Supersonic flow around the trailing edge of a thin profile

The method of mergeable asymptotic expansions has recently been used effectively in investigations devoted to the study of boundary layer interaction with an external inviscid flow at high subcritical Reynolds numbers Re. The asymptotic analysis permits obtaining a limit pattern of the flow around a solid as Re þ, and determining the similarity and quantitative regularity laws which are in good agreement with experimental results. Thus by using the method of mergeable asymptotic expansions it is shown in [1–4] that near sites with high local curvature of the body contour and flow separation and attachment points, an interaction domain appears that has a small length on the order of Re^-3/8. In this flow domain, which has a three-layer structure, the pressure distribution in a first approximation already depends on the change in boundary-layer displacement thickness, while the induced pressure gradient, in turn, influences the flow in the boundary layer. An analogous situation occurs in the neighborhood of the trailing edge of a flat plate where an interaction domain also appears [5, 6]. The flow in the neighborhood of the trailing edge of a flat plate around which a supersonic viscous gas flows was examined in [7]. Numerical results in this paper show that the friction stress on the plate surface remains positive everywhere in the interaction domain, and grows on approaching the trailing edge. The supersonic flow around the trailing edge of a flat plate at a small angle of attack was investigated in [8, 9], Supersonic flow of a viscous gas in the neighborhood of the trailing edge of a flat plate at zero angle of attack is examined in [10], but with different velocity values in the inviscid part of the flow on the upper and lower sides of the plate. The more general problem of the flow around the trailing edge of a profile with small relative thickness is investigated in this paper.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 36–42, May–June, 1981.     

Calculation of three-dimensional boundary layer on a triangular plate of finite length in a hypersonic stream

The results are given of the calculation of a three-dimensional boundary layer on a triangular plate of finite length in a regime of strong viscous interaction with an external hypersonic stream for both symmetric flow as well as in the presence of an angle of slip. The influence of the change in the pressure on the trailing edge of the plate on the boundary layer characteristics is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 46–52, September–October, 1982.     

Isothermal velocity and turbulence measurements downstream of a model multilobed turbofan mixer

An isothermal experimental investigation of the three dimensional flow field downstream of a model multilobed turbofan forced mixer is presented. LDA measurements of the three mean velocities and corresponding turbulence intensities were obtained in the downstream duct where the turbine (primary) and fan (secondary) streams emerging from the lobes mix together. The flow development in the near field was quantified by measuring the cross plane velocities. These were found to consist of large radial flows, of order 15% of the mean axial velocities at the lobe inlet, with changing sign depending on location. The cross-plane flow is consistent with a large scale axial vortex pair (per lobe) which persists throughout the downstream duct and enhances mixing in this region. Turbulence generation and anisotropy of the turbulence structure were evident mainly in the shear layers formed as the fan and turbine streams emanated from the lobe trailing edge. Spatial uniformity in the mean and turbulent fields was measured as little as five heights downstream of the mixer exit, indicating the rapid mixing achievable in these systems.     

A vorticity formulation for computational fluid dynamic and aeroelastic analyses of viscous flows

The paper deals with a novel computational formulation for the analysis of viscous flows past a solid body. The formulation is based upon a convenient decomposition of the flow field into potential and rotational velocity contributions, which has the distinguishing feature that the rotational velocity vanishes in much of, if not all, the region in which the vorticity is negligible. Contrary to related formulations implemented by the authors in the past, in the proposed approach, discontinuities of the potential and rotational velocity fields across a prescribed surface emanating from the trailing edge (such as the wake mid-surface) are eliminated, thereby facilitating numerical implementations. However, the main novelty is related to the application of the boundary condition: first, the expression for the velocity used for the condition on the body boundary is consistent with that for the velocity in the field; also—contrary to related formulations used by the authors in the past—in the proposed approach, the condition on the body boundary does not require the evaluation of the total vorticity (inside and outside the computational domain). The proposed algorithm, valid for three-dimensional compressible flows, is validated—as a first step—for the case of two-dimensional incompressible flows. Specifically, numerical results are presented for the aerodynamic analysis of two-dimensional incompressible viscous flows past a circular cylinder and past a Joukowski airfoil. In order to verify the desirable absence of artificial damping, we present also results pertaining to the flutter (i.e., dynamic aeroelastic) analysis of a spring-mounted circular cylinder in a viscous flow, free to move in a direction orthogonal to the unperturbed flow. In both cases (aerodynamics and aeroelasticity), the results are in good agreement with existing literature data.     

