numerical flux function has been used for modeling the convective fluxes and a central differencing approximation is used for modeling the diffusive fluxes. The code has been characterized for standard test cases involving weakly ionized hypersonic reactive thermo-chemical non equilibrium flows and results obtained are in good qualitative agreement with results available in open literature.

Gas kinetic schemes for solving the magnetohydrodynamic equations with pressure anisotropy

In many astrophysical plasmas, the Coulomb collision is insufficient to maintain an isotropic temperature, and the system is driven to the anisotropic regime. In this case, magnetohydrodynamic (MHD) models with anisotropic pressure are needed to describe such a plasma system. To solve the anisotropic MHD equation numerically, we develop a robust Gas-Kinetic flux scheme for non-linear MHD flows. Using anisotropic velocity distribution functions, the numerical flux functions are derived for updating the macroscopic plasma variables. The schemes are suitable for finite-volume solvers which utilize a conservative form of the mass, momentum and total energy equations, and can be easily applied to multi-fluid problems and extended to more generalized double polytropic plasma systems. Test results show that the numerical scheme is very robust and performs well for both linear wave and non-linear MHD problems.

A new method for establishing the total potential energy equations of steel members based on the principle of virtual work

The overall stability of steel members is essentially studied using the energy method. In existing literature, the expression of deformation energy is derived based on the strain energy theory of an element, which involves a complicated derivation process. Owing to the variations between the considered strain energy types and fundamental assumptions, the total potential energy equations of the steel members exhibit certain deviations. Herein, we propose a new method for establishing a total potential energy equation of steel members based on the principle of virtual work. First, an equilibrium differential equation describing the free body of steel members is derived. Subsequently, the internal forces and the restoring forces are multiplied with the corresponding virtual displacements to obtain the internal virtual work and virtual deformation energy. Finally, total potential energy equation of the steel members is obtained. In particular, the derivation process of the proposed method is simple, and each item is endowed with an explicit physical meaning. Additionally, it reveals the internal relationships between the equilibrium differential equation and the energy method, which ensures their unity. Moreover, the total potential energy equation of steel members in various common loading cases is derived based on the principle of virtual work, and the critical load is further obtained. Overall, the loading cases with exact analytical solutions are completely consistent with the exact analytical solutions. Furthermore, for cases without exact analytical solutions, theoretical derivation, finite element analyses and test results of existing literature were conducted to validate the accuracy of proposed method.

Total mechanical energy budget in open-channel transitions with turbulent flows

Abstract The issue of mechanical energy budget has received a lot of attention in open-channel flows which are frequently turbulent. Turbulent motions play a major role in the total mechanical energy of the flow field. It is difficult to evaluate total mechanical energy balance and energy loss for 3-D turbulent flows in irregular open channels due to the formation of vortices and the presence of secondary currents. In 3-D turbulent flows, the effect of these phenomena has often been considered as the empirical coefficient of energy loss. In this study, these details in the total mechanical energy budget are investigated through theoretical analysis and numerical simulation, a new form of the total mechanical energy equation and a relationship for the energy loss in 3-D turbulent flows are derived, and their applications are examined in open-channel transitions. For this purpose, the Navier-Stokes equations are solved numerically, and the results are used to obtain the various terms in the total mechanical energy balance equation and analyze the details. The relationships derived in this study generally support the concept of the total mechanical energy budget to comprehend the turbulence parameters affecting the mechanical energy balance in various open-channel flows.

A relaxation model for the non-isothermal Navier-Stokes-Korteweg equations in confined domains

The Navier-Stokes-Korteweg (NSK) system is a classical diffuse interface model which is based on van der Waals' theory of capillarity. Diffuse interface methods have gained much interest to model two-phase flow in porous media. However, for the numerical solution of the NSK equations two major challenges have to be faced. First, an extended numerical stencil is required due to a third-order term in the linear momentum and the total energy equations. In addition, the dispersive contribution in the linear momentum equations prevents the straightforward use of contact angle boundary conditions. Secondly, any real gas equation of state is based on a non-convex Helmholtz free energy potential which may cause the eigenvalues of the Jacobian of the first-order fluxes to become imaginary numbers inside the spinodal region.In this work, a thermodynamically consistent relaxation model is presented which is used to approximate the NSK equations. The model is complimented by thermodynamically consistent non-equilibrium boundary conditions which take contact angle effects into account. Due to the relaxation approach, the contribution of the Korteweg tensor in the linear momentum and total energy equations can be reduced to first-order terms which enables a straightforward implementation of contact angle boundary conditions in a numerical scheme. Moreover, the definition of a modified pressure function enables to formulate first-order fluxes which remain strictly hyperbolic in the entire spinodal region. The present work is a generalization of a previously presented parabolic relaxation model for the isothermal NSK equations.A high-order discontinuous Galerkin spectral element method which supports curved elements and hanging nodes is employed to discretize the system. The relaxation model and its corresponding boundary conditions are validated using solutions of the original NSK model and analytical results for one-, two- and three-dimensional test cases. The simulation of a spinodal decomposition in a three-dimensional porous structure underlines the capability of the presented approach.

