Additional Advection Research Articles

Suppression of Chemotactic Explosion by Mixing

Chemotaxis plays a crucial role in a variety of processes in biology and ecology. In many instances, processes involving chemical attraction take place in fluids. One of the most studied PDE models of chemotaxis is given by Keller-Segel equation, which describes a population density of bacteria or mold which attract chemically to substance they secrete. Solution of Keller-Segel equation can exhibit dramatic collapsing behavior, where the density concentrates positive mass in a measure zero region. A natural question is whether presence of fluid flow can affect singularity formation by mixing the density thus making concentration harder to achieve. In this paper, we consider parabolic-elliptic Keller-Segel equation in two and three dimensions with additional advection term modeling ambient fluid flow. We prove that for any initial data, there exist incompressible fluid flows such that the solution to the equation stays globally regular. On the other hand, it is well known that when the fluid flow is absent, there exist initial data leading to finite time blow up. Thus presence of fluid can prevent the singularity formation. We discuss two classes of flows that have the explosion arresting property. Both classes are known as very efficient mixers. The first class are relaxation enhancing (RE) flow. These flows are stationary. The second class of flows are Yao-Zlatos optimal mixing flows, which are time dependent. The proof is based on the nonlinear version of the relaxation enhancement construction of the paper by Peter Constantin, Alexander Kiselev, Lenya Ryzhik, Andrej Zlatos, and on some variations of global regularity estimate for Keller-Segel model.

A Curvature-Augmented, REA Approach to the Level Set Method

When solving a moving interface problem, the interface can be tracked using a variety of methods. The level set method captures the interface as an isocontour of a scalar level set function. The method has many advantages, including the ability to express many geometric quantities, such as the interface curvature, as derivatives of the level set. However, the numerically constructed level set function may not be smooth enough to compute the required derivatives. Furthermore, the method overall is known for having a high sensitivity to numerical dissipation. The former of these two shortfalls is addressed by augmenting the traditional level set equations with an explicitly tracked interface curvature. The curvature is then updated alongside the level set through an additional advection equation. The latter shortfall is addressed by combining a new velocity extension, that better maintains the signed-distance property of the level set, with a reconstruct-evolve-average approach to advancing the advection equations. The new approach is shown to have less mass loss (numerical dissipation) and better accuracy than comparable level set approaches. Three scenarios are investigated: an interface moving according to an external velocity field, an interface moving according to the interface curvature (mean-curvature flow), and the air-water interface of a water drop moving according to the curvature-dependent fluid velocity (surface-tension driven flow).

Global regularity of chemotaxis equations with advection

We study the Patlak–Keller–Segel (PKS) equations in 2D that describe chemotaxis with an additional advection term. We show that solutions are globally regular for smooth initial data with subcritical mass as long as the flow has nonpositive divergence. For initial data with supercritical mass, numerical simulations suggest that blow-up might be prevented by imposing some strong incompressible advection term.

Application of the immersed-body method to simulate wave–structure interactions

This study aims at demonstrating the capability of the immersed-body method to simulate wave–structure interactions using a non-linear finite-element model. In this approach, the Navier–Stokes equations are solved on an extended mesh covering the whole computational domain (i.e. fluids and structure). The structure is identified on the extended mesh through a nonzero solid-concentration field, which is obtained by conservatively mapping the mesh discretising the structure onto the extended mesh. A penalty term relaxes the fluid and structural velocities to one another in the regions covered by the structure. The paper is novel in that it combines the immersed-body method with wave modelling and mesh adaptivity. The focus of the paper is therefore on demonstrating the capability of this new methodology in reproducing well-established test cases, rather than investigating new physical phenomena in wave–structure interactions. Two cases are considered for a bottom-mounted pile. First, the pile is placed in a numerical wave tank, where propagating waves are modelled through a free-surface boundary condition. For regular and irregular waves, it is shown that the wave dynamics are accurately modelled by the computational fluid dynamics model and only small discrepancies are observed in the close vicinity of the structure. Second, the structure is subjected to a dam-break wave impact obtained by removing a barrier between air and water. In that case, an additional advection equation is solved for a fluid-concentration field that tracks the evolution of the air–water interface. It is shown that the load associated with the wave impact on the structure compares well with existing numerical and experimental data.

