The problem of the turbulent boundary layer

Summary Existing measurements of low-speed turbulent surface friction on a flat plate, in the absence of pressure gradient and roughness, are shown to be consistent with a simple analysis based on functional similarity in the velocity profile. In particular, the fully developed turbulent boundary layer is found to be unique within the accuracy of the experimental data, with uniqueness defined as the existence of a definite correspondence between local friction coefficient and momentum thickness Reynolds number. The relationships known as the law of the wall and the velocity defect law are found to describe the turbulent velocity profiles accurately for a considerable range of Reynolds numbers, and an effort is made to clarify the physical significance of these formulae. Finally, the proper definition of a length Reynolds number is discussed in terms of the asymptotic local properties of the ideal boundary layer, and numerical values for ideal mean and local friction coefficients are tabulated against Reynolds numbers based on momentum thickness and on distance from the leading edge.

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Zusammenfassung Es wird gezeigt, dass vorhandene Messungen der turbulenten Wandschubspannung an der glatten ebenen Platte in inkompressibler Strömung ohne Druckgradient durch eine einfache Berechnung in Übereinstimmung gebracht werden können. Die Rechnung beruht auf einer funktionellen Ähnlichkeit der Geschwindigkeitsverteilung. Es wird im besonderen gefunden, dass die vollentwickelte turbulente Grenzschicht innerhalb der Messgenauigkeit einem eindeutigen Zusammenhang zwischen dem örtlichen Reibungskoeffizienten und der Reynoldsschen Zahl, bezogen auf die Impulsdicke, folgt. Die Beziehungen, die als Wandgesetz und Mittengesetz bekannt sind, beschreiben die Geschwindigkeitsverteilung genau innerhalb eines erheblichen Bereiches Reynoldsscher Zahlen, und es wird versucht, den physikalischen Inhalt dieser Gesetzmässigkeiten zu vertiefen. Abschliessend wird eine zweckmässige Definition der auf Plattenlänge bezogenen Reynoldsschen Zahl diskutiert, die auf dem asymptotischen örtlichen Zustand der idealen Grenzschicht beruht. Rechenwerte der idealen, mittleren und örtlichen Reibungskoeffizienten, bezogen auf beide obigen Definitionen der Reynoldsschen Zahl, werden tabelliert.

     

Self-similar turbulent boundary layer in pressure gradient

Self-similar flows in a turbulent boundary layer when the free-stream velocity is specified as a power function of longitudinal coordinate are investigated. The self-similar formulation not only simplifies solving of the problem by reducing the equations of motion to ordinary differential equations but also provides a mean for formulating closure conditions. It is shown that for the class of flows under consideration that depend on three governing parameters the dimensionless mixing length is a function of the normalised distance from the wall and the exponent in the law specifying the free-stream velocity distribution in the outer region and a universal function of local Reynolds number in the wall region, the latter corollary being true even when the skin friction vanishes. In calculations this function is set to be independent of pressure gradient, which gives the results very close to experimental data. There exist four different self-similar flow regimes. Each regime is related to its similarity parameter, one of which is the well-known Clauser equilibrium parameter and the other three are established for the first time. In case of adverse pressure gradient when the exponent lies within certain limits, which depend on Reynolds number, the problem has two solutions with different values of the boundary layer thickness and skin friction, which points out the possibility of hysteresis in near-separating flow. Separation occurs not at the minimum value of the exponent that corresponds to the strongest adverse pressure gradient but at a higher one whose dependence on Reynolds number is calculated in the paper. The results of the theory are in good agreement with experimental data. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Three-dimensional turbulent boundary layer on wind turbine blades

The momentum integral technique for predicting the boundary–layer growth in three–dimensional flow has been extended to include the entrainment equation as the closure model. Special attention has been devoted to those terms in the differential equations that change the boundary–layer structure from that of cvasi two–dimensional steady flow, at outboard locations, to a tree–dimensional flow pattern. It is concluded that the stall is delayed due to the boundary–layer reattachment at inboard sections in conjunction with the onset of a spanwise vortex like structure. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact self-similar solutions of the magnetohydrodynamic boundary layer system for power-law fluids

In a particular self-similar case, the magnetohydrodynamic boundary layer system for an electrically conducting power-law fluid together with certain boundary conditions can be transformed into a boundary value problem for a third-order nonlinear ordinary differential equation, only whose (generalized) normal solutions possess the physical meaning of the original problem. Uniqueness, existence and nonexistence results are established for the problem. Representations are also given for all (generalized) normal solutions. The project was supported by the Natural Science Foundation of Fujian Province of China (No. Z0511005) and NNSF of china(No. 10501037).     

