Finite Volume Solution Research Articles

Approximating Catalyst Effectiveness Factors with Reaction Rate Profiles

A novel approximate solution for catalyst effectiveness factors is presented. It is based on carefully selected approximate reaction rate profiles, instead of typical assumption of composition profiles inside the catalyst. This formulation allows analytical solution of the approximate model, leading to a very simple iterative solution for effectiveness factor for general nonlinear reaction stoichiometry and arbitrary catalyst particle shape. The same model can be used with all practical Thiele modulus values, including multicomponent systems with inert compounds. Furthermore, the correct formulation of the underlying physical model equation is discussed. It is shown that an incorrect but often-used model formulation where convective mass transfer has been neglected may lead to much higher errors than the present approximation. Even with a correctly formulated physical model, rigorous discretization of the catalyst particle volume may have unexpectedly high numerical errors, even exceeding those with the present approximate solution. The proposed approximate solution was tested with a number of examples. The first was an equimolar reaction with first order kinetics, for which analytical solutions are available for the standard catalyst particle geometries (slab, long cylinder, and sphere). Then, the method was tested with a second order reaction in three cases: 1) with one pure reactant, 2) with inert present, and 3) with two reactants and non-stoichiometric surface concentrations. Finally, the method was tested with an industrially relevant catalytic toluene hydrogenation including Maxwell-Stefan formulation for the diffusion fluxes. In all the tested systems, the results were practically identical when compared to the analytical solutions or rigorous finite volume solution of the same problem.

Collapse of particle-laden buoyant plumes

Particle loading affects the dynamics of buoyant plumes, since the difference between particle and fluid densities is much greater than that in the fluid alone. In stratified environments, plume rise is density limited; after initial overshoot, the plume reaches a terminal level and spreads radially. Particles dropping from this horizontal intrusion may be re-entrained. This recycling of dense matter reduces plume buoyancy and intrusion height and, for sufficient load, can lead to plume collapse. Entrainment-based formulae yield a steady-state plume rise. We identify a new conserved quantity for such plumes. Integrating paths of particles dropping from the intrusion yields the fraction re-entrained. A simple mathematical model predicts from buoyancy ratio at source ($P=$ negative particle buoyancy divided by positive fluid buoyancy) whether a particle-laden plume will collapse. Under this model, for small settling velocity, a particle-laden plume will not collapse if $P<0.368$. Above this, collapse depends also on the amount of particle-free ambient fluid entrained in the overshoot region. For pure plumes, experimental evidence suggests that this is small. For forced plumes, more substantial overshoot and entrainment is shown to increase the critical ratio. An extension, based on successive recycling, estimates time to collapse. To investigate further we develop a simple computational model, coupling a ‘top-hat’ plume model, an analytical formula for radially decaying concentrations in the intrusion and an axisymmetric finite-volume solution for time-dependent settling and entrainment. The model can predict the impact of particle load on final rise, as well as the occurrence and time scales of plume collapse.

Application of CFD and FEA coupling to predict dynamic behaviour of a flexible barge in regular head waves

The ever increasing size of ships and offshore platforms has resulted in ‘softer’ hull which require hydroelastic effects to be taken into account when predicting fluid-structure interactions. The majority of such investigations are carried out numerically using potential flow solvers. Although nonlinear potential flow methods are also used, RANS/CFD (Reynolds-averaged Navier-Stokes equations) can fully take into account of nonlinearities. It is important, therefore, to verify and validate the results from such numerical predictions. This paper aims to investigate the symmetric motions and responses of flexible barge in regular waves by coupling RANS/CFD and Finite Element software. The two-way interaction between a fluid solver, Star-CCM+, and a structural solver, Abaqus, is applied by exchanging pressures and nodal displacements more than once every time step, namely implicit coupling scheme. A combination of overset and mesh morphing approaches and finite volume solution to allow for the motions of a body at the free surface is used. The computational results compared with experimental measurements and 2-D linear hydroelastic predictions show good agreement. The discrepancies between the 2-D and CFD/FEA cosimulation arise due to the strong influence of bow and stern radiated waves and the agreement between them improves when the nonlinearities are not as dominant.

