Outflow Boundary Research Articles

The 3D Euler equations with inflow, outflow and vorticity boundary conditions

The 3D Euler equations with inflow, outflow and vorticity boundary conditions

Curved-pipe flow with boundary conditions involving Bernoulli pressure

In this article, we study the steady-state flow of the incompressible viscous fluid through a thin distorted pipe with an arbitrary central curve. We prescribe the inflow and outflow boundary conditions involving the Bernoulli pressure with a given pressure drop. Using the multiscale expansion technique with respect to the pipe's thickness, we construct the higher-order asymptotic approximation of the flow given by the explicit formulae for the velocity and pressure. We also perform a detailed error analysis justifying the usage of the proposed solution and indicating its order of accuracy. For more information see https://ejde.math.txstate.edu/Volumes/2024/63/abstr.html

A numerical approach to overcome the very-low Reynolds number limitation of the artificial compressibility for incompressible flows

We propose a numerical approach to solve a long-standing challenge which is the applicability of the artificial compressibility (AC) formulation for solving the incompressible Navier—Stokes equations at very-low Reynolds numbers. A wide range of engineering applications involves very-low Reynolds number flows in Micro-ElectroMechanical Systems (MEMS) and in the fields of chemical-, agricultural- and biomedical engineering. It is known that the already existing numerical methods using the AC approach fail to provide physically correct results at very-low Reynolds numbers (Re ≤ 1). To overcome the limitation of the AC method for these engineering applications, we propose a higher-order Neumann-type pressure outflow boundary condition treatment along with their up to fourth-order numerical approximations. We found that the numerical treatment of the pressure at the outlet boundary plays the main role in overcoming the limitation of the AC method at very-low Reynolds numbers (Re << 1). Therefore, we provide numerical evidence on the accuracy of the AC method beyond its previously reported limitations, e.g., the low Reynolds number Oseen flow (Re << 1) is first presented in this work. A third-order explicit total-variation diminishing (TVD) Runge–Kutta scheme has been employed with standard finite difference spatial discretisation schemes for improving the accuracy of the numerical solution. For modelling strongly viscous flows, the Reynolds number ranges from 10−1 to 10−4. Overall, we found that the accuracy limitation of the AC method below Re < 1 can be overcome with an accurate numerical treatment of the outlet pressure boundary condition instead of using high-order schemes in the governing equations. For the investigated Reynolds number range (10−1 ≤ Re ≤ 10−4), the obtained results show that the relative errors were smaller than 1% for the numerical simulations performed on the configurations of both the two- and three-dimensional, straight microfluidic channels. The imposition of high-order derivative Neumann-type pressure outflow boundary conditions reduced the maximum relative errors of the numerical solutions from 85% and 95% to below than 1% at the outlet section of the two- and three-dimensional, straight microfluidic channel flows, respectively. Taking the advantage of the numerical approach proposed here, two- and three-dimensional benchmark problems employed in the current investigation in comparison with analytical solutions available in the literature, clearly demonstrate that the artificial compressibility can be used beyond its previously known constraints for very-low Reynolds number incompressible flows.

Entropy-stable in- and outflow boundary conditions for the compressible Euler equations

Entropy-stable in- and outflow boundary conditions for the compressible Euler equations

Influence of Low-Level Shear Orientation and Magnitude on the Evolution and Rotation of Idealized Squall Lines. Part I: Storm Morphology and Bulk Updraft/Mesovortex Attributes

