petrov galerkin finite element model for compressible flows a petrov galerkin method for the solution of the compressible euler and navier stokes equations is presented it is based on the introduction of an anisotropic blancing diffusion in the direction of the local direction of propagation of the scalar variables the local direction in which the anisotropic diffusion is introduced is uniquely determined and the magnitude of the balancing diffusion is automatically calculated locally using a criterion that is optimal for one dimensional transport equations the algorithm has been implemented using four noded bilinear elements with forward euler and second order runge kutta methods of integration in time several applications are presented and show the stability and approximation properties of the method

a fixed grid finite element technique for modelling phase change in steady state conduction advection problems a new fixed grid finite element technique for modelling phase change effects in steady state conduction advection problems is described the method is appropriate for materials that transform at a fixed critical temperature and is based on the assumption that the enthalpy experiences a jump discontinuity at this temperature yielding a singular specific heat this singularity is described using a dirac delta function and the appropriate finite element matrix equations are developed for isoparametric elements a simple first order upwinding scheme is used to avoid numerical oscillations in advection dominated problems the utility of the method is illustrated through the solution of a one dimensional test problem using eight node hexahedral isoparametric finite elements the technique may be applied to a wide range of finite element types and offers a number of advantages over other schemes for modelling phase change effects

an efficient method for sound transmission in non uniform circular ducts an efficient method is presented in this paper for predicting the reflection and transmission coefficients of acoustic waves in ducts at the location of a continuous variation in cross sectional area conventional numerical methods yield high dimensionality in the resulting numerical problem due to segmentation of the non uniform section into a number of subsections this paper shows that an ingenious application of spherical coordinates according to the geometry of the non uniform section enables us to determine analytically the system of characteristic functions for various profiles of the non uniform portion this system of characteristic functions can be used as a complete set to express the solution over the non uniform region so that the numerical problem to be solved is of a much lower dimension than that yielded by conventional numerical methods a technique to render the characteristic functions independent of the frequency is also explained to keep the computation time short the present method is applicable to non plane wave modes of propagation by virtue of the use of standard three dimensional curvilinear co ordinates

wave propagation and attenuation in herschel venturi tubes wave propagation and attenuation in a classical flow metering device the herschel venturi tube is investigated analytically and experimentally the analytical results are shown to compare well with the experimental observations for anechoically terminated herschel venturi tubes in the absence of flow the taylor series approximations to the governing differential equations and the stepwise expansion and contraction method are shown to be identical the effect of mean flow on the wave attenuation performance of herschel venturi tubes is also studied in terms of a solution obtained by the runge kutta method and the applicability of the stepwise expansion and contraction method is discussed with the mean flow through the duct

accuracy and stability improvements of integral equation models using the partial element equivalent circuit peec approach the partial element equivalent circuit peec technique is a formulation which transforms an electric field integral equation efie into a full wave equivalent circuit solution in this paper improvements are made to the peec model through the development of a refined method of computing both the partial inductances as well as the coefficients of potential the method does not increase the number of unknowns in addition damping is added to the peec model in order to further reduce nonphysical resonances which may occur above the useful frequency range the observations and solutions presented in this paper are especially important for time domain solvers the effectiveness of the method is illustrated with several examples

a streamline upwind full galerkin method for space time convection dominated transport problems an original space time finite element approach for the solution of the diffusion convection equation is proposed in this paper a slight manipulation of the differential equation suggests that transient transport problems may in fact be seen as steady state space time transport problems accurately and easily soluble by the standard galerkin technique however concerning convective transport involving sharp fronts or coarse discretization it is shown that implementation of dissipation along space time trajectories significantly improves the solutions classical comparative test problems are run to establish the performances of this method and to show the limits of the more sophisticated petrov and taylor galerkin schemes evocation of a possible space time anisotropy generated by usual finite difference time stepping procedures as well as comparative analysis of amplification matrices help to understand the accuracy and the robustness of the proposed approach

stability of the standard crank nicolson galerkin scheme applied to the diffusion convection equation some new insights a stability analysis of the classical crank nicolson galerkin cng scheme applied to the one dimensional solute transport equation is proposed on the basis of two fairly different approaches using a space time eigenvalue analysis it is shown at least for subsurface hydrology applications that the cng scheme is theoretically stable under the condition pecr 2 where pe and cr are the mesh peclet and courant numbers then to assess the computational stability of the scheme the amplification matrix is constructed and its norm is displayed in the pe cr space the results indicate that the norm of the amplification matrix is never smaller than unity and exhibits a hyperbolic nature analogous to the above theoretical condition