Transient Galerkin finite volume solution of dynamic stress intensity factors

Transient Galerkin finite volume method (GFVM) is developed to solve time-dependent problems and analysis of the dynamic stress intensity factors (DSIFs) for cracked problem. An interesting feature of the developed method is its matrix free operations; therefore, it obviously reduces the computation workloads for dynamic cases with small time marching. The two-point displacement extrapolation method is used for calculating the stress intensity factors (SIFs). To show the ability of this method, the structural problem, such as a beam under dynamic load, is considered as the first case study. The computed transient deflections are used for evaluating the accuracy of the GFVM in comparison with the results of the explicit finite element method (explicit-FEM) and meshless method solvers. A comparison of the CPU time consumption of the GFVM and explicit-FEM solvers shows that the GFVM entails lesser time consumption than the explicit-FEM, without reducing the accuracy of the results. In the second case study, the SIFs are computed for plate with inner crack under constant loading. For the third and fourth case study, the ability of the proposed GFVM solver to cope with DSIFs for a plate with an edge crack and L-shape plate with an inclined crack under dynamic load were tested. The comparison indicates that the GFVM not only provide compatible accuracy close to other common numerical solvers, also offers considerable CPU-time consumption, in comparison with the methods that requires matrix manipulations.

Mesoscopic-based finite volume solutions for waterhammer flows

ABSTRACTIt is becoming evident that high resolution finite volume (FV) numerical schemes for multi-dimensional waterhammer problems are needed for the development of an accurate transient-based condition assessment of pipelines. As a prerequisite, FV methods for one-dimensional waterhammer flows need to be developed. Such models can then be applied to multi-dimensional problems via directional splitting. In this paper, two FV schemes (one uses Bhatnagar–Gross–Krook Boltzmann (BGK) and the other uses kinetic flux vector splitting (KFVS) in the flux approximation) for one-dimensional waterhammer problems are formulated and applied. It is found that the KFVS and BGK schemes correctly capture discontinuity fronts in a classical reservoir-pipe-valve system. An oscillation-free collision time formulation has been proposed, tested and found to be robust. The stability of the proposed schemes is guaranteed when . Comparison between the BGK, KFVS, fixed-grid MOC, and first and second order Godunov schemes reveals that the second-order Godunov performs best, followed closely by the BGK scheme. However, the BGK should not be quickly dismissed since it is known that its real power becomes evident when dealing with complex physics, multi-scale and multi-dimensional problems.

Closure to “Finite-Volume Solutions to the Water-Hammer Equations in Conservation Form Incorporating Dynamic Friction Using the Godunov Scheme” by Aboudou Seck, Musandji Fuamba, and René Kahawita

Closure to “Finite-Volume Solutions to the Water-Hammer Equations in Conservation Form Incorporating Dynamic Friction Using the Godunov Scheme” by Aboudou Seck, Musandji Fuamba, and René Kahawita

Discussion of “Finite-Volume Solutions to the Water-Hammer Equations in Conservation Form Incorporating Dynamic Friction Using the Godunov Scheme” by Aboudou Seck, Musandji Fuamba, and René Kahawita

Discussion of “Finite-Volume Solutions to the Water-Hammer Equations in Conservation Form Incorporating Dynamic Friction Using the Godunov Scheme” by Aboudou Seck, Musandji Fuamba, and René Kahawita

Steady supercritical flow in a straight-wall open-channel contraction

ABSTRACTExperimental and numerical modelling of steady flow in an open channel contraction is conducted. A two-dimensional depth-averaged model based on a finite volume solution of the shallow water equations is developed. A three-dimensional model of the Reynolds-averaged Navier–Stokes equations is adapted for free surface flows by incorporating the volume of fluid model. The numerical models are applied to resolve the discrepancy between earlier simulations of shallow water equation models and a benchmark dataset on supercritical open channel contraction flow. The experiment is repeated in a Plexiglas flume. Measurements are done at several flow rates. The measurements and simulations demonstrate that the original data have measurement or reporting errors, and shallow water equation models cannot reproduce observed steady supercritical flow in a straight-wall contraction. The 3D model satisfactorily predicts the water level in the contraction and the maximum water depth but under-predicts the amplitudes of the standing wave troughs in the downstream prismatic section.

Finite elements with node dependent kinematics and scalable accuracy for the analysis of Stokes flows

This paper extends the use of one-dimensional elements with node-dependent kinematics to the analysis of Stokes flows. According to the Carrera Unified Formulation, the primary variables of the flow, velocity and pressure, are expressed as arbitrary expansions of the generalized unknowns. The novel implementation proposed in this work allows to increase the accuracy of the model only in the areas of the domain where it is required; i.e. close to boundaries, barriers or sudden expansion. Refined one-dimensional models based on Taylor and Lagrange expansions are used in this work and some typical applications are proposed to assess this novel technique, including Stokes flows in cylindrical and non-conventional domains. For each numerical example, different combinations of one-dimensional models have been considered to account for different kinematic approximations of flows, and the results, compared with analytical or finite volume solutions, highlight the capabilities of the proposed approach to handle non-conventional boundary conditions with ease and in preserving the computational cost without any accuracy loss.

