Numerical solution and stability analysis of transient MHD duct flow

Numerical simulation of liquid metal MHD flows in a conducting rectangular duct with triangular strips

Sparse radial basis function approximation with spatially variable shape parameters

A numerical study of magnetohydrodynamic flows and heat transfer past a circular cylinder in a duct under a strong magnetic field parallel to the cylinder axis is presented. In this configuration, the flow is quasi-two-dimensional and the modified Navier–Stokes equations are solved in a two–dimensional domain. The numerical simulations have been performed over a range of parameters including the Reynolds number 50 ≤ Re ≤ 3000, modified Hartmann number 50 ≤ Ha⋆ . 500, blockage ratio 0.1 ≤ β ≤ 0.5, offset ratio 0.25 ≤ γ ≤ 1, velocity amplitude 0 ≤ A ≤ 3 and forcing frequency 0 ≤ Ste ≤ 10. The primary aim of this study is to understand the fundamental mechanism that governs transition to unsteady flow in this system and exploit this for further improvements in heat transfer for MHD cooling duct flows. With this aim in mind, a spectral-element method is employed to compute the MHD flow and heat transfer past a confined circular cylinder in a rectangular duct. Meshes have been constructed to deal with the significant number of geometric flow parameters combinations. Thorough validation and grid resolution studies have been performed to ensure adequate domain sizes, and spatial and temporal resolutions to accurately resolve all flow and thermal features for the reported flow variable ranges. Studies show that the reported flow parameters are converged to better than 0.3% in terms of spatial and temporal accuracy and 1% with respect to the domain size. For the optimal growth studies, the dependence of energy growth on upstream domain length is also considered through the calculation of the energy growth over a fixed time span. This verifies that the effect of truncating the upstream length from 32d to 8d causes an error of less than 3% in the growth rate prediction. The critical Reynolds number Rec for the transition from steady to periodic flow is determined as a function of Ha⋆ and β, and this is found to increase with increasing Ha⋆ and β. In addition, the variation of the wake recirculation length in the steady flow regime is determined as a function of Reynolds number, Hartman number and blockage ratio, and a universal expression is proposed. The characteristics of heat transfer depend strongly on the proximity of the cylinder to the heated wall. For small blockage ratios, it increases significantly as the gap ratio decreases from 1 to 0.25. However, there is a substantial drop in heat transfer for high blockage ratio. Downstream cross-stream mixing induced by the cylinder wake is found to increase the heat transfer augmentation by more than a factor of two in some cases. The maximum gain in heat transfer generated by placing the cylinder in the channel near to the wall with that at the centerline was obtained for the lowest blockage ratio β = 0.1, as the cylinder is further approached the heated wall. For all β, a very significant transient energy growth is found in the subcritical regime below the onset of vortex shedding. This suggests a potential for the design of an actuation mechanism to invoke vortex shedding and then enhance heat transfer in these ducts. The energy amplification of the disturbances is found to decrease significantly with increasing Hartmann number and the peak growth shifts towards smaller times while it increases significantly with increasing blockage ratio. The structure of the optimal initial disturbance is found to be consistent across the all blockage ratios being tested. In line with similar problems, it convects along the separating region being amplified to the peak growth state downstream of the recirculation bubble. The maximum time for maximum energy growth τmax is found to increase significantly as recirculation length increases which demonstrates the amplifying nature of the separated shear layers in the wake. The critical Reynolds number for the onset of positive growth at different Hartmann numbers and blockage ratios is determined. It is found that it increases rapidly with increasing Hartmann number and blockage ratio. For all β, the peak energy amplification grows exponentially with Re from low Hartmann numbers. Direct numerical simulation studies in which the inflow is perturbed by random white noise confirms the predictions arising from the transient growth analysis: that is, the perturbation excites and feeds energy into the global mode. A considerable increase in heat transfer from the heated channel wall occurs from rotational oscillation of the cylinder, with a maximum enhancement of approximately 22% observed at higher amplitude over that for steady flow. The oscillation frequency range for effective enhancement is widened in both directions, while the frequency at which the peak of the Nusselt number occurs is shifted slightly to a lower frequency as A is increased. It is found that as the amplitude was reduced, the forcing frequency approaches the global frequency mode.

