Kinetic Energy Integrals Research Articles

Integral methods for friction decomposition and their extensions to rough-wall flows

A primary objective of integral methods, such as the momentum integral method, is to discern the physical processes contributing to skin friction. These methods encompass the momentum, kinetic energy and angular momentum integrals. This paper reformulates existing integrals based on the double-averaged Navier–Stokes equations, and extends their application to flows over rough walls. Our derivation yields distinct decompositions for the bottom-wall viscous friction coefficient, denoted as $C_S$ , and the roughness element drag coefficient $C_R$ . The decompositions comprise three terms: a viscous term, a turbulent term and a roughness (dispersive) term – regardless of the flow configuration, be it channel or boundary layer. Notably, when these integrals are evaluated for laminar flow scenarios, only the viscous term remains significant. In addition, we elucidate the spatial distributions of the terms within these decompositions. To demonstrate the practicality of our formulations, we apply them to analyse data from direct numerical simulations of turbulent half-channel flows. These flows feature aligned and staggered cubical roughness at various packing densities. Our analyses, based on kinetic-energy-oriented decompositions, reveal that when the surface coverage density $\lambda _p$ is small, the dominant terms within the decompositions are the viscous and turbulent terms. With increasing $\lambda _p$ , the viscous dissipation term decreases, while the turbulent production term increases and then decreases. These variations arise from a subdued near-wall cycle and the development of a shear layer at the height of the cubes.

Application of PIV for Virtual Mass Coefficient Measurement

The particle image velocimetry (PIV) technique is employed for the measurement of the virtual mass of a submerged object, an important parameter in the two-fluid model, particularly so for reactor thermal-hydraulic and safety analyses. Instead of carrying out the measurement through traditional transient processes that mix steady-state drag, virtual mass force, and Basset force, a new PIV approach is developed for steady-state flows through the integration of the fluid kinetic energy around the object. The Basset force, an inseparable transient force in viscous flows, is eliminated in the new approach, making virtual mass quantification possible. This new method has been applied to the virtual mass measurement of a solid cylinder, and although the measurement uncertainty from the flow’s random fluctuations is substantial, the results are very encouraging. The results suggest that the existence of drag force in viscous flow affects the virtual mass, as the flow field is different from the ideal potential flow. When the measurement method was applied to the quantification of air bubbles, no reliable data were obtained due to complications from bubble lateral motions and deformation. Further study is needed for the PIV method to be employed for bubble virtual mass force measurements under steady-state flow conditions.

Pressure-driven tearing and thermal transport in finite-beta reversed field pinch computations

Nonlinear resistive-magnetohydrodynamics (MHD) computation with heating and anisotropic transport is applied to examine the interaction between thermal energy and magnetic fluctuations in inductively driven reversed-field pinches (RFPs). The magnetic fluctuations underlie magnetic field reversal through dynamo-like correlations, and they enhance thermal energy transport through fluctuations of parallel heat flux density. With the unfavorable magnetic curvature that exists across the RFP profile, thermal energy also affects the magnetic fluctuations. Computations with the NIMROD code [Sovinec et al., J. Comput. Phys. 195, 355–386 (2004)] integrate nonlinear MHD dynamics with energy transport and reproduce an RFP state with experimentally relevant values of plasma-β. Equilibria constructed from results of the 3D computations are analyzed to assess the sources of free energy in the saturated nonlinear state. Linear computations for these profiles show unstable modes of tearing parity. Their eigenfunctions are used to evaluate and compare stabilizing and destabilizing contributions to the kinetic energy integral. An assessment of the drives in the integral reveals that the pressure gradient drive is of comparable magnitude to the parallel current drive, and only the sum of the two surpasses the stabilizing contributions. Correlation of magnetic and parallel heat flux density fluctuations in the nonlinear computations shows that fluctuation-induced thermal conduction is the dominant mode of energy loss, as expected from experimental evidence. Decomposition of the fluctuating heat flux density shows that second-order correlations, alone, do not explain the total energy transport. Higher-order correlations are also important.

