Time-dependent Finite-difference Method Research Articles

Finite-difference model of mode shape changes of the Myanmar pat wain drum circle with tuning paste on drum head

The tuning process of drums differ considerably between Western and Southeast Asian drums. While Western drums, like tom-toms, snare, or bass drums of a rock or jazz drum kit, are tuned by changing the tension of the drum head, many drums in Southeast Asia are tuned by adding a paste consisting of rice and ash. As an example, the tuning of the Myanmar pat wain, a unique drum circle of a hsain wain orchestra is investigated. The pat wain has 20 pitched drums and is played with great virtuosity. It was investigated by the author during a field trip in northern Myanmar during its tuning process. The drum head is modeled using a time-dependent Finite-Difference Method (FDM) to estimate pitch and mode changes due to adding the paste. The results are compared to the field recordings which show considerable damping and pitch glides with low-pitched drums. Also the tuning of the drum circle is compared to different tuning systems showing similarities to a tuning pattern of Cambodian roneat deik metalophone.

Analysis of relaxation times on the human head using the thermal wave model

Cellular phones and bluetooth devices have become essential electronic devices for many people in today’s society. However, these devices emit low radio frequencies that may be harmful to both the skin and brain tissue. The thermal wave model is used to predict an unsteady temperature distribution of microwaves as it transmits through the human head. A time dependent finite difference method models a multilayer system depicting this external heat source passing through three mediums: skin, bone, and brain. This model accounts for the heat generated within the body from the chemical reactions of the tissue and a relaxation time in order to account for the response time the tissue takes caused by the perturbation. Previous human head studies took these values to be negligible. Parameters of the head for both the adult and child are shown. Various parameters, including frequency, power density, and relaxation time were varied. It was shown that the relaxation parameters was important; however, changing other variables such as frequency and power density within a certain range are ineffective. This manuscript is aimed to identify the potential threats on the human body caused by the everyday use of today’s technologies such as cellular phones and bluetooth devices.

Process Modeling of the Batch Acetylation of Cotton Linter Pulp

Following previous kinetic investigations of the acetylation of cotton linter pulp (Luo et al. 2013), a mathematical model was set up based on the mass and energy balances to simulate the batch acetylation process; a particular goal was to predict the temperature profile. The equations were discretized using a time-dependent finite difference method. The parameters for the model, including the kinetic parameters and heat transfer rate, were well estimated from the literature and our previous work. The model using the least-mean-square-error criterion optimizes the unknown parameters. The proposed model provides an accurate prediction of the process, including the temperature profile, peak temperature, and DS value under the peak temperature. The content of the catalyst, sulfuric acid, has a dramatic effect on the temperature profile. A slight increase in sulfuric acid content will lead to a faster dissolution process for the sulfated cellulose fibers, whereas the reaction rate of the sulfated fibers with acetic anhydride in the liquid phase is not affected. The optimized values of the activation energy are 11.0 and 7.6 kJ/mol for the dissolution processes with low and high sulfuric acid contents, respectively.

A new time-dependent finite-difference method for relativistic shock acceleration

We present a new approach to calculate the particle distribution function about relativistic shocks including synchrotron losses using the method of lines with an explicit finite difference scheme. A steady, continuous, one dimensional plasma flow is considered to model thick (modified) shocks, leading to a calculation in three dimensions plus time, the former three being momentum, pitch angle and position. The method accurately reproduces the expected power law behaviour in momentum at the shock for upstream flow speeds ranging from 0.1c to 0.995c (1 < \Gamma < 10). It also reproduces approximate analytical results for the synchrotron cutoff shape for a non-relativistic shock, demonstrating that the loss process is accurately represented. The algorithm has been implemented as a hybrid OpenMP--MPI parallel algorithm to make efficient use of SMP cluster architectures and scales well up to many hundreds of CPUs.

