Approximate solution of the problem of the three-dimensional boundary layer in an incompressible fluid

A method of successive approximations is proposed for the solution of the equations of the three-dimensional incompressible boundary layer on bodies of arbitrary shape. A coordinate system connected with the streamlines of the external nonviscous flow is used. It is assumed that the velocity across the external streamlines is small. When the intensity of secondary flow is low the equations describing the boundary layer in an incompressible fluid are reduced to a form analogous to the equations for the boundary layer on axially symmetrical bodies. An approximate analytical solution is obtained for the velocity and for the friction in the form of equations which can be used for any problems of a three-dimensional incompressible boundary layer. The method developed was applied to the problem of the three-dimensional boundary layer at a plate with a cylindrical obstacle in the presence of a slip angle.     

Uniqueness of the self-similar solutions of three-dimensional laminar boundary-layer equations

The equations of the three-dimensional laminar boundary layer on lines of flow outflow and inflow are studied for conical outer flow under the assumption that the Prandtl number and the productρμ are constant. It is shown that in the case of a positive velocity gradient of the secondary flow (α₁>0) the additional conditions which result from the physical flow pattern determine a unique solution of the system of boundary-layer equations. For a negative velocity gradient of the secondary flow (α₁≤0) these conditions are satisfied by two solutions. An approximate solution is obtained for the boundary layer equations which is in rather good agreement with the numerical integration results. Compressible gas flow in a three-dimensional laminar boundary layer is described by a system of nonlinear differential equations whose solution is not unique for given boundary conditions. Therefore additional conditions resulting from the physical pattern of the gas flow are imposed on the resulting solution. In the solution of problems with a negative pressure gradient these additional conditions are sufficient for a unique selection of the solution of the boundary-layer equations. However, in the case of a positive pressure gradient the solution of the boundary-layer equations satisfying the boundary and additional conditions may not be unique. In particular, in [1] in a study of a three-dimensional laminar boundary layer in the vicinity of the stagnation point it was shown that for $$c = {{\frac{{\partial v_e }}{{\partial y}}} \mathord{\left/ {\vphantom {{\frac{{\partial v_e }}{{\partial y}}} {\frac{{\partial u_e }}{{\partial x}}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial u_e }}{{\partial x}}}} > 0$$ the solution is unique, while for c<0 there are two solutions. In the present paper we study the question of the uniqueness of the self-similar solution of the three-dimensional laminar boundary-layer equations on lines of flow outflow and inflow for a conical outer flow.     

Three-dimensional turbulent boundary layer on a body of complex shape

Flow and heat transfer problems associated with three-dimensional compressible gas flow past a body of complex shape at a small angle of attack are investigated on the basis of a finite-difference calculation. The results of a numerical solution of the equations of the three-dimensional turbulent boundary layer are presented. The effect of the leading parameters on three-dimensional flow development and heat transfer is analyzed. The characteristic flow regions in the boundary layer are found: lines of divergence and convergence on the surface, separation zones and flow interfaces. The location of the maximum values of the heat flux and friction on the surface is determined, the behavior of the limiting streamlines on the body is described, and the intensity of the secondary flows in the boundary layer is estimated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 25–35, September–October, 1986.     

Numerical Simulation of the 3D Laminar Viscous Flow on a Horizontal-axis Wind Turbine Blade

The aim of this paper is to describe the methodology followed in order to determine the viscous effects of a uniform wind on the blades of small horizontal-axis wind turbines that rotate at a constant angular speed. The numerical calculation of the development of the three-dimensional boundary layer on the surface of the blades is carried out under laminar conditions and considering flow rotation, airfoil curvature and blade twist effects. The adopted geometry for the twisted blades is given by cambered thin blade sections conformed by circular are airfoils with constant chords. The blade is working under stationary conditions at a given tip speed ratio, so that an extensive laminar boundary layer without flow separation is expected. The boundary layer growth is determined on a non-orthogonal curvilinear coordinate system related to the geometry of the blade surface. Since the thickness of the boundary layer grows from the leading edge of the blade and also from the tip to the blade root, a domain transformation is proposed in order to solve the discretized equations in a regular computational 3D domain. The non-linear system of partial differential coupled equations that governs the boundary layer development is numerically solved applying a finite difference technique using the Krause zig-zag scheme. The resulting coupled equations of motion are linearized, leading to a tridiagonal system of equations that is iteratively solved for the velocity components inside the viscous layer applying the Thomas algorithm, procedure that allows the subsequent numerical determination of the shear stress distribution on the blade surface.     

