Moving Boundary Formulation Research Articles

How canyons evolve by incision into bedrock: Rainbow Canyon, Death Valley National Park, United States

Incising rivers may be confined by low-slope, erodible hillslopes or steep, resistant sidewalls. In the latter case, the system forms a canyon. We present a morphodynamic model that includes the essential elements of a canyon incising into a plateau, including 1) abrasion-driven channel incision, 2) migration of a canyon-head knickpoint, 3) sediment feed from an alluvial channel upstream of the knickpoint, and 4) production of sediment by sidewall collapse. We calculate incision in terms of collision of clasts with the bed. We calculate knickpoint migration using a moving-boundary formulation that allows a slope discontinuity where the channel head meets an alluvial plateau feeder channel. Rather than modeling sidewall collapse events, we model long-term behavior using a constant sidewall slope as the channel incises. Our morphodynamic model specifically applies to canyon, rather than river-hillslope evolution. We implement it for Rainbow Canyon, CA. Salient results are as follows: 1) Sediment supply from collapsing canyon sidewalls can be substantially larger than that supplied from the feeder channel on the plateau. 2) For any given quasi-equilibrium canyon bedrock slope, two conjugate slopes are possible for the alluvial channel upstream, with the lower of the two corresponding to a substantially lower knickpoint migration rate and higher preservation potential. 3) Knickpoint migration occurs at a substantially faster time scale than regrading of the bedrock channel itself, underlying the significance of disequilibrium processes. Although implemented for constant climactic conditions, the model warrants extension to long-term climate variation.

Effects of shear rate and time on deposit composition in the cold flow regime under laminar flow conditions

The effects of shear rate and time on solid deposition and deposit composition, from wax–solvent mixtures in the cold flow regime under laminar flow conditions, were investigated experimentally using a cold finger apparatus. The deposition experiments were conducted with a 10 mass% multicomponent wax–solvent mixture at different agitation speeds, with a coolant temperature of WAT–12 °C, at a wax–solvent mixture temperature of WAT–6 °C, and for different deposition times. The experimental results indicated that the deposit mass increased with a decrease in the agitation speed (Re), or the shear rate, and an increase in the deposition time (t). The predictions from a transient model based on the moving boundary formulation agreed with these results. The wax content and carbon number distribution of deposit samples varied with Re and t. The deposit ‘aging’ behaviour was modeled successfully using a previously developed approach, based on the shear-induced deformation of a cubical cage, causing a partial squeezing out of the liquid phase from the deposit, in which the deformation angle (β) of the tilted cubical cage depends on Re and t. A correlation was developed for the effects of Re and t on deposit composition in the cold flow regime under laminar flow conditions. The results of this study further confirm that the deposition process is described very well by heat-transfer considerations and the ‘aging’ of deposit is modeled satisfactorily by considering the effect of shear stress during its formation and growth.

A state-space dynamic model for vapor compression refrigeration system based on moving-boundary formulation

A transient response model for vapor compression refrigeration system has been developed in the paper. The system model contains four sub-models representing condenser, evaporator, compressor and electronic expansion valve (EEV). The condenser and the evaporator are developed based on the moving-boundary formulation. Through linearization, these dynamic models are transformed into state-space representation which is expressed in the form of matrix. The compressor and EEV adopt steady models because their thermal inertia is much smaller compared with the heat exchangers (i.e., condenser and the evaporator). The system model has been validated by experiment in terms of step change of EEV opening degree and heat load as well as ramp increase of inlet temperature of coolant oil of condenser. The results show that the model simulations have a good agreement with the experimental data. The simulation errors compared with the experimental data are mostly less than 10%. Since the model proposed in this study is expressed in the form of state-space matrix, they are featured by strong portability and high computation efficiency. It allows us to investigate the thermal dynamic characteristics of a refrigeration system under any complicated conditions and develop excellent control schemes.

A comparison between finite volume and switched moving boundary approaches for dynamic vapor compression system modeling

Most work in dynamic heat exchanger modeling for control design can be classified as either a finite volume or a moving boundary formulation. These approaches represent fundamentally different discretization approaches and are often characterized as contrasting accuracy with simulation speed. This work challenges that characterization by validating finite volume and moving boundary heat exchanger models with experimental data from a vapor compression system in order to demonstrate that these approaches are capable of achieving similar levels of accuracy. However, there are differences. The moving boundary model is found to have faster simulation speed, while the finite volume model is more flexible for adaptation to heat exchangers of different physical configuration. The formulation of each modeling approach used in this work is described in detail and techniques to increase simulation speed and avoid numerical issues in implementation are discussed.

