NUMERICAL SOLUTION OF FLUID-STRUCTURE INTERACTION IN LIQUID-FILLED PIPES BY METHOD OF CHARACTERISTICS

Fluid-structure interaction （FSI） is essentially a dynamic phenomenon and always exists in fluid-filled pipe system. The four-equation model, which has been proved to be effective to describe and predict the phenomenon of FSI due to friction coupling and Poisson coupling being taken into account, is utilized to describe the FSI of fluid-filled pipe system. Terse compatibility equations are educed by the method of characteristics （MOC） to describe the fluid-filled pipe system. To shorten computing time needed to get the solutions under the condition of keeping accuracy requirement, two steps are adopted, firstly the time step Δt and divided number of the straight pipe are optimized, sec-ondly the mesh spacing Δz close to boundary is subdivided in several submeshes automatically ac-cording to the speed gradient of fluid. The mathematical model and arithmetic are validated by com-parisons between simulation solutions of two straight pipe systems and experiment known from lit-erature.

Application of material-mesh algebraic collapsing acceleration technique in method of characteristics——based neutron transport code

The algebraic collapsing acceleration(ACA)technique maximizes the use of geometric flexibility of the method of characteristics(MOC).The spatial grids for loworder ACA are the same as the high-order transport,which makes the numerical solution of ACA equations costly,especially for large-size problems.To speed-up the MOC transport iterations effectively for general geometry,a coarse-mesh ACA method that involves selectively merging fine-mesh cells with identical materials,called material-mesh ACA(MMACA),is presented.The energy group batching(EGB)strategy in the tracing process is proposed to increase the parallel efficiency for microscopic crosssection problems.Microscopic and macroscopic crosssection benchmark problems are used to validate and analyse the accuracy and efficiency of the MMACA method.The maximum errors in the multiplication factor and pin power distributions are from the VERA-4 B-2 D case with silver-indium-cadmium(AIC)control rods inserted and are 104 pcm and 1.97%,respectively.Compared with the single-thread ACA solution,the maximum speed-up ratio reached 25 on 12 CPU cores for microscopic cross-section VERA-4-2 D problem.For the C5 G7-2 D and LRA-2 D benchmarks,the MMACA method can reduce the computation time by approximately one half.The present work proposes the MMACA method and demonstrates its ability to effectively accelerate MOC transport iterations.

Bearing capacity analysis using the method of characteristics

Using the method of characteristics,the bearing capacity for a strip footing is analyzed.The method of characteristics leads to an exact true limit load when the calculations of the three terms in the bearing capacity formula are consistent with one collapse mechanism and the soil satisfies the associated flow rule.At the same time,the method of characteristics avoids the assumption of arbitrary slip surfaces,and produces zones within which equilibrium and plastic yield are simultaneously satisfied for given boundary stresses.The exact solution without superposition approximation can still be expressed by Terzaghi＇s equation of bearing capacity,in which the bearing capacity factor N γλ is dependent on the dimensionless parameter λ and the friction angle φ.The influence of groundwater on the bearing capacity of the shallow strip footing is considered,which indicates that when the groundwater effect is taken into account,the error induced by the superposition approximation can be reduced as compared with dry soil condition.The results are presented in the form of charts which give the modified value（Nwγλc /Nγλc） of bearing capacity factor.Finally,an approximated analytical expression,which provides results in close agreement with those obtained by numerical analysis in this paper,has been suggested for practical application purposes.

DOUBLE METHOD OF CHARACTERISTICS TO ANALYZE HYDRAULIC-THERMAL TRANSIENTS OF PIPELINE FLOW

The hydraulic and thermal transients in pipeline flow were studied. The method of characteristics for hydraulic transient analysis of batch transport of pipeline flow had been improved. The thermal transient equation, in which the term with v 3 was involved, had been inferred, while the corresponding method of characteristics was constructed. The double method of characteristics, which can be used to study the coherent hydraulic-thermal transients of batch transport of pipeline flow, was developed.

