Corner Boundary Layer Research Articles

Review of research on streamwise corner boundary layer

This paper is a brief review of recent research on the streamwise corner boundary layer as it affects the component performance evaluation of both the theoretical and practical aircraft design. Typical examples include aircraft wing–body junction, rectangular air intakes, and turbine-hub flow. The paper addresses the questions of what we know and do not know about the streamwise corner boundary layer. Streamwise corner flows are characterized by the presence of secondary flows in the cross-stream planes, which are driven by the normal and secondary-shear components of the Reynolds stress tensor. Extensive studies of analysis for Prandtl's second kind of secondary flows have promoted the understanding of characteristics and formation of streamwise corner boundary layer. However, compared to the flat plate boundary layer, the research on the streamwise corner boundary layer is still far from enough, especially in the similarity solution, the instability, and transition mechanism. In recent years, a significant progress has been achieved in the study on the streamwise corner boundary layer in turbulent flow through direct numerical simulation and stress−ω Reynolds stress model.

Effect of wall temperature in streamwise supersonic corner flow

The interaction between corner flow and heat transfer is very common in the internal and external flow of hypersonic vehicle surface, but it is not clear how the wall thermal boundary condition affects the supersonic corner flow boundary layer. In the present study, the Reynolds stress model is used to solve the Reynolds-averaged Navier–Stokes equations. The results show that when the symmetrical wall temperature Tw is less than or equal to the recovery temperature Taw, the mean velocity profile in the corner zone is similar to the plate zone, and the temperature is higher in the corner zone. At heating wall condition of Tw>Taw, both velocity and temperature profile of outer corner boundary layer are different from that in the plate zone. The near-wall temperature distribution increases relatively, and the temperature profile is no longer similar to the law of the wall when Tw rises. Along the spanwise wall, the deviation of the velocity–temperature quadratic curve and the generalized Reynolds analogy (GRA) relation is within 10%. In the supersonic corner flow, the velocity–temperature relation under non-adiabatic wall condition satisfies the GRA relation. The effect of wall temperature on corner boundary layer velocity and temperature is significant. It is necessary to consider wall thermal boundary condition in the supersonic corner flow boundary layer analysis.

Natural transition of the supersonic streamwise corner flow

The laminar-to-turbulence transition of a streamwise corner flow is recognized to occur first at the corner based on the stability analysis, but there is no persuasive experimental evidence to back it up, especially for supersonic flow. In this work, natural transition in a supersonic corner boundary layer has been experimentally studied using a nanoparticle-based planar laser scattering technique. It is inspiring to observe that the natural transition position of the corner boundary layer shows a random behavior among the corner side, flat-plate side, and their combination. Based on an intermittent factor analysis, these stochastic transitions show a dominant preference for transitions occurring near the corner region.

АСИМПТОТИКА РЕШЕНИЯ ПЕРВОЙ КРАЕВОЙ ЗАДАЧИ ДЛЯ СИНГУЛЯРНО ВОЗМУЩЕННОГО ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ В ЧАСТНЫХ ПРОИЗВОДНЫХ ВТОРОГО ПОРЯДКА ПАРАБОЛИЧЕСКОГО ТИПА

The article constructs a complete uniform asymptotic expansion in a small parameter of the solution of the first boundary value problem. The first boundary value problem is posed for a singularly perturbed linear inhomogeneous second-order partial differential equation with two independent variables of parabolic type. The problem is investigated on a rectangle. The peculiarities of the problem are the presence of a small parameter in front of the heat conduction operator, the existence of corner boundary layers at the lower corners of the rectangle. It is required to construct a uniform asymptotic expansion of the solution of the first boundary value problem on a rectangle, with any degree of accuracy, as the small parameter tends to zero. The asymptotic expansion of the solution in terms of a small parameter is constructed by the Vishik–Lyusternik method. When solving the problem, we use: methods of integration of ordinary differential equations, the classical method of a small parameter, the Vishik–Lyusternik boundary function method, and the maximum principle. As usual, the problem is solved in two stages: in the first stage, a formal expansion of the solution of the first boundary value problem is constructed; and in the second stage, the remainder of the resulting expansion is estimated and this proves that the resulting expansion is indeed asymptotic over the entire rectangle. In the first stage, a formal asymptotic solution is sought in the form of a sum of six functions (solutions): an external solution defined on the entire rectangle; boundary layer solution in a small neighborhood of the lower side of the rectangle; two lateral boundary layer solutions in a small neighborhood of the lateral sides of the rectangle and two corner boundary layer solutions in the neighborhood of the lower vertices of the rectangle. All these boundary layer solutions exponentially decrease outside the boundary layers.

