A New Reduced-Dimension Iteration Two-Grid Crank–Nicolson Finite-Element Method for Unsaturated Soil Water Flow Problem

This paper presents a proper orthogonal decomposition (POD) method that uses dynamic basis functions. The dynamic functions are of a prescribed form and do not explicitly depend on time but rather on parameters associated with flow unsteadiness. This POD method has been developed for modeling nonlinear flows with deforming meshes but can also be applied to fixed meshes. The method is illustrated for subsonic and transonic flows in channels with fixed and deforming meshes. This method properly captured flow nonlinearities and shock motion for cases in which the classical POD method failed.

Using the Richard's equation in the Fokker‐Planck nonlinear diffusion form, unsaturated soil water flow may be treated as a diffusion‐convection wave process. If ∂θ/∂z is assumed a function of θ alone, the unsaturated flow equation may be solved by the method of characteristics, and when ∂θ/∂z becomes sufficiently small, the Peclet number is assumed large enough to treat unsaturated flow kinematically. Changes in θ with depth in the soil profile are treated as waves, moving downward. Advancing and receding “waves” are treated differently in the approximate analytical technique described here, with advancing wetting fronts described by kinematic “shocks.” The method is compared to the complete solution to Richard's equation for a complex rain pattern and found to predict well the location of deeper moving fronts and also general θ patterns. The kinematic method is also shown to apply to root water extraction zones and to layered soil situations.

The tangential impingement of turbulent circular jets on the cylindrical surface in an annular chamber is numerically investigated based on the SAS-SST model. The main purpose is to understand the role of the coherent structures on the temperature field inside the chamber and the heat transfer coefficient field on the hub surface. Based on the jet diameter and inlet velocity, the jet Reynolds numbers are given as 285000, 332000, and 392000, respectively. The accuracy of the numerical simulation is verified by the experimental data. The spectrum information of the velocity magnitude at the monitor point is obtained by the fast Fourier transform (FFT) method. An improvement of the proper orthogonal decomposition (POD) method is known as the extended POD (EPOD) method, which is mainly suitable for analyzing the correlation between the flow field and the synchronized scalar or vector fields, such as the correlation between the velocity and the heat transfer coefficient fields. In present study, both the POD method and the EPOD method are applied and compared. The POD method allows to visualize the main modes in terms of velocity, temperature, and heat transfer coefficient fluctuations. With the help of the EPOD method, the contribution of velocity fluctuations to temperature fluctuations in the chamber and heat transfer fluctuations on the hub surface are further analyzed. For Rej = 392000, the first two velocity POD modes and the velocity EPOD modes that contribute the most to the temperature fluctuations are very similar, reflecting the same large-scale coherent structures. The first two velocity EPOD modes that contribute the most to the heat transfer coefficient fluctuations have a high proportion of energy in the flow field, corresponding to the deflection motion of the jet.

We analyze the combined effects of the spatial variability of soil hydraulic properties and the water uptake by plant roots on unsaturated water flow. For this analysis, we use a simplified macroscopic root water uptake model which is usually applied only for homogeneous or layered soil and therefore we also determine whether it is applicable for multidimensional heterogeneous media. Analytical solutions for mean and variance of pressure head (first‐order second‐moment approximations) in layered media and numerical solutions of two‐dimensional (2‐D) autocorrelated multi‐Gaussian and non multi‐Gaussian parameter fields are analyzed for steady state and transient flow conditions. For non‐Gaussian topological features, that have little influence on the mean and the variance of the pressure field if root water uptake is ignored, we test whether the influence is significant if root water uptake is accounted for. The results reveal that, in field structures with large patches of coarse material, local regions with pressure head values at the wilting point develop; these are surrounded by wet material. Without a compensation mechanism for local stress, the global transpiration demand is not met if local wilting occurs. Various compensation mechanisms are tested that depend, respectively, on the saturation, the relative conductivity or a strategy where the deficit in the global uptake rate is equally distributed to unstressed locations. The strategies lead to a global actual transpiration rate at the potential value and attenuate the formation of locally wilted areas. Wilted regions can, however, still occur, and may be an artifact of the simplified model concept as root‐soil interactions are neglected. Therefore simplified macroscopic models for root water uptake should be used with caution in heterogeneous media.

