On the initial phase of the triple deck stage in marginally separated flows

The method of matched asymptotic expansions is used to investigate marginally separated boundary layer flows (laminar or alternatively transitional separation bubbles) at high Reynolds numbers. Typical examples include, among others, the flow past slender airfoils at small to moderate angels of attack and channel flows with suction. As is well-known, classical (hierarchical) boundary layer computations usually break down under the action of an adverse pressure gradient on the flow, a scenario associated with the appearance of the Goldstein separation singularity. If, however, the parameter controlling the strength of the pressure gradient (the angle of attack or the relative suction rate in the examples mentioned above) is adjusted accordingly, the application of a local viscous-inviscid interaction strategy is capable of describing localized boundary layer separation. Moreover, taking into account unsteady effects and flow control devices allows the investigation of the conditions leading to forced or self-sustained vortex generation and the subsequent evolution process culminating in bubble bursting. Within the asymptotic formulation of this stage bubble bursting is associated with the formation of finite time singularities in the solution of the underlying equations and a corresponding break down. The distinct blow-up structure gives rise to a fully non-linear triple deck interaction stage featuring shorter spatio-temporal scales characteristic of the successive vortex evolution process. The paper will focus on the numerical treatment of the initial phase of the latter stage. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic description of incipient separation bubble bursting

The appearance of short laminar separation bubbles in high Reynolds number (Re) wall bounded flows due to appropriate adverse pressure gradient conditions is usually associated with minor effects on global flow properties (e.g. lift force). However, localized reverse flow regions are known to react very sensitively to perturbations and in further consequence may trigger the laminar-turbulent transition process or even cause global separation. The present investigation of marginally separated boundary layer flows is based on an asymptotic approach Re → ∞. Special emphasis is placed on solutions of the corresponding model equations which blow up within finite time indicating the ejection of a vortical structure and the emergence of shorter spatio-temporal scales reminiscent of the early transition scenario (‘ bubble bursting’ ). Within the framework of marginal separation theory, an alternative adjoint operator method is used to formulate evolution equations governing the viscous-inviscid interaction process in leading and higher order correction required for the study of later stages of the flow development. Their blow up structure specifies the initial condition of and the match to the subsequent triple deck stage. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transonic viscous inviscid interactions in narrow channels

Unsteady as well as steady transonic flows through channels which are so narrow that the classical boundary layer approach fails are considered. As a consequence the properties of the inviscid core and the viscosity dominated boundary layer region can no longer be determined in subsequent steps but have to be calculated simultaneously. The resulting interaction problem for laminar flows is formulated for both perfect and dense gases under the requirement that the channel is sufficiently narrow so that the flow outside the viscous wall layers becomes one-dimensional in the leading order approximation. The latter allows an interpretation of the flow in the core region by means of the theory of one-dimensional transonic inviscid flow through a Laval nozzle while preserving the essential features of the interaction problem associated with the internal structure of pseudoshocks. The sensitivity of a separation bubble caused by a pseudoshock of sufficient strength to perturbations under the condition of choked flow will be demonstrated. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The asymptotics of blow-up in inviscid Boussinesq flow and related systems

We consider the approach to blow-up in two-dimensional inviscidflows with stagnation-point similitude, in particular a buoyancy-drivenflow resulting from a horizontally quadratic density variationin a horizontally unbounded slab. The blow-up, which is onlypossible because the flow has infinite energy, proceeds by intensificationof the vorticity and density gradient in a layer adjacent tothe upper boundary, while the remainder of the flow tends towardsirrotationality. The governing Boussinesq flow equations arefirst solved numerically, and the results suggest scalings whichare then used in an asymptotic analysis as 0, where is thetime remaining until blow-up. The structure of the asymptoticsolution, involving exponential orders as well as powers andlogarithms of the small parameter, is suggested by the analysisof a simpler related problem for which an exact solution isavailable. The expansion is uniformly valid across the upperboundary layer and the outer region, but there is a layer adjacentto the lower boundary where the flow remains dependent on theinitial conditions and is undetermined by the asymptotics.     