Inverse analysis of heat transfer across a multilayer composite wall with Cauchy boundary conditions

In many technical applications, reconstruction of the scattered data is often task. For big scattered dataset in n-dimensional space, the using some meshless method such as the radial basis function (RBF) approximation is appropriate. RBF approximation is based on the distance computation, and therefore, it is dimensionally non-separable. This approximation can be converted to an overdetermined linear system of equations which has to be solved.

The method of radial basis functions (RBFs) is a method of scattered data interpolation that has many applications in different fields. Based on the RBFs approach, two simple and accurate methods are presented in this paper to favor the solution of fractional optimal control problems (FOCPs) which are nominated as indirect and direct methods. In the first one, the considered FOCP is converted into a system of fractional differential equations (FDEs) that is called two-point boundary value problem (TPBVP) of fractional order. Then, both the context of minimization the total error and a joint application of Volterra integral equation will be used to rewrite this problems as as an unconstrained optimization problem that can be solved by using RBFs approximation. In direct one, we rewrite the FOCP as a classical static optimization problem that can be easily solved using known formulas for computing fractional derivatives of RBFs. Numerical example are given to prove the accuracy, effectiveness and feasibility of these methods.

Several technological applications involve the flow of liquid metals in ducts under a magnetic field, for instance, the coolants of fusion reactors. In this paper, using a magnetohydrodynamic MHD formulation based on the electric potential, we obtain an analytical solution for the flow of a liquid metal in a rectangular duct with two insulating walls and two perfectly conducting walls perpendicular to the applied uniform magnetic field. As the Hartmann number increases, the flow displays high velocities in the boundary layers attached to the insulating walls and a quasi-stagnant flow at the core. The effect of this flow pattern on the forced convection heat transfer is then explored numerically considering a uniform heat flux on either the conducting or insulating walls. Compared to the hydrodynamic case, the MHD flow enhances the heat transfer as the Hartmann number increases only in the case when the heat flux is applied at the insulating walls where high velocities are present. The increase of the local Nusselt number as the Péclet number grows indicates an efficient heat removal from the heated wall.

Radial basis function partition of unity methods for pricing vanilla basket options

For the design of the liquid-metal blanket in a fusion reactor, numerical calculations have been carried out on liquid-metal magnetohydrodynamic flows in rectangular ducts with sudden contractions. Conservation equations of fluid mass and fluid momentum and the Poisson equation for electrical potential have been solved numerically. The numerical calculations have been conducted for a Hartmann number of ~10 000; a Reynolds number of ~10 000; and contraction ratios (CRs) of 2, 3, and 4. The pressure loss through the contraction has been estimated by the loss coefficient ζ divided by the interaction parameter N, i.e. ζ/N. The loss coefficient ζ/N through the contraction parallel to the magnetic field is much larger than that through the corresponding contraction perpendicular to the magnetic field. The loss coefficient ζ/N increases consistently with the CR and does not change very much with N. While ζ/N also does not change very much with the wall conductance ratio for the contraction parallel to the magnetic field, ζ/N increases gradually with the wall conductance ratio for the contraction perpendicular to the magnetic field.

In this paper, the accuracy of soft computing technique in solar radiation prediction based on series of measured meteorological data (monthly mean sunshine duration, monthly mean maximum and minimum temperature) taking from Iseyin meteorological station in Nigeria was examined. The process, which simulates the solar radiation with support vector regression (SVR), was constructed. The inputs were monthly mean maximum temperature (Tmax), monthly mean minimum temperature (Tmin) and monthly mean sunshine duration ( $$ \bar{n} $$ ). Polynomial and radial basis functions (RBF) are applied as the SVR kernel function to estimate solar radiation. According to the results, a greater improvement in estimation accuracy can be achieved through the SVR with polynomial basis function compared to RBF. The SVR coefficient of determination R 2 with the polynomial function was 0.7395 and with the radial basis function, the R 2 was 0.5877.

