Effect of moving wall on the three-dimensional separation of an incompressible fluid in front of an obstacle

The problem of the laminar boundary layer formed on the surface of a semiinfinite plate with a perpendicular semi-infinite circular cylinder in a uniform steady incompressible flow normal to the leading edge is considered. Near its sharp edge the plate has a stationary part and, located at a finite distance further downstream, a part of the surface moving downstream at a constant velocity. The first-order boundary layer equations are solved numerically by an implicit finite-difference method. The effect of the moving wall on the variation of the dimensions of the separation zone ahead of the obstacle over a broad range of the governing parameters and flow characteristics is investigated. The flow in the laminar boundary layer on the surface of a plate ahead of such an obstacle was calculated in [1, 2] without motion of the wall. Data on the structure of the separated flow are given in [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 49–53, November–December, 1990.     

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.     

The turbulent boundary layer of dissociated gas in the initial section of a tube

Many theoretical and experimental papers [1–4] have been devoted to investigating the turbulent boundary layer in the initial section of a channel. For the most part, however, the flow of an incompressible fluid with constant parameters is considered. There are many practical cases in which it is of interest to treat the development of the turbulent boundary layer of gas in the initial section of a pipe when conditions are strongly nonisothermal. A solution of a problem of this type, based on the theory of limit laws, is given in paper [1]. The present article extends this solution to the case of the flow of a high-enthalpy gas when the effect of gas dissociation on the turbulent boundary layer characteristics must be taken into account. We shall consider the flow of a mixture of i gases which is in a frozen state inside the boundary layer, and in an equilibrium state on its boundaries. Formulas are derived for the laws of friction and heat exchange, and a solution is given for the turbulent boundary layer equations in the initial section of the pipe when the wall temperature is constant and the gas flows at a subsonic velocity.Finally the authors are grateful to S. S. Kutateladze for discussing the paper.     

Time dependent shear stress and temperature distribution over an insulated flat plate moving at hypersonic speed

The laminar two-dimensional flow over a stepwise accelerated flat plate moving with hypersonic speed at zero angle of attack is analysed. The governing equations in the self-similar form are linearized and solved numerically for small times. The solutions obtained are the deviations of the velocity and the temperature profiles from those of steady state. The presented results may be used to find the first order boundary layer induced pressure on the plate.     

The effects of temperature-dependent fluid properties on free convective flow along a semi-infinite vertical plate by the presence of radiation

The effects of temperature-dependent density, viscosity and thermal conductivity on the free convective steady laminar boundary layer flow by the presence of radiation for large temperature differences, are studied. The fluid density and the thermal conductivity are assumed to vary linearly with temperature. The fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature. The usual Boussinesq approximation is neglected due to the large temperature difference between the plate and the fluid. The nonlinear boundary layer equations, governing the problem under consideration, are solved numerically by applying an efficient numerical technique based on the shooting method. The effects of the density/temperature parameter n, the thermal conductivity parameter , the viscosity/temperature parameter _r and the radiation parameter F are examined on the velocity and temperature fields as well as the coefficient of heat flux and the shearing stress at the plate.     

Universal equations of three-dimensional boundary layer theory

We consider a parametric method for investigating three-dimensional laminar motion of an incompressible fluid in a boundary layer on a curved surface. It is found that the problem solution in the general case depends on four series of parameters, constructed from two components of the outer flow velocity and the two Lamé coefficients characterizing the shape of the immersed surface. From the general equations of the three-dimensional boundary layer we obtain a system of two universal equations which do not contain the characteristics of the outer flow. This system may be solved once and for all. As an example we consider the problem of the laminar boundary layer on the walls of an axisymmetric channel in the case of swirling outer flow. For this case we obtain numerical solutions of the system of universal equations in the local two-parameter approximation.     

An unsteady-state thermal boundary layer on a flat isothermal plate

The Kármán-Polhausen integral method is used to investigate the problem of an unsteady-state thermal boundary layer on an isothermal plate with a stepwise change in the conditions of flow around the plate; analytical expressions are obtained for the thickness of the thermal boundary layer. A dependence is found for the rate of movement of the boundary between the steady-state and unsteady-state regions of the solution on the Prandtl number. A similar problem was solved in [1, 2] for a dynamic layer, Goodman [3] discusses the more partial problem of an unsteady-state thermal boundary layer under steady-state flow conditions. Rozenshtok [4] considers the problem in an adequate statement but, unfortunately, he permitted errors of principle to enter into the writing of the system of characteristic equations; this led to absolutely invalid results. In an evaluation of the advantages and shortcomings of the integral method under consideration, given in [4], it must only be added that the method is applicable to problems in which the initial conditions differ from zero since, in this case, approximation of the velocity and temperature profiles by polynomials is not admissible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 64–69, July–August, 1970.     

