Convective Terms Research Articles

A pressure-free long-time stable reduced-order model for two-dimensional Rayleigh-Bénard convection.

The present work presents a stable proper orthogonal decomposition (POD)-Galerkin based reduced-order model (ROM) for two-dimensional Rayleigh-Bénard convection in a square geometry for three Rayleigh numbers: 104 (steady state), 3×105 (periodic), and 6×106 (chaotic). Stability is obtained through a particular (staggered-grid) full-order model (FOM) discretization that leads to a ROM that is pressure-free and has skew-symmetric (energy-conserving) convective terms. This yields long-time stable solutions without requiring stabilizing mechanisms, even outside the training data range. The ROM's stability is validated for the different test cases by investigating the Nusselt and Reynolds number time series and the mean and variance of the vertical temperature profile. In general, these quantities converge to the FOM when increasing the number of modes, and turn out to be a good measure of accuracy. However, for the chaotic case, convergence with increasing numbers of modes is relatively difficult and a high number of modes is required to resolve the low-energy structures that are important for the global dynamics.

Steady flow of non-Newtonian fluids: Momentum and heat transfer around a rectangular cylinder

This research aims to explore the impact of cross-sectional aspect ratio (AR), power-law index (n), Reynolds number (Re), and Prandtl number (Pr) on convective heat transfer rates using the Nusselt number (Nu) and coefficient of drag (CD) as a key metric. Numerical simulations employ the ANSYS FLUENT solver with the finite volume method. The SIMPLE scheme couples pressure and velocity, and the QUICK differencing scheme discretizes convective terms. This study varies the aspect ratio by adjusting the length-to-width ratio (b/D) to explore different configurations of AR = 1–5 and Pr = 0.7, 1, and 7, which investigate 5 ≤ Re ≤ 10 and 0.2 ≤ n ≤ 1, respectively. An increase in the Reynolds number is observed to diminish the thermal boundary layer thickness, resulting in an escalation of heat transfer rates. Additionally, the Nusselt number exhibits an upward trend with an increase in the Prandtl number. The investigation reveals a decrease in Nu with an increase in both AR and n. Specifically, higher aspect ratios and power-law indices are associated with reduced heat transfer rates. The AR emerges as a significant factor, influencing the coefficient of drag, with CD values showing an increase as AR rises, particularly at low Re. Moreover, CD is found to increase with an elevation in the n value. This study explores new ground by investigating power-law fluid behavior and heat transfer phenomenon across varied aspect ratios. These insights not only enhance the design of heat transfer systems with non-Newtonian fluids but also pave the way for future research in this less explored domain.

Wall-modeled large eddy simulation of 90° bent pipe flows with/without particles: A comparative study

Wall-modeled large eddy simulation (WMLES) has been proven to be a cost-effective approach capable of resolving turbulence up to certain resolutions. Among the simplest wall models used are the equilibrium wall models, assuming the pressure gradient and convective terms balance out in the momentum equations. There is a lack of studies to assess the performance of these standard wall models in internal turbulent flows including separation regions with/without particles. Regarding this research gap, we have conducted WMLES of incompressible turbulent flows, to the authors’ knowledge, for the first time, in 90° bent pipes with and without particles using an algebraic equilibrium wall model (Spalding’s function). A pipe flow simulation was conducted to confirm the simulation setup and assess the sensitivity with respect to the modeling parameters. In each case, comparisons are made with experiment or direct numerical simulation (DNS), and depending on the case, with other existing simulation methods in the literature: WMLES, standard (wall-resolving) LES, and Reynolds stress model (RSM) for Reynolds-averaged Navier-Stokes (RANS) simulations. Despite the controversy on the performance of equilibrium wall models in nonequilibrium flows, our results show acceptable accuracy of this type of wall models. Specifically in the bent pipe flow with particles, WMLES succeeded in predicting particle deposition efficiency at Stokes numbers greater than 0.5, but obtained less accurate results for smaller Stokes numbers. The WMLES errors were, however, on par with those of the standard LES employed with a tenfold higher grid cell count. Improved results would be expected if combined with auxiliary mechanisms such as stochastic models.

A numerical simulation of natural convection in a porous enclosure with nanofluid

In day-to-day life, heat transfer has several applications and hence we have many research areas of interest to learn new things. Nanoparticles improve the characteristics of heat transfer due to the random motion of particles. In this study, a computational approach is used to study natural convection in a porous enclosure with nanofluid. The convective term is solved by the QUICK scheme, and the finite difference method is used to solve the stream function with a 60 × 60 mesh size. CuO-water nanofluid is used in this work since nanoparticles are widely used in almost all engineering fields. In this study, we considered a two-dimensional complex cavity which is solved mathematically by using the Boussinesq approximation method. For validation of the software code, the obtained results are compared with the existing results in the literature and are found to be in good agreement. We see distortion of isotherms increases with an increase in the Rayleigh number and the nanoparticle content. That is, heat transfer takes place by natural convection very fast. Nusselt number increases with increasing nanofluid particles and the Rayleigh number. Also, it can be concluded that the heat transfer is a strong function of the Darcy number and The Rayleigh number.

