Structure of the flow of a viscous gas near the trailing edge of a flat plate

Solution of the complete system of Navier-Stokes equations forms the basis for a study of the nature of flow of a viscous heat-conducting gas in the neighborhood of a trailing edge of a flat plate. The problem was solved in accordance with a difference scheme of the third order of accuracy [1]. The calculation was carried out under the same conditions as the experiment of [2], in which a plate of finite dimensions (L = 12 cm) had supersonic M = 2, Re, = 1000 gas flow round it. In order to obtain a thickness of the boundary layer which was acceptable for the purpose of making the measurements (of the order of 2 cm), the unperturbed gas was slightly rarefied. In the study of such problems [3–7] it is necessary to use the complete system of Navier-Stokes equations, since in the immediate neighborhood of the trailing edge one of the important assumptions in the theory of the boundary layer, ²u/y^{2 2}u/x², does not hold. As a result the flow upstream near the trailing edge of the plate will depend on the flow immediately behind the edge, since the perturbations propagate both upstream and downstream in this case. The rarefaction of the gas creates additional difficulties in the formulation of the boundary conditions on the plate with flow round it when this problem is studied numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 27–30, March–April, 1987.     

Stochastic estimation of flow near the trailing edge of a NACA0012 airfoil

A stochastic estimation technique has been applied to simultaneously acquired data of velocity and surface pressure as a tool to identify the sources of wall-pressure fluctuations. The measurements have been done on a NACA0012 airfoil at a Reynolds number of Re _c  = 2 × 10⁵, based on the chord of the airfoil, where a separated laminar boundary layer was present. By performing simultaneous measurements of the surface pressure fluctuations and of the velocity field in the boundary layer and wake of the airfoil, the wall-pressure sources near the trailing edge (TE) have been studied. The mechanisms and flow structures associated with the generation of the surface pressure have been investigated. The “quasi-instantaneous” velocity field resulting from the application of the technique has led to a picture of the evolution in time of the convecting surface pressure generating flow structures and revealed information about the sources of the wall-pressure fluctuations, their nature and variability. These sources are closely related to those of the radiated noise from the TE of an airfoil and to the vibration issues encountered in ship hulls for example. The NACA0012 airfoil had a 30 cm chord and aspect ratio of 1.     

Flow in the neighborhood of the trailing edge of a plate in a transonic flow of viscous gas

An investigation is made into the influence of the Mach number and the viscosity on the flow in the neighborhood of the trailing edge of a plate. The Mach number is assumed to satisfy m_∞ ² ? 1 = 0(R^?l/5), which corresponds to the regime of transonic interaction. It is shown that if the Mach number is such that ¦M_∞ ² ? 1¦ > O(R^?1/5) the problem in the region of free interaction can be reduced by an appropriate transformation to the already known solutions for an incompressible fluid [5] and supersonic flow [7].     

A double vortex sheet model of the viscous flow near the trailing edge of a lifting aerofoil

Classical definitions of boundary layer mass and momentum flux deficiency thicknesses can lead to gross errors when applied to measurements near a trailing edge where the flow curvature in the free stream is appreciable. This paper presents a double vortex sheet model as a development from the single vortex sheet model of Helmholtz and others. Two bound vortex sheets define a potential function which can describe a flow with the same mass and momentum flux deficiencies as the viscous regions. The bound nature of these sheets allows the modelling of the integral properties of these regions while retaining the advantages of a potential flow. The application to the flow near the trailing edge of a lifting aerofoil is given     

Solution of the discretized incompressible Navier-Stokes equations with the GMRES method

We describe some experiences using interative solution methods of GMRES type to solve the discretized Navier-Stokes equations. The discretization combined with a pressure correction scheme leads to two different systems of equations: the momentum equations and the pressure equation. It appears that a fast solution method for the pressure equation is obtained by applying the recently proposed GMRESR method, or GMRES combined with a MILU preconditioner. The diagonally scaled momentum equations are solved by GMRES(m), a restarted version of GMRES.     

用混合网格求解三维可压雷诺平均Navier—Stokes方程

用混合网格求解了三维紊流 N-S方程。在物面附近采用三棱柱网格 ,其它区域则采用四面体网格。方程的求解采用 Jamson的有限体积法 ,紊流模型采用两层 Baldwin-Lomax代数紊流模型。数值算例表明 ,用混合网格求解三维紊流 Navier-Stokes是非常有效的。     

Turbulent flow of a fluid in the neighborhood of the trailing edge of a plate at zero angle of attack

The laminar flow regime of an incompressible fluid at the trailing edge of a plate was studied by Stewartson and Messiter [1, 2] by means of the method of matched asymptotic expansions. In. the present paper, this method is used to analyze the same problem, but in the case of turbulent flow in the boundary layer and the wake. A system of linear equations of elliptic type with variable coefficients is obtained for the averaged values of the flow parameters in the main part of the boundary layer and the wake that is responsible for the change in the displacement thickness. A solution of this system is constructed by the Fourier method in the case of a power law of the velocity in front of the interaction region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 17–23, November–December, 1983.     

