Stress Boundary Layers Research Articles

A Boundary-layer Stress Analysis of Laminated Composite Beams via a Computational Asymptotic Method and Papkovich-Fadle Eigenvector

A Boundary-layer Stress Analysis of Laminated Composite Beams via a Computational Asymptotic Method and Papkovich-Fadle Eigenvector

Research on Mathematical Model of Shale Oil Reservoir Flow

There are complex pore structures in shale reservoirs, and the nonlinear flow characteristics of shale reservoirs are complicated by stress sensitivity and boundary layer effects. In this research, a new flow mathematical model was established based on capillary model, stress sensitivity and boundary layer effect. The model was verified by the experimental results of Jilin shale oil reservoir, and the influencing factors of flow characteristics were analyzed. The results show that the new shale oil flow model has good fitting and high reliability with the experimental results. The boundary layer effect and stress sensitivity are the main factors affecting the seepage characteristics. With the increase in the pressure gradient, the influence of the boundary layer effect on the flow is gradually enhanced, and the influence of the stress sensitivity on the flow is gradually weakened. At the same time, this study developed a method for the evaluation of effective displacement distance of shale oil reservoir with CO2 injection through the limited injection-production well spacing method, and obtained the nonlinear flow curve of displacement distance with reservoir permeability under different injection and production pressure difference. This research deeply studies the flow characteristics of shale oil reservoir and provides a theoretical basis for the enhanced oil recovery of shale oil with CO2 injection.

Numerical verification of sharp corner behavior for Giesekus and Phan-Thien–Tanner fluids

We verify numerically the theoretical stress singularities for two viscoelastic models that occur at sharp corners. The models considered are the Giesekus and Phan-Thien–Tanner (PTT), both of which are shear thinning and are able to capture realistic polymer behaviors. The theoretical asymptotic behavior of these two models at sharp corners has previously been found to involve an integrable solvent and polymer elastic stress singularity, along with narrow elastic stress boundary layers at the walls of the corner. We demonstrate here the validity of these theoretical results through numerical simulation of the classical contraction flow and analyzing the 270° corner. Numerical results are presented, verifying both the solvent and polymer stress singularities, as well as the dominant terms in the constitutive equations supporting the elastic boundary layer structures. For comparison at Weissenberg order one, we consider both the Cartesian stress formulation and the alternative natural stress formulation of the viscoelastic constitutive equations. Numerically, it is shown that the natural stress formulation gives increased accuracy and convergence behavior at the stress singularity and, moreover, encounters no upper Weissenberg number limitation in the global flow simulation for sufficiently large solvent viscosity fraction. The numerical simulations with the Cartesian stress formulation cannot reach such high Weissenberg numbers and run into convergence failure associated with the so-called high Weissenberg number problem.

Investigation of onset of velocity transition in free convection over an inclined flat plate by PIV

In this work, the experimental investigation of onset of velocity transition in free convection boundary layer over a 45° inclined plate has been carried out. Experiments were performed on a 150 mm long flat plate with uniform heat flux ranging from 100 W/m2 to 5000 W/m2. Particle image velocimetry (PIV) technique was used to determine the flow characteristics over an inclined plate. The impact of various hydrodynamic parameters such as shear stress and velocity boundary layer thickness for defining or identifying the onset of velocity transition has been discussed. Also, the influence of various values of the heat fluxes on the onset of transition has been examined. It has been observed that the initiation of transition is first manifested as an increase in the overall velocity boundary layer thickness. Therefore, the thickness of the velocity boundary layer has been used as a hydrodynamic measure to determine the streamwise distance from the leading edge where the transition begins. This distance decreases with an increase of heat flux. The critical Rayleigh number for the onset of the transition under various heating conditions has been reported. For example, the critical Rayleigh numbers for the beginning of transition at heat fluxes of 100 W/m2 and 5000 W/m2 are 6.07 × 108 and 2.73 × 109 respectively.

