Numerical Evaluation of the Three-Dimensional Searchlight Problem in a Half-Space

A spatial and angular moments analysis of the monoenergetic pencil beam problem

(1907). Summation of Certain Infinite Series. The American Mathematical Monthly: Vol. 14, No. 3, pp. 41-46.

Analytic solutions to a planetary boundary layer (PBL) model with an eddy-diffusivity profile (i.e., a K profile) and nonlocal fluxes are presented for the quasi-steady regime. The solutions demonstrate how different processes contribute to the quasi-steady profiles of heat and/or other scalars in the convective boundary layer. It is shown that for a standard cubic form of the K profile, and flux scales based on the surface fluxes, the nondimensional nonlocal term should be less than six; larger values can cause scalar profiles of water vapor to increase with height in the upper portion of the PBL and can produce weakly superadiabatic layers in the upper PBL temperature profiles. Solutions are also shown to be sensitive to the choice of flux scale: fluxes scaled by their vertically averaged values imply that nondimensional profiles of top-down scalars will have a neutral point somewhere in the PBL, a result in conflict with previous work on the subject, and the predictions of the same model with fluxes scaled by their surface values. The analysis also shows that allowing K to go to zero with the square of the distance from the PBL top results in nonconvergent profiles; in general K should reduce to some positive value at the top of the PBL, or go to zero less rapidly. It is further shown that the class of models investigated here may be physically interpreted as relaxation models, that is, they tend to relax profiles of scalars in the PBL to implicitly defined similarity profiles on a convective timescale. Finally, analysis of a 1-yr integration of a climate model, interpreted in light of the author’s analytic results, suggests that a dynamically important aspect of the nonlocal term is its role in ventilating the surface layer, and thereby indirectly affecting the diagnoses of PBL depth in many models.

Original objectives and revisions The original objectives of this research, as stated in the approved proposal were: 1. To establish guidelines for the use of turbulent transport techniques as accurate and reliable tool for continuous measurements of whole canopy ET and other scalar fluxes (e.g. heat and CO2) in large agricultural structures. 2. To conduct a detailed experimental study of flow patterns and turbulence characteristics in agricultural structures. 3. To derive theoretical models of air flow and scalar fluxes in agricultural structures that can guide the interpretation of TT measurements for a wide range of conditions. All the objectives have been successfully addressed within the project. The only modification was that the study focused on screenhouses only, while it was originally planned to study large greenhouses as well. This was decided due to the large amount of field and theoretical work required to meet the objectives within screenhouses. Background In agricultural structures such as screenhouses and greenhouses, evapotranspiration (ET) is currently measured using lysimeters or sap flow gauges. These measurements provide ET estimates at the single-plant scale that must then be extrapolated, often statistically or empirically, to the whole canopy for irrigation scheduling purposes. On the other hand, turbulent transport techniques, like the eddy covariance, have become the standard for measuring whole canopy evapotranspiration in the open, but their applicability to agricultural structures has not yet been established. The subject of this project is the application of turbulent transport techniques to estimate ET for irrigation scheduling within large agricultural structures. Major conclusions and achievements The major conclusions of this project are: (i) the eddy covariance technique is suitable for reliable measurements of scalar fluxes (e.g., evapotranspiration, sensible heat, CO2) in most types of large screenhouses under all climatic conditions tested. All studies resulted with fair energy balance closures; (ii) comparison between measurements and theory show that the model is capable in reliably predicting the turbulent flow characteristics and surface fluxes within screenhouses; (iii) flow characteristics within the screenhouse, like flux-variance similarity and turbulence intensity were valid for the application of the eddy covariance technique in screenhouses of relatively dilute screens used for moderate shading and wind breaking. In more dense screens, usually used for insect exclusions, development of turbulent conditions was marginal; (iv) installation of the sensors requires that the system’s footprint will be within the limits of the screenhouse under study, as is the case in the open. A footprint model available in the literature was found to be reliable in assessing the footprint under screenhouse conditions. Implications, both scientific and agricultural The study established for the first time, both experimentally and theoretically, the use of the eddy covariance technique for flux measurements within agricultural screenhouses. Such measurements, along with reliable theoretical models, will enable more accurate assessments of crop water use which may lead to improved crop water management and increased water use efficiency of screenhouse crops.

Closure models for the turbulent scalar flux are an important source of uncertainty in Reynolds-averaged-Navier-Stokes (RANS) simulations of scalar transport. This paper presents an approach to quantify this uncertainty in simulations of complex engineering flows. The approach addresses the uncertainty in modeling the pressure scrambling (PS) effect, which is the primary mechanism balancing the productions in scalar flux dynamics. Inspired by the two classical phenomenological theories of return-to-isotropy (RI) and isotropization-of-production (IP), we assume that the most likely directions of the PS term are around a fan-shaped region bounded by the RI and IP directions. Subsequently, we propose a strategy that requires two additional simulations, defining perturbations of the PS directions towards the RI and IP limits. The approach is applied to simulations of forced heat convection in a complex pin-fin array configuration, and shows favorable monotonic properties and bounding behaviors for various quantities of interest (QoIs) relevant to heat transfer. To conclude, the results are analyzed from the perspective of transverse scalar transport in a shear flow; the analysis indicates that the proposed approach is likely to exhibit monotonic behaviors in a wide range of scalar transport problems.