Numerical prediction of three-dimensional time-dependent viscoelastic extrudate swell using differential and algebraic models

This study investigates the numerical simulation of three-dimensional time-dependent viscoelastic free surface flows using the Upper-Convected Maxwell (UCM) constitutive equation and an algebraic explicit model. This investigation was carried out to develop a simplified approach that can be applied to the extrudate swell problem. The relevant physics of this flow phenomenon is discussed in the paper and an algebraic model to predict the extrudate swell problem is presented. It is based on an explicit algebraic representation of the non-Newtonian extra-stress through a kinematic tensor formed with the scaled dyadic product of the velocity field. The elasticity of the fluid is governed by a single transport equation for a scalar quantity which has dimension of strain rate. Mass and momentum conservations, and the constitutive equation (UCM and algebraic model) were solved by a three-dimensional time-dependent finite difference method. The free surface of the fluid was modeled using a marker-and-cell approach. The algebraic model was validated by comparing the numerical predictions with analytic solutions for pipe flow. In comparison with the classical UCM model, one advantage of this approach is that computational workload is substantially reduced: the UCM model employs six differential equations while the algebraic model uses only one. The results showed stable flows with very large extrudate growths beyond those usually obtained with standard differential viscoelastic models.

The flux creep effect in superconducting magnetic bearings levitation force

The main characteristic of superconducting magnetic bearings (SMB's) is the levitation force between the permanent magnet rotor and the superconducting stator. This force can relax with time due to the flux creep effect in the superconductor. In order to investigate the effect of the flux creep in the levitation force, several dynamic levitation force measurements were made in simple axis-symmetric SMB's. These measurements were conducted with different magnet approaching speeds and with the magnet stopped after the movement at a fixed position. The SMB's were composed of superconducting YBCO seeded melt textured cylinders and Nd-Fe-B cylindrical permanent magnets, with different YBCO diameters, heights and pinning characteristics. The measurements were compared with simulations that have taken the flux creep into account. The simulation algorithm finds the current density profile inside the superconductor at each magnet approaching time step. The current density profile was found by solving an integral inside the superconductor and using a power-law for calculating the electric field. This power-law relationship can be derived from the Anderson-Kim model. The time evolution was obtained using time dependent finite difference method and the levitation force was calculated from the Lorentz force. The simulation results show a good agreement with measurements.

Analysis of sound fields in three demensional space by the time-dependent finite-difference method based on the leap frog algorithm

Analysis of sound fields in three demensional space by the time-dependent finite-difference method based on the leap frog algorithm

Laminar impingement jet Mach number and temperature effects on heat transfer

A laminar, axisymmetric jet flow of an ideal gas impinging on an infinite flat plate has been studied for various jet exit Mach numbers, jet height to radius ratios, and jet exit temperature to wall temperature ratios by a time-dependent finite-difference method applying the Lax-Wendroff single-step scheme. From the field calculations of temperature and flow, the friction coefficient and Stanton number at the wall are evaluated. Stagnation wall pressures are compared with those behind a normal shock for a supersonic jet and with an isentropic compression for a subsonic jet. All of the flow parameters, the friction coefficient, and the heat transfer are dependent on the jet exit Mach number, the jet height to radius ratio, the jet exit to wall temperature ratio, the Prandtl number, and the jet exit Reynolds number.

Particle Impaction in Axially Symmetric Supersonic Flow

The results of computations to predict the particle collection efficiency in axially symmetric supersonic flow are presented. In particular, a supersonic gas impinging on a truncated circular cylinder aligned parallel to the freestream has been analyzed. Inviscid solutions are obtained by a slight variant to MacCormack's time-dependent finite difference method. In imposing outflow boundary conditions of fixed Mach number, linear extrapolation of remaining gas properties, and a reflection technique on the cylinder walls, reasonable agreement is demonstrated for the shock detachment distance and stagnation pressure loss. It is demonstrated that the capture efficiency is influenced by particle inertia, departure from Stokes drag, and gas compressibility. By defining an effective Stokes number in terms of the particle stopping distance downstream of the shock that includes the effects of non-Stokesian drag and slip flow, and rescaling the Stokes number with the shock detachment distance, the collection efficiency is shown to be correlated with a single universal curve.