Abstract Observational and modeling efforts have explored the formation and maintenance of mesovortices, which contribute to severe hazards in quasi-linear convective systems (QLCSs). There exists an important interplay between environmental shear and cold-pool-induced circulations which, when balanced, allow for upright QLCS updrafts with maximized lift along storm outflow boundaries. Numerical simulations have primarily tested the sensitivity of squall lines to zonally varying low-level (LL) shear profiles (i.e., purely line-normal, assuming a north–south-oriented system), but observed near-storm environments of mesovortex-producing QLCSs exhibit substantial LL hodograph curvature (i.e., line-parallel shear). Therefore, previous QLCS simulations may fail to capture the full impacts of LL shear variability on mesovortex characteristics. To this end, this study employs an ensemble of idealized QLCS simulations with systematic variations in the orientation and magnitude of the ambient LL shear vector, all while holding 0–3-km line-normal shear constant. This allows for a nuanced examination of how line-parallel shear modulates system structure, as well as mesovortex strength, size, and longevity. Results indicate that hodographs with LL curvature support squall lines with prominent bowing segments and wider, more intense rotating updrafts. Shear orientation also impacts mesovortex characteristics, with curved hodographs favoring cyclonic vortices that are stronger, wider, deeper, and longer-lived than those produced with straight-line wind profiles. These results provide a more complete physical understanding of how LL shear variability influences the generation of rotation in squall lines. Significance Statement Research related to linear storms has largely focused on vertical changes in winds (i.e., shear) oriented perpendicular to squall lines given its ability to balance storm cold pools and keep updrafts upright, thus promoting long-lived storms that presumably can go on to produce rotation. However, squall lines that produce a great deal of rotation often have a component of low-level shear oriented parallel to storms. This study gauges the sensitivity of simulated squall lines to changes in the direction and strength of shear close to the surface. We find that shear oriented parallel to linear storms creates stronger and larger updrafts that in turn support the development of intense and persistent rotation with characteristics supportive of tornadoes. These insights have impacts on both our physical understanding and prediction of the rotation and associated hazards of linear storms.

Adaptive least-squares methods for convection-dominated diffusion-reaction problems

This paper studies adaptive least-squares finite element methods for convection-dominated diffusion-reaction problems. The least-squares methods are based on the first-order system of the primal and dual variables with various ways of imposing outflow boundary conditions. The coercivity of the homogeneous least-squares functionals are established, and the a priori error estimates of the least-squares methods are obtained in a norm that incorporates the streamline derivative. All methods have the same convergence rate provided that meshes in the layer regions are fine enough. To increase computational accuracy and reduce computational cost, adaptive least-squares methods are implemented and numerical results are presented for some test problems.

Elastically modulated wavy vortex flow

We investigate the transition pathway of low elasticity fluids (El=0.003−0.008) in a Taylor-Couette configuration using low-molecular-weight polyacrylamide (PAAM) and visualisation experiments in the Reynolds range from 0 to 300. We report here for the first time an elastically modified wavy vortex flow state with altered spectral and structural characteristics, that precedes the onset of the traditional (inelastic) Newtonian wavy instability. This new wavy regime is characterised by oscillations of both the inflow and outflow boundaries, associated with a weakening of the outflow regions due to low hoop stresses. The modification of the boundaries persists at higher Reynolds numbers, where the spectral characteristics are unaltered compared to the inelastic, Newtonian case. In addition, a hysteretic behaviour is observed for increasing elasticity, as instabilities are shifted towards lower critical Reynolds numbers, confirming the importance of even vanishing elasticity on the stability of Taylor-Couette flows. At higher fluid elasticity (El=0.06), the amplitude of inflows/outflows oscillations increases, and momentum is transferred axially between adjacent vortices, which may contribute to the emergence of Rotating Standing Waves.

On the Outflow Boundary Opening Size for Internal Erosion Testing of Widely Graded Sand and Gravel

Abstract Permeameter testing of a widely graded soil to evaluate its potential for internal instability requires the opening size of the outflow boundary to allow unstable finer fraction particles to pass while retaining the stable coarser fraction. Results from a parametric study, on three widely graded soils and five outflow boundaries, yielded paired-data for opening size Om and d85,e of eroded soil mass passing through the outflow boundary. Gradation shape analysis established d85′ of the finer fraction of each soil used in testing. An opening size ratio 5 ≤ Om/d85′ ≤ 10 correlates with a grain size ratio 1 ≤ d85,e/d85′ ≤ 2 and thus an eroded soil mass believed representative of the finer fraction of the widely graded soil. The novel Om/d85′ criterion for sizing the outflow boundary in permeameter testing is found consistent with an alternate criterion using concepts of base-filter incompatibility.

Singularly perturbed elliptic problems of convection–diffusion type with non-smooth inflow/outflow boundary conditions

A singularly perturbed elliptic problem, with non-smooth inflow or non-smooth outflow boundary conditions, is examined. A decomposition of the continuous solution is constructed, whose components identify the various types of layer functions that can exist in the solution. Parameter-explicit pointwise bounds on the first three partial derivatives of these components are established. An appropriate Shishkin mesh is identified for the problem and this is combined with upwinding to form a numerical method. Parameter-uniform error bounds are deduced. Numerical results are presented to illustrate the performance of the numerical method.