The authors discuss time-dependent integral transport equation single-collision kernels for one-dimensional geometries corresponding to the steady-state single-collision kernels found in the available literature calculated by making use of the Laplace transform technique, simple geometric transformation relationships, and point kernel integrations. Using the convolution theorem, the time-dependent scalar flux is obtained by convoluting the single-collision kernel with the time-dependent source. Using the multiple collision formulation of the integral transport solution isotropic sources thar are delta distributions in time are considered in several examples. Analytical solutions for the uncollided and first-collided scalar fluxes are obtained for a boundary source having an isotropic angular distribution directed into a semi-infinite medium and into a slab of thickness b and for a point source at the origin of an infinite medium and finite sphere of radius a.

The quasidiffusion (QD) method is an established and efficient iterative technique for solving particle transport problems. Each QD iteration consists of a high-order SN sweep, followed by a low-order QD calculation. QD has two defining characteristics: (1) its iterations converge rapidly for any spatial grid and (2) the converged scalar fluxes from the high-order SN sweep and the low-order QD calculation differ, by spatial truncation errors, from each other and from the scalar flux solution of the SN equations. In this paper, we show that by including a transport consistency factor in the low-order equation, the converged high-order and low-order scalar fluxes become equal to each other and to the converged SN scalar flux. However, the inclusion of the transport consistency factor has a negative impact on the convergence rate. We present numerical results that demonstrate the effect of the transport consistency factor on stability.

The analysis of characteristic methods in Cartesian geometry performed by M. L. Adams, T. A. Wareing and W. F. Walters showed that characteristic methods on triangular cells meet the asymptotic diffusion limit. We currently working on the development of a linear characteristic method in RZ geometry on triangular grids. The grid is formed on the basis of a rectangular grid in (r, z) space. Each rectangular cell of this grid is divided into four triangular cells. The discretization method approximates the angular flux in each triangular cell with linear functions along the cell edge and in the interior of the cell. The unknowns are edge-average and cell-average angular fluxes, first spatial moments of the angular flux over the cell and edges. The scattering term is interpolated with a linear function in each cell by means of the cell-average scalar flux and its linear spatial moments. The main research issue is the calculation of linear spatial moments of the scalar flux based on the linear spatial moments of the angular flux in a cell in (ξ, z) coordinates that will enable one to construct a linear approximation of the cell-wise scalar flux with desired properties.

<strong class="journal-contentHeaderColor">Abstract.</strong> Spatially representative estimates of surface energy exchange from field measurements are required for improving and validating Earth system models and satellite remote sensing algorithms. The scarcity of flux measurements can limit understanding of ecohydrological responses to climate warming, especially in remote regions with limited infrastructure. Direct field measurements often apply the eddy covariance method on stationary towers, but recently, drone-based measurements of temperature, humidity, and wind speed have been suggested as a viable alternative to quantify the turbulent fluxes of sensible (H) and latent heat (LE). A data assimilation framework to infer uncertainty-aware surface flux estimates from sparse and noisy drone-based observations is developed and tested using a turbulence-resolving large eddy simulation (LES) as a forward model to connect surface fluxes to drone observations. The proposed framework explicitly represents the sequential collection of drone data, accounts for sensor noise, includes uncertainty in boundary and initial conditions, and jointly estimates the posterior distribution of a multivariate parameter space. Assuming typical flight times and observational errors of light-weight, multi-rotor drone systems, we first evaluate the information gain and performance of different ensemble-based data assimilation schemes in experiments with synthetically generated observations. It is shown that an iterative ensemble smoother outperforms both the non-iterative ensemble smoother and the particle batch smoother in the given problem, yielding well-calibrated posterior uncertainty with continuous ranked probability scores of <span class="inline-formula">12</span> W m<span class="inline-formula">⁻²</span> for both H and LE, with standard deviations of <span class="inline-formula">37</span> W m<span class="inline-formula">⁻²</span> (H) and <span class="inline-formula">46</span> W m<span class="inline-formula">⁻²</span> (LE) for a <span class="inline-formula">12</span> min vertical step profile by a single drone. Increasing flight times, using observations from multiple drones, and further narrowing the prior distributions of the initial conditions are viable for reducing the posterior spread. Sampling strategies prioritizing space–time exploration without temporal averaging, instead of hovering at fixed locations while averaging, enhance the non-linearities in the forward model and can lead to biased flux results with ensemble-based assimilation schemes. In a set of <span class="inline-formula">18</span> real-world field experiments at two wetland sites in Norway, drone data assimilation estimates agree with independent eddy covariance estimates, with root mean square error values of <span class="inline-formula">37</span> W m<span class="inline-formula">⁻²</span> (H), <span class="inline-formula">52</span> W m<span class="inline-formula">⁻²</span> (LE), and <span class="inline-formula">58</span> W m<span class="inline-formula">⁻²</span> (H+LE) and correlation coefficients of <span class="inline-formula">0.90</span> (H), <span class="inline-formula">0.40</span> (LE), and <span class="inline-formula">0.83</span> (H+LE). While this comparison uses the simplifying assumptions of flux homogeneity, stationarity, and flat terrain, it is emphasized that the drone data assimilation framework is not confined to these assumptions and can thus readily be extended to more complex cases and other scalar fluxes, such as for trace gases in future studies.