An analytical solution for linear gravity and sound waves on the sphere as a test for compressible, non-hydrostatic numerical models

An analytical solution for the expansion of gravity and sound waves for the linearized form of the fully compressible, non-hydrostatic, shallow atmosphere Euler equations on the sphere is derived. The waves are generated by a weak initial temperature and density perturbation of an isothermal atmosphere. The derived analytical solution can be used as a benchmark to assess dynamical cores of global models based on the above-mentioned (in general nonlinear) equation system. Three different test configurations, with or without Coriolis force or additional advection, are discussed. Convergence studies of the newly developed ICOsahedral Non-hydrostatic global model (ICON) against this solution are performed.

Well-posedness and qualitative properties of a dynamical model for the ideal free distribution

Understanding the spatial distribution of populations in heterogeneous environments is an important problem in ecology. In the case of a population of organisms that can sense the quality of their environment and move to increase their fitness, one theoretical description of the expected distribution of the population is the ideal free distribution, where individuals locate themselves to optimize fitness. A model for a dynamical process that allows a population to achieve an ideal free distribution was proposed by the Cosner (Theor Popul Biol 67:101-108, 2005). The model is based on a reaction-diffusion-advection equation with nonlinear diffusion which is similar to a porous medium equation with additional advection and population growth terms. We establish that the model is well-posed, show that solutions stabilize, determine the stationary states, discuss their stability, and describe the biological interpretation of the results.

Consistent numerical diffusion terms for simulating compressible multicomponent flows

An interface-capturing method using a high-order central difference scheme is presented for simulations of compressible multicomponent flows. The present method adds consistent numerical diffusion terms to robustly capture interface discontinuities, while maintaining the velocity, pressure, and temperature equilibriums at interfaces. Analysis of the numerical errors generated at the interfaces leads to a proposal of new consistent numerical diffusion terms. The method solves a fully conservative form of the total mass, momentum, total energy, and species-mass, and an additional advection form of the specific heats ratio to preserve the pressure oscillation-free property at the interfaces. Several one- and two-dimensional problems are used to verify the proposed method.

Magnetohydrodynamic drift equations: from Langmuir circulations to magnetohydrodynamic dynamo?

AbstractWe derive the closed system of averaged magnetohydrodynamic (MHD) equations for general oscillating flows. The used small parameter of our asymptotic theory is the dimensionless inverse frequency, and the leading term for a velocity field is chosen to be purely oscillating. The employed mathematical approach combines the two-timing method and the notion of a distinguished limit. The properties of commutators are used to simplify calculations. The derived averaged equations are similar to the original MHD equations, but surprisingly (instead of the commonly expected Reynolds stresses) a drift velocity plays a part of an additional advection velocity. In the special case of a vanishing magnetic field $\mathbi{h}\equiv 0$, the averaged equations produce the Craik–Leibovich equations for Langmuir circulations (which can be called ‘vortex dynamo’). We suggest that, since the mathematical structure of the full averaged equations for $\mathbi{h}\not = 0$ is similar to those for $\mathbi{h}\equiv 0$, these full equations could lead to a possible mechanism of MHD dynamo, such as the generation of the magnetic field of the Earth.

Free surface flow simulations on GPGPUs using the LBM

In this paper, we present the implementation of a volume-of-fluid-(VOF)-based algorithm for the simulation of free-surface flow problems on general purpose graphical processing units (GPGPUs). For the solution of the flow field and the additional advection equation for the VOF fill level, the lattice Boltzmann method on the basis of an MRT collision operator is used. A Smagorinsky LES model serves to capture the small-scale turbulent structures of the flow. We show that despite the additional non-local operations near the phase interface, we end up with an algorithm with good overall performance, which is suitable for the simulation of demanding real-world engineering applications.

