Cartesian Velocity Components Research Articles

On the formulation and implementation of the stress-free boundary condition over deformed bathymetry using a spectral-element-method-based incompressible Navier–Stokes equations solver

Various strategies are proposed for enforcing stress-free boundary conditions on deformed domains using a weak-form-based discretization in a Cartesian frame of reference. Due to the irregularity of the computational domain and the particular type of boundary condition, a coupling between the Cartesian velocity components is introduced. As such, a different computational kernel is required for the numerical solution of the associated vector Helmholtz equation, as contrasted to what is used for the scalar unknowns of the equations. Three approaches are presented, aimed towards the exact or approximate implementation of zero tangential stress (traction) at the deformed boundary while ensuring a decoupling of the velocity components in the solution of the vector Helmholtz equation. Two of these strategies, those which approximate the free slip boundary condition, are applied to the propagation of an internal solitary wave (ISW) of depression over a deformed bathymetry. The spatial structure and amplitude of the resultant pseudo-traction, which is accurately predicted by a simple scaling estimate, are explored as a function of the ISW-based Reynolds number, Re. For the Re values considered, the pseudo-traction is negligible with respect to the corresponding no-slip tangential shear stress. The pseudo-traction-induced, time-integrated loss of ISW energy is found to be significantly weaker than the associated viscous dissipation in the interior water column. Although, at laboratory-scale or oceanic Re, an approximate free-slip boundary condition is found to yield negligible pseudo-traction, this might not be the case when an elevated eddy viscosity is used in this context as a surrogate for no-slip turbulent bottom boundary layer dynamics.

Theoretical analysis of wind flow characteristics to investigate the mass and momentum parameters using a novel computational fluid dynamics-based approach

In this article, an experimental study of a wind turbine in a wind tunnel is performed. The objective has been to present a novel analytical computational fluid dynamics (CFD)-based approach through considering the residual levels of the mass and momentum parameters under effect of different air flow characteristics surrounding the wind turbine, which have an effect on the power losses, turbine’s performance and the economic viability. The involved decision variables are considered to be the wind velocity, the pressure and the turbulence. Evaluation of the convergence showed that the residual level for the maximum method is estimated to be approximately 10–1 to 10–3 times higher than the root mean square. Results also concluded that between two studied turbulence models, the turbulence eddy frequency is found to be more efficient compared with turbulence kinetic energy. In higher iterations compared with the initial iterations, a significant difference between the pressure and the Cartesian velocity components has occurred and the residual level of the velocity components indicated a more efficient convergence compared with the pressure. The overall environmental analysis concluded that on the basis of the CFD residual values, it would be possible to adequately determine the CFD efficiency of the wind energy system in a wind tunnel. It has been demonstrated that, among different decision variables, velocity components of the mass and momentum parameters and the turbulence eddy frequency were determined to produce further accurate results in comparison with the pressure and the turbulence kinetic energy.

Numerical Computation of Low Reynolds Number Viscous Flow Past Bluff Bodies

This article presents a two-dimensional steady viscous flow simulation past circular and square cylinders at low Reynolds numbers (based on the diameter) by the finite volume method with a non-orthogonal body-fitted grid. Diffusive fluxes are discretized using central differencing scheme, and for convective fluxes upwind and central differencing schemes are blended using a ‘deferred correction’ approach. A simplified pressure correction equation is derived, and proper under-relaxation factors are used so that computational cost is reduced without adversely affecting the convergence rate. The governing equations are expressed in Cartesian velocity components and solution is carried out using the SIMPLE algorithm for collocated arrangement of variables. The mesh yielding grid-independent solution is then utilized to study, for the very first time, the effect of the Reynolds number on the separation bubble length, separation angle, and drag coefficients for both circular and square cylinders. Finally, functional relationships between the computed quantities and Reynolds number (Re) are proposed up to Re = <i>40</i>. It is found that circular cylinder separation commences between Re= <i>6.5-6.6</i>, and the bubble length, separation angle, total drag vary as Re, Re<i>−0.5</i>, Re<i>−0.5</i> respectively. Extrapolated results obtained from the empirical relations for the circular cylinder show an excellent agreement with established data from the literature. For a square cylinder, the bubble length and total drag are found to vary as Re and Re<i>−0.666</i>, and are greater than these for a circular cylinder at a given Reynolds number. The numerical results substantiate that a square shaped cylinder is more bluff than a circular one.

