a chebyshev method for the solution of boundary value problems abstract an expansion procedure using the chebyshev polynomials as base functions is proposed the method yields more accurate results than either of the galerkin or tau methods as indicated from solving the orr sommerfeld equation for both the plane poiseuille flow and the blasius velocity profile the chebyshev approximation is also applied to resolve the radial dependence of the flow field for a circular cylinder or a sphere in a uniform flow

a symmetric formulation for computing transient shallow water flows abstract the shallow water equations fall into the category of symmetric advective diffusive systems with source terms these types of equations can be very effectively solved using stabilized methods such as supg and gls a semi discrete finite element method based on these ingredients is presented for the computation of transient shallow water flows special care has been taken in the design of the operators in order to improve the performance for unsteady calculations the solution is advanced in time via a predictor multi corrector algorithm which includes as special cases the second order trapezoidal rule a first order explicit method and a first order implicit method equivalent to the constant in time element of the space time formulation

a conservative formulation of the multidimensional upwind residual distribution schemes for general nonlinear conservation laws in the present paper we consider the numerical solution of systems of general nonlinear hyperbolic conservation laws on unstructured grids by means of the residual distribution method we propose a new formulation of the first order linear optimal positive n scheme relying on a contour integration of the convective fluxes over the boundaries of an element full conservation is achieved for arbitrary flux functions while the robustness and the monotone shock capturing of the original n scheme is retained the new variant of the n scheme is combined with the conservative second order linear lda scheme to obtain a nonlinear second order monotone b scheme the performance of the new residual distribution schemes is evaluated on problems governed by the euler equations as an application to a more complex system of conservation laws lacking an exact conservative linearization we solve the ideal magnetohydrodynamics equations in two spatial dimensions

accurate navier stokes investigation of transitional and turbulent flows in a circular pipe a new fast accurate and roundoff error robust numerical technique for integrating unsteady incompressible navier stokes equations in cylindrical coordinates is presented the algorithm is based on a special change of dependent variables which avoids the singularity problem and provides high accuracy and computational efficiency accuracy and stability of the method are thoroughly tested for the model problem of transitional and turbulent flows in an infinite circular pipe verification of the algorithm includes two issues first spectral characteristics of the hagen poiseuille flow stability problem are compared with those of the discrete linearized navier stokes operator secondly the results of direct navier stokes simulation of all stages of laminar turbulent transition in a circular pipe at reynolds number of 4000 are presented time evolution of finite amplitude disturbances of laminar flow was calculated until the statistically stationary turbulent flow regime was established in addition to common statistical analysis the possibility of turbulence description by means of velocity fields having certain symmetries is examined thus the algorithm presented seems to be a ready to use robust tool for accurate investigation and further parametric studies of both transition mechanisms and fully developed turbulent flow regimes

a pseudospectral based method of lines for solving integro differential boundary layer equations application to the mixed convection over a heated horizontal plate abstract the method of lines is well suited for solving numerically parabolic boundary layer equations because it avoids the numerical difficulties associated to the integration of the continuity equation which is subsumed into the momentum equations as an integral of the main velocity component to deal with these integrals as well as with any other integral operator entering the boundary layer equations in some particular problems it is very efficient to discretize the transversal coordinate using pseudospectral methods the resulting ordinary differential equations odes can be then written in a very compact form suitable for general purpose methods and software developed for the numerical integration of odes we present here such a numerical method applied to the boundary layer equations governing the mixed convection over a heated horizontal plate these parabolic equation can be written in such a way that the natural convection appears as an integro differential term in the usual horizontal momentum equation so that the discretization by pseudospectral methods of the vertical coordinate derivative is very appropriate several matlab based solvers are compared to integrate the resulting odes to validate the numerical results they are compared with analytic solutions valid near the leading edge of the boundary layer

