Concerning the existence of wall terms in a three-parameter model of turbulence

We show the existence of a wall term in the equation for «u₁u₂» in a three-parameter model of turbulence by analogy with the known terms in an “E-ε” model. We suggest a convenient system of differential equations for the boundary layer on a flat plate in fluid flow at small Reynolds numberse. We find the limiting calculated value of the turbulent Reynolds number at which wall terms must be taken into consideration.     

The mode III full-field solution in elastic materials with strain gradient effects

The near-tip asymptotic field and full-field solution are obtained for a mode III crack in an elastic material with strain gradient effects. The asymptotic analysis shows that, even though the near-tip field is governed by a single parameter B (similar to the mode III stress intensity factor), the near-tip field is very different from the classical K_III field; stresses have r ^-3/2 singularity near the crack tip, and are significantly larger than the classical K _III field within a zone of size l to the crack tip, where l is an intrinsic material length, depending on microstructures in the material. This high-order stress singularity, however, does not violate the boundness of strain energy around a crack tip. The parameter B of the near-tip asymptotic field has been determined for two anti-plane shear loadings: the remotely imposed classical K _III field, and the arbitrary shear stress tractions on crack faces. The mode III full-field solution is obtained analytically for an elastic material with strain gradient effects subjected to remotely imposed classical K _III field. It shows that the near-tip asymptotic field dominates within a zone of size 0.5 l to the crack tip, while strain gradient effects are clearly observed within 5l. It is also shown that the conventional way to evaluate the crack tip energy release rate would lead to an incorrect, infinite value. A new evaluation gives a finite crack tip energy release rate, and is identical to the J-integral. This revised version was published online in August 2006 with corrections to the Cover Date.     

Interface roughness effect on slow cyclic annular shear of granular materials

We experimentally investigate the mechanical behaviour in cyclic shear of a granular material near a solid wall in a pressure controlled annular shear cell. The use of a model system (glass beads and saw-tooth shaped solid surface) enables the study of the influence of the wall roughness. After an initial shakedown procedure ensuring reproducible results in subsequent tests, wall shear stress S, volumetric variation ΔV, and the displacement field of the sample bottom surface, are recorded as functions of wall displacement. A dimensionless roughness parameter R _n is shown to control the interface response. The local grain-level or mesoscale behaviour is directly correlated to the global one on the scale of the whole sample.     

Analysis of a turbulent boundary layer subjected to a strong adverse pressure gradient

We define two non-dimensional parameters $\text{Λ =}\text{τ}{}_{\mathrm{w}}\text{p}{}_{\mathrm{x}}\text{δ}$ and $\text{R}{}_{\mathrm{p}}\text{=}\text{U}{}_{\mathrm{p}}\text{δ}\text{ν}$ where τ_w is the wall stress, p_x(?0) is the pressure gradient to which the turbulent boundary layer (of thickness δ) is subjected, ν is the kinematic viscosity, $\text{U}{}_{\mathrm{p}}\text{= (}\text{νp}{}_{\mathrm{x}}\text{p}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$ is a characteristic velocity and p is the density. The limit corresponding to the strong adverse pressure gradient is formulated as Λ → 0, R_p → ∞, ΛR_p finite. Using appropriate inner and outer asympcotic expansions, both above a wall layer possibly scaling with τ_w and ν, it is found by an application of Millikan's argument that there is an inertial sublayer where the streamwise velocity distribution obeys a half-power law, whose slope depends on Λ, and intercept on ΛR_p. Indeed comparison with available experimental data shows the inner law to be well represented by $\text{u}\text{Up}\text{= (3.5 + 19Λ)(}\text{yU}{}_{\mathrm{p}}\text{ν}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+ 2.5ΛR}{}_{\mathrm{p}}$. The outer flow obeys a generalized defect law; use of constant eddy viscosity closure yields results in good agreement with experiment.     

Global existence and blow-up analysis for a nonlinear diffusion equation with inner absorption and boundary flux

This article deals with a nonlinear diffusion equation with inner absorption u _t ?=?(log^σ(1?+?u)u _x ) _x ???λ(1?+?u)log ^p (1?+?u), in ?₊?×?(0,?+∞), subject to a logarithmic boundary flux ? log^σ(1?+?u)u _x (0,?t)?=?(1?+?u)log ^q (1?+?u)(0,?t), t?∈?(0,?+∞). We establish the critical global existence curve and give the asymptotic behaviour close to the blow-up time.     