Theoretical and Numerical Investigation of Gas Lubrication Processes on the Basis of Aerodynamic Equations

The dimensionless parameters of the complete system of Navier-Stokes equations of a compressible gas are estimated with reference to a typical gas bearing. It is found that the three-dimensional compressible boundary layer equations should be used as the determining equations for describing gas lubrication processes. After introducing certain assumptions with respect to the dimensionless parameters in the determining equations, an equation for the pressure, the generalized Reynolds equation, is obtained.Use of the spectral method of analysis makes it possible to transform the generalized Reynolds equation into a system of ordinary differential equations. An analytic solution of the entire boundary value problem is obtained for a journal bearing with fairly small eccentricity. By comparing the numerical results obtained using both the solution of the generalized Reynolds equation and the traditional theory it is possible to estimate the effect of the inertia forces, dissipation processes, and heat transfer.     

NONLINEAR THREE-DIMENSIONAL ANALYSIS OF COMPOSITELAMINATED PLATES

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On Local Perturbations of a Three-Dimensional Boundary Layer

A high-Reynolds steady-state viscous incompressible fluid flow is investigated in the neighborhood of a small three-dimensional irregularity located on the smooth surface of a body and oriented almost along the skin-friction lines. The regime in which quasi-two-dimensional flow with a given pressure gradient is realized on the irregularity scale is studied in detail. The numerical solution of the corresponding boundary value problem for the boundary layer equations is obtained. It is shown that, as distinct from the plane case, this solution is unique.     

Stability of a compressible boundary layer

The problem of stability in a compressible boundary layer, as opposed to an incompressible layer, involves many parameters and requires consideration of three-dimensional perturbations. The transverse component of the velocity, the thermal regime at the wall, etc., take on great significance. Investigation of all aspects of this problem requires systematic calculations performed by electronic computers. There do exist a few calculations of stability of a compressible boundary layer with respect to three-dimensional disturbances for particular cases. It follows from those studies (see, for example, [1]) that consideration of three-dimensional perturbations and of the transverse component of the basic flow velocity is important. Many aspects of this problem remain uninvestigated. Aside from the sheer cumbersomeness of the problem, there exist purely mathematical difficulties connected with the presence of a small parameter with higher derivatives in the differential equations for the perturbations, which causes losses in accuracy of calculation. In this present study an algorithm will be developed for solution of the problem of stability of a compressible boundary layer relative to three-dimensional disturbances with consideration of the transverse component of the basic velocity. Calculations are performed for a boundary layer on a plane thermally insulating plate, and the effects of the transverse velocity component and the three-dimensionality of the perturbations on stability at various Mach numbers are demonstrated.     

Method of calculation of interaction of wall flows

A flow of viscous compressible fluid in the neighborhood of the line of interaction of wall flows is considered. A method of calculating the line of interaction and the direction of the self-induced secondary flow is developed. Papers [1–3] are devoted to the simulation of a separation flow with singularities in the neighborhood of singular lines and points, where boundary-layer equations are invalid. However, the theories of local separation used at present have mainly been developed only for two-dimensional problems, while the models of viscous-inviscid interaction have restrictions in application for turbulent flows with developed separation. The interaction of three-dimensional wall turbulent flows is considered below. It is assumed that the thickness of the boundary layers and the scales of the interaction zones are small in comparison with the characteristic dimension of the system, while the line of discontinuity of the solutions of the three-dimensional boundary layer equations is the same as the line of interaction of the wall flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 53–59, March–April, 1987.The author is grateful to G. Yu. Stepanov and V. N. Ershov for their interest in my work and their valuable remarks.     

Solution of a Three-Dimensional Elastic Problem for a Layer

A closed system of differential equations for stresses and boundary and integral equilibrium and compatibility conditions for the components of the stress tensor are derived in solving a three-dimensional elastic problem for an unbounded layer. These equations are proposed to integrate directly.     

Law of transverse sections for the three-dimensional boundary layer on a thin wing in a hypersonic stream

The investigation of three-dimensional flows in boundary layers is important to determine the aerodynamic characteristics of wings such as the heat fluxes and friction drag. However, the circumstance that interaction of the boundary layer and the wake with an inviscid stream can play a governing role for the formation of the flow diagram as a whole is more important. The three-dimensional flow on a thin delta wing in a hypersonic stream is investigated in this paper. An important singularity of hypersonic flow is the low value of the gas density in the boundary layer as compared with the density on its outer boundary. It is shown that in the general case when the pressure in the wing span direction varies mainly by an order, high transverse velocities originate because of the smallness of the density within the boundary layer. This circumstance permits expansion of the solution for smallspan wings in a series in an appropriate small parameter. The equations in each approximation depend on two variables, while the third—longitudinal—variable enters as a parameter. The zero approximation can be considered as the formulation of the law of transverse plane sections for a three-dimensional boundary layer. As a comparison with the exact solutions calculated for delta wings with power-law distributions of the wing thickness has shown, the first approximation yields a very good approximation. Furthermore, flow modes with a different direction of parabolicity on the whole wing, as well as zones in which interaction with the external stream should absolutely be taken into account, are found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 75–84, March–April, 1976.     

