Three-dimensional turbulent boundary layer on wind turbine blades

The momentum integral technique for predicting the boundary–layer growth in three–dimensional flow has been extended to include the entrainment equation as the closure model. Special attention has been devoted to those terms in the differential equations that change the boundary–layer structure from that of cvasi two–dimensional steady flow, at outboard locations, to a tree–dimensional flow pattern. It is concluded that the stall is delayed due to the boundary–layer reattachment at inboard sections in conjunction with the onset of a spanwise vortex like structure. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the existence of incompressible current–vortex sheets: study of a linearized free boundary value problem

We study the initial boundary value problem resulting from the linearization of the equations of ideal incompressible magnetohydrodynamics and the jump conditions on the hypersurface of tangential discontinuity (current–vortex sheet) about an unsteady piecewise smooth solution. Under some assumptions on the unperturbed flow, we prove an energy a priori estimate for the linearized problem. Since the so‐called loss of derivatives in the normal direction to the boundary takes place even for the constant coefficients linearized problem, for the variable coefficients problem and non‐planar current–vortex sheets the natural functional setting is provided by the anisotropic weighted Sobolev space W₂^1,σ. The result of this paper is a necessary step to prove the local in time existence of solutions of the original non‐linear free boundary value problem. The uniqueness of the regular solution of this problem follows already from the a priori estimate we obtain for the linearized problem. Copyright © 2005 John Wiley & Sons, Ltd.     

Stabilization of periodic Stokesian Hele–Shaw flows of ferrofluids

We consider an incompressible ferrofluid in a vertical Hele–Shaw cell and develop a proper analytic framework for the free interface and the velocity potential of the fluid in a periodic geometry. The flow is assumed to obey a non-Newtonian Darcy law. The forces influencing the fluid are gravity, surface tension and the response to a magnetic field induced by a current. In addition, the flow is stabilized at the lower boundary component by an external source b. We prove a well-posedness result for the flow near flat solutions. Moreover, we find conditions on the parameters and on the slope of b for the exponential stability and instability of flat interfaces. Furthermore, we identify values for the current's intensity ι where critical bifurcation of nontrivial finger-shaped solutions from the branch of trivial (flat) solutions takes place.     

Asymptotic and numerical solutions of three‐dimensional boundary‐layer flow past a moving wedge

We consider a laminar boundary‐layer flow of a viscous and incompressible fluid past a moving wedge in which the wedge is moving either in the direction of the mainstream flow or opposite to it. The mainstream flows outside the boundary layer are approximated by a power of the distance from the leading boundary layer. The variable pressure gradient is imposed on the boundary layer so that the system admits similarity solutions. The model is described using 3‐dimensional boundary‐layer equations that contains 2 physical parameters: pressure gradient (β) and shear‐to‐strain‐rate ratio parameter (α). Two methods are used: a linear asymptotic analysis in the neighborhood of the edge of the boundary layer and the Keller‐box numerical method for the full nonlinear system. The results show that the flow field is divided into near‐field region (mainly dominated by viscous forces) and far‐field region (mainstream flows); the velocity profiles form through an interaction between 2 regions. Also, all simulations show that the subsequent dynamics involving overshoot and undershoot of the solutions for varying parameter characterizing 3‐dimensional flows. The pressure gradient (favorable) has a tendency of decreasing the boundary‐layer thickness in which the velocity profiles are benign. The wall shear stresses increase unboundedly for increasing α when the wedge is moving in the x‐direction, while the case is different when it is moving in the y‐direction. Further, both analysis show that 3‐dimensional boundary‐layer solutions exist in the range −1<α<∞. These are some interesting results linked to an important class of boundary‐layer flows.     

