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.     

Mixing Layers in Sink Flow: Effect of Length of Flight on Mixing in a Channel Downstream

We have experimentally and analytically studied transport of a passive scalar in the wake of a thin flat plate located at the centerline of a planar contraction with flat walls. The constant Launder parameter in the contraction, K = 6.25 ×10^− 6, was twice the value required for a turbulent boundary layer to relaminarize. In addition to the mixing analysis inside the contraction, layer mixing is also investigated downstream, where the flow continues inside a constant cross-section channel. In order to generate the passive scalar, the airflow above the plate was heated and the temperature stratification in the wake was traced by measuring the temperature field using constant current anemometry. Using different plate lengths, we found that the degree of mixing, obtained at a given position in the straight channel, is a function of the distance from the plate trailing edge to the contraction outlet. For a plate which does not protrude into the straight channel, we demonstrate the existence of an optimal trailing edge-contraction outlet distance that results in the lowest possible degree of mixing at a given downstream position in the straight channel. This finding is also supported by a semi-empirical relationship based on our developed self-similar solution for mixing layers in planar contractions.     

Development of quasi two-dimensional structures in a shallow free-surface mixing layer

 The large quasi two-dimensional turbulence structures that emerge in a shallow mixing layer are studied experimentally using Laser Doppler Anemometry. Velocity profiles and turbulence intensities are measured in the first two meters downstream of the splitter plate. In contradistinction with previous experiments, it is shown that the initial growth rate and the turbulence intensities of the shallow mixing layer compare well with deep-water plane mixing layers. Two-point measurements allowed for the determination of spatial correlations of the fluctuating velocity components. The large eddies were found to extend from one tenth of the water depth up to the free surface while the streamwise size of the eddies was found to be three times the mixing layer width. The two-dimensional character of the large structures and the associated reversed energy cascade is inferred from the power spectra of the lateral velocity component. Received: 2 April 1997/Accepted: 25 August 1997     

Experimental investigation of the effects of blowing conditions and Mach number on the unsteady behavior of coolant ejection through a trailing edge cutback

The present paper shows the results of an experimental investigation into the unsteadiness of coolant ejection at the trailing edge of a highly loaded nozzle vane cascade. The trailing edge cooling scheme features a pressure side cutback with film cooling slots, stiffened by evenly spaced ribs in an inline configuration. Cooling air is also ejected through two rows of cylindrical holes placed upstream of the cutback. Tests were performed with a low inlet turbulence intensity level (Tu₁ = 1.6%), changing the cascade operating conditions from low speed (M_2is = 0.2) up to high subsonic regime (M_2is = 0.6), and with coolant to main stream mass flow ratio varied within the 0.5–2.0% range. Particle Image Velocimetry (PIV) and flow visualizations were used to investigate the unsteady mixing process taking place between coolant and main flow downstream of the cutback, up to the trailing edge. For all the tested conditions, the results show the presence of large coherent structures, which presence is still evident up to the trailing edge. Their shape and direction of rotation change with injection conditions, as a function of coolant to mainstream velocity ratio, strongly influencing the thermal protection capability of the injected coolant flow. The Mach number increase is only responsible for a positioning of such vortical structures closer to the wall, while the Strouhal number almost remains unchanged.     

Large-scale structures visualization in a high Reynolds number,turbulent flat-plate wake at supersonic speed

A visual study is performed in a supersonic, two-dimensional wake; the high value of the Reynolds number ensures that the wake is turbulent from the trailing edge. The flow is seeded by fluid vaporization in one boundary layer upstream of the trailing edge; a light sheet is generated by a Q-switched, high energy ruby laser. The set of photographs taken from the trailing edge up to the far wake is then processed after digitization of the pictures. A progressive contamination of the lower part of the wake by the fluid initially present in the upper part can be observed. In the far wake region, well organized large scale structures can be visualized. Statistics are performed and the results are compared with previous hot-wire measurements and discussed in terms of downstream wake behaviour.This paper was presented at the 9th Symposium on turbulence, University of Missouri-Rolla, October 1–3, 1984     