Atmospheric Conditions within Big Telescope Alt-Azimuthal Region and Possibilities of Astronomical Observations

The paper presents the results of analysis of astroclimatic conditions in the Big Telescope Alt-azimuthal (BTA) region (40°N–50°N; 35°E–55°E). Using data from the European center for medium-range weather forecast ReAnalysis (ERA-5), we estimated the averaged spatial distributions in total cloud cover, vertical integral of mean kinetic energy, vertical component of wind speed, and wind speed shears, as well as inverse values of Richardson number 1/Ri. An extensive region with the development of atmospheric flows is formed south and southeast of BTA in winter. High inverse values of the Richardson number, spatial heterogeneities in vertical wind speed, and significant wind speed shears in the lower atmosphere are observed in this region. In terms of turbulence development over BTA, the best time for astronomical observations falls in summer, when vertical shears of wind speed are weakened in the lower atmospheric layers. The situation is opposite in the upper troposphere. In winter, BTA is in the region of moderate vertical wind shears. In summer, a region with increased vertical wind speed shears is formed. Taking into account that the intensity of optical turbulence decreases rapidly with height, better image quality can be expected in summer. Such structure of the atmosphere does not allow one to directly apply atmospheric models in order to describe turbulence based on the turbulence strength as function of its ground values, or to use the classical model describing the turbulence velocity as function of air flow velocity at the height corresponding to the 200 hPa level.

JRAF: A Julia package for computation of relativistic molecular auxiliary functions

The evaluation of relativistic molecular integrals over exponential−type spinor orbitals requires the use of relativistic auxiliary functions in prolate spheroidal coordinates, and has been recently achieved (Bağcı and Hoggan (2015) [14]). This process is used in the solution of the molecular Dirac equation for electrons moving in a Coulomb potential. A series of papers on a method for fully analytical evaluation of relativistic auxiliary functions has been published [2, 3, 4] From the perspective of computational physics, these studies demonstrate how to deal with the integrals of the product of power functions with non−integer exponents and incomplete gamma functions. The computer program package used to calculate these auxiliary functions with high accuracy is presented. It is designed using the Julia programming language and yields highly accurate results for molecular integrals over a wide range of orbital parameters and quantum numbers. Additionally, the program package facilitates the efficient calculation of the angular momentum coefficients that arise from the product of two normalized Legendre functions centered at different atomic positions, and the determination of the rotation angular functions used for both complex and real spherical harmonics. Sample calculations are performed for two−center one−electron integrals over non−integer Slater−type orbitals, and the results prove the robustness of the package. Program summaryProgram Title: JRAFCPC Library link to program files:https://doi.org/10.17632/942xsbvfdf.1Developer's repository link:https://github.com/abagciphys/JRAF.jlLicensing provisions: MITProgramming language:Julia programming languageSupplementary material: An experimental version of the computer program package written in Mathematica programming language [5].External routines/libraries:Nemo computer algebra package for the Julia programming language [6], Cuba multidimensional numerical integration using different algorithms in Julia [7].Nature of problem: Relativistic molecular auxiliary function integrals result from the expression of a two−center two−electron Coulomb energy associated with a charge density. The Coulomb energy is transformed into kinetic energy integrals using Poisson's equation and the single−center potential, considering that the Laplace expansion for the Coulomb interactions is expressed in terms of normalized non−integer Slater−type orbitals [1]. Using the resulting expression for the two−center two−electron integrals, relativistic auxiliary function integrals are derived in prolate ellipsoidal coordinates. These auxiliary functions are generalized to the entire set of physical potential operators for the Coulomb potential case.The integral of the relativistic auxiliary functions have no closed−form solutions except that their parameters are integers. As such, the analytical evaluation of these functions is challenging. They are used in the solution of the matrix form representation of the molecular Dirac−Fock self−consistent field (SCF) equation.Solution method: A criterion that considers the symmetry properties of two−center two−electron molecular integrals is initially proposed [2]. This obviates the need for the computation of incomplete and complementary incomplete gamma functions, and utilizes their sum (P+Q=1). The resulting form of the integral of the relativistic molecular auxiliary functions is expressed in terms of the convergent series representation of incomplete beta functions. Recurrence relationships are then derived for each of these sub−functions [3]. The algorithm for computation of the auxiliary functions is based on the vectorization procedure defined in [4].

The kinocardiograph for assessment of changes in haemodynamic load in patients with chronic heart failure with reduced ejection fraction.