acoustics in variable area duct finite element and finite difference comparisons to experiment plane wave sound propagation without flow in a rectangular duct with a converging diverging area variation is studied experimentally and theoretically the area variation was of sufficient magnitude to produce large reflections and induce modal scattering the rms pressure and phase angle on both the flat and curved surface were measured and tabulated the steady state finite element theory of astley and eversman and the transient finite difference theory of white are in good agreement with the data it is concluded that numerical finite difference and finite element theories appear ideally suited for handling duct propagation problems which encounter large area variations f h j k l p p p0 s t u

upwind techniques via variational principles the amount of upwind or magnitude of off centering needed in the numerical solution of second order differential equations with significant first derivatives is justified more rigorously in this paper by requiring the satisfaction of a variational principle it will be shown that for a discrete solution a set of variational principles can be found which allow the problem to be solved as a self adjoint system the technique presented here will give a clear indication of those regions where i upwind techniques are not required ii upwind techniques are necessary and sufficient and iii upwind techniques are insufficient and artificial viscosity is required

acoustic transmission in non uniform ducts with mean flow part i the method of weighted residuals 3n this and a companion paper the problem of transmission of sound through non uniform ducts carrying a high speed subsonic compressible flow is approached using the method of weighted residuals mwr in the form of a modified galerkin method and the finite element method fem the intent of the investigation is to carry out a careful evaluation of these methods in this computationally difficult problem to this end both mwr and fem have been limited to in core computer implementations to generate useful results with relatively modest computational requirements this paper part i details the mwr formulation and presents numerical results establishing the degree of accuracy of mwr as compared to exact eigenvalue calculations and approximate one dimensional transmission calculations the comparison of mwr and fem results is carried out in the companion paper part ii

optimal design for polymer extrusion part ii sensitivity analysis for weakly coupled nonlinear steady state systems the die design methodology presented in part i is extended to include performance measures that are functions of material residence time we solve a hyperbolic differential equation using the velocity field computed from the hele shaw pressure analysis to evaluate the material residence time in polymer melts the residence time governing equation lacks natural diffusion therefore we employ the streamline upwind petrov galerkin supg finite element method to compute a spatially stable residence time field in the design problem we derive design sensitivities for steady state nonlinear weakly coupled systems and include design variables that parameterize essential boundary conditions design sensitivities are derived via the direct differentiation and adjoint methods special consideration is given to the supg weighting function since it is a function of the design to demonstrate the methodology sheet extrusion dies are designed to simultaneously minimize the exit velocity variation and the exit residence time variation

acoustic transmission in non uniform ducts with mean flow part ii the finite element method this second paper in a two part series describes the implementation of the finite element method for the solution of the problem of acoustic transmission through a non uniform duct carrying a high speed subsonic compressible flow a finite element scheme based on both the galerkin method and the residual least squares method and with eight noded isoparametric elements is described multi modal propagation is investigated by coupling of the solution in the duct non uniform section to modal expansions in uniform sections the accuracy of the finite element results for both the eigenvalue and transmission problems is assessed by comparison with exact solutions and with results from the method of weighted residuals in the form of a modified galerkin method as introduced in part i of this pair of papers the results of calculations show that modal interactions particularly in transmitted modes become increasingly important with increasing duct flow mach number power transmission coefficient calculations for the geometries studied reveal no indication of a linear basis for the phenomenon of subsonic acoustic choking

the time dependent finite difference procedure for propagation of sound in a non uniform lined duct a comparison with experiments abstract a time dependent finite difference procedure is developed for the calculation of acoustic fields which incorporate finite impedance and rigid wall boundaries a numerical mapping technique is evoked to transform the solution domain into a rectangular geometry and the procedure is used to solve for frequencies where cross modes can propagate in the straight sections of duct numerical results are corroborated with experiments

on longitudinal acoustic propagation in convergent and divergent nozzle flows abstract the longitudinal propagation of sound in quasi one dimensional low mach number nozzle flow is considered in section 1 the solution in the ray approximation section 2 1 is used to transform the wave equation into a schrodinger form which is studied for the family of power law ducts including as a particular case the conical nozzle it is shown that the coincidence of flow sources sinks with sound sources can lead to appearance of essential singularities section 2 2 which can be removed by using a riccati transformation section 2 4 the exact solutions of the acoustic equations for the parabolic figure 1 and hyperbolic figure 2 nozzles are obtained in terms of bessel functions section 2 3 respectively of complex order and argument the general formulas together with limiting forms in the compactness ray and asymptotic approximations section 3 1 are used to establish properties of the acoustic velocity and pressure section 3 2 kinetic and compression energies and energy flux and wave action section 3 3 for example it is shown that the equipartition of energies for moderate variations in cross section gives way table 2 to a predominance of kinetic compression energies respectively near blockages openings the effects of non uniform mean flow table 1 are discussed by comparing horns with nozzles section 3 4 e g it is shown that the duality principle in three alternative forms does not extend from horn to nozzles and the acoustic equations have no elementary solutions for the latter in contrast with the former

