Three-dimensional laminar boundary layer on a blunt body

We consider the Prandtl laminar boundary layer which occurs with stationary flow about a blunted cone at an angle of attack. The solution of the Prandtl equations is sought using a finite difference method. It is found that a smooth solution of the problem exists only in the vicinity of the rounded nose of the body, while far from the nose the solutions acquire a singularity; in the problem symmetry plane (on the downwind side) there is a discontinuity of the first derivatives of the velocity components and the density. In the study of the Prandtl boundary layer in the problem of stationary flow about a pointed cone at an angle of attack, it has been shown [1] that the self-similar solution (dependent on two independent variables) of the Prandtl equations has a discontinuity of the first derivatives in the problem symmetry plane (on the downwind side of the cone). The suggestion has been made that in the three-dimensional problem of flow about a blunt cone at an angle of attack the solutions of the Prandtl equations may also be discontinuous. The present study was carried out to clarify the nature of the behavior of the solutions of the three-dimensional Prandtl equations. To this end we considered stationary supersonic flow of an ideal gas past a blunted cone. The results of this study (as well as those of [1]) were obtained using a numerical, finite-difference method. However, an analysis of the numerical results (investigation of the scheme stability, reduction of step size, etc.) shows that the properties of the solutions of the finite-difference equations are not in this case a result of numerical effects, but reflect the behavior of the solutions of the differential equations. The mathematical problem on the boundary layer which is considered in this study will be formulated in §2; this formulation is due to K. N. Babenko.     

The turbulent boundary layer on the moving surface of a cylindrical body

A study is made of the problem of a two-dimensional turbulent boundary layer on the moving surface of a cylindrical body (a Rankine oval with a relative elongation of four) moving at constant velocity in an incompressible fluid. For the numerical simulation of the turbulent flow of the fluid, the boundary layer is divided into exterior and interior regions in accordance with a two-layer model, using different expressions for the coefficients of turbulent transfer for each region. A study was nade of the development of the boundary layer on the body at different speeds of the body surface and different Reynolds numbers. The following integral characteristics were found by numerical calculation: the work of friction as the body is displaced; the work expended on the movement of its surface; and, for a flow regime with separation, the work of the pressure force. In this case the following model of separation flow is assumed: beyond the singular point in the solution of the boundary layer equations that indicates the appearance of a region of reverse flow, the pressure and friction stress on the wall are constant and are determined by their values at the singular point.Translated from Izvestiya Akademii Nauk SSSH, Mekhanika Zhidkosti i Gaza, No. 5, pp. 61–67, September–October, 1984.Finally, the author would like to thank G. G. Chernyi and Yu. D. Shevelev for useful discussions and for their interest in this work.     

Three-dimensional visualization of large structures in the turbulent boundary layer

 A new method of visualizing the coherent structures in the boundary layer is used to develop insight into how these structures form and to provide information on the relative frequency of typical shapes noticed in the near-wall flow. These results were achieved in a water channel using a recently developed tracer which remains as a moving dye streak while conforming to the convoluted motions in the boundary layer. The tracer is formulated from a surfactant–polymer–emulsion mixture which retains its capabilities as a marker of evolving flow motions in the boundary layer for a meter or more before eventually dispersing. Three-dimensional, continuous visualization of the structures can be obtained as they move along a flat plate. Photos and video frames demonstrate the evolution and properties of the most widely discussed boundary-layer structure, the Theodorsen (horseshoe) vortex. Received: 16 November 1999/Accepted: 24 May 2000     

Kinematics of a turbulent boundary layer

In the article an attempt is made, within the framework of the Navier-Stokes equations, to describe the field of the instantaneous velocities of a liquid in the region of a turbulent flow near the wall. It is assumed that the velocities of the liquid are determined by the field of the eddies arising in regions of ejections under the action of pressure pulses in the region near the wall.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 34–40, September–October, 1973.     

Three-dimensional waves in a boundary layer

As is known [1], two-dimensional waves develop in the boundary layer and then become three-dimensional waves with increase of the Reynolds number R. Since Squire [2] has shown that the linear growth of three-dimensional waves is more intense than that of the two-dimensional, it is natural that the behavior of three-dimensional waves in the boundary layer is explained by nonlinear intersection [3], However, Gaster [4] has noted that although disturbances which increase with time are usually considered, experimentally we observe disturbances which grow in space. (Squire's proof does not extend to this case.) It has been shown that the spatially growing disturbances cannot explain the occurrence of the three-dimensional waves (in the linear formulation).The author wishes to thank his scientific advisor G. I. Petrov and also A. A. Zaitsev for valuable discussions of the study.     

