primitive variable strongly implicit calculation procedure for viscous flows at all speeds a coupled solution procedure is described for solving the compressible form of the time dependent twodimensional navier stokes equations in body fitted curvilinear coordinates this approach employs the strong conservation form of the governing equations but uses primitive variables m v p t rather than the more traditional conservative variables p pw pv et as unknowns a coupled modified strongly implicit procedure cmsip is used to efficiently solve the newton linearized algebraic equations it appears that this procedure is effective for mach numbers ranging from the incompressible limit mx 0 01 to supersonic generally smoothing was not needed to control spatial oscillations in pressure for subsonic flows despite the use of central differences dual time stepping was found to further accelerate convergence for steady flows sample calculations including steady and unsteady low mach number internal and external flows and a steady shock boundarylayer interaction flow illustrate the capability of the present solution algorithm

three dimensional supersonic viscous flow over a cone at incidence abstract the viscous supersonic flow over a sharp cone at incidence is examined numerically with a coupled strongly implicit algorithm for the properties in the plane normal to the cone axis the navier stokes equations are considered in a boundary layer like or parabolized manner and global relaxation is considered for the pressure interaction it is shown that departure effects can be effectively eliminated by forward differencing for the axial pressure gradient moreover this approximation retains the implicit free pressure interaction required for geometries where axial flow separation is possible author

natural frequencies and mode shapes of deterministic and stochastic non homogeneous rods and beams abstract natural frequencies and mode shapes of non homogeneous deterministic and stochastic rods and beams are studied the solution is based on the functional perturbation method fpm the frequencies and mode shapes are considered as functionals of the non homogeneous properties the natural frequency and mode shape of the k th order is obtained analytically to any desired degree of accuracy once the functional derivatives with respect to the non uniform property have been found the solution for any morphology is obtained by direct integration without resolving the differential equation several examples with different non homogeneous properties are solved and compared with exact solutions as an accuracy check the fpm accuracy range for the frequency and the mode shape is less than 1 even for high heterogeneities in the stochastic case the accuracy of the natural frequencies depends on the stochastic information used given on the correlation distance roughly the grain size on the function around which the perturbation is executed and on whether we are interested in the properties of or of 2 moreover all frequency modes have the same response to heterogeneity as long as their wave length is of the order of the heterogeneity s characteristic distance in addition the heterogeneity effect on the average natural frequencies is minimal for the fundamental mode and may serve as a design tool

pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations a calculation procedure for viscous flows at all mach numbers has been described the scheme has been developed for a nonorthogonal coordinate system in order to handle both incompressible and compressible flows pressure has been selected as a primary dependent variable in preference to density the pressure field is evaluated using the compressible form of the simpler algorithm the accuracy of the proposed scheme has been assessed by comparing the results with experimental data or other numerical results available in the literature

vibrations of non homogeneous rectangular membranes abstract an exact solution for the title problem is obtained for the case where the two subdomains of the non homogeneous membrane are of rectangular shape it is shown that the fundamental frequency coefficient obtained by means a of the classical kantorovish method and b the optimized kantorovich method are in very good agreement with the result predicted by the exact method

axisymmetric vibrations of solid circular and annular membranes with continuously varying density abstract the present paper reports the results from a series of numerical experiments dealing with the determination of the lower natural frequencies for the transverse vibration of annular membranes including the special case of a solid circular membrane when the mass per unit area varies linearly quadratically and cubically with the radial co ordinate the frequency coefficients have been determined using 1 the differential quadrature method 2 the finite element technique 3 an optimized and or improved rayleigh quotient method and 4 a lower bound based on the stodola vianello method excellent agreement is achieved among the different methodologies

application of the godunov method and its second order extension to cascade flow modeling the godunov method and a new second order accurate extension of the method are used for the solution of two dimensional euler equations both numerical schemes are described in detail their performances in the subsonic transonic and supersonic flow regimes are first tested on the problem of flow in a channel with a circular arc bump the niethods are then applied to calculate the transonic flow through a supercritical com pressor cascade designed by j sanzo for this case the solution with the second order extension of the godunov method gives verygood agreement with the design distribution of parameters given by sanzo

numerical determination of the density profile of an inhomogeneous membrane solution of the inverse vibration problem given one or more vibrational modes of a membrane the free vibration equation can be applied to infer the mass surface density this paper considers determining the surface density of an inhomogeneous membrane from digitized holographic projections interferograms of the modeshapes spatially discrete numerical models of the membrane surface are presented which can be used to solve both forward and inverse vibration problems the accuracy of the discrete models is examined for exactly solvable free vibration problems involving inhomogeneous membranes for the solution of the inverse problem error estimates are given for the mass surface density deduced from modeshape interferograms the practicability of the method is investigated using simulated experimental data for membranes with composite and continuously inhomogeneous density profiles strategies are discussed for reducing errors in the reconstructed densities

