Outflow Boundary Conditions Research Articles

Compressible Navier–Stokes system with hard sphere pressure law and general inflow–outflow boundary conditions

We prove the existence of a weak solution to the compressible Navier–Stokes system with hard sphere possibly non-monotone pressure law involving, in particular, the Carnahan–Starling model [2] largely employed in various physical and industrial applications. We take into account large velocities prescribed at the boundary of a bounded piecewise C2 domain and large densities prescribed at the inflow boundary without any restriction neither on the shape of the inflow/outflow boundaries nor on the shape of the domain.

A versatile algorithm for the treatment of open boundary conditions in Smoothed particle hydrodynamics GPU models

An open boundary algorithm for weakly compressible Smoothed particle hydrodynamics (WCSPH) numerical models is presented. Open boundary conditions are implemented by means of buffer regions whereby physical quantities are either imposed or extrapolated from the fluid region using a first-order accurate SPH interpolation. A unique formulation has been developed which can be used for inflow, outflow, and mixed open boundary conditions. The extrapolation process from the fluid domain encompasses quantities such as velocity, density, pressure and also free-surface elevation. The algorithm has been parallelized for both CPU and general-purpose on graphics processing units (GPGPU) and it has been tested against the 2-D reference solutions of flow past a cylinder and open channel flow. Finally, its capability to simulate 2-D and 3-D complex flows such as water waves and flow past a surface-piercing extraterrestrial submarine is demonstrated.

A novel, simple 3D/2D outflow boundary model for blood flow simulations in compliant arteries

Outflow boundary condition is one of the most challenging issues in the simulation of blood flow in arteries. It has a great impact on the solution within computational domain. Our aim in the present study is to develop a new outflow boundary model which can be easily implemented in commercially available software packages and capable of capturing the three main features of downstream: resistance, compliance and wave reflection. The new boundary model is an extension of a 2D/3D elastic tube to the end of the main computational domain which is partially filled with porous material. Wave reflection characteristics of our model are investigated by applying it to the end of a straight vessel and an axisymmetric model of an aorta. The sensibility of the solution to model parameters is also studied. Pressure, velocity, and wave intensity curves resulted from numerical simulations, are presented. All main characteristics of pressure and flow waves were observed in results of simulations with our boundary model. Computed pressures were in agreement with those previously published in the literature. It is shown that this model provides accurate results while it avoids complexities of other models such as Windkessel and structured tree.

Investigation of the backflows and outlet boundary conditions for computations of the patient-specific aortic valve flows.

Simulation divergence due to the backflow through the outlet boundary is a common, but not fully addressed challenge in patient-specific simulations of the aortic valve flows. The purpose of this study is to develop the outlet boundary conditions aiming to improve convergence of the patient-specific aortic valve computations and to control the backflow in the case of partial reversal of the flow through the outlet. Haemodynamic analysis of the aortic valve flows governed by the Navier-Stokes equations is performed by using the finite volume method. The pressure distribution, obtained from the convergent computations driven by the outflow boundary condition, is approximated by the parabolic surface of revolution and prescribed on the outlet as the novel pressure boundary condition. Various types of outlet boundary conditions are investigated to evaluate their influence on the resulting backflows. The outflow boundary condition produces a smaller backflow than other investigated boundary conditions, but it causes the solution divergence. The proposed outlet boundary condition allows for continuing the solution and preserving the expected backflow. The proposed outlet boundary condition helps to achieve a convergent solution and to conserve the observed backflow by varying the convexity of the specified pressure surface.

Numerical prediction of a flashing flow of saturated water at high pressure

Abstract Transient fluid velocity and pressure fields in a pressurized water reactor (PWR) steam generator (SG) secondary side during the blowdown period of a feedwater line break (FWLB) accident were numerically simulated employing the saturated water flashing model. This model is based on the assumption that compressed water in the SG is saturated at the beginning and decompresses into the two-phase region where saturated vapor forms, creating a mixture of steam bubbles in water by bulk boiling. The numerical calculations were performed for two cases of which the outflow boundary conditions are different from each other; one is specified as the direct blowdown discharge to the atmosphere and the other is specified as the blowdown discharge to an extended calculation domain with atmospheric pressure on its boundary. The present simulation results obtained using the two different outflow boundary conditions were discussed through a comparison with the predictions using a simple non-flashing model neglecting the effects of phase change. In addition, the applicability of each of the non-flashing water discharge and saturated water flashing models for the confirmatory assessments of new SG designs was examined.