Additional flux arising from unresolved scales in eddying ocean models

An analysis is presented of snapshot data (eastward and northward velocity components: u and v; tracer such as potential temperature: τ) from an eddy-resolving (Rgrid: 1/12°) ocean model experiment, in order to explore a method for improving eddy-permitting model performance. Horizontal 3 × 3 R-grid averages give the eddy-permitting grid (P-grid: 1/4°) variables: 〈u〉, 〈v〉, and 〈τ〉, where 〈〉 denotes the spatial P-grid scale average. The difference between the horizontal tracer flux across the boundary face of a P-grid and that across the corresponding faces of R-grids is estimated as F2E. It is found that the correlations among the gradients of u, v, and τ give a good approximation F2C to the estimated flux F2E. The approximated flux is a function of these gradients and the grid size. A method is presented for implementing the F2C for density to an eddying ocean model as an additional advection. Practical experiments were conducted with a realistic configuration. It is shown that the zonal mean isotherms in the Kuroshio extension region are more flattened in the run using the proposed method than in another run using the conventional horizontal biharmonic operator, suggesting that the additional flux correction leads to an enhancement of sub-basin scale mixing.

Verification of a dust transport model against theoretical solutions in multidimensional advection diffusion problems

Potentially explosive dust aerosol mobilization in the vacuum vessel is an important safety issue of the ITER facility, especially in scenarios of loss of vacuum accidents. Therefore dust mobilization modeling is ongoing in Research Center Karlsuhe. At first the aerosol particle model in the GASFLOW computer code is introduced briefly. To verify the particle model, a series of particle diffusion problems are simulated in one-, two- and three-dimensions. In each problem a particle source is initially exposed to an advective gas flow. Then a dust cloud is formed in the down stream. To obtain the theoretical solution about the particle concentration in the dust cloud, the governing diffusion partial differential equations with an additional advection term are solved by using Green's function method. Different spatial and temporal characters about the particle sources are also considered, e.g., instantaneous or continuous sources, line, or volume sources and so forth. The GASFLOW simulation results about the particle concentrations and the corresponding Green's function solutions are compared case by case. Very good agreements are found between the theoretical solutions and the GASGLOW simulations, when the drag force between the micron-sized particles and the conveying gas flow meets the Stokes’ law about resistance. This situation is corresponding to a very small Reynolds number based on the particle diameter, with a negligible inertia effect of the particles. This verification work shows that the particle model of the GASFLOW code can reproduce numerically particle transport and diffusion in a good way.

Influence of the Nonlinear Equation of State on Global Estimates of Dianeutral Advection and Diffusion

Abstract Recent work on the global overturning circulation and its energetics assumes that processes caused by nonlinearities of the equation of state of seawater are negligible. Nonlinear processes such as cabbeling and thermobaricity cause diapycnal motion as a consequence of isopycnal mixing. The nonlinear equation of state also causes the helical nature of neutral trajectories; as a consequence of this helical nature, it is not possible to define a continuous “density” surface that aligns with neutral tangent planes. The result is an additional diapycnal advection, which needs to be accounted for in water-mass analysis. In this paper, the authors take advantage of new techniques in constructing very accurate continuous density surfaces to more precisely estimate isopycnal and diapycnal processes caused by the nonlinear equation of state. They then quantify the diapycnal advection due to each of these nonlinear processes and show that they lead in total to a significant downward diapycnal advection, particularly in the Southern Ocean. The nonlinear processes are therefore another source of dense water formation in addition to high-latitude convection. To maintain the abyssal stratification in the global ocean, these dense water masses have to be brought back toward surface layers, and this can occur by either diabatic or adiabatic processes. Including these nonlinear processes into the advection–diffusion balance, the authors show that observed diapycnal diffusivities are unlikely to be able to support the amount of dense water produced in the global ocean, thus placing more importance on the adiabatic way of bringing the deep waters back to the surface.

