Axisymmetric Boundary Layer Equations Research Articles

The boundary layer flow induced above the torsional motion of a disk

The boundary layer flow above a disk in torsional motion with azimuthal velocity proportional to rm is investigated for all values of m ≥ 1; here, r is the radial coordinate measured from the center of the disk. The resulting flow is a fully three-dimensional exact solution to the steady, axisymmetric boundary layer equations in the form of similarity solutions. We compute wall shear stresses in the radial and azimuthal directions as a function of the torsional exponent m, as well as the flow induced in the far field. The large m asymptotics of the problem are computed and compared with numerical solutions. The induced radial velocity profiles have a “wall-jet” structure, and it is found that both the radial and azimuthal velocity components become confined close to the surface of the disk as m increases.

Transformations, properties, and exact solutions of unsteady axisymmetric boundary layer equations for non-Newtonian fluids

Unsteady axisymmetric boundary layer equations for power-law non-Newtonian fluids are analyzed. A number of new exact solutions containing arbitrary functions and free parameters are constructed using generalized or functional separation of variables. The solutions are obtained using a Crocco-type transformation reducing the order of the equations examined and simpler point transformations. Along with the exact solutions to axisymmetric boundary layer equations, some new exact solutions to planar boundary layer equations for non-Newtonian fluids are constructed. Several properties have been discovered that allow the exact solutions of the unsteady axisymmetric boundary layer equations to be generalized by including additional arbitrary functions therein. All results refer to an arbitrarily shaped streamlined solid of revolution.

One-dimensional reductions and functional separable solutions to unsteady plane and axisymmetric boundary-layer equations for non-Newtonian fluids

The paper deals with equations describing the unsteady axisymmetric boundary layer of power-law non-Newtonian fluids on a body of revolution. The axisymmetric boundary-layer equation for the stream function is shown to reduce to a single third-order PDE of the formwtz+wzwxz−wxwzz=κrn+1(x)wzzn−1wzzz+F(t,x),where n is a rheological parameter of the fluid; the function r(x), determining the shape of the body, is assumed arbitrary. For this non-linear PDE, we describe one-dimensional reductions as well as a number of new generalized and functional separable solutions, which depend on two to five arbitrary functions. The solutions are obtained with the direct method of functional separation of variables by using particular solutions to an auxiliary ODE and systems of first-order PDEs. Many of the solutions are expressed in terms of elementary functions, which, together with significant arbitrariness, makes them especially useful for solving certain model problems and testing numerical and approximate analytical methods in fluid dynamics. Apart from power-law fluids, the paper looks at three-parameter polynomial and generalized Sisko models of non-Newtonian fluids. The unsteady plane boundary-layer equations for the general non-Newtonian fluid model are shown to admit a reduction to an ODE. The paper presents new exact solutions to the plane boundary-layer equations for power-law fluids as well as some other rheologically complex fluids.

Exact solutions of unsteady boundary layer equations for power-law non-Newtonian fluids

A number of new exact solutions (with the generalized and functional separation of variables) of unsteady equations of a planar and asymmetric boundary layer of power-law non-Newtonian fluids are described. To find the solutions, the Crocco transformation reducing the order of the equations considered and simpler point transformations are used. Two theorems allowing one to generalize exact solutions of the unsteady axisymmetric boundary layer equations including additional arbitrary functions into them are proven.

Transformations and exact solutions of unsteady-state axisymmetric boundary layer equations

Transformations and exact solutions of unsteady-state axisymmetric boundary layer equations

Transformations, properties, and exact solutions of nonstationary axisymmetric boundary-layer equations

A nonstationary axisymmetric boundary-layer equation with a pressure gradient, which is written for the stream function, has been studied. It has been shown that this equation can be reduced to a twodimensional boundary-layer equation with variable viscosity that depends on a longitudinal coordinate. A series of new exact generalized and functional separable solutions that admit representation in elementary functions has been described. All of the solutions contain several (two to five) arbitrary functions. Formulas have been given that make it possible to generalize exact solutions to nonstationary axisymmetric boundarylayer equations by introducing additional arbitrary functions. The results are valid for any shape of a body of revolution (or a circular tube with a variable section) in a fluid flow.

Unsteady axisymmetric boundary-layer equations: Transformations, properties, exact solutions, order reduction and solution method

The paper deals with equations describing the unsteady axisymmetric boundary layer on a body of revolution. The shape of the body is assumed to be arbitrary. The axisymmetric boundary-layer equation for the stream function is shown to reduce to a plane boundary-layer equation with a streamwise-coordinate-dependent viscosity of the formwtz+wzwxz−wxwzz=νr2(x)wzzz+F(t,x).We describe a number of new generalized and functional separable solutions to this non-linear equation, which depend on two to five arbitrary functions. The solutions are obtained with a new method (direct method of functional separation of variables) based on using particular solutions to an auxiliary ODE. Many of the solutions are expressed in terms of elementary functions, provided that the arbitrary functions are also elementary. Two theorems are stated that enable one to generalize exact solutions of unsteady axisymmetric boundary-layer equations by including additional arbitrary functions. Furthermore, we specify a von Mises-type transformation that reduces the unsteady axisymmetric boundary-layer equation to a non-linear second-order PDE. We also present several new exact solutions to the plane boundary-layer equation and solve a boundary layer problem for a non-uniformly heated flat plate in a unidirectional fluid flow with temperature dependent viscosity.The method proposed in this paper can also be effective for constructing exact solutions to many other non-linear PDEs.

