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.     

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.     

Preconditioning of the Euler and Navier-Stokes equations in low-velocity flow simulation on unstructured grids

Low-velocity inviscid and viscous flows are simulated using the compressible Euler and Navier-Stokes equations with finite-volume discretizations on unstructured grids. Block preconditioning is used to speed up the convergence of the iterative process. The structure of the preconditioning matrix for schemes of various orders is discussed, and a method for taking into account boundary conditions is described. The capabilities of the approach are demonstrated by computing the low-velocity inviscid flow over an airfoil.     

特殊坐标系中特殊非牛顿流边界层方程的相似解

给出了在一个特殊坐标系中三阶流体的二维定常运动方程组．该坐标系中由无粘流体的势流确定，即以环绕任意物体的非粘性流动的流线为Ф-坐标，速度势线为ψ-坐标，构成正交曲线坐标系．结果表明，边界层方程与浸没在流体中的物体的形状无关．第一次近似假定第二梯度项与粘性项和第三梯度项相比，可以忽略不计．第二梯度项的存在，将防碍第三梯度流相似解的比例变换的导出．利用李群方法计算了边界层方程的无穷小生成元．将边界层方程组变换为常微分方程组．利用Runge-Kutta法结合打靶技术求解了该非线性微分方程组的数值解．     

Exact solutions of problems of the three-dimensional flow of ideal viscous incompressible fluids near cylindrical surfaces

Three-dimensional flows of an incompressible fluid, the parameters of which depend on two coordinates and time, are considered. The stream surfaces of such flows are cylindrical. The equations of continuity and the Navier-Stokes equations can be transformed to relations, one of which is the equation for the stream function the other is the integral of the equations relating the pressure and the stream function, and the third is a linear equation for the projection of the velocity vector onto the axis parallel to the generatrix of the cylindrical surfaces. The problems of modelling the flows are considered on the basis of the exact solutions of the Navier-Stokes equations and Euler's equations using examples. Relations for the distribution of the flow parameters in the channel created by hyperbolical cylinders are derived for the case of unsteady inviscid flow. The streamlines of these flows are situated on the side surfaces of the hyperbolical cylinders and intercept the generatrices of the cylinders at certain indirect angles. The flow around a circular cylinder and the flow of fluid inside an elliptic cylinder are considered in the case of steady inviscid flow. The streamlines on the circular cylinder are arranged transverse to the cylinder (the projection of the velocity vector onto the coordinate axis, parallel to the generatrix of the cylinder, is equal to zero). Far from the cylinder the streamlines are also situated on a cylindrical surfaces, but not transverse to the cylinder, making certain indirect angles with the generatrix. Viscous three-dimensional flows, possessing a certain symmetry, are considered. In the case of radial symmetry the streamlines are helical lines. The non-planar Couette flow between parallel moving planes is characterized by the fact that the velocity vectors, being situated in the same plane, are collinear, while the velocity vectors in parallel planes are not collinear. Relations for viscous steady three-dimensional flows, using well-known relations, obtained for the stream function of two-dimensional flows, are given.     

Local Interaction Theory for Laminar Transonic Flows in Slender Channels

Transonic flows through channels 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 viscous inviscid interaction problem for weakly three dimensional laminar flows is formulated for perfect gases under the requirement that the channel is sufficiently narrow so that the flow outside the viscous wall layers becomes planar in the leading order approximation. Representative solutions for subsonic as well as for supersonic flows disturbed by three dimensional surface mounted obstacles will be presented. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interactive computing methods for aeroelastic analysis of turbomachinery

An interaction viscous‐inviscid method for efficiently computing steady and unsteady viscous flows is presented. The inviscid domain is modeled using a finite element discretization of the full potential equation. The viscous region is modeled using a finite difference boundary layer technique. The two regions are simultaneously coupled using the transpiration approach. A time linearization technique is applied to this interactive method. For unsteady flows, the fluid is assumed to be composed of a mean or steady flow plus a harmonically varying small unsteady disturbance. Numerically exact nonreflecting boundary conditions are used for the far field conditions. Results for some steady and unsteady, laminar and turbulent flow problems are compared to linearized Navier‐Stokes or time‐marching boundary layer methods. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of unsteady flows with transient regimes

Self-oscillatory flows in aerodynamics and astrophysics are studied. The two-dimensional compressible gas equations are solved using the implicit Runge-Kutta scheme of the third order with respect to the inviscid terms and of the second order with respect to the viscous terms. An algebraic Cebeci-Smith turbulence model is used. Weakly unsteady and strongly unsteady flow regimes are observed. The former occur in a supersonic flow past a cylinder with a front projection and in the heliosphere. Such flows became stable when the turbulent diffusion is taken into account. The latter flows occur when a supersonic jet meets an obstacle and when such a jet penetrates a cavity. In these flows, the amplitude of oscillations slightly decreases when the turbulent diffusion is taken into account.     

On the Inviscid Limit Problem of the Vorticity Equations for Viscous Incompressible Flows in the Half‐Plane

We consider the Navier‐Stokes equations for viscous incompressible flows in the half‐plane under the no‐slip boundary condition. By using the vorticity formulation we prove the local‐in‐time convergence of the Navier‐Stokes flows to the Euler flows outside a boundary layer and to the Prandtl flows in the boundary layer in the inviscid limit when the initial vorticity is located away from the boundary. © 2014 Wiley Periodicals, Inc.     