Quantifying the Consequences of the Ill-Defined Nature of Neutral Surfaces

Abstract In the absence of diapycnal mixing processes, fluid parcels move in directions along which they do not encounter buoyant forces. These directions define the local neutral tangent plane. Because of the nonlinear nature of the equation of state of seawater, these neutral tangent planes cannot be connected globally to form a well-defined surface in three-dimensional space; that is, continuous “neutral surfaces” do not exist. This inability to form well-defined neutral surfaces implies that neutral trajectories are helical. Consequently, even in the absence of diapycnal mixing processes, fluid trajectories penetrate through any “density” surface. This process amounts to an extra mechanism that achieves mean vertical advection through any continuous surface such as surfaces of constant potential density or neutral density. That is, the helical nature of neutral trajectories causes this additional diasurface velocity. A water-mass analysis performed with respect to continuous density surfaces will have part of its diapycnal advection due to this diasurface advection process. Hence, this additional diasurface advection should be accounted for when attributing observed water-mass changes to mixing processes. Here, the authors quantify this component of the total diasurface velocity and show that locally it can be the same order of magnitude as diasurface velocities produced by other mixing processes, particularly in the Southern Ocean. The magnitude of this diasurface advection is proportional to the ocean’s neutral helicity, which is observed to be quite small in today’s ocean. The authors also use a perturbation experiment to show that the ocean rapidly readjusts to its present state of small neutral helicity, even if perturbed significantly. Additionally, the authors show how seasonal (rather than spatial) changes in the ocean’s hydrography can generate a similar vertical advection process. This process is described here for the first time; although the vertical advection due to this process is small, it helps to understand water-mass transformation on density surfaces.

A lattice Boltzmann approach for free-surface-flow simulations on non-uniform block-structured grids

In this paper, we present a hybrid volume-of-fluid-based algorithm for the simulation of free-surface-flow problems. For the solution of the flow field, the lattice Boltzmann method is used. The additional advection equation for the volume-of-fluid (VOF) fill level is discretized with a classical finite volume method. For the interface reconstruction, a piecewise linear interface reconstruction in 3D has been implemented. The free-surface-tracking algorithm is embedded into the 3D, non-uniform, lattice-Boltzmann-based solver VirtualFluids ; Freudiger et al. (2009) [1] , Freudiger (2009) [2] . The advection algorithm is verified and validated with well-known advection test cases. For the validation of the free-surface algorithm, we present simulations of a breaking-dam benchmark.

A Lattice‐Boltzmann model to simulate the growth of dendritic and eutectic microstructures under the influence of fluid flow

AbstractWe present a new formulation of a composite phase‐field and Lattice‐Boltzmann model to analyze the effect of fluid flow on the formation of solidification microstructures. The phase‐field model is based on a description suitable to treat phase tansformation processes and moving boundary problems in multiphase and multicomponent systems as well as in polycrystalline structures. The flow of the liquid is simulated by a Lattice‐Boltzmann‐BGK (LBGK) method. Interactions of the flow field with thermal and solutal convection are incorporated by additional advection terms in the mass and heat diffusion equations. At the boundaries between solid and liquid phases, the flow is modelled by a diffuse bounce back approach.In two‐ and three‐dimensional numerical simulations, we apply the model to investigate flow driven changes of the dendritic morphology in the Ni–Cu system. In particular, we impose different flow velocities at the boundary of the computational domain and investigate the different growth rates of the dendritic tip towards the direction of flow and on the shielded side. Furthermore, the influence of a fluid field on lamellar and rod‐like eutectic growth structures is studied for Al–Cu alloys. Since new solute is supplied by the fluid, a more pronounced growth is observed in the direction of the flow for both cases, dendrites and eutectics. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Extension and evaluation of sensitivity analysis capabilities in a photochemical model

The decoupled direct method in three dimensions (DDM-3D) provides an efficient and accurate approach for probing the sensitivity of atmospheric pollutant concentrations to various changes in photochemical model inputs. The implementation of DDM-3D for the widely used Community Multiscale Air Quality (CMAQ) model was updated to account for recent changes in the base model and to include additional chemical mechanisms and advection schemes. The capabilities of CMAQ–DDM-3D were extended to enable execution using multiple processors in parallel and the computation of sensitivities to chemical reaction rate constants. The resulting direct sensitivity modeling system was tested for statistical agreement with the traditional difference method for calculating sensitivities, considering a summer episode in a domain covering the continental United States. Sensitivities to domain-wide and sector specific emissions, initial/boundary conditions, and chemical reaction rates were compared and found to be in good correlation for both primary and secondary air pollutants. The scalability of CMAQ–DDM-3D to the number of processors used in parallel was also examined. Sensitivity calculations were found to scale in a similar way to the base model, where the benefit to model runtime of adding more processors diminished for simulations that used more than eight processors.

