Integral Momentum Balance Research Articles

Simulating the processes controlling ice-shelf rift paths using damage mechanics

Abstract Rifts are full-thickness fractures that propagate laterally across an ice shelf. They cause ice-shelf weakening and calving of tabular icebergs, and control the initial size of calved icebergs. Here, we present a joint inverse and forward computational modeling framework to capture rifting by combining the vertically integrated momentum balance and anisotropic continuum damage mechanics formulations. We incorporate rift–flank boundary processes to investigate how the rift path is influenced by the pressure on rift–flank walls from seawater, contact between flanks, and ice mélange that may also transmit stress between flanks. To illustrate the viability of the framework, we simulate the final 2 years of rift propagation associated with the calving of tabular iceberg A68 in 2017. We find that the rift path can change with varying ice mélange conditions and the extent of contact between rift flanks. Combinations of parameters associated with slower rift widening rates yield simulated rift paths that best match observations. Our modeling framework lays the foundation for robust simulation of rifting and tabular calving processes, which can enable future studies on ice-sheet–climate interactions, and the effects of ice-shelf buttressing on land ice flow.

A unified thermodynamic/Lamb-vector-based analysis of the aerodynamic force

The Lamb vector, cross-product of flow vorticity and velocity, is at the basis of different far-field methods developed in the last decades for the aerodynamic force analysis and decomposition, as an alternative to the nowadays well-assessed thermodynamic methods. We here propose a mixed approach, where exact Lamb-vector-based force formulas are used in combination with a thermodynamic-based calculation of the Lamb vector through Crocco's equation. In computational fluid dynamics, this way of calculating the Lamb vector, therefore, inherits from the numerical form of the flow momentum equation and discretely satisfies the local (and integral) momentum balance on which far-field methods rely. The resulting hybrid method, which does not require an explicit vorticity calculation, provides results in far better agreement with regard to near-field force data when compared to standard vorticity-based approaches, especially in the presence of shock waves, where inaccuracies of domain integrals involving the Lamb vector were systematically reported by different authors. In addition, it overcomes the limitations of previous thermodynamic methods, which only compute the drag force.

The Integral Momentum Balance Applied to Metering Orifices to Obtain the Pressure Drop and the Flow Rate With High Accuracy

Abstract The established method of calculating the pressure loss and the flow rate of flows through metering orifices requires empirical correction coefficients. In this paper, a completely momentum-based approach is presented to predict the flow rate and the pressure loss. It is shown that the proposed approach is as precise as the established method but without any empirical coefficient. The required physical coefficients solely base on physics. They pinpoint the sources of the inaccuracy that effects the employment of an empirical correction coefficient to correct the established method—and confirm the empirical correction coefficient on a physical foundation. Ultimately, the physical coefficients evade the effort to experimentally calibrate the theory with the experiments. The integral momentum balance is applied to a control volume comprising the orifice reaching up to the respective pressure metering cross section. Depending on the location of the control volume's open boundaries, the pressure loss or the flow rate can be obtained. To substitute the integral with averaged expressions, the introduced physical coefficients are deduced directly from numerical simulations and parametrized by simple analytical fit functions to complete the momentum-based approach. The numerical model and the proposed approach are both verified by the comparison with the results obtained by the standard ISO 5167:2003. The agreement with the classical calculation method as per ISO 5167:2003 is very high and within the specified limits, which shows that no empirical coefficient is necessary for high-accuracy flow metering using the momentum balance.

A non-equilibrium slip wall model for large-eddy simulation with an immersed boundary method

A non-equilibrium wall model for large-eddy simulation with the immersed boundary (IB) method is proposed to reduce the required number of grid points in simulating wall-bounded turbulence. The proposed wall model is presented as an appropriate slip velocity on the wall. The slip velocity is constructed by integrating the simplified turbulent boundary layer (TBL) equation along the wall-normal direction, which enhances the integral momentum balance near the wall on a coarse grid. The effect of pressure gradient on the near wall flow is taken into account by retaining the pressure gradient term in the simplified TBL equation. The proposed model is implemented in the form of a direct-forcing IB method with moving-least-square reconstruction near the wall. The benchmarks of plane channel turbulence and the flows over a backward-facing step are used for validation. The proposed model improves the wall stresses and velocity profiles in the region where the pressure gradient dominates the near wall flows.

