Boundary Layer Method Research Articles

Merging of Asymptotics in the Illuminated Part of the Fock Domain

Investigation of the shortwave diffraction by elongated bodies of revolution requires a detailed consideration of matching of local asymptotics in the illuminated part of the Fock domain. In the paper, that task is solved by means of a straightforward construction of the reflected wave with the help of the ray method. The main problem on the way, which was judged by V. A. Fock as a rather complicated one, is the calculation of the eikonal and the geometric spreading in curvilinear coordinates used in the boundary layer method in the vicinity of the light-shadow zone. Bibliography: 9 titles.

Steady solution and spatial stability of gravity-driven thin-film flow: reconstruction of an uneven slippery bottom substrate

Steady solutions of an inverse problem relevant to gravity-driven film flows over an undulated slippery bottom are considered. Given a target free surface shape, the goal is to obtain the corresponding bottom topography of a slippery substrate which causes the specified free surface shape for a film flowing over it. The approaches followed by Sellier (Phys Fluids 20:062106, 2008) for creeping films and by Heining and Aksel (Phys Fluids 21:083605, 2009) for inertial films for reconstructing a rigid bottom topography for a target free surface profile are extended to the reconstruction of a slippery bottom topography. The model equations for film thickness above the bottom topography are derived for creeping flows under lubrication approximation and for inertial films using the weighted-residual integral boundary layer method and are solved numerically. The influence of inertia, slip parameter and surface tension on the shape of the reconstructed bottom topography is analyzed for different prescribed free surface shapes (sinusoidal, trench and bell-shaped). It is observed that the nonlinearities that appear in the reconstructed rigid bottom substrate with no slip at the substrate are suppressed by seeking the bottom substrate to be reconstructed as a slippery substrate. A spatial linear stability analysis of the corresponding direct problem is examined using Floquet theory, and the results reveal that the slip parameter and surface tension have a high influence on the critical Reynolds number. The results provide a strategy for controlling surface defects in a gravity-driven film over a substrate; namely, in order to achieve a target free surface profile, one can design the bottom substrate to be an undulated rough/textured/grooved or a superhydrophobic surface which can be modelled as an undulated smooth substrate with velocity slip at the substrate.

Atmospheric Boundary Layer for a Full-Size Wind Field Based on Large Eddy Simulation

AbstractTo study the numerical simulation method for an atmospheric boundary-layer fluctuating wind field, an ideal randomly fluctuating wind was generated by arranging structural measures, such as the roughness element, the spoiler lever, and the grille in a full-size wind field model. Some of the main parameters, such as the height of the roughness element, the spacing of the spoiler lever, and the arrangement of the grille, were analyzed. All of these parameters affected the numerical simulation results, confirmed the effect law, and suggested the possibility of a combined method of the corresponding numerical model by investigating a class C landscape atmospheric boundary-layer wind field. Simulation results satisfied the requirements of the wind structure calculation, achieved a full simulation of the boundary-layer transition, and provided a valuable stream generation method for the subsequent large eddy simulation of the flow around a structure.

Application Of Boundary Layer Theory In Open Channel Flume Design

Abstract In accordance with the problem of the need for calculation formula of the straight wall flume flow in open channel by experiment, the method of thee boundary layer is proposed; Analysis with boundary layer displacement thickness is studied, and the energy equation, continuous equation and the theory and critical depth of flume flow coefficient, velocity coefficient is deduced using the boundary layer theory and the flow calculation formula is given; Through iterative algorithm, the flow can be calculated theoretically. With four different kinds of parameters in the U-shaped channel flume algorithm are verified, the results show that the theoretical calculation corresponds to the actual observation. Therefore, the boundary layer theory to calculate the flow of the flume is not only studied the amount of water problems in theory, simplifies the complicated test calibration, but also has important practical value.

