Lagrangian Boundary Research Articles

Experimental and Numerical Study of the Trench Fire Spread Rule over a Sloped Uniform Fuel Bed: Rate of Fire Spread, Flame Morphology, and Heat Flux

Trench fires on sloped terrain are always complicated due to the corresponding flame dynamics and heat transfer mechanisms. Flame attachment may increase the rate of fire spread (ROS) by enlarging the heating area of unburned vegetation. In addition, variations in radiative and convective heat flux are of great importance to fire behavior characteristics. In this work, trench fire tests under different slopes (θ) and inclined sidewalls (A) were performed by numerical simulations based on the Lagrangian Particle Model (LPM) and Boundary Fuel Model (BFM) in the Fire Dynamics Simulator (FDS) and small-scale experiments, and the ROS, flame characteristics, and radiative/convective heat flux of the fire front are discussed in detail. The results indicate that the flame tends to adhere to the fuel bed with increasing slope angle and sidewall inclination. In particular, the flame becomes fully attached with a greater pressure difference than the buoyancy, which is caused by the unequal air entrainment between the front and behind the flame. When A = 90°, the critical slope angle of the flame adhesion (from slight tilt to full attachment) is identified as ~20°. The ROS (θ ≤ 15°) predicted by the BFM and LPM are closer to the small-scale experiments. The heat fluxes based on the experiments confirm the predominant mechanism of radiative heat transfer in trench fires at low slopes (θ ≤ 20°). Furthermore, convective heat transfer is more significant than radiative and becomes the main heating mechanism for θ ≥ 20°.

A Lagrangian free-stream boundary condition for weakly compressible smoothed particle hydrodynamics

This paper proposes the first Lagrangian free-stream boundary condition for weakly-compressible smoothed particle hydrodynamics (WCSPH), significantly decreasing the domain-size effect for simulating flows around static or moving geometries and increasing computational efficiency. The boundary condition is implemented based on several numerical techniques, as follows. First, a spatio-temporal identification and an ensuring technique precisely identify surface particles without misjudgment. Second, the far-field corrections of density and velocity precisely target surface particles. Third, the improved in- and outlet boundaries guarantee the smooth inflow profile and free-outflow particles without the issue of the truncated kernel. The test cases of flows over a flat plate and around a circular cylinder have validated the accuracy and computational efficiency of the present method. In order to demonstrate its applicability, the method has also been tested with more complex examples, such as a fluid-structure interaction (FSI) problem and a 3-D flow around a sphere.

Comparison of Machining Simulations of Aerospace Alloy Al6061-T6 Using Lagrangian and Smoothed Particle Hydrodynamics Techniques

This research focuses on the study of the simulation capabilities of the lagrangian (LAG) model and Smoothed Particle Hydrodynamics (SPH) model for the orthogonal dry machining of aluminum alloy Al6061-T6. A three-dimensional finite element model was developed and verified using experimental data from the published literature. The numerical models were developed using lagrangian boundary conditions via finite element modeling in ABAQUS/Explicit 6.14. The cutting simulations were carried out at low and medium cutting speeds. Johnson–Cook material constitutive law and Johnson–Cook damage law were used in both models. The numerical methodologies are compared based on cutting forces, chip morphology, shear angle, chip separation criterion, and chip thickness. The findings of the present work show that the LAG model is good for predictions regarding cutting forces and chip morphology, while the SPH model is good for predictions regarding the shear angle and chip thickness. The difference between results generated by both models mainly occurred due to the friction coefficient. The comparative study shown here offers a guidance approach for various numerical models for appropriate parameter analysis.

A compactness theorem for SO(3) anti-self-dual equation with translation symmetry

Motivated by the Atiyah–Floer conjecture, we consider SO(3) anti-self-dual instantons on the product of the real line and a three-manifold with cylindrical end. We prove a Gromov–Uhlenbeck type compactness theorem, namely, any sequence of such instantons with uniform energy bound has a subsequence converging to a type of singular objects which may have both instanton and holomorphic curve components. In particular, we prove that holomorphic curves that appear in the compactification must satisfy the Lagrangian boundary condition, a claim which has been long believed in the literature. This result is the first step towards constructing a natural bounding cochain proposed by Fukaya for the SO(3) Atiyah–Floer conjecture.

Generalized Gelfand–Yaglom formula for a discretized quantum mechanic system

The Gelfand–Yaglom formula relates the regularized determinant of a differential operator to the solution of an initial value problem. Here, we develop a generalized Gelfand–Yaglom formula for a Hamiltonian system with Lagrangian boundary conditions in the discrete and continuous settings. Later, we analyze the convergence of the discretized Hamilton–Jacobi operator and propose a lattice regularization for the determinant.