Volume average of two-fluid RANS equations and a priori estimation of subgrid terms on subcooled boiling experiments

A statistic-volume averaged two-fluid model, convenient for subchannel and system codes in nuclear industry, is established from RANS equations of Neptune_CFD code and leads to the appearance of subgrid terms. A database of simulations is made on experiments with flow representative of nuclear reactor cores with cylindrical (KIT) and subchannel (PSBT) configurations. The subgrid terms are then compared to the convection terms on the database of 82 test cases. Concerning momentum and energy equations, subgrid terms reach respectively 7.5% and 16% of convection terms. Overall, the comparison shows that dispersion terms are always greater than turbulence terms, but both remain in the same order of magnitude, and are not negligible compared to convection. Moreover, the geometry has an impact on the dispersion terms. This observation suggests that caution should be exercised when using correlation established on a different geometry.

The Upwind Finite Volume Element Method for Two-Dimensional Time Fractional Coupled Burgers’ Equation

The finite volume element method for approximating a two-dimensional time fractional coupled Burgers' equation is presented. The linear finite volume element method is used for spatial discretization and the upwind technique is used for the nonlinear convective term to get the semi-discrete scheme. Further, the time-fractional derivative term is approximated by using L1 formula and the nonlinear convection term is treated by linearized upwind technique to get the fully discrete scheme. We prove that the semi-discrete scheme is convergent with one-order accuracy in space and the fully discrete scheme is convergent with one-order accuracy both in time and space in L^2-norm. Numerical experiments are presented finally to validate the theoretical analysis.

Well-posedness and error estimates for coupled systems of nonlocal conservation laws

Abstract This article deals with the error estimates for numerical approximations of the entropy solutions of coupled systems of nonlocal hyperbolic conservation laws. The systems can be strongly coupled through the nonlocal coefficient present in the convection term. A fairly general class of fluxes is being considered, where the local part of the flux can be discontinuous at infinitely many points, with possible accumulation points. The aims of the paper are threefold: (1) Establishing existence of entropy solutions with rough local flux for such systems, by deriving a uniform $\operatorname {BV}$ bound on the numerical approximations; (2) Deriving a general Kuznetsov-type lemma (and hence uniqueness) for such systems with both smooth and rough local fluxes; (3) Proving the convergence rate of the finite volume approximations to the entropy solutions of the system as $1/2$ and $1/3$, with homogeneous (in any dimension) and rough local parts (in one dimension), respectively. Numerical experiments are included to illustrate the convergence rates.

Existence and uniqueness for renormalized solutions to noncoercive nonlinear parabolic equations with unbounded convective term

We study the existence and uniqueness of renormalized solutions for noncoercive nonlinear parabolic equations in the presence of an unbounded convective term.

Pressure stabilization method for the 2D/3D time‐dependent natural convection equations with salt concentration

AbstractIn this paper, the pressure stabilization method for the 2D/3D time‐dependent natural convection equations with salt concentration (NCSC) is considered. The artificial pressure diffusion term is introduced to reduce the constraint of the incompressible condition. The first‐order backward difference formula is adopted to approximate the temporal derivative terms, the diffusion terms in NCSC are treated implicitly, the convective terms in momentum equation are treated semi‐implicitly, the convective terms in energy equation and salt concentration equation are treated implicitly. The error estimates for the semi‐discrete pressure stabilization method are given. Some numerical experiments are presented to verify the theoretical results and show the reliability of the proposed method.

Urban hydrological model (UHM) developed for an urban flash flood simulation and analysis of the flood intensity sensitivity to urbanization

An urban hydrological model (UHM) forced by radar-observed rainfalls was developed for producing rainstorm-related urban inundation maps for obtaining flash flood forecasts. The parameter sets of the land use and land cover (LULC) and urban drainage capacity derive from satellite multispectral images and high spatial resolution GIS datasets, relating to the urban hydrology and hydraulic properties. The hydrodynamics model was based on the simplified shallow water equation (SWE) neglecting the convective acceleration and pressure terms in the St. Venant equation. The intense convective rainstorm that induced heavy flash floods on 21 Jul. 2012 and the large-scale stratiform precipitation that occurred from 19 Jul. to 21 Jul. 2016 in Beijing were deliberately selected to perform flood-simulation and model-validation case studies. The simulation of severe flooding scenarios in the 2012 event was fairly reproduced and verified using media reports. In the case studies, the sensitivity tests verified that the flood intensity (1) would be enhanced slightly on pure impervious surfaces, (2) would be reduced significantly on pure pervious surfaces, and (3) would increase by 30%–60% without a drainage system. No false alarm could be perceived in the simulation of the 2016 event in a prolonging stratiform precipitation. The case studies confirmed the perspectives of using the hydrological model fed by QPEs/QPFs for flash flood forecasting and concluded that the hourly precipitation surpassing 30 mm/h would be indicative in impending flash floods in Beijing metropolis.

