Nonlinear Convective Terms Research Articles

Numerically stable formulations of convective terms for turbulent compressible flows

A systematic analysis of the discrete conservation properties of non-dissipative, central-difference approximations of the compressible Navier–Stokes equations is reported. A generalized splitting of the nonlinear convective terms is considered, and energy-preserving formulations are fully characterized by deriving a two-parameter family of split forms. Previously developed formulations reported in literature are shown to be particular members of this family; novel splittings are introduced and discussed as well. Furthermore, the conservation properties yielded by different choices for the energy equation (i.e. total and internal energy, entropy) are analyzed thoroughly. It is shown that additional preserved quantities can be obtained through a suitable adaptive selection of the split form within the derived family. Local conservation of primary invariants, which is a fundamental property to build high-fidelity shock-capturing methods, is also discussed in the paper. Numerical tests performed for the Taylor–Green Vortex at zero viscosity fully confirm the theoretical findings, and show that a careful choice of both the splitting and the energy formulation can provide remarkably robust and accurate results.

Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient

In this paper, we study a nonlinear Dirichlet problem of p-Laplacian type with combined effects of nonlinear singular and convection terms. An existence theorem for positive solutions is established as well as the compactness of solution set. Our approach is based on Leray–Schauder alternative principle, method of sub-supersolution, nonlinear regularity, truncation techniques, and set-valued analysis.

Discrete Energy-Conservation Properties in the Numerical Simulation of the Navier–Stokes Equations

Nonlinear convective terms pose the most critical issues when a numerical discretization of the Navier–Stokes equations is performed, especially at high Reynolds numbers. They are indeed responsible for a nonlinear instability arising from the amplification of aliasing errors that come from the evaluation of the products of two or more variables on a finite grid. The classical remedy to this difficulty has been the construction of difference schemes able to reproduce at a discrete level some of the fundamental symmetry properties of the Navier–Stokes equations. The invariant character of quadratic quantities such as global kinetic energy in inviscid incompressible flows is a particular symmetry, whose enforcement typically guarantees a sufficient control of aliasing errors that allows the fulfillment of long-time integration. In this paper, a survey of the most successful approaches developed in this field is presented. The incompressible and compressible cases are both covered, and treated separately, and the topics of spatial and temporal energy conservation are discussed. The theory and the ideas are exposed with full details in classical simplified numerical settings, and the extensions to more complex situations are also reviewed. The effectiveness of the illustrated approaches is documented by numerical simulations of canonical flows and by industrial flow computations taken from the literature.

A nonlinearity lagging method for non-steady diffusion equations with nonlinear convection terms

We analyze an iterative procedure for solving nonlinear algebraic systems arising from the discretization of nonlinear, non-steady reaction-convection-diffusion equations with non-constant (and, in general, nonlinear) velocity terms. The basic idea underlying the procedure consists in lagging the diffusion and the velocity terms of the discretized system, which is thus partly linearized. After analyzing the discretized system and proving some results on the monotonicity of the operators and on the uniqueness of the solution, we prove sufficient conditions that ensure the convergence of this lagged method. We also describe the inner iteration and show how the weakly nonlinear systems arising at each lagged iteration can be solved efficiently. Finally, we analyze numerically the entire solution process by several numerical experiments.

H\xf6lder inequality applied on a non-Newtonian fluid equation with a nonlinear convection term and a source term

Consider a non-Newtonian fluid equation with a nonlinear convection term and a source term. The existence of the weak solution is proved by Simon’s compactness theorem. By the Hölder inequality, if both the diffusion coefficient and the convection term are degenerate on the boundary, then the stability of the weak solutions may be proved without the boundary value condition. If the diffusion coefficient is only degenerate on a part of the boundary value, then a partial boundary value condition is required. Based on this partial boundary, the stability of the weak solutions is proved. Moreover, the uniqueness of the weak solution is proved based on the optimal boundary value condition.

Blow-up rates and uniqueness of entire large solutions to a semilinear elliptic equation with nonlinear convection term

In this paper, we analyze the blow-up rates and uniqueness of entire large solutions to the equation Delta u=a(x)f(u)+ mu b(x) |nabla u|^{q}, xin mathbb{R}^{N} (Ngeq 3), where mu > 0, q > 0 and a, bin mathrm {C}^{alpha }_{mathrm{loc}}(mathbb{R}^{N}) (alpha in (0, 1)). The weight a is nonnegative, b is able to change sign in mathbb{R}^{N}, and fin C^{1}[0, infty ) is positive and nondecreasing on (0, infty ) and rapidly or regularly varying at infinity. Additionally, we investigate the uniqueness of entire large solutions.