A modified wall-adapting local eddy-viscosity model for large-eddy simulation of compressible wall-bounded flow

The wall-adapting local eddy-viscosity (WALE) model in large-eddy simulation can well predict wall-bounded flows but it is also well known for excessive dissipation. In this study, we apply the minimum-dissipation model to constrain the WALE model in compressible flows and obtain the coefficient of the WALE model. Through this process, the dissipation of WALE model can be lower while it still maintains strong stability. In the modified WALE model, the isotropic part of the subgrid-scale (SGS) stress is also reconstructed. In the filtered total energy equation, all of the extra SGS unclosed terms (besides SGS stress and SGS heat flux) are modeled instead of neglecting some SGS terms, such as the SGS viscous diffusion. The modified WALE model is tested in a compressible turbulent channel flow and a supersonic turbulent boundary layer over a compression corner. The new model can well predict the mean velocity, the mean temperature, the Reynolds stress, and the separation bubble.

Radiative Coefficients and Their Influence on In-Depth Heating of Porous Ablators

High entry speeds and exotic planetary gases can result in significant radiative heat loads on space capsules. The mechanism behind the transport of radiative signatures is fundamentally different from the conductive mode of energy transport, and penetration of radiative signatures depends on the radiative coefficients of the thermal protection system (TPS) material that protects the space capsule. The radiative coefficients of carbon-based and silica-based fibrous materials have been computed as functions of wavelength using the photon path length Monte Carlo method by explicitly accounting for the microstructure of the material. Significant variations in the radiative coefficients are observed at wavelengths that are relevant to shock-layer emissions. Although carbon-based fibrous materials exhibit higher absorption coefficients in comparison to silica-based systems, the absorption coefficients of carbon-based material drop by two orders of magnitude in the range of 100–200 nm. The radiative coefficients of carbon-based fibrous material are seen to be dominated by scattering and absorption with minimal transmission. However, the transmission coefficients for the silica system dominated the radiative coefficients in the range of 100–1000 nm, which corresponds to most shock-layer emissions. The radiative coefficients are used to solve the radiative transfer equation using the P-1 approximation to obtain the in-depth radiative heat flux. The total energy equation for decomposing porous TPS materials is solved with the radiative heat flux from the P-1 approximation and the conductive heat flux using the Fourier law. It is observed that peak temperatures inside the material are higher when radiative transport is explicitly accounted for through the P-1 approximation. Small variations in the absorption coefficient of the silica-based materials also affected the in-depth temperature profiles. Additionally, a broader temperature distribution is obtained inside the material with a low absorption coefficient, and the charring density profiles are influenced by the radiative heat flux. This study demonstrates that it is important to include radiative transport in material response solvers, and radiative coefficients must be accurately computed by accounting for the microstructure of the material.

Reconciling and Improving Formulations for Thermodynamics and Conservation Principles in Earth System Models (ESMs)

AbstractThis paper provides a comprehensive derivation of the total energy equations for the atmospheric components of Earth System Models (ESMs). The assumptions and approximations made in this derivation are motivated and discussed. In particular, it is emphasized that closing the energy budget is conceptually challenging and hard to achieve in practice without resorting to ad hoc fixers. As a concrete example, the energy budget terms are diagnosed in a realistic climate simulation using a global atmosphere model. The largest total energy errors in this example are spurious dynamical core energy dissipation, thermodynamic inconsistencies (e.g., coupling parameterizations with the host model) and missing processes/terms associated with falling precipitation and evaporation (e.g., enthalpy flux between components). The latter two errors are not, in general, reduced by increasing horizontal resolution. They are due to incomplete thermodynamic and dynamic formulations. Future research directions are proposed to reconcile and improve thermodynamics formulations and conservation principles.

Simulation of Low-Speed Buoyant Flows with a Stabilized Compressible/Incompressible Formulation: The Full Navier–Stokes Approach versus the Boussinesq Model

This paper compares two strategies to compute buoyancy-driven flows using stabilized methods. Both formulations are based on a unified approach for solving compressible and incompressible flows, which solves the continuity, momentum, and total energy equations in a coupled entropy-consistent way. The first approach introduces the variable density thermodynamics of the liquid or gas without any artificial buoyancy terms, i.e., without applying any approximate models into the Navier–Stokes equations. Furthermore, this formulation holds for flows driven by high temperature differences. Further advantages of this formulation are seen in the fact that it conserves the total energy and it lacks the incompressibility inconsistencies due to volume changes induced by temperature variations. The second strategy uses the Boussinesq approximation to account for temperature-driven forces. This method models the thermal terms in the momentum equation through a temperature-dependent nonlinear source term. Computer examples show that the thermodynamic approach, which does not introduce any artificial terms into the Navier–Stokes equations, is conceptually simpler and, with the incompressible stabilization matrix, attains similar residual convergence with iteration count to methods based on the Boussinesq approximation. For the Boussinesq model, the SUPG and SGS methods are compared, displaying very similar computational behavior. Finally, the VMS a posteriori error estimator is applied to adapt the mesh, helping to achieve better accuracy for the same number of degrees of freedom.