Diffusion and Electrical Processes in the Turbulent Boundary Layer and in the Neighborhood of the Stagnation Point

The problem of electric current (engine current) formation in aircraft jet engine ducts as a result of the development of electrical diffusion boundary layers on the surfaces of the duct and internal engine components is investigated. It is assumed that the outer flow containing electrons and positive ions is quasi-neutral and that the electrical quasi-neutrality is violated (and the electric engine current develops) in the wall flow zone as a result of the difference between the electron and ion diffusion coefficients. The problem of the development of an electrical diffusion boundary layer inside the turbulent gasdynamic boundary layer on a plane surface is formulated and solved. The engine current distribution along the duct is found for various values of a turbulent viscosity on the boundary of the gasdynamic boundary layer which affect the laminar-turbulent transition point.The electrical diffusion processes that occurs when an electrically quasi-neutral hydrodynamic stream impinges on a plane surface (simulation of the flow in the neighborhood of a stagnation point) is studied. In this case the Navier-Stokes equations have a self-similar solution. It is shown that the system of electrohydrodynamic equations also has a self-similar solution. The electrical parameter fields are determined and the engine current is found on the basis of this solution.     

Hall effects on the magnetohydrodynamic shear flow past an infinite porous flat plate subjected to uniform suction or blowing

An analysis is made of Hall effects on the steady shear flow of a viscous incompressible electrically conducting fluid past an infinite porous plate in the presence of a uniform transverse magnetic field. It is shown that for suction at the plate, steady shear flow solution exists only when S²<Q, where S and Q are the suction and magnetic parameters, respectively. The primary flow velocity decreases with increase in Hall parameter m. But the cross-flow velocity first increases and then decreases with increase in m. Similar results are obtained for variation of the induced magnetic field with m. It is further found that for blowing at the plate, steady shear flow solution exists only when , where S₁ is the blowing parameter.     

On automodel numerical and asymptotic solutions of boundary-layer equations for high rates of injection

Automodel solutions of the equations of a laminar, multicomponent, isothermal boundary layer are considered for high rates of injection. The asymptotic velocity profiles and the thickness of the boundary layer are given for various negative pressure gradients (>0), A numerical solution is presented for the boundary-layer equations when injection involves the flow of a gas mixture comprising hydrogen, nitrogen, and carbon dioxide around the surface. The asymptotic solution is compared with the numerical solution, and its ranges of applicability are established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 49–52, May–June, 1971.     

Flow of an oldroyd fluid in a circular pipe with time dependent pressure gradient

The problem of pulsating flow superimposed on the steady laminar flow in a circular tube is investigated for the fluid characterized by the Oldroyd's constitutive equations. The governing equations are solved in an exact manner and the solution is obtained in terms of two non-dimensional fluid parameters. Several interesting illustrations are provided comparing the behaviour of Newtonian fluid and Oldroyd fluids regarding the velocity field, sectional mean velocity, surface friction and balance of force. The flow for small and large frequencies of vibration are obtained as special cases. For Oldroyd fluids also the flow is basically parabolic for small frequencies while it possesses a boundary layer character at large frequencies. The solution for second order fluids and Maxwell fluids can be obtained by appropriately choosing the two fluid parameters.     

Fluid Flow Within Permeable Boundaries

A classification of mathematical models of laminar boundary layer suction is presented. The conditions of singular separation near the tip of a plate are analyzed and the optimal ejection rate distribution necessary for preventing this separation is found. It is shown that, by using inviscid-fluid ejection, it is possible to eliminate the pressure-gradient singularity, which is the cause of the singular separation, and the optimal ejection rate distribution near the edge necessary for this is found. With reference to an exact solution of the Navier-Stokes equations, the unseparated flow about a circle with ejection at the back holder is demonstrated. The flow structure in the presence of fluid injection into the boundary layer is discussed.     

Mixed convection of an incompressible viscous fluid in a porous medium past a hot vertical plate

This paper analyses steady two-dimensional mixed convection of an imcompressible viscous fluid in a porous medium past a hot vertical plate. Assuming Darcy-Brinkman model for the flow in a porous medium, the boundary layer equations are integrated numerically to obtain the non-similar solution for the velocity and temperature distribution for several values of the permeability and viscous dissipation parameters. It is shown that for a fixed value of Prandtl number Pr and dissipation parameter E, the skin-friction at the plate decreases with increase in the permeability parameter K₁. However for the same value or Pr and E, the heat transfer rate at the plate increases with increasing K₁. The dimensionlcss velocity and temperature functions in the flow are plotted for several values of E and K₁ with Pr = 0.73. It is also shown that for fixed values of K₁, and KPr, the skin-friction increases with increase in the dissipation parameter E.     

Analysis of a laminar boundary layer flow over a flat plate with injection or suction

An analysis is performed to study a laminar boundary layer flow over a porous flat plate with injection or suction imposed at the wall. The basic equations of this problem are reduced to a system of nonlinear ordinary differential equations by means of appropriate transformations. These equations are solved analytically by the optimal homotopy asymptotic method (OHAM), and the solutions are compared with the numerical solution (NS). The effect of uniform suction/injection on the heat transfer and velocity profile is discussed. A constant surface temperature in thermal boundary conditions is used for the horizontal flat plate.     