Corner Boundary Layer in Boundary Value Problems with Nonlinearities Having Stationary Points

Corner Boundary Layer in Boundary Value Problems with Nonlinearities Having Stationary Points

Optical diagnostics in a detonation-driven direct-connected circular combustor fueled with hydrogen for mach 10 scramjet

Various optical methods are applied to study flame characteristics of a circular-cross-section combustor for a Mach 10 scramjet. Experiments are performed on a direct-connected test facility fueled by hydrogen and driven using hydrogen/oxygen detonation. The circular-cross-section combustor is advantageous in solving thermal inhomogeneity, corner boundary layer and aircraft integration problems. However, it is difficult to use traditional optical measurement methods, which require large-sized measurement windows. In this study, tunable diode laser spectroscopy and a multi-view imaging system are used with small windows. High repetition rate measurements allow the effective time of facility, which is approximately 5 ms in this case, to be obtained. The dynamic flame characteristics are diagnosed upstream and downstream of the cavity with/without strut injection. When combined with proper orthogonal decomposition to obtain the flame pulsation state during the effective time period, the method is expected to be effective for engine performance evaluation and numerical simulation verification.

Strange wave formation and detonation onset in narrow channels

Experiments are conducted in a smooth 10 × 10 mm square cross-section, 1-m long channel, closed at the ignition end and open at the other end. Simultaneous two-direction schlieren visualization is used to investigate the three-dimensional dynamics of transition to detonation for a stoichiometric H2–O2 mixture. Results show the existence of two distinct structures before detonation onset: (i) asymmetric, composed of an oblique shock trailed by a flame, that runs preferentially along the wall, and seems to get ignited inside the boundary layer developed by the precursor shock; (ii) symmetric, referred to as strange wave in literature, propagating roughly at the speed of sound in combustion products. The combined effect of shock induced preheating and viscous heating near walls seem to be responsible for the formation of the complex flame-shock interactions observed. A simple thermodynamic analysis applied to the strange wave using the experimentally measured wave speed yield a pressure ratio of ~ 15 during its steady propagation; furthermore, an estimate of the total energy losses required to thermodynamically realize such propagation regime revealed that approximately half of the energy released by combustion should be dissipated (i.e. momentum and heat losses). Finally, simultaneous two-direction optical access allows to map the exact location of detonation onset, showing that 78 % of cases exploded in corners, highlighting the role of corner flows and boundary layers in transition to detonation at this scale.

Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Cubic Nonlinearities

Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Cubic Nonlinearities

Formation and turbulent breakdown of large-scale vortical structures behind an obstacle in a channel at moderate Reynolds numbers

This paper deals with experimental investigation and direct numerical simulation of three-dimensional separated laminar and transitional flows behind a semicircular spanwise rib on a bottom wall of a rectangular channel at Reynolds numbers of up to 480. Particular emphasis is given to the formation mechanism of quasiperiodic large-scale vortex clouds in the mixing layer behind the rib. Vortical structures near the channel axis are formed due to pairing of spiral vortices emerging close to the vertical walls when the corner boundary layers impact on the rib. The effect of the Reynolds number and normalized channel size on the spiraling motion, generation, and shedding of large-scale vortex clouds has been estimated.

Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Nonmonotonic Nonlinearities

Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Nonmonotonic Nonlinearities

Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Nonlinearities

A singularly perturbed parabolic equation $${{\varepsilon }^{2}}\left( {{{a}^{2}}\frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}} - \frac{{\partial u}}{{\partial t}}} \right) = F(u,x,t,\varepsilon )$$ is considered in a rectangle with the boundary conditions of the first kind. At the corner points of the rectangle, the monotonicity of the function $$F$$ with respect to the variable $$u$$ in the interval from the root of the degenerate equation to the boundary value is not required. The asymptotic approximation of the solution is constructed under the assumption that the principal term of the corner part exists. A complete asymptotic expansion of the solution as $$\varepsilon \to 0$$ is constructed, and its uniformity in a closed rectangle is proved.

Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Monotonic Nonlinearity

For a singularly perturbed parabolic equation $${{\epsilon }^{2}}\left( {{{a}^{2}}\frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}} - \frac{{\partial u}}{{\partial t}}} \right) = F(u,x,t,\epsilon )$$ in a rectangle, a problem with boundary conditions of the first kind is considered. It is assumed that, at the corner points of the rectangle, the function $$F$$ is cubic in the variable $$u$$ . A complete asymptotic expansion of the solution at $$\varepsilon \to 0$$ is constructed, and its uniformity in a closed rectangle is substantiated.

Singularly Perturbed System of Parabolic Equations in the Critical Case

We examine a system of singularly perturbed parabolic equations in the case where the small parameter is involved as a coefficient of both time and spatial derivatives and the spectrum of the limit operator has a multiple zero point. In such problems, corner boundary layers appear, which can be described by products of exponential and parabolic boundary-layer functions. Under the assumption that the limit operator is a simple-structure operator, we construct a regularized asymptotics of a solution, which, in addition to corner boundary-layer functions, contains exponential and parabolic boudary-layer functions. The construction of the asymptotics is based on the regularization method for singularly perturbed problems developed by S. A. Lomov and adapted to singularly perturbed parabolic equations with two viscous boundaries by A. S. Omuraliev.

Singularly Perturbed Parabolic Problems with Multidimensional Boundary Layers

The first boundary value problem for a multidimensional parabolic differential equation with a small parameter e multiplying all derivatives is studied. A complete (i.e., of any order with respect to the parameter) regularized asymptotics of the solution is constructed, which contains a multidimensional boundary layer function that is bounded for x = (x1, x2) = 0 and tends to zero as e → +0 for x ≠ 0. In addition, it contains corner boundary layer functions described by the product of a boundary layer function of the exponential type by a multidimensional parabolic boundary layer function.

Quantitative description of non-equilibrium turbulent phenomena in compressors

A description of non-equilibrium turbulence in compressors is introduced by using the equation of Lagrangian velocity gradient correlation. This description is consistent to other traditional definitions in homogeneous isotropic turbulence and can be easily extended to complex anisotropic flows in compressors. We then show by post-processing in a linear cascade, that the non-equilibrium phenomena are obvious in the regions of corner separation, wake and boundary layer. Theoretical predictions show good agreement in magnitude with numerical results. We also show that the non-equilibrium phenomena cannot be eliminated even when the flow is far downstream, which indicates the importance of non-equilibrium phenomena in compressors. The aim of the present contribution is to provide a method to quantitatively describe the non-equilibrium phenomena in compressors, and to inspire future improvement on turbulence models by considering these non-negligible non-equilibrium properties in the future.

A discontinuous Galerkin method with Lagrange multipliers for spatially-dependent advection–diffusion problems

A higher-accuracy discontinuous Galerkin method with Lagrange multipliers (DGLM) is presented for the solution of the advection–diffusion equation with a spatially varying advection field in the high Péclet number regime, where the classical polynomial finite element method (FEM) produces spurious oscillations in the solution at practical mesh resolutions. The proposed DGLM method is based on discontinuous polynomial shape functions that are attached to an element rather than its nodes. It overcomes the aforementioned spurious oscillation issue by enriching these functions with approximate free-space solutions of homogeneous equations derived from an asymptotic analysis of the governing partial differential equation inspired by Prandtl’s boundary layer theory. These enrichment functions are capable of resolving exponential, parabolic, and corner boundary layers at relatively coarse mesh resolutions. The proposed method enforces a weak continuity of the solution approximation across inter-element boundaries using polynomial Lagrange multipliers, which makes it a hybrid method. However, unlike other hybrid methods, it operates directly on the second-order form of the advection–diffusion equation and does not require any stabilization. Its intrinsic performance and its superiority over the higher-order polynomial FEM are demonstrated for several test problems at Péclet numbers ranging from one thousand to one billion.