DNS of Mixing and Reaction of Two Species in a Turbulent Channel Flow: A Validation of the Conditional Moment Closure

We consider the chemical reaction in a turbulent flow for the case that the time scale of turbulence and the time scale of the reaction are comparable. This process is complicated by the fact that the reaction takes place intermittently at those locations where the species are adequately mixed. This is known as spatial segregation. Several turbulence models have been proposed to take the effect of spatial segregation into account. Examples are the probability density function (PDF) and the conditional moment closure (CMC) models. The main advantage of these models is that they are able to parameterize the effects of turbulent mixing on the chemical reaction rate. As a price several new unknown terms appear in these models for which closure hypothesis must be supplied. Examples are the conditional dissipation 〈 χ ∣ φ 〉, the conditional diffusion 〈 κ ∇² φ ∣ u, φ 〉 and the conditional velocity 〈 u ∣ φ 〉. In the present study we investigate these unknown terms that appear in the PDF and CMC model by means of a direct numerical simulation (DNS) of a fully developed turbulent flow in a channel geometry. We present the results of two simulations in which a scalar is released from a continuous line source. In the first we consider turbulent mixing without chemical reaction and in the second we add a binary reaction. The results of our simulations agree very well with experimental data for the quantities on which information is available. Several closure hypotheses that have been proposed in the literature, are considered and validated with help of our simulation results. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical solution of the problem of the mixing of the boundary layers shed from the trailing edge of a wing

During the mixing of viscous incompressible flows with different velocities, in the vicinity of a trailing edge an interaction region with a three-layer structure is formed, similar to that in the case of symmetric shedding with equal velocities. The boundary layers developing on the upper and lower sides of the airfoil form a viscous mixing layer, or vortex sheet, which separates the flows downstream of the trailing edge. The boundary value problem corresponding to the flow in the viscous sublayer in the vicinity of the trailing edge of a flat plate is solved for high Reynolds numbers using an efficient numerical method for solving the equations of asymptotic interaction theory.     

Stochastic estimation of flow near the trailing edge of a NACA0012 airfoil

A stochastic estimation technique has been applied to simultaneously acquired data of velocity and surface pressure as a tool to identify the sources of wall-pressure fluctuations. The measurements have been done on a NACA0012 airfoil at a Reynolds number of Re _c  = 2 × 10⁵, based on the chord of the airfoil, where a separated laminar boundary layer was present. By performing simultaneous measurements of the surface pressure fluctuations and of the velocity field in the boundary layer and wake of the airfoil, the wall-pressure sources near the trailing edge (TE) have been studied. The mechanisms and flow structures associated with the generation of the surface pressure have been investigated. The “quasi-instantaneous” velocity field resulting from the application of the technique has led to a picture of the evolution in time of the convecting surface pressure generating flow structures and revealed information about the sources of the wall-pressure fluctuations, their nature and variability. These sources are closely related to those of the radiated noise from the TE of an airfoil and to the vibration issues encountered in ship hulls for example. The NACA0012 airfoil had a 30 cm chord and aspect ratio of 1.     

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.     

Experimental investigation of a blunt trailing edge flow field with application to sound generation

The unsteady lift generated by turbulence at the trailing edge of an airfoil is a source of radiated sound. The objective of the present research was to measure the velocity field in the near wake region of an asymmetric beveled trailing edge in order to determine the flow mechanisms responsible for the generation of trailing edge noise. Two component velocity measurements were acquired using particle image velocimetry. The chord Reynolds number was 1.9 × 10⁶. The data show velocity field realizations that were typical of a wake flow containing an asymmetric periodic vortex shedding. A phase average decomposition of the velocity field with respect to this shedding process was utilized to separate the large scale turbulent motions that occurred at the vortex shedding frequency (i.e., those responsible for the production of tonal noise) from the smaller scale turbulent motions, which were interpreted to be responsible for the production of broadband sound. The small scale turbulence was found to be dependent on the phase of the vortex shedding process implying a dependence of the broadband sound generated by the trailing edge on the phase of the vortex shedding process.     