A balanced outflow boundary condition for swirling flows

In open flow simulations, the dispersion characteristics of disturbances near synthetic boundaries can lead to unphysical boundary scattering interactions that contaminate the resolved flow upstream by propagating numerical artifacts back into the domain interior. This issue is exacerbated in flows influenced by real or apparent body forces, which can significantly disrupt the normal stress balance along outflow boundaries and generate spurious pressure disturbances. To address this problem, this paper develops a zero-parameter, physics-based outflow boundary condition (BC) designed to minimize pressure scattering from body forces and pseudo-forces and enhance transparency of the artificial boundary. This “balanced outflow BC” is then compared against other common BCs from the literature using example axisymmetric and three-dimensional open swirling flow computations. Due to centrifugal and Coriolis forces, swirling flows are known to be particularly challenging to simulate in open geometries, as these apparent forces induce non-trivial hydrostatic stress distributions along artificial boundaries that cause scattering issues. In this context, the balanced outflow BC is shown to correspond to a geostrophic hydrostatic stress correction that balances the induced pressure gradients. Unlike the alternatives, the balanced outflow BC yields accurate results in truncated domains for both linear and nonlinear computations without requiring assumptions about wave characteristics along the boundary.

A Conceptual Model for the Development of Tornadoes in the Complex Orography of the Po Valley

Abstract The Po Valley in northern Italy is a hotspot for tornadoes in Europe in spite of being surrounded by two mountain ridges: the Alps in the north and the Apennines in the southwest. The research focuses on the case study of 19 September 2021, when seven tornadoes (four of them rated as F2) developed in the Po Valley in a few hours. The event was analyzed using observations and numerical simulations with the convection-permitting Modello Locale in Hybrid Coordinates (MOLOCH) model. Observations show that during the event in the Po Valley, there were two surface boundaries that created a triple point: an outflow boundary generated by convection triggered in the Alpine foothills and a dryline generated by downslope winds from the Apennines, while warm and moist air advected westward from the Adriatic Sea east (ahead) of the boundaries. Tornadoes developed about 20 km northeast of the triple point. Numerical simulations with 500-m grid spacing suggest that the development of supercells and drylines in the Po Valley was sensitive to the elevation of the Apennines. Simulated vertical profiles show that the best combination of instability and wind shear for the development of tornadoes was attained within a narrow area located ahead of the dryline. A conceptual model for the development of tornadoes in the Po Valley is proposed, and the differences between tornado environments over a flat terrain and over a region with complex terrain are discussed. Significance Statement The Po Valley is a highly populated area where some of the most violent tornadoes in Europe have developed. We investigated a tornado outbreak that occurred on 19 September 2021 in this region, in order to identify its main environmental characteristics. High-resolution numerical simulations revealed that values of instability and wind shear were compatible with the development of several tornadoes only in a narrow area close to the intersection of two surface boundaries (a triple point). Moreover, the atmospheric environment during the tornado outbreak was strongly influenced by the presence of mountain ridges surrounding the plain. We have summarized our results in a conceptual model that can potentially be used for forecasting applications.

DeHNSSo: The Delft Harmonic Navier-Stokes Solver for nonlinear stability problems with complex geometric features

