Boundary Layer Model Research Articles

Further investigation of two-parameter J-A estimation for clamped SET specimens

Based on the J-A two-parameter characterization, this paper presents a systematic study of the crack-tip constraint for clamped single-edge notched tension (SET) specimens. Extensive finite element analyses (FEA) are conducted to obtain the A solutions for the modified boundary layer (MBL) model and the SET specimen. The material used in the analysis is the Ramberg-Osgood power-law material, with varying yield stress and strain hardening exponents. Based on previous studies, the T-stress-related A solutions for MBL problems under small-scale yielding conditions are obtained from FEA results, and the A solutions for the SET specimen are found to scale with normalized applied load under large-scale yielding conditions. By summing up these two solutions, the total A for the SET specimen can be estimated, covering the entire range from small-scale to large-scale yielding conditions. A key geometry factor is introduced as a function of material properties, crack depths and applied load. Some fitting equations are proposed in this estimation method, and a Python program is developed to quickly and accurately obtain the total A for the SET specimen. The results of the present study show good agreement for all the simulated cases. The proposed solutions of constraint parameter A can be used to establish the constraint-corrected J-R curve for SET specimens.

Unsteady 3D MHD Boundary Layer Stream for Non-Newtonian Power-Law Fluid Near Stagnation Point of Moving Surfaces

An unsteady three-dimensional MHD boundary layer is a fluid flow region near a surface where magnetic fields are present and interact with the fluid flow, causing it to become unsteady. This type of flow is commonly found in various astrophysical and technological applications, such as in plasmas and fusion reactors. The 3D nature of the flow introduces additional complexities to the flow dynamics, making the study and modeling of unsteady MHD boundary layers a challenging and active area of research. The unsteady boundary layer flow of fluid over a moving stagnation surface is theoretically examined in the current work with the impression of a magnetic field. The exact outcomes of the governing equations for the flow domain are obtained by utilizing the shooting phenomena. The specified analytical outcomes are also obtained for some cases. Detailed discussions of the parameters involved are confirmed both physically and graphically. Numerical results for both profiles are presented graphically. The study and modeling of unsteady 3D MHD boundary layers is imperative for a thorough understanding of various physical phenomena, improving the performance of technological systems, and advancing our knowledge of fluid dynamics.

Influence of heterogeneity due to toughness variations on weakest-link modeling for brittle failure

The effect of heterogeneous microstructures on the macroscopic probability of failure is studied by use of weakest-link modeling. Heterogeneity is here associated with a local variation of toughness, where a size scale characteristic of this variation defines a length parameter. The ratio between this length parameter and the size of the active fracture process zone, defined as the heterogeneity ratio, is key to evaluating the impact of a heterogeneous microstructure. Two extremes are identified; small-scale heterogeneity (SSH) and large-scale heterogeneity (LSH). For these cases, it is possible to formulate analytical expressions based on the weakest-link concept, and references are made to existing models in the literature. Typically, heterogeneity along the crack front, where gradients of the mechanical fields are small, falls under the category of SSH. On the other hand, the effect of heterogeneity in a plane perpendicular to the crack front depends strongly on the heterogeneity ratio. Cases that can neither be identified with SSH nor LSH must be addressed with care. How this can be done is discussed, and examples are given for four different microstructure configurations of interest. The investigation is carried out by numerical analysis of a modified boundary layer model. The cumulative probability of failure by cleavage fracture is evaluated in a post-processing step, where two different statistical models are examined; the Beremin model and the Kroon–Faleskog model. Both models render the same conclusion about the alteration of the overall failure probability distributions caused by heterogeneity.