Effect of local forcing on a turbulent boundary layer

An experimental study is performed to analyze flow structures behind local suction and blowing in a flat-plate turbulent boundary layer. The local forcing is given to the boundary layer flow by means of a sinusoidally oscillating jet issuing from a thin spanwise slot at the wall. The Reynolds number based on the momentum thickness is about Re _θ =1700. The effects of local forcing are scrutinized by altering the forcing frequency (0.011 ≤ f ⁺≤ 0.044). The forcing amplitude is fixed at A ₀=0.4. It is found that a small local forcing reduces the skin friction and the skin friction reduction increases with the forcing frequency. A phase-averaging technique is employed to capture the large-scale vortex evolution. An organized spanwise vortical structure is generated by the local forcing. The cross-sectional area of vortex and the time fraction of vortex are examined by changing the forcing frequency. An investigation of the random fluctuation components reveals that turbulent energy is concentrated near the center of vortical structures. Received: 17 March 2000/Accepted: 3 April 2001     

Bubble migration in a turbulent boundary layer

The wall void peaking distribution observed in an upward turbulent bubbly boundary layer along a flat plate is generated by bubbles that move towards the plate, come into contact with the wall and then slide along it. This transverse ‘migration’ has been studied using flow visualization, high speed video and particle tracking techniques to measure the trajectories of mono-disperse air bubbles at very low void fractions. Investigations have been performed at four Reynolds numbers in the range 280 < Re_θ < 3000, covering both the laminar and turbulent regimes, with mono-disperse bubbles of mean equivalent diameter between 2 mm and 6 mm. Lagrangian statistics calculated from hundreds of trajectories show that the migration only occurs in the turbulent regime and for bubble diameters below some critical value: 3.5 mm < d_eqcrit < 4 mm. Above this size (We > 3), the interface deformation is such that bubbles do not remain at the wall, even when they are released at the surface. Also, bubble migration is shown to be non-systematic, to have a non-deterministic character in the sense that trajectories differ significantly, to increase with Reynolds number and to take place on a short time scale. A series of experiments with isolated bubbles demonstrates that these results are not influenced by bubble–bubble interactions and confirm that two-way coupling in the flow is limited. Flow visualizations show that the migration originates with the capture of bubbles inside the large turbulent structures of the boundary layer (‘bulges’). The bubbles begin to move towards the wall as they cross these structures, and the point at which they reach the wall is strongly correlated with the position of the deep ‘valleys’ which separate the turbulent ‘bulges’. The analysis of the mean Lagrangian trajectories of migrating bubbles confirms these observations. Firstly, the average time of migration calculated from these trajectories coincides with the mean transit time of the bubbles across the structures. Secondly, once the trajectories have been scaled by this transit time and the boundary layer thickness δ, they all have the same form in the region y/δ < 0.4, independent of the Reynolds number.     

Low-Reynolds-number effects in a turbulent boundary layer

Low-Reynolds-number effects in a zero pressure gradient turbulent boundary layer have been investigated using a two-component LDV system. The momentum thickness Reynolds number R is in the range 400 to 1320. The wall shear stress is determined from the mean velocity gradient close to the wall, allowing scaling on wall variables of the inner region of the layer to be examined unambiguously. The results indicate that, for the present R range, this scaling is not appropriate. The effect of R on the Reynolds normal and shear stresses is felt within the sublayer. Outside the buffer layer, the mean velocity is more satisfactorily described by a power-law than by a logarithmic distribution.The support of the Australian Research Council is gratefully acknowledged     