Computation of Gas Flowfields in Supersonic Particle Probes

The results of computations to predict the properties of gas flowfields in supersonic particle probe inlets are presented. In particular, a supersonic gas impinging on a thin walled cylindrical probe aligned parallel to the freestream has been analyzed. Inviscid solutions are obtained by a slight variant to MacCormack’s time-dependent finite difference method. Imposing outflow boundary conditions of fixed static pressure at the probe outlet, linear extrapolation of remaining gas properties, and a reflection technique on the probe wall including the tip, reasonable agreement is demonstrated for the shock detachment distance, stagnation pressure loss and gas ingestion rate. With some spacial smoothing and initial conditions characteristic of gas properties back of the shock, the numerical method converged in reasonable computation times, yielding a shock thickness of 2.5 grid spacings with no oscillations near the shock front.

Flowfield calculations past an entry probe with ablated nose shape

The effects of ablated nose shapes on the flowfield solutions are studied, using a time-dependent finite-difference method developed by Kumar, et al. (1979). Solutions are obtained for the laminar flow of a radiating mixture of H-He in chemical equilibrium past a blunt axisymmetric body at zero angle of attack. The freestream conditions correspond to a point on a typical Jovian entry trajectory, and the initial probe shape is a 45-deg half-angle spherically blunted cone. It is found that as nose bluntness increases, the following occur: in the nose region, shock standoff distances and radiative heating rates increase substantially; surface pressure level increases, but convective heating rates decrease.

Calculation of intense blowing on a blunt body and an airfoil

Various aspects of the problem of intense blowing through the surface of bodies have, been theoretically studied by a number of authors, within the framework of inviscid flow theory. A detailed bibliography on this topic is given, e.g., in [1, 2]. The well-known approaches to solution of this problem have a limited area of application. For example, asymptotic methods can be used for hypersonic flow regimes only at relatively low levels of the blown gas momentum (Λ = Ω2 = ϱovo2/ ϱ∞V∞2 ≪ 1). The same limitation applies to the numerical method of straight lines [2]. The forward Eulerian calculation schemes [3, 4] smear the contact discontinuity severely, and cannot handle the case where the blown gas and the gas in the incident flow have different thermodynamic properties (γo ≠ γ∞). This paper presents results of a numerical investigation of supersonic flow over two-dimensional and axisymmetric bodies with intense blowing on the forward surface, performed using a time-dependent finite-difference method [5] with an explicit definition of the contact interface between the two cases. The calculations encompass a family of elliptic cylinders with semiaxis ratio 0.5 ≤ δ ≤ 4, a flat-face cylinder, and a flat plate with rounding near the midsection, with variations in the blowing law, the incident flow Mach number M∞ (3 ≤ M∞ ≤ 10), the adiabatic indices, and the blowing parameter 0 ≤ Ω ≤ 0.5.

Comparison of a two-dimensional shock impingement computation with experiment

Results of computations of two-dimensional viscous blunt-body flowfields with an impinging shock wave, with a time-dependent finite-difference method employed to solve the complete set of Navier-Stokes equations, are compared with experimental results. The experimental results were obtained in a 20-inch hypersonic tunnel with a planar shock impinging on the cylindrical leading edge of a fin, hence with the shock parallel to the centerline of the leading edge, so that type III and type IV interference patterns were generated. Close agreement is found. The overall effects of smoothing and grid size on the calculations are determined. A 31 x 51 mesh is adequate for wall pressure values (except in peaked regions).

Numerical Computation of Two-Dimensional Viscous Blunt Body Flows with an Impinging Shock

Two-dimensional viscous blunt body flows with an impinging shock have been computed using a time-dependent finite-difference method which solves the complete set of Navier-Stokes equations for a compressible flow. For low Reynolds number flows, the entire flow field, including the bow shock and impinging shock, has been captured in the computation. For higher Reynolds number flows, the bow shock is treated as a discontinuity across which the Rankine-Hugoniot equations are applied, while the boundary layer and interaction regions are captured as before. Using this latter shock-fitting approach, a Type III shock interaction flow field has been computed with flow conditions corresponding to the space shuttle orbiter freestream conditions at 61 km (200,000 ft).