Plasticiser leaching from polyvinyl chloride microplastics and the implications for environmental risk assessment

Microplastics in aquatic environments is a growing concern, particularly due to the leaching of chemical additives such as plasticisers. To develop comprehensive environmental risk assessments (ERAs) of high-concern polymers and plasticisers, an understanding of their leachability is required. This work investigated diethylhexyl phthalate (DEHP) and bisphenol A (BPA) leaching from polyvinyl chloride (PVC) microplastics (average diameter = 191 μm) under simulated marine conditions. Leaching behaviours were quantified using gel permeation chromatography (GPC) and thermal gravimetric analysis (TGA), and the polymer's physiochemical properties analysed using differential scanning calorimetry (DSC), Fourier Transform-Infrared Spectroscopy (FT-IR) and optical microscopy. Experimental data were fitted to a diffusion and boundary layer model, which found that BPA leaching was temperature-dependent (diffusion-limited), whereas DEHP leaching was controlled by surface rinsing. Model predictions also highlighted the importance of microplastic size on leaching dynamics. These data contribute towards greater accuracy in ERAs of microplastics, with implications for water quality and waste management, including decommissioning of plastic infrastructure.

Near-wall model for compressible turbulent boundary layers based on an inverse velocity transformation

In this work, a near-wall model, which couples the inverse of a recently developed compressible velocity transformation (Griffin et al., Proc. Natl Acad. Sci., vol. 118, 2021, p. 34) and an algebraic temperature–velocity relation, is developed for high-speed turbulent boundary layers. As input, the model requires the mean flow state at one wall-normal height in the inner layer of the boundary layer and at the boundary-layer edge. As output, the model can predict mean temperature and velocity profiles across the entire inner layer, as well as the wall shear stress and heat flux. The model is tested in an a priori sense using a wide database of direct numerical simulation high-Mach-number turbulent channel flows, pipe flows and boundary layers (48 cases, with edge Mach numbers in the range 0.77–11, and semi-local friction Reynolds numbers in the range 170–5700). The present model is significantly more accurate than the classical ordinary differential equation (ODE) model for all cases tested. The model is deployed as a wall model for large-eddy simulations in channel flows with bulk Mach numbers in the range 0.7–4 and friction Reynolds numbers in the range 320–1800. When compared to the classical framework, in the a posteriori sense, the present method greatly improves the predicted heat flux, wall stress, and temperature and velocity profiles, especially in cases with strong heat transfer. In addition, the present model solves one ODE instead of two, and has a computational cost and implementation complexity similar to that of the commonly used ODE model.

Expected Changes in Evaporation in the Taiga Zone of European Russia in the XXI Century

Model estimates were made for changes in the actual evaporation of boreal forests in the taiga zone of European Russia caused by climate evolution in the XXI century. Climate changes were forecasted using the results of calculations by an MGO regional climate model. The space and time specification of the forecast data was made with the use of a model of the atmospheric boundary layer. Variations of the forecasted changes due to the use of different methods to calculate evaporation were studied. Calculations based on an additional (complementary) relationship, determining the actual evaporation with the use of standard meteorological characteristics were compared with observations and calculations based on a detail model of energy transport in a forest massif. The model values of the actual evaporation in the XXI century increase all over the European Russia. The effect of the intensity of dry periods on the evolution of actual evaporation is analyzed. It is shown that, despite the arid climate, the effect of dry periods on taiga forests will be moderately negative.

A Parametric Model of Tropical Cyclone Surface Winds for Sea and Land

Abstract Parametric models of tropical cyclone winds are widely used for risk assessment. Although tropical cyclones often present their worst wind risk to humanity during landfall, parametric models that represent land–sea differences are rare. This paper presents a parametric model with explicit representation of land–sea differences. Statistical models were developed over each surface of the frictional wind speed reduction from gradient level to 10 m, and of the surface inflow angle, based on about 1200 simulations with a three-dimensional dynamical boundary layer model. The wind profile of Willoughby et al. is used to represent the gradient flow, and a maximum likelihood scheme used to fit this profile to best track data. The mean RMS difference between the statistical and dynamical surface winds within 100 km of the storm center is 0.78 m s−1 and 4.26° over sea, and 1.04 m s−1 and 4.59° over land. During landfall, the use of a common gradient-level structure, but different surface roughnesses, provides dynamical consistency between the estimated winds over sea and land. A simple representation of internal boundary layers is applied near the coast. Analysis of the dynamical simulations revealed substantial consistency with observational studies of the tropical cyclone boundary layer, including that the azimuth of the surface wind maximum is on average 65° from the front of the storm, in the left-forward quadrant in the Southern Hemisphere. There was, however, substantial variability around this figure, with the maximum occurring in the opposite forward quadrant in storms that were intense, and/or had a relatively rapid decrease in wind speed outside of the radius of maximum winds.