Two-Dimensional Finite Element Method for Modeling Edge Effects in Cross-Ply Laminates

A 2-dimensional finite element (FE) method, based on an approximate analytical solution, is developed to define accurately stress fields and boundary layer width in symmetrically laminated composites. A better insight into the edge effect and delamination failure of cross-ply laminates is attained through a physical explanation of the mathematical FE solution. While some other solutions failed, the present FE solution obtains accurate results using a limited number of elements, and therefore reduces computer storage and computation time. A perturbation technique is employed to derive shape functions for the free edge boundary region, which properly describes the physics of sudden exponential growth of stress field in the vicinity of the edge. An original and comprehensive FE computer program is developed, using these specific shape functions, special numerical integration, and mesh generation algorithms. Results for bidirectional laminates under uniaxial extension are presented and compared with finite difference, FE, and approximate analytical solution.
 HIGHLIGHTS
 
 Defining accurately stress field in free edge region is the main challenge
 Shape functions based on perturbation solution yield accurate results
 Boundary layer width is proportional to geometric ratio of laminates
 Finer mesh in boundary region is required for small geometric ratio laminates
 A huge number of finite elements are needed to determine narrow boundary width
 
 GRAPHICAL ABSTRACT

Eddy viscosity from bottom Ekman veering profiles

ADCP profiles near the sea bed, at a 130 m depth continental shelf, were used to estimate the mean veering of the velocity vectors using a complex correlation coefficient. An analytic expression for the veering vertical gradient (∂θ/∂z) as function of turbulent Ekman layer thickness (δE) was developed and, using constant eddy viscosity assumptions, sea bottom stress and additional turbulent boundary layer features were evaluated. Further statistics on the time series yields median values for δE of about 13 m, eddy viscosity Az = 4.2×10−3 m2s−1 and median bottom stress of 9.2×10−2 Pa, thus leading to high Reynolds numbers. Very good agreement is obtained between observations and bottom Ekman balance solutions according to determination coefficients assessed (R≥0.99).

High Weissenberg number boundary layer structures for UCM fluids

We describe three distinct stress boundary layer structures that can arise in the high Weissenberg number limit for the upper convected Maxwell (UCM) model. One is a single layer structure previously noted by M. Renardy, High Weissenberg number boundary layers for the upper convected Maxwell fluid J. Non-Newtonian Fluid Mech.68 (1997), 125–132. The other two are double layer structures. These latter two structures extend the core flows that can be accommodated by the UCM model in the high Weissenberg regime. The three structures taken together, represent the main dominant balances that occur for the UCM equations near solid boundaries. For each structure, the leading order equations are derived in each region together with particular exact solutions when available. Importantly, the matching conditions between respective regions for each structure are also derived and explained. These stress boundary layers can arise in order one Reynolds number flows and are independent of the velocity boundary layers that can arise in high Reynolds number flows.

Surface Stress and Atmospheric Boundary Layer Response to Mesoscale SST Structure in Coupled Simulations of the Northern California Current System

Abstract The influence of mesoscale sea surface temperature (SST) variations on wind stress and boundary layer winds is examined from coupled ocean–atmosphere numerical simulations and satellite observations of the northern California Current System. Model coupling coefficients relating the divergence and curl of wind stress and wind to downwind and crosswind SST gradients are generally smaller than observed values and vary by a factor of 2 depending on planetary boundary layer (PBL) scheme, with values larger for smoothed fields on the 0.25° observational grid than for unsmoothed fields on the 12-km model grid. Divergence coefficients are larger than curl coefficients on the 0.25° grid but not on the model grid, consistent with stronger scale dependence for the divergence response than for curl in a spatial cross-spectral analysis. Coupling coefficients for 10-m equivalent neutral stability winds are 30%–50% larger than those for 10-m wind, implying a correlated effect of surface-layer stability variations. Crosswind surface air temperature and SST gradients are more strongly coupled than downwind gradients, while the opposite is true for downwind and crosswind heat flux and SST gradients. Midlevel boundary layer wind coupling coefficients show a reversed response relative to the surface that is predicted by an analytical model; a predicted second reversal with height is not seen in the simulations. The relative values of coupling coefficients are consistent with previous results for the same PBL schemes in the Agulhas Return Current region, but their magnitudes are smaller, likely because of the effect of mean wind on perturbation heat fluxes.