The effect of inlet and outlet boundary conditions in image-based CFD modeling of aortic flow

BackgroundComputational modeling of cardiovascular flow is a growing and useful field, but such simulations usually require the researcher to guess the flow’s inlet and outlet conditions since they are difficult and expensive to measure. It is critical to determine the amount of uncertainty introduced by these assumptions in order to evaluate the degree to which cardiovascular flow simulations are accurate. Our work begins to address this question by examining the sensitivity of flow to several different assumed velocity inlet and outlet conditions in a patient-specific aorta model.MethodsWe examined the differences between plug flow, parabolic flow, linear shear flows, skewed cubic flow profiles, and Womersley flow at the inlet. Only the shape of the inlet velocity profile was varied—all other parameters were identical among these simulations. Secondary flow in the form of a counter-rotating pair of vortices was also added to parabolic axial flow to study its effect on the solution. In addition, we examined the differences between two-element Windkessel, three element Windkessel and the outflow boundary conditions. In these simulations, only the outlet boundary condition was varied.ResultsThe results show axial and in-plane velocities are considerably different close to the inlet for the cases with different inlet velocity profile shapes. However, the solutions are qualitatively similar beyond 1.75D, where D is the inlet diameter. This trend is also observed in other quantities such as pressure and wall shear stress. Normalized root-mean-square deviation, a measure of axial velocity magnitude differences between the different cases, generally decreases along the streamwise coordinate. The linear shear inlet velocity boundary condition and plug velocity boundary condition solution exhibit the highest time-averaged wall shear stress, approximately 8% higher than the parabolic inlet velocity boundary condition. Upstream of 1D from the inlet, adding secondary flow has a significant impact on temporal wall shear stress distributions. This is especially observable during diastole, when integrated wall shear stress magnitude varies about 26% between simulations with and without secondary flow. The results from the outlet boundary condition study show the Windkessel models differ from the outflow boundary condition by as much as 18% in terms of time-averaged wall shear stress. Furthermore, normalized root-mean-square deviation of axial velocity magnitude, a measure of deviation between Windkessel and the outflow boundary condition, increases along the streamwise coordinate indicating larger variations near outlets.ConclusionIt was found that the selection of inlet velocity conditions significantly affects only the flow region close to the inlet of the aorta. Beyond two diameters distal to the inlet, differences in flow solution are small. Although additional studies must be performed to verify this result, the data suggest that it is important to use patient-specific inlet conditions primarily if the researcher is concerned with the details of the flow very close to the inlet. Similarly, the selection of outlet conditions significantly affects the flow in the vicinity of the outlets. Upstream of five diameters proximal to the outlet, deviations between the outlet boundary conditions examined are insignificant. Although the inlet and outlet conditions only affect the flow significantly in their respective neighborhoods, our study indicates that outlet conditions influence a larger percentage of the solution domain.

Inflow and outflow boundary conditions for 2D suspension simulations with the immersed boundary lattice Boltzmann method

In- and outflow boundary conditions for 2D immersed boundary lattice Boltzmann suspension simulations, applied to cell based blood flow models, are presented. The inlet is constructed with an one-way coupling to a periodic domain containing a correct distribution of suspended particles. This provides an inflow of particles that has a correct distribution and is decoupled from any phenomena in the flow domain. An outflow boundary for the particles that does not influence the distribution of particles in the flow domain is also constructed. With this a method to run long ( > 1 s) cell based blood flow simulations within any type of domain is provided. These boundary conditions are then used for a simulation of blood flow in a curved vessel with an aneurysm.

Experimental insight for flood flow repartition in urban areas

Modeling floods in urban areas remains a challenge. To understand flow patterns in urban geometries better and constrain models, an experimental rig representing a 1/200 scale urban geometry with various street widths and angles is presented. Measurements of hydraulic variables for flow conditions ranging from moderate to extreme flooding were performed. Over this range, accurate inflow and outflow boundary condition measurements allow the geometry effect on inlet–outlet discharge conservation to be studied for each street. Froude numbers are found to be independent of the total flowrate. Interestingly, the flow distribution among all streets remains comparable over the range of boundary conditions. Moreover, three behaviors have been identified depending on street response as a function of the evolution of the upstream discharge distribution. Future measurements with high spatiotemporal sampling would allow possible coupling of flow features and energy dissipation to be studied at various scales and other flow configurations and district geometries to be characterized.