A free-streamline boundary-layer model for small-amplitude oscillation regime of square cylinder under vortex-induced rotation

Numerical studies about vortex-induced rotation (VIR) have shown that six regimes exist for the dynamics of square cylinder under VIR at small Reynolds numbers; however, theoretical explanations are still rare. Aiming at stepping a little further, we select one of these regimes, that is, small-amplitude oscillation regime, to develop an analytical model. In the present model, we expand the symmetric free streamline theory to an inviscid flow pasting an inclined plate and introduce the Blasius boundary layer theory in the calculation of viscous torque on a square cylinder. This model is supported by comparing with numerical results using the immersed boundary method. Results explain the mechanism of small-amplitude oscillation regime: the main sources of the out-of-phase torque come from the imbalanced distribution of pressure on the windward face and the asymmetric distribution of pressure and viscous force on the top/bottom face. An alternative way to determine the local flow parameters by global dynamic quantities is also proposed based on this model.

Cascade sensitivity tests to model deep convective systems in complex orography with WRF

The goal of the present study is to improve the analysis of the performance of the Weather Research and Forecasting (WRF) model, Version 4.3.3, under different settings, in the prediction of precipitation in case of deep convective events. This is done through a wide cascade sensitivity test involving parameterizations expected to play a major role in the description of precipitation for deep convective events: cloud microphysics (CM), planetary boundary layer (PBL), surface layer (SL) and land-surface (L-S) model. Four significant precipitation events, which were associated to deep convective systems and occurred between 2019 and 2021 in Lombardia and Liguria regions (Northern Italy), have been simulated according to 45 different WRF settings. High resolution simulations are required when dealing with small scale deep convective systems as well as a suitable verification method. So, traditional statistical indexes have been integrated with new-generation verification methods, based on the identification of precipitation patterns in forecast and observed fields and on the evaluation of their similarity through the calculation of a coupling index. This analysis method has been exploited to evaluate the quality of the high-resolution simulations performed, and to individuate the best-performing WRF setting in simulating intense convective events. From the present work emerges how the unique CM parameterization characterized by a triple-moment treatment for cloud ice significantly outperforms all other CM schemes underlying the main role of cloud ice in the description of the precipitation case studies. Furthermore, PBL and L-S schemes show a leading role, at least as much as CM, in the description of the events.

Stochastic determination of thermal reaction rate coefficients for air plasmas.

This work deals with the stochastic inference of gas-phase chemical reaction rates in high temperature air flows from plasma wind tunnel experimental data. First, a Bayesian approach is developed to include not only measurements but also additional information related to how the experiment is performed. To cope with the resulting computationally demanding likelihood, we use the Morris screening method to find the reactions that influence the solution to the stochastic inverse problem from a mechanism comprising 21 different reactions for an air mixture with seven species: O2, N2, NO, NO+, O, N, e-. A set of six reactions, mainly involving nitrogen dissociation and exchange, are the ones identified to impact the solution the most. As such, they are assumed to be uncertain and estimated along with the boundary conditions of the experiment and the catalytic recombination parameters of the materials involved in the testing. The remaining 15 reactions are set to their nominal values. The posterior distribution is then propagated through the proposed boundary layer model to produce the posterior predictive distributions of the temperature and mass fraction profiles along the boundary layer stagnation line. It is identified that NO concentrations have the largest increase in uncertainty levels compared to cases where the inference problem is carried out for fixed chemical model parameter values. This allows us to inform a new experimental campaign targeting the reduction of uncertainties affecting the chemical models.

