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.

Uniformly Convergent Numerical Scheme for Singularly Perturbed Parabolic PDEs with Shift Parameters

In this article, singularly perturbed parabolic differential difference equations are considered. The solution of the equations exhibits a boundary layer on the right side of the spatial domain. The terms containing the advance and delay parameters are approximated using Taylor series approximation. The resulting singularly perturbed parabolic PDEs are solved using the Crank–Nicolson method in the temporal discretization and nonstandard finite difference method in the spatial discretization. The existence of a unique discrete solution is guaranteed using the discrete maximum principle. The uniform stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. The scheme converges uniformly with the order of convergence O N − 1 + Δ t 2 , where N is number of subintervals in spatial discretization and Δ t is mesh length in temporal discretization. Two test numerical examples are considered to validate the theoretical findings of the scheme.

A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

A space-time fully decoupled wavelet integral collocation method (WICM) with high-order accuracy is proposed for the solution of a class of nonlinear wave equations. With this method, wave equations with various nonlinearities are first transformed into a system of ordinary differential equations (ODEs) with respect to the highest-order spatial derivative values at spatial nodes, in which all the matrices in the resulting nonlinear ODEs are constants over time. As a result, these matrices generated in the spatial discretization do not need to be updated in the time integration, such that a fully decoupling between spatial and temporal discretization can be achieved. A linear multi-step method based on the same wavelet approximation used in the spatial discretization is then employed to solve such a semi-discretization system. By numerically solving several widely considered benchmark problems, including the Klein/sine–Gordon equation and the generalized Benjamin–Bona–Mahony–Burgers equation, we demonstrate that the proposed wavelet algorithm possesses much better accuracy and a faster convergence rate than many existing numerical methods. Most interestingly, the space-associated convergence rate of the present WICM is always about order 6 for different equations with various nonlinearities, which is in the same order with direct approximation of a function in terms of the proposed wavelet approximation scheme. This fact implies that the accuracy of the proposed method is almost independent of the equation order and nonlinearity.

Deconvolution of induced spatial discretization filters subgrid modeling in LES: application to two-dimensional turbulence

The paper presents a new approximate deconvolution subgrid model for Large Eddy Simulation in which corrections to implicit filtering due to spatial discretization are integrated explicitly. The top-hat filter implied by second-order central finite differencing is a key example, which is discretised using the discrete Fourier transform involving all the mesh points in the computational domain. This discrete filter kernel is inverted by inverse Wiener filtering. The inverse filter obtained in this way is used to deconvolve the resolved scales of the implicitly filtered velocity field on the computational grid. Subgrid stresses are subsequently calculated directly from the deconvolved velocity field. The model was applied to study decaying two-dimensional turbulence. Results were compared with predictions based on the Smagorinsky model and the dynamic Germano model. A posteriori testing in which Large Eddy Simulation is compared with filtered Direct Numerical Simulation obtained with a Fourier spectral method is included. The new model presented strictly speaking applies to periodic problems. The idea of recovering a high-order inversion of the numerically induced filter kernel can be extended to more general non-periodic problems, also in three spatial dimensions.

Spatio-Temporal Discretization Uncertainty of Distributed Hydrological Models

Quantifying the uncertainty linked to the degree to which the spatio-temporal variability of the catchment descriptors (CDs), and consequently calibration parameters (CPs), represented in the distributed hydrology models and its impacts on the simulation of flooding events is the main objective of this paper. Here, we introduce a methodology based on ensemble approach principles to characterize the uncertainties of spatio-temporal variations. We use two distributed hydrological models (WaSiM and Hydrotel) and six catchments with different sizes and characteristics, located in southern Quebec, to address this objective. We calibrate the models across four spatial (100, 250, 500, 1000 $m^2$) and two temporal (3 hours and 24 hours) resolutions. Afterwards, all combinations of CDs-CPs pairs are fed to the hydrological models to create an ensemble of simulations for characterizing the uncertainty related to the spatial resolution of the modeling, for each catchment. The catchments are further grouped into large ($>1000 km^2$), medium (between 500 and 1000 $km^2$) and small ($<500km^2$) to examine multiple hypotheses. The ensemble approach shows a significant degree of uncertainty (over $100\%$ error for estimation of extreme streamflow) linked to the spatial discretization of the modeling. Regarding the role of catchment descriptors, results show that first, there is no meaningful link between the uncertainty of the spatial discretization and catchment size, as spatio-temporal discretization uncertainty can be seen across different catchment sizes. Second, the temporal scale plays only a minor role in determining the uncertainty related to spatial discretization. Third, the more physically representative a model is, the more sensitive it is to changes in spatial resolution. Finally, the uncertainty related to model parameters is dominant larger than that of catchment descriptors for most of the catchments. Yet, there are exceptions for which a change in spatio-temporal resolution can alter the distribution of state and flux variables, change the hydrologic response of the catchments, and cause large uncertainties.

Spatial Discretization for Stochastic Semi-Linear Subdiffusion Equations Driven by Fractionally Integrated Multiplicative Space-Time White Noise

Spatial discretization of the stochastic semi-linear subdiffusion equations driven by fractionally integrated multiplicative space-time white noise is considered. The nonlinear terms f and σ satisfy the global Lipschitz conditions and the linear growth conditions. The space derivative and the fractionally integrated multiplicative space-time white noise are discretized by using the finite difference methods. Based on the approximations of the Green functions expressed by the Mittag–Leffler functions, the optimal spatial convergence rates of the proposed numerical method are proved uniformly in space under some suitable smoothness assumptions of the initial value.

Two Efficient Time-Marching Explicit Procedures Considering Spatially/Temporally-Defined Adaptive Time-Integrators

In this paper, two explicit time-marching techniques are discussed for the solution of hyperbolic models, which are based on adaptively computed parameters. In both these techniques, time integrators are locally and automatically evaluated, taking into account the properties of the spatially/temporally discretized model and the evolution of the computed responses. Thus, very versatile solution techniques are enabled, which allows computing highly accurate responses. Here, the so-called adaptive [Formula: see text] method is formulated based on the elements of the adopted spatial discretization (elemental formulation), whereas the so-called adaptive [Formula: see text] method is formulated based on the degrees of freedom of the discretized model (nodal formulation). In this context, each adaptive procedure can be better applied according to the specific features of the focused spatial discretization technique. At the end of the paper, numerical results are presented, illustrating the excellent performance of the discussed adaptive formulations.