Lagrangian–Eulerian methods for uniqueness in hydrodynamic systems

We present a Lagrangian–Eulerian strategy for proving uniqueness and local existence of solutions of limited smoothness for a class of incompressible hydrodynamic models including Oldroyd-B type complex fluid models and zero magnetic resistivity magneto-hydrodynamics equations.     

An Eulerian–Lagrangian method for coupled parabolic-hyperbolic equations

We present an Eulerian–Lagrangian method for the numerical solution of coupled parabolic-hyperbolic equations. The method combines advantages of the modified method of characteristics to accurately solve the hyperbolic equations with an Eulerian method to discretize the parabolic equations. The Runge–Kutta Chebyshev scheme is used for the time integration. The implementation of the proposed method differs from its Eulerian counterpart in the fact that it is applied during each time step, along the characteristic curves rather than in the time direction. The focus is on constructing explicit schemes with a large stability region to solve coupled radiation hydrodynamics models. Numerical results are presented for two test examples in coupled convection-radiation and conduction–radiation problems.     

Summarizing an Eulerian–Lagrangian model for subsea gas release and comparing release of CO2 with CH4

A subsea gas release is a concern for both safety and environment. This can be assessed by mathematical models. The development of an Eulerian–Lagrangian modelling concept to study subsea gas release has taken place over many years and the piecewise enhancements have been documented in the open literature. The model in its current state is summarized in this article. Model simulations are shown to be consistent with different experiments varying in depth from 7 to 138 m. The model can be applied to estimate how gas surfaces into the atmosphere from a subsea source. This is vital input to risk assessments. Due to recent interest in subsea CO₂ storage and transport, a comparison of CO₂- and CH₄-releases has been performed. Model results show that a much smaller fraction of released CO₂ reaches the atmosphere than CH₄ due to the high solubility of CO₂ in water.     

Development of a slurry abrasion model using an Eulerian–Eulerian ‘two-fluid’ approach

The aim of the present study was to develop a numerical model that predicts the quantity and location of erosion damage in slurry systems susceptible to erosive wear. Unlike many forms of erosion which often need to be prevented, hydro-erosion is deliberately introduced during the manufacturing process of automotive diesel injector nozzles to smooth out imperfections in the spray-hole geometry. This model was integrated into a commercial CFD code, ANSYS CFX^®, and took into account the change in geometry by dynamically updating the mesh to model the removal of material. A slurry jet erosion apparatus was developed to determine experimentally the parameters influencing the erosion process. Simplified planar geometries with four different angles of inclination were investigated and subject to typical hydro-erosive conditions similar to that used during the smoothing process of injector spray-hole geometries. In addition, the effect of slurry viscosity, particle size and concentration were studied. Results were used to calibrate the developed erosion model which showed encouraging trends in comparison with experimental studies for predicting the location and quantity of erosive wear.     

An Eulerian–Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs

We present a numerical method for solving tracking-type optimal control problems subject to scalar nonlinear hyperbolic balance laws in one and two space dimensions. Our approach is based on the formal optimality system and requires numerical solutions of the hyperbolic balance law forward in time and its nonconservative adjoint equation backward in time. To this end, we develop a hybrid method, which utilizes advantages of both the Eulerian finite-volume central-upwind scheme (for solving the balance law) and the Lagrangian discrete characteristics method (for solving the adjoint transport equation). Experimental convergence rates as well as numerical results for optimization problems with both linear and nonlinear constraints and a duct design problem are presented.     

An Eulerian–Lagrangian mixed discrete least squares meshfree method for incompressible multiphase flow problems

Mixed discrete least squares meshfree (MDLSM) method has been developed as a truly meshfree method and successfully used to solve single-phase flow problems. In the MDLSM, a residual functional is minimized in terms of the nodal unknown parameters leading to a set of positive-definite system of algebraic equations. The functional is defined using a least square summation of the residual of the governing partial differential equations and its boundary conditions at all nodal points discretizing the computational domain. Unlike the discrete least squares meshfree (DLSM) which uses an irreducible form of the governing equations, the MDLSM uses a mixed form of the original governing equations allowing for direct calculation of the gradients leading to more accurate computational results. In this study, an Eulerian–Lagrangian MDLSM method is proposed to solve incompressible multiphase flow problems. In the Eulerian step, the MDLSM method is used to solve the governing phase averaged Navier–Stokes equations discretized at fixed nodal points to get the velocity and pressure fields. A Lagrangian based approach is then used to track different flow phases indexed by a set of marker points. The velocities of marker points are calculated by interpolating the velocity of fixed nodal points using a kernel approximation, which are then used to move the marker points as Lagrangian particles to track phases. To avoid unphysical clustering and dispersing of the marker points, as a common drawback of Lagrangian point tracking methods, a new approach is proposed to smooth the distribution of marker points. The hybrid Eulerian and Lagrangian characteristics of the approach used here provides clear advantages for the proposed method. Since the nodal points are static on the Eulerian step, the time-consuming moving least squares (MLS) approximation is implemented only once making the proposed method more efficient than corresponding fully Lagrangian methods. Furthermore, phases can be simply tracked using the Lagrangian phase tracking procedure. Efficiency of the proposed MDLSM multiphase method is evaluated using several benchmark problems and the results are presented and discussed. The results verify the efficiency and accuracy of the proposed method for solving multiphase flow problems.     

