Comparison of two methods for stress relaxation data presentation of solid foods

Published exponential relaxation equations, derived from Maxwellian models, were used to generate data for linear representation in the form ofP(0) ·t/(P(0) —P(t)) =k ₁ +k ₂t whereP(t) is the decaying parameter (force, stress or modulus),P(0) its initial value (att = 0) andk ₁ andk ₂ constants. The computer plots indicated that the fit of this normalized and linearized form was excellent for equations containing at least three exponential decay terms. The fit was not as good for some of the two-term exponential equations mainly due to the lack of accurate account for the initial stage of the relaxation process. In all the cases, however, the linear representation could clearly reveal the general rheological character of the analysed materials in terms of the relative degree of solidity.     

High‐order 𝒞1 finite‐element interpolating schemes—Part II: Nonlinear semi‐Lagrangian shallow‐water models

The finite‐element, semi‐implicit, and semi‐Lagrangian methods are used on unstructured meshes to solve the nonlinear shallow‐water system. Several ??¹ approximation schemes are developed for an accurate treatment of the advection terms. The employed finite‐element discretization schemes are the P–P₁ and P₂–P₁ pairs. Triangular finite elements are attractive because of their flexibility for representing irregular boundaries and for local mesh refinement. By tracking the characteristics backward from both the interpolation and quadrature nodes and using ??¹ interpolating schemes, an accurate treatment of the nonlinear terms and, hence, of Rossby waves is obtained. Results of test problems to simulate slowly propagating Rossby modes illustrate the promise of the proposed approach in ocean modelling. Copyright © 2007 John Wiley & Sons, Ltd.     

Effects of element order and interface reconstruction in FEM/volume‐of‐fluid incompressible flow simulation

In previous studies, the moment‐of‐fluid interface reconstruction method showed dramatic accuracy improvements in static and pure advection tests over existing methods, but this did not translate into an equivalent improvement in volume‐tracked multimaterial incompressible flow simulation using low‐order finite elements. In this work, the combined effects of the spatial discretization and interface reconstruction in flow simulation are examined. The mixed finite element pairs, Q₁Q₀ (with pressure stabilization) and Q₂P _{? 1} are compared. Material order‐dependent and material order‐independent first and second‐order accurate interface reconstruction methods are used. The Q₂P _{? 1} elements show significant improvements in computed flow solution accuracy for single material flows but show reduced convergence using element‐average piecewise constant density and viscosity in volume‐tracked simulations. In general, a refined Q₁Q₀ grid, with better material interface resolution, provided an accuracy similar to the Q₂P _{? 1} element grid with a comparable number of degrees of freedom. Moment‐of‐fluid shows more benefit from the higher‐order accurate flow simulation than the LVIRA, Youngs', and power diagram interface reconstruction methods, especially on unstructured grids, but does not recover the dramatic accuracy improvements it has shown in advection tests. Published 2012. This article is a US Government work and is in the public domain in the USA.     

Finite difference analysis of laminar free convection flow past a non isothermal vertical cone

A finite-difference analysis for the transient free convection flow of an incompressible viscous fluid past a vertical cone with variable wall surface temperature T _w^′ (x) = T _∞^′ + a x ⁿ varying as power function of distance from the apex (x = 0) is presented here. The dimensionless governing equations of the flow that are unsteady, coupled and non-linear partial differential equations are solved by an efficient, accurate and unconditionally stable finite difference scheme of Crank-Nicolson type. The velocity and temperature fields have been studied for various parameters such as Prandtl number and n (exponent in power law variation in surface temperature). The local as well as average skin-friction and Nusselt number are also presented and analyzed graphically. The present results are compared with available results in literature and are found to be in good agreement.     

New stabilized finite element method for time‐dependent incompressible flow problems

A new stabilized finite element method is considered for the time‐dependent Stokes problem, based on the lowest‐order P₁?P₀ and Q₁?P₀ elements that do not satisfy the discrete inf–sup condition. The new stabilized method is characterized by the features that it does not require approximation of the pressure derivatives, specification of mesh‐dependent parameters and edge‐based data structures, always leads to symmetric linear systems and hence can be applied to existing codes with a little additional effort. The stability of the method is derived under some regularity assumptions. Error estimates for the approximate velocity and pressure are obtained by applying the technique of the Galerkin finite element method. Some numerical results are also given, which show that the new stabilized method is highly efficient for the time‐dependent Stokes problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Thermoeconomic design and analysis of constant cross-sectional area fins