One modified method of characteristics used to analyze the multiconductor transmission lines

To solve the coupling effect of multiconductor transmission lines excited by external electromagnetic wave, the modified method of characteristics is proposed. The modified method of characteristics which can compute the terminal induced voltages excited by the external electromagnetic wave when the terminal networks or intereonnection networks contain the dynamic elements is introduced. The simulation results indicate that the modified method can analyze the terminal induced voltages when the terminal networks or the interconnection networks contain the dynamic elements excited by the external electromagnetic wave. And the results are compared with the results acquired by FDTD method, the two results are completely same. So one effective modified method is implemented to compute the transmission lines.

Preliminary study on CAD-based method of characteristics for neutron transport calculation

Our new method makes use of a CAD-based automatic modeling tool, MCAM, for geometry modeling and ray tracing of particle transport in method of characteristics （MOC）. It was found that it could considerably enhance the capability of MOC to deal with more complicated models for neutron transport calculation. In our study, the diamond-difference scheme was applied to MOC to reduce the spatial discretization errors of the fiat flux approximation. Based on MCAM and MOC, a new 2D MOC code was developed and integrated into the SuperMC system, which is a Super Multi-function Computational system for neutronics and radiation simulation. The numerical results demonstrated the feasibility and effectiveness of the new method for neutron transport calculation in MOC.

LOCAL ERROR ESTIMATES FOR METHODS OF CHARACTERISTICS INCORPORATING STREAMLINE DIFFUSION

Allen and Liu (1995) introduced a new method for a time-dependent convection dominated diffusion problem, which combines the modified method of characteristics and method of streamline diffusion. But they ignored the fact that the accuracy of time discretization decays at half an order when the characteristic line goes out of the domain. In present paper, the author shows that, as a remedy, a simple lumped scheme yields a full accuracy approximation. Forthermore, some local error bounds independent of the small viscosity axe derived for this scheme outside the boundary layers.

Simulating advance distance in border irrigation systems based on the improved method of characteristics

Improving the simulation accuracy of the advance distance based on the method of characteristics is essential to develop numerical solutions for simulating surface irrigation.Instead of volume balance in the traditional method of characteristics(T-MC),the position of critical flow is determined to simulate the advance distance in the improved method of characteristics(I-MC),which is used in border irrigation systems with rapid variation in inflow discharge in the current research.Specifically,the zones of both subcritical and supercritical flow were firstly distinguished to determine the position of the critical flow point,which was also the upstream boundary of the wetting front region,and then the advance distance was calculated by applying the diffusive wave equation in the wetting front region.The results showed that the I-MC accurately simulated the advance distance with high determination coefficients(0.984-0.998)and low errors(root mean square error of 0.35-1.56 min and coefficient of residual mass of 0.01-0.06),which performed much better than the T-MC.The I-MC provided a suitable and simple numerical simulation tool to improve the establishment of numerical surface irrigation models.

An Accurate Numerical Scheme for Maxwell Equation with CIP-Method of Characteristics

A new multi-dimensional scheme for the Maxwell equations is established by the CIP method in combination with the method of characteristics(CIP-MOC).In addition,the CIP-MOC can be extended to arbitrary grid system by the Soroban grid without losing the third-order accuracy.With the accuracy fixed,the grid points required for the CIP are 40 times less than the conventional schemes like the FDTD in three dimensions.Numerical solutions obtained by the CIP-MOC are compared with analytical solution and the FDTD in plane-wave scattering by a perfectly-conducting circular cylinder,and the CIP-MOC agrees very well with analytical solutions.The Soroban grid is also applied to the Vlasov equation that describes the kinematics of plasmas that is frequently combined with the Maxwell equation.The adaptively moving points in velocity space are similar to the particle codes but can provide accurate solutions.

HIGH SPEED FLOW DESIGN USING THE THEORY OF OSCULATING CONES AND AXISYMMETRIC FLOWS

An inverse method of characteristics was introduced into the design concept of using osculating cones (OC) in the supersonic flow, which can extend the domain of options for generating the aerospace vehicle configurations with supersonic leading edge as well as inlet diffusers. Some more practical waverider shapes with higher volumetric efficiency can be obtained through using the concept of osculating axisymmetric (OA) flows with rotationality in the post shock flow field by inputting curved shocks.