Optimal control of three-dimensional unsteady partial differential equations with convection term in continuous casting

In this paper, we investigate an optimal control problem (OCP) for three-dimensional (3D) unsteady partial differential equations with convection term (UPDECT), extensively applied to the solidification process of billets. Firstly, we introduce the mathematical model of OCP and adopt a life cycle model method (LCMM) to numerically solve the 3D UPDECT. Secondly, the Lipschitz continuity of the cost function gradient is proved using the adjoint approach as a numerical method for designing optimal control systems. The cost function gradient of OCP for 3D UPDECT is analyzed based on a Hamiltonian costate system. Thirdly, we propose a modified trinomial conjugate gradient method (MTCGM) to solve the OCP for 3D UPDECT and prove the global convergence of MTCGM. Finally, the MTCGM is applied to the experimental simulation of continuous casting. According to experiment results, the optimizer based on MTCGM can significantly accelerate the convergence speed and reduce the iterative numbers, and the MTCGM provides a more stable temperature control effect.

On anisotropic parabolic equation with nonstandard growth order

ABSTRACT In this paper, the existence and the uniqueness of an evolutionary anisotropic p i ( x ) -Laplacian equation with a damping term are studied. If the damping term is with a subcritical index, by the Di Giorgi iteration technique, the L ∞ -estimate of the weak solutions can be obtained. The existence of weak solution is proved by the renormalized solution method, and how the anisotropic characteristic of the considered equation affect the L ∞ -estimate of the weak solutions is revealed. The uniqueness is true strongly depending on subcritical index of the damping term, and this result goes beyond previous efforts in the literature (Bertsch M, Dal Passo R, Ughi M: Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798; Li Z, Yan B, Gao W. Existence of solutions to a parabolic p ( x ) − Laplace equation with convection term via L ∞ − Estimates. Electron J Differ Equ. 2015;46:1–21; Zhang Q, Shi P. Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal. 2010;72:2744–2752; Zhou W, Cai S. The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin University (Natural Sci.). 2004;42:341–345), etc.

Existence results for Schrödinger type double phase variable exponent problems with convection term in $ \mathbb R^{N} $

<abstract><p>This paper was concerned with a new class of Schrödinger equations involving double phase operators with variable exponent in $ \mathbb R^{N} $. We gave the corresponding Musielak-Orlicz Sobolev spaces and proved certain properties of the double phase operator. Moreover, our main tools were the topological degree theory and Galerkin method, since the equation contained a convection term. By using these methods, we derived the existence of weak solution for the above problems. Our result extended some recent work in the literature.</p></abstract>

Singular quasilinear convective systems involving variable exponents

The paper deals with the existence of solutions for quasilinear elliptic systems involving singular and convection terms with variable exponents. The approach combines the sub-supersolutions method and Schauder's fixed point theorem.

Специальная разностная схема для решения жестких краевых задач конвективно-диффузионного переноса

The convective-diffusion transfer equation is often found in problems of hydromechanics and heat and mass transfer. The dominance of convection over diffusion and the change in sign of the coefficient at the first derivative lead to the formation of boundary and internal layers with high gradients of the function. This creates serious difficulties in numerical analysis of the problem using traditional difference schemes. The traditional method of approximating the first derivative using central differences at high Peclet numbers can lead to oscillations and violate the monotonicity of the numerical solution. To avoid this problem, it is necessary to significantly reduce the size of grid cells in narrow areas with large gradients of the unknown function. The use of one-sided differences significantly smears the desired solution, due to the viscosity of the scheme, and leads to loss of accuracy. The practical need to solve stiff boundary value problems requires the development and use of computational technologies that guarantee monotonicity, accuracy, and cost-effectiveness in numerical analysis. In this paper, a new special difference scheme is proposed for the numerical solution of a stiff equation of convective-diffusion transfer. The dominant convective term is eliminated from explicit consideration by transforming the equation into self-adjoined form, which permits the use of well-known numerical approximation techniques. The control volume method is used to construct a difference analogue of a differential equation on a three-point template. The resulting scheme is monotonic and conservative. The test examples show great possibilities of the proposed difference scheme for large Peclet numbers on coarse grids in solving stiff boundary value problems of convective diffusion transfer.

