Nonlinear Convective Terms Research Articles

Fluid flow between two parallel active plates

This paper investigates the fluid flow phenomenon arising from the combined action of two parallel plates, which can expand/squeeze, absorb/inject, and stretch/shrink at different rates. These physical mechanisms are incorporated into the governing unsteady Navier–Stokes equations, which are then reduced to a fourth-order nonlinear differential equation with boundary conditions reflecting the imposed wall constraints. By letting the permeable Reynolds number (controlling the nonlinear convective terms) limit to zero, we demonstrate the existence of exact solutions expressed in terms of advanced mathematical functions. Additionally, in the absence of wall expansion/contraction, elementary exponential solutions are obtained under particular relationships between the stretching/shrinking and permeability parameters. A shear-like exact solution with broader applicability across various physical parameters is also identified. For moderate values of the expansion/squeezing parameters and permeable Reynolds numbers, we propose an efficient double-expansion perturbation analysis to approximate the flow behavior. Otherwise, for general physical parameters, a comprehensive mathematical analysis is provided and numerical simulations are employed to extract insights into the complex fluid motion between the parallel plates.

Squeeze force of a Maxwell fluid between circular smooth surfaces with simple harmonic motion

The force and mechanical power required to maintain the simple harmonic motion (SHM) of the upper circular surface squeezing a viscoelastic fluid film is analyzed. The amplitude of the displacement of the upper surface is very small compared to the gap width as a function of time. The smoothness of the upper and lower surfaces is characterized by the slip model with two constant parameters, a slip length and a critical surface shear stress. The nonlinear convection terms in the momentum equation are neglected since the viscous forces dominate the inertial forces. The acceleration and deceleration terms are retained since the upper plate oscillates harmonically and the velocity in the fluid is strictly periodic. An exact solution of the governing equations is found as a function of the Deborah number, the Womersley number, the slip length, and the critical surface shear stress. A circular region without slip condition, bounded by a time-dependent radius, appears when the shear stress of the fluid does not exceed a critical surface shear stress. In addition, an annular region with slip up to the radius of the disk appears when the critical surface shear stress is exceeded. Our results show that viscoelastic and hydrophobic effects together with the Womersley number and a critical surface stress cause changes in the amplitude and phase lag of the waveform of the time-dependent radius and the force acting on the wall surface to maintain the SHM of the upper disk.

An artificial neural network model for recovering small-scale velocity in large-eddy simulation of isotropic turbulent flows

Small-scale motions in turbulent flows play a significant role in various small-scale processes, such as particle relative dispersion and collision, bubble or droplet deformation, and orientation dynamics of non-sphere particles. Recovering the small-scale flows that cannot be resolved in large eddy simulation (LES) is of great importance for such processes sensitive to the small-scale motions in turbulent flows. This study proposes a subgrid-scale model for recovering the small-scale turbulent velocity field based on the artificial neural network (ANN). The governing equations of small-scale turbulent velocity are linearized, and the pressure gradient and the nonlinear convection term are modeled with the aid of the ANN. Direct numerical simulation (DNS) and filtered direct numerical simulation (FDNS) provide the data required for training and validating the ANN. The large-scale velocity and velocity gradient tensor are selected as inputs for the ANN model. The linearized governing equations of small-scale turbulent velocity are numerically solved by coupling the large-scale flow field information. The results indicate that the model established by the ANN can accurately recover the small-scale velocity lost in FDNS due to filtering operation. With the ANN model, the flow fields at different Reynolds numbers agree well with the DNS results regarding velocity field statistics, flow field structures, turbulent energy spectra, and two-point, two-time Lagrangian correlation functions. This study demonstrates that the proposed ANN model can be applied to recovering the small-scale velocity field in the LES of isotropic turbulent flows at different Reynolds numbers.