Numerical Analysis of Turbulent Flow Around Two-Dimensional Bodies Using Non-Orthogonal Body-Fitted Mesh

This paper deals with the numerical simulation of a turbulent flow around two-dimensional bodies by the finite volume method with non-orthogonal body-fitted grid. The governing equations are expressed in Cartesian velocity components and solution is carried out using the SIMPLE algorithm for collocated arrangement of scalar and vector variables. Turbulence is modeled by the <i>k</i>- ε turbulence model and wall functions are used to bridge the solution variables at the near wall cells and the corresponding quantities on the wall. A simplified pressure correction equation is derived and proper under-relaxation factors are used so that computational cost is reduced without adversely affecting the convergence rate. The numerical procedure is validated by comparing the computed pressure distribution on the surface of NACA 0012 and NACA 4412 hydrofoils for different angles of attack with experimental data. The grid dependency of the solution is studied by varying the number of cells of the C-type structured mesh. The computed lift coefficients of NACA 4412 hydrofoil at different angles of attack are also compared with experimental results to further substantiate the validity of the proposed methodology.

C2 continuous time-dependent feedrate scheduling with configurable kinematic constraints

We present a configurable trajectory planning strategy on planar paths for offline definition of time-dependent C2 piecewise quintic feedrates. The scheduler can be applied to any path with a sufficiently smooth piecewise parametric representation and it can be used in different formulations subdivided in strict and relaxed. In the first case, control on chord tolerance, as well as on the Cartesian components of velocity, acceleration and jerk is ensured. Since the relaxed formulations can usually still ensure the desired bounds while simultaneously producing faster motions, the configurability feature is useful not only when reduced motion control is required but also when full kinematic control has to be guaranteed. When Pythagorean-hodograph spline curves are considered, the corresponding accurate and efficient CNC interpolator algorithms can be exploited, as shown in the numerical experiments.

Confined active Brownian particles: theoretical description of propulsion-induced accumulation

The stationary-state distribution function of confined active Brownian particles (ABPs) is analyzed by computer simulations and analytical calculations. We consider a radial harmonic as well as an anharmonic confinement potential. In the simulations, the ABP is propelled with a prescribed velocity along a body-fixed direction, which is changing in a diffusive manner. For the analytical approach, the Cartesian components of the propulsion velocity are assumed to change independently; active Ornstein–Uhlenbeck particle (AOUP). This results in very different velocity distribution functions. The analytical solution of the Fokker–Planck equation for an AOUP in a harmonic potential is presented and a conditional distribution function is provided for the radial particle distribution at a given magnitude of the propulsion velocity. This conditional probability distribution facilitates the description of the coupling of the spatial coordinate and propulsion, which yields activity-induced accumulation of particles. For the anharmonic potential, a probability distribution function is derived within the unified colored noise approximation. The comparison of the simulation results with theoretical predictions yields good agreement for large rotational diffusion coefficients, e.g. due to tumbling, even for large propulsion velocities (Péclet numbers). However, we find significant deviations already for moderate Péclet number, when the rotational diffusion coefficient is on the order of the thermal one.