Finite Volume Solution of Non-Newtonian Casson Fluid Flow in A Square Cavity

A two dimensional unsteady flow of non-Newtonian fluid in a square cavity is investigated numerically by using finite volume method based on staggered grids. The discretized equations are integrated by using second order Adams-Bashforth time advancement scheme togather with pressure correction approach. Error history for velocities and pressure are recorded for high Reynolds number when grid resolution is \(128\times128\). The results are also compared with already published work for special case. An excellent agreement is observed. The behavior of velocity components are studied for different values of non-Newtonian parameter $\beta$, the Casson fluid parameter.

Finite-volume homogenization and localization of nanoporous materials with cylindrical voids. Part 2: New results

The finite-volume homogenization theory with surface-elasticity effects based on the Gurtin-Murdoch model developed in Part 1 is employed to investigate little explored aspects of nanoporous materials' response. New results illustrate the effects of pore array architecture, aspect ratio and mean radius of elliptical porosities on local stress fields and homogenized moduli in an admissible range of porosity volume fractions. These results highlight the importance of adjacent pore interactions neglected in the classical micromechanics models, quantified herein for the first time. The theory is also shown to capture the highly oscillatory stress fields associated with surface-elasticity induced solution instabilities in this class of materials with negative surface moduli without ill-conditioning problems. Differences and similarities between comparable finite-element and finite-volume solutions of the unit cell boundary-value problem are delineated, including identification of pore radii, and associated aspect ratios and volume fractions, at which instabilities initiate. Consistent with reported and herein generated finite-element based results, the solution instability is also shown to depend on finite-volume mesh refinement. Hence care is required to identify the admissible range of parameters in the calculation of homogenized moduli. The new theory provides an alternative and independent means of identifying stable solution ranges, and hence is a good tool in assessing the finite-element method's predictive capability of generating stable solutions. Comparison with molecular dynamics simulations is included in further support of the theory's potential to capture both the initial homogenized response and local stress fields that may lead to failure.

Adaptive stabilized finite volume method and convergence analysis for the Oseen equations

In this paper, based on the pressure project method, we consider an adaptive stabilized finite volume method for the Oseen equations with the lowest equal order finite element pair. Firstly, we develop the discrete forms in both finite element and finite volume methods, and establish the existence and uniqueness of numerical solutions by establishing the equivalence of linear terms in finite element and finite volume methods. Secondly, a residual type a posteriori error estimator is designed, and the computable global upper and local lower bounds between the exact solutions and the finite volume solutions are established. Thirdly, a discrete local lower bound between two successive finite volume solutions is obtained, convergence analysis of the adaptive stabilized finite volume method is also performed. Finally, some numerical results are presented to verify the performances of the developed error estimators and confirm the established theoretical findings.

High-order fully implicit SIMPLE-based model for fully implicit simulation of upward two-phase flow

The drift-flux model has a practical importance in two-phase flow analysis. In this study, a finite volume solution is developed for a transient four-equation drift-flux model through the staggered mesh, leading to the development of a fully implicit discretization method. The main advantage of the fully implicit method is its unconditional stability. Newton’s scheme is a popular method of choice for the solution of a nonlinear system of equations arising from fully implicit discretization of field equations. However, the lack of convergence robustness and the construction of Jacobian matrix have created several difficulties for the researchers. In this paper, a fully implicit model is developed based on the SIMPLE algorithm for two-phase flow simulations. The drawbacks of Newton’s method are avoided in the developed model. Different limiter functions are considered, and the stabilized method is developed under steady and transient conditions. The results obtained by the numerical modeling are in good agreement with the experimental data. As expected, the results prove that the developed model is not restricted by any stability limit.

Simulation of impacts on elastic\u2013viscoplastic solids with the flux-difference splitting finite volume method applied to non-uniform quadrilateral meshes

The flux-difference splitting finite volume method (Leveque in J Comput Phys 131:327–353, 1997; Leveque in Finite volume methods for hyperbolic problems. Cambridge: Cambridge University Press, 2002) is here employed to perform numerical simulation of impacts on elastic–viscoplastic solids on bidimensional non-uniform quadrilateral meshes. The formulation is second order accurate in space through flux limiters, embeds the corner transport upwind method, and uses a fractional-step method to compute the relaxation operator. Elastic–viscoplastic constitutive models falling within the framework of generalized standard materials (Halphen and Nguyen in J Mech 14:667–688, 1975) in small strains are considered. Many test cases are proposed and two particular viscoplastic constitutive models are studied, on which comparisons with finite element solutions show a very good accuracy of the finite volume solutions, both on stresses and viscoplastic strains.