Deconvolution-based stabilization of the incompressible Navier–Stokes equations

The numerical simulation of the incompressible Navier–Stokes equations may suffer from instabilities due to the energy cascades activating small-scale dynamics even when high-frequency components are not in the initial conditions. The energy cascade is generally triggered by the non-linear convective term. However, the process can be triggered by particular geometries, such that the instabilities occur even with relatively low Reynolds numbers. While instabilities for high convective fields are well known and investigated (for instance, in the framework of the Variational Multiresolution formulation designed by T.J. Hughes and his collaborators), numerical stabilization of low-convection instabilities is less investigated. In this paper, we present a novel method where the backbone of classical stabilizations is merged with a localization of the potentially unstable regions by means of a deconvolution-filter indicator inspired by the Large Eddie Simulation (LES) turbulence modeling advocated by W. Layton and his collaborators. We introduce the method for steady incompressible Navier–Stokes problems, we motivate the design of the method in two different variants, the Streamline-diffusion one and the strongly consistent one. We provide some analysis of the approach and numerical results proving the improved performances in comparison with classical schemes.

A dynamical study of certain nonlinear diffusion–reaction equations with a nonlinear convective flux term

We explore the dynamics of quadratic and quartic nonlinear diffusion–reaction equations with nonlinear convective flux term, which arise in well-known physical and biological problems such as population dynamics of the species. Three integration techniques, namely the \(({G^\prime }/{G})\)-expansion method, its generalised version and Kudryashov method, are adopted to solve these equations. We attain new travelling and solitary wave solutions in the form of Jacobi elliptic functions, hyperbolic functions, trigonometric functions and rational solutions with some constraint relations that naturally appear from the structure of these solutions. The travelling population fronts, which are the general solutions of nonlinear diffusion–reaction equations, describe the species invasion if higher population density corresponds to the species invasion. This effort highlights the significant features of the employed algebraic approaches and shows the diversity in the constructed solutions.

Stability of the ale space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains

The paper is concerned with the analysis of the space-time discontinuous Galerkin method (STDGM) applied to the numerical solution of nonstationary nonlinear convection-diffusion initial- boundary value problem in a time-dependent domain. The problem is reformulated using the arbitrary Lagrangian-Eulerian (ALE) method, which replaces the classical partial time derivative by the so-called ALE derivative and an additional convective term. The problem is discretized with the use of the ALE- space time discontinuous Galerkin method (ALE-STDGM). In the formulation of the numerical scheme we use the nonsymmetric, symmetric and incomplete versions of the space discretization of diffusion terms and interior and boundary penalty. The nonlinear convection terms are discretized with the aid of a numerical flux. The main attention is paid to the proof of the unconditional stability of the method. An important step is the generalization of a discrete characteristic function associated with the approximate solution and the derivation of its properties.

A two-grid discontinuous Galerkin method for a kind of nonlinear parabolic problems

A discontinuous Galerkin approximation for the space variables with the backward Euler time discretisation for a kind of parabolic problems with the nonlinear diffusion, convection and source terms is investigated. To solve the strongly nonlinear algebra system, a two-grid method is proposed. With this algorithm, solving a nonlinear system on the fine discontinuous finite element space is reduced into solving a nonlinear problem on a coarse gird of size and solving a linear problem on a fine grid of size. Convergence estimates in H1-norm are obtained. The numerical experiments are provided to confirm our theoretical analysis.

A divergence-free semi-implicit finite volume scheme for ideal, viscous, and resistive magnetohydrodynamics.