Self-similar solutions of the magnetohydrodynamic boundary layer system for a non-dilatable fluid

A rigorous mathematical analysis is given for a magnetohydrodynamic boundary layer problem, which arises in the study of self-similar solutions of the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting non-dilatable fluid (i.e., a Newtonian fluid or a pseudo-plastic one) along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. For this problem, only a normal solution has the physical meaning. The uniqueness, existence, and nonexistence results for normal solutions are established. Also the asymptotic behavior of the normal solution at the infinity is displayed. Received: January 10, 2007; revised: September 6, 2007, April 21, 2008     

On a singular parabolic p-biharmonic equation with logarithmic nonlinearity

This paper is concerned with the well-posedness and asymptotic behavior of Dirichlet initial boundary value problem for a singular parabolic p-biharmonic equation with logarithmic nonlinearity. We establish the local solvability by the technique of cut-off combining with the methods of Faedo–Galerkin approximation and multiplier. Meantime, by virtue of the family of potential wells, we use the technique of modified differential inequality and improved logarithmic Sobolev inequality to obtain the global solvability, infinite and finite time blow-up phenomena, and derive the upper bound of blow-up time as well as the estimate of blow-up rate. Furthermore, the results of blow-up with arbitrary initial energy and extinction phenomena are presented.     

On the similarity solutions of magnetohydrodynamic flows of power-law fluids over a stretching sheet

A rigorous mathematical analysis is given for a magnetohydrodynamics boundary layer problem, which arises in the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluid along a stretching flat sheet in the presence of an exterior magnetic field orthogonal to the flow. In the self-similar case, the problem is transformed into a third-order nonlinear ordinary differential equation with certain boundary conditions, which is proved to be equivalent to a singular initial value problem for an integro-differential equation of first order. With the aid of the singular initial value problem, the uniqueness and existence results for (generalized) normal solutions are established and some properties of these solutions are explored.     

一类非局部反应扩散方程组解的爆破性质

该文研究一类带非局部源项的反应扩散方程组. 作者证明了初值充分大时解在有限时刻爆破, 建立了爆破解的爆破速率估计以及边界层估计.     

Multiple absorption-related blow-up rates to a coupled heat system

This paper deals with asymptotic behavior of solutions to a heat system with absorptions and coupling positive multi-nonlinearities. It is known that although absorption mechanisms may affect such as blow-up criteria, blow-up time, and initial data required for blow-up solutions, they cannot change blow-up rates of solutions in general. It has been reported in the current literature that blow-up rates for scalar equations with absorptions are all absorption-independent. In a previous paper of the authors, four absorption-independent simultaneous blow-up rates were obtained already for the same problem under weak absorptions. The present paper will furthermore prove that if the absorptions are unbalanced in the model (i.e., the absorption is stronger for one component and weaker for another), then there are in addition eight possible absorption-related blow-up rates for the model, besides the four absorption-independent ones. This exposes a significant difference between scalar and coupled nonlinear parabolic equations with absorptions.     

Asymptotic properties of singular solutions in degenerate parabolic equation with boundary flux

ABSTRACT

This paper deals with blow-up and quenching solutions of degenerate parabolic problem involving m-Laplacian operator and nonlinear boundary flux. The blow-up and quenching criteria are classified under the conditions on the initial data but with less conditions on the relationship among the exponents, respectively. Moreover, asymptotic properties including singular rates, set and time estimates are determined for the blow-up solutions and the quenching solutions, respectively.     

具有非局部边界的退化抛物方程组的爆破解

本文研究了具有非局部边界条件和非局部源的退化抛物方程组的弱解问题.利用基于比较原理的上下解的方法,在权函数和初始条件的假设下,获得了该方程组问题的爆破临界指标.此外,还获得了同时爆破解趋于爆破时间时的渐近行为,推广了已有的结果.     

An Asymptotic-Numerical Method for a Class of Weakly Coupled System of Singularly Perturbed Convection-Diffusion Equations

A weakly coupled convection dominated system of m-equations is analyzed. A higher order accurate asymptotic-numerical method is presented. The solutions of convection dominated problem are known to exhibit multi-scale character. There exist narrow region across the boundary of the domain where the solution exhibit steep gradient. This region is termed as boundary layer region and the solution of problem is said to have a boundary layer. Outside of this region, the solution of system behaves smoothly. To capture this multi-scale nature given system is factorized into two explicit systems. The degenerate system of initial value problems (IVPs), obtained by setting ??=?0, corresponds to the smooth solution, which lies outside of boundary layers. For solution inside boundary layers, a system of boundary value problems (BVPs) is obtained using stretching transformation. Regardless of this simple factorization, solutions of these systems preserve the key features of the given coupled system. Runge–Kutta method is used to solve the degenerate system of IVPs, whereas the system of BVPs is solved analytically. Stability and consistency of the proposed method is established. A uniform convergence of higher order is obtained. Possible extension to differential difference equations are also brought to attention. A comparative study of the present method with some state of art existing numerical schemes is carried out by means of several test problems. The results so obtained demonstrate the effectiveness and potential of present approach.     