Modelling of Solidification Processes in Magma Chambers Cooled from above

Moving Boundary Formulation is used to analyze one dimensional model of magma chamber which gradually cools and solidifies after emplacement in the region. Cooling of a layer of pure melt (magma chamber) from above leads to a temperature profile that increases with depth and is thermally unstable. The vigor of convection thus produced, is determined by Rayleigh-Nusselt number based on the finite depth of liquid layer remaining at a particular time and some other parameters like initial heat flow and Stefan number etc. We solve the nonlinear coupled set of differential equations by Runge-Kutta-Fehlberg method and results for different parameters are compared for phase boundary movement.

A transient model for the heat exchange in a solar thermal once through cavity receiver

This paper presents a dynamic model of a once-through-to-superheat solar steam receiver for electricity generation. The receiver is a mono-tube cavity boiler mounted at the focal point of a 500m2 paraboloidal dish concentrator at the Australian National University. The dynamic model is derived from physical principles of mass and energy conservation, and uses a moving boundary formulation, coupled with a switching approach, to represent outlet flow, ranging from sub-cooled liquid to superheated steam. A method to compute outlet mass flow rate for all three receiver outlet flow conditions is included. This modelling approach results in a compact state-space representation of the receiver which is useful in the development of model-based control strategies for the operation of the receiver in a concentrator plant. The model is implemented in TRNSYS 16 and validated with experimental data from the Australian National University 500m2 dish system.

An accurate moving boundary formulation in cut-cell methods

A cut-cell method for Cartesian meshes to simulate viscous compressible flows with moving boundaries is presented. We focus on eliminating unphysical oscillations occurring in Cartesian grid method...

An accurate moving boundary formulation in cut-cell methods

A cut-cell method for Cartesian meshes to simulate viscous compressible flows with moving boundaries is presented. We focus on eliminating unphysical oscillations occurring in Cartesian grid methods extended to moving-boundary problems. In these methods, cells either lie completely in the fluid or solid region or are intersected by the boundary. For the latter cells, the time dependent volume fraction lying in the fluid region can be so small that explicit time-integration schemes become unstable and a special treatment of these cells is necessary. When the boundary moves, a fluid cell may become a cut cell or a solid cell may become a small cell at the next time level. This causes an abrupt change in the discretization operator and a suddenly modified truncation error of the numerical scheme. This temporally discontinuous alteration is shown to act like an unphysical source term, which deteriorates the numerical solution, i.e., it generates unphysical oscillations in the hydrodynamic forces exerted on the moving boundary. We develop an accurate moving boundary formulation based on the varying discretization operators yielding a cut-cell method which avoids these discontinuities. Results for canonical two- and three-dimensional test cases evidence the accuracy and robustness of the newly developed scheme.

On the moving boundary formulation for parabolic problems on unbounded domains

The aim of this paper is to propose an original numerical approach for parabolic problems whose governing equations are defined on unbounded domains. We are interested in studying the class of problems admitting invariance property to Lie group of scalings. Thanks to similarity analysis the parabolic problem can be transformed into an equivalent boundary value problem governed by an ordinary differential equation and defined on an infinite interval. A free boundary formulation and a convergence theorem for this kind of transformed problems are available in [R. Fazio, A novel approach to the numerical solution of boundary value problems on infinite intervals, SIAM J. Numer. Anal. 33 (1996), pp. 1473–1483]. Depending on its scaling invariance properties, the free boundary problem is then solved numerically using either a noniterative, or an iterative method. Finally, the solution of the parabolic problem is retrieved by applying the inverse map of similarity.

Bread baking as a moving boundary problem. Part 1: Mathematical modelling

A mathematical model for the bread baking process is developed in this work. Experimental data (temperature, water content, weight loss, crust thickness) obtained during baking is used to well understand the simultaneous heat and mass transfer occurring during the process. The evaporation–condensation mechanism is responsible for the rapid heating of the porous matrix and takes place either in a closed (dough) or open (crumb) structure. The existence of a moving evaporation front inside bread, which is a determining step of baking, is incorporated in a model applying a moving boundary formulation with equivalent thermophysical properties. The approach proposed here can be extended to other similar processes such as baking of other products (e.g. biscuit, cake), high-temperature drying, cooking and roasting.