Inflow turbulence generation using an equivalent boundary layer model

Eddy-resolved simulation of external flow usually requires inflow boundary conditions representing a turbulent boundary layer (TBL) flow, and the quality of the inflow turbulent fluctuation directly impact the accuracy and the cost of the simulation. The present study proposes a new method to generate TBL inflow turbulence, i.e., the equivalent boundary layer (EBL) model. Based on the open-channel model, EBL approximates TBL flow at a given Reynolds number by recovering the mean momentum balance with driving force. It simulates streamwise homogeneous turbulence, applying periodic boundary conditions and, thus, overcomes the complexity and artificiality incurred by the classic recycling–rescaling methods. The current paper discusses the difference between turbulent channel and boundary layer flows from the equation point of view and designs the driving force corresponding to the mean inertial force of boundary layer. Also, the total shear stress models for obtaining the driving force are validated both a priori and a posteriori. Direct numerical simulations (DNS) are carried out for EBLs at Reθ=1000,1420, and 2000 (where Reθ is the Reynolds number based on the momentum thickness), showing that the EBL model well reflects the statistical characteristics of TBL at corresponding Reynolds numbers. The application of the EBL model for the generation of inflow turbulence is also demonstrated by DNS of turbulent boundary layers with inlet Reθ=1000,1420, and 2000. The computational results agree well with generally acknowledged DNS data published in the literature, in terms of streamwise developing statistics, and profiles and energy spectra at characteristic cross sections. Judging from the mean velocity, the adjustment section is shorter than one boundary layer thickness.

A coupled equilibrium boundary layer model with stable water isotopes and its application to local water recycling

The contribution of evapotranspiration (ET) to regional precipitation, known as “local water recycling”, is a key process in the water cycle that can affect water management. However, the amount of planetary boundary layer (PBL) moisture that arises from ET is highly uncertain due to complex atmosphere and land surface conditions. In this study, an idealized two-layer equilibrium boundary layer model was coupled with a stable water isotope module including HDO and H218O to constrain PBL growing season water transport processes. The model was validated using turbulent heat fluxes and isotope ratios of water vapor (δm) and precipitation (δP) measured at a cropland site and a nearby tall tower in the Upper Midwest, United States. The results show that the PBL equilibrium features of δm and δP are well-constrained by thermal and moisture equilibrium in the PBL. For this study region, the summer values of rain evaporation fraction (f) and local water recycling ratio (LRR) are estimated to be 0.09 and 0.29 ± 0.12, respectively. The summer LRR values for the years 2006–2010 were 0.35, 0.36, 0.17, 0.29, and 0.29, respectively. The small value of LRR in 2008 corresponded to a drought condition with the lowest precipitation and second lowest ET among the five years. The summer magnitude of the amount effect is −2.8‰ (mm day−1)−1 and −0.8‰ (mm day−1)−1 for HDO and H218O, respectively. The local water recycling is identified as a significant factor influencing the continental isotope effect. Cropland has likely changed the regional LRR by −7.6 to 19.5% under different pre-agriculture land use scenarios. The feedback processes revealed here indicate that local water recycling is expected to be weakened under drought conditions, but it will be enhanced if irrigation is applied more intensely with more frequent drought events as the climate continues to warm.

Cracking direction in graphene under mixed mode loading

Crack branching in graphene under complex stresses is investigated by molecular dynamics simulations and boundary-layer models. Using the maximum energy release rate (MERR) criterion, an analogy of Wulff curve is plotted considering the anisotropic fracture toughness of hexagonal graphene, which shows the direction of crack kinking (local bump on the curve) is consistent with that of the weak fracture toughness (along zigzag edge). The angle of crack kinking prefers 0°/30°/60°, rather than the angle predicted by maximum circumferential stress criterion. The T-stress along crack front, obtained by the over-deterministic method, can indicate the possibility of kink by its transformation between positive and negative, especially for zigzag cracks. This study incorporates Wulff-like curve and MERR criterion to provide an accurate prediction of nanocrack paths in graphene.