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

stabilized residual distribution for shallow water simulations we propose a stabilized residual distribution rd scheme for the simulation of shallow water flows the final discretization is obtained combining the stabilized rd approach proposed in abgrall j comp phys 214 2006 and ricchiuto and abgrall iccfd4 springer verlag 2006 with the conservative formulation already used in ricchiuto et al j comp phys 222 2007 to simulate shallow water flows the scheme proposed is a nonlinear variant of a lax friedrichs type discretization it is well balanced it actually yields second order of accuracy in smooth areas and it preserves the positivity of the height of the water in presence of dry areas this is made possible by the residual character of the discretization by properly adapting the stabilization operators proposed in abgrall j comp phys 214 2006 and ricchiuto and abgrall iccfd4 springer verlag 2006 and thanks to the positivity preserving character of the underlying lax friedrichs scheme we demonstrate the properties of the discretization proposed on a wide variety of tests

numerical aspects of searching convective absolute instability transition an overview of various numerical techniques used to determine the spatio temporal character of instabilities in fluid flows is given the advantageous features of various previously known individual techniques are discussed and a practical procedure combining them is suggested for a specific task of determining the complete boundary between linearly convectively and absolutely unstable regimes in a multi parameter space in problems with a fully numerical dispersion relation special attention is paid to aspects of automatization of computations as this is a crucial condition for their efficiency the suggested procedure is successfully used and is shown to provide a high degree of automatism in the physical example of non boussinesq mixed convection in a vertical channel this example comprises most of the major numerical difficulties found in various spatio temporal instability studies of two dimensional fluid flows which previously could not be handled without frequent human intervention and visual inspection of intermediate results this paper focuses on the general numerical aspects of the computations leaving the detailed discussion of the obtained physical results for a separate publication

an explicit residual based approach for shallow water flows we describe fully explicit residual based discretizations of the shallow water equations with friction on unstructured grids the schemes are obtained by properly adapting the explicit construction proposed in ricchiuto and abgrall 2010 57 in particular previous work on well balanced integration ricchiuto 2011 56 and preservation of the depth non negativity ricchiuto and bollermann 2009 60 is reformulated in the context of a genuinely explicit time stepping still based on a weighted residual approximation the paper discusses in depth how to achieve in this context an exact preservation of all the simple known steady equilibria and how to obtain a super consistent approximation for smooth non trivial moving equilibria the treatment of the wetting drying interface is also discussed giving formal conditions for the preservation of the non negativity of the depth for a particular case based on a nonlinear variant of a lax friedrichs type scheme the approach is analyzed and tested thoroughly the quality of the numerical results shows the interest in the proposed approach over previously proposed schemes in terms of accuracy and efficiency

numerical methods for hypersonic boundary layer stability various numerical methods for the solution of linear stability equations for compressible boundary layers are compared both the global and local eigenvalue methods for temporal stability analysis are discussed global methods are used to compute all the eigenvalues of the discretized system when a guess for the desired eigenvalue is available local methods may be used both to purify the eigenvalue and compute the associated eigenfunctions the extension to spatial stability analysis is also considered the discretizations studied include a second order finite difference method a fourth order accurate two point compact difference scheme and a chebyshev spectral collocation method eigenvalue results are presented for mach numbers up to 10 as the mach number increases the performance of the spectral method deteriorates due to the outward movement of the critical layer to alleviate this problem a multi domain spectral collocation method is developed which exhibits better convergence the overall performance of the fourth order compact scheme is excellent our results also indicate that in the limit of vanishing mach number there exist stable discrete modes in addition to the discrete modes of the orr sommerfeld equation

unstructured finite volume discretisation of bed friction and convective flux in solute transport models linked to the shallow water equations the finite volume discretisation of the shallow water equations has been the subject of many previous studies most of which deal with a well balanced conservative discretisation of the convective flux and bathymetry however the bed friction discretisation has not been so profusely analysed in previous works while it may play a leading role in certain applications of shallow water models in this paper we analyse the numerical discretisation of the bed friction term in the two dimensional shallow water equations and we propose a new unstructured upwind finite volume discretisation for this term the new discretisation proposed improves the accuracy of the model in problems in which the bed friction is a relevant force in the momentum equation and it guarantees a perfect balance between gravity and bed friction under uniform flow conditions the relation between the numerical scheme used to solve the hydrodynamic equations and the scheme used to solve a scalar transport model linked to the shallow water equations is also analysed in the paper it is shown that the scheme used in the scalar transport model must take into consideration the scheme used to solve the hydrodynamic equations the most important implication is that a well balanced and conservative scheme for the scalar transport equation cannot be formulated just from the water depth and velocity fields but has to consider also the way in which the hydrodynamic equations have been solved

effect of curvature on three dimensional boundary layer stability the problem of the stability of a three dimensional laminar boundary layer formed on a curved surface is considered a calculation scheme which takes account of the curvature of the flow streamlines and of the surface is proposed for the prediction of the development of small amplitude temporal disturbances computations have been performed for the flow over the windward face of an infinitely long yawed cylinder and comparisons have been made with experimental data these indicate that the theory correctly predicts many of the features of the unstable laminar flow the theory also suggests that transition in this particular situation is dominated by traveling disturbance waves and that at the experimentally observed transition location the wave which has undergone greatest total amplification has an amplitude ratio of approximately e to the 11th when the effects of curvature are omitted the maximum amplitude ratio at transition is about e to the 17th