Impact of uncertainties in outflow boundary conditions on the predictions of hemodynamic simulations of ascending thoracic aortic aneurysms

The impact of outflow boundary conditions on the results of numerical simulations of the flow inside ascending thoracic aortic aneurysms is investigated. The adopted outflow conditions are based on the lumped three-element Windkessel model, in which it is assumed that the pressure at the outflow is related to the flow rate and to a terminal constant pressure, through an electric circuit analogue that has a proximal resistance in series with a parallel arrangement of a capacitance and of a distal resistance. The values of the Windkessel model parameters must be a-priori specified. The impact on the numerical simulation results of uncertainties in the values of the Windkessel model parameters is quantified here through a stochastic approach. The propagation of the considered uncertainties is evaluated, in particular, for the instantaneous and time-averaged wall shear stress. The generalized Polynomial Chaos is used as a surrogate model to obtain continuous response surfaces of the quantities of interest in the parameter space starting from a few deterministic simulations. A patient-specific geometry, obtained through in-vivo imaging, is considered. The analysis is carried out for both rigid and compliant vessel walls. Our results show that, although the uncertainties in the selected outflow parameters may give significant variability of the instantaneous shear stresses in regions characterized by flow recirculation or large streamline curvature, the impact on cycle-averaged shear stresses is moderate. Taking into account the wall compliance seems to reduce the impact of the uncertainties in the outflow parameters compared to the rigid case.

Minimized Capillary End Effect During CO2 Displacement in 2‐D Micromodel by Manipulating Capillary Pressure at the Outlet Boundary in Lattice Boltzmann Method

AbstractWe propose a new outflow boundary condition to minimize the capillary end effect for a pore‐scale displacement simulation. The Rothman‐Keller lattice Boltzmann method with multi‐relaxation time is implemented to manipulate a nonflat wall and inflow‐outflow boundaries with physically acceptable fluid properties in 2‐D microfluidic chip domain. Introducing a mean capillary pressure acting at ‐water interface to the nonwetting fluid at the outlet effectively prevents injection pressure from suddenly dropping upon breakthrough such that the continuous invasion and the increase of saturation are allowed. This phenomenon becomes most pronounced at capillary number of logCa = −5.5, while capillary fingering and massive displacement of prevail at low and high capillary numbers, respectively. Simulations with different domain length in homogeneous and heterogeneous domains reveal that capillary pressure and saturation near the inlet are reproducible compared with those with a proposed boundary condition. The residual saturation uniquely follows the increasing tendency with increasing capillary number, corroborated by experimental evidences. The determination of the mean capillary pressure and its sensitivity are also discussed. The proposed boundary condition is commonly applicable to other pore‐scale simulations to accurately capture the spatial distribution of nonwetting fluid and corresponding displacement ratio.

An Outflow Boundary Condition Model for Noninvasive Prediction of Fractional Flow Reserve in Diseased Coronary Arteries.

This paper reports on a new boundary condition formulation to model the total coronary myocardial flow and resistance characteristics of the myocardial vascular bed for any specific patient when considered for noninvasive diagnosis of ischemia. The developed boundary condition model gives an implicit representation of the downstream truncated coronary bed. Further, it is based on incorporating patient-specific physiological parameters that can be noninvasively extracted to account for blood flow demand to the myocardium at rest and hyperemic conditions. The model is coupled to a steady three-dimensional (3D) collocated pressure-based finite volume flow solver and used to characterize the "functional significance" of a patient diseased coronary artery segment without the need for predicting the hemodynamics of the entire arterial system. Predictions generated with this boundary condition provide a deep understanding of the inherent challenges behind noninvasive image-based diagnostic techniques when applied to human diseased coronary arteries. The overall numerical method and formulated boundary condition model are validated via two computational-based procedures and benchmarked with available measured data. The newly developed boundary condition is used via a designed computational methodology to (a) confirm the need for incorporating patient-specific physiological parameters when modeling the downstream coronary resistance, (b) explain the discrepancies presented in the literature between measured and computed fractional flow reserve (FFRCT), and

Computational simulation of postoperative pulmonary flow distribution in Alagille patients with peripheral pulmonary artery stenosis.

