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.     

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.

     

A Lagrangian simulation of particle deposition in a turbulent boundary layer in the presence of thermophoresis

Knowledge of particle deposition in turbulent flows is often required in engineering situations. Examples include fouling of turbine blades, plate-out in nuclear reactors and soot deposition. Thus it is important for numerical simulations to be able to predict particle deposition. Particle deposition is often principally determined by the forces acting on the particles in the boundary layer. The particle tracking facility in the CFD code uses the eddy lifetime model to simulate turbulent particle dispersion, no specific boundary layer being modelled. The particle tracking code has been modified to include a boundary layer. The non-dimensional yplus, y⁺, distance of the particle from the wall is determined and then values for the fluid velocity, fluctuating fluid velocity and eddy lifetime appropriate for a turbulent boundary layer used. Predictions including the boundary layer have been compared against experimental data for particle deposition in turbulent pipe flow. The results giving much better agreement. Many engineering problems also involve heat transfer and hence temperature gradients. Thermophoresis is a phenomena by which small particles experience a force in the opposite direction to the temperature gradient. Thus particles will tend to deposit on cold walls and be repulsed by hot walls. The effect of thermophoresis on the deposition of particles can be significant. The modifications of the particle tracking facility have been extended to include the effect of thermophoresis. A preliminary test case involving the deposition of particles in a heated pipe has been simulated. Comparison with experimental data from an extensive experimental programme undertaken at ISPRA, known as STORM (Simplified Tests on Resuspension Mechanisms), has been made.     

Near-separating,self-similar turbulent boundary layer

The self-similar flow in a turbulent boundary layer, which is in a state close to separation as a result of the effect of adverse pressure gradient, is investigated. Such a boundary layer has a triple-deck asymptotic structure. Between outer and near-wall regions above the logarithmic sublayer, i. e. the constant-stress layer, an intermediate region — the gradient sublayer — is formed, where the shear stress varies linearly due to adverse longitudinal pressure gradient. In the external part of the gradient sublayer, the velocity profile obeys the square-root law. The velocity profile obtained from the solution for the outer region satisfies a slip condition on the wall. The slip value decreases as the similarity parameter increases and vanishes at the value of Ω = 0.0911, which corresponds to separation, here δ^* is the displacement thickness, and U and U′ are the free-stream velocity and its derivative with respect to the longitudinal coordinate. In this case, the exponent m in the law specifying the free-stream self-similar velocity distribution increases, with separation occurring not at the minimal value of m = −1/3, which corresponds to the strongest adverse pressure gradient, but at the value m = 0.228. (© 2013 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)     

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)     

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.     

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)     

Friction law for turbulent boundary layers on a flat plate with suction

A universal friction law for turbulent boundary layers on a flat plate with suction is established. Experimental skin-friction distributions obtained for various suction factors and Reynolds numbers are described in similarity variables by a single universal curve. The law is valid for the entire range of suction velocities from zero one till the limiting values corresponding to asymptotic-suction boundary layers. The analysis is not based on any particular turbulence model. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Laminar boundary layer on a flat plate at high Prandtl number

Zusammenfassung Das Problem der Laminarströmung einer kompressiblen Flüssigkeit längs einer ebenen Platte bei grossen Prandtlzahlen wird mit Hilfe der Methode der inneren und äusseren Expansion gelöst unter der Annahme, dass das Produkt aus Dichte und Viskosität quer zur Grenzschicht konstant ist. Die Tabellen des Anhanges ermöglichen die Berechnung einer vollständigen Temperaturverteilung in der Grenzschicht für isolierte und nichtadiabatische Wände. Es erweist sich, dass die für den Rückgewinnungsfaktor gewonnene Formelr=1,922 ^1/3–1,341 eine wesentliche Verbesserung gegenüber früheren Ergebnissen ist, selbst bei =10³. Es wird eine Interpolationsformel vorgeschlagen, die in die exakte asymptotische Lösung bei grossem übergeht und beir=1 ebenfalls =1 ergibt.     

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).     

Drag reduction via wall-normal oscillations in turbulent flat plate boundary layer flow at very high Reynolds number

The effects of wall-normal single point oscillations in turbulent boundary layers at very high Reynolds number are investigated by numerical simulation. The impact on the friction drag and on the turbulent structures is analyzed. (© 2016 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. Project supported by the National Natural Science Foundation of China (Grant No. 19732005) and National Climbing Project.     

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.