Viscous and inviscid spatial stability analysis of compressible swirling mixing layers

We examine the viscous and inviscid spatial stabilities of circular swirling mixing layers that differ in swirl intensity, Mach number, and Reynolds number. The corresponding base flows are numerical solutions of the axisymmetric boundary-layer equations in cylindrical coordinates. A high-order numerical discretization scheme based on Chebyshev collocation and a global eigenvalue search are employed to solve the corresponding stability equations. The two disturbance modes that are observed are of centrifugal (or Rayleigh) type and of shear-instability (or Kelvin–Helmholtz) type. The most amplified shear instabilities with higher azimuthal wavenumber possess a long-axial-wavelength character and are fed by the axial vorticity arising from the swirl component. Moreover, secondary and higher unstable modes corresponding to the centrifugal instability are observed. Viscosity has a slightly stabilizing effect on the shear-instability disturbances. Centrifugal-instability disturbances with short-azimuthal and short axial wavelengths are stabilized by viscous effects.

Unsteady flow and heat transfer of viscous incompressible fluid with temperature-dependent viscosity due to a rotating disc in a porous medium

This paper studies the effect of a porous medium and temperature-dependent viscosity on the unsteady flow and heat transfer for a viscous laminar incompressible fluid due to an impulsively started rotating infinite disc. The unsteady axi-symmetric boundary layer equations are solved using three methods, namely, (i) perturbation solution for small time, (ii) asymptotic analysis for large time and (iii) the finite difference method together with the Keller box elimination technique for intermediate times. The solutions are obtained in terms of local radial skin friction, local tangential skin friction and local rate of heat transfer at the surface of the disc, for different values of the pertinent parameters: the Prandtl number Pr, the viscosity variation parameter e and porosity parameter m. The computed dimensionless velocity and temperature profiles for Pr = 0.72 are shown graphically for different values of e and m.

Axisymmetric wake and jet solutions in decelerating streams

New solutions of the axisymmetric boundary-layer equations are presented. The solutions are associated with free shear layers within non-uniform streams. Both jet- and wake-like velocity profiles have been found. In the appropriate limit it is demonstrated that the jet-like profiles asymptote to the round jet solution in a quiescent ambient.

Unsteady flow of viscous incompressible fluid with temperature-dependent viscosity due to a rotating disc in presence of transverse magnetic field and heat transfer

This paper studies the effect of a magnetic field and temperature-dependent viscosity on the unsteady flow and heat transfer for a viscous laminar incompressible and electrically conducting fluid due to an impulsively started rotating infinite disc. The unsteady axisymmetric boundary layer equations are solved using three methods, namely, (i) perturbation solution for small time, (ii) asymptotic analysis for large time and (iii) finite difference method together with Keller box elimination technique for intermediate times. The solutions are obtained in terms of local radial skin friction, local tangential skin friction, and local rate of heat transfer at the surface of the disc, for different values of the pertinent parameters: the Prandtl number Pr, the viscosity variation parameter ε and magnetic field parameter m. The computed dimensionless velocity and temperature profiles for Pr=0.72 are shown graphically for different values of ε and m.

Effect of nose shape on three-dimensional streamlines and heating rates

A new method for calculating the three-dimensional inviscid surface streamlines and streamline metrics using Cartesian coordinates and time as the independent variable of integration has been developed. The technique calculates the streamline from a specified point on the body to a point near the stagnation point by using a prescribed pressure distribution in the Euler equations. The differential equations, which are singular at the stagnation point, are of the two-point boundary-value-problem type. Laminar heating rates are calculated using the axisymmetric analog concept for three-dimensional boundary layers and approximate solutions to the axisymmetric boundary-layer equations. Results for elliptic conic forebody geometries indicate that the location of the point of maximum heating depends on the angle of attack of the conic and the type of conic in the plane of symmetry. This location is in general different from the stagnation point. The new method was found to give smooth predictions of heat transfer in the nose region where previous methods gave oscillatory results. a b by e

A spectral collocation method for compressible, non‐similar boundary layers

AbstractAn efficient and highly accurate algorithm based on a spectral collocation method is developed for numerical solution of the compressible, two‐dimensional and axisymmetric boundary layer equations. The numerical method incorporates a fifth‐order, fully implicit marching scheme in the streamwise (timelike) dimension and a spectral collocation method based on Chebyshev polynomial expansions in the wall‐normal (spacelike) dimension. The discrete governing equations are cast in residual form and the residuals are minimized at each marching step by a preconditioned Richardson iteration scheme which fully couples energy, momentum and continuity equations. Preconditioning on the basis of the finite difference analogues of the governing equations results in a computationally efficient iteration with acceptable convergence properties. A practical application of the algorithm arises in the area of compressible linear stability theory, in the investigation of the effects of transverse curvature on the stability of flows over axisymmetric bodies. The spectral collocation algorithm is used to derive the non‐similar mean velocity and temperature profiles in the boundary layer of a ‘fuselage’ (cylinder) in a high‐speed (Mach 5) flow parallel to its axis. The stability of the flow is shown to be sensitive to the gradual streamwise evolution of the mean flow and it is concluded that the effects of transverse curvature on stability should not be ignored routinely.