Natural Convection from Discrete Heat Sources Placed in Wavy Enclosure

The effect of sinusoidal corrugated upper surface for enclosure on the heat transfer by natural convection from two heat sources placed on the bottom surface of the enclosure has been studied. The two heat sources constant and high temperature, length of each heat source (10%) from the total length of bottom surface, the distances that do not contain heat source on the bottom surface thermally insulated, temperature of verticals walls and upper surface low and constant. corrugating upper surface was change the dimensional corrugation amplitude between (0.1-0.3), for number of corrugations (3), the dimensional distances between the two heat sources were change between (0.2-0.8), the Rayleigh numbers ranged between (103 ≤ Ra ≤ 105), the fluid inside the enclosure was air Prandtl number for it (0.7). were analyzed flow and temperature fields inside the enclosure numerically by solving Navier Stokes and Energy Equations, have been used Cartesian velocity components and pressure on a collocated grid are used as dependent variables in the momentum equations which discretized by finite volume method, body fitted coordinates are used to represent the geometry shape of the problem accurately, were grid generated technique based on elliptic partial differential equations, SIMPLE algorithm is used to adjust the velocity field to satisfy the conservation of mass. The results showed the increase in the magnitude of dimensional distance between two heat sources lead to increased average Nusselt number for all values of Rayleigh numbers (103 ≤ Ra ≤ 105), as are the highest value of the average Nusselt number at dimensional distance (0.8)and less value at the dimensional distance (0.2), the dimensional distance (0.6) is a special case when the Rayleigh number equal (105) as at least average Nusselt number from the remaining values of the two dimensional distances (0.2) and (0.4), as the average Nusselt number increases with increasing dimensional corrugation amplitude for the number of corrugations (N=3) for dimensional distance (0.8), as well as comparing numerical results showed good agreement with published results.

Propagation of a finite-amplitude pulse in a bar of berea sandstone: The mechanisms of classical nonlinearity, conditioning, and hysteresis

We study the propagation of a finite-amplitude pulse in a slender bar of Berea sandstone. The center frequency of the pulse and aspect ratio of the bar are such that the problem can be adequately described by the propagation of a longitudinal wave in a 1D system. The evolution of the three Cartesian components of the particle velocity on the surface of the bar as functions of the propagation distance and source amplitude is carefully monitored without contact using a 3D laser Doppler vibrometer. In these experiments, we evidence simultaneously the effects from classical nonlinearity, hysteresis, and conditioning (i.e., elastic softening) in the impulsive waveforms, as the pulse propagates away from the source. Traditionally, this type of experiments has been conducted to quantify only classical nonlinearity, through the parameter β, based on the amplitude growth of the second harmonic as a function of the propagation distance. In this work, we also use these experiments to quantify conditioning, through the parameter of nonclassical nonlinearity α, based on the relative change in the arrival time of the pulse as a function of strain and distance from the source.

DETECTABILITY OF COSMIC DARK FLOW IN THE TYPE IA SUPERNOVA REDSHIFT–DISTANCE RELATION

ABSTRACT We reanalyze the detectability of large-scale dark flow (or local bulk flow) with respect to the CMB background based upon the redshift–distance relation for SN Ia. We made two independent analyses: one based upon identifying the three Cartesian velocity components; and the other based upon the cosine dependence of the deviation from Hubble flow on the sky. We apply these analyses to the Union2.1 SN Ia data and to the SDSS-II supernova survey. For both methods, results for low redshift, , are consistent with previous searches. We find a local bulk flow of v bf ∼ 300 km s−1 in the direction of (l, b) ∼ (270, 35)°. However, the search for a dark flow at is inconclusive. Based upon simulated data sets, we deduce that the difficulty in detecting a dark flow at high redshifts arises mostly from the observational error in the distance modulus. Thus, even if it exists, a dark flow is not detectable at large redshift with current SN Ia data sets. We estimate that a detection would require both significant sky coverage of SN Ia out to z = 0.3 and a reduction in the effective distance modulus error from 0.2 mag to ≲0.02 mag. We estimate that a greatly expanded data sample of ∼104 SN Ia might detect a dark flow as small as 300 km s−1 out to z = 0.3 even with a distance modulus error of 0.2 mag. This may be achievable in a next generation large survey like LSST.

Mathematical model for two-dimensional dry friction modified by dither

A new dynamic two-dimensional friction model is developed that is based on the bristle theory. Actually, it is the Reset Integrator Model converted into a two-dimensional space. Usually, two-dimensional friction models are in fact one-dimensional models that are rotated into a slip velocity direction. However, this common approach cannot be applied to the bristle model. That is why the idea of a two-dimensional bristle is presented. The bristle’s deformation is described using polar coordinates. The carried-out numerical simulation of a planar oscillator has proved that the new model correctly captures the mechanism of smoothing dry friction by dither applied in both a perpendicular and co-linear way regarding body velocity. Furthermore, the introduced mathematical model captures two-dimensional stick-slip behaviour. Cartesian slip velocity components are the only inputs to the model. In addition, our proposed model allows one to describe friction anisotropy using bristle parameters. The paper contains the results of an experimental verification of the new friction model, conducted with a special laboratory rig employed to investigate a two-dimensional motion in the presence of dither as well as to validate our numerical results.