A closed-form model for the second-law-based thermoeconomic optimization of constant cross-sectional area fins, is discussed with an example problem. In this approach, different monetary values are attached to the irreversible losses caused by the finite temperature difference heat transfer (T _{0Δ T} ) and pressure drop (T _{0Δ P} ) in the fin application. In addition, a simplified closed-form solution is presented for the case when the capital cost of the fin is negligible and only operational costs are considered. To illustrate the usefulness of the present analytical approach, the simplified cost optimized results are compared with the numerical results obtained from Poulikakos and Bejan's analysis, who have assumed same monetary values for T _{0Δ T} and T _{0Δ P} . Furthermore, the influence of important fin thermal, physical, geometrical and cost parameters on the optimum Reynolds numbers Re _Dopt and Re _Lopt are presented in algebraic forms, and also graphical results are shown for the case of pin and plate fin, as examples. Received on 26 January 1998     

Heat transfer in the flow of a viscoelastic fluid over a stretching surface

An analysis is made of heat transfer in the boundary layer of a viscoelastic fluid flowing over a stretching surface. The velocity of the surface varies linearly with the distance x from a fixed point and the surface is held at a uniform temperature T _w higher than the temperature T _∞ of the ambient fluid. An exact analytical solution for the temperature distribution is found by solving the energy equation after taking into account strain energy stored in the fluid (due to its elastic property) and viscous dissipation. It is shown that the temperature profiles are nonsimilar in marked contrast with the case when these profiles are found to be similar in the absence of viscous dissipation and strain energy. It is also found that temperature at a point increases due to the combined influence of these two effects in comparison with its corresponding value in the absence of these two effects. A novel result of this analysis is that for small values of x, heat flows from the surface to the fluid while for moderate and large values of x, heat flows from the fluid to the surface even when T _w >T _∞. Temperature distribution and the surface heat flux are determined for various values of the Prandtl number P, the elastic parameter K ₁ and the viscous dissipation parameter a. Numerical solutions are also obtained through a fourth-order accurate compact finite difference scheme. Received on 14 October 1997     

Modelling of rotational factor in notched bend specimen under general and local yield situation

The location of the plastic hinge axis in a three point SEN bend specimen is a highly controversial issue. An unambiguous and reliable estimation of rotational factor (r_p) is very essential for the accurate determination of CTOD data. In contrast to the numerous studies reported on the r_p determination in a cracked situation, limited information is available for a blunt notch situation, although many engineering structures do contain notchlike defects with finite root radius. An attempt is made to determine r_p for two situations, namely well below the general yield and around the general yield. The work is based on a theoretical estimation of the plastic zone size using the stress concentration factor and the elastic as well as the elastic-plastic stress distribution. A theoretical estimation of r_p in both the pseudo-elastic and the elastic-plastic situation is estimated through analytical modelling involving factors like plastic zone size, bend angle and notch opening displacement. The values of the rotational factor are found to increase from a small value to around 0.29 in a well below general yield situation to 0.53 to 0.54 in a general yield situation with continued loading. A wide discrepancy in the P/P_GY ratio separating the two situations, i.e. well below general yield and around general yield, is observed. Consideration of the elastic and the elasto-plastic stress distribution indicates a much smaller value of P/P_GY as compared to the ratio obtained from experimental load-displacement plots.     

Structure of the flow of gas escaping from a gap in a pipeline

A study is made of the two-dimensional steady flow of gas escaping from a circular gap formed when the ends of pipes move apart along their common symmetry axis. A study is made of the rearrangement of the flow from the shockless flow in the neighborhood of the symmetry axis to the shocked flow. A numerical investigation gave the value H_* of the width of the gap at which the transition takes place from a shockless flow structure to one with a shock. A study is made of the influence on H_* of the pressure ratio ?_a (?_a = P_a/P₀, where P_a is the pressure in the ambient space and p₀ is the stagnation pressure) and the specific—heat ratio. Godunov's scheme [1] was used for the numerical realization.     

