Unsteady Three-dimensional Boundary Layer Research Articles

Numerical study of a dynamic resonant wall shear stress sensor with spectral element method

A numerical study with spectral element method for a dynamic resonant shear stress sensor concept is presented in this paper, in which the numerical model of the sensor consisted of an unsteady three-dimensional boundary layer model for the flow and a simple mechanical model for the sensor itself. Spectral element method was used to explore fluid flow properties around the sensor in the three-dimensional boundary layer model. The three-dimensional unsteady spectral element method code was first verified with Blasius solutions in a flat plate boundary layer flow. The sensor's sensitivity to wall shear stress was then numerically determined in a laminar boundary layer. Finally, the physical mechanism of the dynamic resonant shear stress sensor was analyzed by using the verified model. The results showed that the sensitivity of the dynamic resonant shear stress sensor was due to the energy lost produced by the oscillating interaction between the sensor and fluid flow.

Unsteady three-dimensional boundary layer flow due to a permeable shrinking sheet

The unsteady viscous flow over a continuously permeable shrinking surface is studied. Similarity equations are obtained through the application of similar transformation techniques. Numerical techniques are used to solve the similarity equations for different values of the unsteadiness parameter, the mass suction parameter, the shrinking parameter and the Prandtl number on the velocity and temperature profiles as well as the skin friction coefficient and the Nusselt number. It is found that, different from an unsteady stretching sheet, dual solutions exist in a certain range of mass suction and unsteadiness parameters.

Unsteady boundary layers with small cross flows

Despite the intensive development of computer technology and methods of solving the Navier-Stokes and Reynolds equations, the unsteady problems of the three-dimensional boundary layer are of significant interest in aerodynamics. So far these problems have been little studied as a result of objective difficulties related with the large dimensionality of the system of equations and the complexity of its investigation [{xc1}]. Therefore, analytic results in this field are important. In the present study the unsteady three-dimensional boundary layer equations are investigated in the case of small cross flows using the perturbation method. An intermediate system of equations, which includes the basic three-dimensional effects but is significantly simpler than the initial system is derived. The features of the formulation considered are studied in relation to the important practical problems of boundary layer flow past slender wings and weakly asymmetric bodies at small angles of attack.

Turbulent flow between a rotating and a stationary disk

Turbulent flow between a rotating and a stationary disk is studied. Besides its fundamental importance as a three-dimensional prototype flow, such flow fields are frequently encountered in rotor–stator configurations in turbomachinery applications. A direct numerical simulation is therefore performed by integrating the time-dependent Navier–Stokes equations until a statistically steady state is reached and with the aim of providing both long-time statistics and an exposition of coherent structures obtained by conditional sampling. The simulated flow has local Reynolds number r2ω/v = 4 × 105 and local gap ratio s/r = 0.02, where ω is the angular velocity of the rotating disk, r the radial distance from the axis of rotation, v the kinematic viscosity of the fluid, and s the gap width.The three components of the mean velocity vector and the six independent Reynolds stresses are compared with experimental measurements in a rotor–stator flow configuration. In the numerically generated flow field, the structural parameter a1 (i.e. the ratio of the magnitude of the shear stress vector to twice the mean turbulent kinetic energy) is lower near the two disks than in two-dimensional boundary layers. This characteristic feature is typical for three-dimensional boundary layers, and so are the misalignment between the shear stress vector and the mean velocity gradient vector, although the degree of misalignment turns out to be smaller in the present flow than in unsteady three-dimensional boundary layer flow. It is also observed that the wall friction at the rotating disk is substantially higher than at the stationary disk.Coherent structures near the disks are identified by means of the λ2 vortex criterion in order to provide sufficient information to resolve a controversy regarding the roles played by sweeps and ejections in shear stress production. An ensemble average of the detected structures reveals that the coherent structures in the rotor–stator flow are similar to the ones found in two-dimensional flows. It is shown, however, that the three-dimensionality of the mean flow reduces the inter-vortical alignment and the tendency of structures of opposite sense of rotation to overlap. The coherent structures near the disks generate weaker sweeps (i.e. quadrant 4 events) than structures in conventional two-dimensional boundary layers. This reduction in the quadrant 4 contribution from the coherent structures is believed to explain the reduced efficiency of the mean flow in producing Reynolds shear stress.

