Truncated Taylor Series Expansion Research Articles

Skin friction and pressure: the “footprints” of turbulence

The problems of exact state reconstruction and approximate state estimation based on wall information in a wall-bounded incompressible unsteady flow are addressed. It is shown that, if in an arbitrarily small neighborhood of time t precise measurements are made of the two components of wall skin friction and the wall pressure, all terms in the Taylor-series expansions of the unsteady flow state near the wall at time t may be determined (in the linear setting, this determination may be made based on skin-friction measurements alone). Combining this fact with the analyticity of solutions of the nonlinear Navier–Stokes equation and the unique continuation theorem for analytic functions, in theory complete reconstruction of a fully-developed turbulent flow in a channel at any Reynolds number at time t is possible given only information about the unsteady flow available at the wall in a neighborhood of time t, without knowledge of the initial conditions of the flow. Thus, skin-friction and pressure measurements on the wall in a neighborhood of time t provide a unique “footprint” of the entire unsteady turbulent flow state; no other flow can have the same footprint. Indeed, higher-order terms are shown to uniformly improve the correlation of truncated Taylor-series expansions with the DNS of a turbulent flow near the wall. However, such series extrapolations amplify measurement noise, as they require differentiation in both space and time of the measurements, and the radius of convergence of the Taylor series expansions is less than 10 wall units. The so-called linear stochastic estimation technique, in which the polynomials forming the basis of the series expansion are replaced by well-behaved functions (such as POD modes) on the entire flow domain also demonstrates very poor convergence. In light of these limitations on direct extrapolations from measurements in the practical setting, an adjoint-based algorithm is presented and numerically tested for estimating the state of an entire turbulent channel-flow system based on a time history of noisy measurements at the wall. This algorithm effectively uses the unsteady nonlinear Navier–Stokes equation itself as a filter to find the flow solution that is most consistent with the available measurements.

Synchronization sensitivity of multicarrier systems

AbstractIn this paper, we give an overview of the sensitivity of a number of multicarrier systems to carrier and clock synchronization errors. A comparison is made between orthogonal frequency division multiplexing/multiple access (OFDM(A)) and two combinations of the orthogonal multicarrier technique and the code‐division multiple access (CDMA) technique, i.e. multicarrier CDMA (MC‐CDMA), where the spreading is accomplished in the frequency domain and multicarrier direct‐sequence CDMA (MC‐DS‐CDMA), where the spreading is done in the time domain. We evaluate the effect of small synchronization errors on the BER performance by deriving simple analytical expressions for the BER degradation that are based upon truncated Taylor series expansions. To allow a fair comparison, all considered systems are able to accommodate the same number of users and each user operates at the same data rate. Under these conditions, we show that all multicarrier systems exhibit the same sensitivity to carrier phase jitter and timing jitter. Further, when the number of carriers is equal to the maximum number of users, the different multicarrier systems also are equally affected by a carrier frequency offset or a clock frequency offset. Copyright © 2004 AEI

Effect of measurement volume size on turbulent flow measurement using ultrasonic Doppler method

This paper presents the results of an investigation on the effects of measurement volume size on the mean velocity profile and the Reynolds stress for fully developed turbulent pipe flows. The study employs the ultrasonic velocity profile method, which is based on the ultrasonic Doppler method. The ultrasonic Doppler method offers many advantages over conventional methods for flow rate measurement in the nuclear power plant piping system. This method is capable of measuring the instantaneous velocity profile along the measuring line and is applicable for opaque liquids and opaque pipe wall materials. Furthermore, the method has the characteristic of being non-intrusive. Although it is applicable to various flow conditions, it requires a relatively large measurement volume. The measurement volume of the present method has a disk-shape determined by the effective diameter of the piezoelectric element and the number of the wave cycles of the ultrasonic pulse. Considering this disk-shaped measurement volume and expressing the time-averaged velocity in a truncated Taylor series expansion around the value at the center of the measuring control volume, the value of the velocity can be obtained. The results are then compared with the data obtained from DNS and LDA measurements. The result shows that the effect of the measurement volume size appears in the buffer region and viscous sublayer.