A Study on Stability Analysis of 3-D Shear Deformable Isotropic Plate Elastically Restrained against Rotation and Simply Supported in the two Adjacent Edges using Exact Displacement Potential

In this paper, a three-dimensional (3-D) stability analysis of rectangular deformable plate was performed to find the solution to the buckling problem of a uniaxially compressed plate in which the two adjacent loaded edges is clamped and the other, simply supported (CSCS). A three dimensional kinematics and constitutive relations were used to obtain the equation of total energy functional. The general and direct variation of the total potential energy function was done to get the general and direct governing equation of the plate by considering the effect of shear deformation. The solution of the general governing equation gave the deflection of the plate which is a product of the coefficient of deflection and shape function of the plate. The shape function is derived in terms of polynomial and trigonometric function and solved to get the exact deflection of the plate. The expression for the critical buckling load and other formulae was obtained by the direct variation of the total potential energy equation. This was done by minimizing the energy functional with respect to the coefficients of deflection after including the deflection and shear deformation rotation functions in it. The span to thickness ratio and aspect ratios were varied to ascertain the buckling behavior of different type of plate under uniformly distributed load. The outcome of the numerical analysis revealed that increase in the span- thickness ratio led to the increased value of the critical buckling load which implies that the plate structure is safe when the plate thickness is increased. The result showed that the critical buckling loads from the present study using polynomial are slightly higher than those obtained using trigonometric theories signifying the more exactness of the latter. The overall average percentage differences between the two functions recorded are 2.4%. This shows that at about 98% both approaches are the same and can be applied with confidence in the stability analysis of any type of plate with such boundary condition. The result of the present study using the established 3-D model for both functions is satisfactory and were found to follow an identical pattern, but quite distinct in validation which shows the credibility of the derived relationships.

Analytical and numerical investigation of mechanical energy balance and energy loss of three-dimensional steady turbulent flows in open-channels

Abstract Study about the mechanical energy balance and the energy loss of 3-D turbulent flows in open-channels has its own complexities. The governing equation of the mechanical energy in turbulent flows has been previously known and includes turbulence parameters that their calculations or measurements are not easy. In this study, a form of the total mechanical energy equation that leads to a number of significant physical insights is analytically investigated, from which analytical relationships for the energy loss estimation in 3-D turbulent flows are defined. The effect of different turbulence parameters is reflected on the new relationships and analyzed by equalizations replacing unknown correlations with closure approximations using the numerical turbulence simulation. In order to investigate the application of the analytical relationships, numerical simulations are performed by using OpenFOAM software to solve the Navier-Stokes equations with the RSM turbulence model in open-channels with different geometries. Then, the contribution of the turbulence parameters to the total mechanical energy balance is evaluated in uniform and nonuniform turbulent flows and their difference is analyzed, that leads to identify the parameters affecting the friction and local losses. The results demonstrate that the magnitudes of the turbulent diffusion, the work done by the viscous stresses pertaining to the mean motion and the viscous diffusion of the turbulence energy are substantially smaller than the other terms of the total energy equation for turbulent flows in open-channels with different geometries, while the effect of the variations of the turbulence kinetic energy and the work done by the turbulence stresses, that has not been considered in the previous mechanical energy equations, is more important in complex flows. From a practical viewpoint, in order to study the details of the total mechanical energy balance and the energy loss in 3-D turbulent flows with the presence of the secondary currents, the proposed method can be useful.

Buckling stability analysis of underpinning piles during basement excavation beneath existing buildings

In the present paper, the pile-soil system’s total potential energy equation of underpinning piles was established based on the Winkler elastic foundation beam theory. This energy equation was used to explore the effect of basement excavation beneath existing buildings on the underpinning pile’ buckling stability. Utilizing the minimum potential energy theory, the expression of the critical buckling load for underpinning piles’ stability during the excavation project was obtained. Moreover, the influencing factors of the critical buckling load were investigated. It was found that the underpinning pile’s critical load converged with the augment of half-wave number. Moreover, the pile skin friction and deadweight had an insignificant influence on it. In addition, the critical load of underpinning piles decreased sharply with the increasing excavation depth and gradually increased with the augment of pile diameter. The results of this study provides a basis for the design of adding piles in similar projects and reduces the hidden danger of excavation instability.