a three dimensional incompressible navier stokes flow solver using primitive variables an implicit finite difference computer code has been developed to solve the incompressible navier stokes equations in a three dimensional curvilinear coordinate system the pressure field solution is based on the pseudocompressibility approach in which a time derivative pressure term is introduced into the mass conservation equation the solution procedure employs an implicit approximate factorization scheme the reynolds stresses which are uncoupled from the implicit scheme are lagged by one time step to facilitate implementing various levels of the turbulence model test problems for external and internal flows are computer and the results are compared with existing experimental data the application of this technique for general three dimensional problems is then demonstrated

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

coupled space marching method for the navier stokes equations for subsonic flows an implicit space marching finite difference procedure is described for solving the compressible form of the steady two dimensional navier stokes equations in body fitted curvilinear coordinates the coupled system of equations is solved for the primitive variables of velocity and pressure by making multiple sweeps of the space marching procedure a new pressure correction method has been developed that significantly accelerates convergence of the iterative process the scheme is used to compute incompressible flows by taking the low mach number limit of the compressible formulation computed results are compared with other numerical predictions for low reynolds number channel inlet flow flow over a rearward facing step in a channel and external flow over a cylinder number of different methods have been developed to solve the navier stokes equations in primitive variables for steady subsonic flows many of these algorithms are described in anderson et al 1 the present methods are not entirely adequate for all problems and advanced techniques continue to be developed new procedures and recent im provements to existing methods are briefly reviewed in the following paragraphs the most frequently used primitive variable algorithms for incompressible flows solve the momentum equations for the velocity components in an uncoupled segregated manner holding pressure fixed although different in detail the segre gated solution schemes e g refs 2 4 use the continuity equation indirectly in the formulation of a separate poisson equation that is solved for the pressure since the velocity components and pressure are computed from different algo rithms rather than in a coupled manner convergence is slowed furthermore the solution of the elliptic poisson equa tion consumes a large part of the total computaion time for each global iteration van doormaal et al 5 recently have eval uated improved algorithms for solving the pressure poisson equation and rhie4 has applied a multigrid method to acceler ate the solution of the momentum equations and the pressure equation several strategies have been advanced for solving the cou pled momentum and continuity equations for the primitive variables including the pressure the schemes all share the advantage that a separate procedure for imposing the continu ity constraint and determining the pressure is not required the choice of variables and the algebraic approach used to solve the system of nonlinear equations distinguish the direct time marching and space marching algorithms vanka and leaf6 and recently patankar et al 7 have com pared solutions for two dimensional flows obtained by direct and segregated methods a large sparse matrix solver is used to invert a system of equations spanning the entire flow domain

the use of domain decomposition in accelerating the convergence of quasihyperbolic systems this paper proposes an alternate form of the active domain method k nakahashi and e saitoh aiaa j 35 1280 1997 that is applicable to streamwise separated flows named the marching window the algorithm consists of performing pseudo time iterations on a minimal width subdomain composed of a sequence of cross stream planes of nodes the upstream boundary of the subdomain is positioned such that all nodes upstream exhibit a residual smaller than the user specified convergence threshold the advancement of the downstream boundary follows the advancement of the upstream boundary except in zones of significant streamwise ellipticity where a streamwise ellipticity sensor ensures its continuous progress compared to the standard pseudo time marching approach the marching window decreases the work required for convergence by up to 24 times for flows with little streamwise ellipticity and by up to eight times for flows with large streamwise separated regions storage is reduced by up to six times by not allocating memory to the nodes not included in the computational subdomain the marching window satisfies the same convergence criterion as the standard pseudo time stepping methods hence resulting in the same converged solution within the tolerance of the user specified convergence threshold the algorithm is not restricted to a discretization stencil and pseudo time stepping scheme in particular and is used here with the yee roe scheme and block implicit approximate factorization solving the favre averaged navier stokes fans equations closed by the wilcox k turbulence model the eigenstructure of the fans equations is also presented

non extensive statistical mechanics and generalized fokker planck equation a non linear generalized fokker planck gfp equation is studied whose exact stationary solutions are the maximum entropy distributions introduced by tsallis in his generalization of statistical mechanics in the case of a constant or linearly varying drift the time dependent solutions of the gfp equation are seen to obey a suitable form of the celebrated h theorem the temporal changes of tsallis entropies are seen to be given in terms of the fisher s information measure particular time dependent solutions of the gfp equation of the maximum tsallis entropy form are obtained