Singularity in the solution of the equations of a three-dimensional boundary layer due to the collision of wall jets

The problem of the propagation of a three-dimensional jet of viscid incompressible fluid flowing from a narrow curved slot into a fluid-filled space along a rigid plane is considered within the framework of the equations of a steady laminar boundary layer. A class of initial conditions at the slot outlet which generates in the jet a velocity field without secondary flows is identified. Within this class the boundaryvalue problem for the three-dimensional boundary layer can be divided into two problems of lower dimensionality: a dynamic and a kinematic problem. As a result of the analysis of the kinematic problem the general structure of the regions of existence and uniqueness of the solution is determined. An investigation of the dynamic problem shows that as the boundaries of the region of existence are approached a singularity characterized by an infinite increase in the thickness of the jet is formed in the solution of the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 75–81, July–August, 1991.     

Three-dimensional desingularized boundary integral methods for potential problems

The concept of desingularization in three-dimensional boundary integral computations is re-examined. The boundary integral equation is desingularized by moving the singular points away from the boundary and outside the problem domain. We show that the desingularization gives better solutions to several problems. As a result of desingularization, the surface integrals can be evaluated by simpler techniques, speeding up the computation. The effects of the desingularization distance on the solution and the condition of the resulting system of algebraic equations are studied for both direct and indirect versions of the boundary integral method. Computations show that a broad range of desingularization distances gives accurate solutions with significant savings in the computation time. The desingularization distance must be carefully linked to the mesh size to avoid problems with uniqueness and ill-conditioning. As an example, the desingularized indirect approach is tested on unsteady non-linear three-dimensional gravity waves generated by a moving submerged disturbance; minimal computational difficulties are encountered at the truncated boundary.     

Solution of the elastic boundary value problems for a layer with tunnel stress raisers

A novel procedure for solving three-dimensional problems for elastic layer weakened by through-thickness tunnel cracks has been developed and is presented in this paper. This procedure reduces the given boundary value problem to an infinite system of one-dimensional singular integral equations and is based on a system of homogeneous solutions for a layer. Integral representations of single- and double-layer potentials are used for metaharmonic and harmonic functions entering in the singular integral equations. These representations provide a continuous extendibility of the stress vector while allowing a jump in the displacement vector in the transition through the cut.Expanding the potential and biharmonic solutions in the Fourier series over the thickness coordinate yields the integral representations of the displacement vector and stress tensor. The problem of reducing a denumerable set of the integral equations of the given boundary value problem to one-to-one correspondence with the set of unknown densities appearing in the Fourier’s coefficient representations has been settled efficiently. Numerical investigations show a rapid convergence of the proposed reduction procedure as applied to the solution of the infinite system of one-dimensional integral equations. Numerical examples illustrate the proposed method and demonstrate its advantages.     

Effects of g-jitter and radiation on three-dimensional double diffusion stagnation point nanofluid flow

The unsteady double diffusion of the boundary layer with the nanofluid flow near a three-dimensional (3D) stagnation point body is studied under a microgravity environment. The effects of g-jitter and thermal radiation exist under the microgravity environment, where there is a gravitational field with fluctuations. The flow problem is mathematically formulated into a system of equations derived from the physical laws and principles under the no-slip boundary condition. With the semi-similar transformation technique, the dimensional system of equations is reduced into a dimensionless system of equations, where the dependent variables of the problem are lessened. A numerical solution for the flow problem derived from the system of dimensionless partial differential equations is obtained with the Keller box method, which is an implicit finite difference approach. The effects studied are analyzed in terms of the physical quantities of principle interest with the fluid behavior characteristics, the heat transfer properties, and the concentration distributions. The results show that the value of the curvature ratio parameter represents the geometrical shape of the boundary body, where the stagnation point is located. The increased modulation amplitude parameter produces a fluctuating behavior on all physical quantities studied, where the fluctuating range becomes smaller when the oscillation frequency increases. Moreover, the addition of Cu nanoparticles enhances the thermal conductivity of the heat flux, and the thermal radiation could increase the heat transfer properties.     

