Improved artificial dissipation schemes for the Euler equations

The influence of artificial dissipation schemes on the accuracy and stability of the numerical solution of compressible flow is extensively examined. Using an implicit central difference factored scheme, an improved form of artificial dissipation is introduced which highly reduces the errors due to numerical viscosity. A function of the local Mach number is used to scale the amount of numerical damping added into the solution according to the character of the flow in several flow regimes. The resulting scheme is validated through several inviscid flow test cases.     

A method for deriving hydrodynamic equations for complicated systems

A general algorithm is proposed for constructing a uniformly valid asymptotic solution of the kinetic equations under conditions when the number of slowly varying macroscopic variables is greater than the number of integral invariants of the collision operator. The case of a chemically reacting gas mixture is considered, and a method for constructing the asymptotic solution for this case is described. The hydrodynamic equations for reacting and relaxing gas mixtures are described in general form and it is noted that consistent allowance for the disequilibrium of the reaction and relaxation processes leads to the appearance in the hydrodynamic equations of a number of additional terms, which describe the dependence of the rates of these processes on the spatial derivatives of the hydrodynamic variables.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 96–105, May–June, 1981.     

A global approach to error estimation and physical diagnostics in multidimensional computational fluid dynamics

An approach for simultaneously assessing numerical accuracy and extracting physical information from multidimensional calculations of complex (engineering) flows is proposed and demonstrated. The method is based on global balance equations, i.e. volume-integrated partial differential equations for primary or derived physical quantities of interest. Balances can be applied to the full computational domain or to any subdomain down to the single-cell level. Applications to in-cylinder flows in reciprocating engines are used for illustration. It is demonstrated that comparison of the relative magnitude of the terms in the balances provides insight into the physics of the flow being computed. Moreover, for quantities that are not conserved at the cell or control volume level in the construction of the numerical scheme, the imbalance allows a direct assessment of numerical accuracy in a single run using a single mesh. The mean kinetic energy imbalance is shown to be a particularly sensitive indicator of numerical accuracy. This simple and powerful diagnostic approach can be implemented for finite-difference, finite-volume or finite-element methods.     

Large eddy simulation of a confined square cavity with natural convection based on compressible flow equations

Large eddy simulation of natural convection in a confined square cavity is described. The use of a complex compressible code with an artificial acoustic stiffness correction method, allows the use of higher time steps for a faster time and statistical convergence. We consider a broadly studied experimental case, consisting of a natural convective flow in a confined square cavity, with vertical walls heated at different rates (active walls), set at Ra = 1.58 × 10⁹. Turbulent boundary layers developing on the active walls and a vertical stable stratification characterize the mean flow. It is shown here that the results of this study match the experimental results reported in literature; for instance, mean velocity results. Although results for rms velocity fluctuations are barely over-predicted, the peak region is properly represented, while the greatest disagreements are found in the turbulent heat flow rate (velocity–temperature correlations). Turbulent structures were identified using different visualization methods and statistical studies. The authors found that the boundary layers on the active walls almost reach the fully turbulent regime, tending toward the laminar regime along the horizontal walls.     

Linearized dynamics for step jumps of velocity and displacement of shearing flows of a simple fluid

We consider linearized dynamics associated with step jumps in the velocity or displacement of the boundary of a fluid in a shearing motion. The discontinuity will propagate into the interior with a speed ( is the density) if the initial valuesG(0) andG(0) of the fading memory kernels are bounded, 0 <G(0) < , – <G (0) < 0. IfG(0) butG(0) = – , then the boundary of the support of the solution still propagates with the speedC. However, the solutions on both sides of the boundary match together in aC -fashion. IfG(0) butG(0) = 0, the amplitude of the discontinuity will not damp as in a purely elastic fluid. IfG(0) = , the step change is felt immediately throughout the fluid, without shocks, as in Navier-Stokes fluids. This same type of parabolic behavior can be achieved by a small Newtonian contribution added to the integral form of the stress but if this contribution is small, a smooth transition layer around the shock will propagate with the speedC. In the case of step displacement, from rest to rest, singular surfaces of infinite velocity can propagate into the interior with speed of propagationC. The singular surfaces undergo multiple reflections off bounding walls, but the final steady state reached asymptotically is in universal form independent of material.     

Shock and wave dynamics in cavitating compressible liquid flows in injection nozzles

Due to the exceptional high inlet pressures up to 2,000 bar flow dynamics and efficiency of modern injection systems are controlled by high frequency wave dynamics of the compressible liquid flow. Corresponding to alternating shock and expansion waves the liquid fluid evaporates and recondenses instantaneously. Here we present CFD simulations of the time accurate evolution of cavitating flows in 2-D plane and in six-hole injection nozzles with focus on the wave dynamics just after initialisation of the flow and within the time scale Δt ≤ 10^?4 s of pilot and multi-point injection. Due to shock reflections at the bottom of the sack hole the instantaneous maximum pressure increases more than three times higher as compared with the prescribed pressure at the nozzle inlet. For instance, in case of an inlet pressure of 600 bar the maximum pressure in the sack and therefore ahead of the nozzle bore holes reaches about 2,100 bar. It is quite reasonable that this amplification of the pressure affects the evolution of the convective flow and therefore the mass flow through the nozzle bore holes.     