On the use of kinetic energy balance for the volumetric identification of drag sources of a blunt body. Application to road vehicles

A volumetric identification of drag sources is proposed using the Reynolds equation for the integral mean momentum and mean flow kinetic energy balance. This methodology works independently of the turbulence model used in CFD and relies on a physical interpretation. The approach, developed in the context of automotive engineering, is of general use. A road vehicle is indeed a complex multi-scale geometry and all the flow regions are interacting. As far as the kinetic energy of the mean flow is concerned, the main volumetric loss is production of turbulent kinetic energy at all relevant sub-scales of the body. The quantitative link to the power of the mean drag force is derived from first principles. For practical applications, the volume integral used to compute the volumetric losses can be decomposed in user-defined regions and integrated into optimization loops. Indeed, if losses in a given flow region are minimized, consequences can be precisely and quantitatively evidenced for all other regions. Applications to a model two-dimensional test case and a real car geometry are discussed.

Determining the Solids Mass Flow Rate in a Gas–Solids Riser Using an Integral Momentum Balance

Determining the Solids Mass Flow Rate in a Gas–Solids Riser Using an Integral Momentum Balance

A Lagrangian relaxation towards equilibrium wall model for large eddy simulation

A large eddy simulation wall model is developed based on a formal interpretation of quasi-equilibrium that governs the momentum balance integrated in the wall-normal direction. The model substitutes the law-of-the-wall velocity profile for a smooth surface into the wall-normal integrated momentum balance, leading to a Lagrangian relaxation towards equilibrium (LaRTE) transport equation for the friction–velocity vector ${\boldsymbol u}_\tau (x,z,t)$ . This partial differential equation includes a relaxation time scale governing the rate at which the wall stress can respond to imposed fluctuations due to the inertia of the fluid layer from the wall to the wall-model height. A priori tests based on channel flow direct numerical simulation (DNS) data show that the identified relaxation time scale ensures self-consistency with assumed quasi-equilibrium conditions. The new approach enables us to formally distinguish quasi-equilibrium from additional, non-equilibrium contributions to the wall stress. A particular model for non-equilibrium contributions is derived, motivated by laminar Stokes layer dynamics in the viscous sublayer when applying fast-varying pressure gradients. The new wall modelling approach is first tested in standard equilibrium channel flow in order to document various properties of the approach. The model is then applied in large eddy simulation of channel flow with a suddenly applied spanwise pressure gradient (SSPG). The resulting mean wall-stress evolution is compared with DNS with good agreement. At the onset of the SSPG, the laminar Stokes layer develops rapidly while the LaRTE portion of the stress has a delayed response due to its inherent relaxation dynamics. Results also highlight open challenges such as modelling the response of near-wall turbulence occurring above the viscous sublayer and at time scales faster than quasi-equilibrium conditions.

The Application of the Integral Momentum Balance on the Pressure Drop of a Sudden Contraction

Abstract A new approach based on the momentum balance to calculate the pressure drop in turbulent flow through sharp-edged axisymmetric sudden contractions is presented. The momentum balance needs the velocity as well as the pressure distributions on the boundaries of the control volume. These distributions are obtained by a series of numerical simulations with different settings for the discharge, as well as the contraction ratio. The numerical model itself is validated by the comparison of the simulated and measured pressure drops in a laboratory experiment at different positions. To get easily applicable hydraulic formulations for the pressure drop depending on the discharge and the contraction ratio, the missing momentum and pressure coefficients are determined from the simulated velocity and pressure distributions. Only the pressure coefficient shows a dependency on the contraction ratio. After fitting the dependency by a simple analytical expression, a new formulation for the hydraulics of a sharp-edged sudden contraction based solely on momentum balance was obtained. The comparison with own experimental results as well as the classical parameterization of Idelchik show in both cases very good agreement.

Letter: Convergence rate of individual and global quantities in direct numerical simulations

Typical individual quantities in direct numerical simulations of statistically steady flows are converging at a rate of 1/T, where T is the averaging time of the simulation. However, global quantities that represent the integral momentum balance in the computational domain can exhibit a faster convergence rate of 1/T. This faster convergence rate is analysed and explained. Theoretical predictions are supported with a direct numerical simulation.