The core flow approximation for liquid metal (LM), magnetohydrodynamic (MHD) duct flow is a method that ignores the contributions of viscous forces in the fluid. For a fully developed, steady state flow situation, this approximation leaves the magnetic forces to be balanced only by the pressure gradient and results in a greatly simplified momentum equation. The velocity field predicted by the core flow equations is obtained much more easily than that described by the full solution, which usually requires a numerical approach. For this reason it is desirable to use the core flow method for flow situations in which viscosity has little effect. Developed here is an analytic core flow solution for a square duct in an obliquely incident magnetic field which omits any special treatment of boundary layers. This solution is compared to the full solution method developed in the code MH2D and a parametric comparison is performed. This general analytic approach can be expanded to consider rectangular ducts or walls of different thickness and electrical conductivity. The latter, however, will greatly complicate the equations presented here.

Scaling radial basis functions via euclidean distance matrices

The radial basis function (RBF) approximation is a rapidly developing field of mathematics. In the paper, we are concerned with the measurement of scalar physical quantities at nodes on sphere in the 3D Euclidean space and the spherical RBF interpolation of the data acquired. We employ a multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 considered in Cartesian coordinates. Attention is paid to geodesic metrics that define the distance of two points on a sphere. The choice of a particular geodesic metric function is an important part of the construction of interpolation formula.We show the existence of an interpolation formula of the type considered. The approximation formulas of this type can be useful in the interpretation of measurements of various physical quantities. We present an example concerned with the sampling of anisotropy of magnetic susceptibility having extensive applications in geosciences and demonstrate the advantages and drawbacks of the formulas chosen, in particular the strong dependence of interpolation results on condition number of the matrix of the system considered and on round-off errors in general.

Monitoring of heart rate (HR) and its dynamics is necessary in ambulatory and telemedicine for diagnosis and treatment of diseases. Remote photoplethysmography (rPPG) allows for HR and its dynamics monitoring by video analysis of facial skin blood flow, and is ofparticular importance for patients with delicate skin such as infants, the elderly, or those with severe burn injuries. Unlike other contactless HR measurement methods, rPPG does not require special hardware, but allows to monitor HR on the basis of analyzing a sequence of video images of a person's face. rPPG involves a multi-step process including signal extraction, dimensionality reduction to estimate the photoplethysmographic (PPG) signal, and subsequent HR estimation. However, the presence of high-amplitude spikes due to subject movement, facial expressions, lighting fluctuations, video compression artifacts, ROI tracking errors, among others, can interfere the useful PPG signal, leading to inaccurate HR estimations. A method has been developed that allows to increase the accuracy of HR estimation in the rPPG problem due to its robustness to interferences inevitable during video recording. The proposed approach tackles the rPPG challenge by applying a pre-processing approximation of the signal finite difference using a single-layer neural network with radial basis function (RBF) inner layer. Transitioning to the signal finite difference helps reduce the amplitudes of irrelevant low-frequency peaks within the HR search range, thus avoiding their masking effect on the HR-related spectral peaks. The neural network's RBF approximation further diminishes irrelevant high-frequency spectral peaks when the number of RBF nodes is less than half the signal sample count. The correctness of the solutions is confirmed by numerical experiments carried out on the Mahnob-HCI public database.

The results of a theoretical and experimental investigation regarding the static and dynamic characteristics of shear walls are described. Solutions were obtained for the natural frequencies, mode shapes, and response quantities for an appropriate mathematical model, consisting of a distributed mass shear panel having a concentrated roof load mass at its top. Damping for such a structure may be represented in terms of an experimentally determined equivalent viscous damping ratio, veq. Using numerical analysis techniques, appropriate equations were derived and programmed, and a digital computer used to obtain the desired response quantitites resulting from either harmonic or arbitrary forcing functions. The 1940 El Centro earthquake, N-S component ground accelerations were utilized for most of the latter calculations. Using experimentally obtained static-cyclic load-deflection curves, values of veq were found to average 0.08-0.10 for low design-type loads, a range which was further verified by vibration generator and “snap” test dynamic testing. It was thus deemed satisfactory to recommend constant damping ratios of 0.07-0.08 for response calculations. Numerical accuracy of the computer solutions was investigated, with it being noted the acceleration values provided a good criterion for stability of solutions. Being least accurate because of the numerical procedure used, the accelerations always became unstable at a time prior to that of the displacements, thus giving a simple check as to the time span of correct solutions. Another effective procedure simply involved making two computer runs with the same input data, but different time increments, and then comparing results. If results were basically the same, the larger time increment was adequate; otherwise, another run was made with an even smaller time increment, answers compared, etc., until results essentially agreed.