Up to 90% of individuals with Alagille syndrome have congenital heart diseases. Peripheral pulmonary artery stenosis (PPS), resulting in right ventricular hypertension and pulmonary flow disparity, is one of the most common abnormalities, yet the hemodynamic effects are ill-defined, and optimal patient management and treatment strategies are not well established. The purpose of this pilot study is to use recently refined computational simulation in the setting of multiple surgical strategies, to examine the influence of pulmonary artery reconstruction on hemodynamics in this population. Based on computed tomography angiography and cardiac catheterization data, preoperative pulmonary artery models were constructed for 4 patients with Alagille syndrome with PPS (all male, age range: 0.6-2.9 years), and flow simulations with deformable walls were performed. Surgeon directed virtual surgery, mimicking the surgical procedure, was then performed to derive postoperative models. Postoperative simulation-derived hemodynamics and blood flow distribution were then compared with the clinical results. Simulations confirmed substantial resistance, resulting from preoperative severe ostial stenoses, and the use of newly developed adaptive outflow boundary conditions led to excellent agreement with in vivo measurements. Relief of PPS decreased pulmonary artery pressures and improved pulmonary flow distribution both in vivo and in silico with good correlation. Using adaptive outflow boundary conditions, computational simulations can estimate postoperative overall pulmonary flow distribution in patients with Alagille syndrome after pulmonary artery reconstruction. Obstruction relief along with pulmonary artery vasodilation determines postoperative pulmonary flow distribution and newer methods can incorporate these physiologic changes. Evolving blood flow simulations may be useful in surgical or transcatheter planning and in understanding the complex interplay among various obstructions in patients with peripheral pulmonary stenosis.

DNS of MILD combustion with mixture fraction variations

Direct numerical simulations of Moderate or Intense Low-oxygen Dilution combustion inside a cubical domain are performed. The computational domain is specified with inflow and outflow boundary conditions in one direction and periodic conditions in the other two directions. The inflowing mixture is constructed carefully in a preprocessing step and has spatially varying mixture fraction and reaction progress variable fields. Thus, this mixture includes a range of thermo-chemical states for a given mixture fraction value. The combustion kinetics is modelled using a 58-step skeletal mechanism including a chemiluminescent species, OH*, for methane–air combustion. The study of reaction zone structures in the physical and mixture fraction spaces shows the presence of ignition fronts, lean and rich premixed flames and non-premixed combustion. These three modes of combustion are observed without the typical triple-flame structure and this results from the spatio-temporally varying mixture fraction field undergoing turbulent mixing and reaction. The flame index and its pdf are analysed to estimate the fractional contributions from these combustion modes to the total heat release rate. The lean premixed mode is observed to be quite dominant and contribution of non-premixed mode increased from about 11% to 20% when the mean oxygen mole fraction in the inflowing mixture is reduced from about 2.7% to 1.6%. Also, the non-premixed contribution increases if one decreases the integral length scale of the mixture fraction field. All of these results and observations are explained on physical basis.

Nonreflecting outlet boundary conditions for incompressible flows using SPH

In this paper we implement a simple strategy, based on Jin and Braza’s method, to deal with nonreflecting outlet boundary conditions for incompressible Navier-Stokes flows using the method of smoothed particle hydrodynamics (SPH). The outflow boundary conditions are implemented using an outflow zone downstream of the outlet, where particles are moved using an outgoing wave equation for the velocity field so that feedback noise from the outlet boundary is greatly reduced. For unidirectional flow across the outlet, this condition reduces to Orlanski’s wave equation. The performance of the method is demonstrated through several two-dimensional test problems, including unsteady, plane Poiseuille flow, flow between two inclined plates, the Kelvin–Helmholtz instability in a channel, and flow in a constricted conduit, and in three-dimensions for turbulent flow in a 90° section of a curved square pipe. The results show that spurious waves incident from the outlet are effectively absorbed and that steady-state laminar flows can be maintained for much longer times compared to periodic boundary conditions. In addition, time-dependent anisotropies in the flow, like fluid recirculations, are convected across the outlet in a very stable and accurate manner.

On the performance of the DG method with a directional do-nothing boundary condition

We present two numerical experiments concerning application of a recent outflow boundary condition proposal, by Dong et al. (J Comput Phys 261:83–105, 1), to discontinuous Galerkin method (DG) solution of incompressible laminar flows. This new boundary condition (BC) is tailored out for outflow boundaries and its rationale is based on energy influx control at this boundary surface, as described in Braack and Mucha (J Comput Math 32(5):507–521, 2). The authors applied it to various incompressible test-flow examples in a spectral and classic finite-element contexts, and a major result achieved by them was this new outflow boundary’s approach allows us to significantly reduce computational domain size without generating significant errors. Accordingly, due to its already known capabilities and established mathematical basis, it is a natural issue to ask about the DG behaviour over this all-useful achievement. In this work, the DG method is tested in two different flow instances: (1) Kovasznay flow, where convergence rates are measured and (2) laminar incompressible flow around cylinder inside rectangular channel. In this case, drag and lift coefficients are computed, further of the dimensionless Strouhal number. Conclusions are traced to show the readiness of DG to embody this new boundary condition technique.