Some applications of fractional calculus involving summation of infinite series

Summation of Certain Infinite Series

The stress-focusing effect is the phenomenon that, under an instantaneous heating, stress waves reflected from the free surface of the cylinder result in very high stresses at the center. Ho solved the problem by using Laplace transform on time and presented the solution as infinite series summations, which converge very slowly for certain combinations of time and the radius of a cylinder. However, adopting a concept of the ray theory, the solution of stress waves in the cylinder is sorted out into rays according to the ray path of multiply reflected waves. The results expressed in an infinite series reveal that stresses peak out periodically at a constant period and the order of singularity of the stresses in a cylinder is O(ρ−2).

We derive the exact theoretical symbol error rate (SER) of M-ary phase-shift keying (M-PSK), differentially encoded M-ary PSK (DE-M-PSK) and M-ary quadrature amplitude modulation (M-QAM), for EGC with two correlated diversity branches in Nakagami-m fading. The starting point is the derivation of the probability density function (pdf) of the sum of two correlated Nakagami-m variables, for which no closed-form expression was previously available in the communications literature. Several novel expressions are then obtained, either in the form of a single-dimensional integral, or in closed-form as infinite series summations. Some closed-form solutions for the SER of dual-branch MRC can also be derived by using an alternative expression for the pdf of the sum of the squares of two correlated Nakagami-m variables.

For a few particular waveguide problems, standard variational expressions have previously been shown to be upper or lower bounds on the quantities of interest. However, bounds have not previously been obtained for any truly three-dimensional problem; that is, where the fields cannot be derived from a single scalar potential. An example is a three-dimensional obstacle which contacts only one waveguide surface. As one consequence, no straightforward procedure exists for improving the approximations. Recently, Kato devised a rather general method for bounding the cotangent of the phase shift for a given angular momentum in a quantum mechanical central potential scattering problem. This method should be applicable whenever a system can be analyzed in terms of uncoupled standing waves each characterizable by one real phase shift. The method is here adapted to waveguides, including truly three-dimensional problems. The obstacle must be symmetric about a plane perpendicular to the waveguide axis, with certain exceptions only one mode should propagate, and the system should be lossless. Analysis is then possible in terms of ηe and η0, the real uncoupled phase shifts associated with the even and odd standing waves, respectively. The bounds obtained on cotηe and cotη0 determine bounds on the equivalent network elements.

Bellman, Kalaba, and Lockett recently proposed a numerical method for inverting the Laplace transform. The method consists in first reducing the infinite interval of integration to a finite one by a preliminary substitution of variables, and then employing an $n$-point Gauss-Legendre quadrature formula to reduce the inversion problem (approximately) to that of solving a system of $n$ linear algebraic equations. Luke suggests the possibility of using Gauss-Jacobi quadrature (with parameters $\alpha$ and $\beta$) in place of Gauss-Legendre quadrature, and in particular raises the question whether a judicious choice of the parameters $\alpha$, $\beta$ may have a beneficial influence on the condition of the linear system of equations. The object of this note is to investigate the condition number cond $(n,\alpha ,\beta )$ of this system as a function of $n$, $\alpha$, and $\beta$. It is found that cond $(n,\alpha ,\beta )$ is usually larger than cond $(n,\beta ,\alpha )$ if $\beta > \alpha$, at least asymptotically as $n \to \infty$. Lower bounds for cond $(n,\alpha ,\beta )$ are obtained together with their asymptotic behavior as $n \to \infty$. Sharper bounds are derived in the special cases $n$, $n$ odd, and $\alpha = \beta = \pm \frac {1} {2}$, $n$ arbitrary. There is also a short table of cond $(n,\alpha ,\beta )$ for $\alpha$, $\beta = - .8(.2)0,.5,1,2,4,8,16,\beta \leqq \alpha$, and $n = 5,10,20,40$. The general conclusion is that cond $(n,\alpha ,\beta )$ grows at a rate which is something like a constant times ${(3 + \surd 8)^n}$, where the constant depends on $\alpha$ and $\beta$, varies relatively slowly as a function of $\alpha$, $\beta$, and appears to be smallest near $\alpha = \beta = - 1$. For quadrature rules with equidistant points the condition grows like $(2\surd 2/3\pi ){8^n}$.