A comparison of moving-boundary and finite-volume formulations for transients in centrifugal chillers

Dynamic models of vapor compression systems are useful tools in developing feedback controllers and are gaining increasing attention in recent years. The dominant dynamics are typically those of the evaporator and condenser, which are difficult to model. Accuracy and execution speed of system models therefore are highly dependent on the modeling approach of the heat exchangers. The two common approaches are the finite-volume (FV) and the moving-boundary (MB) methods. Both have been used successfully and reported in the literature, but there is little discussion presented for either choice. This paper presents the development and comparative study of shell-and-tube heat exchanger dynamic models using both the FV and the MB approaches. Detailed model formulations are provided and stability is demonstrated as components and within a complete centrifugal chiller system model. The system models are validated using data from a 300 kW R134a centrifugal chiller test stand. The FV formulation is found to be more robust through start-up and all load-change transients, but executes slower. The moving-boundary method can handle all load-change transients but start-up stability is more sensitive to compressor and expansion valve formulations. The moving-boundary formulation also executes about three times faster than the finite-volume while maintaining nearly identical accuracy. With the homogenous two-phase assumption, charge prediction is seen to be less accurate in the moving-boundary approach.

On the moving boundary formulation for parabolic problems on unbounded domains

AbstractA parabolic problem defined on unbounded domains and admitting invariance property to Lie group of scalings can be transformed into an equivalent boundary value problem governed by an ODE defined on an infinite interval. A free boundary formulation and a convergence theorem for this kind of transformed problems are available in [Fazio, SIAM J. Numer. Anal., 33 (1996), pp. 1473‐1484]. Depending on its scaling invariance properties, the free boundary problem is then solved numerically using either an iterative or a non‐iterative method. Finally, the solution of the parabolic problem is retrieved applying the inverse map of similarity. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modeling of Deposition from “Waxy” Mixtures in a Pipeline under Laminar Flow Conditions via Moving Boundary Formulation

A mathematical model, based on the moving boundary approach, is developed for the growth of the deposit from paraffinic mixtures due to heat transfer in a pipeline. The model extends a recent study on the deposition from “waxy” mixtures by including unsteady-state energy balance and heat transfer in both radial and axial directions for hydrodynamically established laminar flow. Numerical solutions were obtained for the growth of deposit with time, both radially and axially, from a binary eutectic mixture of n-C16H34 and n-C28H58. Two different scenarios for the initiation of the deposition process were investigated. Even though the transient results for the temperature profile and deposit thickness differed significantly, identical steady-state predictions were obtained for the two scenarios. For a constant pipe-wall temperature, the steady-state deposit thickness was predicted to increase with the pipe length. The predicted deposit thickness was lower for a higher inlet mixture temperature, pipe-wall temperature, and Reynolds number. The trends in model predictions were compared with the published results from deposition experiments performed on similar waxy mixtures.

A moving boundary formulation for recursive plastic heat release during friction stir welding

In this paper, a model for the growth of third region during friction stir welding is presented. The domain of interest is the heat and deformation affected zone under the tool shoulder. The model integrates a moving boundary framework for the growth of a stirring induced plastic flow region within the domain and formulates the heat generated by recursive plastic work as an enthalpy jump condition across the moving interface. From global energy balance considerations, the approach provides the means to estimate the time required to develop the stirring induced plastic flow region as a function of tool geometry (shoulder radius and pin radius).

Modeling of Deposit Formation from “Waxy” Mixtures via Moving Boundary Formulation: Radial Heat Transfer under Static and Laminar Flow Conditions

A mathematical formulation, based on heat transfer considerations, is presented for solids deposition from “waxy” or paraffinic mixtures. The proposed unsteady-state model uses the moving boundary problem approach for investigating the deposit-layer growth in a circular pipe from binary eutectic mixtures of n-C16H34 and n-C28H58. The model equations were solved numerically to explore the deposition behavior and the growth of deposit layer with time in the radial direction under both static and laminar flow conditions. The deposit-layer growth was predicted to be dependent on the rate of heat transfer at the liquid−deposit interface as well as in the liquid and deposit regions. For cooling under static conditions, complete pipe gelling was predicted to be faster for relatively lower values of mixture temperature, pipe-wall temperature, concentration of C28 in the mixture, and pipe diameter. For cooling under laminar flow conditions, higher values of mixture temperature, pipe-wall temperature and/or heat transfer coefficient yielded a thinner deposit layer with a faster approach to thermal steady state.