An Eulerian Vlasov-Fokker–Planck algorithm for spherical implosion simulations of inertial confinement fusion capsules

We present a numerical algorithm that enables a phase-space adaptive Eulerian Vlasov–Fokker–Planck (VFP) simulation of inertial confinement fusion (ICF) capsule implosions. The approach relies on extending a recent mass, momentum, and energy conserving phase-space moving-mesh adaptivity strategy to spherical geometry. In configuration space, we employ a mesh motion partial differential equation (MMPDE) strategy while, in velocity space, the mesh is expanded/contracted and shifted with the plasma’s evolving temperature and drift velocity. The mesh motion is dealt with by transforming the underlying VFP equations into a computational (logical) coordinate, with the resulting inertial terms carefully discretized to ensure conservation. To deal with the spatial and temporally varying dynamics in a spherically imploding system, we have developed a novel nonlinear stabilization strategy for MMPDE in the configuration space. The strategy relies on a nonlinear optimization procedure that optimizes between mesh quality and the volumetric rate change of the mesh to ensure both accuracy and stability of the solution. Implosions of ICF capsules are driven by several boundary conditions: (1) an elastic moving wall boundary; (2) a time-dependent Maxwellian Dirichlet boundary; and (3) a pressure-driven Lagrangian boundary. Of these, the pressure-driven Lagrangian boundary driver is new to our knowledge. The implementation of our strategy is verified through a set of test problems, including the Guderley and Van-Dyke implosion problems — the first-ever reported using a Vlasov–Fokker–Planck model.

Holomorphic disks and the disk potential for a fibered Lagrangian

We consider a fibered Lagrangian $L$ in a compact symplectic fibration with small monotone fibers, and develop a strategy for lifting $J$-holomorphic disks with Lagrangian boundary from the base to the total space. In case $L$ is a product, we use this machinery to give a formula for the leading order potential and formulate an unobstructedness criteria for the $A_\infty$ algebra. We provide some explicit computations, one of which involves finding an embedded 2n+k dimensional submanifold of Floer-non-trivial tori in a 2n+2k dimensional fiber bundle.

Lagrangian analysis of the fluid transport induced by the interaction of two co-axial co-rotating vortex rings

In this paper, the fluid transport in the interaction of two co-axial co-rotating vortex rings are investigated. Vortex rings are generated using the piston-cylinder apparatus, and the resulting velocity fields are measured using digital particle image velocimetry. The interaction process is analysed by means of vorticity contour, as well by investigation of the Lagrangian coherent structures (LCSs) defined by the ridges of the finite-time Lyapunov exponent (FTLE). Experimental results demonstrate that two types of vortex interaction are identified, namely strong and weak interactions, respectively. For the strong interaction, the Lagrangian boundaries of the two vortex rings are merged together and form a flux window for fluid transport. For weak interaction, only the Lagrangian drift induced by the motion of the front vortex ring is observed and affects the Lagrangian boundary of the rear vortex ring. Moreover, the fluids transported in the strong interaction carry considerable momentum but no circulation. By contrast, there are nearly no fluxes occurring in the weak interaction. By tracking the variations of circulation and impulse occupied by the separated regions distinguished by the LCSs, it is found that the circulation nearly has no change, but the impulse occupied by vortex core region has significant change. In the strong interaction, the impulse of rear vortex ring decreases but the impulse of the front vortex ring increases. Based on the impulse law, it is speculated that the fluid force generated by the formation of the rear vortex rings can be enhanced. Therefore, the strong interaction between wake vortices can actually improve the propulsive efficiency of the biological systems by operating the formation of large-scale vortices.

Effects of liquid fuel/wall interaction on thermoacoustic instabilities in swirling spray flames

Computational prediction of thermoacoustic instabilities arising in gas turbine and aero-engine combustors still remains a challenge especially if fuel is injected in a liquid spray form. This study shows that, in LES of such a combustor, the treatment of the liquid fuel film created on the walls of the injection system affects the mean flame weakly, but modifies the flame dynamics strongly. The configuration used for this work is the experimental setup SICCA-spray available at EM2C laboratory in Paris. First steady spray flame measurements are used to validate the LES Euler-Lagrange approach. Two modelling strategies for the interaction between the liquid fuel and the injector walls are tested with a negligible impact on the flame shape and structure. In the second part the same comparison is applied to another operating condition where a self-sustained thermo-acoustic limit-cycle is experimentally observed. In that case resonant coupling is achieved with LES, confirming the adequacy of the approach but only when the film layer is taken into account. Indeed, contrarily to the stable configuration, the difference between the two Lagrangian boundary conditions is shown to have a major impact on the feedback mechanism leading to the thermoacoustic oscillation.