On a Turbulent Mixing Layer Created Downstream of a “Λ” Notch Simulating One Wavelength of a Chevron Nozzle

Measurements were carried out in a turbulent mixing layer formed downstream of a splitter plate, that had a Λ-shaped trailing edge. The results revealed that the center of the mixing layer shifts toward the high-speed flow while its sides bend toward the low speed stream at larger distances from the splitter plate. This suggests the existence of a counter rotating streamwise eddies that dominate the flow and substantially increase its level of turbulence relative to the classical plane mixing layer. The change in the orientation of the vorticity, emanating from a chevron nozzle, decreases the susceptibility of the flow to spanwise uniform periodic excitation relative to a classical plane mixing layer.     

Spatial structures and scaling in the Convective Boundary Layer

We performed an investigation on spatial features of the Convective Boundary Layer (CBL) of the atmosphere, which was simulated in a laboratory model and analyzed by means of image analysis techniques. This flow is dominated by large, anisotropic vortical structures, whose spatial organization affects the scalar transport and therefore the fluxes across the boundary layer. With the aim of investigating the spatial structure and scaling in the Convective Boundary Layer, two-dimensional velocity fields were measured, on a vertical plane, by means of a pyramidal Lucas–Kanade algorithm. The coherent structures characterizing the turbulent convection were educed by analyzing the Finite-Time Lyapunov Exponent fields, which also revealed interesting phenomenological features linked to the mixing processes occurring in the Convective Boundary Layer. Both velocity and vorticity fields were analyzed in a scale-invariance framework. Data analysis showed that normalized probability distribution functions for velocity differences are dependent on the scale and tend to become Gaussian for large separations. Extended Self Similarity holds true for velocity structure functions computed within the mixing layer, and their scaling exponents are interpreted well in the phenomenological framework of the Hierarchical Structure Model. Specifically, β parameter, which is related to the similarity between weak and strong vortices, reveals a higher degree of intermittency for the vertical velocity component with respect to the horizontal one. On the other hand, the analysis of circulation structure functions shows that scaling exponents are fairly constant in the lowest part of the mixed layer, and their values are in agreement with those reported in Benzi et al. (Phys Rev E 55:3739–3742, 1997) for shear turbulence. Moreover, the relationship between circulation and velocity scaling exponents is analyzed, and it is found to be linear in the bottom part of the mixing layer. The investigation of the CBL spatial features, which has seldom been studied experimentally, has important implications for the comprehension of the mixing dynamics, as well as in turbulence closure models.     

On the vorticity characteristics of lobe-forced mixer at different configurations

Lobe-forced mixer is one typical example of the passive flow controllers owing to its corrugated trailing edge. Besides the spanwise Kelvin–Helmholtz vortex shedding, streamwise vortices are also generated within its mixing layer. The geometrical configuration of the lobe significantly affects these two types of vortices, which in turn affects the mixing performance of the mixer. In the present investigation, characteristics of mixers with five different configurations were examined and evaluated for two velocity ratios (r = 1, 0.4). The mixers have only one lobe in order to eliminate any possible interactions between neighboring vortices generated by the adjacent lobes. Hot-wire anemometer was used to examine the Kelvin–Helmholtz vortices via the spectrum analysis while laser Doppler anemometer was employed to examine the streamwise vortices. It was found that there were two main frequencies for the Kelvin–Helmholtz vortices in the wake of the mixer; and the Strouhal numbers approached their respective maximum values at high Reynolds number. The rectangular mixer had similar mixing performance with the semicircular one; and both of them were better than the triangular mixer. The scalloping modification enhanced mixing by generating additional streamwise vortices while the scarfing modification could not improve the mixing performance.     

