Additional Source Term Research Articles

Eulerian–Lagrangian model for traffic dynamics and its impact on operational ventilation of road tunnels

This contribution deals with a computational Eulerian–Lagrangian model that simulates movement of cars inside a road tunnel and its impact on operational ventilation. The model simulates moving cars as discrete objects that “fly” through the tunnel. The objects are treated with a Lagrangian momentum equation and their velocity is solved along their trajectories that are determined by the shape of the roadway. The flow of the ambient air is solved with a commercial CFD code StarCD. Due to drag force, the cars virtually change their velocity, but the latter is continuously re-set to its original value. The momentum equation for the continuous phase contains an additional source term that results from the net efflux of momentum of cars when they enter and leave a particular control volume of the solution domain. The model by Jicha et al. (Int. J. Environ. Monit. Asses. 65 (2000) 343) can simulate cars moving at different speeds and traffic rates in individual traffic lanes. As a result we obtain flow rate generated by moving vehicles as a function of traffic speed and traffic rates. Turbulence was modelled using standard k– ε model with three different formulas for extra sources of the kinetic energy of turbulence that account for additional turbulence generated by moving vehicles. The traffic-induced turbulence shows a non-negligible effect on the total flow rate inside the tunnel. The model was validated with experimental data from Chen et al. (Int. J. Wind Eng. Ind. Aerod. 73 (1998) 99), where the small-scale tunnel 1:20 was investigated. The experiments were carried out with a moving belt carrying small car-like objects. The tunnel length was 20 m, the height 36.5 cm and the tunnel had two parallel lanes. Several traffic densities and speeds were simulated, namely 10,000, 20,000 and 30,000 cars/h per lane with speeds of 20 km/h and 40 km/h.

Zone conditional modeling of premixed turbulent flames at a high Damköhler number

The zone conditioning approach is applied to generic mass and momentum conservation equations in unburned and burned gas. Conditionally averaged equations include additional source terms with flame surface density to reflect interfacial transports between two zones. Zone conditional averaging does not involve countergradient diffusion, which is due to description in terms of mixture properties in Favre averaging. Asymptotic analysis shows that flame propagation characteristics are determined by gradient diffusion by turbulence and an externally imposed pressure gradient at the leading edge. Two-dimensional CFD simulations are performed for two typical premixed turbulent flames stabilized in an impinging jet. With adjusted constants to match the c̄ profiles, conditional axial velocities show good agreement with experimental data for the two test flames. The zone conditional model provides an intuitive set of governing equations suitable for a solution procedure of premixed combustion devices at a high Damköhler number.

COMPUTATION OF THREE-DIMENSIONAL DEVELOPING PRESSURE-DRIVEN LIQUID FLOW IN A MICROCHANNEL WITH EDL EFFECT

Advances in miniature fabrication technology have allowed the possibilities of using microchannels in ultracompact, very efficient heat exchangers, which capitalize on the large surface area-to-volume ratio of the channels, to transport high heat fluxes with small thermal resistance. This article describes a developing flow model by using 3-D Navier-stoke equations with flow pressure as a driving force to simulate the developing liquid flow phenomenon in microchannels. The concept of electric double layer (EDL) is introduced to explain the microscale deviation between flow in microscale channels and large-scale channels. Governing equations are stated for developing rectangular microchannel flows. An additional source term, related to the electric potential and resulting from the EDL effect, is introduced in the conventional momentum equation as a body force, thereby modifying the flow and heat transfer characteristics. The electric potential distribution is simulated using a Poisson-Boltzmann equation. A finite-volume scheme is used to solve the differential equations. Results comparing the performance of the microchannel under different factors such as Reynolds number, nondimensionalized electric potential, etc., are shown and discussed. Nusselt numbers for each case, with or without EDL effect, are also included, so that the actual effectiveness of the microchannel can be understood.

