Fourier Solution Research Articles

Some Observations on the TLM Numerical Solution of the Laplace Equation

This paper describes progress on the TLM modelling of the Laplace equation, in particular, how the rate of convergence is influenced by the choice of scattering parameter for a particular discretisation. The hypothesis that optimum convergence is achieved when the real and imaginary parts for the lowest harmonic in a Fourier solution cancel appears to be upheld. The Fourier solution for the problem has been advanced by a better understanding of the nature of the initial excitation. The relationship between the form of the initial condition used in this and many other numerical solutions of the Laplace equation and oscillatory behavior in the results is given a firmer theoretical basis. A correlation between TLM numerical results and those obtained from matrix spectral radius calculations has confirmed much previous work.

Accurate heat transfer measurements using thermochromic liquid crystal. Part 2: Application to a rotating disc

Encapsulated thermochromic liquid crystal (TLC) can accurately measure surface temperature in a variety of heat transfer and fluid-flow experiments. In Part 1 of this two-part paper, two narrow-band liquid crystals were specifically calibrated for application to experiments on a disc rotating at high speed (∼5000 rpm). Part 2 describes how these crystals were used to measure the surface temperature on the disc in a transient experiment that models the flow of internal cooling air in a gas turbine. The TLC was viewed through the transparent polycarbonate disc using a digital video camera and strobe light synchronised to the disc frequency. The convective heat transfer coefficient, h, was subsequently calculated from the one-dimensional solution of Fourier’s conduction equation for a semi-infinite wall. The analysis accounted for the exponential rise in the air temperature driving the heat transfer, and for experimental uncertainties in the measured values of h. The paper focuses on the method used, and sample experimental results are provided to demonstrate the accuracy and potency of the technique.

Heat problems for a starlike shaped plate

The Fourier solution of the heat problem for a circular plate is generalized to a starlike shaped plate. We show that the classical solution can be used even in this more general case, provided that a suitable change of variables in the polar co-ordinate system is performed.

Practical Application of Two-Turning-Point Theory to Mountain-Wave Transmission through a Wind Jet

Abstract A Fourier method is used to model mountain waves that have nearby turning points in a wind jet. In Fourier space, the propagation equations are solved by ray theory. To correct for the ray singularity at a turning point without time-consuming special-function evaluations, the ray solution is linearly interpolated across the breakdown region. The Fourier solutions for the spatial wavefield are compared with mesoscale model simulations in two cases: two-dimensional flow over idealized topography with uniform stratification and a sech-squared wind profile and three-dimensional flow over the island of Jan Mayen with stratification and wind profiles taken from radiosonde measurements. The latter case reveals the partial transmission of trapped mountain waves into the stratosphere.

A Fourier Approach to Model Electromagnetic Fields Scattered by a Buried Rectangular Cavity

We consider the problem of a two‐dimensional rectangular cavity in a PEC half plane covered by layers of material with uniform thickness. The rectangular geometry allows for an application of Fourier methods to solve the problem. The paper will also discuss how to compute the far field scattering once the solution is found. The fast mode matching approximation technique developed by Morgan and Schwering will be applied to this layered cavity problem setting. Numerical results will show that using the Fourier solution on the layered problem combined with Morgan and Schwering′s technique can produce good approximations while using much less computation time than would be required for the entire solution.

A COMPARISON OF DIFFUSION MODELED BY TWO MIXED DERIVATIVE EQUATIONS

Fourier and Bessel function solutions of two mixed derivative equations are investigated. For the appropriate sign of the material constants in the derivation of the mixed derivative equation, we obtain both Fourier and Bessel function solutions that tend to the corresponding solutions of the phenomenological diffusion equation. For the opposite sign of the material constants, the solutions diverge.

A quick method for thermal diffusivity estimation: Application to several foods

A reliable, quick and userfriendly method for thermal diffusivity estimation has been developed. An appropriate software tool, based on least square optimization of a finite difference solution of Fourier’s equation , has been created and an appropriate measuring cell has been designed and made in order to decrease the systematic error in probe positioning. The method has been experimentally validated and its results have been compared with those obtained by three other available methods. Several foods (tomato products, low-acid pasta sauces, olive pate, confectioner’s custard and apricot jam) were tested and in every case the method proved to be effective. The developed software also allowed estimation of thermal diffusivity via heat penetration curves obtained by variable temperature treatments. So it has been also possible to exclude the contribution of container material from the estimation of thermal diffusivity of liquid packed foods. The proposed method turns out to be a useful tool for scientific design of several processes, such as sterilization and pasteurization, and for correct control of transport, storage and distribution of foods.