Modeling of fluid flow in periodic cell with porous cylinder using a boundary element method

The problem of viscous incompressible flow past a periodic array of porous cylinders (a model of flow in an aerosol filter) is solved. The approximate periodic cell model of Kuwabara is used to formulate the fluid flow problem. The Stokes flow model is then adopted to model the flow outside the cylinder and the Darcy law of drag is applied to find the filtration velocity field inside the porous cylinder. The boundary value problems for biharmonic and Laplace equations for stream functions outside and inside the porous cylinder are solved using a boundary elements method. A good agreement of numerical and analytical models is shown. The analytical formulas for the integrals in the expressions for the stream function, vorticity and Cartesian velocity components are obtained. It is shown that the use of analytical integration gives considerable advantage in computing time.

Fundamental free-space solutions of a steady axisymmetric MHD viscous flow

This work obtains the velocity and pressure fields of two fundamental axisymmetric MHD viscous flows of a conducting Newtonian liquid due to axial and radial distributions of forces on a ring in the presence of a uniform axial magnetic field The worked out procedure rests on the analytical determination of the pressure and of some of the Cartesian velocity components of another general three-dimensional fundamental MHD flow. The derived axisymmetric fundamental flows are found to deeply depend upon the nature (axial or radial) of the forces distributed on the ring, the ring size and the Hartmann layer thickness This is illustrated by computing a few flow patterns using a suitable numerical treatment.

Effect of base elongation of circular cylinder on its wake characteristics

The present numerical study aims at the control of vortex shedding and drag reduction by shaping the rear geometry of a circular cylinder. The ratio of the semi-major/minor axis dimension (A) of the rear half to the radius of the circular part (R) is varied over a range of values between 0.0 and 4.0. A two-dimensional flow solver has been developed using body-fitting coordinate transformation with Cartesian velocity components as dependent variables and solved using a modified MAC algorithm. The flow solver offers a fast and accurate procedure for modelling viscous incompressible flows in complex geometries. The effect of elongation on the wake characteristics and hydrodynamic coefficients has been investigated for different values of Reynolds number (Re) ranging from 40 to 200. The predicted results indicate that vortex suppression occurs primarily due to reduction in the amplitude of fluctuation than by reduction in the vortex shedding frequency. The critical value of Re for the onset of vortex shedding has been correlated with elongation ratio ER, based on the numerical predictions.

Helmholtz Fermi surface harmonics: an efficient approach for treating anisotropic problems involving Fermi surface integrals

We present a new efficient numerical approach for representing anisotropic physical quantities and/or matrix elements defined on the Fermi surface (FS) of metallic materials. The method introduces a set of numerically calculated generalized orthonormal functions which are the solutions of the Helmholtz equation defined on the FS. Noteworthy, many properties of our proposed basis set are also shared by the FS harmonics introduced by Philip B Allen (1976 Phys. Rev. B 13 1416), proposed to be constructed as polynomials of the cartesian components of the electronic velocity. The main motivation of both approaches is identical, to handle anisotropic problems efficiently. However, in our approach the basis set is defined as the eigenfunctions of a differential operator and several desirable properties are introduced by construction. The method is demonstrated to be very robust in handling problems with any crystal structure or topology of the FS, and the periodicity of the reciprocal space is treated as a boundary condition for our Helmholtz equation. We illustrate the method by analysing the free-electron-like lithium (Li), sodium (Na), copper (Cu), lead (Pb), tungsten (W) and magnesium diboride (MgB).