An LES Turbulent Inflow Generator using A Recycling and Rescaling Method

The present paper describes a recycling and rescaling method for generating turbulent inflow conditions for Large Eddy Simulation. The method is first validated by simulating a turbulent boundary layer and a turbulent mixing layer. It is demonstrated that, with input specification of mean velocities and turbulence rms levels (normal stresses) only, it can produce realistic and self-consistent turbulence structures. Comparison of shear stress and integral length scale indicates the success of the method in generating turbulent 1-point and 2-point correlations not specified in the input data. With the turbulent inlet conditions generated by this method, the growth rate of the turbulent boundary/mixing layer is properly predicted. Furthermore, the method can be used for the more complex inlet boundary flow types commonly found in industrial applications, which is demonstrated by generating non-equilibrium turbulent inflow and spanwise inhomogeneous inflow. As a final illustration of the benefits brought by this approach, a droplet-laden mixing layer is simulated. The dispersion of droplets in the near-field immediately downstream of the splitter plate trailing edge where the turbulent mixing layer begins is accurately reproduced due to the realistic turbulent structures captured by the recycling/rescaling method.     

Propagation of perturbations upstream with interaction between a hypersonic flow and a boundary layer

It is shown that, for hypersonic flows with moderate and strong degrees of interaction, perturbations brought about, for example, by a bottom opening or by any other sort of obstacle are propagated up to the leading edge of a solid body. Local regions with very large pressure gradients cannot arise in the flow. This is connected with the possibility of the development of breakaway zones with a length on the order of magnitude of the size of the solid body, described in the first approximation by the equations of the boundary layer. From a mathematical point of view the problem comes down to establishing the nonsingular nature of the solution near the leading edge, and to finding eigensolutions which make it possible to satisfy the boundary conditions at the trailing edge of the solid body. It is shown that, with a weak interaction between the hypersonic flow and the boundary layer, there may arise short flow regions with free interaction and locally nonviscous flows with large pressure gradients, within the limits of which the perturbations may move upstream.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 40–49, July–August, 1970.In conclusion, the author thanks V. V. Sychev for his evaluation of the problem.     

A double vortex sheet model of the viscous flow near the trailing edge of a lifting aerofoil

Classical definitions of boundary layer mass and momentum flux deficiency thicknesses can lead to gross errors when applied to measurements near a trailing edge where the flow curvature in the free stream is appreciable. This paper presents a double vortex sheet model as a development from the single vortex sheet model of Helmholtz and others. Two bound vortex sheets define a potential function which can describe a flow with the same mass and momentum flux deficiencies as the viscous regions. The bound nature of these sheets allows the modelling of the integral properties of these regions while retaining the advantages of a potential flow. The application to the flow near the trailing edge of a lifting aerofoil is given     

Contribution to the theory of thin-profile trailing edge separation

The conditions of nonsymmetric trailing edge flow with separation are investigated. Solutions of the equations for the interaction zone in the neighborhood of the trailing edge of a thin profile at an angle of attack of the order O(Re^–1/16) in the separated flow regime are constructed numerically. It is shown that for this zone a solution exists up to a certain angle of attack. In all the regimes the value of the friction on the upper surface at the very end of the trailing edge remains a positive quantity. The solution of the equations in the separated flow regimes is found to be nonunique. The flow over the leading edge is assumed to be unseparated, and the separation at the trailing edge, if present, is assumed to be localized in the interior of the boundary layer. The flow over a Kutta profile at zero angle of attack is taken as an example. In this case the satisfaction of the Chaplygin-Joukowsky condition at the trailing edge ensures smooth flow over both the trailing and leading edges.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 55–59, July–August, 1989.     