Experimental validation of an analytical model for predicting the thermal and hydrodynamic capabilities of flat micro heat pipes

An analytical model by Lefèvre and Lallemand [F. Lefèvre, M. Lallemand, Coupled thermal and hydrodynamic models of flat micro heat pipes for the cooling of multiple electronic components, Int. J. Heat Mass Transfer 49 (2006) 1375–1383] that couples a 2D hydrodynamic model for both the liquid and the vapor phases inside a flat micro heat pipe (FMHP) and a 3D thermal model of heat conduction inside the FMHP wall has been modified. It consists of superposing two independent solutions in order to take into account the impact of evaporation or condensation on the equivalent thermal conductivity of the capillary structure. The temperature, pressure and velocity fields can be determined using Fourier solutions. The model has been experimentally validated based on literature data from a grooved FMHP. Two new correlations for the equivalent thermal conductivities during evaporation and condensation inside rectangular micro-grooves have been proposed based on a numerical database. The influence of the saturation temperature and geometry on the maximum heat flux transferred by the system is presented.

Application of the principle of symmetry to the solution of the slichter problem

This problem was first studied in [1] under the assumption that the water depth in channels is negligible. However, the use of the Fourier solution method results in the following fact. The convergence of the Fourier series obtained for components of the filtration rate becomes very slow as the ratio L/T increases, i.e., in the most cases which are of interest for practice. In [2] (P. 181) one solves a similar problem with h = 0 by the method of conformal mappings. In [3] one solves the problem by the same method, taking into account the presence of the water in channels. However, the introduction of an auxiliary parametric variable for the conformal mapping of the flow domain and the plane of the velocity hodograph curve results in rather intricate formulas. For this reason, no numeric calculations were performed and the influence of the parameters onto the character of the flow was not studied. Let us apply the Riemann–Schwarz symmetry method ([4], P. 147) and the known properties of elliptic functions ([5], pp. 69–73). Below we show that this allows us to avoid difficulties connected with the use of the method of conformal mappings and to obtain an exact analytic solution to the problem by comparatively simple calculations (using the known special or elementary functions).

Analytical Modelling of Voltage Balance Dynamics for a Flying Capacitor Multilevel Converter

This paper presents a strategy for the analytic determination of the natural voltage balancing dynamics of flying capacitor converters. The approach substitutes double Fourier series representations of the pulsewidth modulation (PWM) switching signals into a nonlinear dynamic circuit model of the converter. The result reduces to a linearized state-space model that can be readily solved, with the Fourier solution coefficients defining the state-space matrix terms. The solution can be readily developed for converters of any level, and allows rapid analytical investigation of the dynamic (and static) balancing behavior over a wide range of conditions. Furthermore, the approach allows powerful strategies such as root locus to be used to investigate the converter's performance as a function of changes in parameters such as modulation index and load. The analysis approach has been fully verified by comparing it against experimental results on a low voltage prototype converter.

Explicit solutions for boundary value problems in diffusion of heat and mass

The Fourier's infinite series solutions of transient diffusion in finite solids, with convection boundary conditions, contain eigenvalues, which are defined by implicit transcendental equations. The eigenvalue equations have to be evaluated numerically, making the use of the Fourier's solution difficult. Using the successive substitution method, explicit expressions were derived for the first eigenvalue λ 1, in plates, cylinders and spheres, as a function of the Biot number. The derived expressions for λ 1( Bi), together with the one-term Fourier series solution, yield simple explicit solutions for temperature or concentration profiles in plates, cylinders and spheres. The explicit solutions are valid for Fo>0.2.

Evaluating constitutive scaling models for application to compressible microflows

We demonstrate here, for the first time, the constitutive scaling approach applied to simulate a fully compressible, non-isothermal gas microflows within a mainstream computational physics framework. First, the physics underlying constitutive-relation scaling models is discussed, including the effects of velocity slip, temperature jump and the Knudsen layer. Results for Couette-type flows in micro-channels, including heat transfer effects, are then reported and we show comparisons with both traditional Navier–Stokes–Fourier solutions and independent numerical studies. We discuss the limitations of the constitutive scaling process, such as the breakdown of the model as the Knudsen number increases and the influence of the wall interaction model on the numerical results. Advantages of the constitutive scaling technique are described, with particular reference to the practicality of using it for microscale engineering design.

Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise

One-dimensional wave equations with cubic power law perturbed by Q-regular additive space-time random noise are considered. These models describe the displacement of nonlinear strings excited by state-independent random external forces. The presented analysis is based on the representation of its solution in form of Fourier-series expansions along the eigenfunctions of Laplace operator with continuous, Markovian, unique Fourier coefficients (the so-called commutative case). We shall discuss existence and uniqueness of Fourier solutions using energy-type methods based on the construction of Lyapunov-functions. Appropriate truncations and finite-dimensional approximations are presented while exploiting the explicit knowledge on eigenfunctions of related second order differential operators. Moreover, some nonstandard partial-implicit difference methods for their numerical integration are suggested in order to control its energy functional in a dynamically consistent fashion. The generalized energy $\cE$ (sum of kinetic, potential and damping energy) is governed by the linear relation $\E [\varepsilon(t)] = \E [\varepsilon(0)] + b^2 trace (Q) t / 2$ in time $t \ge 0$, where $b$ is the scalar intensity of noise and $Q$ its covariance operator.

Central Ideas of Buckingham (1907): A Century Later

A century ago, Edgar Buckingham presented data and a theoretical conceptualization of soil moisture movement. His work constitutes a milestone in the history of soil physics and more generally, of movement of multiple fluid phases in porous media. Starting from first principles, Buckingham formulated a conceptual model to make rational sense of long‐term observations of evaporation from soil columns. Central to his model were the notion of a capillary potential, soil moisture retention curve, and potential‐dependent hydraulic conductivity. Buckingham recognized that whereas heat capacity and thermal conductivity were independent of temperature in Fourier's heat equation, in the case of soil moisture, the slope of the soil‐moisture retention curve (analogous to specific heat) and capillary conductivity were both strong functions of capillary potential. Noting that available solutions of Fourier's linear differential equation did not apply to moisture movement in soils, Buckingham was skeptical that the nonlinear problem could be solved mathematically. This is perhaps why he did not present a partial differential equation for soil‐moisture movement. Such an equation would be gvien in 1931 by Richards. Despite considerable efforts, analytical solutions to Richards' equation can be obtained only under simplifying assumptions. While these solutions give valuable insights into patterns of soil moisture movement, they cannot adequately address problems of the natural soil environment. Although Buckingham's model remains the only workable physical‐mathematical conceptualization for studying moisture movement in soils, his own skepticism of its ability to reliably describe moisture movement in soils is still valid. More profound, his skepticism captures the limitations inherent in precisely describing the behavior of earth systems. This paper examines Buckingham's central ideas in light of developments in groundwater hydrology and soil mechanics and reflects on the limits of our ability to quantitatively understand moisture movement in unsaturated soils.

Wide-Angle Shift-Map PE for a Piecewise Linear Terrain—A Finite-Difference Approach

A wide-angle parabolic wave equation solution is presented using shift-map and finite-difference techniques. The corresponding split-step Fourier solution is well known. The solution using finite-difference technique, where the standard parabolic wave equation is modified into the so-called Claerbout equation allowing propagation angles up to 45deg from the paraxial direction, is also well known. Here, we present an extension to that solution in which the shift-map technique is incorporated into the finite-difference scheme allowing a varying terrain to be considered. The result is a solution that corresponds to the well known split-step solution, which is believed to perform well for terrain slopes up to 10deg-15deg and discontinuous slope changes on the order of 15deg-20deg. This solution is a first-order one with respect to the terrain slope. However, when using the finite-difference technique, it is also possible to find a second-order solution with respect to the terrain slope. This new solution performs well for slopes up to about 15deg and discontinuous slope changes up to about 30deg, which is an improvement.

Overall thermal diffusivity of materials using a 3D cartesian model. Effect of sample geometry on the measurements

In this work, we analyse the influence of the sample shape on the experimental determination of the thermal diffusivity of materials used in several engineering fields. By means of a simple method we obtain good estimations of the overall thermal diffusivity for homogeneous or inhomogeneous materials. The method is based on the combination of simple temperature measurements with the analytical solution of the 3D differential Fourier’s equation applied to cartesian geometry. The classical transient analytical solution based on the separation of space and time variables is adapted to particular thermal boundary conditions. By associating these solutions with the measurements obtained on a simple experimental bench, the thermal characterization of materials can be achieved with a satisfactory precision. The present 3D study attempts to complete previous works dealing with cartesian and cylindrical 1D and 2D models. Based on the same procedure, it sheds further light on the fast multidimensional surface phenomena of heat exchange. Although we conclude that the 3D model does not provide the directional values typical of anisotropic materials, our results revealed that thermal diffusivity can be useful in other respects, being advantageous in the overall thermal characterization needed for sizing certain systems.