On the minimum entropy production in steady state heat conduction processes

On the basis of minimum entropy generation principle, a new formulation of the boundary value problems is proposed. Applying Euler–Lagrange variational formalism, a new mathematical form of heat conduction equation with additional heat source terms has been derived. To obtain a unique solution a special mathematical form of boundary conditions for 2D and 3D problems is required. As a result, entropy generation rate of the process can significantly be reduced, which leads to the decrease of the irreversibility ratio according to the Gouy–Stodola theorem. Minimization of entropy generation in heat conduction process is always possible by introducing additional heat sources.

The Simon and Simon–Mars tensors for stationary Einstein–Maxwell fields

Modulo conventional scale factors, the Simon and Simon–Mars tensors are defined for stationary vacuum spacetimes so that their equality follows from the Bianchi identities of the second kind. In the nonvacuum case one can absorb additional source terms into a redefinition of the Simon tensor so that this equality is maintained. Among the electrovacuum class of solutions of the Einstein–Maxwell equations, the expression for the Simon tensor in the Kerr–Newman–Taub–NUT spacetime in terms of the Ernst potential is formally the same as in the vacuum case (modulo a scale factor), and its vanishing guarantees the simultaneous alignment of the principal null directions of the Weyl tensor, the Papapetrou field associated with the timelike Killing vector field, the electromagnetic field of the spacetime and even the Killing–Yano tensor.

Verification of Euler/Navier–Stokes codes using the method of manufactured solutions

AbstractThe method of manufactured solutions is used to verify the order of accuracy of two finite‐volume Euler and Navier–Stokes codes. The Premo code employs a node‐centred approach using unstructured meshes, while the Wind code employs a similar scheme on structured meshes. Both codes use Roe's upwind method with MUSCL extrapolation for the convective terms and central differences for the diffusion terms, thus yielding a numerical scheme that is formally second‐order accurate. The method of manufactured solutions is employed to generate exact solutions to the governing Euler and Navier–Stokes equations in two dimensions along with additional source terms. These exact solutions are then used to accurately evaluate the discretization error in the numerical solutions. Through global discretization error analyses, the spatial order of accuracy is observed to be second order for both codes, thus giving a high degree of confidence that the two codes are free from coding mistakes in the options exercised. Examples of coding mistakes discovered using the method are also given. Copyright © 2004 John Wiley & Sons, Ltd.

Modeling instantaneous discharge of unsorted particle cloud in ambient water by an Eulerian-Lagrangian method

The mixing characteristics of waste particles (e.g. dredged sediments) of variable size discharged into ambient water is studied numerically by an Eulerian-Lagrangian method. A three-dimensional (3D) numerical model using the modified κ-e parameterization of turbulence for the fluid phase (water) and a Lagrangian method for the solid phase (particles) has been developed. In the model the wake turbulence induced by particles has been included as additional source and sink terms in the κ-e model; and the variable drift velocities of the particles are treated efficiently by the Lagrangian method in which the particles are tracked explicitly and the diffusion process is approximated by a random walk model. The hydrodynamic behavior of dumping a cloud of particles is governed by the total buoyancy of the cloud, and the drag force on each particle. The computed results show a roughly linear relationship between the displacement of the frontal position and the longitudinal width of the particle cloud. Radial vor...

Modeling instantaneous discharge of unsorted particle cloud in ambient water by an Eulerian—Lagrangian method

The mixing characteristics of waste particles (e.g. dredged sediments) of variable size discharged into ambient water is studied numerically by an Eulerian–Lagrangian method. A three-dimensional (3D) numerical model using the modified k–ϵ parameterization of turbulence for the fluid phase (water) and a Lagrangian method for the solid phase (particles) has been developed. In the model the wake turbulence induced by particles has been included as additional source and sink terms in the k–ϵ model; and the variable drift velocities of the particles are treated efficiently by the Lagrangian method in which the particles are tracked explicitly and the diffusion process is approximated by a random walk model. The hydrodynamic behavior of dumping a cloud of particles is governed by the total buoyancy of the cloud, and the drag force on each particle. The computed results show a roughly linear relationship between the displacement of the frontal position and the longitudinal width of the particle cloud. Radial vortices appear before the particles reach the bottom. The particle size distribution in the cloud affects the rate of increase of the cloud size, as well as the rate of change of frontal position. The largest particles in the cloud dominate the flow behavior. The computed results are compared with the results of laboratory experiments and satisfactory agreement is obtained.