SummaryIn this paper, we present a novel pressure‐based semi‐implicit finite volume solver for the equations of compressible ideal, viscous, and resistive magnetohydrodynamics (MHD). The new method is conservative for mass, momentum, and total energy, and in multiple space dimensions, it is constructed in such a way as to respect the divergence‐free condition of the magnetic field exactly, also in the presence of resistive effects. This is possible via the use of multidimensional Riemann solvers on an appropriately staggered grid for the time evolution of the magnetic field and a double curl formulation of the resistive terms. The new semi‐implicit method for the MHD equations proposed here discretizes the nonlinear convective terms as well as the time evolution of the magnetic field explicitly, whereas all terms related to the pressure in the momentum equation and the total energy equation are discretized implicitly, making again use of a properly staggered grid for pressure and velocity. Inserting the discrete momentum equation into the discrete energy equation then yields a mildly nonlinear symmetric and positive definite algebraic system for the pressure as the only unknown, which can be efficiently solved with the (nested) Newton method of Casulli et al. The pressure system becomes linear when the specific internal energy is a linear function of the pressure. The time step of the scheme is restricted by a CFL condition based only on the fluid velocity and the Alfvén wave speed and is not based on the speed of the magnetosonic waves. Being a semi‐implicit pressure‐based scheme, our new method is therefore particularly well suited for low Mach number flows and for the incompressible limit of the MHD equations, for which it is well known that explicit density‐based Godunov‐type finite volume solvers become increasingly inefficient and inaccurate because of the more and more stringent CFL condition and the wrong scaling of the numerical viscosity in the incompressible limit. We show a relevant MHD test problem in the low Mach number regime where the new semi‐implicit algorithm is a factor of 50 faster than a traditional explicit finite volume method, which is a very significant gain in terms of computational efficiency. However, our numerical results confirm that our new method performs well also for classical MHD test cases with strong shocks. In this sense, our new scheme is a true all Mach number flow solver.

An energy-conserving ultra-weak discontinuous Galerkin method for the generalized Korteweg–de Vries equation

Abstract We propose an energy-conserving ultra-weak discontinuous Galerkin (DG) method for the generalized Korteweg– de Vries (KdV) equation in one dimension. Optimal a priori error estimate of order k + 1 is obtained for the semi-discrete scheme for the KdV equation without the convection term on general nonuniform meshes when polynomials of degree k ≥ 2 are used. We also numerically observe optimal convergence of the method for the KdV equation with linear or nonlinear convection terms. It is numerically observed for the new method to have a superior performance for long-time simulations over existing DG methods.

Inlet flow disturbance effects on direct numerical simulation of incompressible round jet at Reynolds number 2500

This letter reports inlet flow disturbance effects on direct numerical simulation of incompressible round jet at Reynolds number 2500. The simulation employs an accurate projection method in which a sixth order biased upwind difference scheme is used for spatial discretization of nonlinear convective terms, with a fourth order central difference scheme used in the discretization of the divergence of intermediate velocity. Carefully identifying reveals that the inlet flow disturbance has some influences on the distribution pattern of mean factor of swirling strength intermittency. With the increase of inlet disturbance magnitude jet core cone slightly shortens, observable differences occur in the centerline velocity and its fluctuations, despite the negligible impacts on the least square fitted centerline velocity decay constant (Bu) and distribution parameter (Ku) for velocity profile in self-similar region.

A posteriori sub-cell finite volume limiting of staggered semi-implicit discontinuous Galerkin schemes for the shallow water equations

We propose a novel family of a posteriori sub-cell finite volume limiters for spatially high order accurate semi-implicit discontinuous Galerkin (DG) schemes on staggered Cartesian grids for the solution of the shallow water equations expressed in conservative form in one and two space dimensions.We start from the unlimited arbitrary high order accurate staggered semi-implicit DG scheme proposed by Dumbser and Casulli (2013). In this method, the continuity equation and the momentum equations are integrated using a discontinuous finite element strategy on staggered control volumes, where the discrete free surface elevation is defined on the main grid and the discrete momentum is defined on edge-based staggered dual control volumes. In the semi-implicit approach, pressure terms are discretized implicitly, while the nonlinear convective terms are discretized explicitly. Inserting the momentum equations into the discrete continuity equation leads to a well conditioned block penta-diagonal linear system for the free surface elevation which can be efficiently solved with modern iterative methods. However, according to Godunov's theorem, any unlimited high order scheme inevitably produces spurious oscillations in the vicinity of discontinuities and strong gradients.In the present paper, we therefore propose to extend the successful family of a posteriori subcell finite volume limiters recently introduced by Dumbser et al. (2014) for explicit DG schemes also to semi-implicit time discretizations. At time tn the unlimited DG scheme is run in order to produce a so-called candidate solution for time tn+1. Then, the cells characterized by a non-admissible candidate solution are found by using physical and numerical detection criteria based on the positivity of the solution, the absence of floating point errors and the use of a relaxed discrete maximum principle (DMP) according to the MOOD strategy of Clain, Loubère and Diot (2013). In all the cells that are flagged as troubled control volumes a more robust semi-implicit finite volume (FV) method is then applied on a sub-grid composed of 2P+1 cells, where P denotes the polynomial degree used for approximating the discrete solution within the DG scheme. Then, after having identified the troubled cells, the linear system for the new free surface elevation is assembled and solved again, where unlimited cells use the high order semi-implicit DG scheme and limited cells are evolved via the more robust finite volume method. Finally, from the subcell finite volume averages a higher order DG polynomial is reconstructed and the scheme proceeds with the next time step.We apply the new semi-implicit staggered DG method with a posteriori subcell FV limiter to classical benchmarks such as Riemann problems in 1D and circular dam-break problems in 2D with shock waves, showing that the new subcell finite volume limiter is able to resolve shocks accurately without producing spurious oscillations. Moreover, if the solution is smooth, the detector does not find any troubled cells, as expected; consequently, the limiter is not activated and the method corresponds to the unlimited staggered semi-implicit DG scheme. In addition, we carry out numerical tests which show that the new scheme is well-balanced and able to deal with wet and dry fronts.