Analytic Three-Dimensional Model of a Double-Shielded Giant Magnetoresistive Perpendicular Head Using a Singular Function Expansion

By matching a singular function approximation to the Fourier solution at the air-bearing surface (ABS), we have derived a three-dimensional (3-D) analytic model of a shielded giant magnetoresistive (GMR) head, with side shields, for perpendicular replay. We present an explicit expression for the potential in the ABS and give values for the parameters in that expression for a range of practical head dimensions. Using only a few terms of this singular potential model, we show that the vertical field is accurate to within 1% of the sensor potential in the region of the medium for the majority of head dimensions suitable for magnetic recording.

LOCAL HEAT TRANSFER MEASUREMENT FROM A PAIR OF LONGITUDINAL VORTICES USING A TRANSIENT LIQUID CRYSTAL TECHNIQUE

In transient experiments, it is common practice to employ thermochromatic liquid crystal (TLC) to measure the local heat transfer coefficients on the wall surface. This method depends on the solution of Fourier's conduction equation with the boundary conditions of the air temperature. It is impossible to create a step-change in the air temperature because of thermal inertia in the heater, or in the ducting upstream of the test section. Therefore, the local heat transfer coefficient must be calculated from a more general solution to the one-dimensional conduction equation. The present study describes a calibration method in the TLC, a design of a mesh heater, and a calculation procedure of the local heat transfer coefficient from the general solution to the one-dimensional conduction for a semi-infinite wall when the air temperature rises exponentially through the mesh heater. Also, an experiment is presented as an application of the present method to measure the local heat transfer distributions on the wall surface cooled by air flow and disturbed by vortex generators.

TH-E-L100J-01: Ultrasound Reflectivity Imaging with a Split-Step Fourier Propagator for Cancer Detection and Diagnosis in Heterogeneous Breasts

Purpose: To improve resolution and reduce speckle in ultrasound breast images by accounting for ultrasound scattering from breast heterogeneities during reflectivity image reconstruction. Method and Materials: X-ray mammography often fails to detect cancers in dense breasts, while breast ultrasound has the potential to detect them. Breast heterogeneities, particularly in dense breasts, generate significant ultrasound scattering. Properly handling ultrasound scattering is critical for reliable cancer detection and diagnosis in dense breasts. Ultrasound wave propagation in the breast is governed by the acoustic-wave equation in heterogeneous media, which can be decomposed into two one-way wave equations describing wave propagation in opposite directions. A split-step Fourier solution of a one-way wave equation is used for backpropagation of reflected ultrasound waves. The backpropagation consists of two steps: one phase-shift step in the frequency-wavenumber domain, and another phaseshift step in the frequency-space domain. During the backpropagation of ultrasound wavefields, heterogeneous breast sound-speed models obtained from transmission ultrasound tomography are used to approximately account for ultrasound wave scattering. The reflectivity imaging method based on the split-step Fourier propagator is applied to computer-generated ultrasound data and in-vivo ultrasound breast data acquired using a ring transducer array. The ultrasound images are compared with those obtained using a uniform sound-speed model. Results: Comparison of ultrasound reflectivity images obtained using heterogeneous breast sound-speed models with those obtained with a uniform model shows that ultrasound scattering of breast heterogeneities needs to be taken into account to obtain high-resolution and high-quality breast images. Conclusion: Using heterogeneous sound-speed models for ultrasound wave backpropagation during reflectivity image reconstruction significantly improves image resolution and reduces speckle. The resolution and quality of ultrasound reflectivity images are further enhanced with increasing accuracy and resolution of transmission ultrasound tomography.

Effect of inplane boundary constraints on the response of thick general (unsymmetric) cross-ply plates

A hitherto unavailable HSDT (higher-order shear deformation theory)-based boundary-discontinuous Fourier solution to the class of problems of general cross-ply plates that cannot be solved using the classical Navier or Levy type approach is presented. A self-adjoint differential system of five highly coupled fourth-order linear partial differential equations, subjected to SS1 and SS4-type simply supported boundary conditions prescribed at all four edges, is solved analytically using the boundary-discontinuous double Fourier series technique. The numerical accuracy of the solution is ascertained by studying the convergence characteristics of deflections and moments of an antisymmetric cross-ply plate. The primary focus of the present study is to investigate the effect of inplane boundary constraints on the response of a thick asymmetrically laminated cross-ply plate, while keeping the remaining end constraints unaltered. Important numerical results presented include sensitivity of the predicted response quantities of interest to lamination, material property, thickness effects and inplane end constraint as well as their interactions.