A linear second-order maximum bound principle preserving finite difference scheme for the generalized Allen–Cahn equation

In this paper, we consider the numerical method for generalized Allen–Cahn equation with nonlinear mobility and convection term. We propose a linear second-order finite difference scheme which preserves the discrete maximum bound principle (MBP). The scheme is discretized by stabilized Crank–Nicolson in time, upwind scheme for convection term and central-difference scheme for diffusion term. We show that the proposed scheme preserves the discrete MBP under some constraints on temporal step size and stabilizing parameter. Optimal L∞-error estimate is obtained for our scheme. Several numerical experiments are performed to validate our theoretical results.

A new class of efficient high order semi-Lagrangian IMEX discontinuous Galerkin methods on staggered unstructured meshes

In this paper we present a new high order semi-implicit discontinuous-Galerkin (DG) scheme on two-dimensional staggered triangular meshes applied to different nonlinear systems of hyperbolic conservation laws such as advection-diffusion models, incompressible Navier-Stokes equations and natural convection problems. While the temperature and pressure field are defined on a triangular main grid, the velocity field is defined on a quadrilateral edge-based staggered mesh. A semi-implicit time discretization is proposed, which separates slow and fast time scales by treating them explicitly and implicitly, respectively. The nonlinear convection terms are evolved explicitly using a semi-Lagrangian approach, whereas we consider an implicit discretization for the diffusion terms and the pressure contribution. High-order of accuracy in time is achieved using a new flexible and general framework of IMplicit-EXplicit (IMEX) Runge-Kutta schemes specifically designed to operate with semi-Lagrangian methods. To improve the efficiency in the computation of the DG divergence operator and the mass matrix, we propose to approximate the numerical solution with a less regular polynomial space on the edge-based mesh, which is defined on two sub-triangles that split the staggered quadrilateral elements. This allows for a fast computation of the DG operators and matrices without any need to store them for each mesh element. Due to the implicit treatment of the fast scale terms, the resulting numerical scheme is unconditionally stable for the considered governing equations because the semi-Lagrangian approach is the only explicit discretization for convection terms that does not need any stability restriction on the maximum admissible time step. Contrarily to a genuinely space-time discontinuous-Galerkin scheme, the IMEX discretization permits to preserve the symmetry and the positive semi-definiteness of the arising linear system for the pressure that can be solved at the aid of an efficient matrix-free implementation of the conjugate gradient method. We present several convergence results, including nonlinear transport and density currents, up to third order of accuracy in both space and time.

Robust discontinuous Galerkin-based scheme for the fully-coupled nonlinear thermo-hydro-mechanical problem

Abstract We present and analyze a discontinuous Galerkin method for the numerical modeling of the nonlinear fully-coupled thermo-hydro-mechanic problem. We propose an arbitrary-order weighted symmetric interior penalty scheme that supports general polytopal grids and is robust with respect to strong heterogeneities in the model coefficients. We focus on the treatment of the nonlinear convective transport term in the energy conservation equation and we propose suitable stabilization techniques that make the scheme robust for advection-dominated regimes. The stability analysis of the problem and the convergence of the fixed-point linearization strategy are addressed theoretically under mild requirements on the problem data. A complete set of numerical simulations is presented in order to assess the convergence and robustness properties of the proposed method.

On Error Estimates of a discontinuous Galerkin Method of the Boussinesq System of Equations

Abstract In this paper, we propose and analyze a discontinuous Galerkin finite element method for solving the transient Boussinesq incompressible heat conducting fluid flow equations. This method utilizes an upwind approach to handle the nonlinear convective terms effectively. We discuss new a priori bounds for the semidiscrete discontinuous Galerkin approximations. Furthermore, we establish optimal a priori error estimates for the semidiscrete discontinuous Galerkin velocity approximation in L 2 \mathbf{L}^{2} and energy norms, the temperature approximation in L 2 L^{2} and energy norms and pressure approximation in L 2 L^{2} -norm for t > 0 t>0 . Additionally, under the smallness assumption on the data, we prove uniform in time error estimates. We also consider a backward Euler scheme for full discretization and derive fully discrete error estimates. Finally, we provide numerical examples to support the theoretical conclusions.