Sediment flux based model of instantaneous sediment transport due to pure velocity-skewed oscillatory sheet flow with boundary layer stream

Velocity-skewed oscillatory sheet flow transport is studied by an approximate model for instantaneous sediment transport resulting from exponential profiles of sediment concentration and velocity over a mobile bed. The phase lag effect, and asymmetry in wave friction factor, oscillatory flow orbital amplitude, roughness height, bed shear stress and wave boundary layer thickness are modelled so that the net boundary layer stream of pure velocity-skewed oscillatory flow can be obtained. The proposed model is found consistent with the previous classical formulas, and qualitatively refines a series of exponents of velocity power function for the approximation of instantaneous sediment transport rate under different flow conditions. The sediment fluxes can be well predicted with accuracy as that of a two-phase numerical model. The net boundary layer stream and sediment transport rates are also well predicted, especially the negative net stream and rates with large phase residual. The phase residual is shown a useful but insufficient factor in the net sediment transport in pure velocity-skewed flows. In addition, the net boundary layer stream resulting from velocity-skewed wave boundary layer development difference between onshore and offshore flow durations is an essential factor as well.

Stresses of PTT, Giesekus, and Oldroyd-B fluids in a Newtonian velocity field near the stick-slip singularity

We characterise the stress singularity of the Oldroyd-B, Phan-Thien–Tanner (PTT), and Giesekus viscoelastic models in steady planar stick-slip flows. For both PTT and Giesekus models in the presence of a solvent viscosity, the asymptotics show that the velocity field is Newtonian dominated near to the singularity at the join of the stick and slip surfaces. Polymer stress boundary layers are present at both the stick and slip surfaces. By integrating along streamlines, we verify the polymer stress behavior of r−4/11 for PTT and r−5/16 for Giesekus, where r is the radial distance from the singularity. These asymptotic results for PTT and Giesekus do not hold in the limit of vanishing quadratic stress terms for Oldroyd-B. However, we can consider the Oldroyd-B model in the fixed kinematics of a prescribed Newtonian velocity field. In contrast to PTT and Giesekus, this is not the correct balance for the momentum equation but does allow insight into the behavior of the Oldroyd-B equations near the singularity. A three-region asymptotic structure is again apparent with now a polymer stress singularity of r−4/5. The high Weissenberg boundary layer equations are found to manifest themselves at the stick surface and are of thickness r3/2. At the slip surface, dominant balance between the upper convected stress and rate-of-strain terms gives a slip boundary layer of thickness r2. The solution of the slip boundary layer shows that the polymer stress is now singular along the slip surface. These results are supported through numerical integration along streamlines of the Oldroyd-B equations in a Newtonian velocity field. The Oldroyd-B model thus extends the point singularity at the join of the stick and slip surfaces to the whole of slip surface. As such, it does not have a physically meaningful solution in a Newtonian velocity field. We would expect a similar stress behavior for this model in the true viscoelastic velocity field.

A high-order perturbation solution for the steady sedimentation of a sphere in a viscoelastic fluid