Transport equation with boundary conditions of hysteresis type

This paper is an advanced extension of the work reported in (Nonlinear Anal. 2005; 63 :1467–1473). A transport equation that describes the propagation of a substance in a moving fluid or gas is considered. The equation contains the transient, convection, and diffusion terms. The problem is formulated in a bounded domain provided with an inlet and an outlet for the fluid or gas flow. The crucial point of the problem setting is a hysteresis‐type condition posed on an active part of the boundary. This condition reflects the nondecreasing accumulation with saturation of the transported substance at each point of the active boundary part. We prove the existence and uniqueness of solutions to this problem, study the regularity properties of solutions, and perform numerical simulations that clarify the behavior of the model. Comparing with the results of (Nonlinear Anal. 2005; 63 :1467–1473), the advancement of this work consists in accounting for the motion of the fluid or gas and posing inlet and outlet boundary conditions. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical Solution of Newtonian and Non-Newtonian Flows in Bypass

This paper deals with a numerical solution of laminar incompressible steady flows of Newtonian and non–Newtonian fluids through bypass of a restricted vessel. Blood flow is considered to be Newtonian in the case of vessels of large diameters as aorta. On the other hand, with decreasing diameter of a vessel the non–Newtonian behavior of blood can play a significant role. One could describe these problems using Navier–Stokes equations and continuity equation as a model. In the case of Newtonian fluids one considers constant viscosity compared to non–Newtonian fluids where viscosity varies and can depend on the tensor of deformation. In order to find numerical solution, the system of equations is completed using an artificial compressibility method. The space derivatives are discretised using a cell centered finite volume method and arising system of ordinary differential equations is solved using an explicit multistage Runge–Kutta method with given steady boundary conditions. The steady solution is achieved for time t→∞ and steady boundary conditions. The results can be used in the field of cardiovascular research. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence analysis and computational testing of the finite element discretization of the Navier–Stokes alpha model

This report performs a complete analysis of convergence and rates of convergence of finite element approximations of the Navier–Stokes‐α (NS‐α) regularization of the NSE, under a zero‐divergence constraint on the velocity, to the true solution of the NSE. Convergence of the discrete NS‐α approximate velocity to the true Navier–Stokes velocity is proved and rates of convergence derived, under no‐slip boundary conditions. Generalization of the results herein to periodic boundary conditions is evident. Two‐dimensional experiments are performed, verifying convergence and predicted rates of convergence. It is shown that the NS‐α‐FE solutions converge at the theoretical limit of O(h²) when choosing α = h, in the H¹ norm. Furthermore, in the case of flow over a step the NS‐α model is shown to resolve vortex separation in the recirculation zone. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Boundary value problems on manifolds with fibered boundary

We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah–Patodi–Singer problems in subspaces (it contains both as special cases). The boundary conditions in this theory are taken as elements of the C *‐algebra generated by pseudodifferential operators and families of pseudodifferential operators in the fibers. We prove the Fredholm property for elliptic boundary value problems and compute a topological obstruction (similar to Atiyah–Bott obstruction) to the existence of elliptic boundary conditions for a given elliptic operator. Geometric operators with trivial and nontrivial obstruction are given. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Low-Regularity Solutions of the Periodic Camassa–Holm Equation

We prove local existence and uniqueness of weak solutions of the Camassa–Holm equation with periodic boundary conditions in various spaces of low-regularity which include the periodic peakons. The proof uses the connection of the Camassa–Holm equation with the geodesic flow on the diffeomorphism group of the circle with respect to the L ² metric.     

Non‐homogeneous Navier–Stokes systems with order‐parameter‐dependent stresses

We consider the Navier–Stokes system with variable density and variable viscosity coupled to a transport equation for an order‐parameter c. Moreover, an extra stress depending on c and ?c, which describes surface tension like effects, is included in the Navier–Stokes system. Such a system arises, e.g. for certain models of granular flows and as a diffuse interface model for a two‐phase flow of viscous incompressible fluids. The so‐called density‐dependent Navier–Stokes system is also a special case of our system. We prove short‐time existence of strong solution in L^q‐Sobolev spaces with q>d. We consider the case of a bounded domain and an asymptotically flat layer with a combination of a Dirichlet boundary condition and a free surface boundary condition. The result is based on a maximal regularity result for the linearized system. Copyright © 2010 John Wiley & Sons, Ltd.     