an edge based unstructured mesh discretisation in geospherical framework an arbitrary finite volume approach is developed for discretising partial differential equations governing fluid flows on the sphere unconventionally for unstructured mesh global models the governing equations are cast in the anholonomic geospherical framework established in computational meteorology the resulting discretisation retains proven properties of the geospherical formulation while it offers the flexibility of unstructured meshes in enabling irregular spatial resolution the latter allows for a global enhancement of the spatial resolution away from the polar regions as well as for a local mesh refinement a class of non oscillatory forward in time edge based solvers is developed and applied to numerical examples of three dimensional hydrostatic flows including shallow water benchmarks on a rotating sphere

high degree efficient symmetrical gaussian quadrature rules for the triangle gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method a brief review of existing quadrature rules for the triangle is given and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed new quadrature rules of degree 12 20 are presented and a short fortran program is included

separation of variables in the hydrodynamic stability equations the problem of variable separation in the linear stability equations which govern the disturbance behaviour in viscous incompressible fluid flows is discussed the so called direct approach in which a form of the ansatz for a solution with separated variables as well as a form of reduced odes are postulated from the beginning is applied the results of application of the method are the new coordinate systems and the most general forms of basic flows which permit the postulated form of separation of variables then the basic flows are specified by the requirement that they themselves satisfy the navier stokes equations calculations are made for the 1 3 dimensional disturbance equations written in cartesian and cylindrical coordinates the fluid dynamics interpretation and stability properties of some classes of the exact solutions of the navier stokes equations defined as flows for which the stability analysis can be reduced via separation of variables to the eigenvalue problems of ordinary differential equations are discussed the eigenvalue problems are solved numerically with the help of the spectral collocation method based on chebyshev polynomials for some classes of perturbations the eigenvalue problems can be solved analytically those unique examples of exact explicit solution of the nonparallel unsteady flow stability problems provide a very useful test for numerical methods of solution of eigenvalue problems and for methods used in the hydrodynamic stability theory in general

three dimensional solutions of trailing vortex flows using parabolized equations flows of practical significance exist like systems of trailing vortices which are inhomogeneous in two directions and weakly dependent along the third spatial direction exploiting these characteristics an integration of the navier stokes equations using the parabolic navier stokes concept is proposed for the recovery of steady solutions that might be used subsequently in the scope of primary instability analyses the parabolic navier stokes equations are first formulated in a cylindrical coordinate frame and used to calculate the solution of an isolated axisymmetric nonparallel axially developing vortex then the fully three dimensional flow corresponding to a counter rotating pair of nonparallel vortices is obtained by parabolic navier stokes formulated in cartesian coordinates stable high order finite difference based numerical schemes are used for the spatial discretization to exploit the benefits of using a sparse direct solver for the inversion of the large matrices resulting from the discre

application of spectral collocation techniques to the stability of swirling flows abstract a chebyshev spectral collocation method for the temporal and spatial stability of swirling flows is presented the linearized stability equations in cylindrical coordinates are solved using the method and eigenvalues obtained by employing the qz routine the developed algorithm is found to be robust and easily adaptable to various flow configurations including internal and external flows with only minor changes in the application of the boundary conditions the accuracy and efficiency of the spectral method is tested for plane poiseuille flow annular flow rotating pipe flow and a trailing line vortex

differential eigenvalue problems in which the parameter appears nonlinearly several methods are examined for determining the eigenvalues of a system of equations in which the parameter appears nonlinearly the equations are the result of the discretization of differential eigenvalue problems using a finite chebyshev series two global methods are considered which determine the spectrum of eigenvalues without an initial estimate a local iteration scheme with cubic convergence is presented calculations are performed for a model second order differential problem and the orr sommerfeld problem for plane poiseuille flow

entropy functions for symmetric systems of conservation laws abstract using a simple symmetrizability criterion we show that symmetric systems of conservation laws are equipped with a one parameter family of entropy functions