Concave solutions of a general self-similar boundary layer problem for power-law fluids

In this paper, we give a rigorous mathematical analysis for a third order nonlinear boundary value problem. The boundary value problem can be applied to steady free convection around a vertical impermeable flat plate in a fluid-saturated porous medium, or steady flow of a power-law fluid induced by impermeable stretching walls in the frame of boundary layer approximation. We establish the uniqueness, existence and nonexistence of (normal) concave solutions or generalized concave solutions to the problem, and obtain some results about boundedness and asymptotic behavior of the (normal) concave solution or the generalized concave solution.     

A self-similar premixed turbulent flame model

This paper is devoted to premixed combustion modelling in turbulent flow. First, we derive a model for the turbulent flame velocity based on the observed self-similarity of the turbulent flame. The model uses the local flame brush width as a fundamental parameter and, therefore, we show how it can be retrieved for numerical implementation. The diffusive property of the brush width is treated in such a way as to theoretically let the brush have a clearly defined boundary propagating at finite velocity. The model, implemented in Star-CD CFD software through user programming, is then numerically tested on three configurations for which another model, the Turbulent Flame Closure model, is known to give very good agreement. Some effects of numerics are commented and results for both models are compared. While based on very different approaches the two models lead to substantially similar results. In this way, we have shown that the local brush width can effectively be used, giving an additional degree of freedom for premixed turbulent combustion modelling.     

Direct simulation of the turbulent boundary layer on a plate

A numerical method for the integration of three-dimensional Navier–Stokes equations for compressible fluid as applied to direct numerical simulation is proposed. By way of example, the boundary layer on a plate is simulated. The computations were carried out for Re_θ = 1500. The computational grid consisted of a half billion nodes. The flow region includes the laminar, transitional, and turbulent zones. The numerically obtained distributions of average velocity, friction, and pulsations are compared with experimental data and available numerical solutions.     

Dynamical structure of the turbulent boundary layer on rough surface

The experiments on structure of turbulent boundary layer on the plane rough wall without pressure gradient are presented. Sand roughness of the wall is considered. Measurements are carried out using Time-Resolved PIV technique in planes parallel and perpendicular to the wall. The results on rough wall are compared with the base case of boundary layer on smooth wall. Hairpin vortices have been detected. Topology and typical size of those structures substantially differ in the cases in question. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An asymptotic approach for shock-wave/transpired turbulent boundary layer interactions

Summary In this work perturbation techniques are used to study the problem of the interaction between a shock wave and a transpired turbulent boundary layer at transonic speeds. In the case considered here, the Mach number is assumed to be high enough for the sonic line to penetrate deep into the boundary layer so that it ends close to the wall. The flow region is divided into a region of strong interaction and regions of weak interaction. For the regions of weak interaction, upstream and downstream of the shock, a classical two-deck structure is assumed to hold for the boundary layer. Solutions chosen for these regions must account for the effects of blowing or suction. The strong interaction region on the other hand is shown to consist of three decks. A detailed analysis of the whole flow field is carried out and solutions valid in the double limit as Reynolds number tends to infinity and Mach number tends to one are proposed. Solutions of adjacent layers are shown to match so providing a smooth solution for the entire flow region. The analysis yields solutions for the pressure and skin-friction profiles.

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Zusammenfassung In dieser Veröffentlichung wird mit Methoden der Störungsrechnung das Problem der Wechselwirkung zwischen einer Schockwelle und einer turbulenten Grenzschicht mit Blas- und Saugeffekten beim Übergang zu Überschallgeschwindigkeiten untersucht. In dem hier betrachteten Fall ist die Machzahl so hoch, daß die Schockwellenfront tief in die Grenzschicht eindringt und nahe bei der Wand endet. Das Strömungsgebiet wird in ein Gebiet starker Wechselwirkung und in ein Gebiet schwacher Wechselwirkung eingeteilt. Für die Gebiete schwacher Wechselwirkung (vor und hinter der Schockwellenfront) wird angenommen, daß die Grenzschicht eine klassische zwei-Deck Struktur besitzt. Lösungen, die für diese Gebiete gewählt werden, müssen die Blas- oder Saugeffekte erklären. Es wird gezeigt, daß das Gebiet starker Wechselwirkung dagegen aus drei Decks besteht. Eine ausführliche Berechnung des gesamten Strömungsfeldes wird durchgeführt, und es werden Lösungen angegeben, die gültig sind für den doppelten Grenzfall, daß die Reynoldszahl gegen unendlich und die Machzahl gegen eins strebt. Es zeigt sich, daß die Lösungen für benachbarte Schichten aneinander passen, und daß sich damit eine stetige Lösung für das gesamte Strömungsgebiet ergibt. Die Lösungen für die Druck- und Wandreibungsprofile werden bestimmt.