Thin boundary layer model underestimates greenhouse gas diffusion from inland waterways

Inland waters are significant sources of atmospheric greenhouse gas (GHG) emissions. The thin boundary layer (TBL) model is often employed as a means of estimating GHG diffusion in inland waters based on gas transfer velocity (k) at the air-water interface, with k being subject to regulation by near-surface turbulence that is primarily driven by wind speed in many cases. This wind speed-based estimation of k (wind-k), however, can introduce substantial uncertainty for turbulent waterways where wind speed does not accurately represent overall turbulence. In this study, GHG diffusion in the Beijing-Hangzhou Grand Canal (China), the first and longest man-made canal in the world, was estimated using the TBL model, revealing that this model substantially underestimated GHG diffusion when relying on wind-k. Strikingly, the carbon dioxide, methane, and nitrous oxide diffusions were respectively underestimated by 159%, 162%, and 124% when using this model. These findings are significant for developing more reliable approaches to evaluate GHG emissions from inland waterways.

Convection effect on solidification microstructure: dendritic arm spacing and microsegregation

Convection affect primary crystalline structure, particularly, the dendrite crystal velocity and dendrite tip radius. The present work aims on the influence of the convective flow on the primary dendrite spacing between neighbouring crystals within the dendrite ensemble. Solidification of a binary alloy is considered within the model of stagnant boundary layer, under imposed thermal gradient which influence on the crystal microstructure and chemical microsegregation. Chemical composition in the solidifying liquid and crystalline solid is derived from the solution of the solute diffusion transport equation taking into account the convective flow. The model results are consistent with the Scheil-Brody-Flemings model, experimental data and computational results.

Enhanced ocean wave modeling by including effect of breaking under both deep- and shallow-water conditions

Abstract. Accurate description of the wind energy input into ocean waves is crucial to ocean wave modeling and a physics-based consideration on the effect of wave breaking is absolutely necessary to obtain such an accurate description. This study evaluates the performance of an improved formula recently proposed by Xu and Yu (2020), who took into account not only the effect of breaking but also the effect of airflow separation on the leeside of steep wave crests in a reasonably consistent way. Numerical results are obtained through coupling an enhanced atmospheric wave boundary layer model with the ocean wave model WAVEWATCH III (v5.16). The coupled model has been extended to be valid in both deep and shallow waters. Duration-limited waves under controlled normal conditions and storm waves under practical hurricane conditions are studied in detail to verify the improved model. Both the representative wave parameters and the parameters characterizing the wave spectrum are discussed. It is shown that the improved source-term package for the wind energy input and the wave energy dissipation leads to more accurate results under all conditions. It performs evidently better than other standard source-term options of ST2, ST4 and ST6 embedded in WAVEWATCH III. It is also demonstrated that the improvement is particularly important for waves at their early development stage and waves in shallow waters.

Numerical Study of Steam–CO2 Mixture Condensation over a Flat Plate Based on the Solubility of CO2

In order to successfully study the condensation and separation of a steam–CO2 mixture, a boundary layer model was applied to the mixture condensation of steam and CO2 on horizontal and vertical plates. The modified condensation boundary layer model of steam and CO2, given the CO2 solubility in the condensate, was established, numerically solved, and verified with existing experimental data. Different condensation data of steam–air and steam–CO2 mixtures were compared, and the effect of CO2 solubility on the mixed gas condensation was analyzed under multiple pressure conditions (1 atm–10 MPa). The simulation data show that the presence of CO2 will deteriorate the condensation heat transfer, just like air. Given that CO2 is slightly soluble, some CO2 can pass through the gas–liquid interface to enter the condensate film and reduce the accumulated CO2 on the gas–liquid interface, which improves the condensation. However, the solubility of CO2 is only significant under high-pressure conditions, inducing its effects on condensation. A comparison of the condensation coefficients of the steam–CO2 mixture shows the lower impact of CO2 condensation on the horizontal plate compared to that on the vertical plate. For most conditions, the steam–CO2 mixture gas condensation heat transfer coefficient on the vertical plate surface is still larger than that on the horizontal plate surface, and the improvement in the condensation heat transfer coefficient caused by low CO2 solubility (2 or 10%) at 10 MPa on the vertical plate is also larger than that of the horizontal plate.