Free boundary problem of a reaction–diffusion equation with nonlinear convection term

In this paper, we consider the free boundary problem of a reaction diffusion equation with nonlinear convection term in one dimensional space. Our study contains three parts: in the first part we establish the existence and uniqueness of global solution, in the second part we obtain the spreading–vanishing dichotomy, and in third part, we obtain some estimations of the asymptotic speed of free boundaries when spreading happens.

Direct Numerical Simulation of Particles in Spatially Varying Electric Fields †

A numerical scheme is developed to simulate the motion of dielectric particles in the uniform and nonuniform electric fields of microfluidic devices. The motion of particles is simulated using a distributed Lagrange multiplier method (DLM) and the electric force acting on the particles is calculated by integrating the Maxwell stress tensor (MST) over the particle surfaces. One of the key features of the DLM method used is that the fluid-particle system is treated implicitly by using a combined weak formulation, where the forces and moments between the particles and fluid cancel, as they are internal to the combined system. The MST is obtained from the electric potential, which, in turn, is obtained by solving the electrostatic problem. In our numerical scheme, the domain is discretized using a finite element scheme and the Marchuk-Yanenko operator-splitting technique is used to decouple the difficulties associated with the incompressibility constraint, the nonlinear convection term, the rigid-body motion constraint and the electric force term. The numerical code is used to study the motion of particles in a dielectrophoretic cage which can be used to trap and hold particles at its center. If the particles moves away from the center of the cage, a resorting force acts on them towards the center. The MST results show that the ratio of the particle-particle interaction and dielectrophoretic forces decreases with increasing particle size. Therefore, larger particles move primarily under the action of the dielectrophoretic (DEP) force, especially in the high electric field gradient regions. Consequently, when the spacing between the electrodes is comparable to the particle size, instead of collecting on the same electrode by forming chains, they collect at different electrodes.

Numerical study on the propagation of solitary waves in the near-shore

The propagation and transformation of water waves in the near-shore region lead complicated wave-current field, which plays a significant role in the study of the mechanism of sediment transport and coastal evolution. Solitary wave is supposed as an initially incoming wave because of its widespread use in the near-shore. The Reynolds-averaged Navier-Stokes (RANS) equations were incorporated into a k-ε turbulence model and applied to simulate the run-up and hydrodynamic characteristics of solitary waves under the influences of different incident wave height and beach slope. The simulation results were validated with theoretical and experimental results from the literature. The effects of the non-linear terms associated with the curl of the fluid, Reynolds stresses and viscous stresses in the governing equations of fluid as well as the turbulent production term, convection term and diffusion term in the k-ε model were analyzed. The analysis shows that wave breaking is of great importance to the hydrodynamic characteristics of the wave-current field in the near-shore. The run-up heights of breaking solitary waves increase with beach slope, and the velocity and vorticity reach the maximum after wave breaking. The non-linear (curl and convection) terms and turbulent production term contribute most to the nonlinearities of the near-shore hydrodynamics, and increase with beach slope under these conditions. Nevertheless, under the conditions of wave non-breaking, the wave run-up height, the maximum of velocity and vorticity decrease with increasing beach slope. Meanwhile, the effects of fluid diffusion and Reynolds stresses increase gradually with increasing beach slope.