Development of flow and heat transfer of a viscous fluid in the stagnation-point region of a three-dimensional body with a magnetic field

The development of flow and heat transfer of a viscous electrically conducting fluid in the stagnation point region of a three-dimensional body with an applied magnetic field is studied when the external stream is set into an impulsive motion from rest and at the same time the surface temperature is suddenly raised from that of the surrounding fluid. This analysis includes both short time solution (Rayleigh-type of solution) and the steady-state solution as time tends to infinity (Falkner-Skan type of solution). The unsteady three-dimensional boundary layer equations represented by a system of parabolic partial differential equations are solved numerically using an implicit finite-difference scheme. For certain particular cases analytical solutions are obtained. In the absence of the magnetic field, the reverse flow occurs in the transverse component of the velocity in a certain portion of the saddle-point region (−1≦c<−0.4, wherec=b/a is the ratio of the velocity gradients in they- andx-directions at the edge of the boundary layer). The magnetic field delays or prevents the reverse flow. The surface shear stresses in the principal and transverse directions and the surface heat transfer increase with the magnetic field both in nodal point (0≦c≦1) and saddle point (−1≦c<0) regions. For a fixed magnetic field, the surface shear stress inx-direction and the surface heat transfer increase with time in nodal and saddle point regions, but the surface shear stress in the transverse direction increases with time for 0<c≦1 and decreases with increasing time for −1≦c<0.

Symmetry reductions of unsteady three-dimensional boundary layers of some non-Newtonian fluids

Three-dimensional, unsteady, laminar boundary layer equations of a general model of non-Newtonian fluids are treated. In this model, the shear stresses are considered to be arbitrary functions of velocity gradients. Using Lie Group analysis, the infinitesimal generators accepted by the equations are calculated for the arbitrary shear stress case. The extension of the Lie algebra, for the case of Newtonian fluids, is also presented. A general boundary value problem modeling the flow over a moving surface with suction or injection is considered. The restrictions imposed by the boundary conditions on the generators are calculated. Assuming all flow quantities to be independent of the z-direction, the three-independent-variable partial differential system is converted first into a two-independent-variable system by using two different subgroups of the general group. Lie Group analysis is further applied to the resulting equations, and final reductions to ordinary differential systems are obtained.

Unsteady three-dimensional boundary layer flow due to a stretching surface

In this numerical study, the unsteady laminar incompressible boundary-layer flow over a continuously stretching surface has been investigated when the velocity of the stretching surface varies arbitrarily with time. Both the nodal and the saddle point regions of flow have been considered for the analysis. Also, constant wall temperature/concentration and constant heat/mass flux at the stretching surface have been taken into account. The quasilinearisation method with an implicit finite-difference scheme is used in the nodal point region (0≦c≦1) wherec denotes the stretching ratio. This method fails in the saddle point region (−1≦c≦0) due to the occurrence of reverse flow in they-component of velocity. In order to overcome this difficulty, the method of parametric differentiation with an implicit finite-difference scheme is used, where the values atc=0 are taken as starting values. Results have been obtained for the stretching velocities which are accelerating and decelerating with time. Results show that the skin friction, the heat transfer and the mass transfer parameters respond significantly to the time dependent stretching velocities. Suction (A>0) is found to be an important parameter in obtaining convergent solution in the case of the saddle point region of flow. The Prandtl number and the Schmidt number strongly affect the heat and mass transfer of the diffusing species, respectively.

A Multizone Time-Marching Technique for Unsteady Separating Three-Dimensional Boundary Layers and Its Application to the Symmetry-Plane Solution of an Impulsively Started Prolate Spheroid

Recent interest in unsteady separation and separated flows brings up the need of an accurate and efficient computational scheme for general unsteady three-dimensional boundary-layer flows. Resolution of the singular behavior at separation is a delicate problem. The task is further complicated by the geometrical singularity and the nonstationary stagnation point. The present paper proposes a numerical scheme to sidestep these difficulties. At the first stage of development, the simpler problem of the symmetry-plane solution of the laminar boundary-layer over an impulsively-started prolate spheroid is calculated. Results show that the present Eulerian calculation satisfactorily captures the singular behavior of the boundary layer when separation is approached. Comparison with Xu and Wang’s recent results and those for the two-dimensional elliptic cylinder calculated by the Lagrangian method are also made. Discussions of the results for unsteady separation at zero, small and large incidences are presented.

Interaction of a laminar boundary layer with external turbulence

Transition in the boundary layer on a flat plate in a turbulent flow is investigated experimentally and theoretically. It is established that over a broad range of flow conditions (variation of the intensity and scale of the external turbulence, the angle of attack, the shape of the leading edge, etc.) transition takes place without the formation of Tollmien-Schlichting waves, and its initial stages, including the amplification of disturbances, are described by the linearized unsteady three-dimensional boundary layer equations without a pressure gradient.