The steady, creeping and isothermal sedimentation of a rigid sphere in an incompressible viscoelastic fluid which follows the exponential Phan–Thien and Tanner constitutive equation is studied analytically. The solution of the governing equations is expanded as a regular perturbation series for small values of the Deborah number, and the resulting sequence of two-dimensional partial differential equations is solved analytically up to eighth order. Although the domain is unbounded, the solution is able to resolve features of the flow that cannot be revealed by a low-order theoretical analysis, such as very fine flow structures and a stress boundary layer close to the surface of the sphere. The calculation of the drag force exerted on the sphere by the fluid was done by developing two formulas. The first is based on the flow-field close to the sphere and the second is based on the far flow-field. Both formulas produce the same analytical expressions verifying the correctness and consistency of the series solution. At small Deborah numbers, a decrease of the drag force is predicted (i.e., an increase of the sedimentation velocity), followed by a significant drag enhancement at higher Deborah numbers. Investigation of the solution for the existence of a negative wake close to the rear stagnation point did not reveal such a phenomenon when physical constraints on the solutions were posed. On the contrary, a negative wake is predicted only as an artifact due to the loss of positive definiteness of the conformation tensor. It is also demonstrated that the effect of viscoelasticity is maximized in a region around the sphere with a radius which is about ten times the radius of the sphere. Last, it is shown that the fluid disturbances due to the viscoelasticity of the matrix fluid decay very slowly with the distance from the sphere, depending on the magnitude of the Deborah number. Streamlines, extra-stress and pressure contours as well as the extension of the polymer molecules are also presented and discussed.

Boundary-layer flow and bed shear stress under a solitary wave: revision

AbstractWe address two shortcomings in the article by Liu, Park & Cowen (J. Fluid Mech., vol. 574, 2007, pp. 449–463), which gave a theoretical and experimental treatise of the bottom boundary-layer under a solitary wave.

Flow and Mass Transfer for an Unsteady Stagnation-Point Flow Over a Moving Wall Considering Blowing Effects

In this paper, the flow and mass transfer of a two-dimensional unsteady stagnation-point flow over a moving wall, considering the coupled blowing effect from mass transfer, is studied. Similarity equations are derived and solved in a closed form. The flow solution is an exact solution to the two-dimensional unsteady Navier–Stokes equations. An analytical solution of the boundary layer mass transfer equation is obtained together with the momentum solution. The examples demonstrate the significant impacts of the blowing effects on the flow and mass transfer characteristics. A higher blowing parameter results in a lower wall stress and thicker boundary layers with less mass transfer flux at the wall. The higher wall moving parameters produce higher mass transfer flux and blowing velocity. The Schmidt parameters generate a local maximum for the mass transfer flux and blowing velocity under given wall moving and blowing parameters.

Unsteady flows of a class of novel generalizations of the Navier–Stokes fluid

In this short paper we study the counterpart, within the context of a general class of fluids, of two famous unsteady flows originally studied by Stokes, within the context of Navier–Stokes fluid, namely Stokes’ first and second problems. The class of fluids considered, stress power-law fluids, are capable of stress thinning or stress thickening and can describe phenomena that the classical power-law fluids are incapable of modeling. Within the context of the problems considered, we are able to find solutions wherein stress boundary layers develop.

Boundary layer flow dynamics at a cold-water coral reef

The Tisler cold-water coral reef is a 2km long reef in the north-eastern Skagerrak, Norway. The reef is comprised principally of Lophelia pertusa at depths between 70 and 160m. Velocity shear and boundary layer shear stresses have been measured at Tisler Reef to quantify the effect of the reef structure on the benthic boundary layer (BBL) dynamics.Two different approaches to estimating the magnitude of the near seabed stress were employed: using a logarithmic (constant stress) boundary layer approach and direct Reynolds stress measurements. Resultant estimates of near seabed stresses using both methods were comparable.Using the logarithmic layer approach to estimate seabed stresses both inside and out of the reef structure demonstrated that, for any particular impinging flow strength, higher shear stresses were observed within the live coral region than in the dead coral rubble region with no live coral stands. Bottom shear stresses of up to 3.5Nm−2 were measured within the reef complex and 1.2Nm−2 in the rubble region outside the live reef. This difference is due to large roughness length scales inside the rough living coral area relative to the smaller scales in the coral rubble.Low frequency acoustic backscatter data, used as a proxy for relative suspended particulate matter concentrations, suggested that both local re-suspension and advection of suspended material most likely occur at, and through, the reef system. The high stresses measured inside the living reef may favour corals by increasing the degree of re-suspension for a given current speed and providing more particulates to the filter feeding polyps.