A nonlinear Harmonic Navier-Stokes (HNS) framework is introduced for simulating instabilities in laminar spanwise-invariant shear layers, featuring sharp and smooth wall surface protuberances. While such cases play a critical role in the process of laminar-to-turbulent transition, classical stability theory analyses such as parabolized or local stability methods fail to provide (accurate) results, due to their underlying assumptions. The generalized incompressible Navier-Stokes (NS) equations are expanded in perturbed form, using a spanwise and temporal Fourier ansatz for flow perturbations. The resulting equations are discretized using spectral collocation in the wall-normal direction and finite-difference methods in the streamwise direction. The equations are then solved using a direct sparse-matrix solver. The nonlinear mode interaction terms are converged iteratively. The solution implementation makes use of a generalized domain transformation to account for geometrical smooth surface features, such as humps. No-slip conditions can be embedded in the interior domain to account for the presence of sharp surface features such as forward- or backward-facing steps. Common difficulties with Navier-Stokes solvers, such as the treatment of the outflow boundary and convergence of nonlinear terms, are considered in detail. The performance of the developed solver is evaluated against several cases of representative boundary layer instability growth, including linear and nonlinear growth of Tollmien-Schlichting waves in a Blasius boundary layer and stationary crossflow instabilities in a swept flat-plate boundary layer. The latter problem is also treated in the presence of a geometrical smooth hump and a sharp forward-facing step at the wall. HNS simulation results, such as perturbation amplitudes, growth rates, and shape functions, are compared to benchmark flow stability analysis methods such as Parabolized Stability Equations (PSE), Adaptive Harmonic Linearized Navier-Stokes (AHLNS), or Direct Numerical Simulations (DNS). Good agreement is observed in all cases. The HNS solver is subjected to a grid convergence study and a simple performance benchmark, namely memory usage and computational cost. The computational cost is found to be considerably lower than high-fidelity DNS at comparable grid resolutions. Program summaryProgram Title: DeHNSSoCPC Library link to program files:https://doi.org/10.17632/9bnms99kk2.1Developer's repository link:https://github.com/SvenWesterbeek/DeHNSSoLicensing provisions: GPLv3Programming language: MatlabSupplementary material: The supplementary material contains the code as well as a user manual.Nature of problem: Fluid flows are subject to laminar-to-turbulent transition following the growth of instabilities. To avoid computationally demanding Direct Numerical Simulations (DNS), perturbation theory is often applied to their analysis. However, classical stability methods based on the Orr-Sommerfeld equation or the Parabolized Stability Equations neglect the influence of streamwise gradients in varying degrees. The validity of these assumptions is difficult to estimate a priori.Solution method: The Delft Harmonic Navier-Stokes Solver (DeHNSSo) solves the harmonic Navier-Stokes equations nonlinearly on domains featuring sharp and smooth spanwise invariant surface features using a generalized grid approach in combination with an embedded boundary method. This allows the user to include the effects of streamwise gradients on flow at a fraction of the cost of DNS.Additional comments including restrictions and unusual features: In DeHNSSo, the equations are solved using direct matrix solvers. As such, memory is treated as a scarce resource. The problem is formulated in a mode-independent manner such that left-hand side matrices need only be computed and stored once despite incorporating nonlinear terms. Additionally, DeHNSSo offers the user the possibility to prescribe inhomogeneous boundary conditions to introduce and solve the receptivity problem or manipulate instabilities. Due to the double Fourier expansion, the solver is restricted to spanwise and temporally periodic problems.

Open boundary conditions for atmospheric large-eddy simulations and their implementation in DALES4.4

Abstract. Open boundary conditions were developed for atmospheric large-eddy simulation (LES) models and implemented into the Dutch Atmospheric Large-Eddy Simulation model. The implementation was tested in a “Big Brother”-like setup, in which the simulation with open boundary conditions was forced by an identical control simulation with periodic boundary conditions. The results show that the open boundary implementation has minimal influence on the solution. Both the mean state and the turbulent structures are close to the control simulation, and disturbances at the in- and outflow boundaries are negligible. To emulate a setup in which the LES is coupled to a coarser model, the influence of coarse boundary input was tested by smoothing the output of the periodic control simulation both temporally and spatially before feeding it as input to the simulation with open boundary conditions. When smoothing is applied over larger spatial and longer temporal scales, disturbances start to form at the inflow boundary and an area exists where turbulence needs to develop. Adding synthetic turbulence to the smoothed input reduces the size of this area and the magnitude of the disturbances.

Analysis of Uncertainties and Associated Convective Processes in Simulations of Extreme Precipitation Over Cities With a Regional Earth System Model: A Case Study