Simulation numérique directe de l'influence de la forme aval d'une plaque séparatrice sur une couche de mélangeDirect-numerical simulation of the splitting-plate downstream-shape influence upon a mixing layer

In this Note, we present direct numerical simulation results of a spatial mixing layer generated behind an upstream plate separating two boundary layers. The effect of the shape of the trailing edge of the plate is considered through comparisons between flows obtained from a bevelled or a blunt plate. In the former case, a spatial mixing layer consistent with previous experimental and numerical observations is obtained. In the latter case, the self-excited state that establishes in the near wake region dominates primary and secondary instability mechanisms while understating the importance of inflow perturbations. This behaviour is interpreted in terms of convective or absolute instability. The effects on turbulent statistics are also discussed. To cite this article: S. Laizet, E. Lamballais, C. R. Mecanique 334 (2006).     

Heat transfer in the neighborhood of a point of inflection in the leading edge of a plate in hypersonic flight

A study is made of the three-dimensional flow of a viscous gas around a flat plate with an inflection in the generator of the leading edge in the case of strong interaction between the exterior hypersonic flow and the boundary layer. Numerical solutions to the problem are obtained. It is shown that near points of inflection of the profile of the leading edge of a flat wing strong self-induced secondary flows can be formed together with associated local peaks of the heat fluxes and the friction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 40–45, May–June, 1980.     

Development of secondary flows in the stator of a model turbine

Measurements are described which show the development of the secondary flows through the stator blades in a single stage model turbine. The build of the turbine is intended to simulate a stage in the HP cylinder of a steam turbine and three different thicknesses of inlet boundary layers are studied. Flow visualisation of the inlet boundary fluid shows the development of the horseshoe vortex round the blade leading edge. Probe measurements upstream and downstream of the trailing edge plane show how the passage vortices are developed and how they influence the distribution of losses across the annulus.     

Thermal chaotic mixing of power-law fluids in a mixer with alternately rotating walls

We investigate the enhancement of both mixing and heat transfer in a two-rod mixer for highly viscous non-Newtonian fluids. The mixer is composed of two vertical circular rods in a cylindrical tank. Chaotic flows are obtained by imposing the temporal modulations of the rotational velocities of the walls. We study the effects of different stirring protocols, which lead to non-chaotic and chaotic flows, on the efficiency of both mixing and heat transfer for three different rheological fluid behaviors: shear-thinning, Newtonian and shear-thickening. For this purpose, we use statistical indicators that characterize the mean value of the fluid temperature and its homogenization. We find that chaotic mixing is suitable for shear-thickening fluids for which we observe a clear enhancement of the thermal mixing (heat extraction and homogenization). This is due to the increase in the apparent fluid viscosity in the vicinity of the rotating walls. This aspect confirms the relevance of chaotic mixing for highly viscous fluids.     

Evaluation of an energy relaxation method for the simulation of unsteady,viscous, real gas flows

This study investigates a new energy relaxation method designed to capture the dynamics of unsteady, viscous, real gas flows governed by the compressible Navier–Stokes equations. We focus on real gas models accounting for inelastic molecular collisions and yielding temperature‐dependent heat capacities. The relaxed Navier–Stokes equations are discretized using a mixed finite volume/finite element method and a high‐order time integration scheme. The accuracy of the energy relaxation method is investigated on three test problems of increasing complexity: the advection of a periodic set of vortices, the interaction of a temperature spot with a weak shock, and finally, the interaction of a reflected shock with its trailing boundary layer in a shock tube. In all cases, the method is validated against benchmark solutions and the numerical errors resulting from both discretization and energy relaxation are assessed independently. Copyright © 2004 John Wiley & Sons, Ltd.     

Fluid mixing induced by vibrating walls

Recent progress in micro-fluid dynamics has identified an increased demand for efficient mixing of highly viscous fluids in small channels and cavities. One way to do this is through the steady streaming generated by the vibration of solid boundaries. In this paper we investigate the mixing properties of such streaming flows in an infinite channel. A Newtonian fluid is confined within flexible walls with transverse motion in the form of standing waves of small amplitude. The velocity field is determined using a perturbation approach with the slope of the wall as a small parameter [Phys. Fluids 16 (2004) 1822]. Streaming occurs at second order with the formation of cellular flow patterns in the channel. The Lagrangian velocities were found to mimic the Eulerian except for flows at large channel half-widths and low frequencies. Most effective mixing is observed for flows at channel half-widths of similar, or lower, order than the vibratory wavelength and for sufficiently high frequencies.     