Method for solving the equations describing the interaction of a three-dimensional boundary layer with an outer inviscid flow

A new method is developed for solving the three-dimensional time-independent equations describing the interaction of a laminar boundary layer with an outer inviscid flow. The method also applies to the interaction of plane flows. By applying the method, the problem of the three-dimensional viscous supersonic gas flow over a roughness element (a hump and a cavity) is solved for the first time within the framework of the classical triple-deck theory. The asymptotic height of the roughness element corresponding to the nonseparated flow is determined, and separated flow patterns are constructed.     

Convex solutions of a general similarity boundary layer equation for power-law fluids

This paper is devoted to a general similarity boundary layer equation for power-law fluids, which includes many important similarity boundary layer problems such as the Falker-Skan equation and the magnetohydrodynamic boundary layer equation which arises in the study of self-similar solutions of the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluids along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. By a rigorous mathematical analysis, the uniqueness, existence and nonexistence results for convex solutions, normal convex solutions and generalized convex solutions to the general similarity boundary layer equation are established. Also the asymptotic behavior of the normal convex solutions at the infinity are displayed.     

Analytic and Experimental Studies of the Errors in Numerical Methods for the Valuation of Options

The value of a European option satisfies the Black-Scholes equation with appropriately specified final and boundary conditions.We transform the problem to an initial boundary value problem in dimensionless form.There are two parameters in the coefficients of the resulting linear parabolic partial differential equation.For a range of values of these parameters,the solution of the problem has a boundary or an initial layer.The initial function has a discontinuity in the first-order derivative,which leads to the appearance of an interior layer.We construct analytically the asymptotic solution of the equation in a finite domain.Based on the asymptotic solution we can determine the size of the artificial boundary such that the required solution in a finite domain in x and at the final time is not affected by the boundary.Also,we study computationally the behaviour in the maximum norm of the errors in numerical solutions in cases such that one of the parameters varies from finite (or pretty large) to small values,while the other parameter is fixed and takes either finite (or pretty large) or small values. Crank-Nicolson explicit and implicit schemes using centered or upwind approximations to the derivative are studied.We present numerical computations,which determine experimentally the parameter-uniform rates of convergence.We note that this rate is rather weak,due probably to mixed sources of error such as initial and boundary layers and the discontinuity in the derivative of the solution.     

Transonic viscous inviscid interactions in a slender Laval nozzle

Transonic, high Reynolds number flows through a Laval nozzle, which is so narrow that the classical boundary layer correction can no longer be considered to be an effect of higher order, are considered. As a consequence the properties of the inviscid core and the viscosity dominated boundary layer region can no longer be determined in subsequent steps but have to be calculated simultaneously. The resulting interaction problem for laminar flows in a small nozzle is presented for perfect gases. Representative solutions including the internal structure of pseudo-shocks forming in the diffuser part of the nozzle and being strongly associated with the chocking phenomenon will be presented. The linear stability of the various flow regimes observed in the nozzle will be discussed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A boundary layer problem arising in gravity-driven laminar film flow of power-law fluids along vertical walls

A rigorous mathematical analysis is given for a boundary layer problem for a third-order nonlinear ordinary differential equation, which arises in gravity-driven laminar film flow of power-law fluids along vertical walls. Firstly, the problem is transformed into a singular nonlinear two-point boundary value problem of second order. Next, the latter is proved to have a unique positive solution, for which some estimates are established. Finally, the result above-mentioned is turned over to the original problem. The conclusion of this paper is that the boundary layer problem has a unique normal solution if the power-law index is less than or equal to one and a generalized normal solution if the power-law index is greater than one. Also the asymptotic behavior of the normal solution at the infinity is displayed.The work was supported by NNSF of China.