explicit runge kutta residual distribution schemes for time dependent problems second order case in this paper we construct spatially consistent explicit second order discretizations for time dependent hyperbolic problems starting from a given residual distribution rd discrete approximation of the steady operator we review the existing knowledge on consistent rd mass matrices and highlight the relations between different definitions we then introduce our explicit approach which is based on three main ingredients first recast the rd discretization as a stabilized galerkin scheme then use a shifted time discretization in the stabilization operator and lastly apply high order mass lumping on the galerkin component of the discretization the discussion is particularly relevant for schemes of the residual distribution type 18 3 which we will use for all our numerical experiments however similar ideas can be used in the context of residual based finite volume discretizations such as the ones proposed in 14 12 the schemes are tested on a wide variety of classical problems confirming the theoretical expectations

a spectral collocation method for the navier stokes equations abstract a fourier chebyshev spectral method for the incompressible navier stokes equations is described it is applicable to a variety of problems including some with fluid properties which vary strongly both in the normal direction and in time in this fully spectral algorithm a preconditioned iterative technique is used for solving the implicit equations arising from semi implicit treatment of pressure mean advection and vertical diffusion terms the algorithm is tested by applying it to hydrodynamic stability problems in channel flow and in external boundary layers with both constant and variable viscosity

unsteady free surface flow simulation over complex topography with a multidimensional upwind technique in the context of numerical techniques for solving unsteady free surface problems finite element and finite volume approximations are widely used a class of upwind methods which attempts to model the equations in a genuinely multidimensional manner has been recently introduced as an alternative multidimensional upwind schemes mus were developed initially for the approximation of steady state solutions of the two dimensional euler equations on unstructured grids although they can be applicable to any system of hyperbolic conservation laws such as the shallow water equations the formal analogy between the two systems of equations is useful for simple cases however in practical applications of interest in hydraulics complex geometries and bottom slope variation can lead to important numerical errors produced by an inadequate source term discretization this problem has been analyzed and in this work the necessity of a multidimensional upwind discretization of the source terms is justified the basis of the numerical method is stated and the particular adaptation to unsteady shallow water flows over irregular geometry is described as test cases laboratory experimental data are used together with academic tests for validation

linear stability of incompressible flow using a mixed finite element method we analyzed the linear stability for incompressible flow using the mixed finite element method first by means of the fractional step finite element method the two dimensional base flows were numerically computed over a range of reynolds numbers and then they were perturbed with three dimensional disturbances by using linear stability and normal mode analysis we obtained the partial differential equations governing the evolution of perturbation from linearized navier stokes equations with slight compressibility in terms of the mixed finite element discretization in which six node quadratic lagrange triangular elements with quadratic interpolation for velocities p2 and three node linear lagrange triangular elements for pressure p1 were used a nonsingular generalized eigenproblem was formulated from these equations whose solution gives the dispersion relationship between the complex growth rate and the wave number we examined the stability of lid driven cavity flow in this study the stability results obtained from thep2p1elements were compared with those fromp1p1elements the spatial distribution of the pressure disturbance can be smoothed by using ap2p1element such as that in the simulation of incompressible flows by the mixed finite element method however the choice of elemental type has little influence on the stability results this study presents stability curves to identify the critical reynolds number and the critical wavelength of the neutral mode and discusses the instability mechanism the taylor goertler like vortices in the cavity were obtained by means of the reconstruction of three dimensional flows the stability results and the reconstructed three dimensional flows were in agreement with the observed ones and the other numerical stability results respectively

a numerical method to study the dynamics of capillary fluid systems we propose a numerical approach to study both the nonlinear dynamics and linear stability of capillary fluid systems in the nonlinear analysis the time dependent fluid region is mapped onto a fixed numerical domain through a coordinate transformation the hydrodynamic equations are spatially discretized with the chebyshev spectral collocation technique while an implicit time advancement is performed using second order backward finite differences the resulting algebraic equations are solved with the iterative newton raphson technique the most novel aspect of the method is the fact that the elements of the jacobian of the discretized system of equations are symbolic functions calculated before running the simulation these functions are evaluated numerically in the newton raphson iterations to find the solution at each time step which reduces considerably the computing time besides this numerical procedure can be easily adapted to solve the eigenvalue problem which determines the linear global modes of the capillary system therefore both the nonlinear dynamics and the linear stability analysis can be conducted with essentially the same algorithm we validate this numerical approach by studying the dynamics of a liquid bridge close to its minimum volume stability limit the results are virtually the same as those obtained with other methods the proposed approach proves to be much more computationally efficient than those other methods finally we show the versatility of the method by calculating the linear global modes of a gravitational jet