Mean and turbulence measurements in the boundary layer and wake of a symmetric aerofoil

Detailed measurements of two-dimensional profiles of static pressure, mean velocity, turbulence intensity and Reynolds shear stress were carried out with conventional pressure probes and hot wire probes at preselected streamwise stations in the boundary layer and wake of a 12.5% thick, 600 mm chord two-dimensional symmetric aerofoil mounted at zero incidence in a low speed wind tunnel. The chord Reynolds number was one million and the wake measurements extended up to three chord lengths (or nearly 660 trailing edge momentum thicknesses) downstream of the trailing edge. The data indicate rapid interaction of the wall layers immediately behind the trailing edge, leading to significant changes in the flow parameters close to the trailing edge. The relaxation of the wake is preceded by initial ‘overshoot’ in the streamwise profiles of mean-flow parameters and peak values of turbulence components. Further growth of the wake towards similarity/equilibrium is discussed.     

Development of a Two-velocities Hybrid RANS-LES Model and its Application to a Trailing Edge Flow

The flow around a trailing edge is computed with a new hybrid method designed to more clearly separate the effects of total and sub-grid turbulent stress-modelling on the time-averaged and instantaneous velocity fields, and in turn, mean momentum and kinetic energy balances. These two velocity fields independently define Reynolds averaged and sub-grid-scale viscosities, and distinct stresses, at the same location. In particular, resolved eddies can emerge, or sweep in and out of the Reynolds averaged near wall layer, without being dampened by higher levels of the viscosity in this RANS dominated layer. The two-field hybrid model, first tested on channel flows, gives accurate predictions of mean velocities and stresses for different Reynolds numbers and coarse meshes. For the trailing edge flow the results of the hybrid model are close to the reference fine LES for mean velocity and turbulent content, whereas the DES-SST on the same coarse mesh gives too early separation.     

The mixing layer between non-parallel streams

Compared to the classical two-dimensional plane mixing layer, the mixing layer between non-parallel streams has an additional degree of freedom: the angle between the streams and the direction perpendicular to the trailing edge. Consequently the mean vorticity vector, which depends on these angles, is no longer by necessity parallel to the trailing edge of the flow. The ensuing coherent structures are generally helices with components normal to the trailing edge. They can be controlled by different mechanisms, depending on the velocity vectors.     

Generation of Three-Dimensional Disturbances in a Boundary Layer on Strong Interaction with a Hypersonic Flow

Laminar boundary layer flow over an infinite-span, finite-length flat plate is investigated in the regime of strong interaction with a hypersonic gas flow. Under the assumption that an additional condition dependent on the transverse coordinate can be imposed on the trailing edge of the plate the flow functions are expanded in power series in the vicinity of the leading edge. It is shown that these expansions include an indefinite function dependent on the transverse coordinate. The corresponding boundary value problems are formulated and solved and the eigenvalues are determined. It is established that in this case the two-dimensional boundary layer can rearrange itself into a three-dimensional boundary layer.     

Analysis of turbulent near-wake development behind an infinitely yawed flat plate

An asymptotic analysis of the turbulent near-wake flow behind an infinitely yawed flat plate with sharp trailing edge has been formulated. The feature that the near-wake which is dominated by mixing of oncoming turbulent boundary layers, retains to a large extent the memory of the turbulent structure of the upstream boundary layer has been exploited to develop the analysis. This analysis leads to three regions of wake flow (the inner near-wake, the outer near-wake and the far-wake) for which the governing equations are derived. The matching conditions amongst these regions lead to logarithmic variations for both the components (i.e. components perpendicular to and along the trailing edge) of velocity vector in both normal and downstream directions in the overlapping region surrounding the inner near-wake. These features are validated by the available experimental data. Similarity solutions for both the components of the velocity vector (which satisfy the required matching conditions) in the inner near-wake and outer near-wake regions have been obtained by making appropriate eddy-viscosity assumptions. Uniformly valid solutions for both the components of the velocity vector have been constructed for the near-wake. The solutions show good agreement with available experimental data.