On the transient behaviour of a laminar rotor–stator cavity

The unsteady laminar flow between two large rotating disks when one of them is impulsively started is described using the von Kármán similarity analysis to reduce the solution of the Navier–Stokes equations to a set of ordinary differential equations. It is found that the transient phenomenon consists of three distinct phases. Firstly, the development of the Ekman boundary layer, where a quasi-steady Stewartson-type of flow is created. Secondly, angular momentum is initially diffused in the axial direction until the inward radial convection of angular momentum becomes dominating. Thirdly, a Batchelor flow is created once the Bödewadt boundary layer is developed and the entrainment of disk and stator boundary layers are balanced. The dependences of the characteristic dimensionless times on the Reynolds number based on the cavity gap for the second and third stages have been identified numerically and analytically. It is found that the diffusive part of the transient is bypassed if the flow, initially at rest, is perturbed with a small circumferential velocity. The flow and heat transfer transient during a ramp of finite duration has been computed numerically. The integral angular momentum and energy balances of the cavity have been performed in order to understand the energy and momentum budget of the cavity. It is concluded that the spin-up and the spin-down process of a rotor–stator cavity using a quasi-stationary approximation of the fluid, where the time derivatives are neglected, is questionable for realistic gas turbine applications. Finally, the self-similar solution is compared against two-dimensional axisymmetric, steady and unsteady, laminar simulations to assess the limitations and validity of the self-similar analysis. It has been identified that in a closed squared cavity the outer shroud modifies the physics of the transient, but the general conclusions drawn from the one-dimensional model are still valid.

Dual integral porosity shallow water model for urban flood modelling

With CPU times 2 to 3 orders of magnitude smaller than classical shallow water-based models, the shallow water equations with porosity are a promising tool for large-scale modelling of urban floods. In this paper, a new model formulation called the Dual Integral Porosity (DIP) model is presented and examined analytically and computationally with a series of benchmark tests. The DIP model is established from an integral mass and momentum balance whereby both porosity and flow variables are defined separately for control volumes and boundaries, and a closure scheme is introduced to link control volume- and boundary-based flow variables. Previously developed Integral Porosity (IP) models were limited to a single set of flow variables. A new transient momentum dissipation model is also introduced to account for the effects of sub-grid scale wave action on porosity model solutions, effects which are validated by fine-grid solutions of the classical shallow-water equations and shown to be important for achieving similarity in dam-break solutions. One-dimensional numerical test cases show that the proposed DIP model outperforms the IP model, with significantly improved wave propagation speeds, water depths and discharge calculations. A two-dimensional field scale test case shows that the DIP model performs better than the IP model in mapping the floods extent and is slightly better in reproducing the anisotropy of the flow field when momentum dissipation parameters are calibrated.

Macroscopic characteristics of heavy particle resuspension obtained from direct numerical simulations of pressure driven channel flow

Three-dimensional direct numerical simulations are presented of the flow of fluid containing heavy spherical particles in a channel. The simulations resolve the motion of individual particles and the details of the flow at the particle level. They are based on the finite element method and the Distributed Lagrange Multiplier/ Fictitious Domain Method. The problem is characterized by four parameters, namely a Reynolds and an Archimedes number, a density ratio and a dimensionless particle size. The last three are fixed and the effects of the Reynolds number are studied over a range covering the onset of particle resuspension and the transition of flow to significant unsteadiness. Encouraging estimates for the onset of resuspension and the particle flux as a function of the excess Shields number are found. The results are analyzed in terms of the vertical distribution of horizontally averaged macroscopic quantities, obtained by applying integral momentum balances. On the basis of such balances, particle stresses and interfacial forces between the fluid and particle phases are estimated and compared to existing models, developed mainly for Stokes flow conditions. The qualitative agreement is encouraging but points to the need for taking into account several effects, notwithstanding those of finite particle Reynolds numbers. Provided that such information becomes available the set of four equations derived from the suspension and particle phase momentum equations could describe the average equilibrium distribution, the flow of both phases and the pressure in the fluid.

Dynamics of wind‐driven upwelling off the northeastern coast of Hainan Island

AbstractBoth observational and reanalysis sea surface temperature data reveal that upwelling occurs frequently off the northeastern coast of Hainan Island (downstream of the change in topography off Tongluo Cape), which cannot be attributed to the along‐shelf wind alone. To identify dynamics of the upwelling, we conduct a numerical experiment using an idealized topography that is simplified from the actual topography off the eastern and northeastern coasts of Hainan Island. The result indicates that the upwelling downstream of the change in topography is associated with onshore cross‐isobath transport. Analysis of the vertically integrated momentum balance shows that the upwelling‐linked onshore transport is primarily intensified by the along‐isobath barotropic pressure gradient force (PGT), but is weakened by the along‐isobath baroclinic pressure gradient force (PGC). The along‐isobath PGT is linked to the advection of relative vorticity, the bottom stress curl and the gradient of momentum flux in vorticity equation. On the other hand, the PGC‐related process is diagnosed by potential vorticity (PV) balance. Similar to the negative PV term from wind stress, the negative PV terms of the joint effect of baroclinicity and relief and the baroclinic bottom pressure torque weaken the upwelling‐linked onshore transport downstream of the change in topography. The onshore transport is enhanced by the positive PV from bottom stress. In addition, the cross‐isobath forces play an important role in upwelling intensification in the shallow nearshore region.