MSW-resonant fermion mixing during reheating

We study the dynamics of reheating in which an inflaton field couples two flavor fermions through Yukawa-couplings. When two fermions have a mixing term with a constant coupling, we show that the Mikheyev–Smirnov–Wolfenstein (MSW)-type resonance emerges due to a time-dependent background in addition to the standard fermion creation via parametric resonance. This MSW resonance not only alters the number densities of fermions generated by a preheating process but also can lead to the larger energy transfer from the inflaton to fermions. Our mechanism can provide additional source terms for the creation of superheavy fermions which may be relevant for the leptogenesis scenario.

A Note on k - ε Modelling of Vegetation Canopy Air-Flows

The k - e turbulence model is a standard of computational software packages for engineering, yet its application to canopy turbulence has not received comparable attention. This is probably due to the additional source (and/or sink) terms, whose parameterization remained uncertain. This model must include source terms for both turbulent kinetic energy (k) and the viscous dissipation rate (e), to account for vegetation wake turbulence budget. In this note, we show how Kolmogorov's relation allows for an analytical solution to be calculated within the portion of a dense and homogeneous canopy where the mixing length does not vary. By substitution within model equations, this solution allows for a set of constraints on source term model coefficients to be derived.Those constraints should meet both Reynolds averaged Navier–Stokes equationsand large-eddy simulation sub-grid scale turbulence modelling requirements.Although originating from within a limited portion of the canopy, the predictedcoefficients values must be valid elsewhere in order to make the model capable of predicting the whole canopy-layer flow with a single set of constants.

Seston dynamics in a tidal inlet with shellfish aquaculture: a model study using tracer equations

A process-oriented modelling study is used to examine biophysical control of the distribution of particulate organic matter, or seston, in a tidal embayment with shellfish aquaculture. The focus is on the spatio-temporal dynamics of seston as influenced by the processes of water motion and mixing, internal primary production of seston, and the clearance of the water volume by the grazing activity of a large bivalve population. A fluid dynamical framework is used wherein seston is treated as a non-conservative tracer in an advection–diffusion equation with additional source and sink terms. An idealized one-dimensional (1D) tidal inlet is first used to examine the sensitivity of tidally averaged seston concentration and flux to variations in tidal transport, internal production, and shellfish grazing. This model is then applied to Tracadie Bay, a tidal inlet off Canada's east coast, to illustrate temporal variability in seston level and flux for a more complex tidal regime. The results of this study suggest that seston flux is mainly under physical control, with its spatial distribution set by tidal transport processes. Seston level, on the other hand, is affected by both grazing and production, with the magnitude of these effects being spatially dependent as dictated by the tidal currents. Grazing and production effects on seston are most pronounced near the head of the inlet, which depends on internal, or local, processes. More seaward areas are buffered against these changes due the advection of seston from the adjacent open ocean. Variation in the spatial distribution of grazing activity demonstrates how local processes have inlet-wide effects. The temporal response of the inlet to tidal changes in the incoming far-field seston flux resembles a low-pass filter with a phase lag; temporal changes in seston at the head of the inlet are highly dampened and occur later than the forcing flux at the mouth. The implications of these results for marine bivalve aquaculture in terms of growth potential (seston level) and carrying capacity (seston flux) are discussed.

On spatial and material settings of thermo-hyperelastodynamics for open systems

The present treatise aims at deriving a general framework for the thermodynamics of open systems typically encountered in chemo- or biomechanical applications. Due to the fact that an open system is allowed to constantly gain or lose mass, the classical conservation law of mass has to be recast into a balance equation balancing the rate of change of the current mass with a possible in- or outflux of matter and an additional volume source term. The influence of the generalized mass balance on the balance of momentum, kinetic energy, total energy and entropy is highlighted. Thereby, special focus is dedicated to the strict distinction between a volume specific and a mass specific format of the balance equations which is of no particular relevance in classical thermodynamics of closed systems. The change in density furnishes a typical example of a local rearrangement of material inhomogeneities which can be characterized most elegantly in the material setting. The set of balance equations for open systems will thus be derived for both, the spatial and the material motion problem. Thereby, we focus on the one hand on highlighting the remarkable duality between both approaches. On the other hand, special emphasis is placed on deriving appropriate relations between the spatial and the material motion quantities.