Aeroelastic predictions for steady and unsteady flow characteristics of the HIRENASD wing

In this study, we focus on static and dynamic aeroelastic analyses of the HIRENASD wing based on reference experimental data for two different flight conditions for Aeroelastic Prediction Workshop-1. The major anticipations from the HIRENASD project are to improve the knowledge about aero-structural dynamics, and to get experimental and computational data in a wide range of flight conditions. The experiments have been formerly conducted in cryogenic medium to investigate steady and unsteady aeroelastic responses in transonic regime for low and high Reynolds numbers. For current aeroelastic computations, first, a free vibration analysis is performed in Nastran using the latest structural model provided by NASA. Sequentially, the modal solution is imported from the finite element solver to Zeus software which is an Euler equations based aeroelastic solver coupled with integral boundary layer method. The aerodynamic model and fluid structure interaction parameters are constructed in Zeus. Steady and unsteady aeroelastic results are examined for the specified stations along the wing span, and interpolated for chordwise direction so as to match them with the wind tunnel test points. The designated output parameters such as steady aerodynamic lift, moment and drag coefficients, and steady and unsteady pressure distributions along the chordwise direction are compared to the experimental data and NASA's FUN3D code results which are provided in literature.

Ultrafiltration of charge-stabilized dispersions at low salinity.

We present a comprehensive study of cross-flow ultrafiltration (UF) of charge-stabilized suspensions, under low-salinity conditions of electrostatically strongly repelling colloidal particles. The axially varying permeate flux, near-membrane concentration-polarization (CP) layer and osmotic pressure profiles are calculated using a macroscopic diffusion-advection boundary layer method, and are compared with filtration experiments on aqueous suspensions of charge-stabilized silica particles. The theoretical description based on the one-component macroion fluid model (OCM) accounts for the strong influence of surface-released counterions on the renormalized colloid charge and suspension osmotic compressibility, and for the influence of the colloidal hydrodynamic interactions and electric double layer repulsion on the concentration-dependent suspension viscosity η, and collective diffusion coefficient Dc. A strong electro-hydrodynamic enhancement of Dc and η, and likewise of the osmotic pressure, is predicted theoretically, as compared with their values for a hard-sphere suspension. We also point to the failure of generalized Stokes-Einstein relations describing reciprocal relations between Dc and η. According to our filtration model, Dc is of dominant influence, giving rise to an only weakly developed CP layer having practically no effect on the permeate flux. This prediction is quantitatively confirmed by our UF measurements of the permeate flux using an aqueous suspension of charged silica spheres as the feed system. The experimentally detected fouling for the largest considered transmembrane pressure values is shown not to be due to filter cake formation by crystallization or vitrification.

Approximating the traveling wavefront for a nonlocal delayed reaction-diffusion equation

In this paper a boundary layer method is combined with an asymptotic expansion method to approximate the traveling wave solution of a nonlocal delayed reaction-diffusion model. In particular, assuming that the diffusion coefficients of the mature and immature populations are small, the wave solution is approximated in three steps. First, the model is reduced by considering the Dirac delta function as the kernel function of the integral term. Second, a boundary layer method is employed to approximate the wave solution of the reduced model. Third, using this result and the generalized Watson’s lemma, the wave solution of the general model is approximated. By considering various birth functions, the approximate wave solutions are numerically compared with the exact wave solutions.

STATE-DEPENDENT BOUNDARY LAYER METHOD FOR ATTITUDE CONTROL OF SATELLITE

Sliding mode control is known to be robust against parameter uncertainties and external disturbances. Based on the dynamic equation of motion, a sliding mode controller is designed to solve this problem. However, for the sliding surface to be attractive, a switching function is used in the control law, which caused chattering of the control signal. In order to avoid this, a boundary layer method is considered in the modified controller. So this paper proposes new boundary layer designs that resolve the problem in control accuracy and control signal smoothness in sliding mode control. The propose design improve the system state to almost zero with no chattering in the control signals.