Transient modeling of fiber spinning with filament pull-out

Flow instabilities of wet-spun fibers in the form of draw resonance can result in radius fluctuations which impose limitations on either fiber quality or production rate. Also, at high winding velocities, if the fluid strength is sufficiently high, a filament can pull out of the capillary. Filament pull-out complicates the stability analysis of fiber spinning in the sense that the upstream boundary conditions now depend on the position in the spinneret. In addition, the filament length varies in time. Intuitively, one would expect that including filament pull-out will stabilize the spinning process since the change in filament length adjusts the spinning force to result in a process similar to spinning with a constant take-up force, which is always stable. However, from a linear stability analysis on a thin filament fiber spin model including filament pull-out, we obtained the least stable solution in a straight capillary with minimal restriction on the movement of the detachment position, see van der Walt et al. (2012) [1]. In this paper we extend the stability analysis to a fully axisymmetric transient isothermal fiber spin model including filament detachment from the capillary wall using the eXtended Pom-Pom (XPP) constitutive model. Using finite elements, we develop a numerically efficient algorithm for the dynamics of the detachment point. A comparison between the stability results of the axisymmetric and the thin filament model shows that the axisymmetric model is more stable than the thin filament pull-out model but less stable than the thin filament fixed length model for the same De (Deborah) number. We are also not able to confirm from the axisymmetric model the assumption of the force being a function of the detachment position in the thin filament model. Revisiting the integral momentum balance at the detachment point derived by Bulters and Meijer (1990) [2] we find that their force balance is not valid. There is a transient region near the detachment position from the integrated normal stress to the spinning force.

Integral momentum balance on a growing bubble

The integral momentum balance on a growing boiling bubble is investigated. All forces acting on the bubble are detailed, and the methods and assumptions used to calculate their integral resultants are discussed. The momentum balance computation is then performed using experimental data of bubbles growing on an artificial nucleation site in a controlled environment. The relative magnitude of each force component is compared, showing negligible dynamic forces, upwards forces composed mainly of the buoyancy and contact pressure components, and downwards forces being exclusively due to surface tension and adhesion. The difficulty encountered in measuring the apparent contact angle due to mirage effects has been highlighted; a new method, fitting numerically simulated bubble profile to the contour measurements has been proposed and used to correct the effects of refraction on the bubble profile determination. As all forces acting on the bubble were measured, it was possible to estimate the residuals of the momentum balance. Their small value validated both the expressions used for the forces and the methodology to evaluate their value.

Deformation of leaky-dielectric fluid globules under strong electric fields: boundary layers and jets at large Reynolds numbers

AbstractIn Taylor’s analysis of electrohydrodynamic drop deformation by a uniform electric field (Proc. R. Soc. Lond. A, vol. 291, 1966, pp. 159–166) inertia is neglected at the outset, resulting in fluid velocities that scale as the square of the applied-field magnitude. For large (i.e. millimetric) drops, with increasing field strength the Reynolds number predicted by this scaling may actually become large, suggesting the need for a complementary large-Reynolds-number investigation. Balancing viscous stresses and electrical shear forces in this limit reveals a different velocity scaling, with the $4/ 3$-power of the applied-field magnitude. For simplicity, we focus upon the flow about a spherical gas bubble. It is essentially confined to two boundary layers propagating from the poles to the equator, where they collide to form a radial jet. The transition occurs over a small deflection region about the equator where the flow is effectively inviscid. The deviation of the bubble shape from the original sphericity is quantified by the capillary number given by the ratio of a characteristic Maxwell stress to Laplace’s pressure. At leading order the bubble deforms owing to: (i) the surface distribution of the Maxwell stress, associated with the familiar electric-field profile; (ii) the hydrodynamic boundary-layer pressure, engendered here by centrifugal forces; and (iii) the intense pressure distribution acting over the narrow equatorial deflection zone, appearing on the bubble scale as a concentrated load. Remarkably, the unique flow topology and associated scalings allow the obtaining of a closed-form expression for the deformation through the mere application of integral mass and momentum balances. On the bubble scale, the concentrated pressure load is manifested in the appearance of a non-smooth equatorial dimple.

