Assessment of Different CFD Modeling and Solving Approaches for a Supersonic Steam Ejector Simulation

The effects of different modeling and solving approaches on the simulation of a steam ejector have been investigated with the computational fluid dynamics (CFD) technique. The four most frequently used and recommended turbulence models (standard k-ε, RNG k-ε, realizable k-ε and SST k-ω), two near-wall treatments (standard wall function and enhanced wall treatment), two solvers (pressure- and density-based solvers) and two spatial discretization schemes ( the second-order upwind scheme and the quadratic upstream interpolation for convective kinematics (QUICK) of the convection term have been tested and compared for a supersonic steam ejector under the same conditions as experimental data. In total, more than 185 cases of 17 different modeling and solving approaches have been carried out in this work. The simulation results from the pressure-based solver (PBS) are slightly closer to the experimental data than those from the density-based solver (DBS) and are thus utilized in the subsequent simulations. When a high-density mesh with y+ < 1 is used, the SST k-ω model can obtain the best predictions of the maximum entrainment ratio (ER) and an adequate prediction of the critical back pressure (CBP), while the realizable k-ε model with the enhanced wall treatment can obtain the best prediction of the CBP and an adequate prediction of the ER. When the standard wall function is used with the three k-ε models, the realizable k-ε model can obtain the best predictions of the maximum ER, and the three k-ε models can gain the same CBP value. For a steam ejector with recirculation inside the diffuser, the realizable k-ε model or the enhanced wall treatment is recommended for adoption in the modeling approach. When the spatial discretization scheme of the convection term changes from a second-order upwind scheme to a QUICK scheme, the effect can be ignored for the maximum ER calculation, while only the CBP value from the standard k-ε model with the standard wall function is reduced by 2.13%. The calculation deviation of the ER between the two schemes increases with the back pressure at the unchoked flow region, especially when the standard k-ε model is adopted. The realizable k-ε model with the two wall treatments and the SST k-ω model is recommended, while the standard k-ε is more sensitive to the near-wall treatment and the spatial discretization scheme and is not recommended for an ejector simulation.

Nonlinear Nonhomogeneous Obstacle Problems with Multivalued Convection Term

In this paper, a nonlinear elliptic obstacle problem is studied. The nonlinear nonhomogeneous partial differential operator generalizes the notions of p-Laplacian while on the right hand side we have a multivalued convection term (i.e., a multivalued reaction term may depend also on the gradient of the solution). The main result of the paper provides existence of the solutions as well as bondedness and closedness of the set of weak solutions of the problem, under quite general assumptions on the data. The main tool of the paper is the surjectivity theorem for multivalued functions given by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone one.

The Green's function for Caputo fractional boundary value problem with a convection term

<abstract><p>By using the operator theory, we establish the Green's function for Caputo fractional differential equation under Sturm-Liouville boundary conditions. The results are new, the method used in this paper will provide some new ideas for the study of this kind of problems and easy to be generalized to solving other problems.</p></abstract>

Implicit-explicit multistep formulations for finite element discretisations using continuous interior penalty

We consider a finite element method with symmetric stabilisation for the discretisation of the transient convection--diffusion equation. For the time-discretisation we consider either the second order backwards differentiation formula or the Crank-Nicolson method. Both the convection term and the associated stabilisation are treated explicitly using an extrapolated approximate solution. We prove stability of the method and the $\tau^2 + h^{p+{\frac12}}$ error estimates for the $L^2$-norm under either the standard hyperbolic CFL condition, when piecewise affine ($p=1$) approximation is used, or in the case of finite element approximation of order $p \ge 1$, a stronger, so-called $4/3$-CFL, i.e. $\tau \leq C h^{4/3}$. The theory is illustrated with some numerical examples.

Investigation on the effects of vertical baffles on liquid sloshing based on a particle method

This study adopts the numerical simulations of Moving Particle Semi-Implicit Methods (MPS), which are meshless methods based on Lagrange particles. Using Lagrange particle has an advantage that it can avoid numerical dissipation problems without directly discretizing the convection term in the governing equation. First of all, a numerical model of a liquid sloshing tank without baffles is used to confirm the effectiveness of the MPS by comparing the numerical results with the experimental data of Kang and Li. And the pressure curves obtained with MPS method were in good agreement with the experimental findings, which confirmed its effectiveness. On that basis, simulations of liquid sloshing movements with one baffle, two symmetrical baffles, and three baffles are performed, respectively. The results indicate that the addition of vertical baffles in the tanks effectively enhanced the ability to reduce liquid sloshing.

Degenerated and Competing Dirichlet Problems with Weights and Convection

This paper focuses on two Dirichlet boundary value problems whose differential operators in the principal part exhibit a lack of ellipticity and contain a convection term (depending on the solution and its gradient). They are driven by a degenerated (p,q)-Laplacian with weights and a competing (p,q)-Laplacian with weights, respectively. The notion of competing (p,q)-Laplacians with weights is considered for the first time. We present existence and approximation results that hold under the same set of hypotheses on the convection term for both problems. The proofs are based on weighted Sobolev spaces, Nemytskij operators, a fixed point argument and finite dimensional approximation. A detailed example illustrates the effective applicability of our results.

Homogenization of a nonlinear elliptic system with a transport term in L 2

We consider the homogenization of a non-linear elliptic system of two equations related to some models in chemotaxis and flows in porous media. One of the equations contains a convection term where the transport vector is only in L 2 and a right-hand side which is only in L 1 . This makes it necessary to deal with entropy or renormalized solutions. The existence of solutions for this system has been proved in reference (Comm. Partial Differential Equations 45(7) (2020) 690–713). Here, we prove its stability by homogenization and that the correctors corresponding to the linear diffusion terms still provide a corrector for the solutions of the non-linear system.

On the Estimates in Various Spaces to the Control Function of the Extremum Problem for Parabolic Equation

For the minimization problem with pointwise observation governed by a one-dimensional parabolic equation with a free convection term and a depletion potential, we formulate a result on the existence and uniqueness of a minimizer from a prescribed set. We use a weight quadratic cost functional showing the temperature deviation. We obtain estimates for the norm of control functions in terms of the value of the quality functional in different functional spaces. It gives us a possibility to estimate the required internal energy of the system. To prove these results we establish the positivity principle.

SUPG-stabilized virtual elements for diffusion-convection problems: a robustness analysis

The objective of this contribution is to develop a convergence analysis for SUPG-stabilized Virtual Element Methods in diffusion-convection problems that is robust also in the convection dominated regime. For the original method introduced in [Benedetto et al, CMAME 2016] we are able to show an “almost uniform” error bound (in the sense that the unique term that depends in an unfavourable way on the parameters is damped by a higher order mesh-size multiplicative factor). We also introduce a novel discretization of the convection term that allows us to develop error estimates that are fully robust in the convection dominated cases. We finally present some numerical result.