A Simple and Robust Subsonic Boundary Layer Method to Be Used with the Panel Method for Wing Calculations Using a Wing Strip Approach

This paper will present the development of a simple subsonic boundary layer method suitable to be used coupled with panel methods in order to estimate the aerodynamic characteristics, including viscous drag and maximum lift coefficient, of 3D wings. The proposed method does not require viscous-inviscid iterations and is based on classical integral bi-dimensional boundary layer theory using Thwaites and Head ́s models with bi-dimensional empirical corrections applied to each wing strip being therefor robust and efficient to be used in the early conceptual stage of aircraft design. Presented results are compared to the Modified CS Method in an IBL scheme and experimental data and are shown to provide good results.

The invariant integral of physical mesomechanics as the foundation of mathematical physics: Some applications to cosmology, electrodynamics, mechanics and geophysics

Phe general invariant integral based on the energy conservation law is introduced into physical mesomechanics, with taking into account the cosmic, gravitational, mass, elastic, thermal and electromagnetic energy of matter. Phe physical mesomechanics thus becomes a mega-mechanics embracing most of the scales of nature. Some basic laws following from the general invariant integral are indicated, including Coulomb’s law of electricity generalized for moving electric charges, Newton’s law of gravitation generalized for coupled gravitational/cosmic field, the Archimedes’ law of buoyancy generalized for bodies partially submerged in water, and others. Using the invariant integral the temperature track behind moving cracks and dislocations is found out, and the coupling of elastic and thermal energies is set up in fracturing and plastic flow, namely for opening mode cracks and edge dislocations. For porous materials saturated with a fluid or gas, the notion of binary continuum is used to introduce the corresponding invariant integrals. As applied to the horizontal drilling and hydrofracturing of boreholes in the Earth’ crust, the field of pressure and flow rate as well as the fluid output from both a horizontal borehole and a diskshape fracture issuing the borehole, are derived in the fluid extraction regime. A theory of fracking in shale gas/oil reservoirs is suggested for three basic regimes of the drill mud permeation into the multiply fractured rock region, with calculating the shape and volume of this region in terms of the geometry parameters and pressures of rock, drill mud and shale gas. Phe method of functional equations in the theory of a complex variable and the boundary layer method are also used to solve these problems.

Thermophoresis of aerosol particles in near-critical vapor: An inverse size effect

When subjected to a temperature gradient, the liquid film formed on a particle surface in near-critical vapor may undergo osmotic flow. Such a flow can induce a normal vapor flux onto (away from) the particle surface to compensate the loss (release the excess) of liquid as a result of condensation (evaporation) and produce significant thermophoretic mobility due to the extreme fluid compressibility during the phase change. Based on this principle, we have formulated this unique process and solved for the phoretic mobility by applying the boundary layer method. Unlike the classic mechanisms, this term of phase transition-induced phoretic mobility scales reversely with particle size, which becomes dominant for nanoparticles.

High-Frequency Asymptotics of a Solution to a Linear System with the Stokes Operator in the Principal Part

We consider a linear system of partial differential equations with the Stokes operator in the principal part and coefficients that rapidly oscillate in time. In the case where the limiting (averaged) stationary part of the system degenerates, we use the boundary layer method for constructing a complete formal asymptotic expansion of a time-periodic solution.

Self-consistent parabolized stability equation (PSE) method for compressible boundary layer

The parabolized stability equation (PSE) method has been proven to be a useful and convenient tool for the investigation of the stability and transition problems of boundary layers. However, in its original formulation, for nonlinear problems, the complex wave number of each Fourier mode is determined by the so-called phase-locked rule, which results in non-self-consistency in the wave numbers. In this paper, a modification is proposed to make it self-consistent. The main idea is that, instead of allowing wave numbers to be complex, all wave numbers are kept real, and the growth or decay of each mode is simply manifested in the growth or decay of the modulus of its shape function. The validity of the new formulation is illustrated by comparing the results with those from the corresponding direct numerical simulation (DNS) as applied to a problem of compressible boundary layer with Mach number 6.