Attenuation analysis of hydraulic transients with laminar-turbulent flow alternations

An improved compound mathematical model is established to simulate the attenuation of hydraulic transients with laminar-turbulent alternations,which usually occur when the pipeline flow velocity fluctuates near the critical velocity.The laminar friction resistance and the turbulent friction resistance are considered respectively in this model by applying different resistance schemes to the characteristics method of fluid transient analysis.The hydraulic transients of the valve closing process are simulated using the model.A more reasonable attenuation of hydraulic transients is obtained.The accurate attenuation is more distinct than that obtained from the traditional mathematical model.The research shows that the hydraulic transient is a type of energy waves,and its attenuation is caused by the friction resistance.The laminar friction resistance is more important than the turbulent friction resistance if the flow velocity is smaller than the critical velocity.Otherwise the turbulent friction resistance is more important.The laminar friction resistance is important in the attenuation of hydraulic transients for the closing process.Thus,it is significant to consider the different resistances separately to obtain more accurate attenuation of hydraulic transients.

Supersonic Two-Dimensional Minimum Length Nozzle Design at High Temperature. Application for Air

When the stagnation temperature of a perfect gas increases, the specific heat ratio does not remain constant any more, and start to vary with this temperature. The gas remains perfect, its state equation remains always valid, except it will name in more calorically imperfect gas or gas at High Temperature. The goal of this work is to trace the profiles of the supersonic Minimum Length Nozzle with centered expansion when the stagnation temperature is taken into account, lower than the threshold of dissociation of the molecules and to have for each exit Mach number several nozzles shapes by changing the value of the temperature. The method of characteristics is used with a new form of the Prandtl Meyer function at high temperature. The resolution of the obtained equations is done by the second order of fmite differences method by using the predictor corrector algorithm. A study on the error given by the perfect gas model compared to our model is presented. The comparison is made with a calorically perfect gas for goal to give a limit of application of this model. The application is for the air.

Flux Vector Splitting Schemes for Water Hammer Flows in Pumping Supply Systems with Air Vessels

To solve water hammer problems in pipeline systems,many numerical simulation approaches have been developed. This paper improves a flux vector splitting( FVS) scheme whose grid is the same as the fixedgrid MOC scheme. The proposed FVS scheme is used to analyze water hammer problems caused by a pump abrupt shutdown in a pumping system with an air vessel. This paper also proposes a pump-valve-vessel model combining a pump-valve model with an air vessel model. The results show that the data obtained by the FVS scheme are similar to the ones obtained by the fixed-grid method of characteristics( MOC). And the results using the pump-valve-vessel model are almost the same as the ones using both the pump-valve model and the air vessel model. Therefore,it is effective that the proposed FVS scheme is used to solve water hammer problems and the pump-valve-vessel model replaces both the pump-valve model and the air vessel model to simulate water hammer flows in the pumping system with the air vessel.

A SECOND ORDER MODIFIED CHARACTERISTICS VARIATIONAL MULTISCALE FINITE ELEMENT METHOD FOR TIME-DEPENDENT NAVIER-STOKES PROBLEMS

In this paper, by combining the second order characteristics time discretization with the variational multiscale finite element method in space we get a second order modified characteristics variational multiscale finite element method for the time dependent Navier- Stokes problem. The theoretical analysis shows that the proposed method has a good convergence property. To show the efficiency of the proposed finite element method, we first present some numerical results for analytical solution problems. We then give some numerical results for the lid-driven cavity flow with Re = 5000, 7500 and 10000. We present the numerical results as the time are sufficient long, so that the steady state numerical solutions can be obtained.

Riemann problem for one-dimensional binary gas enhanced coalbed methane process

With an extended Langmuir isotherm, a Riemann problem for one-dimensional binary gas enhanced coalbed methane (ECBM) process is investigated. A new analytical solution to the Riemann problem, based on the method of characteristics, is developed by introducing a gas selectivity ratio representing the gas relative sorption affinity. The influence of gas selectivity ratio on the enhanced coalbed methane processes is identified.