On-line measurement of elasticity and viscosity in flowing polymeric liquids

A new slit-die rheometer (the Stressmeter) for on-line and sample measurement of the viscosity, , and the first normal stress difference, N ₁, in steady shear flow for molten polymers and other high-viscosity liquids is described. Two liquid-filled transverse slots, located in one die wall near the center station, give pressures P ₂ and P ₃ from whose difference the wall shear stress is calculated. In the other die wall at a location opposite the center of the P ₂ slot is a flush-mounted transducer, giving a pressure P ₁. N ₁ is calculated from the hole pressure P ^* = P ₁–P ₂. A metering pump, used to measure the flow rate Q, is supplied with melt either from an extruder (online mode) or from a pressurized sample cylinder (sample mode). The wall shear rate is calculated from Q and ; the Weissenberg-Rabinowitsch correction and a new small-viscous-heating-correction algorithm (affecting ) are used. Viscous heating corrections are small; entrance and exit errors are negligible. The instrument is tested by comparing its results with those obtained from cone-plate and capillary rheometers. Measurement ranges extend to = 200 kPa, = 3000 s^–1, and temperature = 250°C.Dedicated to Prof. Dr. J. Meissner on the occasion of his retirement from the chair of Polymer Physics at the Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland     

Signal-processing techniques for low signal-to-noise ratio laser Doppler velocimetry signals

A variety of methods have been developed to obtain acurate frequency estimates from laser Doppler velocimetry (LDV) signals. Rapid scanning and fiber optic LDV systems require robust methods for extracting accurate frequency estimates with computational efficiency from data with poor signal-to-noise ratios. These methods typically fall into two general categories, time domain parametric techniques and frequency domain techniques. The frequency domain approach is initiated by transforming the Doppler bursts into the frequency domain using the fast Fourier transform (FFT). From this basic transformation a variety of interpolation procedures (parabolic, Gaussian, and centroid fits) have been developed to optimize the frequency estimation accuracy. The time domain approaches are derived from the parametric form of a sinusoid. The estimation of constants in this relationship is performed to satisfy specific constraints, typically a minimization of a variance expression. A comparison of these techniques is presented using simulated signals and additive Gaussian and Poisson white noise. The statistical bias and random errors for each method are presented from 200 signal simulations at each condition. Frequency estimation via the FFT with zero-padding and a Gaussian interpolation scheme was found to produce the lowest bias and random errors.List of symbols A(z) eigenfilter or characteristic polynomial - a _{m + 1} eigenvector - f frequency, Hz - f normalized frequency f = f/f _s - _d Doppler frequency estimate - f _i frequency of FFT spectral bin - f ^s sampling frequency - N number of sample points in data set - P _i ith power spectral line from PSD - r _xx (i) autocorrelation coefficient for time lag i - R_{Mn + 1} autocorrelation matrix of order M+1 - T sampling period - f spectral resolution for FFT, f = 1/N t - t sampling interval     

Efficient preconditioning techniques for finite‐element quadratic discretization arising from linearized incompressible Navier–Stokes equations

We develop an efficient preconditioning techniques for the solution of large linearized stationary and non‐stationary incompressible Navier–Stokes equations. These equations are linearized by the Picard and Newton methods, and linear extrapolation schemes in the non‐stationary case. The time discretization procedure uses the Gear scheme and the second‐order Taylor–Hood element P₂?P₁ is used for the approximation of the velocity and the pressure. Our purpose is to develop an efficient preconditioner for saddle point systems. Our tools are the addition of stabilization (penalization) term r?(div(·)), and the use of triangular block matrix as global preconditioner. This preconditioner involves the solution of two subsystems associated, respectively, with the velocity and the pressure and have to be solved efficiently. Furthermore, we use the P₁?P₂ hierarchical preconditioner recently proposed by the authors, for the block matrix associated with the velocity and an additive approach for the Schur complement approximation. Finally, several numerical examples illustrating the good performance of the preconditioning techniques are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

Global nonlinear stability in the Bénard problem for a mixture near the bifurcation point

The nonlinear stability of the motionless state of a binary fluid mixture heated and salted from below, in the Oberbeck-Boussinesq scheme, for stress-free and rigid-rigid boundary conditions and Schmidt numbers P_C greater than Prandtl numbers P_T, is studied in the region around the bifurcation point of linear instability. An improvement of the results in Mulone [11] is found for small values of p = P _C /P _T and P_T. For p sufficiently large the critical nonlinear Rayleigh number is very close to the linear one (with relative difference less than in the sea water case) Received December 12, 2002 / Published online April 23, 2003 RID="a" ID="a" e-mail: mbasurto@dmi.unict.it RID="b" ID="b" e-mail: lombardo@dmi.unict.it ID="Communicated by Brian Straugham, Durham"     

A discontinuous Galerkin‐based sharp‐interface method to simulate three‐dimensional compressible two‐phase flow