Macroscopic finite-difference scheme and modified equationsof the general propagation multiple-relaxation-time lattice Boltzmann model.

In this paper we first present the general propagation multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the corresponding macroscopic finite-difference (GPMFD) scheme on conservative moments. Then based on the Maxwell iteration method, we conduct the analysis on the truncation errors and modified equations(MEs) of the GPMRT-LB model and GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear anisotropic convection-diffusion equation(NACDE) and Navier-Stokes equations(NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model and GPMFD scheme. In particular, for the one-dimensional convection-diffusion equation(CDE) with the constant velocity and diffusion coefficient, we can develop a fourth-order GPMRT-LB (F-GPMRT-LB) model and the corresponding fourth-order GPMFD (F-GPMFD) scheme at the diffusive scaling. Finally, three benchmark problems, the Gauss hill problem, the CDE with nonlinear convection and diffusion terms, and the Taylor-Green vortex flow in two-dimensional space, are used to test the GPMRT-LB model and GPMFD scheme, and it is found that the numerical results not only are in good agreement with corresponding analytical solutions, but also have a second-order convergence rate in space. Additionally, a numerical study on one-dimensional CDE also demonstrates that the F-GPMRT-LB model and F-GPMFD scheme can achieve a fourth-order accuracy in space, which is consistent with our theoretical analysis.

A high-order upwind compact difference scheme for solving the streamfunction-velocity formulation of the unsteady incompressible Navier–Stokes equations

In this paper, we propose an upwind compact difference method with fourth-order spatial accuracy and second-order temporal accuracy for solving the streamfunction-velocity formulation of the two-dimensional unsteady incompressible Navier–Stokes equations. The streamfunction and its first-order partial derivatives (velocities) are treated as unknown variables. Three types of compact difference schemes are employed to discretize the first-order partial derivatives of the streamfunction. Specifically, these schemes include the fourth-order symmetric scheme, the fifth-order upwind scheme, and the sixth-order symmetric scheme derived by combining the two parts of the fifth-order upwind scheme. As a result, the fourth-order spatial discretization schemes are established for the Laplacian term, the biharmonic term, and the nonlinear convective term, along with the Crank–Nicolson scheme for the temporal discretization. The unconditional stability characteristic of the scheme for the linear model is proved by discrete von Neumann analysis. Moreover, six numerical experiments involving three test problems with the analytic solutions, and three flow problems including doubly periodic double shear layer, lid-driven cavity flow, and dipole-wall interaction are carried out to demonstrate the accuracy, robustness, and efficiency of the present method. The results indicate that the present method not only has good numerical performance but also exhibits quite efficiency.

A Semi-implicit Finite Volume Scheme for Incompressible Two-Phase Flows

This paper presents a mass and momentum conservative semi-implicit finite volume (FV) scheme for complex non-hydrostatic free surface flows, interacting with moving solid obstacles. A simplified incompressible Baer-Nunziato type model is considered for two-phase flows containing a liquid phase, a solid phase, and the surrounding void. According to the so-called diffuse interface approach, the different phases and consequently the void are described by means of a scalar volume fraction function for each phase. In our numerical scheme, the dynamics of the liquid phase and the motion of the solid are decoupled. The solid is assumed to be a moving rigid body, whose motion is prescribed. Only after the advection of the solid volume fraction, the dynamics of the liquid phase is considered. As usual in semi-implicit schemes, we employ staggered Cartesian control volumes and treat the nonlinear convective terms explicitly, while the pressure terms are treated implicitly. The non-conservative products arising in the transport equation for the solid volume fraction are treated by a path-conservative approach. The resulting semi-implicit FV discretization of the mass and momentum equations leads to a mildly nonlinear system for the pressure which can be efficiently solved with a nested Newton-type technique. The time step size is only limited by the velocities of the two phases contained in the domain, and not by the gravity wave speed nor by the stiff algebraic relaxation source term, which requires an implicit discretization. The resulting semi-implicit algorithm is first validated on a set of classical incompressible Navier-Stokes test problems and later also adds a fixed and moving solid phase.