Computational study of a turbulent wall jet flow on an oblique surface

Purpose – The purpose of the present study is to investigate the flow and turbulence characteristics of a turbulent wall jet flowing over a surface inclined with the horizontal and to investigate the effect of variation of the angle of inclination of the wall on the flow structure of the wall jet. Design/methodology/approach – The high Reynolds number two-equation κ− model with standard wall function is used as the turbulence model. The Reynolds number considered for the present study is 10,000. The Reynolds averaged Navier-Stokes (RANS) equations are used for predicting the turbulent flow. A staggered differencing technique employing both contravariant and Cartesian components of velocity has been applied. Results for distribution of wall static pressure and skin friction, decay of maximum streamwise velocity, streamwise variation of integral momentum and energy flux have been compared for the cases of α=0°, 5°, and 10°. Findings – Flow field has been represented in terms of streamwise and lateral velocity contours, static pressure contour, vorticity contour and streamwise velocity and static pressure profiles at different locations along the oblique offset plate. Distribution of Reynolds stresses in terms of spanwise, lateral and turbulent shear stresses, and turbulent kinetic energy and its dissipation rate have been presented to describe the turbulent characteristics. Similarity of streamwise velocity and the velocity parallel to the oblique wall has been observed in the developed region of the wall jet flow. A decaying trend is observed in the variation of total integral momentum flux in the developed region of the wall jet which becomes more evident with increase in oblique angle. Developed flow region has indicated trend of similarity in profiles of streamwise velocity as well as velocity component parallel to the oblique wall. A depression in wall static pressure has been observed near the nozzle exit when the wall is inclined and the depression increases with increase in inclination. Effect of variation of oblique angles on skin friction coefficient has indicated that it decreases with increase in oblique angle. Growth of the outer and inner shear layers and spread of the jet shows linear variation with distance along the oblique wall. Decay of maximum streamwise velocity is found to be unaffected by variation in oblique angle except in the far downstream region. The streamwise variation of spanwise integral energy shows increase in oblique angle and decreases the magnitude of energy flux through the domain. In the developed flow region, streamwise variation of centreline turbulent intensities shows increased values with increase in oblique angle, while turbulence intensities along the jet centreline in the region X<12 remain unaffected by change in oblique angles. Normalized turbulent kinetic energy distribution highlights the difference in turbulence characteristics between the wall jet and reattached offset jet flow. Near wall velocity distribution shows that the inner region of boundary layer of the developed oblique wall jet follows a logarithmic profile, but it shows some difference from the standard logarithmic curve of turbulent boundary layers which can be attributed to an increase in skin friction coefficient and a decrease in thickness of the wall attached layer. Originality/value – The study presents an in-depth investigation of the interaction between the jet and the inclined wall. It is shown that due to the Coanda effect, the jet follows the nearby wall. The findings will be useful in the study of combined flow of wall jet and offset jet and dual offset jet on oblique surfaces leading to a better design of some mechanical jet flow devices.

Simplified Biomechanics for a Possible Explanation of the Ancient Greek Long Jump Using Halteres

In this paper closed form analytical expressions were derived in order to simulate the possible action of halteres used in the ancient Greek long jump. For the sake of simplicity, elementary theory of rigid body dynamics is used, which however is capable of simulating the motion of a hypothetical rigid jumper for whom the Cartesian components of the initial velocity at the take-off and the angular velocity of rotating arms are prescribed. Particular attention is paid on the initial position and the direction of arms' rotation as well as on the role of the amount of masses due to the halteres. It was found that if at the take-off the upper limbs are upwards, also rotate forwards, whereas at the landing they are almost downwards, the length of the jump increases as the weight of the halters.