Effect of Volume Energy Supply on the Stability of a Subsonic Vortex Flow

Calculations of the stability of an axisymmetric vortex flow of viscous heat-conducting gas with volume energy supply are presented. The unperturbed axisymmetric vortex flow was found numerically using a quasi-cylindrical approximation of the Navier-Stokes equations under the assumption of constant peripheral-velocity circulation in the ambient co-current flow. The volume energy supply in the viscous vortex core was modeled by an additional source term in the energy equation. The stability characteristics of the viscous vortex flow in a longitudinal vortex with respect to both axisymmetric and non-axisymmetric three-dimensional waves traveling along the vortex axis and corresponding to both positive and negative values of the azimuthal wave number were found using the time-dependent formulation of the linear stability theory for compressible three-dimensional plane-parallel flows.

ETUDE COMPARATIVE DE MODELES A BAS NOMBRE DE REYNOLDS DANS LA PREDICTION D'UN ECOULEMENT A POINT DE STAGNATION

A comparative study is presented of turbulent models simulating the round jet flow impinging normally on the flat solid surface. The models are a high and low-Reynolds-Number version of the k-e model. Predictions for the local convective heat-transfer rate to the wall are compared to available experimental data. It has been found that results obtained with the low-Reynolds-Number k-e models are in better agreement with experimental data. An additional source term in the e equation was necessary for reducing near-wall length scales and improving the heat-transfer predictions by means of the low-Reynolds-Number k-e turbulence models. Predictions for the velocity, temperature and turbulence characteristics field in the near wall region are presented and discussed. It is concluded that the two low-Reynolds-Numbers k-e models to be successful are the Chien model and the Lam model. © 2002 Elsevier Science Ltd.

CFD ANALYSIS OF DOUBLE-LAYER MICROCHANNEL CONJUGATE PARALLEL LIQUID FLOWS WITH ELECTRIC DOUBLE-LAYER EFFECTS

This work deals with a code development using a finite-volume scheme on a double-layer microchannel conjugate heat transfer problem consisting of the simultaneous determination of the temperature field in both the solid substrate and the fluid, with streaming potential as the driving force. The concept of electric double layer (EDL) was introduced to explain the microscale deviation. Governing equations were derived for fully developed rectangular microchannels' pressure-driven flows. Toward a realistic modeling of the problems, we conducted a conjugate analysis that solves both the solid and liquid regions. An additional source term resulting from the EDL effects was introduced in the conventional momentum equation, thereby modifying the flow and heat transfer characteristics. Analysis concerning the comparison of the double-layer configuration versus the conventional single-layer microchannel heat sink were included. The computed results reveal significant deviations in the velocity and temperature profiles under EDL effects, in particular, when the hydraulic diameter of the channel is less than 40 u m.

NUMERICAL SIMULATION OF CONDENSATION ON A CAPILLARY GROOVED STRUCTURE

Condensation in a capillary groove is investigated using the volume of fluid (VOF) model. The governing equations are written in a generalized form and are applicable to both liquid and vapor phases. Condensation on the fin top and at the meniscus is modeled by introducing additional source terms in the continuity, VOF, and energy equations. The effects of temperature drop, contact angle, surface tension, and fin thickness on the condensation heat transfer are also investigated.

A numerical strategy for efficient modeling of the equatorial wave guide

Convection in the tropics is observed to involve a wide-ranging hierarchy of scales from a few kilometers to the planetary scales and also has a profound impact on short-term climate. The mechanisms responsible for this behavior present a major unsolved problem. A promising emerging approach to address these issues is cloud-resolving modeling. Here a family of numerical models is introduced specifically to model the feedback of small-scale deep convection on tropical planetary waves and tropical circulation in a highly efficient manner compatible with the approach through cloud-resolving modeling. Such a procedure is also useful for theoretical purposes. The basic idea in the approach is to use low-order truncation in the meriodonal direction through Gauss--Hermite quadrature projected onto a simple discrete radiation condition. In this fashion, the cloud-resolving modeling of equatorially trapped planetary waves reduces to the solution of a small number of purely zonal two-dimensional wave systems along a few judiciously chosen meriodonal layers that are coupled only by some additional source terms. The approach is analyzed in detail with full mathematical rigor for linearized equatorial primitive equations with source terms.