A multi-region analysis of unsteady Oseen's equation for accelerating flow past a circular cylinder

It is difficult to find a valid high-order (of Reynolds number) analytical approximation to the uniform viscous flow past an infinite circular cylinder by applying perturbation techniques. In this study, we show that by accelerating flow past a circular cylinder with taking unsteady Stokes' solution as an initial approximation, higher-order approximation solutions to unsteady Oseen's equation can be obtained by an iteration scheme of perturbation, which exactly satisfy the boundary conditions. As a matter of fact, the nonlinear (convective) term linearized by Oseen's approximation is overweighted especially near the body surface. To eliminate the overweight, a multi-region analysis is proposed in this study to improve analytical unsteady Oseen's solution so as to extend the valid range of Reynolds numbers. In other words, the flow region is hypothetically divided into several annular regions, and the linearized convective term in each region is modified by multiplying with a Carrier's coefficient c (0 < c ≤ 1). The results show that, when the accelerating parameter in the range of 0.5 ≤ a ≤ 4, the maximum effective Reynolds number Reeff of the fourth-order five-region solution is both much larger than that of Stokes' and Oseen's ones. For example, when a = 0.5, Reeff = 22.26 is roughly eight times that of Stokes' solution (Reeff = 2.67), and nine times that of Oseen's solution (Reeff = 2.41). Moreover, when the instantaneous Reynolds number Re(t) is smaller than the Reeff, the flow separation angle and the wake length are both consistent with the numerical results obtained by accurately solving the full Navier-Stokes equations. In addition, the flow properties, including the drag coefficients, streamline patterns, and the pressure coefficients, as well as the vorticity distributions also agree well with the numerical results. This study represents a significant extension of our previous study for unsteady Stokes' equation [Xu et al., Phys. Fluids 35, 033608 (2023)] to unsteady Oseen's equation and its generalization.

A generalized differential quadrature approach to the modelling of heat transfer in non-similar flow with nonlinear convection

This study aims to investigate heat transfer and entropy production in the non-similar flow of an incompressible fluid, with nonlinear convection and viscous dissipation. To obtain precise solutions, the Sparrow-Quack-Boerner local non-similarity method is implemented. The non-similarity arises from the nonlinear convection term present in the momentum equation. The non-similarity arises from the nonlinear convection term present in the momentum equation. Consequently, the inclusion of non-similarity terms into the energy equation is achieved through the interconnection of momentum and energy equations. Entropy generation is explored by employing the second law of thermodynamics. Numerical results from several truncation levels are presented in tabular form, and equations for both first and second-level truncations are generated. The one- and two-equation models are solved by implementing the Generalized Differential Quadrature Method (GDQM). Comparison with a second level of truncation reveals significantly greater inaccuracies in numerical results obtained from the first level. This discrepancy arose due to the omission of non-similar terms in the primary governing equations at the first truncation stage. The validity and accuracy of the derived numerical solutions are further demonstrated by the application of the GDQM and the midpoint method with Richardson extrapolation. Additionally, a plot of the numerical data produced from the second level of truncations is presented together with a discussion of the various physical factors. Increasing the mixed convection parameter accelerates fluid velocity, also when the viscous dissipation parameter increases, temperature and the Bejan number increase. The Prandtl number and the thickness of the thermal boundary layer are observed to be inversely related. Moreover, the quantity of entropy generated at the stretching surface and inside the boundary layer rises in proportion to the increases in the Prandtl and Eckert numbers.

An improved partitioned time stepping method based on modified characteristic FEM for the evolutionary dual-porosity-Navier–Stokes model

In this paper, we study an improved partitioned time stepping method based on the modified characteristic finite element method for the evolutionary dual-porosity-Navier–Stokes coupling model with the Beavers–Joseph interface condition. By referring to the idea of partitioned time stepping, we lead into the half-time steps to separate the original coupling problem into three subproblems, implement two-level decoupling to simplify the complexity of solving coupling systems. Before that, the matrix and microfracture equations on the half-time steps in the dual-porosity media are calculated first, with the aim of replacing the initial values of the decoupling equations. Besides, for the nonlinear convection term in Navier–Stokes equations, the modified characteristic finite element method is chosen to avoid the problem of low computational efficiency. The convergence of the decoupling algorithm is rigorously derived by using the idea of mathematical induction. Finally, the reliability of the theoretical analysis and the rationality of the scheme are verified by numerical experiments.

