A Viscous Velocity Potential/Stream Function

The motion of fluids presents interesting phenomena including flow separation, wakes, turbulence etc. The physics of these are enshrined in the continuity equation and the NSE. Therefore, their studies are important in mathematics and physics. They also have engineering applications. These studies can either be carried out experimentally, numerically, or theoretically. Theoretical studies using classical potential theory (CPT) have some gaps when compared to experiments. The present publication is part of a series introducing refined potential (RPT) that bridges these gaps. It leverages experimental observations, physical deductions and the match between CPT and experimentally observed flows in the theoretical development. It analytically imitates the numerical source/vortex panel method to describe how wall bounded eddies in a three-dimensional cylinder crossflow are linked to the detached wake eddies. Unlike discrete and arbitrary vortices/sources on the cylinder surface whose strengths are numerically determined in the panel method, the vortices/sources/sinks in RPT are mutually concentric and continuously distributed on the cylinder surface. Their strengths are analytically determined from CPT using physical deductions starting from Reynolds number dependence. This study results in the incompressible Kwasu function which is a Eulerian velocity potential/stream function that captures vorticity, boundary layer, shed wake vortices, three-dimensional effects, and turbulence. This Eulerian Kwasu function also theorizes streaklines. The Lagrangian form of the function is further exploited to obtain flow pathlines.

On the use of outgoing longwave radiation data in incorporating divergent part of the wind in analysis

In the present study, kinematic divergence computed using ECMWF grid point data at 850 hPa is enhanced by using the relationship between OLR and divergence. This new enhanced divergence is used to compute the velocity potential and then, the divergence part of the wind is obtained from velocity potetial. To obtain the rotational part of wind, we computed the vorticity from wind data, and subsequently stream function and obtained and the rotational part of the wind from the stream function. The total wind is the combination of divergent part obtained from modified velocity potential (using OLR data) and rotational part from unmodified stream function. This total wind field is used as initial guess for univariate objective analysis by optimum interpolation scheme so that Initial Guess field contained the more realistic divergent part of the wind. Consequently, the analysed field also will contain the divergent part of the wind.

A Closed-Form Analytical Solution to the Complete Three-Dimensional Unsteady Compressible Navier-Stokes Equations

The three main approaches in fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. Numerical simulations and theoretical solutions are based on the continuity equation and Navier-Stokes equations (NSE) that govern experimental observations of fluid dynamics.Theoretical solutions can offer huge advantages over numerical solutions and experiments in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, theoretical solutions have been limited by the prized NSE problem that seeks a physically consistent solution than what classical potential theory (CPT) offers. Therefore, the current author refined CPT. He introduced refined potential theory (RPT) that provides a viscous potential/stream function as a physically consistent solution to the NSE problem. This function captures observable unsteady flow features including separation, wake, vortex shedding, compressibility, turbulence, and Reynolds-number-dependence. It appropriately combines the properties of a three-dimensional potential function that satisfy the inertia terms of NSE and the features of a stream function that satisfy the continuity equation, the viscous vorticity equation, and the viscous terms of NSE. RPT has been verified and validated against experimental and numerical results of incompressible unsteady sub-critical Reynolds number flows on stationary finite circular cylinder, sphere, and spheroid.

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.

A virtual element discretization for the time dependent Navier-Stokes equations in stream-function formulation

In this work, a new Virtual Element Method (VEM) of arbitrary order $k \geq 2$ for the time dependent Navier-Stokes equations in stream-function form is proposed and analyzed. Using suitable projection operators, the bilinear and trilinear terms are discretized by only using the proposed degrees of freedom associated with the virtual space. Under certain assumptions on the computational domain, error estimations are derived and shown that the method is optimally convergent in both space and time variables. Finally, to justify the theoretical analysis, four benchmark examples are examined numerically.

Methods: Of Stream Functions and Thin Wires: An Intuitive Approach to Gradient Coil Design

The design of gradient coils is sometimes perceived as complex and counterintuitive. However, a current density is connected to a stream function in fact by a simple relation. Here we present an intuitive open source code collection to derive stream functions from current densities on simple surface geometries. Discrete thin wires, oriented orthogonally to the main magnetic field direction are used to describe a surface current density. An inverse problem is solved and stream functions are derived to find coil designs in the current and stream function domains. The flexibility of the design method is demonstrated by deriving gradient coil designs on several different surface topologies. This collection is primarily intended for teaching, as well as for demonstrating all gradient coil design steps with openly available software tools.

Indices of the Hadley circulation strength and associated circulation trends

This study compares the trends of Hadley cell (HC) strength using different HC measures applied to the ECMWF ERA5 and ERA-Interim reanalyses in the period 1979–2018. The HC strength is commonly evaluated by indices derived from the mass-weighted zonal-mean stream function. Other measures include the velocity potential and the vertical velocity. Six known measures of the HC strength are complemented by a measure of the average HC strength, obtained by averaging the stream function in the latitude-pressure (φ-p) plane, and by the total energy of unbalanced zonal-mean circulation in the normal-mode function decomposition. It is shown that measures of the HC strength, which rely on point values in the φ-p plane, produce unreliable long-term trends of both the northern and southern HCs, especially in ERA-Interim; magnitudes and even the signs of trends depend on the choice of HC strength measure. The two new measures alleviate the vertical and meridional inhomogeneities of the trends in the HC strength. In both reanalyses, there is a positive trend in the total energy of zonal-mean unbalanced circulation. The average HC strength measure also shows a positive trend in ERA5 in both hemispheres, while the trend in ERA-Interim is insignificant.