Mechanism of three-dimensional boundary-layer receptivity

Boundary-layer receptivity is always a hot issue in laminar-turbulent transition. Most actual laminar-turbulent transitions belong to three-dimensional flows. An infinite back-swept flat-plate boundary layer is a typical three-dimensional flow. Study of its receptivity is important both in theory and applications. In this paper, a freestream turbulence model is established. A modified fourth-order Runge-Kutta scheme is used for time marching, and compact finite difference schemes are used for space discretization. On these bases, whether unsteady cross-flow vortices can be excited in the three-dimensional boundary layer(the infinite back-swept flat-plate boundary layer) by free-stream turbulence is studied numerically. If so, effects of the level and the direction of free-stream turbulence on the three-dimensional boundary-layer receptivity are further studied. Differences of the three-dimensional boundary-layer receptivity are then discussed by considering the non-parallel effect, influence of the leading-edge stagnation point of the flat plate, and variation of the back-swept angle separately. Intensive studies on the three-dimensional boundary-layer receptivity will benefit the development of the hydrodynamic stability theory, and provide a theoretical basis for prediction and control of laminar-turbulent transition.     

Boundary-layer eigen solutions for multi-field coupled equations in the contact interface

The dissipative equilibrium dynamics studies the law of fluid motion under constraints in the contact interface of the coupling system. It needs to examine how con- straints act upon the fluid movement, while the fluid movement reacts to the constraint field. It also needs to examine the coupling fluid field and media within the contact in- terface, and to use the multi-scale analysis to solve the regular and singular perturbation problems in micro-phenomena of laboratories and macro-phenomena of nature. This pa- per describes the field affected by the gravity constraints. Applying the multi-scale anal- ysis to the complex Fourier harmonic analysis, scale changes, and the introduction of new parameters, the complex three-dimensional coupling dynamic equations are transformed into a boundary layer problem in the one-dimensional complex space. Asymptotic analy- sis is carried out for inter and outer solutions to the perturbation characteristic function of the boundary layer equations in multi-field coupling. Examples are given for disturbance analysis in the flow field, showing the turning point from the index oscillation solution to the algebraic solution. With further analysis and calculation on nonlinear eigenfunctions of the contact interface dynamic problems by the eigenvalue relation, an asymptotic per- turbation solution is obtained. Finally, a boundary layer solution to multi-field coupling problems in the contact interface is obtained by asymptotic estimates of eigenvalues for the G-N mode in the large flow limit. Characteristic parameters in the final form of the eigenvalue relation are key factors of the dissipative dynamics in the contact interface.     

NON-AXISYMMETRIC MIXED BOUNDARY VALUE PROBLEM FOR DYNAMIC INTERACTION BETWEEN SATURATED POROUS FOUNDATION AND RECTANGULAR PLATE$^{\bf 1)}$

在同一界面的不同区域具有多种边界条件, 称之为混合边界, 这是一个熟知的力学问题. 对这类问题进行精确分析时, 必须要进行混合边值问题的求解. 而对于一般的三维非轴对称情形, 混合边值问题的求解往往存在数学困难. 本文利用Hilbert定理和双重Fourier变换, 给出了一种求解三维非轴对称混合边值问题的解析方法, 利用该方法对具有混合透水边界的饱和多孔地基上矩形板的振动弯曲进行了解析研究(板与地基接触面为不透水边界, 其余为透水边界). 首先, 基于Kirchhoff理论和Biot多孔介质理论建立矩形板与饱和多孔地基的动力控制方程, 进行耦合求解. 针对板土接触面和非接触面的混合边值问题, 采用双重Fourier变换构造出两对二维对偶积分方程, 以接触应力和接触面孔隙压力为基本未知量, 用Jacobi正交多项式将未知量展开, 再利用Schmidt法对二维对偶积分方程完成求解, 最终推导出板土系统在动力作用下的位移和应力解析式. 通过将本文计算模型退化为单一弹性地基, 与已有研究结果进行对比, 验证了本文方法的正确性和有效性. 最后, 通过数值算例, 对饱和多孔地基上矩形板的动力响应及参数影响做出分析和讨论. 此外, 本文提出的解析法具有一般性, 可广泛应用于复杂接触问题和多场耦合问题的求解.     

Nonlinear dynamical stability analysis of the circular three-dimensional frame

The three-dimensional frame is simplified into flat plate by the method of quasiplate. The nonlinear relationships between the surface strain and the midst plane displacement are established. According to the thin plate nonlinear dynamical theory, the nonlinear dynamical equations of three-dimensional frame in the orthogonal coordinates system are obtained. Then the equations are translated into the axial symmetry nonlinear dynamical equations in the polar coordinates system. Some dimensionless quantities different from the plate of uniform thickness are introduced under the boundary conditions of fixed edges, then these fundamental equations are simplified with these dimensionless quantities. A cubic nonlinear vibration equation is obtained with the method of Galerkin. The stability and bifurcation of the circular three-dimensional frame are studied under the condition of without outer motivation. The contingent chaotic vibration of the three-dimensional frame is studied with the method of Melnikov. Some phase figures of contingent chaotic vibration are plotted with digital artificial method.