Simplified Discretization of Systems of Hyperbolic Conservation Laws Containing Advection Equations

The high speed flow of complex materials can often be modeled by the compressible Euler equations coupled to (possibly many) additional advection equations. Traditionally, good computational results have been obtained by writing these systems in fully conservative form and applying the general methodology of shock-capturing schemes for systems of hyperbolic conservation laws. In this paper, we show how to obtain the benefits of these schemes without the usual complexity of full characteristic decomposition or the restrictions imposed by fully conservative differencing. Instead, under certain conditions defined in Section 2, the additional advection equations can be discretized individually with a nonconservative scheme while the remaining system is discretized using a fully conservative approach, perhaps based on a characteristic field decomposition. A simple extension of the Lax–Wendroff Theorem is presented to show that under certain verifiable hypothesis, our nonconservative schemes converge to weak solutions of the fully conservative system. Then this new technique is applied to systems of equations from compressible multiphase flow, chemically reacting flow, and explosive materials modeling. In the last instance, the flexibility introduced by this approach is exploited to change a weakly hyperbolic system into an equivalent strictly hyperbolic system, and to remove certain nonphysical modeling assumptions.

River-plume sedimentation modeling for sequence stratigraphy: application to the Eel margin, northern California

Sediment deposition in the coastal zone from riverine input is of key importance in understanding the construction of sediment sequences through time. A turbulent-jet model is used to simulate sediment input to the continental margin from a riverine source. Turbulent-jet models track the momentum, which is the first-order force carrying most of the sediment flowing into the ocean. The Froude number at the river mouth remains supercritical during floods of the Eel River, justifying the use of a jet model for the initial dispersion of the sediment-laden plume. Sediment is deposited from the plume prior to the centerline Froude number decreasing to its critical value. Empirically derived first-order removal-rate constants are used for each grain size in an advection–diffusion equation to predict the areal extent of sediment deposition. One strength of using a jet model is its ability to simulate deposits on a daily time-step over intermediate time scales (thousands of years) using a limited number of input parameters. A number of simulations of the Eel River plume are shown for conditions of the present day and 18,000 yr BP, during the last glacial maximum. The model results for modern conditions show that a majority of the deposition occurs within 20 km north of the river mouth and is typically confined to within 10 km of the coastline, corresponding well with the shelf sands on the Eel margin. However, the initial mud deposit from a 1995 flood was farther offshore and northward. Additional advection of the mud fraction must occur after it leaves the surficial plume and prior to its initial deposition. The model runs of 18,000 yr BP (with 100 m lower sea level, wetter and colder climate, and the river mouth assumed to be at the canyon) show that the plume is able to transport sand, silt and clay out of the canyon and onto the continental slope.

Nuclear flow in consistent Boltzmann algorithm models

We investigate the stochastic Direct Simulation Monte Carlo method (DSMC) for numerically solving the collision-term in heavy-ion transport theories of the Boltzmann-Uehling-Uhlenbeck (BUU) type. The first major modification we consider is changes in the collision rates due to excluded volume and shadowing/screening effects (Enskog theory). The second effect studied by us is the inclusion of an additional advection term. These modifications ensure a non-vanishing second virial and change the equation of state for the scattering process from that of an ideal gas to that of a hard-sphere gas. We analyse the effect of these modifications on the calculated value of directed nuclear collective flow in heavy ion collisions, and find that the flow slightly increases.

Parameterizing Eddy-Induced Tracer Transports in Ocean Circulation Models

Abstract It is shown that the effects of mesoscale eddies on tracer transports can be parameterized in a large-scale model by additional advection and diffusion of tracers. Thus, tracers are advected by the effective transport velocity, which is the sum of the large-scale velocity and the eddy-induced transport velocity. The density and continuity equations are the familiar equations for adiabatic, Boussinesq, and incompressible flow with the effective transport velocity replacing the large-scale velocity. One of the main points of this paper is to show how simple the parameterization of Gent and McWilliams appears when interpreted in terms of the effective transport velocity. This was not done in their original 1990 paper. It is also shown that, with the Gent and McWilliams parameterization, potential vorticity in the planetary geostrophic model satisfies an equation close to that for tracers. The analogy of this parameterization with vertical mixing of momentum is then described. The effect of the Gent ...