the fundamental frequency of non homogeneous rectangular membranes the present study deals with the case in which the density of the membrane is a linear function of the x variable two independent solutions are obtained using the optimized galerkin kantorovich approach and the differential quadrature method

closed form solutions for the free longitudinal vibration of inhomogeneous rods this paper aims at presenting a family of exact solutions for the longitudinal vibration of variable area rods area variations that give solutions in terms of the confluent hypergeometric function are being sought for and the governing differential equation is appropriately reduced to the confluent hypergeometric differential equation using a generic transformation the eigenfrequencies of rods with certain area variations subjected to classical boundary conditions are obtained and the parametric space of the solutions obtained is studied these solutions are also highly useful in other topics of study such as torsional vibration of rods and wave propagation in ducts with variable cross sectional areas since the governing differential equations are very similar

vortical solutions in supersonic corner flows vortical solutions are investigated for inviscid supersonic steady flows described by the steady state euler equations that occur in the corner region generated by two orthogonal ramps complex shock interactions appear with the formation of a vorticity field in the corner region and the vorticity itself converging into spiral singularities symmetric or asymmetric flow configurations are generated within symmetric corners the investigation is carried out by a numerical technique based on a space marching procedure with a finite volume approximation the integration of the conservation laws allows for the correct numerical capturing of shocks and contact surfaces a flux differenc e splitting procedure is used for the calculation of the fluxes on the side walls of the volumes based on the hyperbolicity of the euler equations for steady supersonic regimes a high order accuracy scheme is introduced founded on the essentially nonoscillatory eno scheme numerical results are presented and discussed with reference to similar problems investigated by other authors his paper analyzes inviscid supersonic conical corner flows generated by two intersecting wedges such config urations are typical of supersonic inlet flows and are similar to one of the four corners of the box type inlets the corner flows are conical if the two intersecting walls are plane then all flow variables pressure velocity and entropy will be independent of the spherical radial coordinate centered at the origin of the corner flow and these flows will depend only on two space dimensions moreover the topology of a conical flow is characterized by its conical or crossflow streamlines the projections of the three dimensional streamlines onto a spherical surface centered at the origin of the flow two possible shock configurations could be generated by the intersection of two compressive wedges one is character ized by a regular reflection and the other by the presence of a mach disk irregular reflection at the intersection of the two dimensional shocks produced by each wedge such possibili ties have been investigated for a symmetrical geometry two wedges of equal deflection and the regular reflection shock configuration has been predicted using linearized theory 1 the mach disk configuration has been detected experimentally2 and in the numerical solutions of the euler equations using a shock capturing approach3 and a shock fitting approach 4 figure 1 from ref 4 shows in the range of wall deflections 6 and upstream conditions mo where the regular or the mach disk configuration is expected to occur note that for deflec tions above 5 deg the only possible solution is the mach disk shock configuration contact surfaces are generated at points where shocks interact because of the different entropies pro duced by the wave systems on either side of the interaction however in the present case of symmetrical geometry and for regular interaction the strength of the contact surface van ishes conversely in the cases of irregular reflection two symmetric contact surfaces are generated at the two triple points that bound the mach disk for moderate values of upstream mach number and deflection the two contact sur faces converge on the corner according to the results of the present study when the mach number or the deflection in creases the configuration may drastically change

bounds on natural frequencies of composite circular membranes integral equation methods this paper is concerned with finding upper and lower bounds for the natural frequencies of vibration of a circular membrane with stepped radial density such problems involving discontinuous coefficients in the differential equation may be treated by using classical variational methods however it is shown here that eigenvalue estimation techniques based on an integral equation formulation are more effective integral equation methods provide more accurate upper bounds for a comparable amount of effort and supply improvable lower bounds as well

deformed logarithms and entropies by solving a differential functional equation inposed by the maxent principle we obtain a class of two parameter deformed logarithms and construct the corresponding two parameter generalized trace form entropies generalized distributions follow from these generalized entropies in the same fashion as the gaussian distribution follows from the shannon entropy which is a special limiting case of the family we determine the region of parameters where the deformed logarithm conserves the most important properties of the logarithm and show that important existing generalizations of the entropy are included as special cases in this two parameter class

weakly non local fluid mechanics the schrodinger equation a weakly non local extension of ideal fluid dynamics is derived from the second law of thermodynamics it is proved that in the reversible limit the additional pressure term can be derived from a potential the requirement of the additivity of the specific entropy function determines the quantum potential uniquely the relation to other known derivations of the schrodinger equation stochastic fisher information exact uncertainty is clarified