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.     

Spontaneous rotation in the exact solution of magnetohydrodynamic equations for flow between two stationary impermeable disks

Magnetohydrodynamic (MHD) flow of a viscous electrically conducting incompressible fluid between two stationary impermeable disks is considered. A homogeneous electric current density vector normal to the surface is specified on the upper disk, and the lower disk is nonconducting. The exact von Karman solution of the complete system of MHD equations is studied in which the axial velocity and the magnetic field depend only on the axial coordinate. The problem contains two dimensionless parameters: the electric current density on the upper plate Y and the Batchelor number (magnetic Prandtl number). It is assumed that there is no external source that produces an axial magnetic field. The problem is solved for a Batchelor number of 0–2. Fluid flow is caused by the electric current. It is shown that for small values of Y, the fluid velocity vector has only axial and radial components. The velocity of motion increases with increasing Y, and at a critical value of Y, there is a bifurcation of the new steady flow regime with fluid rotation, while the flow without rotation becomes unstable. A feature of the obtained new exact solution is the absence of an axial magnetic field necessary for the occurrence of an azimuthal component of the ponderomotive force, as is the case in the MHD dynamo. A new mechanism for the bifurcation of rotation in MHD flow is found.     

Melting heat transfer in an axisymmetric stagnation-point flow of the Jeffrey fluid

This investigation explores the characteristics of melting heat transfer in a boundary layer flow of the Jeffrey fluid near the stagnation point on a stretching sheet subject to an applied magnetic field. The governing boundary layer equations are transformed to ordinary differential equations by similarity transformations. Resulting nonlinear problems are solved analytically by the homotopy analysis method. It is noticed that an increase in the melting parameter decreases the dimensionless velocity and temperature, while an increase in the Deborah number increases the velocity and momentum boundary layer thickness.     

Interaction between longitudinal vortices and flat plate boundary layer

The interaction between longitudinal vortices and flat plate boundary layer has been studied numerically for both laminar and turbulent flow situations. The vortices are assumed to be placed in an otherwise two-dimensional boundary layer flow. The flow is assumed to be incompressible and steady. Considering the fact that the velocity, vorticity and temperature gradients in the transverse directions are much larger than the longitudinal (streamwise) gradients for these flows, the original Navier Stokes equations are parabolized in the streamwise direction. A simple model, based on Boussinesq hypothesis, is used for turbulent flow. The discretized equations are then solved step by step in the streamwise direction, using an iterative procedure at each station. Numerical solutions have been obtained for different parameters, such as the Reynolds number, the circulation and the initial position of the vortices. The computed flow patterns and the skin friction coefficient and Stanton number are found to be qualitatively consistent with available experimental results. It is shown that the interaction between the vortices and the boundary layer may severely disturb the boundary layer flow field and thus considerably increase the local skin friction and heat transfer rate on surface of an aircraft.     

Locally similar solutions for hydromagnetic and thermal slip flow boundary layers over a flat plate with variable fluid properties and convective surface boundary condition

This paper presents heat transfer process in a two-dimensional steady hydromagnetic convective flow of an electrically conducting fluid over a flat plate with partial slip at the surface of the boundary subjected to the convective surface heat flux at the boundary. The analysis accounts for both temperature-dependent viscosity and temperature dependent thermal conductivity. The local similarity equations are derived and solved numerically using the Nachtsheim-Swigert iteration procedure. Results for the dimensionless velocity, temperature and ambient Prandtl number within the boundary layer are displayed graphically delineating the effect of various parameters characterizing the flow. The results show that momentum boundary layer thickness significantly depends on the surface convection parameter, Hartmann number and on the sign of the variable viscosity parameter. The results also show that plate surface temperature is higher when there is no slip at the plate compared to its presence. For both slip and no-slip cases surface temperature of the plate can be controlled by controlling the strength of the applied magnetic field. In modelling the thermal boundary layer flow with variable viscosity and variable thermal conductivity, the Prandtl number must be treated as a variable irrespective of flow conditions whether there is slip or no-slip at the boundary to obtain realistic results.     

Thermal radiation effects on MHD flow past a semi-infinite inclined plate in the presence of mass diffusion

MHD mixed free-forced heat and mass convective steady incompressible laminar boundary layer flow of a gray optically thick electrically conducting viscous fluid past a semi-infinite inclined plate for high temperature and concentration differences is studied. A uniform magnetic field is applied perpendicular to the plate. The density of the fluid is assumed to reduce exponentially with temperature and concentration. The usual Boussinesq approximation is neglected due to the high temperature and concentration differences between the plate and the ambient fluid. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The boundary layer equations governing the flow are reduced to ordinary differential equations, which are numerically solved by applying an efficient technique. The effects of the density/temperature parameter n, the density/concentration parameter m, the local magnetic parameter M_x and the radiation parameter R are examined on the velocity, temperature and concentration distributions as well as the coefficients of skin-friction, heat flux and mass flux.