Steady solution of an inverse problem in gravity-driven shear-thinning film flow: Reconstruction of an uneven bottom substrate

We consider a thin film of a power-law fluid flowing over an undulated substrate under the action of gravity. Instead of determining the free surface position as in the case of a direct problem, we focus on the inverse problem, where for a specific free surface shape, we find the corresponding bottom topography which causes the free surface profile. As an asymptotic approach for thin films and moderate Reynolds numbers, we apply the weighted-residual integral boundary-layer method (WRIBL) which enables us to derive a set of two evolution equations for the film thickness h and the flow rate q. We obtain the steady solutions of the above model equation for the inverse problem for weakly undulated free surface profile by a perturbation method. We examine the influence of viscosity of fluid, inertia, film thickness, hydrostatic pressure and surface tension on the reconstructed bottom topography for a shear-thinning fluid. For a moderately undulated free surface shape, we solve the model equation numerically and obtain the bottom topography. We perform spatial linear stability analysis of the corresponding direct problem using Floquet theory. The results show that the critical Reynolds number is influenced by the shear-thinning rheology, surface tension effects and the amplitude of the free surface of the target profile. The analysis provides a strategy for control of free surface instabilities that arise in gravity-driven shear-thinning films over inclined undulated substrates and it corresponds to reconstruction of bottom undulated substrate that causes a target free surface.

The Boundary Layer Approach to the Description of an Interference Head Wave

The boundary layer method is used for constructing a mathematical description of an interference head wave.

Study Based on the AIAA Aerodynamic Design Optimization Discussion Group Test Cases

Three model problems associated with aerodynamic drag minimizations are studied. These test cases have been proposed by the aerodynamic design optimization discussion group, and they include an inviscid NACA0012 nonlifting airfoil, a viscous RAE2822 lifting airfoil, and a viscous lifting wing based on the NASA Common Research Model. Various optimization methods are used, including MDOPT, TRANAIR, SYN83, and SYN107. The resulting designed and associated baseline geometries are cross analyzed by several computational fluid dynamics codes, including OVERFLOW, TRANAIR, GGNS, and FLO82. Pathological issues are unveiled in both of the simple airfoil model problems. Designed geometries for the inviscid symmetric test case exhibit strong tendencies to permit nonsymmetric flow solutions. Designed airfoils for the viscous lifting case also support nonunique solutions and hysteresis loops at or near the design point in Reynolds-averaged Navier–Stokes and integral boundary-layer method simulations. These results provide further evidence that single-point aerodynamic optimization is often ill posed. In extreme cases, it can yield designs with very undesirable aerodynamic characteristics, at least as analyzed by Reynolds-averaged Navier–Stokes and integral boundary-layer methods occurring at offdesign, and even ondesign, conditions. These examples are used to further document the multiple-solution and pseudosolution phenomena for steady-state Reynolds-averaged Navier–Stokes. This provides evidence that, even in practical engineering settings, numerical methods to assess stability and uniqueness of steady-state solutions and/or predict the bifurcations of these solutions have value. The single-point wing design problem is likewise ill posed in the spanwise direction. A multipoint design with a potentially large number of points or the inclusion of inequality constraints can regularize the problem.

Numerical Modeling of Heat Transfer and Flow Phenomena in an Axial Rotating Rotor Cascade

This paper presents a study of heat transfer in a rotating axial rotor. The study was done on the basis of computational fluid dynamics simulations and validated with an infrared thermocamera experimental setup. The influence of full turbulence vs. Menter's boundary layer method was studied in terms of heat transfer and flow phenomena. Additionally it was found out that a thin paint layer has a major influence on heat transfer phenomena and acts as insulation. For comparison of numerical results with experimental, a model of the actual rotor was established. Results were found to be in good agreement, so the effect of rotation on heat transfer was studied. Results are presented in terms of Nusselt number distribution and an empirical model of heat transfer.