Assessment of turbulence modeling for gas flow in two-dimensional convergent–divergent rocket nozzle

In the present study, the turbulent gas flow dynamics in a two-dimensional convergent–divergent rocket nozzle is numerically predicted and the associated physical phenomena are investigated for various operating conditions. The nozzle is assumed to have impermeable and adiabatic walls with a flow straightener in the upstream side and is connected to a plenum surrounding the nozzle geometry and extended in the downstream direction. In this integrated component model, the inlet flow is assumed a two-dimensional, steady, compressible, turbulent and subsonic. The physics based mathematical model of the considered flow consists of conservation of mass, momentum and energy equations subject to appropriate boundary conditions as defined by the physical problem stated above. The system of the governing equations with turbulent effects is solved numerically using different turbulence models to demonstrate their numerical accuracy in predicting the characteristics of turbulent gas flow in such complex geometry. The performance of the different turbulence models adopted has been assessed by comparing the obtained results of the static wall pressure and the shock position with the available experimental and numerical data. The dimensionless shear stress at the nozzle wall and the separation point are also computed and the flow field is illustrated. The various implemented turbulence models have shown different behavior of the turbulent characteristics. However, the shear-stress transport (SST) k–ω model exhibits the best overall agreement with the experimental measurements. In general, the proposed numerical procedure applied in the present paper shows good capability in predicting the physical phenomena and the flow characteristics encountered in such kinds of complex turbulent flow.     

Numerical simulation of sphere water entry problem using Eulerian–Lagrangian method

This paper looks at the hydrodynamic’s numerical simulation of a free-falling sphere impacting the free surface of water by using the coupled Eulerian–Lagrangian (CEL) formulation included in the commercial software ABAQUS. A 3D model of a sphere with an unsteady viscous transient flow condition is used for numerical simulation. The simulation is performed for sphere with different density. The simulation results are verified by showing the computed shape of the air cavity, displacement of sphere, pinch-off time and depth that agree well with experimental results.     

Modelling and Simulation of Milling Forces Using an Arbitrary Lagrangian–Eulerian Finite Element Method and Support Vector Regression

A new method is presented to predict milling forces synthetically. Firstly, the 3D simulation model of the milling process is established using the arbitrary Lagrangian–Eulerian finite element method. And the simulated accuracy is calibrated by milling tests. Then the simulation model is taken as a virtual milling test system to replace extensive real milling experiments. Secondly, the specific cutting coefficients in the mechanistic milling forces model are identified by the support vector regression method using the training sample generated from the established virtual milling test system. Lastly, this methodology was validated by the slot milling operation of 2024-T3 aluminum sheets. The results show that this new approach can dramatically eliminate the experimental machining work and achieve good estimation accuracy.     

Traveling wave solutions for an autocatalytic reaction–diffusion model

In this paper we consider an autocatalytic reaction–diffusion model which has many applications. We extend previous results using qualitative analysis and show the existence of an exponentially decaying traveling wave front for a minimum speed and algebraically decaying wave fronts for large speeds. Further, the wave front profiles are calculated and the minimum speed is accurately determined using different numerical methods.     

Augmented Lagrangian preconditioners for the Oseen–Frank model of nematic and cholesteric liquid crystals

BIT Numerical Mathematics - We propose a robust and efficient augmented Lagrangian-type preconditioner for solving linearizations of the Oseen–Frank model arising in nematic and cholesteric...     

Global well-posedness for an inhomogeneous LSW–model in unbounded domains

The LSW–theory of domain coarsening describes the evolution of the size distribution of a system of particles evolving by diffusional mass exchange to reduce their total surface area. We prove existence and uniqueness of solutions to an inhomogeneous extension of the LSW–model in unbounded domains. This model arises naturally as a homogenization limit of the underlying free boundary problem in the case of a system of particles for which the screening length (the effective range of particle interactions) is smaller than the system size. The crucial ingredients in the analysis are, first, to establish the screening property by showing that the corresponding Greens function decreases exponentially over the relevant distances. Second, we have to control the mass fluxes between different regions to prevent aggregation of mass in few particles which would result in blow–up of the solution.Mathematics Subject Classification (2000):82C26, 35Q72, 35D05Received: 16, December 2001     

Hopf bifurcation and steady-state bifurcation for an autocatalysis reaction–diffusion model

This paper is concerned with an autocatalysis model subject to no-flux boundary conditions. The existence of Hopf bifurcation are firstly obtained. Then by the center manifold theory and the normal form method, the direction of Hopf bifurcation and the stability of spatially homogeneous and inhomogeneous periodic solutions are established. On the other hand, the steady-state bifurcations from simple and double eigenvalues are intensively studied. The techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalues. Finally, some numerical simulations are shown to verify the analytical results.     

Well posedness of an integrodifferential kinetic model of Fokker–Planck type for angiogenesis

Tumor induced angiogenesis processes including the effect of stochastic motion and branching of blood vessels can be described coupling a (nonlocal in time) integrodifferential kinetic equation of Fokker–Planck type with a diffusion equation for the tumor induced angiogenic factor. The chemotactic force field depends on the flux of blood vessels through the angiogenic factor. We develop an existence and uniqueness theory for this system under natural assumptions on the initial data. The proof combines the construction of fundamental solutions for associated linearized problems with comparison principles, sharp estimates of the velocity integrals and compactness results for this type of kinetic and parabolic operators.