Supersonic flow around the trailing edge of a thin profile

The method of mergeable asymptotic expansions has recently been used effectively in investigations devoted to the study of boundary layer interaction with an external inviscid flow at high subcritical Reynolds numbers Re. The asymptotic analysis permits obtaining a limit pattern of the flow around a solid as Re þ, and determining the similarity and quantitative regularity laws which are in good agreement with experimental results. Thus by using the method of mergeable asymptotic expansions it is shown in [1–4] that near sites with high local curvature of the body contour and flow separation and attachment points, an interaction domain appears that has a small length on the order of Re^-3/8. In this flow domain, which has a three-layer structure, the pressure distribution in a first approximation already depends on the change in boundary-layer displacement thickness, while the induced pressure gradient, in turn, influences the flow in the boundary layer. An analogous situation occurs in the neighborhood of the trailing edge of a flat plate where an interaction domain also appears [5, 6]. The flow in the neighborhood of the trailing edge of a flat plate around which a supersonic viscous gas flows was examined in [7]. Numerical results in this paper show that the friction stress on the plate surface remains positive everywhere in the interaction domain, and grows on approaching the trailing edge. The supersonic flow around the trailing edge of a flat plate at a small angle of attack was investigated in [8, 9], Supersonic flow of a viscous gas in the neighborhood of the trailing edge of a flat plate at zero angle of attack is examined in [10], but with different velocity values in the inviscid part of the flow on the upper and lower sides of the plate. The more general problem of the flow around the trailing edge of a profile with small relative thickness is investigated in this paper.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 36–42, May–June, 1981.     

用任意不规则网格求解N-S方程

引入辅助点法开发了新的通量近似计算方法,建立了采用任意不规则畸变网格作为控制体积的单元中心有限体积求解Navier—Stokes的方法。它以同位网格作为变量布置方式,压力一速度耦合采用SIMPLE方法。数值算例表明,该算法对高度不规则的畸变网格适应性强;其改进了传统算法在不规则网格下计算的困难,保证了模型在高扭曲度的网格下的整体计算精度不受网格拉伸畸变和剪切畸变的影响。     

Invariant solutions of the Navier-Stokes equations

We examine certain invariant solutions of the Navier-Stokes equations. We prove theorems concerning the existence of solutions of boundary-value problems of the corresponding S/H systems.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 56–64, November–December, 1972.In conclusion the author thanks V. V. Pukhnachev for a discussion of the results and for his advice.     

Contribution to the theory of thin-profile trailing edge separation

The conditions of nonsymmetric trailing edge flow with separation are investigated. Solutions of the equations for the interaction zone in the neighborhood of the trailing edge of a thin profile at an angle of attack of the order O(Re^–1/16) in the separated flow regime are constructed numerically. It is shown that for this zone a solution exists up to a certain angle of attack. In all the regimes the value of the friction on the upper surface at the very end of the trailing edge remains a positive quantity. The solution of the equations in the separated flow regimes is found to be nonunique. The flow over the leading edge is assumed to be unseparated, and the separation at the trailing edge, if present, is assumed to be localized in the interior of the boundary layer. The flow over a Kutta profile at zero angle of attack is taken as an example. In this case the satisfaction of the Chaplygin-Joukowsky condition at the trailing edge ensures smooth flow over both the trailing and leading edges.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 55–59, July–August, 1989.     

Three-dimensional wake dynamics of a blunt and divergent trailing edge airfoil

The wake dynamics of an airfoil with a blunt and divergent trailing edge is investigated experimentally at relatively high Reynolds. The near wake topology is examined versus different levels of free stream turbulence FST and angles of attack, while the downstream wake evolution is characterized at various levels of FST. The FST is found to have a significant effect on the shapes of turbulence profiles and on the downstream location where the flow reaches its quasi-asymptotic behavior. Streamwise vortices (ribs) corresponding to spanwise variations of turbulence quantities are identified in the near wake region. Simultaneous multi-point hot-wire measurements indicate that their spatial arrangement is similar to Williamson’s (Ann Rev Fluid Mech 29:477–539, 1996) mode B laminar wake flow topology. The results suggest that the statistical spanwise distribution of ribs is independent of FST effects and angle of attack as long as the vortex shedding Strouhal number remains approximately similar.     

Effect of trailing edge shape on hydrodynamic damping for a hydrofoil

Flow induced vibration on a hydrofoil may be significantly reduced with a slight modification of the trailing edge without alteration of the hydrodynamic performance. Particularly, the so called Donaldson trailing edge shape gave remarkable results and is being used in a variety of industrial applications. Nevertheless, the physics behind vibration reduction is still not understood. In the present study, we have investigated the hydrodynamic damping of a 2D hydrofoil with Donaldson trailing edge shape. The results are compared with the same hydrofoil with blunt trailing edge. The tests are carried out in EPFL high speed cavitation tunnel and two piezoelectric patches are used for the hydrofoil excitation in non-intrusive way. It was found that the hydrodynamic damping is significantly increased with the Donaldson cut. Besides, as the flow velocity is increased, the hydrodynamic damping is found to remain almost constant up to the hydrofoil resonance and then increases linearly, for both tested trailing edge shapes and for both first bending and torsion modes.     

Solution of multiple crack problem in a finite plate using coupled integral equations

This paper studies a numerical solution of multiple crack problem in a finite plate using coupled integral equations. After using the principle of superposition, the multiple crack problem in a finite plate can be converted into two problems: (a) the multiple crack problem in an infinite plate and (b) a usual boundary value problem for the finite plate. For the former problem, the Fredholm integral equation is used. For the latter problem, a BIE based on complex variable is suggested in which a Cauchy singular kernel exists. For the proposed BIE, after using the inverse matrix technique, the dependence of the traction at a domain point from the boundary tractions is formulated indirectly. This is a particular advantage of the present study. Several numerical examples are provided and the computed results for stress intensity factor and T-stress at crack tips are given.