A bounce back-immersed boundary-lattice Boltzmann model for curved boundary

In this paper, a bounce back-immersed boundary-lattice Boltzmann model(BB-IB-LBM) is proposed for curved boundary.In the present model, a modified density distribution function is proposed for curved boundary including stationary and moving curved boundaries. A special treatment is also developed to satisfy no-slip boundary condition for the curved boundary with large curvature. On Lagrangian boundary points, the modified distribution functions are implemented to replace the artificial correction force in conventional immersed boundary-lattice Boltzmann methods(IB-LBMs). Numerical experiments are given to illustrate the accuracy and efficiency of present BB-IB-LBM. The drag coefficient of the test cases predicted by the present model is in better agreement with the results of experimental results than that of the previous IB-LBMs. It is also concluded that the average drag coefficient of present model are consistent with the experimental results. Comparing with conventional IB-LBMs, the present model eliminates the non-physical vortex at the tail of an airfoil. Simulation of flow over a sphere also proves the extensibility of present method in three-dimensional simulation.

A Degree Theory for Lagrangian Boundary Value Problems

We study those nonlinear partial differential equations which appear as Euler-Lagrange equations of variational problems. On defining weak boundary values of solutions to such equations we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to Lagrangian problems

Hill-type formula for Hamiltonian system with Lagrangian boundary conditions

In this paper, we build up Hill-type formula for linear Hamiltonian systems with Lagrangian boundary conditions, which include standard Neumann, Dirichlet boundary conditions. Such a kind of boundary conditions comes from the N-reversible symmetry periodic orbits in n-body problem naturally, where N is an anti-symplectic orthogonal matrix with N2=I. The Hill-type formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions. Consequently, we derive the Krein-type trace formula and give nontrivial estimation for the eigenvalue problem. Combined with the Maslov-type index theory, we give some new stability criteria for the N-reversible symmetry periodic solutions of Hamiltonian systems. As an application, we study the linear stability of elliptic relative equilibria in planar 3-body problem.

Hamiltonian Boundary Value Problems, Conformal Symplectic Symmetries, and Conjugate Loci

In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we present a convenient, coordinate free framework to analyse separated Lagrangian boundary value problems which include classical Dirichlet, Neumann and Robin boundary value problems. The framework is then used to prove the existence of obstructions arising from conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems. Under non-degeneracy conditions, a group action by conformal symplectic symmetries has the effect that the flow map cannot degenerate in a direction which is tangential to the action. This imposes restrictions on which singularities can occur in boundary value problems. Our results generalise classical results about conjugate loci on Riemannian manifolds to a large class of Hamiltonian boundary value problems with, for example, scaling symmetries.

Error estimates for projection-based dynamic augmented Lagrangian boundary condition enforcement, with application to fluid–structure interaction

In this work, we analyze the convergence of the recent numerical method for enforcing fluid–structure interaction (FSI) kinematic constraints in the immersogeometric framework for cardiovascular FSI. In the immersogeometric framework, the structure is modeled as a thin shell, and its influence on the fluid subproblem is imposed as a forcing term. This force has the interpretation of a Lagrange multiplier field supplemented by penalty forces, in an augmented Lagrangian formulation of the FSI kinematic constraints. Because of the non-matching fluid and structure discretizations used, no discrete inf-sup condition can be assumed. To avoid solving (potentially unstable) discrete saddle point problems, the penalty forces are treated implicitly and the multiplier field is updated explicitly. In the present contribution, we introduce the term dynamic augmented Lagrangian (DAL) to describe this time integration scheme. Moreover, to improve the stability and conservation of the DAL method, in a recently-proposed extension we projected the multiplier onto a coarser space and introduced the projection-based DAL method. In this paper, we formulate this projection-based DAL algorithm for a linearized parabolic model problem in a domain with an immersed Lipschitz surface, analyze the regularity of solutions to this problem, and provide error estimates for the projection-based DAL method in both the [Formula: see text] and [Formula: see text] norms. Numerical experiments indicate that the derived estimates are sharp and that the results of the model problem analysis can be extrapolated to the setting of nonlinear FSI, for which the numerical method was originally proposed.

The Competing Effects of Stability and Humidity on Subtropical Stratocumulus Entrainment and Cloud Evolution from a Lagrangian Perspective

Abstract The evolution of subtropical stratocumulus clouds and the boundary layer is studied on daily time scales from the Lagrangian perspective, following the flow. Measures of humidity above the boundary layer and of inversion strength are obtained from reanalysis data, and their effects on the Lagrangian evolution of cloud cover and the boundary layer are compared. An analysis that disentangles these variables and tests their effects independently is developed. Increased inversion strength and increased humidity above the boundary layer lead to anomalously persistent cloud cover and slower Lagrangian deepening of the boundary layer. These parameters affect the stratocumulus boundary layer in different ways: inversion strength controls the buoyancy difference across the inversion, while humidity differences affect both the radiation balance and rate of cloud drop evaporation at cloud top. The relative strengths of the two effects of humidity are compared using two products: the entraining humidity in the layer directly above the inversion and the radiating humidity, which is the mean humidity in the column above the entraining humidity. Results show that the variability in the radiating humidity is the primary driver of Lagrangian boundary layer depth changes, but entraining humidity variation is mostly responsible for altering cloud lifetime.