A data-parallel approach to multiblock flow computations

Multiblock methods are often employed to compute flows in complex geometries. While such methods lend themselves in a natural way to coarse-grain parallel processing by the distribution of different blocks to different processors, in some situations a fine-grain data-parallel implementation may be more appropriate. A study is presented of the resolution of the Euler equations for compressible flow on a block-structured mesh, illustrating the advantages of the data-parallel approach. Particular emphasis is placed on a dynamic block management strategy that allows computations to be undertaken only for blocks where useful work is to be performed. In addition, appropriate choices of initial and boundary conditions that enchance solution convergence are presented. Finally, code portability between five different massively parallel computer systems is examined and an analysis of the performance results obtained on different parallel systems is presented.     

A new multigrid algorithm for non-linear equations in conjunction with time-marching procedures

Full approximate storage (FAS) multigrid algorithm is the most commonly used multigrid algorithm for non-linear equations. The algorithm initially developed for steady-state equations was later extended to obtain steady-state solutions employing unsteady equations. In extending the FAS algorithm for the steady-state non-linear equations to unsteady non-linear equations, the FAS algorithm does not to take into account that the governing equations are typically linearized in time before they are solved. Thus, there is a scope to develop a new multigrid algorithm to apply the multigrid technique to the equations linearized in time. In the present work, such an algorithm is developed exploring this possibility and is implemented for two-dimensional incompressible and compressible flows coupled with explicit time marching procedures. The results of the new algorithm compare favourably with those of the FAS multigrid method and single grid.     

A concise method for determining a valve flow coefficient of a valve under compressible gas flow

This work presents a novel method of determining a valve flow coefficient for a valve and transient mass flow rate of compressible gas discharged from a reservoir. The proposed method consists of a set of equations to express the physical phenomena and measurement equipment to measure the indispensable data used in the above equations. Regardless of the kind of valve, the valve flow coefficient can be obtained efficiently and feasibly. The results of this study indicate not only that the valve flow characteristics of the diaphragm valve significantly differ from those of the ball valve, but also that the C_v flow equation conventionally used is no longer valid for the diaphragm valve. The valve flow coefficient of the ball valve determined by the proposed method is about 48 and the representative one proposed by the ANSI/ISA is about 56. In addition, the mass flow rate of gas flow through a valve under transient process can be estimated without using a flow meter. Moreover, the cumulated masses discharged predicted by the method proposed herein are consistent well with those of the experimental results. The deviations are smaller than 6%.     

A collocated finite volume method for predicting flows at all speeds

An existing two-dimensional method for the prediction of steady-state incompressible flows in complex geometry is extended to treat also compressible flows at all speeds. The primary variables are the Cartesian velocity components, pressure and temperature. Density is linked to pressure via an equation of state. The influence of pressure on density in the case of compressible flows is implicitly incorporated into the extended SIMPLE algorithm, which in the limit of incompressible flow reduces to its well-known form. Special attention is paid to the numerical treatment of boundary conditions. The method is verified on a number of test cases (inviscid and viscous flows), and both the results and convergence properties compare favourably with other numerical results available in the literature.     

Finite element solutions of axisymmetric Euler equations for an incompressible and inviscid fluid

In this paper we present a finite element method for the numerical solution of axisymmetric flows. The governing equations of the flow are the axisymmetric Euler equations. We use a streamfunction angular velocity and vorticity formulation of these equations, and we consider the non-stationary and the stationary problems. For industrial applications we have developed a general model which computes the flow past an annular aerofoil and a duct propeller. It is able to take into account jumps of angular velocity and vorticiy in order to model the flow in the presence of a propeller. Moreover, we compute the complete flow around the after-body of a ship and the interaction between a ducted propeller and the stern. In the stationary case we have developed a simple and efficient version of the characteristics/finite element method. Numerical tests have shown that this last method leads to a very fast solver for the Euler equations. The numerical results are in good agreement with experimental data.     

A method for determining valve coefficient and resistance coefficient for predicting gas flowrate

A method is introduced to determine the valve flow coefficient and resistance coefficient with the experiment of air discharging from a reservoir, and with the least squares method to fit the cumulative molar quantities discharged. The test valve is an angle-seat valve (Type 2632, Bürkert) with different apertures. At pressure difference of about 6 bar, the choked flow occurs when the valve aperture over 60%. Both the valve coefficient and resistance coefficient model can exactly predict the flowrate for the non-choked flow, while there are larger deviations for the choked flow. The modified equation for the choked flow can improve the prediction. In the resistance coefficient model, the value of resistance coefficient and the discharged cumulative molar quantities obtained with both the compressible and incompressible assumption are very close. The compressibility of air is negligible within the experimental pressure difference of about 6 bar. The additivity of the resistance coefficient makes the model more convenient to use.     