Unsteady three-dimensional boundary layer flow over a flat plate with attached cylinder

The unsteady laminar incompressible three-dimensional boundary layer flow and heat transfer on a flat plate with an attached cylinder have been studied when the free stream velocity components and wall temperature vary inversely as linear and quadratic functions of time, respectively. The governing semisimilar partial differential equations with three independent variables have been solved numerically using a quasilinear finite-difference scheme. The results indicate that the skin friction increases with parameter λ which characterizes the unsteadiness in the free stream velocity and the streamwise distance x ̄ , but the heat transfer decreases. However, the skin friction and heat transfer are found to change little along z ̄ . The effect of the Prandtl number on the heat transfer is found to be more pronounced when λ is small, whereas the effect of the dissipation parameter is more pronounced when λ is comparatively large.

Interaction of an unsteady three-dimensional boundary layer with a hypersonic flow near a rapidly heated section of the surface

A study is made of unsteady three-dimensional flows with self-induced pressure realized near a rapidly heated section of the surface of a flat plate in a hypersonic flow.

On the unsteady three-dimensional boundary layer freely interacting with the external stream

Asymptotic equations that define unsteady processes in a three-dimensional boundary layer with self-induced pressure are derived. The pressure gradient under conditions of free interaction is, as usually, calculated not by the solution of the external problem of flow over a body, but on the assumption that it is due to growth of streamline displacement thickness near the body surface. Besides the principal terms, terms of second order of smallness are retained in asymptotic sequencies. If the characteristic dimensions of the free interaction region are the same in all directions in the plane tangent to the body surface, the system of equations defining the thin layer next to the wall must be integrated together with the system which defines the nonviscous stream.

On the nature of unsteady three-dimensional laminar boundary-layer separation

Solutions have been obtained for a family of unsteady three-dimensional boundary-layer flows which approach separation as a result of the imposed pressure gradient. These solutions have been obtained in a co-ordinate system which is moving with a constant velocity relative to the body-fixed co-ordinate system. The flows studied are those which are steady in the moving co-ordinate system. The boundary-layer solutions have been obtained in the moving co-ordinate system using the technique of semi-similar solutions. The behaviour of the solutions as separation is approached has been used to infer the physical characteristics of unsteady three-dimensional separation.In the numerical solutions of the three-dimensional unsteady laminar boundary-layer equations, subject to an imposed pressure distribution, the approach to separation is characterized by a rapid increase in the number of iterations required to obtain converged solutions at each station and a corresponding rapid increase in the component of velocity normal to the body surface. The solutions obtained indicate that separation is best observed in a co-ordinate system moving with separation where streamlines turn to form an envelope which is the separation line, as in steady three-dimensional flow, and that this process occurs within the boundary layer (away from the wall) as in the unsteady two-dimensional case. This description of three-dimensional unsteady separation is a generalization of the two-dimensional (Moore-Rott-Sears) model for unsteady separation.

Similarity solutions of unsteady laminar incompressible boundary layer equations for flow, heat and mass transfer in non-Newtonian fluids around axisymmetric bodies

The possible existence of similarity solutions for the unsteady three-dimensional boundary layer flows with heat and mass transfer around stationary axisymmetric bodies which are fully immersed in purely viscous moving non-Newtonian fluids has been searched in general by the application of transformations, involving a single linear parameter. In particular, the cases involving rotational flows around stationary bodies and rotating bodies have been discussed as corollaries of the main analysis. The main analysis shows that the similarity solutions are possible only for the bodies for which $$\bar r \propto \bar x^n $$ where $$\bar r$$ is a cross-sectional radius; and $$\bar x$$ is the longitudinal distance from the nose point to the cross section. In case of rotating bodies, similarity solutions exist only for cones and disks. The analysis, as an example, has successfully been applied to the Powell-Eyring model. It is seen that for the same rate of shear, expenditure of energy for maintaining the rotation of the solid body is comparatively higher for a flow with a higher Eyring number where the Eyring numberEy=1/µBE. µ, B, andE are the material functions of the Powell-Eyring fluid.

Three-dimensional periodic boundary layers

The method of successive approximations used by Eichelbrenner and Askovic1 for the study of unsteady three-dimensional boundary layer flow has been extended to analyse the periodic boundary layers in three dimensions. The analysis, which is valid for oscillations of small amplitude, shows some special features such as “steady streaming” flow in the first-order cross-flow similar to the one that has been predicted and observed by Schlichting for two-dimensional periodic boundary layers.