AimsThe kinocardiograph (KCG) is an unobtrusive device, consisting of a chest sensor, which records local thoracic vibrations produced in result of cardiac contraction and ejection of blood into the great vessels [seismocardiography (SCG)], and a lower back sensor, which records micromovements of the body in reaction to blood flowing through the vasculature [ballistocardiography (BCG)]. SCG and BCG signals are translated to the integral of cardiac kinetic energy (iK) and cardiac maximum power (Pmax), which might be promising metrics for future telemonitoring purposes in heart failure (HF). As a first step of validation, this study aimed to determine whether iK and Pmax are responsive to exercise‐induced changes in the haemodynamic load of the heart in HF patients.Methods and resultsFifteen patients with stable HF with reduced ejection fraction performed a submaximal exercise protocol. KCG and cardiac ultrasound measurements were obtained both at rest and at submaximal exercise. BCG iK over the cardiac cycle (CC) increased significantly (0.0026 ± 0.0017 to 0.0052 ± 0.0061 mJ.s.; P = 0.01) during exercise, in contrast to a non‐significant increase in SCG iK CC. BCG Pmax CC increased significantly (0.92 ± 0.89 to 2.03 ± 1.95 mJ/s; P = 0.02), in contrast to a non‐significant increase in SCG Pmax CC. When analysing the systolic phase of the CC, similar patterns were found. Cardiac output (CO) ratio (i.e. CO exercise/CO rest) showed a moderate, significant correlation with BCG Pmax CC ratio (r = +0.65; P = 0.008) and with SCG Pmax CC ratio (r = +0.54; P = 0.04).Conclusions iK and Pmax measured with the KCG, preferentially using BCG, are responsive to changes in the haemodynamic load of the heart in HF patients. The combination of the BCG and SCG sensor might be of added value to fully understand changes in haemodynamics and to discriminate between an HF patient and a healthy individual.

Numerical Simulation Research Based on Plasma Line-Tied Instability

In order to study the linear instability and turbulence of the boundary plasma, this paper uses the BOUT++ numerical simulation tool. Mainly analyze the driving mechanism of the instability of the ideal balloon model, and the instability suppression mechanism of the ion diamagnetic effect, and compare with the analysis results of the dispersion relationship. Subsequently, the integral dispersion relationship is used to analyze the suppression mechanism of the shear flow. In addition, the dispersion relationship is not suitable for analyzing the global effect of the shear flow. Due to the locality of the dispersion relationship, the integral dispersion relationship uses the numerical integration of the mode structure to resolve the dispersion relationship. Used to analyze shear flow. Then, use the numerical integration of kinetic energy in the whole space to study the contribution of these effects to free energy. Finally, using the above linear analysis method, the physical mechanism of these effects under the EAST divertor configuration is studied.

Finite-Horizon Kinetic Energy Optimization of a Redundant Space Manipulator

The minimization of energy consumption is of the utmost importance in space robotics. For redundant manipulators tracking a desired end-effector trajectory, most of the proposed solutions are based on locally optimal inverse kinematics methods. On the one hand, these methods are suitable for real-time implementation; nevertheless, on the other hand, they often provide solutions quite far from the globally optimal one and, moreover, are prone to singularities. In this paper, a novel inverse kinematics method for redundant manipulators is presented, which overcomes the above mentioned issues and is suitable for real-time implementation. The proposed method is based on the optimization of the kinetic energy integral on a limited subset of future end-effector path points, making the manipulator joints to move in the direction of minimum kinetic energy. The proposed method is tested by simulation of a three degrees of freedom (DOF) planar manipulator in a number of test cases, and its performance is compared to the classical pseudoinverse solution and to a global optimal method. The proposed method outperforms the pseudoinverse-based one and proves to be able to avoid singularities. Furthermore, it provides a solution very close to the global optimal one with a much lower computational time, which is compatible for real-time implementation.

Kinocardiography Derived from Ballistocardiography and Seismocardiography Shows High Repeatability in Healthy Subjects.

Recent years have witnessed an upsurge in the usage of ballistocardiography (BCG) and seismocardiography (SCG) to record myocardial function both in normal and pathological populations. Kinocardiography (KCG) combines these techniques by measuring 12 degrees-of-freedom of body motion produced by myocardial contraction and blood flow through the cardiac chambers and major vessels. The integral of kinetic energy (iK) obtained from the linear and rotational SCG/BCG signals, and automatically computed over the cardiac cycle, is used as a marker of cardiac mechanical function. The present work systematically evaluated the test–retest (TRT) reliability of KCG iK derived from BCG/SCG signals in the short term (<15 min) and long term (3–6 h) on 60 healthy volunteers. Additionally, we investigated the difference of repeatability with different body positions. First, we found high short-term TRT reliability for KCG metrics derived from SCG and BCG recordings. Exceptions to this finding were limited to metrics computed in left lateral decubitus position where the TRT reliability was moderate-to-high. Second, we found low-to-moderate long-term TRT reliability for KCG metrics as expected and confirmed by blood pressure measurements. In summary, KCG parameters derived from BCG/SCG signals show high repeatability and should be further investigated to confirm their use for cardiac condition longitudinal monitoring.