A numerical method for the simulation of compressible two‐phase flows is presented in this paper. The sharp‐interface approach consists of several components: a discontinuous Galerkin solver for compressible fluid flow, a level‐set tracking algorithm to follow the movement of the interface and a coupling of both by a ghost‐fluid approach with use of a local Riemann solver at the interface. There are several novel techniques used: the discontinuous Galerkin scheme allows locally a subcell resolution to enhance the interface resolution and an interior finite volume Total Variation Diminishing (TVD) approximation at the interface. The level‐set equation is solved by the same discontinuous Galerkin scheme. To obtain a very good approximation of the interface curvature, the accuracy of the level‐set field is improved and smoothed by an additional P_NP_M‐reconstruction. The capabilities of the method for the simulation of compressible two‐phase flow are demonstrated for a droplet at equilibrium, an oscillating ellipsoidal droplet, and a shock‐droplet interaction problem at Mach 3. Copyright © 2015 John Wiley & Sons, Ltd.     

An adaptive domain decomposition method for simulation of transport in porous media

A Galerkin finite element method is used along with a self-adaptive strategy of domain discretisation to model dispersion in an axisymmetric cylindrical porous medium. A solution strategy is proposed based on the use of a Gear scheme for the time stepping and partial vectorisation of the code. The domain is highly discretised in the area of the sharp transient front, while the remainder is coarsely discretised. The area covered by the fine mesh is determined by the value of the local concentration gradients. Numerical results are presented for the one and two dimensional cases.Nomenclature a _L Longitudinal dispersivity - a _T Transverse dispersivity - [A] Global matrix - C Concentration - D Dispersion coefficient - D _o Diffusion coefficient - {F} Solicitation vector - {G} Solicitation vector - h Finite element length - k Constant - [K] Global matrix of coefficients - [M] Mass matrix - n Outward pointing normal vector - P Pressure - Pe Peclet number - t Time - v Interstitial velocity Greek Symbols Boundary - Test function - Domain     

Detonability of THDCPD-exo–air mixtures

In the frame of industrial risk and propulsive application, the detonability study of JP10–air mixtures was performed. The simulation and measurements of detonation parameters were performed for THDCPD-exo/air mixtures at various initial pressure (1 bar < P ₀ < 3 bar) and equivalence ratio (0.8 < Φ < 1.6) in a heated tube (T ₀ ~ 375 K). Numerical simulations of the detonation were performed with the STANJAN code and a detailed kinetic scheme of the combustion of THDCPD. The experimental study deals with the measurements of detonation velocity and cell size λ. The measured velocity is in a good agreement with the calculated theoretical values. The cell size measurements show a minimum value for Φ ~ 1.2 at every level of initial pressure studied and the calculated induction length L _i corresponds to cell size value with a coefficient k = λ/L _i = 24 at P ₀ = 1 bar. Based on the comparison between the results obtained during this study and those available in the literature on the critical initiation energy E _c, critical tube diameter d _c and deflagration to detonation transition length L _DDT, we can conclude that the detonability of THDCPD–air mixtures corresponds to that of hydrocarbon–air mixtures.

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This paper is based on the work presented at the 33rd International Pyrotechnics Seminar, IPS 2006, Fort Collins, July 16–21, 2006.     

Impact of mass lumping on gravity and Rossby waves in 2D finite‐element shallow‐water models

The goal of this study is to evaluate the effect of mass lumping on the dispersion properties of four finite‐element velocity/surface‐elevation pairs that are used to approximate the linear shallow‐water equations. For each pair, the dispersion relation, obtained using the mass lumping technique, is computed and analysed for both gravity and Rossby waves. The dispersion relations are compared with those obtained for the consistent schemes (without lumping) and the continuous case. The P₀?P₁, RT₀ and P?P₁ pairs are shown to preserve good dispersive properties when the mass matrix is lumped. Test problems to simulate fast gravity and slow Rossby waves are in good agreement with the analytical results. Copyright © 2008 John Wiley & Sons, Ltd.     