AbstractThis study utilizes the Weather Research and Forecasting model coupled with an atmospheric chemistry model, a multi‐layer urban canopy model (UCM), and a building energy model to simulate the extreme rainfall event influencing the Guangzhou city in South China on 7 May 2017. By employing small variations in the longwave emissivity of buildings within the UCM, 11 convective‐permitting experiments are conducted. The maximal 18‐hr and hourly rainfall accumulation vary from a 2‐year return period to as high as a 20 or 40‐year return period with notable spatial differences. Comparisons between the more accurate group (GOOD) and less accurate group (POOR) simulations highlight that some minor differences in the near‐surface air thermodynamic conditions in urban area could lead to substantial differences in local convection and its impacts on subsequent convective systems. With persistent transportation of warm, moist airflows from the northern South China Sea, formation of a slow‐moving mesoscale outflow boundary to the north of the urban agglomeration leads to the development of a quasi‐stationary, compactly structured meso‐γ‐scale rainstorm in the GOOD simulations. The stronger low‐level to near‐surface convergence and mid‐level cyclonic shear within this system substantially enhance low‐level updrafts, leading to increased microphysical production and stronger horizontal advection of rainwater within the system. These findings offer some process‐based understanding about the uncertainties in simulating urban extreme rainfall and underscore the need to develop ensemble forecasting methods for convection‐permitting numerical models that incorporate increasingly complicated representations of anthropogenic influences.

A vortex damping outflow forcing for multiphase flows with sharp interfacial jumps

Outflow boundaries play an important role in multiphase fluid dynamics simulations that involve transition between liquid and vapor phases. These flows are dominated by low Weber numbers and a sharp jump in pressure, velocity and temperature. Inadequate treatment of these jumps at the outlet generates undesirable fluid disturbances that propagate upstream and lead to instabilities within the computational domain. To mitigate these disturbances, we introduce a forcing term that can be applied to incompressible Navier-Stokes equations to enforce stability in the numerical solution. The forcing term acts as a damping mechanism to control vortices that are generated by droplet/bubbles in multiphase flows, and is designed to be a general formulation that can be coupled with a fixed pressure outflow boundary condition to simulate a variety of multiphase flow problems. We demonstrate its applicability to simulate pool and flow boiling problems, where bubble-induced vortices during evaporation and condensation present a challenge at the outflow. Validation and verification cases are chosen to quantify accuracy and stability of the proposed method in comparison to established benchmarks and reference solutions, along with detailed performance analysis for three-dimensional simulations on leadership supercomputing platforms. Computational experiments are performed using Flash-X, which is a composable open-source software instrument designed for multiscale fluid dynamics simulations on heterogeneous architectures.

Mixed convection flow over a horizontal plate and the horizontal wake far downstream

The laminar mixed convection flow over a heated or cooled horizontal plate of finite length is a surprisingly intriguing problem. The hydrostatic pressure jump at the trailing edge and across the wake is to be compensated by induced circulation in the outer potential flow, similar to a plate at a non-zero angle of attack. The mixed convection flows over semi-infinite and finite horizontal plates, respectively, are thus substantially different. As the hydrostatic pressure jump remains finite along the infinitely extended wake, the induced outer potential flow around the plate would grow beyond bounds in a domain extending to infinity. Thus, boundary conditions must be imposed at the boundaries of a finite domain. However, the numerical solution of the Navier-Stokes equations remains a challenging problem due to the lack of appropriate outflow conditions. The traditional outflow condition cannot comply with the finite hydrostatic pressure jump across the wake. Thus, an asymptotic expansion for the wake far downstream from the plate is performed. Remarkably, the perturbation of the flow does not decay with increasing distance from the plate as in a classical wake. The asymptotic solution is implemented as an outflow boundary condition for the numerical solution of the Navier-Stokes equations. The results are compared with a boundary-layer solution obtained in previous work for the limiting case of vanishing Prandtl number.

Investigating the use of 3-component-2-dimensional particle image velocimetry fields as inflow boundary condition for the direct numerical simulation of turbulent channel flow