Assessment of left ventricular twist by 3D ballistocardiography and seismocardiography compared with 2D STI echocardiography in a context of enhanced inotropism in healthy subjects

Ballistocardiography (BCG) and Seismocardiography (SCG) assess the vibrations produced by cardiac contraction and blood flow, respectively, by means of micro-accelerometers and micro-gyroscopes. From the BCG and SCG signals, maximal velocities (VMax), integral of kinetic energy (iK), and maximal power (PMax) can be computed as scalar parameters, both in linear and rotational dimensions. Standard echocardiography and 2-dimensional speckle tracking imaging echocardiography were performed on 34 healthy volunteers who were infused with increasing doses of dobutamine (5–10–20 μg/kg/min). Linear VMax of BCG predicts the rates of left ventricular (LV) twisting and untwisting (both p < 0.0001). The linear PMax of both SCG and BCG and the linear iK of BCG are the best predictors of the LV ejection fraction (LVEF) (p < 0.0001). This result is further confirmed by mathematical models combining the metrics from SCG and BCG signals with heart rate, in which both linear PMax and iK strongly correlate with LVEF (R = 0.7, p < 0.0001). In this setting of enhanced inotropism, the linear VMax of BCG, rather than the VMax of SCG, is the metric which best explains the LV twist mechanics, in particular the rates of twisting and untwisting. PMax and iK metrics are strongly associated with the LVEF and account for 50% of the variance of the LVEF.

Effects of acquisition device, sampling rate, and record length on kinocardiography during position-induced haemodynamic changes

BackgroundKinocardiography (KCG) is a promising new technique used to monitor cardiac mechanical function remotely. KCG is based on ballistocardiography (BCG) and seismocardiography (SCG), and measures 12 degrees-of-freedom (DOF) of body motion produced by myocardial contraction and blood flow through the cardiac chambers and major vessels.ResultsThe integral of kinetic energy ( iK) obtained from the linear and rotational SCG/BCG signals was computed over each dimension over the cardiac cycle, and used as a marker of cardiac mechanical function. We tested the hypotheses that KCG metrics can be acquired using different sensors, and at 50 Hz. We also tested the effect of record length on the ensemble average on which the metrics were computed. Twelve healthy males were tested in the supine, head-down tilt, and head-up tilt positions to expand the haemodynamic states on which the validation was performed.ConclusionsKCG metrics computed on 50 Hz and 1 kHz SCG/BCG signals were very similar. Most of the metrics were highly similar when computed on different sensors, and with less than 5% of error when computed on record length longer than 60 s. These results suggest that KCG may be a robust and non-invasive method to monitor cardiac inotropic activity.Trial registration Clinicaltrials.gov, NCT03107351. Registered 11 April 2017, https://clinicaltrials.gov/ct2/show/NCT03107351?term=NCT03107351&draw=2&rank=1.

Cardiovascular adaptation to simulated microgravity and countermeasure efficacy assessed by ballistocardiography and seismocardiography

Head-down bed rest (HDBR) reproduces the cardiovascular effects of microgravity. We tested the hypothesis that regular high-intensity physical exercise (JUMP) could prevent this cardiovascular deconditioning, which could be detected using seismocardiography (SCG) and ballistocardiography (BCG). 23 healthy males were exposed to 60-day HDBR: 12 in a physical exercise group (JUMP), the others in a control group (CTRL). SCG and BCG were measured during supine controlled breathing protocols. From the linear and rotational SCG/BCG signals, the integral of kinetic energy (iK) was computed on each dimension over the cardiac cycle. At the end of HDBR, BCG rotational iK and SCG transversal iK decreased similarly for all participants (− 40% and − 44%, respectively, p < 0.05), and so did orthostatic tolerance (− 58%, p < 0.01). Resting heart rate decreased in JUMP (− 10%, p < 0.01), but not in CTRL. BCG linear iK decreased in CTRL (− 50%, p < 0.05), but not in JUMP. The changes in the systolic component of BCG linear iK were correlated to those in stroke volume and VO2 max (R = 0.44 and 0.47, respectively, p < 0.05). JUMP was less affected by cardiovascular deconditioning, which could be detected by BCG in agreement with standard markers of the cardiovascular condition. This shows the potential of BCG to easily monitor cardiac deconditioning.