Finite volume solution for two-phase flow in a straight capillary

The problem of two-phase flow in straight capillaries of polygonal cross section displays many of the dynamic characteristics of rapid interfacial motions associated with pore-scale displacements in porous media. Fluid inertia is known to be important in these displacements but is usually ignored in network models commonly used to predict macroscopic flow properties. This study presents a numerical model for two-phase flow which describes the spatial and temporal evolution of the interface between the fluids. The model is based on an averaged Navier-Stokes equation and is shown to be successful in predicting the complex dynamics of both capillary rise in round capillaries and imbibition along the corners of polygonal capillaries. The model can form the basis for more realistic network models which capture the effect of capillary, viscous and inertial forces on pore-scale interfacial dynamics and consequent macroscopic flow properties.

Thermo-hydraulic analysis of multi-row cross-flow heat exchangers

This paper presents thermal hydraulic analysis of the cross-flow finned tube heat exchangers for an out-door unit in residential air-conditioning and heat pump applications. Performance of heat exchangers affect significantly the system energy efficiency and size of the air-conditioning and heat pumps. The Navier-Stokes equations and the energy equation are solved for the three dimensional computation domain that encompasses multiple rows of the fin-tube heat exchangers. Rather than solving the flow and temperature fields for the outdoor heat exchanger directly, the fin-tube array has been approximated by the porous medium of equivalent permeability, which is estimated from a three dimensional finite volume solution for the periodic fin element. This information is essential and time-effective in carrying out the global flow field calculation which, in turn, provides the face velocity for the microscopic temperature-field calculation of the heat exchanger. The flow field and associated heat transfer for a wide range of face velocity and fin-tube arrangements are examined and the results are presented compared with experimental data. The predicted pressure drop and heat transfer rate for various inlet velocities are in excellent agreement with the measured data.

Finite volume solution of 2-D hyperbolic conduction with contact resistance

PurposeThe purpose of this paper is to present a numerical methodology for the solution of non-Fourier conduction in two-dimensional (2-D) heterogeneous materials with contact resistance.Design/methodology/approachEnergy and heat flux equations with time lagging constant are combined to form a 2-D hyperbolic conduction equation in conservational form, and the resulting equation is solved by finite volume method.FindingsThe magnitude of contact resistance is inversely proportional to the temperature jump at the contact surface and phonon transmission coefficient between heterogeneous medium. Numerical results show that higher the contact resistance, lower the heat flux through the interface, lower the strength of transmitted wave and higher the strength of reflected wave at the interface. These results are in agreement with physical expectations. Temperature profiles show expected discontinuity at the interface while the heat fluxes are continuous, demonstrating the accuracy of the proposed methodology.Originality/valueIn most available numerical methods for hyperbolic conduction with contact resistance, contact resistances are treated as internal boundaries at which boundary conditions are specified. In the present formulation, contact resistance between two heterogeneous materials is treated as a part of interface transport properties not as an added boundary condition. This approach makes the formulation much simpler and straightforward for multidimensional applications. This approach is never used previously and is original.

Maximum-norms error estimates for high-order finite volume schemes over quadrilateral meshes

In this paper, we perform $$L^\infty $$ and $$W^{1,\infty }$$ error estimates for a class of bi-k finite volume schemes on a quadrilateral mesh for elliptic equations, where $$k\ge 2$$ is arbitrary. We show that the errors of the finite volume solution in both the $$L^\infty $$ and $$W^{1,\infty }$$ norms converge to zero with optimal orders, provided the solution $$u\in W^{k+2,\infty }$$ . Our analysis is based mainly on an estimate of the difference between the finite volume and the corresponding finite element bilinear forms, as well as some techniques derived for $$L^\infty $$ and $$W^{1,\infty }$$ estimates of the finite element method. Our theoretical findings are supported by several numerical examples.

A priori analysis of multilevel finite volume approximation of 1D convective Cahn–Hilliard equation

In this work, we analyze four finite volume methods for the nonlinear convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The methods used are: implicit one-level, explicit one-level, implicit multilevel and explicit multilevel finite volume methods. The existence and uniqueness of solution, convergence and stability of the finite volume solutions are proved. We compute $$L_2$$ - error and rate of convergence for all methods. We then compare the multilevel methods with the one-level methods by means of stability and CPU time. It is shown that the multilevel finite volume method is faster than the one-level method.

Finite-Volume Solutions to the Water-Hammer Equations in Conservation Form Incorporating Dynamic Friction Using the Godunov Scheme

AbstractAlthough derived from the principles of conservation of mass and momentum, the water-hammer equations integrating dynamic friction are almost never expressed in conservative form. This is b...