A normalized gradient descent algorithm for nonlinear adaptive filters using a gradient adaptive step size

A fully adaptive normalized nonlinear gradient descent (FANNGD) algorithm for online adaptation of nonlinear neural filters is proposed. An adaptive stepsize that minimizes the instantaneous output error of the filter is derived using a linearization performed by a Taylor series expansion of the output error. For rigor, the remainder of the truncated Taylor series expansion within the expression for the adaptive learning rate is made adaptive and is updated using gradient descent. The FANNGD algorithm is shown to converge faster than previously introduced algorithms of this kind.

Numerical solution of time-delayed optimal control problems by iterative dynamic programming

This work presents a numerical method to solve the optimal control problem with time-delayed arguments and a fixed terminal time. A series of auxiliary states obtained from the linearly truncated Taylor series expansion are used to represent the status of a time-delayed state at different time intervals. The backward iterative dynamic programming (IDP) technique can thus be directly employed to solve the delay-free optimal control problem with augmented states. Five numerical examples are provided, demonstrating the proposed method's effectiveness in solving the time-delayed optimal control problems. Copyright © 2000 John Wiley & Sons, Ltd.

Progressive genetic algorithm for solution of optimization problems with nonlinear equality and inequality constraints

A new approach, identified as progressive genetic algorithm (PGA), is proposed for the solutions of optimization problems with nonlinear equality and inequality constraints. Based on genetic algorithms (GAs) and iteration method, PGA divides the optimization process into two steps; iteration and search steps. In the iteration step, the constraints of the original problem are linearized using truncated Taylor series expansion, yielding an approximate problem with linearized constraints. In the search step, GA is applied to the problem with linearized constraints for the local optimal solution. The final solution is obtained from a progressive iterative process. Application of the proposed method to two simple examples is given to demonstrate the algorithm.

Solution of the linear shallow water equations by the fourth‐order leapfrog scheme

Numerical schemes of the first and second order of approximation introduce numerical distortion when the wave propagation over a long distance is investigated. To alleviate this problem, the fourth‐order leapfrog scheme is constructed. The standard leapfrog method is based on the truncated Taylor series expansion which depicts an error proportional to the second‐order terms. In the proposed method the numerical solution is corrected for these terms. The space and time corrections work well in diminishing numerical dispersion and dissipation.

Convection in a concentric annulus with heat generating porous media and a permeable inner boundary

The effects of a permeable inner boundary on the maximum temperature and the convective flows were investigated numerically for a two-dimensional horizontal annulus containing a uniformly heat generating porous media. The time-dependent governing equations were nondimensionalized and put into an explicit finite difference numerical form. The finite difference equations were obtained from truncated Taylor series expansions using a central differencing scheme. Nondimensional temperatures and streamlines were obtained for heat generation rates ranging from 20 to 1,500 W/m 3 corresponding to a range of modified Rayleigh numbers of 324 to 24,340 and for several combinations of isothermally heated wall conditions for annuli of radius ratio of 2. It was found that multi-cellular flows occur at the highest Rayleigh numbers investigated.

Unified Analysis of Compressive Packed Beds, Filter Cakes, and Thickeners

Abstract Fluid—particle systems are widely used throughout industry. Using an ad hoc approach to model a particular fluid—solid process lacks generality. A unified approach using continuum theory for multiphase systems is applied here to evaluate a packed bed, filter cake, and continuous gravity thickener. The unified approach has the advantage that the governing equations are obtained by simplifying the generalized continuum equations. The simplifying assumptions are obvious from the equations and do not require intuitive understanding as does the ad hoc approach. Also, for similar localized conditions the same constitutive relations can be applied when the same material is used in several processes. In the packed bed, filter cake, and thickener modeled here, a compressible solid matrix behavior is represented by a truncated Taylor series expansion for the solid phase stress. The packed bed and filter cake behaviors are evaluated for a range of pump powers and bed masses or cake heights. The thickener be...