Reduced open Gromov–Witten invariants on K3 surfaces and multiple cover formula

In the paper, we study the wall-crossing phenomenon of reduced open Gromov–Witten invariants on K3 surfaces with rigid special Lagrangian boundary condition. As a corollary, we derived the multiple cover formula for the reduced open Gromov–Witten invariants, which is an open analog of the Gopakumar–Vafa conjecture of genus zero.

A diffuse interface immersed boundary method for complex moving boundary problems

A diffuse interface immersed boundary method is proposed for problems with arbitrarily moving complex rigid bodies. The solid-fluid interface is replaced by fluid–fluid Lagrangian boundary. A universal equation that combines the momentum conservation and rigid body velocity through the volume fraction is solved in the entire domain. The ever changing volume fraction field is accurately computed locating the object boundary. As the desired boundary conditions are only enforced directly in a volume average sense scope of it spreads over a computing cell. Thus, by reducing size of the computing cell the proposed method approaches the conventional sharp interface methods though it eliminates the spurious force oscillations. Owing to the single universal equation on a fixed Eulerian Cartesian mesh the proposed method retains the simplicity, robustness and versatility of the original Cartesian grid simulation. Convergence study of the transverse oscillation of a cylinder shows second-order convergence with refinement of numerical parameters and significantly diminished force oscillation. The proposed method is parallelized on a mixed shared and distributed memory notions in order to facilitate large scale 2D and 3D simulations. A range of test cases have been carefully chosen to check the quality of prediction of the proposed technique for forced and induced motions of multiple objects. They include in-line, transverse and rotational forced motion of a circular cylinder, motion of a hovering thin wing, vortex induced vibration (VIB) of multiple circular and square cylinder, angular VIB of multiple objects, particle sedimentation, instability driven fluttering of a freely falling plate, settling of a sphere in a quiescent medium, 3D VIB of two cylinders, wake transition due to pitching of a large aspect ratio thin plate and turbulent wake behind a wavy cylinder.

Open Gromov–Witten invariants, mirror maps, and Seidel representations for toric manifolds

Let $X$ be a compact toric K\"ahler manifold with $-K_X$ nef. Let $L\subset X$ be a regular fiber of the moment map of the Hamiltonian torus action on $X$. Fukaya-Oh-Ohta-Ono defined open Gromov-Witten (GW) invariants of $X$ as virtual counts of holomorphic discs with Lagrangian boundary condition $L$. We prove a formula which equates such open GW invariants with closed GW invariants of certain $X$-bundles over $\mathbb{P}^1$ used to construct the Seidel representations for $X$. We apply this formula and degeneration techniques to explicitly calculate all these open GW invariants. This yields a formula for the disc potential of $X$, an enumerative meaning of mirror maps, and a description of the inverse of the ring isomorphism of Fukaya-Oh-Ohta-Ono.

Compactness in the adiabatic limit of disk vortices

This paper is the first input towards an open analogue of the quantum Kirwan map. We consider the adiabatic limit of the symplectic vortex equation over the unit disk for a Hamiltonian G-manifold with Lagrangian boundary condition, by blowing up the metric on the disk. We define an appropriate notion of stable solutions in the limit, and prove that any sequence of disk vortices with energy uniformly bounded has a subsequence converging to such a stable object. We also proved several analytical properties of vortices over the upper half plane, which are new type of bubbles appearing in our compactification.

Local membrane length conservation in two-dimensional vesicle simulation using a multicomponent lattice Boltzmann equation method.

We present a method for applying a class of velocity-dependent forces within a multicomponent lattice Boltzmann equation simulation that is designed to recover continuum regime incompressible hydrodynamics. This method is applied to the problem, in two dimensions, of constraining to uniformity the tangential velocity of a vesicle membrane implemented within a recent multicomponent lattice Boltzmann simulation method, which avoids the use of Lagrangian boundary tracers. The constraint of uniform tangential velocity is carried by an additional contribution to an immersed boundary force, which we derive here from physical arguments. The result of this enhanced immersed boundary force is to apply a physically appropriate boundary condition at the interface between separated lattice fluids, defined as that region over which the phase-field varies most rapidly. Data from this enhanced vesicle boundary method are in agreement with other data obtained using related methods [e.g., T. Krüger, S. Frijters, F. Günther, B. Kaoui, and J. Harting, Eur. Phys. J. 222, 177 (2013)10.1140/epjst/e2013-01834-y] and underscore the importance of a correct vesicle membrane condition.