**Unformatted text preview:**

Ideal fluid Statistical Hydrodynamics Onsager Revisited Raoul Robert in Handbook of Mathematical Fluid Dynamics 2003 The role of the viscosity and the stability of the equilibrium states In real fluids there is of course always some viscosity In the study of two dimensional turbulence it is usually assumed that the viscosity is small enough so that a significant dissipation of energy cannot occur but it causes a decrease of the enstrophy This is due to the fact that the supplementary term 1Re has a filtering effect on the small scale oscillations of the vorticity t x and will soon get it equal to its local mean value Then we may wonder whether the equilibrium state has some stability property What do we get as equilibrium state if repeat the process and take as initial vorticity Let be the unique solution of the general G S E equation given by Proposition 2 1 9 the corresponding vorticity Then it can be shown that Arnold s classical estimates 2 apply and give the following stability result Any bounded initial vorticity 0 gives a solution t of Euler equation which satisfies the inequality 2d 0 2d for all where C 0 is some constant Furthermore we can prove that if the Young measure v is a mixture of such that v then for all x vx x So if we repeat the process starting with we shall again get as equilibrium state Applications of the momentum principle hydraulic jump surge and flow resistance in open channels Hubert Chanson in Hydraulics of Open Channel Flow Second Edition 2004 Introduction In a real fluid flow situation energy is continuously dissipated In open channel flows flow resistance can be neglected over a short transition3 forces as a first approximation and the continuity and Bernoulli equations can be applied to estimate the downstream flow properties as functions of the upstream flow conditions and boundary conditions However the approximation of frictionless flow is no longer valid for long channels Considering a water supply canal extending over several kilometres the bottom and sidewall friction retards the fluid and at equilibrium the friction force counterbalances exactly the weight force component in the flow direction The laws of flow resistance in open channels are essentially the same as those in closed pipes Henderson 1966 In an open channel the calculations of the boundary shear stress are complicated by the existence of the free surface and the wide variety of possible cross sectional shapes Another difference is the propulsive force acting in the direction of the flow In closed pipes the flow is driven by a pressure gradient along the pipe whereas in open channel flows the fluid is propelled by the weight of the flowing water resolved down a slope Ideal fluid CFD Solution Procedure A Beginning Jiyuan Tu Chaoqun Liu in Computational Fluid Dynamics Second Edition 2013 2 3 1 Initialization and Solution Control Step 5 The fifth step of the CFD analysis encompasses two prerequisite processes within the CFD solver initialization and solution control First the underlying physical phenomena in real fluid flows which are generally complex and non linear within such flows usually require treatment of the key phenomena to be resolved through an iterative solution approach An iterative procedure generally requires all the discrete values of the flow properties such as the velocity pressure temperature and other transport parameters of interest to be initialized before calculating a solution In theory initial conditions can be purely arbitrary However in practice there are certain advantages to imposing initial conditions intelligently Good initial conditions are crucial to the iterative procedure Two reasons that a CFD user should undertake the appropriate selection of initial conditions are If the initial conditions are close to the final steady state solution the quicker the iterative procedure will converge and yield results in a shorter computational time If the initial conditions are far away from reality the computations will require longer computational efforts to reach the desired convergence Also improper initial conditions may lead to the iterative procedure s misbehaving and possibly blowing up or diverging Second setting up appropriate parameters in the solution control usually entails the specification of appropriate discretization interpolation schemes and selection of suitable iterative solvers Almost all well established and thoroughly validated general purpose commercial codes adopt the finite volume method Chapter 4 as their standard numerical solution technique The algebraic forms of equations governing the fluid flow within these codes are usually approximated by the application of finite difference type approximations to a finite volume cell in space At each face of the cell volume surface fluxes of the transport variables that are required can be determined through different interpolation schemes Some of the common interpolation schemes are First Order Upwind Second Order Upwind Second Order Central and Quadratic Upstream Interpolation Convective Kinetics QUICK The inability of the Central scheme to identify the flow direction resulted in the formulation of other schemes such as Upwind or QUICK interpolation methods that are biased toward the upstream occurrence of the fluid flow and thus account for the flow direction The choice of a higher order interpolation scheme may achieve the desired level of accuracy for evaluation at the cell faces Solution procedures like the SIMPLE SIMPLEC and PISO algorithms are popular in many commercial codes The SIMPLE SIMPLEC or PISO algorithm is geared toward guaranteeing correct linkage between the pressure and velocity which predominantly accounts for the mass conservation within the flow domain At present our intention is not to dwell on the many underlying numerical properties but simply to present the interpolation schemes and pressure velocity coupling methods that are offered as standard options in many CFD codes It is imperative that some background knowledge of the appropriate selection of these options be acquired before any CFD calculation is performed More discussion of and practical guidance Ideal fluid on the many numerical issues pertaining to the application of interpolation schemes and pressure velocity coupling methods are provided in Chapters 4 and 6 Iterative solvers so called number crunching engines for numerical calculations are employed to resolve the algebraic equations Nowadays robust

View Full Document