Exponentially correlated Hylleraas-configuration interaction non-relativistic energy of the 1S ground state of the helium atom.

A generalization of the Hylleraas-Configuration Interaction method (Hy-CI) first proposed by Wang, et al., the Exponentially Correlated Hylleraas-Configuration Interaction method (E-Hy-CI) in which the single r ij of an Hy-CI wave function is generalized to a form of the generic type , is explored. This type of correlation, suggested by Hirshfelder in 1960, has the right behavior both in the vicinity of the rij cusp as rij goes to 0 and as rij goes to infinity; this work explores whether wave functions containing both linear and exponential r ij factors converge more rapidly than either one alone. The method of calculation of the two-electron E-Hy-CI kinetic energy and electron repulsion integrals in a stable and efficient way using recursion relations is discussed, and the relevant formulas are given. The convergence of the E-Hy-CI wave function expansion is compared with that of the Hy-CI wave function without exponential correlation factors, demonstrating both convergence acceleration and an improvement in the accuracy for the same basis. This makes the application of the E-Hy-CI method to systems with N > 4, for which this formalism with at most a single factor per term leads to solvable integrals, very promising. E-Hy-CI method variational calculations with up to 10080 expansion terms are reported for the ground 1 S state of the neutral helium atom, with a resultant nonrelativistic energy of -2.9037 2437 7034 1195 9831 1084 hartree for the best expansion.

Reducing energy consumption in plasma NO conversion utilizing plasma aerodynamic effect

Improving conversion efficiency with low power consumption is an important target in removing NO by plasma. In this paper, a novel design of surface dielectric barrier discharge (SDBD) reactor is proposed to control the flow fields. The influence of special input energy, inlet NO concentration, relative humidity, and oxygen content on NO conversion was investigated. The velocity and turbulence kinetic energy distribution in the reactor were simulated. The temperature during discharge was tested. The performance of configuration (a) is found better than that of configuration (b) in most cases. For example, the conversion efficiency was 73% and 52%(11.5 kV-50 kHz-300 ppm-2 L/min). And the energy efficiency was about 10.7 g/kWh and 7.7 g/kWh. The reaction time changes, temperatures, and turbulence kinetic energy are discussed. The reaction time changes and temperature are found to have little influence on the difference of results. The possible reason for this phenomenon is the change in turbulent kinetic energy. The products of the reactions are removed faster and the reactants are more uniformly distributed at a higher TKE. Based on simulation results, the integral of turbulent kinetic energy was 20 J/Kg without plasma and was significantly increased over 4956 J/Kg in configuration (a) and 4108 J/Kg in configuration (b)(11.5 kV-50 kHz-300 ppm-2 L/min). It is possibly the reason for a different performance of configuration (a) and (b). The phenomenon could be useful in NOx treatment or other gas-related plasma chemical processes.

Regularization of the Movement of a Material Point Along a Flat Trajectory: Application to Robotics Problems

A control problem of the robot’s end-effector movement along a predefined trajectory is considered. The aim is to reduce the work against resistance forces and improve the comfortability of the motion. The integral of kinetic energy and weighted inertia forces for the whole period of motion is introduced as a cost functional. By applying variational methods, the problem is reduced to a system of quasilinear ordinary differential equations of the fourth order. Numerical examples of solving the problem for movement along straight, circular and elliptical trajectories are presented. For the sake of clarity, the problem is studied for a specific kind of a 3D printer in the 2DoF approximation. However, in the case of negligible masses of moving elements compared the mass of an end-effector, the solution is universal, i.e., it remains the same for given trajectories.

Single super-vortex as a proxy for ocean surface flow fields

Abstract. Empirical flow field data evaluation in a well-studied ocean region along the US west coast revealed a surprisingly strong relationship between the surface integrals of kinetic energy and enstrophy (squared vorticity). This relationship defines a single isolated Gaussian super-vortex, whose fitted size parameter is related to the mean eddy size, and the square of the fitted height parameter is proportional to the sum of the square of all individual eddy amplitudes obtained by standard vortex census. This finding allows very effective coarse-grained eddy statistics with minimal computational efforts. As an illustrative example, the westward drift velocity of eddies is determined from a simple cross-correlation analysis of kinetic energy integrals.