A new finite element formulation for both compressible and nearly incompressible fluid dynamics

A new finite element formulation designed for both compressible and nearly incompressible viscous flows is presented. The formulation combines conservative and non‐conservative dependent variables, namely, the mass–velocity (density * velocity), internal energy and pressure. The central feature of the method is the derivation of a discretized equation for pressure, where pressure contributions arising from the mass, momentum and energy balances are taken implicitly in the time discretization. The method is applied to the analysis of laminar flows governed by the Navier–Stokes equations in both compressible and nearly incompressible regimes. Numerical examples, covering a wide range of Mach number, demonstrate the robustness and versatility of the new method. Copyright © 2000 John Wiley & Sons, Ltd.     

A simulated annealing-based inverse computational fluid dynamics model for unknown parameter estimation in fluid flow problem

In this paper, a simulated annealing (SA)-based optimisation is carried out for simultaneous estimation of the Reynolds number (Re) and the dimensions of the enclosure (l_x, l_y ) from the knowledge of centreline velocity field. For demonstrating the retrieval methodology, the required centreline velocity field is first obtained from a forward method using some known values of the unknowns, which are ultimately estimated by the inverse method. SA is used to optimise the objective function represented by the square of the difference between the known field and an arbitrary guessed field (calculated using some guessed value of the unknowns). For studying the sensitivity of the estimated parameters, the effect of random errors has been investigated and the suitability of the SA has been checked for different initial guesses. The algorithm reported in the present work is useful in estimating the above unknowns (Re, l_x, l_y ) for a given velocity field.     

Towards numerical simulations of supersonic liquid jets using ghost fluid method

A computational tool based on the ghost fluid method (GFM) is developed to study supersonic liquid jets involving strong shocks and contact discontinuities with high density ratios. The solver utilizes constrained reinitialization method and is capable of switching between the exact and approximate Riemann solvers to increase the robustness. The numerical methodology is validated through several benchmark test problems; these include one-dimensional multiphase shock tube problem, shock–bubble interaction, air cavity collapse in water, and underwater-explosion. A comparison between our results and numerical and experimental observations indicate that the developed solver performs well investigating these problems. The code is then used to simulate the emergence of a supersonic liquid jet into a quiescent gaseous medium, which is the very first time to be studied by a ghost fluid method. The results of simulations are in good agreement with the experimental investigations. Also some of the famous flow characteristics, like the propagation of pressure-waves from the liquid jet interface and dependence of the Mach cone structure on the inlet Mach number, are reproduced numerically. The numerical simulations conducted here suggest that the ghost fluid method is an affordable and reliable scheme to study complicated interfacial evolutions in complex multiphase systems such as supersonic liquid jets.     

An asymptotic method for deriving the equations of the Reissner-Mindlin plate theory

The equations of the Kirchhoff-Love and Reissner-Mindlin plate theories are derived with the use of the asymptotic homogenization method.     

The dynamics of a simple model for a thixotropic yield stress fluid

We study the dynamics of a model for thixotropic yield stress fluids which was recently proposed in [1]. This is based on the partially extended convected model (PEC), modified to allow for a non-zero shear stress limit for large shear rates (PECR), and combined with a Newtonian solvent to enable yielding to homogeneous shear flow (PECR-N) under a prescribed shear stress. We define ? to be the ratio of retardation time to relaxation time and focus on the limit ? → 0. Multiple time scales arise and the solutions are investigated with perturbation methods, in conjunction with direct computation of the full set of equations. Both PEC-N and PECR-N capture the experimentally observed phenomenon that the value of yield stress depends on the observation time. For the PECR-N model, we find transient solutions that are not expected from prior work on the PEC-N model, such as blow-up in finite time on the slow manifold, and yielded time-periodic flow.     

Connections and geodesic characteristic of equations of motion for constrained mechanical systems

I.IntroductionSinceEinsteinestablishedgeneralrelativityatthebiginningofthiscentury,differentialgeometry,especiallythemodernditTerentialgeometry,hasbeenextellsivelyappliedtomanyfieldsofphysics.Thestudyofregularholonomicmechanicalsystemsinthemodernsettingofdifferentialgeometryhasahistoryofmorethanthirtyyears.Andtheresearchtendstoperfectgraduallyt'~'l.Sinceearlyin1980'sthegeometrizationaboutconstrainedmechanicalsystemsandsingularmechanicalsystemshasbeenattachedimportanceextensivelyandsomeresult…     

A review of the multiple scale and reductive perturbation methods for deriving uncoupled nonlinear evolution equations

The method of multiple scales and the reductive perturbation method are reviewed and compared for calculating uniformly valid solutions of ion-acoustic waves. It is shown that they differ only in the choice of characteristic scales used in nondimensionalizing the problem. The inconsistent initial scaling used in the reductive method is later corrected by introducing artificially scaled variables to describe the solution in a form which remains uniformly valid in the far field. As a result this solution obeys the same evolution equations in both methods. Uncoupled nonlinear evolution equations describing waves traveling in opposite directions are derived using the multiple scale method for a number of different physical problems including the case of a general Galilean invariant system in conservation form.