Matched Asymptotic Expansions of Unbalanced Adhesive Joints

AbstractA matched asymptotic expansion analysis is used to determine the dependence of shear stress boundary layer thickness on adhesive properties in unbalanced single-lap joints. A uniformly accurate expansion of shear stress, in a small and positive dimensionless parameter e, is shown to contain a pair of adhesive edge boundary layers and an outer zone where the stress is slowly varying between the two layers. An overlap constraint is also found, and if it can be satisfied for e≪1, then there is sufficient overlap and a single boundary layer of width the order of e1/2 at either adhesive end. The analytic results are presented in a generic format allowing their application and/or extension to similar problems. All results are numerically verified using a finite difference approximation.

Boundary layers for the upper convected Maxwell fluid

We discuss boundary layers arising in the high Weissenberg number limit of viscoelastic flows and the scalings that arise in analyzing them. Two quite distinct mechanisms for the formation of viscoelastic boundary layers exist: One mechanism is analogous to that of the Prandtl boundary layer and is linked to the enforcement of the no-slip boundary condition. The other mechanism has nothing to do with the no-slip condition, but is linked to long memory which manifests itself in different stresses near the wall versus some distance from the wall. In certain situations, these stress boundary layers may be embedded within Prandtl type boundary layers.

Free-edge stress analysis of general composite laminates under extension, torsion and bending

In this study, based on the reduced form of elasticity displacement field for a long laminate, an analytical method is established to exactly obtain the interlaminar stresses near the free edges of generally laminated composite plates subjects to extension, torsion, and bending. The constant parameters being in the displacement field, which describe the global deformation of a laminate, are appropriately calculated by using the improved first-order shear deformation theory. Reddy’s layerwise theory is subsequently employed for analytical and numerical examinations of the boundary layer stresses within arbitrary laminated composite plates. Various numerical results are developed for the interlaminar normal and shear stresses along the interfaces and through the thickness of laminates near the free edges. Finally the effects of end conditions of laminates and geometric parameters on the boundary-layer stress are studied.

Interlaminar stress analysis of general composite laminates

In this study, based on the reduced from of elasticity displacement field for a long laminate, an analytical method is established to exactly obtain the interlaminar stresses near the free edges of generally laminated composite plates under the extension and bending. The constant parameters, which describe the global deformation of a laminate, are properly computed by means of the improved first-order shear deformation theory. Reddy's layerwise theory is subsequently utilized for analytical and numerical examinations of the boundary layer stresses within arbitrary laminated composite plates. A variety of numerical results are obtained for the interlaminar normal and shear stresses along the interfaces and through the thickness of laminates near the free edges. Finally the effects of end conditions of laminates on the boundary-layer stress are examined.

Spherical inflation of a class of compressible elastic bodies

Using a special model that belongs to a new class of elastic bodies wherein the Cauchy–Green stretch is given in terms of the Cauchy stress and its invariants, within the context of the spherical inflation of a spherical annulus, we show that interesting phenomena like the development of “stress boundary layers” manifest themselves. We consider two cases of boundary value problems, one in which there is a cavity in a sphere and the other in which there is a rigid spherical inclusion in a sphere. We show that in the case of a rigid inclusion, it is possible for a pronounced “stress boundary” layer to develop, in that the values of the stresses within this boundary layer that is adjacent to a spherical inclusion are much larger than external to it. We also show that in the case of both the cavity and a rigid inclusion, the stress concentration is an order of magnitude higher than the increase in the deformation gradient, that is, the stress and the stretch do not scale in a similar manner. While the stress adjacent to a rigid inclusion can be 2500 times the applied radial stress, the maximum stretch, which occurs at the rigid inclusion is about 10. While the variation in the stresses are linear in thin walled annular regions, we find that in thick walled annular regions, the variation of the stresses is non-linear.