Bathe algorithm converts problem of determining inner infiltration face into a material nonlinearity problem, and Signorini condition coverts outflow boundary into conventional water boundary. Because an unsaturated seepage problem is also a nonlinear flow problem, through the joint application of this 2 methods, solving variational inequalities is avoided. In conventional material nonlinearity finite element framework with small modification, using NR algorithm, the solving of unconfined water seepage problems including unsteady and unsaturated water seepage problems is realized. The followings are discussed: (1) the improvement of convergence of Bathe method and its generalized form in a three-dimensional model; (2) use Switching Algorithm of Signorini condition to solve water seepage problems; (3) the under-relaxation methods of improving the convergence and mass conservation when solving unsaturated and unsteady water seepage problems. At last, through classic calculation examples, the calculating accuracy and application fields of the proposed methods are evaluated.

PurposeWind loading calculations are currently performed according to the ASCE 7 standard. Values in this standard were estimated from simplified models that do not necessarily take into account relevant flow characteristics. Thus, the standard does not have provisions to handle the majority of rooftop photovoltaic (PV) systems. Accurate solutions for this problem can be produced using a full-fledged three-dimensional computational fluid dynamics (CFD) analysis. Unfortunately, CFD requires enormous computation times, and its use would be unsuitable for this application which requires real-time solutions. To this end, a real-time response framework based on the proper orthogonal decomposition (POD) method is proposed.Design/methodology/approachA real-time response framework based on the POD method was used. This framework used beforehand and off-line CFD solutions from an extensive data set developed using a predefined design space. Solutions were organized to form the basis snapshots of a POD matrix. The interpolation network using a radial-basis function (RBF) was used to predict the solution from the POD method given a set of values of the design variables. The results presented assume varying design variables for wind speed and direction on typical PV roof installations.FindingsThe trained POD–RBF interpolation network was tested and validated by performing the fast-algebraic interpolation to obtain the pressure distribution on the PV system surface and they were compared to actual grid-converged fully turbulent 3D CFD solutions at the specified values of the design variables. The POD network was validated and proved that large-scale CFD problems can be parametrized and simplified by using this framework.Originality/valueThe solar power industry, engineering design firms and the society as a whole could realize significant savings with the availability of a real-time in situ wind-load calculator that can prove essential for plug-and-play installation of PV systems. Additionally, this technology allows for automated parametric design optimization to arrive at the best fit for a set of given operating conditions. All these tasks are currently prohibited because of the massive computational resources and time required to address large-scale CFD analysis problems, all made possible by a simple but robust technology that can yield massive savings for the solar industry.

Evaluation of fully-coupled strata deformation and groundwater flow in response to longwall mining

As an initial part of a performance assessment task for the Hanford Waste Management Plan, we have reviewed research done over a number of years on unsaturated water flow at the Hanford site near Richland, Washington. This work will be helpful in assessing the probability that water infiltrating the ground surface may eventually contribute to ground-water recharge at the site. Reports done primarily by Atlantic Richfield Hanford, Rockwell Hanford Operations, and Pacific Northwest Laboratory were reviewed for their pertinence to unsaturated water flow and specifically for information that could help resolve the question of whether recharge of the unconfined aquifer by natural precipitation (meteoric water) is occurring on the Hanford site and, if so, what are the expected ranges of recharge. The reports that we reviewed are primarily technical reports detailing the physical and hydrologic characteristics of soils and sediments at specific locations on the Hanford site, or describing studies that have either monitored or simulated such parameters as surface evaporation, soil moisture storage, and deep drainage (recharge). The appendix of this report provides abstracts and annotations on 24 key technical reports on subjects related to estimation of recharge at Hanford. 66 refs.