     

Effect of pressure gradient on coherent structures in a turbulent boundary layer

By using the idea of resonant triad of the theory of hydrodynamic stability, the effect of pressure gradient on coherent structures in a turbulent boundary layer is investigated. The favorable pressure gradient suppresses the generation of the coherent structure, while the adverse pressure gradient has the opposite effect. The scale, form, as well as the propagation speed of the coherent structures are different from those with zero pressure gradient. The theoretical results are, in general, in agreement with those found from experiments. From the calculated probability density curve of the circulation differences of the nearly streamwise vortex pairs, it is found that the adverse pressure gradient makes the vortex pair more symmetric. Project supported by the National Natural Science Foundation of China.     

Numerical simulation of the turbulent boundary layer equations via a Runge-Kutta algorithm

This paper addresses a hybrid computational procedure for the step-by-step calculation of momentum transfer in turbulent boundary layer flows along flat plates. The proposed procedure relies on a modified method of lines wherein transversal discretizations are carried out by a “control volume” being infinitesimal in the streamwise direction and finite in the transversal direction of the fluid flow. Using mixing length theory and coarse intervals in the transversal direction, the resulting system of ordinary differential equations of first order may be readily integrated on a personal computer utilizing a fourth-order Runge-Kutta algorithm. In general, a maximum number of sixteen lines is necessary at the trailing edge of the flat plate for a typical calculation. As a consequence, computing time and storage for each run were very small when compared to other finite-difference methods. Furthermore, to validate the hybrid procedure involving the method of lines and control volumes (MOLCV), comparisons with experimental data have been done in terms of both velocity distributions and local skin friction coefficients. For all cases tested, the proposed methodology predicts the growth of the boundary layer of gases correctly.     

Modeling of individual coherent structures in wall region of a turbulent boundary layer

Models for individual coherent structures in the wall region of a turbulent boundary layer are proposed. Method of numerical simulations is used to follow the evolution of the structures. It is found that the proposed model does bear many features of coherent structures found in experiments. Project supported by the National Natural Science Foundation of China (Grant No. 19732005) and National Climbing Project.     

Scaling laws for turbulent boundary layer flows in favorable and adverse pressure gradient

An analytical theory is proposed to describe incompressible plane and axisymmetric turbulent boundary layer flows in favorable and adverse pressure gradients for near-equilibrium conditions. Scaling laws for the mean velocity, the Reynolds stress components, and the skin friction have been established. A universal friction law makes it possible to represent the skin friction distributions corresponding to different Reynolds numbers and pressure gradients in terms of a function of one variable. The theory is based on general physical assumptions and does not involve any special hypotheses on the nature of the turbulent motion. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of individual coherent structures in wall region of a turbulent boundary layer

Models for individual coherent structures in the wall region of a turbulent boundary layer are proposed. Method of numerical simulations is used to follow the evolution of the structures. It is found that the proposed model does bear many features of coherent structures found in experiments.     

The heat transfer problem in the wall region of a turbulent boundary layer

The problem of heat transfer in the wall region of a turbulent boundary layer has been investigated. The resonant triad in the theory of hydrodynamic stability was used to obtain the velocity field induced by the coherent structures in the wall region of the turbulent boundary layer, while the small scale turbulence was represented by a simple model. By such a new approach of modeling, the 3-D temperature field is calculated, the mean temperature profile in the wall region and the Nusselt number characterizing the heat flux, which was found to be in good agreement with the experimental observations are obtained. The instantaneous temperature field had streaky structures, thus offering a mechanism for their generation found in numerical simulations. Project supported by the National Natural Science Foundation of China (Grant No. 19132011).     

On the visualization of laminar boundary layer oscillations and the transition to turbulent flow

Zusammenfassung Eine Methode für die Sichtbarmachung periodischer Grenzschichtph?nomene ist beschrieben und angewandt auf die Beobachtung des Umschlages Laminar-Turbulent an einem zweidimensionalen Flügel mit NACA-0012-Profil. Str?mungsbilder, die die verschiedenen Stadien der laminaren Grenzschichtschwingungen und ihren endgültigen Zusammenbruch in Turbulenz zeigen, sind hinzugefügt.