K+ uptake by root systems grown in soil under salinity: II. Sensitivity analysis

A numerical algorithm for the solution of multicomponent transport of Ca²⁺, Mg²⁺, Na⁺, K⁺, Cl^– in soil and their uptake by plant roots has been developed. The model emphasizes adsorptiondesorption due to cation exchange mechanism, dissolution-precipitation of CaCO₃, and pH changes at the root surface controlled by the anion-cation influx balance. A fully implicit finite difference scheme is used for numerical implementation. Sensitivity analysis was conducted to evaluate the effect of each parameter on nutrient uptake. Each parameter (independent of all others) was varied between 0.25 to 4 times its speculated average level. Predicted K⁺ uptake was found to be more sensitive to changes of root radius and the parameter indicating maximal influx of K⁺. Effective diffusion coefficient and soil moisture are less influential. The influence of C_aCO₃ dissolution and different kinds of boundary conditions were also considered.Nomenclature A, B, E matrices of coefficients for finite difference scheme - b _i coefficients of equation for H⁺ concentration - C ^I concentration of theI-component in water - C ₀ ^I initial concentration of theI-component - _I ^r0 concentration of theI-component at the inner side of the root surface - C _I ^r1 concentration of theI-component at the external boundary - C _Na ^cr critical concentration for Na⁺ influx into root - CEC cation exchange capacity - D ^*I effective diffusion coefficient of theI-component in soil - F ^I concentration of theI-component on the exchange complex - G vector of coefficients in finite difference scheme - h Hill's cooperativity index for K⁺ influx - h ⁰ value ofh whenC ^Na=0 - J ^I uptake of theI-component by a root length unit - J _I ^r0 influx at the root surface of theI-component - J _max maximal influx of K⁺ - J _max ⁰ value ofJ ^max whenC ^Na=0 - K _a apparent Michaelis-Menten coefficient for K⁺ influx - K _a ⁰ value ofK _a whenC ^Na=0 - K _i selectivity coefficient of the exchange reaction - P _m ^I permeability of root surface for theI-component - P_CO2 partial pressure of CO₂ - r radial distance - r ₀ root radius - r ₁ half the distance between adjacent root (external boundary) - R ^I retardation factor of the-component in mass balance equation for theI-component - t time - t ₀ initial time - T simulation time - v ₀ water radial velocity at the root surface - x _i coordinate of nodes of finite difference mesh Greek coefficient of linear change of K⁺ influx - coefficient of linear change of Na⁺ influx - _s mass density of soil solid phase - soil porosity - volumetric content of liquid in soil - _i coefficients in formulae for parameters of K⁺ influx - parameter of perturbation in finite difference scheme - gg ^I activity coefficient of theI-component - accuracy of iteration convergence - time step for finite difference scheme - steps of finite difference mesh Special Symbols [...] activity symbol     

One Equation Model for Turbulent Channel Flow with Second Order Viscoelastic Corrections

A modified second order viscoelastic constitutive equation is used to derive a k–l type turbulence closure to qualitatively assess the effects of elastic stresses on fully-developed channel flow. Specifically, the second order correction to the Newtonian constitutive equation gives rise to a new term in the momentum equation involving the time-averaged elastic shear stress and in the turbulent kinetic energy transport equation quantifying the interaction between the fluctuating elastic stress and rate of strain tensors, denoted by P _w , for which a closure is developed and tested. This closure is based on arguments of isotropic turbulence and equilibrium in boundary layer flows and a priori P _w could be either positive or negative. When P _w is positive, it acts to reduce the production of turbulent kinetic energy and the turbulence model predictions qualitatively agree with direct numerical simulation (DNS) results obtained for more realistic viscoelastic fluid models with memory which exhibit drag reduction. In contrast, P _w  < 0 leads to a drag increase and numerical breakdown of the model occurs at very low values of the Deborah number, which signifies the ratio of elastic to viscous stresses. Limitations of the turbulence model primarily stem from the inadequacy of the k–l formulation rather than from the closure for P _w . An alternative closure for P _w , mimicking the viscoelastic stress work predicted by DNS using the Finitely Extensible Nonlinear Elastic-Peterlin fluid model, which is mostly characterized by P _w  > 0 but has also a small region of negative P _w in the buffer layer, was also successfully tested. This second model for P _w leads to predictions of drag reduction, in spite of the enhancement of turbulence production very close to the wall, but the equilibrium conditions in the inertial sub-layer were not strictly maintained.     

A rapid technique for measuring and visualizing the extent of separated flow

A method is described for rapidly measuring and visualizing the extent of separated flow suitable for a wind tunnel environment. The method utilizes a continuously swinging total pressure probe. This technique permits acquiring and presenting graphical records of separated regions in a fraction of the time required by other methods. Typical results indicate the presence of highly complicated three-dimensional separated regions for a typical general aviation twin-engine aircraft at post-stall conditions.List of Symbols c wing mean geometric chord - c_pt total pressure coefficient, (P _T-P_S)/q - P _S static pressure - P _T total pressure - q free stream dynamic pressure - q local dynamic pressure - r radial direction - x streamwise coordinate (x=0 corresponds to leading edge of wing) - angle of attach - azimuth direction