Convection in heat generating porous media in a concentric annulus with a permeable outer boundary

The effects of a permeable outer boundary on the maximum temperature and the convective flows was numerically investigated in a two-dimensional horizontal annulus containing a uniformly heat generating porous media. The time-dependent governing equations were nondimensionalized and put into an explicit finite difference numerical form. The finite difference equations were obtained from truncated Taylor series expansions using a central differencing scheme. Nondimensional temperatures and streamlines were obtained for heat generation rates ranging from 20 to 1,500 W/m 3 (corresponding to a range of modified Rayleigh numbers of 324 to 24,340) and for several combinations of isothermally heated wall conditions for annuli of radius ratio of 2.

Multiple moment expansion of fully coupled aspherical earth seismograms

Summary An asymptotic theory is developed to generate long-period synthetic seismograms on the aspherical earth. For given source and receiver, the method requires only the knowledge of aspherical earth structure along the source-receiver great circle, as well as the transverse gradients and curvature of this structure along the great circle. The theory is developed here for spheroidal modes observed on the vertical component. Each isolated spheroidal mode pSl is represented in the time domain by a truncated Taylor series expansion, with the nth expansion coefficient expressed as the nth moment of the multiplet spectrum about a fiducial frequency. The resulting expansions are valid up to order 1/l and account for all of the effects associated with refraction and focusing of the equivalent surface waves.

The feasibility of motion and structure from noisy time-varying image velocity information

This research addresses the problem of noise sensitivity inherent in motion and structure algorithms. The motion and structure paradigm is a two-step process. First, we measure image velocities and, perhaps, their spatial and temporal derivatives, are obtained from time-varying image intensity data and second, we use these data to compute the motion of a moving monocular observer in a stationary environment under perspective projection, relative to a single 3-D planar surface. The first contribution of this article is an algorithm that uses time-varying image velocity information to compute the observer's translation and rotation and the normalized surface gradient of the 3-D planar surface. The use of time-varying image velocity information is an important tool in obtaining a more robust motion and structure calculation. The second contribution of this article is an extensive error analysis of the motion and structure problem. Any motion and structure algorithm that uses image velocity information as its input should exhibit error sensitivity behavior compatible with the results reported here. We perform an average and worst case error analysis for four types of image velocity information: full and normal image velocities and full and normal sets of image velocity and its derivatives. (These derivatives are simply the coefficients of a truncated Taylor series expansion about some point in space and time.) The main issues we address here are: just how sensitive is a motion and structure computation in the presence of noisy input, or alternately, how accurate must our image velocity information be, how much and what type of input data is needed, and under what circumstances is motion and structure feasible? That is, when can we be sure that a motion and structure computation will produce usable results? We base our answers on a numerical error analysis we conduct for a large number of motions.

Gain-sensitivity augmentation for near-optimal control of linear parameter-dependent plants

This paper develops a parameter-adaptive version of the steady-state linear quadratic Gaussian controller for plants with structured parameter dependencies. Both scalar and vector parameter perturbations are treated. The design method, referred to as gain-sensitivity augmentation, is based on approximating the optimal parameterdependent regulator and filter gain matrices with truncated Taylor series expansions. The coefficient matrices of the Taylor series expansions are referred to as gain-sensitivity matrices and are precomputed off-line. Parameter information, assumed to be available on-line by direct measurement or estimation, is used to adjust the gain matrices to near-optimal values.

Determination of first swing stability limit of multimachine power systems through Taylor series expansions

A new technique of determining first swing stability limit of a power system through checking the existence of peaks of rotor angles of the severely disturbed generators in the postfault period is presented. The existence of peaks is checked by observing the roots of time derivatives of rotor angles of these generators expressed through truncated Taylor series expansions. The stability limit and hence the critical clearing time corresponds to the maximum value of fault duration for which at least one of the roots of the derivative of the angle is real positive. The proposed method has been tested on four different power networks and the critical clearing times obtained were found to be in very good agreement with those obtained by simulation of system dynamics. The method is computationally very fast compared with the existing methods.