Stability and dynamics of a chemotaxis system with deformed free-surface in a shallow chamber

Complex bioconvection patterns have been studied analytically, experimentally, and numerically previously only for a flat free-surface of a suspension of chemotaxis bacteria in a shallow/deep chamber. In this paper, we have considered a two-dimensional chemotaxis-diffusion-convection system with a deformed free-surface. The influence of aggregation of chemotactic cells on the deformed free-surface of a shallow chamber is studied analytically. The aim of this paper is to investigate the nature of the instability in the chemotaxis-diffusion-convection system. We performed a detailed linear stability analysis of a steady-state cell and oxygen concentration distribution. The system becomes dominated by nonlinear convection terms beyond a critical Rayleigh number Raτ, which also depends on the critical wavenumber k as well as the other parameters. We have investigated that how the critical Rayleigh number in this system varies with three different sets of parameters. A weakly nonlinear analysis is carried out as well to determine the relative stability of the pattern formation at the onset of instability. A reactance between rolls, squares, hexagons, and mixed mode pattern is investigated in detail. Further research should link the weakly nonlinear analysis with the bifurcation analysis. Some important direct numerical simulation results have been presented in the support of linear stability analysis. Comparison of the analytical steady-state solution shows good agreement with the numerical result.

Efficient monolithic projection method for time-dependent conjugate heat transfer problems

We propose herein an efficient monolithic projection method (MPM) to solve time-dependent conjugate heat transfer problems involving not only natural convection in the fluid domain and heat conduction in the solid domain, but also the thermal interaction between solid and fluid domains across the solid–fluid interface. We obtain a global discretized linearized system by advancing the buoyancy, nonlinear convection, and linear diffusion terms in time using the Crank–Nicolson scheme and introducing the second-order central finite difference in space along with linearizing the nonlinear convection terms in both momentum and energy equations. The energy equations are simultaneously and implicitly discretized in both solid and fluid domains with the implemented Taylor series expansion for thermal interaction normal to the interface without an involved sub-time step iteration. Approximated lower–upper decompositions and an approximate factorization are also imposed to speed up the computation. Thus, we obtain a non-iterative monolithic projection method over the entire domain. Numerical simulations of two-dimensional (2D) conjugate natural convection and 2D conjugate Rayleigh–Bénard convection and periodic forced flows are performed to investigate the numerical performance of the proposed method. Consequently, the MPM correctly predicts the solution of the conjugate natural convection problem involving strong thermal interactions and provides a more stable and efficient computation than the semi-implicit projection method proposed by Kim and Moin (1985) [21] with a loosely or strongly coupled algorithm for the solid–fluid interface, while preserving the second-order temporal and spatial accuracy. Finally, the proposed method reasonably simulates a typical real-world problem, namely conjugate heat transfer through double-pane windows, by considering 2D heat conduction in each pane of glass for three different climatic conditions. Using the proposed MPM, we also investigate the effects of the air layer thickness ranging from 5 mm to 40 mm on the averaged Nusselt number and the distribution of temperature as well as fluid motion.

Three-dimensional through-flow modelling of axial flow compressor rotating stall and surge

This paper presents a three-dimensional through-flow approach based on the cylindrical Euler equations incorporating a body force method. Blade performance is captured through a mixture of empirical correlations and a novel reverse flow treatment. The code is the first application of a physically correct Godunov solver to three-dimensional rotating stall and surge modelling. This solver ensures the accurate calculation of inter-cell fluxes unlike in typical modern CFD codes in which the non-linear convective terms are linearised. Validation consists of modelling a low speed three-stage axial compressor in all operating regions, recreating the reverse flow, rotating stall and forward flow characteristics with good agreement to experimental data. Additional comparisons are made against rotating stall cell size and speed, to which good agreement is also shown. The paper ends with some full surge cycle simulations modifying both the tank volume after the compressor and the level of inlet distortion applied. Both tank volume and level of distortion have been found to affect the type of instability developed. The development of this code is a step forward in compressor rotating stall and reverse flow modelling and allows recreation of a full compressor map at a significantly low computational cost when compared to commercially available 3D CFD codes.