non linear kinetics underlying generalized statistics abstract the purpose of the present effort is threefold firstly it is shown that there exists a principle that we call kinetical interaction principle kip underlying the non linear kinetics in particle systems independently on the picture kramers boltzmann used to describe their time evolution secondly the kip imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy thirdly the kip allows on one hand to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and on the other hand suggests how we can introduce naturally new distributions finally as a working example of the approach to the non linear kinetics here presented a new non extensive statistics is constructed and studied starting from a one parameter deformation of the exponential function holding the relation f x f x 1

pressure based navier stokes solver using the multigrid method apressure based implicit procedure to solve steadystate navier stokes equations is developed a multistep pressure correction procedure with an implicit density treatment is used to establish the pressure and velocity fields a multigrid relaxation scheme is used to solve the scalar matrices resulting from the finite volume formulation the algorithm is valid for all mach number flows ranging from incompressible to supersonic flow regimes

unique additive information measures boltzmann gibbs shannon fisher and beyond abstract it is proved that the only additive and isotropic information measure that can depend on the probability distribution and also on its first derivative is a linear combination of the boltzmann gibbs shannon and fisher information measures power law equilibrium distributions are found as a result of the interaction of the two terms the case of second order derivative dependence is investigated and a corresponding additive information measure is given

vibration of inhomogeneous strings rods and membranes this paper is concerned with the vibration of inhomogeneous strings rods and membranes with continuously varying properties the work has two main purposes first we provide specific examples for which closed form exact solutions are obtained apart from their intrinsic interest such examples serve as benchmark problems against which the accuracy of approximate analytical or numerical methods may be assessed the second objective is to demonstrate the effectiveness of an integral equation based method for obtaining lower bounds for vibration frequencies

schrodinger equation from an exact uncertainty principle an exact uncertainty principle formulated as the assumption that a classical ensemble is subject to random momentum fluctuations of a strength which is determined by and scales inversely with uncertainty in position leads from the classical equations of motion to the schrodinger equation

an approximate approach to study the free vibration of a string with an arbitrary variation in mass density and tension with attached concentrated masses and its application to hanging chain and rotating cord abstract a multi stepped approximation has been developed to study the free vibration of a string with arbitrary variation in the tension and the mass density to achieve this objective a novel exact approach is developed to study free vibration of a string with n uniform sections of different mass densities under n different tensions with n 1 attached concentrated masses and springs at the ends the two advantages of this approach are that it leads to a single frequency equation for any number of sections and the same computer program can be used to solve different problems using pertinent data this method is applied to cases where tension and or mass density changes continuously such as hanging chain rotating chain or inhomogeneous strings vibrating in either horizontal or vertical position in all cases the results obtained using this approach agreed with previous results obtained using exact and approximate methods it was found that natural frequencies of a fixed string under constant tension t with linear variation in the mass density x 0 1 x l can be approximated using a very simple equation n n l t 0 1 0 5 1 2 and results reasonably agree with previous results for a wide range of new results for many other cases are also presented in what follows

iterative space marching method for compressible sub trans and supersonic flows anewiterativefully coupledimplicitspace marching method isproposedforsolving thetwo dimensionalsteady eulerequationsforcompressiblee owsatmach numbers ranging from subsonic to supersonic aspecial treatment of the streamwise pressure gradient component permits us to calculate both supersonic e ow regions where the euler equations are hyperbolic and subsonic regions where the equations reveal elliptic properties to take into account the elliptic effects of subsonic and transonic e ows space marching sweeps are carried out iteratively a new parabolic pressure correction procedure is developed to accelerate the convergence rate this procedure can be applied for subsonic and transonic regimes and is consistent with the characteristic analysis of the euler equations ateach marching station a newton iterativetechniqueisused tosolvethenonlinearsystem ofequations in a fully coupled manner to resolve strong shocks and contact discontinuities as well as smooth e owe elds with high accuracy implicit symmetric second order total variation diminishing and upwind second order richardson schemes are employed to approximate the transverse and streamwise derivatives respectively the method is tested on theproblem ofthee owin a ductwith a circulararcbump fordifferentmach numberregimes numerical calculations show that the method is accurate is robust and can efe ciently be applied for calculating subsonic transonic and supersonic e ows without streamwise separation

some exact solutions of the vibration of non homogeneous membranes the purpose of this note is to show that the linear density variation case studied by masad has a closed form exact solution also we shall present another exact solution for the vibration of a continuously non homogeneous annular membrane