Multicomponent turbulent boundary layer on a chemically active surface

When a gaseous mixture flows past chemically active surfaces the boundary layer formed on the wetted body may contain a large number of components with different diffusion properties. This leads to the necessity for studying the diffusion of the components in the multicomponent boundary layer.The use of thebinary boundary layer concept in the general case cannot yield satisfactory results, since replacement of the mutual diffusion coefficients D_ij of the various pairs of components by a single diffusion coefficient D in many cases is a rough approximation.In the general case the number of different diffusion coefficients is equal to N(N–1)/2 (N is the number of components). Usually it is possible to identify groups of components with similar molecular weights. Then the number of different diffusion coefficients may be reduced without large error. However, even in the comparatively simple case when it is possible to divide all the components into two groups with similar molecular weights we must take account of three different diffusion coefficients (one diffusion coefficient in each group and also the diffusion coefficient for the components of one group relative to the components of the other group). Only in particular cases when the gaseous mixture consists of only two components with arbitrary molecular weights, or if all the components of the gaseous mixture have similar molecular weights, can we with justification introduce a single diffusion coefficient (if in this case there are no limitations on the direction of the diffusion).Studies have been published covering the laminar multicomponent boundary layer. An analytic method for solving the equations of the laminar multicomponent boundary layer was developed by Tirskii [1]. There are also studies in which concrete results were obtained by numerical methods with the use of computers (for example, [2, 3]).As far as the author knows, for turbulent flow there are studies (for example, [4, 5]) covering flow with chemical reactions only in the case when all the diffusion coefficients are equal (D_ij=D).The present paper presents a method for calculating the turbulent multicomponent boundary layer with account for several different diffusion coefficients.Notation x, y coordinates - u, v velocity components - density - T temperature - h heat content - H enthalpy - c_i mass concentration of the i-th component - c ₁ ⁽¹⁾ element concentrations in solid body - J_i diffusion flux of the i-th component - m molecular weight - dynamic viscosity coefficient - kinematic viscosity coefficient - heat conduction coefficient - c_p specific heat - adiabatic index - D_ij binary diffusion coefficients - P Prandtl number - S_ij Schmidt number - S_t Stanton number - M Mach number - friction - q radiant thermal flux - boundary layer thickness - D rate of displacement of gas-solid interface - degree of gasification - r_ij weight fraction of element i in component j - _ij stoichiometric coefficients - K_i reaction equilibrium constants - l number of components for which I_i0 Indices i, j component number - w quantities for y=0 - * quantities on the edge of the laminar sublayer - ⁽¹⁾ quantities at the solid body - quantities at the outer edge of the boundary layer - molar transport coefficients     

Multiple distortion of a supersonic turbulent boundary layer

Two experiments were performed to study the response of a supersonic turbulent boundary layer to successive distortions. In the first experiment (Case 1), the flow passed over a forward-facing ramp formed by 20° compression corner followed by a 20° expansion corner located about 4_o downstream, where _o is the incoming boundary layer thickness. In the second experiment (Case 2), the forward-facing ramp was constructed of curved compression and expansion surfaces with the same turning angles and total step height as in Case 1. The radii of curvature for the compression and expansion surfaces were equal to 12_o. In both experiments, the flow relaxation was observed over a distance equal to 12_o. In this relaxation region, the mean and turbulent flow behavior of the boundary layer was measured. The mean velocity profile was found to be altered by the distortion. Recovery of the profile began near the wall and occurred rapidly, but in the outer part of the boundary layer, recovery proceeded slowly. Turbulence measurements revealed a dramatic reduction in the turbulence shear stress and a progressively decaying streamwise Reynolds stress profile.     

Influence of local ultrasonic forcing on a turbulent boundary layer

An experimental study was carried out to investigate the effect of local ultrasonic forcing on a turbulent boundary layer. The ultrasonic forcing system was constructed by adhering six ultrasonic transducers to a flat plate over which water was flowed. In this system, the ultrasonic waves projected into the water by the transducers caused cavitation, giving rise to an enormous number of tiny water-vapor bubbles. Stereoscopic particle image velocimetry (SPIV) was used to probe the flow characteristics. The SPIV results showed that imposition of the ultrasonic forcing caused a substantial increase in the mean wall-normal velocity but a decrease in the mean streamwise velocity. The ultrasonic forcing reduced the skin friction coefficient by up to 60% immediately downstream of the transducers; this effect gradually dissipated with moving downstream. The streamwise turbulence intensity was reduced near the wall but increased away from the wall, whereas the wall-normal turbulence intensity was not much affected near the wall but increased away from the wall. The Reynolds shear stress and the production of turbulent kinetic energy were reduced near the wall. Imposition of the ultrasonic forcing shifted the streamwise vortical structures away from the wall, leading to a reduction in skin friction.     

Influence of localised double suction on a turbulent boundary layer

The effects of localised suction applied through a pair of porous wall strips on a turbulent boundary layer have been quantified through the measurements of mean velocity and Reynolds stresses. The results indicate that the use of second strip extends the pseudo-relaminarisation zone but also reduces the overshoot in the longitudinal and normal r.m.s. velocities. While the minimum r.m.s. occurs at x/δ_o=3.0 (one strip) and x/δ_o=12 (two strips), the reduction observed for the latter case is larger. Relative to no suction, the turbulence level is modified by suction and the effect is enhanced with double suction. This increased effectiveness reflects the fact that the second strip acts on a boundary layer whose near-wall active motion has been seriously weakened by the first strip.     