Boundary layer mesh generation on arbitrary geometries

SummaryVolume boundary layer mesh generation on non‐smooth geometries with multiple normals is considered in this work. A new highlight is given to corner and ridge boundary layer mesh generation through the generalized Voronoi diagram in general, and the spherical Voronoi diagram in particular. This provides the keystone to allow for the first time boundary layer mesh generation at arbitrary corner configurations. The work proposed in is revisited and compared with the new approach. A detailed description of the new algorithm is provided. The spherical Voronoi diagram delivers geometrically the optimal normals as well as topologically the optimal connectivities, as far as normality is concerned. Numerical examples illustrate the accuracy and robustness of the method. This approach seems to handle arbitrary geometry boundary layer mesh generation, in theory as well as in practice. Copyright © 2017 John Wiley & Sons, Ltd.

Recent progresses in boundary layer theory

In this article,we review recent progresses in boundary layer analysis of somesingular perturbation problems.Using the techniques of differential geometry,an asymptotic expansion of reaction-diffusion or heat equations in a domain with curved boundaryis constructed and validated in some suitable functional spaces.In addition, we investigate the effect of curvatureas well as thatof an ill-prepared initial data.Concerning convection-diffusion equations, the asymptotic behavior of their solutionsis difficult and delicate to analyze because it largely depends on the characteristics of the corresponding limit problems, which are first order hyperbolic differential equations. Thus, the boundary layer analysis is performed on relatively simpler domains,typically intervals, rectangles, or circles.We consider also the interior transition layers at the turning point characteristics in an interval domain and classical (ordinary), characteristic (parabolic) and corner (elliptic) boundary layers in a rectangular domainusing the technique of correctors and the tools of functional analysis.The validity of our asymptotic expansions is also established in suitable spaces.

Experimental investigation on a high subsonic compressor cascade flow

With the aim of deepening the understanding of high-speed compressor cascade flow, this paper reports an experimental study on NACA-65 K48 compressor cascade with high subsonic inlet flow. With the increase of passage pressurizing ability, endwall boundary layer behavior is deteriorated, and the transition zone is extended from suction surface to the endwall as the adverse pressure gradient increases. Cross flow from endwall to midspan, mixing of corner boundary layer and the main stream, and reversal flow on the suction surface are caused by corner separation vortex structures. Passage vortex is the main corner separation vortex. During its movement downstream, the size grows bigger while the rotating direction changes, forming a limiting circle. With higher incidence, corner separation is further deteriorated, leading to higher flow loss. Meanwhile, corner separation structure, flow mixing characteristics and flow loss distribution vary a lot with the change of incidence. Compared with low aspect-ratio model, corner separation of high aspect-ratio model moves away from the endwall and is more sufficiently developed downstream the cascade. Results obtained present details of high-speed compressor cascade flow, which is rare in the relating research fields and is beneficial to mechanism analysis, aerodynamic optimization and flow control design.

Corner Boundary Layer in Nonlinear Elliptic Problems Containing Derivatives of First Order

In a rectangular domain the first boundary value problem is considered for a singularly perturbed elliptic equation e 2∆u − e α A(x, y) ∂u/∂y = F(u, x, y, e) with a nonlinear on u function F. The complete asymptotic solution expansion uniform in a closed rectangle is constructed for α > 1. If 0 < α < 1, the uniform asymptotic approximation is constructed in zero and first approximations. The features of the asymptotic behavior are noted in the case α = 1 .