Convergence analysis of high-order IMEX-BDF schemes for the incompressible Navier–Stokes equations

In this paper, we consider developing high-order temporal integration schemes for the unsteady incompressible Navier–Stokes equations in bounded two-dimensional domain subjected to the periodic boundary conditions. Utilizing the k-step (k=3,4,5) backward differentiation formula (BDF) coupled with the implicit–explicit (IMEX) treatment of the nonlinear convective term in an anti-symmetry form, a class of IMEX-BDFk schemes up to fifth-order in time are constructed and analyzed. By imposing a zero-mean constrain on the finite-dimensional space for the pressure, the proposed numerical schemes are proven to be uniquely solvable. Based on the recent theoretical framework consisting of a class of discrete orthogonal convolution kernels, rigorous L2 norm error estimates for both the velocity and the pressure are established by using a novel divergence free projection system. The proposed schemes are then implemented in two benchmark experiments, including a Taylor–Green vortex problem and a double shear layer flow at various high Reynolds numbers. Numerical results demonstrate the expected solution accuracy and the computational effectiveness in simulating the realistic flow dynamics.

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.

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>

Exact Solutions of the Modified Nonlinear Burgers' Equation

The expansion approach has been used to solve the modified nonlinear Burgers' equation, which has a nonlinear convection term, a viscosity term, and a time-dependent term in its structure. In this paper, the main focus is to find exact solutions of the modified nonlinear Burgers' equation. The (G′=G)-expansion method is one of the methods used to find exact solutions of nonlinear problems. It requires an appropriate transform equation to convert partial differential equations to ordinary differential equations, making it easier to find the solution. In this work, we choose the traveling wave equation to covert the equation. The results show that the exact solutions of the modified nonlinear Burgers' equation by the (G′=G)-expansion method are decreasing if traveling wave parameter, ω, increases for the first case and the exact solution is increasing if ω is increasing for the second case. It is observed that (G′=G)-expansion method is an advanced and easy tool for finding exact solutions of the modified nonlinear Burgers' equation.

Dynamics of a silicone oil drop submerged in a stratified ethanol-water bath.

We analyze numerically the Marangoni flow around an immiscible droplet submerged in a stably stratified mixture of ethanol and water. The linear stability analysis shows that the base flow undergoes a supercritical Hopf bifurcation that leads to oscillations. The theoretical prediction for the critical droplet radius is consistent with previous experimental results. Ethanol diffusion in water is critical in the flow stability for both low and high droplet viscosity. Direct numerical simulations of the nonlinear oscillatory flow show that the frequency of those oscillations approximately equals that of the critical eigenmode. The nonlinear convective term of the ethanol diffusive-convective transport equationfixes the amplitude of the droplet oscillations. The viscous dissipation associated with the Marangoni flow inside the droplet considerably reduces the oscillation.

A coupling of Galerkin and mixed finite element methods for the quasi-static thermo-poroelasticity with nonlinear convective transport

In this paper, we propose a combined Galerkin and mixed finite element methods to analyze the fully coupled nonlinear thermo-poroelastic model problems. We design the Galerkin finite element method for the temperature, the mixed finite element method for the pressure, the Galerkin finite element method for the elastic displacement. We linearize the nonlinear convective transport term in the energy balance equation and establish the fully discrete finite element schemes. The stability and convergence of the coupled method are obtained. In particular, all previous works have required certain time step restrictions, but we unconditionally prove the optimal error estimates without certain extra restrictions on both time step and spatial meshes. Finally, some numerical examples are presented to illustrate the accuracy of the method and confirm the unconditional stability of the method.