A numerical strategy for efficient modeling of the equatorial wave guide.

Convection in the tropics is observed to involve a wide-ranging hierarchy of scales from a few kilometers to the planetary scales and also has a profound impact on short-term climate. The mechanisms responsible for this behavior present a major unsolved problem. A promising emerging approach to address these issues is cloud-resolving modeling. Here a family of numerical models is introduced specifically to model the feedback of small-scale deep convection on tropical planetary waves and tropical circulation in a highly efficient manner compatible with the approach through cloud-resolving modeling. Such a procedure is also useful for theoretical purposes. The basic idea in the approach is to use low-order truncation in the meriodonal direction through Gauss--Hermite quadrature projected onto a simple discrete radiation condition. In this fashion, the cloud-resolving modeling of equatorially trapped planetary waves reduces to the solution of a small number of purely zonal two-dimensional wave systems along a few judiciously chosen meriodonal layers that are coupled only by some additional source terms. The approach is analyzed in detail with full mathematical rigor for linearized equatorial primitive equations with source terms.

A Positive Conservative Method for Magnetohydrodynamics Based on HLL and Roe Methods

The exact Riemann problem solutions of the usual equations of ideal magnetohydrodynamics (MHD) can have negative pressures, if the initial data has ∇·B≢0. This creates a problem for numerical solving because in a first-order finite-volume conservative Godunov-type method one cannot avoid jumps in the normal magnetic field component even if the magnetic field was divergenceless in the three-dimensional sense. We show that by allowing magnetic monopoles in MHD equations and properly taking into account the magnetostatic contribution to the Lorentz force, an additional source term appears in Faraday's law only. Using the Harten–Lax–vanLeer (HLL) Riemann solver and discretizing the source term in a specific manner, we obtain a method which is positive and conservative. We show positivity by extensive numerical experimentation. This MHD-HLL method is positive and conservative but rather diffusive; thus we show how to hybridize this method with the Roe method to obtain a much higher accuracy while still retaining positivity. The result is a fully robust positive conservative scheme for ideal MHD, whose accuracy and efficiency properties are similar to the first order Roe method and which keeps ∇·B small in the same sense as Powell's method. As a special case, a method with similar characteristics for accuracy and robustness is obtained for the Euler equations as well.

A Turbulent Model for Gas-Particle Jets

This work is concerned with turbulent diffusion in gas-particle flows. The cases studied correspond to dilute flows and small Stokes number, this implies that the mean velocity of the particles is very similar to that of the fluid element. The classical k-ε method is used to model the gas-phase, modified with additional terms for the k and ε equations, that takes into account the effect of particles on the carrier phase. The additional dissipation term included in the equation for k is due to the slip between phases at an intermediate scale, far from both the Kolmogorov and the integral scales. This term has a proportionality constant equal to 3/2 of Kolmogorov constant, C0. In this paper, a value of 3.0 has been used for this constant as suggested by Du et al., 1995, “Estimation of the Kolmogorov Constant C0 for the Langarian Structure Using a Second-Order Lagrangian Model of Grid Turbulence,” Phys. Fluids 7, (12), pp. 3083–3090. The additional source term for the ε equation is taken as proportional to ε/k, as is usually done. In all experiments analyzed the particles increased the dissipation of turbulent kinetic energy. A comparison is made between the results obtained with the model proposed in this work and the experiments of Shuen et al., 1985, “Structure of Particle-Laden Jets: Measurements and Predictions,” AIAA Journal, 23, No. 3, and Hishida et al., 1992, “Experiments on Particle Dispersion in a Turbulent Mixing Layer,” ASME Journal of Fluids Engineering, 119, pp. 181–194. [S0098-2202(00)02103-9]