phase shift and attenuation characteristics of acoustic waves in a flowing gas confined by cylindrical walls abstract wave propagation in flowing ideal gases confined by cylindrical waveguides is described in the low frequency range using an iterative frobenius series expansion method the primary concern is to present a mathematical model enabling any radial dependent flow profile to be analyzed in contrast to previous analytical results the present model is applicable in the general case where cubic and higher order terms in the axial acoustic velocity become important and to examine the influence of a non vanishing radial velocity term as a numerical test case it is found that a gas flow velocity w r for simplifying reasons assumed to be a linear combination of a flat flow profile and a parabolic flow profile corresponding to a mean flow equal to w is well approximated by a flat flow profile of the same mean flow value w at low shear wavenumbers and at higher shear wavenumbers calculations were done for shear wavenumbers up to 8 in actual fact the error introduced by making this mean flow approximation is smaller than the error introduced by neglecting the radial velocity term

a comparison of measured and computed sound pressure levels in a non uniform acoustically lined duct a comparison of measured and numerically calculated acoustical fields is presented for a non uniform lined duct in the absence of appreciable mean flow the frequency range investigated includes the cut on frequencies of several transverse modes in certain portions of the duct measured pressure fields are compared to those predicted by one and two dimensional numerical models the validity of the one dimensional model is confirmed for frequencies below cut on of the first transverse mode for higher frequencies the one dimensional model is clearly unsatisfactory as might be expected the two dimensional model gives reasonable results for frequencies below and above cut on of the transverse modes although it indicates a very strong sensitivity of acoustical fields to small variations in local wall impedance data

plane wave analysis of conical and exponential pipes with incompressible mean flow the general equation for one dimensional wave propagation at low flow mach numbers m less than or equals slant0 2 is derived and is solved analytically for conical and exponential shapes the transfer matrices are derived and shown to be self consistent comparison is also made with the relevant data available in the literature the transmission loss behaviour of conical and exponential pipes and mufflers involving these shapes are studied analytical expressions of the same are given for the case of a stationary medium the mufflers involving conical and exponential pipes are shown to be inferior to simple expansion chambers of similar dimensions at higher frequencies from the point of view of noise abatement as was observed earlier experimentally

the influence of a liquid flow on sound fields confined by conical walls abstract sound propagation in conical waveguides is analyzed for the case where a background flow is maintained through the cone and points along the radial co ordinate r only a general expression for the acoustic pressure is derived and the perturbations of the acoustic field accommodated by the presence of a background flow are found by use of the green function method in the second part of this paper flow measurement properties are discussed the designation of propagation constants such as the wave number and phase speed lose much of their intuitive meaning in a conical waveguide instead the so called pseudo guide wave number and pseudo phase speed are introduced as they are well defined in the cone case and simplifies to the wave number and phase speed in the cylinder case respectively it is shown that changes due to a background flow in pseudo guide wave number pseudo phase speed and zero point crossing times all exhibit an oscillatory behavior as a function of the r co ordinate this is in contrast to the case of a cylinder where such changes become a linear function of the distance from the transmitter the oscillatory behavior in the changes in zero point crossing times as a function of r does not hamper flow measurement in the cone case since the only requirement to be fulfilled is in principle that a one to one correspondence between measured output changes in zero point crossing times and actual flow determined by v 0 at a specific receiver location exists it is shown that this requirement is fulfilled as changes in zero point crossing times depend linearly on the flow coefficient v 0 at a given r co ordinate there are however certain discrete r co ordinate values where this is not the case namely those where changes in zero point crossing times become zero for any value of v 0 in other words if the receiver is positioned near r co ordinates where zero point crossing times are at a maximum or a minimum for a given value of v 0 the truncated cone flow meter sensor is able to measure flow unambiguiosly by detection of changes in zero point crossing times induced by a background flow

petrov galerkin method for multidimensional time dependent convective diffusion equations a petrov galerkin finite element method for two and three dimensional time dependent convection diffusion equations is presented in the context of two space dimensions the method involves perturbed weighting functions in the weighted residuals formulation that are bilinear in space and quadratic in time and depend on two parameters which are calculated according to a local analysis for the one dimensional case the perturbations to the weighting functions can be interpreted as an added local anisotropic balancing diffusion and an added dispersion in the direction of the convective motion the effectiveness of the method is shown through several examples involving the convective and diffusive transport of scalar distributions in known velocity fields