Determination of wall shear stress from velocity profile measurement in the outer part of the boundary layer

The shear stress on a wall is determined through a comparison of experimental and theoretical determinations of velocity distribution using a family of Thompson or Cole profiles.Notation C_f friction coefficient - u current velocity over the thickness of the boundary layer - u₁ velocity at the outer boundary of the boundary layer - _W wall shear stress - dynamic velocity - u^*=u/v_*; y^*=yv_*/; parameter in Coles' wake law - (u/u₁)_in, (u/v*)_in velocity distribution in the wall region [Eqs. (10), (11), (1)] Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 5, pp. 787–792, May, 1981.     

The problem of sharp notch in microstructured solids governed by dipolar gradient elasticity

In this paper, we deal with the asymptotic problem of a body of infinite extent with a notch (re-entrant corner) under remotely applied plane-strain or anti-plane shear loadings. The problem is formulated within the framework of the Toupin-Mindlin theory of dipolar gradient elasticity. This generalized continuum theory is appropriate to model the response of materials with microstructure. A linear version of the theory results by considering a linear isotropic expression for the strain-energy density that depends on strain- gradient terms, in addition to the standard strain terms appearing in classical elasticity. Through this formulation, a microstructural material constant c is introduced, in addition to the standard Lamé constants (λ, μ). The faces of the notch are considered to be traction-free and a boundary-layer approach is followed. The boundary value problem is attacked with the asymptotic Knein-Williams technique. Our analysis leads to an eigenvalue problem, which, along with the restriction of a bounded strain energy, provides the asymptotic fields. The cases of a crack and a half-space are analyzed in detail as limit cases of the general notch (infinite wedge) problem. The results show significant departure from the predictions of the standard fracture mechanics.     

Boundary-layer equations for a power-law shear-driven flow over a plane surface of non-Newtonian fluids

We study the boundary-layer similarity flow driven over a semi-infinite flat plate by a power-law shear with asymptotic velocity profile u _e (y) = βy ^α (y → ∞, β > 0), for fluids both Newtonian and non-Newtonian. Theoretical analysis is reported to derive a range of exponents α and amplitudes β for which similarity solutions exist. The shear stress parameter f′′(0) is determined as a function of α and β.     

The entrainment theorem for wall driven boundary layer flows

Boundary layer flows driven by permeable plane surfaces, stretching with power–law velocities are considered in the presence of an applied lateral mass flux. The relationship between the wall shear stress and the entrainment velocity (the transversal velocity at the outer edge of the boundary layer) as a function of the mass transfer parameter f _w is examined analytically by using the Merkin transformation method. It is shown that at the value of f _w where the wall shear stress vanishes, the entrainment velocity reaches a minimum or maximum value. This relationship between two characteristic quantities at the outer and inner edge of the boundary layer, respectively, is referred to as entrainment theorem. Its physical content is analyzed in the paper in some detail.     

Investigation of laminar-turbulent transition in supersonic boundary layers in an axisymmetric aerophysical flight complex and in a model in a wind tunnel in the presence of heat transfer and suction of air

Analysis is made of the problems associated with laminar-turbulent transition in wall boundary layers, as well as of scale effects observed in the investigation of laminar-turbulent transition in wind tunnels and laminarization of flow. Flight-performance data are given on the Reynolds number and on the gradient criterion of stability at the beginning of transition on the nose part of the Oblako aerophysical complex in the presence of heat transfer for the numbers Re _{L, ∞} ≤ 2 × 10⁷, M_∞ ≤ 2.0, and acceleration a ≤ 12g. Experimental data are given on laminarization of flow past a porous plate in a wind tunnel under the effect of suction for M_∞ = 2.5. The theory of Emmons turbulent spots is generalized to the flight conditions of flow past the nose part of the Oblako aerophysical flight complex in the presence of heat transfer and to the case of laminar-turbulent transition on a porous plate for M_∞ = 2.5 in the presence of suction of air.     