Effect of the Presence of Semi-circular Cylinders on Heat Transfer From Heat Sources Placed in Two Dimensional Channel

The effect of a semi-circular cylinders in a two dimensional channel on heat transfer byforced convection from two heat sources with a constant temperature has been studiednumerically. Each channel contains two heat sources; one on the upper surface of thechannel and the other on the lower surface of the channel. There is semi-circular cylinderunder the source in upper surface and there is semi-circular cylinder above the source inlower surface. The location of the second heat source with its semi-cylinder has beenchanged and keeps the first source with its semi- cylinder at the same location. The flowand temperature field are studied numerically with different values of Reynolds numbersand for different spacing between the centers of the semi-cylinders. The laminar flow fieldis analyzed numerically by solving the steady forms of the two-dimensional incompressibleNavier- Stokes and energy equations. The Cartesian velocity components and pressure on acollocated (non-staggered) grid are used as dependent variables in the momentumequations, which discretized by finite volume method, body fitted coordinates are used torepresent the complex channel geometry accurately, and grid generation technique based onelliptic partial differential equations is employed. SIMPLE algorithm is used to adjust thevelocity field to satisfy the conservation of mass. The range of Reynolds number is (Re=100 – 800) and the range of the spacing between the semi-cylinders is(1-4) and the Prandtlnumber is 0.7.The results showed that increasing the spacing between the semi-cylinders increases theaverage of Nusselt number of the first heat source for all Reynolds numbers. As well as theresults show that the best case among the cases studied to enhance the heat transfer is whenthe second heat source and its semi-cylinder located on at the distance (S=1.5) from thefirst half of the cylinder and the Reynolds number is greater than (Re ≥ 400) because of theseparation behind of the first semi-cylinder where it works to reduce the area of sectionflow, and this is leading to increase the speed and thus increases the rate of heat transfer. Acomparison of numerical results agrees well with published results.

Near wake Reynolds‐averaged Navier–Stokes predictions of the wake behind the MEXICO rotor in axial and yawed flow conditions

ABSTRACTIn the present paper, Reynolds‐averaged Navier–Stokes predictions of the flow field around the MEXICO rotor in yawed conditions are compared with measurements. The paper illustrates the high degree of qualitative and quantitative agreement that can be obtained for this highly unsteady flow situation, by comparing measured and computed velocity profiles for all three Cartesian velocity components along four axial transects and several radial transects. Copyright © 2012 John Wiley & Sons, Ltd.

Natural Convection in Parabolic Enclosure Heated from Below

The effects of vertical parabolic walls on natural convection in a parabolic enclosure have investigated numerically in this paper. The bottom wall is heated isothermally, while the other vertical parabolic walls are maintained at constant cold temperature and the top wall is well insulated. The flow and temperature fields are studied numerically for three values (C = 0.1, 0.5, 1.0) of the parabolic equation constant. The laminar flow field is analyzed numerically by solving the steady, two-dimensional incompressible Navier-Stokes and energy equations. The Cartesian velocity components and pressure on a collocated (non-staggered) grid are used as dependent variables in the momentum equations, which discretized by finite volume method, body fitted coordinates are used to represent the complex parabolic wall geometry accurately, and grid generation technique based on elliptic partial differential equations is employed. SIMPLE algorithm is used to adjust the velocity field to satisfy the conservation of mass. The range of Rayleigh number is (103? Ra ?105) and Prandtl number is 0.7. The results show that the heat transfer rates decrease with increase the parabolic equation constant.

Dexterity measures and their use in quantitative dexterity comparisons

The majority of proposed dexterity measures rely on the use of the condition number of the manipulator’s Jacobian matrix mapping actuator velocities to end effector velocities. Unfortunately, for the vast majority of manipulators, the conventional Jacobian matrix has inconsistent units making the aforementioned measures dependent on units and scale. Recently, a Jacobian matrix mapping the joint velocities to independent Cartesian velocity components of three points on the end effector was proposed. In most cases, this Jacobian matrix has consistent units and in many cases it is dimensionless. This dimensionally-homogeneous Jacobian matrix yields meaningful dexterity measures that allow quantitative dexterity comparisons between manipulators with different architectures. Although these new measures were originally proposed within the context of parallel manipulator design and analysis, they can also be used for serial architectures. In this paper, the analysis of the Tricept manipulator’s dexterous workspace, as a function of architectural variables, is provided through the use of a quantitative metric for dexterity. Furthermore, the generality of this metric is also demonstrated by first employing it to analyze a serial manipulator and then comparing it to a parallel manipulator having the same degrees-of-freedom. The paper also proposes a strategy to deal with singularities introduced mathematically by the novel Jacobian formulation.