Integrals arising in atomic Hylleraas methods

ABSTRACTMethods of integral computation for atomic Hylleraas methods are reviewed and difficulties in evaluating kinetic-energy integrals are noted. This problem can be avoided, at least for Hylleraas-CI, by expressing the kinetic-energy integrals in terms of those for the potential energy using formulas that exploit the properties of vector spherical harmonics. These formulas had probably not been found previously because their discovery uses concepts that do not arise in typical atomic or molecular computations. The validity of the new formulas is confirmed by examining a collection of test computations.

Closed-loop control of boundary layer streaks induced by free-stream turbulence

The central aim of the paper is to carry out a theoretical and numerical study of active wall transpiration control of streaks generated within an incompressible boundary layer by free-stream turbulence. The disturbance flow model is based on the linearized unsteady boundary-region (LUBR) equations, studied by Leib, Wundrow, and Goldstein [J. Fluid Mech. 380, 169 (1999)], which are the rigorous asymptotic limit of the Navier-Stokes equations for low-frequency and long-streamwise wavelength. The mathematical formulation of the problem directly incorporates the random forcing into the equations in a consistent way. Due to linearity, this forcing is factored out and appears as a multiplicative factor. It is shown that the cost function (integral of kinetic energy in the domain) is properly defined as the expectation of a random quadratic function only after integration in wave number space. This operation naturally introduces the free-stream turbulence spectral tensor into the cost function. The controller gains for each wave number are independent of the spectral tensor and, in that sense, universal. Asymptotic matching of the LUBR equations with the free-stream conditions results in an additional forcing term in the state-space system whose presence necessitates the reformulation of the control problem and the rederivation of its solution. It is proved that the solution can be obtained analytically using an extension of the sweep method used in control theory to obtain the standard Riccati equation. The control signal consists of two components, a feedback part and a feed-forward part (that depends explicitly on the forcing term). Explicit recursive equations that provide these two components are derived. It is shown that the feed-forward part makes a negligible contribution to the control signal. We also derive an explicit expression that a priori (i.e., before solving the control problem) leads to the minimum of the objective cost function (i.e., the fundamental performance limit), based only on the system matrices and the initial and free-stream boundary conditions. The adjoint equations admit a self-similar solution for large spanwise wave numbers with a scaling which is different from that of the LUBR equations. The controlled flow field also has a self-similar solution if the weighting matrices of the objective function are chosen appropriately. The code developed to implement this algorithm is efficient and has modest memory requirements. Computations show the significant reduction of energy for each wave number. The control of the full spectrum streaks, for conditions corresponding to a realistic experimental case, shows that the root-mean-square of the streamwise velocity is strongly suppressed in the whole domain and for all the frequency ranges examined.

Kinetic-energy matrix elements for atomic Hylleraas-CI wave functions.

Hylleraas-CI is a superposition-of-configurations method in which each configuration is constructed from a Slater-type orbital (STO) product to which is appended (linearly) at most one interelectron distance rij. Computations of the kinetic energy for atoms by this method have been difficult due to the lack of formulas expressing these matrix elements for general angular momentum in terms of overlap and potential-energy integrals. It is shown here that a strategic application of angular-momentum theory, including the use of vector spherical harmonics, enables the reduction of all atomic kinetic-energy integrals to overlap and potential-energy matrix elements. The new formulas are validated by showing that they yield correct results for a large number of integrals published by other investigators.

Evaluation of Hylleraas–CI atomic integrals by integration over the coordinates of one electron. IV. An improved algorithm for three-electron kinetic energy integrals

An improved algorithm to evaluate the nonrelativistic three-electron Hylleraas-Configuration Interaction (Hy-CI) kinetic energy integrals over Slater orbitals and the Hamiltonian in Hylleraas coordinates is shown. The resulting analytical expressions are general for all quantum numbers of the orbitals. From there, the restriction of employing orbitals with quantum numbers $l \le 2$ of the above algorithm presented in paper I of this series has been removed. With the new algorithm it is possible, in the direct integration method described in this series, to carry out Hy-CI atomic structure calculations including $f$-, $g$-, $\ldots$ , $l$ and higher angular-momentum Slater orbitals and to determine $F$, $G$, $\ldots$ , $L$ and states of higher order symmetry.