General Analysis of Steady Laminar Mixed Convection Heat Transfer on Vertical Slender Cylinders

A general analysis has been developed to study fluid flow and heat transfer characteristics for steady laminar mixed convection on vertical slender cylinders covering the entire range from pure forced to pure natural convection. Two uniquely transformed sets of axisymmetric boundary-layer equations for the constant wall heat flux case and the isothermal surface case are solved using a two-point finite difference method with Newton linearization. Of interest are the effects of the new mixed convection parameter, the cylinder heating/cooling mode, the transverse curvature parameter, and the Prandtl number on the velocity/temperature profiles and on the local skin friction parameter and the heat transfer parameter. The results of the validated computer simulation model are as follows. Depending upon the magnitude and direction of the buoyancy force, i.e., the value of the mixed convection parameter and the heating or cooling mode applied, natural convection can have a significant effect on the thermal flow field around vertical cylinders. Specifically, strong variations of the local skin friction parameter and reversing trends in the heat transfer parameter are produced as the buoyancy force becomes stronger in aiding flow. The skin friction parameter increases with higher curvature parameters and Prandtl numbers. Similarly, the modified Nusselt number is larger for higher transverse curvature parameters; however, this parameter may reverse the impact of the Prandtl number on the Nusselt number for predominantly forced convection.

Boundary Layer Flow Over Long Cylinders With Suction

The full, axisymmetric boundary layer equations over a long circular cylinder have been repeatedly solved in the past by a variety of approximate Pohlhausen-type and series expansion methods. These methods are not suitable however, for axisymmetric solutions with wall suction boundary condition. In the present paper, the basic equations are solved by a simple approximate method, based on successive derivatives of the wall compatibility condition. A family of velocity profiles, which coincides with the asymptotic suction profile at infinity, is assumed, and the wall shear force is found by integrating an ordinary differential equation. The computed boundary layer properties for the impermeable (zero-suction) cylinder case are shown to be in good agreement with solutions of other, more sophisticated techniques. The case of an infinite cylinder with constant suction asymptotically approaches the well-known constant thickness boundary layer solution, in analogy with the case of constant suction on a flat plate. The present method is further explored to exhibit the drag reduction properties of wall suction. An optimal suction profile is found, as a function of free stream conditions and the cylinder fineness ratio. For example, in the case of a cylinder fineness ratio 5, and Re = 25 × 106, the flow can be laminarized with only 80 percent of the mass sucked off, relative to uniform suction.

Analysis of laser-induced convection in unconfined fluids and in vertical cylinders

Laminar natural convection induced in fluids by absorption of energy from a vertical laser beam is analyzed for two different configurations. In the first configuration, the vertical beam propagates in an unconfined fluid, and is modeled as a line source of thermal energy. Ordinary differential equations are obtained by a similarity analysis of the axisymmetric boundary layer equations, and approximate solutions of them are obtained. In the second configuration, the beam propagates in a fluid confined in a tall, isothermal vertical cylinder, and is assumed to have a Gaussian radial distribution of intensity. An exact solution of the Navier-Stokes and thermal energy equations is obtained for the case of parallel flow. Representative numerical results are presented, and estimates are made of the parametric ranges of validity of the analyses.

Free convection heat transfer from vertical cylinders part II: Exponential surface temperature variation

Investigation on laminar free convection heat transfer from vertical cylinders and wires having a surface temperature variation of the form T W − T ∞ = M e mx are presented. As in Part I for power law surface temperature variation, the axisymmetric boundary layer equations of mass, momentum and energy are transformed to more convenient forms and solved numerically. The second approximation refines the results of the first upto a maximum of only 2%. Analysis of the results indicates that cylinders can be classified into the same three categories as in Part I, namely, short cylinders, long cylinders, and wires, heat transfer and fluid flow correlations being developed for each case.

Three-Dimensional Boundary Layers on Cones at Small Angles of Attack

Based on an expression given by Cooke, an equation is derived for a three-dimensional “equivalent radius” for cones at a small angle of attack. This function when used in any of the available axisymmetric boundary-layer equations yields corresponding solutions for yawed cones. Expressions for the streamlines along which the above equations are to be integrated are also derived. The method yields a line of possible incipient separation or wake formation in addition to the boundary-layer properties in both the longitudinal and circumferential direction. Numerical solutions including heat transfer effects are presented for a wind tunnel model and compared with experimental results.

A class of solutions of the axisymmetric boundary-layer equations with mass transfer.

A class of solutions of the axisymmetric boundary-layer equations with mass transfer.