Comparison between closed static chamber method and thin boundary layer method on monitoring air-water CO<sub>2</sub> diffusion flux

模型估算法与静态箱法是水-气界面气体通量监测的主要方法,因原理不同监测结果通常存在一定差异.目前对引起上述差异的主要环境因素仍不清晰.本研究使用自行设计的静态箱对三峡支流澎溪河水-气界面CO₂通量进行监测,并与同步开展的CO₂通量薄边界层模型估算法结果相比较,探讨该水域引起这两种监测方法结果产生差异的主要环境因素.结果表明,瞬时风速、水汽温差及水深均会对静态箱法及模型估算法的监测结果产生影响.风速越强、水汽温差越大、水深越大,这两种方法监测结果的差异就越小;而水域面积对两种方法的差异没有影响.比较发现,两种方法所获通量数据呈显著正相关,但静态箱法所获通量数据离散性显著高于薄边界层模型估算法.从方法的稳定性角度,在峡谷河道型水库水体温室气体监测中薄边界层模型估算法可能更为适宜.;Closed static chamber method and thin boundary layer method are frequently applied in monitoring air-water gas diffusive fluxes. However, there are differences in results between the methods due to theoretical principles which are still unclear. A type of closed static chamber was applied in Pengxi River, Three Gorges Reservoir to monitor air-water CO₂ flux and compared with the results from thin boundary layer method that simultaneously monitored to discuss the causes for the differences. Results indicated that wind speed, air-water temperature difference, and depth of the water were the impact factors related to the difference between the methods above. It was found that the smaller difference between the methods above was created by the higher wind speed, or the greater air-water temperature difference, or the greater water depth. However, water surface area showed no significant impact on the difference between the methods. Through a comparison of CO₂ fluxes data sets from the methods, it could be noted that there was a significant positive correlation on the monitoring results of the CO₂ fluxes.But monitoring results from the closed static chamber method had higher variance compared to thin boundary layer method.From the viewpoint of the stability of the methods, the thin boundary layer method might be more feasible in the river-channel based reservoir area.

Application of the boundary layer flow and heat transfer over a flat plate using collocation method

The aim of this article is to apply the collocation method for boundary layer in unbounded domain. The solution for velocity and temperature are computed by applying the collocation method. It does not need any perturbation, linearization, or small parameter versus homotopy perturbation method and parameter perturbation method. Also the determination of the auxiliary parameter and auxiliary function versus homotopy analysis method is not necessary. The use of a special technique to obtain solutions that are very close to the exact solution of the equation has been attempted. The idea is to transform the equations and boundary conditions into another set of variables. In comparison with previous studies, the solution shows that the results of the present method are in excellent agreement with those of the numerical methods. As an important result, it is depicted that approximation of the physical quantities f ′′(0) and θ′(0) are more accurate in comparison with those obtained using homotopy perturbation method. Also, the results reveal that this method is more suitable for solution of boundary layer problems with infinite boundary values.

Depth-averaged model for undular hydraulic jump

ABSTRACTAn undular hydraulic jump corresponds to the weak transition from super- to subcritical-flow in the form of steady free surface undulations. Previous models on undular hydraulic jumps employed the potential flow theory, i.e., the solitary and cnoidal wave theories. Experimental observations indicate the inadequacy of this theory, which motivated the development of more advanced approximations. Basic flow features including friction effects on the velocity profile, modelling of the bed-shear stress, and Reynolds stresses are considered. However, none of the models currently available include all these aspects. In this study, a general depth-averaged model is developed based on the k-ϵ turbulence closure. The general depth-averaged equations are applied to the undular jump problem, introducing a suitable time-averaged velocity distribution based on a composite power-law model, in which both streamline curvature and vorticity are accounted for. The bed-shear stress closure is included by a boundary layer method. Predictions of the depth-averaged Reynolds averaged Navier-Stokes (RANS) model are shown to be close to the 2D RANS solution and to experimental observations.