Efficient ray tracing through a realistic ionosphere

An algorithm for accurate three‐dimensional ray tracing is presented for realistic, nonspherical ionospheric electron density distributions, ignoring effects of the magnetic field. At the start of each ray path increment a truncated Taylor series expansion of the electron density results in a computationally efficient, closed‐form solution of the Euler‐Lagrange equations for the increment. The technique is applied to computations of ray paths in a nonspherical ionosphere which is specified by a climatological model used for HF over‐the‐horizon (OTH) radar studies. The effects of tilts in a sunrise transition region on signal losses and ray path deviations are found to be substantial. Results of various ionospheric approximations used for single‐site transmitter location are calculated.

Large displacement analysis of plates by a stress-based finite element approach

A mixed-hybrid incremental variational formulation, involving orthogonal rigid rotations and a symmetric stretch tensor, is proposed for finite deformation analysis of thin plates and shells. Isoparametric eight-noded elements are based upon the Kirchhoff-Love hypotheses, the assumption of plane stress, and the moderately large rotations of Von Karman plate theory. Semilinear elastic isotropic material properties are assumed, and the right polar decomposition of the deformation gradient is used. The symmetrized Biot-Luré (Jaumann) stress measure gives a unique complementary energy density and a set of variational principles with a priori satisfaction of linear momentum balance, a posteriori angular momentum balance, and interelement traction reciprocity by means of Lagrange multipliers. The incremental modified Newton-Raphson solution procedure is generated by a truncated Taylor series expansion of the functional in a total Lagrangian formulation. The theory is applied to laterally loaded and buckled thin plates, and numerical results are compared with truncated series solutions.

An Optimization Technique for Lumped-Distributed Two Ports

A frequency domain direct efficient analysis and an optimization technique of a large class of lumped-distributed networks containing active elements are presented. Sensitivity and Hessian matrix calculations are performed using truncated Taylor series expansion of two-port parameters of subnetworks. An interactive computer program was developed to demonstrate the application of the method. Examples of network optimization are included to illustrate the powerfuless of the technique.

Consensus Values and Weighting Factors.

A method is presented for the statistical analysis of sets of data which are assembled from multiple experiments. The analysis recognizes the existence of both within group and between group variabilities, and calculates appropriate weighting factors based on the observed variability for each group. The weighting factors are used to calculate a "best" consensus value from the overall experiment. The technique for obtaining the consensus value is applicable to either the determination of the weighted average value, or to the parameters associated with a weighted least squares regression problem. The calculations are made by using an iterative technique with a truncated Taylor series expansion. The calculations are straightforward, and are easily programmed on a desktop computer. An examination of the observed variabilities, both within groups and between groups, leads to considerable insight into the overall experiment and greatly aids in the design of future experiments.

Analysis and control of time-delay systems using finite dimensional approximations

Analysis and control of time-delay systems governed by differential-difference equations are considered using finite dimensional approximations obtained from truncated Taylor series expansion of the delay terms. Through the use of simple examples a tutorial presentation is given of some of the problems in control and analysis caused by the finite dimensional models.

Interpolation of normal mode eigenvalues in the frequency domain

Direct computation of the Fourier inversion integral to obtain the time dependent pressure field for explosive sources using normal mode theory has been inhibited by the high resolution required in the frequency domain. Very large numbers of mode sets would normally be required even for bandlimited data. This number can be considerably reduced through the use of a sequence of narrow‐band expansions based on the interpolation of eigenvalues between frequencies using a truncated Taylor series expansion. This approach relies on the use of very rapidly computable modal group velocities to obtain the first order term, dkn/dω. This paper shows the errors incurred in interpolating eigenvalues between discrete frequencies for several bandwidths and orders of approximation. Estimates of the second frequency derivative from a three point difference scheme are shown to considerably reduce the errors. Results thus obtained indicate that this approach, based on a sequence of narrow‐band expansions, may make possible the extension of time independent normal mode modeling to the time domain. Also, this technique will have application in octave‐band averaging of time independent propagation loss. [Work supported by the Naval Electronic Systems Command.]