Investigation of a turbulent boundary layer on a hypersonic aircraft model

An algorithm for calculation of a spatial compressible turbulent boundary layer on the surface of a pointed body is developed. The algorithm is based on the numerical solution of three-dimensional equations and algebraic models of turbulence. The flow around a hypersonic aircraft model is calculated, and the resultant Stanton numbers are compared with experimental data. The influence of the Mach number, the angle of attack, and the Reynolds number on the boundary-layer parameters is studied. It is shown that the change in the location of the transition zone has a weak effect on the skin-friction coefficient in the region of developed turbulent flow. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090.¹Technical University, Delft, the Netherlands. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 115–125, January–February, 1999.     

Ignition of hydrogen in a turbulent boundary layer on a plate

The ignition of hydrogen blown into a turbulent supersonic boundary layer on a flat plate is investigated numerically. It is assumed that the mixture consists of six chemically active components H, O, OH, H₂O, O₂, H₂ and inert nitrogen N₂. The boundary layer is divided into outer and inner regions, for which different expressions for the coefficients of turbulent transport are used. The influence of pulsations on the rates of the chemical reactions, and also the back reaction of the chemical processes on the mechanism of turbulent transfer are not taken into account. The surface of the plate is assumed to be absolutely catalytic with respect to the recombination reactions of the H and O atoms. The influence of the blowing intensity, the Mach number in the outer flow, and the pressure on the ignition delay is analyzed. The possibility of effective porous cooling of the surface when there is combustion in the boundary layer is demonstrated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 33–40, November–December, 1979.I thank V. G. Gromov and V. A. Levin for their interest in the work.     

Three-component LDA measurements in a turbulent boundary layer

An experimental study of a three-dimensional, pressuredriven, attached turbulent boundary-layer flow was made at Mach 0.4. Both the mean velocities and the full Reynolds stress tensor were measured simultaneously by a three-component LDA system. Value of the resultant shear stress to turbulent kinetic energy ratio varied between 0.1 and 0.2 and did not remain constant across the boundary-layer. Slopes of the streamwise and azimuthal mixing length distributions in the wall region were around 0.4 and 1.2, respectively. Skew angle of the turbulent shear stress was larger than skew angle of the velocity gradient.     

Low Reynolds number axisymmetric turbulent boundary layer on a cylinder

In the present study, an axisymmetric turbulent boundary layer growing on a cylinder is investigated experimentally using hot wire anemometry. The combined effects of transverse curvature as well as low Reynolds number on the mean and turbulent flow quantities are studied. The measurements include the mean velocity, turbulence intensity, skewness and flatness factors in addition to wall shear stress. The results are presented separately for the near wall region and the outer region using dimensionless parameters suitable for each case. They are also compared with the results available in the open literature.The present investigation revealed that the mean velocity in near wall region is similar to other simple turbulent flows (flat plate boundary layer, pipe and channel flows); but it differs in the logarithmic and outer regions. Further, for dimensionless moments of higher orders, such as skewness and flatness factors, the main effects of the low Reynolds number and the transverse curvature are present in the near wall region as well as the outer region.     

Aerodynamic forces on a tall building in a turbulent boundary layer

The model studies were designed to obtain information concerning wind loads on a tall building by placing the model in a turbulent shear flow simulating expected atmospheric boundary-layer winds. Since current design codes are inadequate for predicting all possible motions of tall buildings, it is important that better knowledge of mean and fluctuating loadings and their distributions becomes available. Experiments were conducted to determine the mean and fluctuating forces and twisting moments at several levels over the surface of a model. By determining the effects at several levels simultaneously, it was possible to correlate forces and moments at five levels with one common level. A single model was tested at varying orientations. Tests were also conducted with an identical model placed upstream so that its wake influenced the flow around the instrumented model. Results are presented in terms of distributions of force and moment coefficients and correlations at different levels. The spectral character of the force and moment components is illustrated for one case. Paper was presented at 1977 SESA Spring Meeting held in Dallas, TX on May 15–20.     

Wake layer in a thermal turbulent boundary layer with pressure gradient

The open equations of thermal turbulent boundary layer subjected to pressure gradient have been analysed by method of matched asymptotic expansions at large Reynolds number. The flow is divided into outer wake layer and inner wall layer. The asymptotic expansions are matched by Millikan-Kolmogorov hypothesis. The temperature profile in overlap region yields composite law which reduce to log. law for moderate pressure gradient and inverse half power law for strong adverse pressure gradient. In case of a shallow thermal wake, the matching result of outer wake layer reduces to composite temperature defect law, which is more general than the classical log. law. The comparison of data for thermal boundary layer with strong adverse pressure gradient is also considered. Received on 26 May 1998