Direct numerical simulation (DNS) of turbulent wall-bounded flows requires long streamwise computational domains to establish the correct spatial evolution of large-scale structures with high fidelity. In contrast, experimental measurements can relatively easily capture large-scale structures but struggle to resolve the dissipative flow scales with high fidelity. One methodology to overcome the shortcomings of each approach is by incorporating experimental velocity field measurements into DNS as an inflow boundary condition. This hybrid approach combines the strengths of DNS and experimental measurements, allowing for a reduction in the streamwise computational domain and accelerated development of large-scale structures in turbulent wall-bounded flows. To this end, this paper reports the results of an investigation to establish the impact of limited spatial resolution and limited near-wall experimental inflow data on the DNS of a wall-bounded turbulent shear flow. Specifically, this study investigates the spatial extent required for the DNS of a turbulent channel flow to recover the turbulent velocity fluctuations and energy when experimental inflow data is typically unable to capture fluctuations down to the viscous sub-layer or the smallest viscous scales (i.e. the Kolmogorov scale or their surrogate viscous scale in wall-bounded turbulent shear slows) is used as the inflow to a DNS. A time-resolved numerically generated experimental field is constructed from a periodic channel flow DNS (PCH-DNS) at Reτ=\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Re_{\ au } =$$\\end{document} 550 and 2300, which is subsequently used as the inflow velocity field for an inflow–outflow boundary conditions DNS. The time-resolved experimental inflow field is generated by appropriately filtering the small scales from the PCH-DNS velocity by integrating over a spatial domain that is representative of a particle image velocimetry interrogation window. This study shows that the recovery of small scales requires a longer domain as the spatial resolution at the inflow decreases with all flow scales recovered and their correct scale-dependent energy is re-established once the flow has developed for 3 channel heights.

Numerical investigation of flow past a cylinder using cumulant lattice Boltzmann method

This paper presents simulations of flow past a circular cylinder within the subcritical Reynolds number (Re) range from 3900 to 2 × 105, utilizing the parameterized cumulant lattice Boltzmann model. In this study, a three-dimensional characteristic boundary condition for incompressible flow has been integrated into the lattice Boltzmann method at the outflow boundary to minimize spurious reflection. The flow field, wake statistics, hydrodynamic force, and power spectra results of Re = 3900 from the cumulant lattice Boltzmann model are exhaustively compared with the laboratory data and other numerical models. Relative to other numerical models employing turbulence closure, the cumulant lattice Boltzmann simulations demonstrate enhanced agreement with the experimental data even with relatively coarser grid resolution. The resolution-spanning feature for the cumulant lattice Boltzmann model in turbulent flows, without using explicit turbulence model, aligns with the previous benchmark case studies. The stability-preserving regularization process in the present model is analyzed. Results indicate that the influence of the regularization parameter is mitigated with improved grid resolution. A specific regularization parameter for flow around cylinder simulations is recommended. Variations in flow properties and hydrodynamic forces within the subcritical Reynolds number range of 3900 to 2 × 105 are analyzed. The results confirm that the parameterized cumulant lattice Boltzmann model can accurately simulate practical engineering flows, characterized by complex separation and recirculation, within the subcritical range. Moreover, the computational efficiency and parallel scalability are compared with other numerical methods.

Simulating the propagation of boundary-layer disturbances by solving boundary-value and initial-value problems

Abstract The article deals with the downstream propagation of small–amplitude disturbances of viscous incompressible laminar boundary layers, using the linearized equations for disturbance amplitudes. Two different methods are proposed. The first one solves a two-dimensional boundary-value problem, using a buffer-domain technique to mimic the outflow boundary condition. The second one solves a streamwise initial-value problem, using a spectral parabolization at each integration step. Both methods show good performance in simulating the propagation of the Tollmien–Schlichting waves and the Görtler vortices and can be applied to compute the spatial optimal disturbances.

Various Flow in a Slender Tube: Mach Number and Domain Shape Influenced by Boundary Obstacles Via Small Disturbance Equations

With the exponential advancements in computer hardware and software in recent years, computational fluid dynamics (CFD) has emerged as a highly efficient alternative to traditional fluid mechanics studies. Its prowess in accurately predicting and analyzing flow mechanics has led to widespread adoption across diverse engineering disciplines. This research delves into a numerical analysis of flow conditions utilizing the small disturbance equation (SDE). The study bridges the gap between mathematical formulations and computational language, exemplified through Python code, by harnessing the successive over-relaxation (SOR) method, Neumann boundary conditions (BC), and the ghost point technique. The research underscores its predictions' precision in subsonic and supersonic scenarios by meticulously evaluating the resultant errors. This accuracy is further corroborated by examining overarching flow patterns and Mach number distributions. However, the program's outputs indicate a discernible lack of precision in the transonic case, with errors surpassing acceptable thresholds. This discrepancy is hypothesized to stem from potential mischaracterizations of points at the inflow and outflow boundaries. This study, thus, offers invaluable insights while also highlighting areas necessitating further refinement.