A novel approach for predicting inlet pressure of aircraft hydraulic pumps under transient conditions

Cavitation caused by insufficient suction is a major factor that influences the life of aircraft pumps. Currently, pressurizing the tank can solve the cavitation problem under steady largeflow conditions. However, this method is not always effective under transient conditions(from zero flow to full flow in a very short time). Moreover, to apply and design other measures, such as a boost impeller, the suction dynamics during the transient period must be investigated. In this paper,a novel approach based on the pressure wave propagation theory is proposed for predicting the inlet pressure of an aircraft pump under transient conditions. First, a dynamic model of a typical aircraft pump is established in the form of differential equations. Then, the transient flow model of the inlet line is described using momentum and continuity equations, and the governing equations are discretized by the method of characteristics and the finite difference method. The simulated results are in good agreement with the results from verification tests. Further simulation analysis indicates that the wave velocity and transient time may influence the inlet and reservoir pressure as well as the size of the inlet line. Finally, solutions for upgrading the inlet pressure are discussed. These solutions provide guidelines for designing inlet installations.

A scheme for improving computational efficiency of quasi-two-dimensional model

The quasi-2D model, taking into account the axial velocity profile in the cross section and neglecting the convective term in the 2-D equation, can more accurately simulate the water hammer than the 1-D model using the cross-sectional mean velocity. However, as compared with the 1-D model, the quasi-2D model bears a higher computational burden. In order to improve the computational efficiency, the 1-D method is proposed to be used to solve directly the pressure head and the discharge in the quasi-2D model in this paper, based on the fact that the pressure head obtained as the solution of the two-dimensional characteristic equation is identical to that solved by the 1-D characteristic equations. The proposed scheme solves directly the 1-D characteristic equations for the pressure head and the discharge using the MOC and solves the 2-D characteristic equation for the axial velocities in order to calculate the wall shear stress. If the radial velocity is needed, it can be evaluated easily by an explicit equation derived from the explicit 2-D characteristic equation. In the numerical test, the accuracy and the efficiency of the proposed scheme are compared with two existing quasi-two-dimensional models using the MOC. It is shown that the proposed scheme has the same accuracy as the two quasi-2D models, but requires less computational time. Therefore, it is efficient to use the proposed scheme to simulate the 2-D water hammer flows.

ON STABILITY OF SYMPLECTIC ALGORITHMS

The stability of symplectic algorithms is discussed in this paper. There are following conclusions. 1. Symplectic Runge-Kutta methods and symplectic one-step methods with high order derivative are unconditionally critically stable for Hamiltonian systems. Only some of them are A-stable for non-Hamiltonian systems. The criterion of judging A-stability is given. 2. The hopscotch schemes are conditionally critically stable for Hamiltonian systems. Their stability regions are only a segment on the imaginary axis for non-Hamiltonian systems. 3. All linear symplectic multistep methods are conditionally critically stable except the trapezoidal formula which is unconditionally critically stable for Hamiltonian systems. Only the trapezoidal formula is A-stable, and others only have segments on the imaginary axis as their stability regions for non-Hamiltonian systems.

One-Dimensional Simulation of the Pressure Wave near the Exit of a High-Speed Train Tunnel

The one dimensional (1 D) unsteady flow induced by a high speed train entering a tunnel is numerically studied by the method of characteristics. The tube area is dependent on time and distance. The energy equation used by Kage et al. is corrected to avoid the conflict with the isentropic assumption. The effect of the tunnel hood on the pressure wave is studied near the tunnel exit. Results show that the tunnel hood is useful in reducing the peak value and the time derivative of the pressure wave.

Simulation of cavitation induced by water hammer

A simulation is carried out for the pressure fluctuation driven by the water hammer, based on a joint use of the onedimensional method of characteristics（MOC） and the three-dimensional finite volume method（FVM）. The three-dimensional visualization of the cavitation induced by the water hammer is implemented, and the temporal and spatial analyses of extreme regions are made. A practical case of the water hammer, with the minimum boundary pressure higher than the saturated vapor pressure condition, is simulated. The simulation prediction that the cavitation would occur in the front of the gasket could serve some guideline for the optimization of industrial designs.