Algebraic model of turbulent boundary layer on a convex curvilinear surface

A simple algebraic model of turbulent boundary layer on convex curvilinear surfaces is suggested that is based on the generalization of the two-layer one-parameter algebraic model for a flat plate [ 1 ]. The model is tested in a wide range of variation of the curvature parameter (0.01 ≤ δ₀/R _w ≤ 0.09, where δ₀ is the thickness of the boundary layer at the initial cross section of the curvilinear region andR _w is the curvature radius of the surface), the results of which are indicative of a good agreement between the experimental and calculated data on the integral characteristics of the boundary layer, namely, the friction coefficientC _f , the displacement thickness δ^* and momentum thickness δ^**, and the form parameterH = δ^*/δ^**. Based on the comparison between the calculated and experimental data on the distribution of tangential turbulent stresses, a conclusion is made that the model predicts a much lower effect of the curvature on the suppression of turbulence in the outer region of boundary layers at a mild curvature of the surface (δ ₀ /R _w = 0.01) than in experiments. However, this difference has a tendency to decrease as the surface curvature increases. An analysis of the calculated and experimental velocity profiles plotted in the variables of the wall law leads to a conclusion that the generalized Townsend wall law is partially realized on a curvilinear surface.     

Failure analysis of multiphase flow corrosion–erosion with three-way injecting water pipe

Water injection is an essential component in a reaction effluent air cooler (REAC) system because its primary function is to dissolve the generated ammonium salt, which leads to deposition or blockage accidents. A damage incident in a three-way pipe made of carbon steel under the multiphase flow field was investigated. The failure analysis was performed by means of scanning electron microscope (SEM) inspection and computational fluid dynamics (CFD) simulation. CFD results show that a large velocity gradient exists near the area of 5–12d at the bottom of the main pipe. This gradient results in a region of low flow velocity, high wall shear stress, and high turbulent kinetic energy. The flow state becomes very chaotic, and the non-uniformity coefficient of velocity is high. The corrosive medium (NH₄Cl or H₂S) dissolving in water increases the causticity of fluid medium and aggravates the flow corrosion. The high risk area from the CFD simulation coincides with the breakage area of the three-way pipe on the spot. This failure incident is attributed to the flow corrosion–erosion.     

A variant of the heat balance integral method and a new proof of the exponentially fast asymptotic behavior of the solutions in heat conduction problems with absorption

We give a new and explicit estimate for the asymptotic behavior of the solutions of the problem u_t − u_xx + u₊^P = 0, x > 0, t> 0, with conditions u(0, t) = 1, t > 0 and u(x, 0) = U₀(x) ≥ 0, x > 0, for a class of functions U₀ and parameter 0 < p < 1. We use an approximate solution given by the heat balance integral method with the innovation property which fixes appropriately the asymptotic limit of the corresponding approximate free boundary.     

A novel boundary element approach for solving the anisotropic potential problems

This presentation is mainly devoted to the research on the regularization of indirect boundary integral equations (IBIEs) for anisotropic potential problems. Based on a new idea, a novel regularization technique is pursued, in which the regularized IBIEs excluding the CPV and HFP integrals are established. The proposed method has many advantages. First, it does not need to calculate multiple integral as the Galerkin method, so it is simple and easy for programming. Second, it can compute boundary quantities ?u/?x_i (i=1,2). Third, the anisotropic problems can be solved directly without transforming them into isotropic ones so that no inverse transform is required. Finally, the gradient BIEs are independent of the potential BIEs and they can provide variously useful equations. Numerical examples show that a better precision and high computational efficiency can be achieved by the present method.     

Momentum transfer on power law stretching plate with free stream pressure gradient

The similarity solution of laminar boundary layer driven by the stretching surface boundary and pressure gradient, each proportional to the same power law of the downstream coordinate, based on composite reference velocity (sum of the velocities of stretching boundary and free stream) has been formulated by single set of equations, containing two parameters: β measuring the stretch rate of the moving boundary, and ? the ratio of free stream velocity to composite reference velocity. The closed form exact and asymptotic solutions in special cases, approximate integral solution and general numerical solutions have been obtained. For β>0 and 0???1 solutions are unique. For β<0 solutions are dual for 0<?<?₀(β), unique solution for ?=0 and ?=?₀(β) and no solution exists ?>?₀(β).