A reduced-dimension method for unknown Crank-Nicolson finite element solution coefficient vectors of elastic wave equation with singular source term

To analyze the complex spatiotemporal evolution law of the high-speed jet flow field in a liquid-ring pump ejector, both the classic and spectral proper orthogonal decomposition (SPOD) methods were introduced to decompose the transient flow field based on the numerical large eddy simulation. The proper orthogonal decomposition (POD) reduced order model (ROM) of the complex flow in the ejector was established to analyze the transient characteristics and reconstruct the flow field. The spatial mode characteristics of density, pressure and vorticity field, and the frequency domain characteristics of mode coefficients were analyzed. The results show that the spatiotemporal decoupling analysis of a high-speed jet flow field can be accomplished by the POD and SPOD methods. A large-scale symmetric coherent structure is formed in the shear layer of the first two modes of vorticity, and a small-scale antisymmetric structure is formed at the trailing edge of the jet flow for the third and fourth modes. This reflects the evolutionary characteristics of the jet shear vortex. There is a significant spatial correlation between the density and pressure modes, reflecting that the evolution of the two structures has important correlation characteristics. The spectral analysis of the mode coefficients shows that the dominant frequencies of 1250 Hz and 18 kHz correspond to the self-excited oscillation frequency of shock waves and the vortex shedding frequency. The POD-ROM can reconstruct the high-speed flow field of the ejector with enough accuracy using the first 30-order modes, and the root mean square error and mean absolute percentage error for the velocity prediction are 0.48% and 0.24%, respectively.

Use of soil physical characteristics from laboratory measurements or standard series for modelling unsaturated water flow

Dynamic response analysis of a double-disc rotor system with rolling bearings based on POD method

We have analyzed the computational dynamics of a complex one-dimensional structural system consisting of a number of alternating stiff and soft subsystems. In particular, the method of Proper Orthogonal Decomposition (POD) in the frequency domain has been applied to analyze the (single-frequency) steady state dynamics in terms of spectral amplitudes and spatial shapes of proper orthogonal modes. It is shown that the dominant POD modes of such a multi-body system are sensitive to imperfections. The processing of the computational dynamics by the spectral-POD method leads to a novel computation of the transfer function of the system.

A proper orthogonal decomposition (POD) method is used to interpolate the flow around an airfoil for various Mach numbers and angles of attack in the transonic regime. POD uses a few numerical simulations, called snapshots, to create eigenfunctions. These eigenfunctions are combined using weighting coefficients to create a new solution for different values of the input parameters. Since POD methods are linear, their interpolation capabilities are quite limited when dealing with flow presenting nonlinearities, such as shocks. In order to improve their performance for cases involving shocks, a new method is proposed using variable fidelity. The main idea is to use POD to interpolate the difference between the CFD solution obtained on two different grids, a coarse one and a fine one. Then, for any new input parameter value, a coarse grid solution is computed using CFD and the POD interpolated difference is added to predict the fine grid solution. This allows some nonlinearities associated with the flow to be introduced. Results for various Mach numbers and angles of attack are compared to full CFD results. The variable fidelity-based POD method shows good improvement over the classical approach.

This study presents experimental investigations of entrainment events of a single sediment particle resting in a small pocket on a smooth bed in an open channel flow. The data were acquired with a tomographic particle tracking velocimetry system. The shake the box algorithm, a spatially high resolved Lagrangian tracking method, was applied to determine time-resolved and three-dimensional flow velocities in a volume during particle entrainment. The proper orthogonal decomposition (POD) method was applied to identify motions in the flow field carrying the most turbulent kinetic energy (TKE). Based on the POD method, the most energetic flow structures were linked with the occurring quadrant events at particle entrainment. It was shown that particle entrainment follows TKE peaks based on low-order POD modes caused by large sweeps. Two streamwise elongated counter rotating vortices with emerging sweeps in between were observed during particle entrainment. Here, most of the TKE exists in the first POD mode. This signature is suggested to be part of very large-scale coherent motions.