Optimizing Water Consumption in Richards’ Equation Framework with Step-Wise Root Water Uptake: A Simplified Model

This work arises from the need of exploring new features for modeling and optimizing water consumption in irrigation processes. In particular, we focus on water flow model in unsaturated soils, accounting also for a root water uptake term, which is assumed to be discontinuos in the state variable. We investigate the possibility of accomplishing such optimization by computing the steady solutions of a $$\theta$$ θ -based Richards equation revised as equilibrium points of the ODEs system resulting from a numerical semi-dicretization in the space; after such semi-discretization, these equilibrium points are computed exactly as the solutions of a linear system of algebraic equations: the case in which the equilibrium lies on the threshold for the uptake term is of particular interest, since the system considerably simplifies. In this framework, the problem of minimizing the water waste below the root level is investigated. Numerical simulations are provided for representing the obtained results. Article Highlights Root water uptake is modelled in a Richards’ equation framework with a discontinuous sink term. After a proper semidiscretization in space, equilibrium points of the resulting nonlinear ODE system are computed exactly. The proposed approach simplifies a control problem for optimizing water consumption.

Two-Dimensional Steady Boussinesq Convection: Existence, Computation and Scaling

This research investigates laser-induced convection through a stream function-vorticity formulation. Specifically, this paper considers a solution to the steady Boussinesq Navier–Stokes equations in two dimensions with a slip boundary condition on a finite box. A fixed-point algorithm is introduced in stream function-vorticity variables, followed by a proof of the existence of steady solutions for small laser amplitudes. From this analysis, an asymptotic relationship is demonstrated between the nondimensional fluid parameters and least upper bounds for laser amplitudes that guarantee existence, which accords with numerical results implementing the algorithm in a finite difference scheme. The findings indicate that the upper bound for laser amplitude scales by O(Re−2Pe−1Ri−1) when Re≫Pe, and by O(Re−1Pe−2Ri−1) when Pe≫Re. These results suggest that the existence of steady solutions is heavily dependent on the size of the Reynolds (Re) and Peclet (Pe) numbers, as noted in previous studies. The simulations of steady solutions indicate the presence of symmetric vortex rings, which agrees with experimental results described in the literature. From these results, relevant implications to thermal blooming in laser propagation simulations are discussed.

Bifurcation scenario for a two-dimensional static airfoil exhibiting trailing edge stall

We numerically investigate stalling flow around a static airfoil at high Reynolds numbers using the Reynolds-averaged Navier–Stokes equations (RANS) closed with the Spalart–Allmaras turbulence model. An arclength continuation method allows to identify three branches of steady solutions, which form a characteristic inverted S-shaped curve as the angle of attack is varied. Global stability analyses of these steady solutions reveal the existence of two unstable modes: a low-frequency mode, which is unstable for angles of attack in the stall region, and a high-frequency vortex shedding mode, which is unstable at larger angles of attack. The low-frequency stall mode bifurcates several times along the three steady solutions: there are two Hopf bifurcations, two solutions with a two-fold degenerate eigenvalue and two saddle-node bifurcations. This low-frequency mode induces a cyclic flow separation and reattachment along the airfoil. Unsteady simulations of the RANS equations confirm the existence of large-amplitude low-frequency periodic solutions that oscillate around the three steady solutions in phase space. An analysis of the periodic solutions in the phase space shows that, when decreasing the angle of attack, the low-frequency periodic solution collides with the unstable steady middle-branch solution and thus disappears via a homoclinic bifurcation of periodic orbits. Finally, a one-equation nonlinear stall model is introduced to reveal that the disappearance of the limit cycle, when increasing the angle of attack, is due to a saddle-node bifurcation of periodic orbits.

The Momentum Conserving Scheme for Two-Layer Shallow Flows

This paper confronts the numerical simulation of steady flows of fluid layers through channels of varying bed and width. The fluid consists of two immiscible fluid layers with constant density, and it is assumed to be of a one-dimensional shallow flow. The governing equation is a coupled system of two-layer shallow water models. In this paper, we apply a direct extension of the momentum conserving scheme previously used for solving the one layer shallow water equations. Computations of various steady-state solutions are used to demonstrate the performance of the proposed numerical scheme. Under the influence of a given flow rate, the numerical steady interface is generated in a channel topography with a hump. The results obtained confirm the analytic steady interface of the two-layer rigid-lid model. Furthermore, the same scheme was used with an additional artificial damping to simulate the maximal exchange flow in channels of varying width. The numerical steady interface agreed well with the analytical steady solutions.