A two-phase flow with a viscous and an inviscid fluid

We study the free boundary between a viscous and an inviscid fluid satisfying the Navier-Stokes and Euler equations respectively. Surface tension is incorporated. We read the equations as a free boundary problem for one viscous fluid with a nonlocal boundary force. We decompose the pressure distribution in the inviscid fluid into two contributions. A positivity result for the first, and a compactness property for the second contribution are dervied. We prove a short time existence theorem for the two-phase problem     

A Method for Determining Linear Solutions in Magnetohydrodynamics

Given any two-dimensional and incompressible flow describedby a set of linear partial differential equations, a methodis presented for determining the solution for the dependentvariables (velocity, pressure, etc.). The method is then usedto investigate the following magnetohydrodynamic flows. (a) The flow of an irrotational and inviscid fluid. (b) The flow of a viscous fluid. (c) The flow of an electrically conducting, inviscid fluid inthe presence of a magnetic field aligned with the flow at infinity. (d) The flow of an electrically conducting, viscous fluid inthe presence of a magnetic field having arbitrary direction.     

Vanishing viscous limits for the 2D lake equations with Navier boundary conditions

The vanishing viscosity limit is considered for the viscous lake equations with Navier friction boundary conditions. We prove that the inviscid limit satisfies the inviscid lake equations, and the results include flows generated by Lp initial vorticity with 1<p?∞.     

A first Integral of Navier­Stokes equations for two-dimensional flows

As wellknown, Bernoulli's equation is obtained as the first integral of Euler's equations in the absence of vorticity. Even in case of non-vanishing vorticity, a first integral from Euler's equations is obtained by using the so called Clebsch transformation [1] for inviscid flows. In contrast to this, a generalisation of this procedure towards viscous flows has not been established so far. In the present paper a first integral of Navier-Stokes equations is constructed in the case of two-dimensional flow by making use of an alternative representation of the fields in terms of complex coordinates and introducing a potential representation for the pressure. The associated boundary conditions are also considered. The first integral is a suitable tool for the development of new analytical methods and numerical codes in fluid dynamics. (© 2010 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)     

Flow past a porous approximate spherical shell

In this paper, the creeping flow of an incompressible viscous liquid past a porous approximate spherical shell is considered. The flow in the free fluid region outside the shell and in the cavity region of the shell is governed by the Navier–Stokes equation. The flow within the porous annulus region of the shell is governed by Darcy’s Law. The boundary conditions used at the interface are continuity of the normal velocity, continuity of the pressure and Beavers and Joseph slip condition. An exact solution for the problem is obtained. An expression for the drag on the porous approximate spherical shell is obtained. The drag experienced by the shell is evaluated numerically for several values of the parameters governing the flow.     

Flow past a porous approximate spherical shell

In this paper, the creeping flow of an incompressible viscous liquid past a porous approximate spherical shell is considered. The flow in the free fluid region outside the shell and in the cavity region of the shell is governed by the Navier–Stokes equation. The flow within the porous annulus region of the shell is governed by Darcy’s Law. The boundary conditions used at the interface are continuity of the normal velocity, continuity of the pressure and Beavers and Joseph slip condition. An exact solution for the problem is obtained. An expression for the drag on the porous approximate spherical shell is obtained. The drag experienced by the shell is evaluated numerically for several values of the parameters governing the flow.     

Viscous fluid flow at all speeds: analysis and numerical simulation

In [43] a finite volume method for reliable simulations of inviscid fluid flows at high as well as low Mach numbers based on a preconditioning technique proposed by Guillard and Viozat [14] is presented. In this paper we describe an extension of the numerical scheme for computing solutions of the Euler and Navier-Stokes equations. At first we show the high resolution properties, accuracy and robustness of the finite volume scheme in the context of a wide range of complicated transonic and supersonic test cases whereby both inviscid and viscous flow fields are considered. Thereafter, the validity of the method in the low Mach number regime is proven by means of an asymptotic analysis as well as numerical simulations. Whereas in [43] the asymptotic analysis of the scheme is focused on the behaviour of the continuous and discrete pressure distribution for inviscid low speed simulations we prove both the physical sensible discrete pressure field for viscous low Mach number flows and the divergence free condition of the discrete velocity field in the limit of a vanishing Mach number with respect to the simulation of inviscid fluid flow.     

Viscous fluid flow at all speeds: analysis and numerical simulation

In [43] a finite volume method for reliable simulations of inviscid fluid flows at high as well as low Mach numbers based on a preconditioning technique proposed by Guillard and Viozat [14] is presented. In this paper we describe an extension of the numerical scheme for computing solutions of the Euler and Navier-Stokes equations. At first we show the high resolution properties, accuracy and robustness of the finite volume scheme in the context of a wide range of complicated transonic and supersonic test cases whereby both inviscid and viscous flow fields are considered. Thereafter, the validity of the method in the low Mach number regime is proven by means of an asymptotic analysis as well as numerical simulations. Whereas in [43] the asymptotic analysis of the scheme is focused on the behaviour of the continuous and discrete pressure distribution for inviscid low speed simulations we prove both the physical sensible discrete pressure field for viscous low Mach number flows and the divergence free condition of the discrete velocity field in the limit of a vanishing Mach number with respect to the simulation of inviscid fluid flow.     

Weakly 3D Transonic Flows in Narrow Channels

Steady transonic flows through channels so slender that the classical boundary layer approach fails are considered. The resulting viscous inviscid interaction problem for weakly three-dimensional laminar flows is formulated for perfect gases under the requirement that the channel is sufficiently narrow so that the flow outside the viscous wall layers becomes two-dimensional in the leading order approximation. The behavior of the flow upstream of a surface mounted three-dimensional obstacle will be demonstrated. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

MOTIONS OF CURVES IN THE GALILEAN SPACE (G)3

In this article, we study the flows of curves in the Galilean 3-space and its equiform geometry without any constraints. We find that the Frenet equations and the intrinsic quantities of the inelastic flows of curves are independent of time. We show that the motion of curves in the Galilean 3-space and its equiform geometry are described by the inviscid and viscous Burgers' equations.