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.     

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.     

Asymptotic solution of the unsteady boundary layer equations

Summary An extension of the Meksyn asymptotic method to unsteady boundary layers in laminar, incompressible flow is investigated. The results indicate that unsteady boundary layers can be calculated by the Meksyn asymptotic method with comparable accuracy to that obtained for steady flows. Several differences from the well developed steady-flow application exist and require further work before general problems can be treated. The calculation technique is more straight-forward for cases involving acceleration because three or four terms in the expansions may then yield sufficient accuracy. The form of the governing equation required by the Meksyn method indicates that it is most useful for unsteady stagnation boundary layers since some basic unsteady flows are not directly accessible in their simplest form from that equation. The effect of unsteadiness on the rate of asymptotic convergence is assessed by detailed comparison of a similar solution for unsteady, stagnation flow with analogous results from the Falkner-Skan equation and of reliable numerical results for both cases.

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Résumé On étudie une extension de la méthode asymptotique de Meksyn aux couches limites instables des écoulements laminaires de fluides incompressibles. Les résultats montrent que les couches limites instables peuvent être calculées à l'aide de la méthode asymptotique de Meksyn avec une précision comparable à celle obtenue pour les écoulements stables. Plusieurs différences existent par rapport à l'application, bien mise au point, aux écoulements stables; elles demandent encore du travail avant que les problèmes généraux puissent être traités. La méthode de calcul est plus directe dans les cas impliquant une accélération, car 3 ou 4 termes dans les développements assurent alors une précision suffisante. La forme de l'équation principale nécessaire à la méthode de Meksyn indique qu'elle est très utile pour les couches limites instables au repos; en effet, certains écoulements instables de base ne peuvent être atteints directement dans leur forme la plus simple à partir de cette équation. L'effet de l'instabilité sur la vitesse de convergence asymptotique est établi grâce à une comparaison détaillée d'une solution analogue pour un écoulement instable stagnant avec les résultats semblables obtenus par l'équation Falkner-Skan, et des résultats numériques sûrs obtenus dans les deux cas.

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This work was supported by the Energy Research and Development Administration.     

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).     

Minimum error solutions of boundary layer equations

The minimum error solutions of boundary layer equations in the least square sense have been studied by employing the Euler-Lagrange equations. To test the method a class of problems,i.e., boundary layer on a flat plate, Hiemenz flow, boundary layer on a moving sheet and boundary layer in non-Newtonian fluids have been studied. The comparison of the results with approximate methods, like Karman-Pohlhuasen, local potential and other variational methods, shows that the present predictions are invariably better.     

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.     

Some Fredholm equations with boundary layer resonance

Certain singularly perturbed differential equations which exhibit boundary layer resonance are difficult to solve by the application of standard asymptotic methods. After reformulation as a singularly perturbed integral equation and treatment by a recently developed asymptotic methodology, the desired solution is obtained in a straightforward manner.     

Monte Carlo simulation of boundary layer transition

A statistical method for simulating a boundary layer transition flow is proposed as based on experimental data on the kinematics and dynamics of turbulent spots (Emmons spots) on a flat plate placed in an incompressible fluid. The method determines intermittency with allowance for overlapping spots, which makes it possible to determine the forces on the plate surface and the flow field near the transition region if the mean streamwise velocity field in a developed turbulent boundary layer is known as a function of the Reynolds number. In contrast to multiparameter transition models, this approach avoids the use of nonphysical parameter values.     

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.     

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.     

A system of steady-state boundary layer equations for a dilatantous medium

The existence of a solution of a system of equations that arise in the hydrodynamics of a Newtonian liquid is proved, and a number of properties of this solution are studied. It is established that the rate of propagation of the perturbations is finite under particular conditions. The uniqueness of the solution where the rate of propagation of the perturbations is finite is demonstrated.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 89–108, 1989.     

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)     

The boundary layer analysis for stokes equations on a half space

The time dependent Stokes equations on a half space are considered. We decompose the solution of these equations in an inviscid solution: a boundary layer solution and a correction. Bounds on these solutions are given, in the appropriate Sobolev spaces, in terms of the norms of the initial and boundary data. The correction is shown to be of the same order of magnitude of the square root of the viscosity.     

Numerical simulation of the three-dimensional Kolmogorov flow in a shear layer

Numerical simulation is used to study the Kolmogorov flow in a shear layer of a compressible inviscid medium. A periodic permanent force applied to the flow gives rise to a vortex cascade of instabilities. The influence exerted by the size of the computational domain, the initial conditions, and the amplitude of the force on the formation of an instability cascade and the transition to turbulence is studied. It is shown that the mechanism of the onset of turbulence has an essentially three-dimensional nature. For the turbulent flows computed, the classical Kolmogorov ?5/3 power law holds in the inertial range.     

Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations

Runge-Kutta methods are studied when applied to stiff differential equations containing a small stiffness parameter . The coefficients in the expansion of the global error in powers of are the global errors of the Runge-Kutta method applied to a differential algebraic system. A study of these errors and of the remainder of the expansion yields sharp error bounds for the stiff problem. Numerical experiments confirm the results.     

Thin-layer version of the moment equations in the boundary layer problem

The two-dimensional problem of a hypersonic kinetic boundary layer developing on a thin body in the case of a monatomic gas is considered. The model of the flow arises from the kinetic theory of gases and, within its accuracy, i.e., in the approximation of a hypersonic boundary layer, takes into account the strong nonequilibrium of the flow with respect to translational degrees of freedom. A method for representing the solution of the problem in terms of the solution of a similar classical (Navier-Stokes) hypersonic boundary layer problem is described. For the kinetic version of the problem, it is shown that the shear stress and the specific heat flux on the body surface are equal to their counterparts in the Navier-Stokes boundary layer.     

The Crocco transformation for the three-dimensional Prandtl boundary layer equations

The three-dimensional fluid flow of a steady or unsteady moving incompressible continuous medium is considered. Several Crocco-type transformations are applied which reduce the system of three equations of the Prandtl boundary layer to a system of two strongly coupled parabolic equations.     

Asymptotic behavior of the unbounded solutions of some boundary layer equations

We give an asymptotic equivalent at infinity of the unbounded solutions of some boundary layer equations arising in fluid mechanics.Received: 24 May 2004