Simulation of fluid dynamics in a microchannel Fischer-Tropsch reactor

Fluid dynamics in a microchannel of a new-generation Fischer-Tropsch reactor for artificial petroleum synthesis is reported. Liquid and gaseous product downflows are modeled in the annular flow approximation. The conjugate flow of gaseous and liquid products is investigated in terms of integral momentum balance equations with capillary effects and microchannel wall roughness taken into consideration. A simulation procedure appropriate for the method of deposition of catalyst particles onto the inner wall is suggested for modeling a random inner surface of the microchannel. The simulation procedure takes into account the variation of the synthesis product composition and the variation of the thermal properties of the liquid and gas phases along the microchannel length. A stable computational procedure has been devised for calculating fluid dynamic parameters of the two-phase flow. The effects of the synthesis gas flow rate and conversion, microchannel diameter, chain propagation constant, temperature, and pressure on the two-phase flow parameters and on the aerodynamic resistance of the gas have been investigated.

Effect of Air Pressure Difference on the Stability and Sensitivity of Simplified Curtain Drawing Flow

Stability and sensitivity of the curtain drawing flow have been investigated by employing the simplified one-dimensional viscocapillary model. A time-dependent model has been derived using the lubrication approximation and integral momentum balance approach of two-dimensional curvilinear Navier–Stokes equations. Process stability and sensitivity have been examined by considering linear stability and frequency response methods for a linearized curtain system, respectively. Among the process conditions affecting the curtain dynamics, the effect of air pressure difference across the curtain has been mainly focused upon. It is shown that the air pressure difference makes the system less stable and more sensitive to disturbances, leading to curtain deflection and a slightly longer trajectory.

Amplification of a Surface-Intensified Eddy Drift along a Steep Shelf in the Eastern Mediterranean Sea

Abstract The datasets of the Eddies and Gyre Path Tracking (EGYPT)/EGITTO program in the eastern Mediterranean Sea reveal a large mesoscale anticyclone traveling along the Libyan shelf. Surface drifter trajectories combined with a CTD transect accurately quantify the horizontal velocity and the vertical structure of this surface-intensified anticyclone. The observed westward drift speed is significantly higher than expected from the beta effect alone. To study the impact of a steep shelf topography on the propagation of compact surface-intensified vortices, the authors used a two-layer beta-plane model with steep continental slope and nearly zonal boundary. A perturbation theory derived by G. Sutyrin for a circular vortex in the upper layer with the lower layer at rest as a basic state is generalized for nonuniform slope in the presence of the image effect. An integral momentum balance is used to derive the drifting velocity of an upper-layer vortex with the main assumption that a stable and steady drifting solution of the two-layer system exists. The interface is described by a steady drifting circular dome at the leading order. This approach allows the problem to be reduced to the calculation of the deep-flow pattern, depending on the interface shape and topography. When the topographic slope beneath the eddy changes rapidly from a steep continental slope to a gentle continental rise, most of the deep-flow pattern is shifted offshore. The corresponding anticyclonic deep-flow feedback provides an additional along-slope propagation, which is proportional to the basic drift speed and the steepness parameter.

Application of a nested‐grid ocean circulation model to Lunenburg Bay of Nova Scotia: Verification against observations

A nested‐grid ocean circulation modeling system is used to study the response of Lunenburg Bay in Nova Scotia, Canada, to local wind‐forcing, tides, remotely generated waves, and buoyancy forcing in the summer and fall of 2003. Quantitative comparisons between observations and model results demonstrate that the modeling system reproduces reasonably well the observed sea level, temperature, salinity, and currents in the bay. Numerical results reveal that the spatial and temporal variability of temperature and salinity in the bay during the study period is mainly forced by the local wind stress and surface heat/freshwater fluxes, with some contribution from tidal circulation. In particular, the local heat balance on the monthly timescale is dominated by cooling due to vertical advection and warming due to horizontal advection and net surface heat flux, while high‐frequency variations (timescales of 1–30 days) are mainly associated with vertical advection, i.e., wind‐induced upwelling and downwelling. There is also a strong baroclinic throughflow over the deep water region outside Lunenburg Bay that is strongly influenced by wind‐forcing. The vertically integrated momentum balance analysis indicates a modified geostrophic balance on the monthly timescale and longer, and is dominated by the pressure term and wind minus bottom stress in the high‐frequency band.