A Computational Study on Fluid Flow and Heat Transfer Through a Rotating Curved Duct with Rectangular Cross Section

The understanding of fluid flow and heat transfer (HT) through a rotating curved duct (RCD) is important for different engineering applications. The available literature improved the understanding of the fluid flow and HT through a large-curvature rotating duct. However, the comprehensive knowledge of fluid flow and HT through an RCD with small curvature is little known. This numerical study aims to perform fluid flow characterization and HT through an RCD with curvature ratio 0.001. The spectral based numerical approach investigates the effects of rotation on fluid flow and HT for the Taylor number −1000≤Tr≤1500. A constant pressure gradient force, the Dean number Dn = 100, and a constant buoyancy force parameter, the Grashof number Gr = 500 are used for the numerical simulation. Fortran code is developed for the numerical computations and Tecplot software is used for the post-processing purpose. The numerical study investigates steady solutions and a structure of two-branches of steady solutions is obtained for positive rotation. The transient solution reports the transitional flow patterns and HT through the rotating duct, and two- to four-vortex solutions are observed. In case of negative rotation, time-dependent solutions show that the Coriolis force exhibits an opposite effect to that of the curvature so that the flow characteristics exhibit various flow instabilities. The numerical result shows that convective HT is increased with the increase of rotation and highly complex secondary flow patterns influence the overall HT from the heated wall to the fluid. To validate the numerical results, a comparison with the experimental data is provided, which shows that a good agreement is attained between the numerical and experimental investigations.

Cross immunity protection and antibody dependent enhancement: A distributed delay dynamic model

Dengue fever is endemic in tropical and sub-tropical countries, and some of the important features of Dengue fever spread continues to pose challenges for mathematical modelling. Here, we propose a system of integro-differential equations (IDE) to study the disease transmission dynamics that involves multiserotypes and cross immunity. Our main objective is to incorporate and analyze the effect of a general time delay term describing acquired cross immunity protection and the effect of antibody dependent enhancement (ADE), both characteristics of Dengue fever. We perform qualitative analysis of the model and obtain results to show the stability of the epidemiologically important steady solutions that is completely determined by the basic reproduction number and the invasion reproduction number. We establish the global dynamics, by constructing suitable Lyapunov functions. We also conduct some numerical experiments to illustrate bifurcation structures, indicating the occurrence of periodic oscillations for specific range of values of a key parameter representing the ADE.

Bifurcations of drops and bubbles propagating in variable-depth Hele-Shaw channels

The steady propagation of air bubbles through a Hele-Shaw channel with either a rectangular or partially occluded cross section is known to exhibit solution multiplicity for steadily propagating bubbles, along with complicated transient behaviour where the bubble may visit several edge states or even change topology several times, before typically reaching its final propagation mode. Many of these phenomena can be observed both in experimental realisations and in numerical simulations based on simple Darcy models of flow and bubble propagation in a Hele-Shaw cell. In this paper, we investigate the corresponding problem for the propagation of a viscous drop (with viscosity $$\nu $$ ν relative to the surrounding fluid) using a Darcy model. We explore the effect of drop viscosity on the steady solution structure for drops in rectangular channels or with imposed height variations. Under the Darcy model in a uniform channel, steady solutions for bubbles map directly on to those for drops with any internal viscosity $$\nu \ne 1$$ ν ≠ 1 . Hence, the solution multiplicity predicted for bubbles also occurs for drops, although for $$\nu >1$$ ν > 1 , the interface shape is reversed with inflection points appearing at the rear rather than the front of the drop. The equivalence between bubbles and drops breaks down for transient behaviour, at the introduction of any height variation, for multiple bodies of different viscosity ratios and for more detailed models which produce a more complicated flow in the interior of the drop. We show that the introduction of topography variations affects bubbles and drops differently, with very viscous drops preferentially moving towards more constricted regions of the channel. Both bubbles and drops can undergo transient behaviour which involves breakup into two almost equal bodies, which then symmetry break before either recombining or separating indefinitely.

Vortex sheet turbulence as solvable string theory

We study steady vortex sheet solutions of the Navier–Stokes in the limit of vanishing viscosity at fixed energy flow. We refer to this as the turbulent limit. These steady flows correspond to a minimum of the Euler Hamiltonian as a functional of the tangent discontinuity of the local velocity parametrized as [Formula: see text]. This observation means that the steady flow represents the low-temperature limit of the Gibbs distribution for vortex sheet dynamics with the normal displacement [Formula: see text] of the vortex sheet as a Hamiltonian coordinate and [Formula: see text] as a conjugate momentum. An infinite number of Euler conservation laws lead to a degenerate vacuum of this system, which explains the complexity of turbulence statistics and provides the relevant degrees of freedom (random surfaces). The simplest example of a steady solution of the Navier–Stokes equation in the turbulent limit is a spherical vortex sheet whose flow outside is equivalent to a potential flow past a sphere, while the velocity is constant inside the sphere. Potential flow past other bodies provide other steady solutions. The new ingredient we add is a calculable gap in tangent velocity, leading to anomalous dissipation. This family of steady solutions provides an example of the Euler instanton advocated in our recent work, which is supposed to be responsible for the dissipation of the Navier–Stokes equation in the turbulent limit. We further conclude that one can obtain turbulent statistics from the Gibbs statistics of vortex sheets by adding Lagrange multipliers for the conserved volume inside closed surfaces, the rate of energy pumping, and energy dissipation. The effective temperature in our Gibbs distribution goes to zero as [Formula: see text] with Reynolds number [Formula: see text] in the turbulent limit. The Gibbs statistics in this limit reduces to the solvable string theory in two dimensions (so-called [Formula: see text] critical matrix model). This opens the way for nonperturbative calculations in the Vortex Sheet Turbulence, some of which we report here.