Noninvasive assessment of carotid artery stenoses by the principle of multiscale modelling of non-Newtonian blood flow in patient-specific models

The concept of geometrical multiscale modelling of non-Newtonian blood flow in patient-specific models is presented with the aim to provide a methodology for the assessment of hemodynamic significance of carotid artery stenoses. The content of the paper is divided into two consequent parts. In the first one, the principle of the fractional flow reserve (FFR) as an indicator of ischemia-inducing arterial stenoses is tested on three large arterial models containing the aortic arch and both left and right carotid arteries. Using the three-element Windkessel model as an outflow boundary condition, the blood flow simulations are carried out on the basis of data taken from the literature due to unavailable information on patient-specific flow and pressure waveforms. In the second part of the paper, the incorporation of real in-vivo measurements into the multiscale simulations is addressed by presenting a sequential algorithm for the estimation of Windkessel parameters. The ability of the described estimation method, which employs a non-linear state estimator (unscented Kalman filter) on zero-dimensional flow models, is demonstrated on two different patient-specific carotid bifurcation models.

Extendable outflow boundary conditions for dissipative particle dynamics simulation

This paper presents a systematic study on a basic kind of outflow boundary condition for Dissipative Particle Dynamics (DPD) simulation, in which the flow near the outlet can be assumed as fully developed. The boundary conditions are expressed in Neumann-type constraints due to the zero gradients of hydrodynamic quantities on the boundary. The flow rate is included as an additional constraint to ensure the stability of a simulation. The boundary conditions are imposed through a reservoir, which is a replica of the simulation domain in the vicinity of the outlet. All the particle configurations are conserved in the reservoir. A velocity adjusting scheme is developed for the reservoir in order to control the deviation of the outflow rate and to remove the pressure discontinuity across the boundary. The effectiveness of the boundary method is examined through the simulations of a uniform flow and a laminar flow in a bifurcated tunnel. The outflow rates, as well as the velocity profile and pressure distribution, all agree well with the Navier–Stokes solution by comparison. The assumption of the fully developed outflow is validated by measuring the gradients of the flow rates. It is expected that this boundary method can be used in the simulation of flows with heterogeneous particle distribution, without having to evaluate the distribution function in advance.

Numerical study of Taylor bubbles rising in a stagnant liquid using a level-set/moving-mesh method

An Arbitrary Lagrangian-Eulerian formulation has been posed to solve the challenging problem of the three-dimensional Taylor bubble, within a Conservative Level Set (CLS) framework. By employing a domain optimization method (i.e. the moving mesh method), smaller domains can be used to simulate rising bubbles, thus saving computational resources. As the method requires the use of open boundaries, a careful treatment of both inflow and outflow boundary conditions has been carried out. The coupled CLS - moving mesh method has been verified by means of extensive numerical tests. The challenging problem of the full three-dimensional Taylor bubble has then been thoroughly addressed, providing a detailed description of its features. The study also includes a sensitivity analyses with respect to the initial shape of the bubble, the initial volume of the bubble, the flow regime and the inclination of the channel.

Robust boundary treatment for open-channel flows in divergence-free incompressible SPH

A robust Incompressible Smoothed Particle Hydrodynamics (ISPH) framework is developed to simulate specified inflow and outflow boundary conditions for open-channel flow. Being purely divergence-free, the framework offers smoothed and structured pressure distribution. An implicit treatment of Pressure Poison Equation and Dirichlet boundary condition is applied on free-surface to minimize error in velocity-divergence. Beyond inflow and outflow threshold, multiple layers of dummy particles are created according to specified boundary condition. Inflow boundary acts as a soluble wave-maker. Fluid particles beyond outflow threshold are removed and replaced with dummy particles with specified boundary velocity. The framework is validated against different cases of open channel flow with different boundary conditions. The model can efficiently capture flow evolution and vortex generation for random geometry and variable boundary conditions.

Outflow Boundary Condition of Coronary Artery Flow Analysis Based on Physiologic Findings

Outflow Boundary Condition of Coronary Artery Flow Analysis Based on Physiologic Findings