The Trailing Edge Problem For Mixed Convection Flow Past a Horizontal Plate

The influence of buoyancy onto the boundary‐layer flow past a horizontal plate aligned parallel to a uniform free stream is characterized by the buoyancy parameter K = Gr/Re^5/2 where Gr and Re are the Grashof and Reynolds number, respectively. An asymptotiy analysis of the complete flow field including potential flow, boundary layer, wake and interaction region is given for small buoyancy parameters and large Reynolds numbers in the distinguished limit KRe^1/4 = O(1). (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Artificial boundary conditions for viscoelastic flows

The steady three-dimensional exterior flow of a viscoelastic non-Newtonian fluid is approximated by reducing the corresponding nonlinear elliptic–hyperbolic system to a bounded domain. On the truncation surface with a large radius R, nonlinear, local second-order artificial boundary conditions are constructed and a new concept of an artificial transport equation is introduced. Although the asymptotic structure of solutions at infinity is known, certain attributes cannot be found explicitly so that the artificial boundary conditions must be constructed with incomplete information on asymptotics. To show the existence of a solution to the approximation problem and to estimate the asymptotic precision, a general abstract scheme, adapted to the analysis of coupled systems of elliptic–hyperbolic type, is proposed. The error estimates, obtained in weighted Sobolev norms with arbitrarily large smoothness indices, prove an approximation of order O(R^−2+ε), with any ε>0. Our approach, in contrast to other papers on artificial boundary conditions, does not use the standard assumptions on compactly supported right-hand side f, leads, in particular, to pointwise estimates and provides error bounds with constants independent of both R and f. Copyright © 2007 John Wiley & Sons, Ltd.     

On the Nonlinear Development of Görtler Vortices in a Compressible Boundary Layer

The nonlinear development of the Görtler instability in compressible boundary layers on curved walls is considered for vortices of asymptotically large wavenumber. The starting point for our calculations lies in the work of Hall and Lakin (Proc. Roy. Soc. London Ser. A 415:421–444), where the incompressible results were formulated. Without neglecting downstream partial derivatives, the initial development of a vortex from the point where it first starts to grow is calculated. It is shown how the same basic structure that occurs in incompressible flow exists, where the disturbance is confined to a core region bounded above and below by thin shear layers, but that the flow in the core region is of more complicated form than that for incompressible flow.     

Viscous Flows in a Half Space Caused by Tangential Vibrations on Its Boundary

The paper is devoted to the studies of viscous flows caused by a vibrating boundary. The fluid domain is a half‐space, its boundary is a nondeformable plane that exhibits purely tangential vibrations. Such a simple geometrical setting allows us to study general boundary velocity fields and to obtain general results. From a practical viewpoint, such boundary conditions may be seen as the tangential vibrations of the material points of a stretchable plane membrane. In contrast to the classical boundary layer theory, we aim to build a global solution. To achieve this goal we employ the Vishik–Lyusternik approach, combined with two‐timing and averaging methods. Our main result is: we obtain a uniformly valid in the whole fluid domain approximation to the global solutions. This solution corresponds to general boundary conditions and to three different settings of the main small parameter. Our solution always include the inner part and outer part that both contain oscillating and non‐oscillating components. It is shown that the nonoscillating outer part of the solution is governed either by the full Navier–Stokes equations or the Stokes equations (both with the unit viscosity) and can be interpreted as a steady or unsteady streaming. In contrast to the existing theories of a steady streaming, our solutions do not contain any secular (infinitely growing with the inner normal coordinate) terms. The examples of the spatially periodic vibrations of the boundary and the angular torsional vibrations of an infinite rigid disc are considered. These examples are still brief and illustrative, while the core of the paper is devoted to the adaptation of the Vishik–Lyusternik method to the development of the general theory of vibrational boundary layers.     

Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions

Analytical solutions are provided for the two- and three-dimensional advection–diffusion equation with spatially variable velocity and diffusion coefficients. We assume that the velocity component is proportional to the distance and that the diffusion coefficient is proportional to the square of the corresponding velocity component. There is a simple transformation which reduces the spatially variable equation to a constant coefficient problem for which there are available a large number of known analytical solutions for general initial and boundary conditions. These solutions are also solutions to the spatially variable advection–diffusion equation. The special form of the spatial coefficients has practical relevance and for divergent free flow represent corner or straining flow. Unlike many other analytical solutions, we use the transformation to obtain solutions of the spatially variable coefficient advection–diffusion equation in two and three dimensions. The analytical solutions, which are simple to evaluate, can be used to validate numerical models for solving the advection–diffusion equation with spatially variable coefficients. For numerical schemes which cannot handle flow stagnation points, we provide analytical solution to the spatially variable coefficient advection–diffusion equation for two-dimensional corner flow which contains an impermeable flow boundary. The impermeable flow boundary coincides with a streamline along which the fluid velocity is finite but the concentration vanishes. This example is useful for validating numerical schemes designed to predict transport around a curved boundary.     

Ritz–Galerkin method for solving a parabolic equation with non‐local and time‐dependent boundary conditions

The paper is devoted to the investigation of a parabolic partial differential equation with non‐local and time‐dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz–Galerkin method, which is a first attempt at tackling parabolic equation with such non‐classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non‐local boundary condition, we use a trick of introducing the transition function G(x,t) to convert non‐local boundary to another non‐classical boundary, which can be handled with the Ritz–Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2015 John Wiley & Sons, Ltd.     

Deformation Driven Homogenization of Fracturing Solids

The paper discusses numerical formulations of the homogenization for solids with discrete crack development. We focus on multi–phase microstructures of heterogeneous materials, where fracture occurs in the form of debonding mechanisms as well as matrix cracking. The definition of overall properties critically depends on the developing discontinuities. To this end, we extend continuous formulations [1] to microstructures with discontinuities [2]. The basic underlying structure is a canonical variational formulation in the fully nonlinear range based on incremental energy minimization. We develop algorithms for numerical homogenization of fracturing solids in a deformation–driven context with non–trivial formulations of boundary conditions for (i) linear deformation and (ii) uniform tractions. The overall response of composite materials with fracturing microstructures are investigated. As a key result, we show the significance of the proposed non–trivial formulation of a traction–type boundary condition in the deformation–driven context. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Long Eddies in Sheared Flows

The characteristic feature of the wide variety of hydraulic shear flows analyzed in this study is that they all contain a critical level where some of the fluid is turned relative to the ambient flow. One example is the flow produced in a thin layer of fluid, contained between lateral boundaries, during the passage of a long eddy. The boundaries of the layer may be rigid, or flexible, or free; the fluid may be either compressible or incompressible. A further example is the flow produced when a shear layer separates from a rigid boundary producing a region of recirculating flow. The equations used in this study are those governing inviscid hydraulic shear flows. They are similar in form to the classical boundary layer equations with the viscous term omitted. The main result of the study is to show that when the hydraulic flow is steady and contained between lateral boundaries, the variation of vorticity ω(ψ) cannot be prescribed at any streamline which crosses the critical level. This variation is, in fact, determined by (1) the vorticity distribution at all streamlines which do not cross the critical level, by (2) the auxiliary conditions which must be satisfied at the boundaries of the fluid layer, and by (3) the dimensions of the region containing the turned flow. If at some instant the vorticity distribution is specified arbitrarily at all streamlines, generally the subsequent flow will be unsteady. In order to emphasize this point, a class of exact solutions describing unsteady hydraulic flows are derived. These are used to describe the flow produced by the passage of a long eddy which distorts as it is convected with the ambient flow. They are also used to describe the unsteady flow that is produced when a shear layer separates from a boundary. Examples are given both of flows in which the shear layer reattaches after separation and of flows in which the shear layer does not reattach. When the shear layer vorticity distribution has the form ωαyⁿ, where y is a distance measure across the layer, the steady flows are of Falkner-Skan type inside, and adjacent to, the separation region. The unsteady flows described in this paper are natural generalizations of these Falkner-Skan flows. One important result of the analysis is to show that if the unsteady flow inside the separation region is strongly sheared, then the boundary of the separation region moves upstream towards the point of separation, forming large transverse currents. Generally, the assumption of hydraulic flow becomes invalid in a finite time. On the other hand, if the flow inside the separation region is weakly sheared, this region is swept downstream and the flow becomes self-similar.