Numerical simulation of the flow in a flat suddenly expanding channel based on a two-fluid turbulence model

The article presents the results of a numerical study of the flow structure in a flat channel in the zone of its sudden expansion in the form of a ledge. The calculations are based on the numerical solution of a system of non-stationary equations using a new two-fluid turbulence model. The results of the profiles of longitudinal velocity and turbulent stress in different sections of the channel before and after the ledge, as well as the change in the coefficient of friction for the lower wall of the channel depending on the distance after the ledge are obtained. For the difference approximation of the initial equations, the control volume method was used, and the relationship between velocities and pressure was found using the SIMPLEC procedure. In this case, the viscous terms were approximated by the central difference, and for convective terms, a second-order scheme of accuracy against the flow was used. To confirm the correctness of the numerical results, a comparison was made with experimental data from the NASA database for the Reynolds number Re=36000. And the influence of the calculated grid in the results of the study was also investigated.

Model predictive control of non-interacting active Brownian particles.

Active matter systems are strongly driven to assume non-equilibrium distributions owing to their self-propulsion, e.g., flocking and clustering. Controlling the active matter systems' spatiotemporal distributions offers exciting applications such as directed assembly, programmable materials, and microfluidic actuation. However, these applications involve environments with coupled dynamics and complex tasks, making intuitive control strategies insufficient. This necessitates the development of an automatic feedback control framework, where an algorithm determines appropriate actions based on the system's current state. In this work, we control the distribution of active Brownian particles by applying model predictive control (MPC), a model-based control algorithm that predicts future states and optimizes the control inputs to drive the system along a user-defined objective. The MPC model is based on the Smoluchowski equation with a self-propulsive convective term and an actuated spatiotemporal-varying external field that aligns particles with the applied direction, similar to a magnetic field. We apply the MPC framework to control a Brownian dynamics simulation of non-interacting active particles and illustrate the controller capabilities with two objectives: splitting and juggling sub-populations, and polar order flocking control.

Natural convection flow and heat transfer of generalized Maxwell fluid with distributed order time fractional derivatives embedded in the porous medium

<p>Numerical simulation was performed for unsteady natural convection flow and heat transfer in a porous medium using the generalized Maxwell model and fractional Darcy's law with distributed order time fractional derivatives. The finite volume method combined with the fractional <italic>L1</italic> scheme was used to solve strongly coupled governing equations with nonlinear fractional convection terms. Numerical solutions were validated via grid independence tests and comparisons with special exact solutions. The effects of porosity, Darcy number, and relaxation time parameters on transport fields are presented. The results illustrate that porosity and permeability have opposite influences on temperature and velocity profiles. Moreover, the relaxation time parameters have remarkable effects on velocity profiles, and the variations possess significant differences.</p>

Convergence of finite element solution of stochastic Burgers equation

<abstract><p>We explore the numerical approximation of the stochastic Burgers equation driven by fractional Brownian motion with Hurst index $ H\in(1/4, 1/2) $ and $ H\in(1/2, 1) $, respectively. The spatial and temporal regularity properties for the solution are obtained. The given problem is discretized in time with the implicit Euler scheme and in space with the standard finite element method. We obtain the strong convergence of semidiscrete and fully discrete schemes, performing the error estimates on a subset $ \Omega_{k, h} $ of the sample space $ \Omega $ with the Gronwall argument being used to overcome the difficulties, caused by the subtle interplay of the nonlinear convection term. Numerical examples confirm our theoretical findings.</p></abstract>

Turbulence characteristics in partially vegetated open channels with alternating sparse and dense patches

A large amount of vegetation in nature exists in the form of heterogeneous vegetation patches, and variations in vegetation characteristics significantly affect water flow structures. The objective of this study is to investigate the effect of alternating sparse and dense patches on turbulence characteristics. Multiple sets of heterogeneous vegetation scenarios were designed for numerical simulation analysis, and a comparison was made with homogeneous vegetation. Results indicate that compared with that of homogeneous vegetation, the arrangement of heterogeneous vegetation alters the distribution of flow velocities in the vegetation zone and the main channel, promoting material exchange between these regions. The vegetation density difference between sparse and dense vegetation patches in heterogeneous vegetation effect on the main channel increases with larger vegetation density differences, but the magnitude of the effect is limited and generally remains within 10% of the vegetation width. In the balance equation of turbulent kinetic energy, the terms are influenced differently by changes in vegetation density. As vegetation density difference increases, the convective term gradually increases, while the production and diffusion terms exhibit a “lower in the middle, higher at both ends” pattern. The dissipation term demonstrates a reduction effect at low-density differences, gradually increases with larger differences, and ultimately exhibits an amplification effect. Furthermore, this study determines that using data from mid-height to represent the entire cross section for heterogeneous vegetation may result in a maximum error of up to 11%.