Hele-Shaw approximation for resin transfer molding

A Hele-Shaw flow is studied as the limit of a Stokes flow. The velocity fields associated to each fluid solve a fluid dynamic problem with time dependent domain. The moving boundary formulation is based on a pseudo-concentration equation. Some existence of solution of this set of equations is given using the classical Leary-Schauder Theorem and the use of space-time integrated least square method for the transport equation.

Study on heat driven pump. Part 2––Mathematical modeling

Literature shows that the study on heat driven pump mainly focused on the experimental aspect. Numerical modeling is scarce. In this paper a numerical model is proposed to simulate the thermal hydraulic characteristics of a heat driven pump based on a variable-node moving boundary formulation. In the two-phase region of the heat driven pump, homogeneous equilibrium flow is assumed. The governing equations are successfully integrated over the single-phases and two-phase time-dependent regions. The simulation results demonstrate that the working principle of the pump could be sub-divided into four distinct stages. Attempts have been made to compare the predictions with experimental data. The detailed thermal hydraulic characteristics, cycle behavior and factors affecting the pump performance are presented and discussed.

Fatigue corrosion crack extension across the interface of an elastic bi-material

In this work crack propagation in a bi-material composed of a thin elastic layer ideally bonded to a large elastic substrate is studied. A flaw is assumed existing on the surface of the layer. Cracks nucleate from the flaw and propagate through the bi-material. The cracks have realistic geometrical shapes, where the crack tip is an integral part of the crack surface. Thus the crack propagation is associated with a crack surface evolution. Material loss due to corrosion of the crack surface is the physical ground for the evolution. A controlling mechanism for the surface advancement is the rupture of a brittle corrosion-protective film, which is continually building-up along the corroding surface. The rate of surface evolution is a function of the degree of protective film damage, caused by the surface straining. This leads to a moving boundary formulation, for which a numerical solution is proposed. Fatigue loading is considered as a suitable way to maintain crack evolution at a constant peak load level. Under the assumed model, the cracks always pass the interface. The elastic mismatch is shown to influence the growth rate variation around the interface. Crack extensions are presented as functions of the elastic mismatch and as functions of the initial flaw size. It is shown how the results can be used in designing bi-material systems. A typical morphology evolution of a crack passing through an interface with a weak–stiff transition is presented. An example of fatigue corrosion fracture is offered, which shows that the crack morphology of the model resembles the one observed in reality. It is concluded that the realistic crack geometry is an effective concept and the moving boundary formulation could be a very successful tool for simulating realistic crack propagation.

Progradational sand-mud deltas in lakes and reservoirs. Part 2. Experiment and numerical simulation

Rivers deposit deltas wherever they reach standing water, i.e. a lake or reservoir. Here the case of a sand-bed river carrying mud as wash load is considered. In general the sand tends to deposit out to form a fluvial topset and an avalanching foreset, and the mud tends to deposit out as a bottomset. During floods, many sand-bed rivers carry sufficiently high concentrations of washload to render the river water heavier than that of the body of standing water. In such a case the mud-laden river fow plunges to form a bottom turbidity current. In a companion paper a 1 -D numerical model of a prograding delta was presented. In this model fluvial, avalanching and turbidity current deposition are all linked in terms of a moving boundary formulation. Here the model is compared against two experiments on delta progradation. The experiments reveal an intriguing interaction between the three elements of the model, with foreset progradation burying the bottomset and pushing its upstream point ever downstream, and with bottomset deposition raising the toe of the foreset, so increasing the rate of foreset progradation as sand is delivered from the topset. The numerical model of the companion paper captures this interaction with a minimum of adjustment of input parameters. The resulting description of delta morphodynamics is very similar to that observed in e.g. the delta in the Colorado River, USA where it reaches Lake Mead.

Thermal-hydraulic modeling and transient analysis of pressurized water reactors

This paper examines a new dynamic moving boundary thermal-hydraulic fuel pin model (FUELPIN) for the transient analysis of a pressurized water reactor (PWR). FUELPIN is developed to accommodate the reactor core thermal-hydraulic model of the fuel pin and adjacent coolant flow channel, with detailed thermal conduction in fuel elements. Transient analyses using a known thermal-hydraulic analysis code, COBRA, and FUELPIN linked with a PWR system analysis code show that the thermal margin gains more by a transient MDNBR approach than the traditional quasi-steady methodology for a PWR. The studies of the nuclear reactor system show that moving boundary formulation is highly suitable for the transient thermal-hydraulic analysis of PWRs.