Magnetic diffusion of time distributed-order Maxwell fluid in boundary layer under the action of induced magnetic field

Hydrogel has been widely used in energy storage devices and flexible electronic equipment. To promote its applications, a laminar boundary layer model is proposed to analyze the velocity and temperature distributions in the mold. Based on the time distributed-order Maxwell and Cattaneo constitutive relations, the research establishes governing equations of unsteady two-dimensional incompressible viscoelastic electrically conducting hydrogel taking into account the induced magnetic field. When the magnetic Reynolds number is large enough, it is necessary to consider the magnetic diffusion in the boundary layer. The effects of thermal radiation and velocity slip are also considered at the same time. The Gauss quadrature rule is used to approximate the distributed-order integral, and then the numerical solution of the model is obtained by using the finite difference method and the L1-algorithm. The analytical solution is constructed to verify the effectiveness of the numerical solution. The results show that the fluid velocity decreases with the increment of magnetic parameter, while the induced magnetic field increases. The boundary layer thickness of velocity, induced magnetic field, and thermal becomes thinner with the enlargement of velocity and temperature relaxation time parameters. Moreover, the distributed-order and the fractional constitutive models are compared through different weight coefficients, and it is found that the fractional constitutive model obtains larger velocity and temperature distributions.

The characteristics of laminar film condensation of pure vapor flow in a vertical tube

In this study, we investigate the characteristics of laminar film condensation of pure vapor flow in a vertical tube, taking into account (i) the vapor flow rate reduction owing to condensation, (ii) the finiteness of the vapor boundary-layer thickness, (iii) the curvature of the vapor layer, and (iv) the pressure variation. An approximate integral two-phase boundary-layer equation model is derived, and a condition that pressure recovery occurs is determined from the asymptotic analysis around the tube inlet. Furthermore, the present model is further simplified to obtain an analytical solution if the degree of subcooling is sufficiently large. The asymptotic solution for sufficiently large degrees of subcooling gives a good approximation to the numerical solution. The tube length of complete condensation is estimated. Comparison with vertical plane condensation demonstrates that the film thickness and the Nusselt number increased and decreased, respectively, by the vapor flow reduction for large degrees of subcooling.

Effect of flow rate on condensation of CO2-water vapor mixture on a vertical flat plate

The presence of non-condensable gas significantly deteriorates the heat transfer of water vapor condensation, and the condensation process of a CO2-water vapor mixture is a key process for CO2 separation and purification. In this study, we fabricated an experimental system to investigate the heat transfer characteristics of the mixed vapor condensation using a CO2 mass fraction of 32–85 % at a flow rate of 0.4–1.2 m/s and surface subcooling of 3–70 K. Liquid film thickness, gas-phase diffusion layer thickness, and interface temperature were calculated based on the double boundary layer model. An increase in the flow rate (0.4–1.2 m/s) improved the mixed vapor heat transfer, especially at high CO2 concentrations, and increased the heat transfer coefficient by 177 % at a surface subcooling of 40 K. The lower the surface subcooling, the greater the effect of the flow rate. As the CO2 concentration increased, the liquid film thickness decreased, whereas the gas-phase diffusion layer thickness increased to 1.7 mm, which is tens and hundreds of times the liquid film thickness. Increasing the flow rate slightly changed the liquid film but significantly reduced the gas-phase diffusion layer thickness and afforded a better heat transfer performance. A new heat transfer correlation equation was fitted, and the error of the predicted value was within ± 30 %. This study provides fundamental insights for the design of related heat exchangers.