Turbulent free convection in a vertical converging channel

This paper presents the results of experimental and numerical investigations of the problem of turbulent natural convection in a converging-plate vertical channel. The channel has two isothermally heated inclined walls and two adiabatic vertical side walls. The parameters involved in this study are the channel geometry represented by the channel width at exit, the inclination of the heated walls and the temperature difference between the heated walls and the ambient. The investigation covered modified Rayleigh numbers up to 10⁸ in the computational study and up to 9.3 × 10⁶ in the experimental work. The experimental measurements focused on the velocity field and were carried out using a PIV system and included measurements of the mean velocity profiles as well as the root-mean-square velocity and shear stress profiles. The experiments were conducted for an inclination angle of 30°, a gap width of 10 mm and two temperature differences (∆T=25.4°C and 49.8°C). The velocity profiles in the lower part of the channel indicated the presence of two distinct layers. The first layer is adjacent to the heated plate and driven by buoyancy forces while the second layer extends from the point of maximum velocity to the channel center plane and driven mainly by shear forces. The velocity profile at the upper portion of the channel has shown the merging of the two boundary layers growing over the two heated walls. The measured values of the Reynolds shear stress and root mean square of the horizontal and vertical velocity fluctuation components have reached their maximum near the wall while having smaller values in the core region. The computational results have shown that the average Nusselt number increases approximately linearly with the increase of the modified Rayleigh number when plotted on log–log scale. The variation of the local Nusselt number indicated infinite values at the channel inlet (leading edge effect) and high values at the channel exit (trailing edge effect). For a fixed value of the top channel opening, the increase of the inclination angle tended to reduce flow velocity at the inlet section while changing the flow structure near the heated plates in such a way to create boundary-layer type flow. The maximum value of the average Nusselt number occurs when θ = 0 and decreases with the increase of the inclination angle. On the other hand, the increase of the channel width at exit for the same inclination angle caused a monotonic increase in the flow velocity at the channel inlet.     

Rheology of poly(methyl methacrylate-<Emphasis Type="Italic">co</Emphasis>-styrene) particles suspended in water: effects of electrostatic surface layer

 Linear and nonlinear viscoelastic properties were examined for aqueous suspensions of monodisperse poly(methyl methacrylate-co-styrene) (MS) particles having the radius a ₀ =45 nm and the volume fractions φ=0.428−0.448. These particles had surface charges and the resulting electrostatic surface layer (electric double layer) had a thickness of t_s=5.7 nm. At low frequencies in the linear viscoelastic regime, the MS particles behaved approximately as the Brownian hard particles having an effective radius a _eff=a ₀ + t_s, and the dependence of their zero-shear viscosity η₀ on an effective volume fraction φ_eff (={a _eff/a ₀}³φ) agreed with the φ dependence of η₀ of ideal hard-core silica suspensions. In a range of φ_eff < 0.63, this φ_eff dependence was well described by the Brady theory. However, the φ_eff dependence of the high-frequency plateau modulus was weaker and the terminal relaxation mode distribution was narrower for the MS suspensions than for the hard-core suspensions. This result suggested that the electrostatic surface layer of the MS particles was soft and penetrable (at high frequencies). In fact, this “softness” was more clearly observed in the nonlinear regime: the nonlinear damping against step strain was weaker and the thinning under steady shear was less significant for the MS suspension than for the hard-core silica suspensions having the same φ_eff. These weaker nonlinearities of the concentrated MS particles with φ_eff∼ 0.63 (maximum volume fraction for random packing) suggested that the surface layers of those particles were mutually penetrating to provide the particles with a rather large mobility. Received: 10 July 2001 Accepted: 2 November 2001     

An empirical model predicting the viscosity of highly concentrated, bimodal dispersions with colloidal interactions

The relationship between particle size distribution and viscosity of concentrated dispersions is of great industrial importance, since it is the key to get high solids dispersions or suspensions. The problem is treated here experimentally as well as theoretically for the special case of strongly interacting colloidal particles. An empirical model based on a generalized Quemada equation is used to describe η as a function of volume fraction for mono- as well as multimodal dispersions. The pre-factor η˜ accounts for the shear rate dependence of η and does not affect the shape of the η vs φ curves. It is shown here for the first time that colloidal interactions do not show up in the maximum packing parameter and φ_max can be calculated from the particle size distribution without further knowledge of the interactions among the suspended particles. On the other hand, the exponent ɛ is controlled by the interactions among the particles. Starting from a limiting value of 2 for non-interacting either colloidal or non-colloidal particles, ɛ generally increases strongly with decreasing particle size. For a given particle system it then can be expressed as a function of the number average particle diameter. As a consequence, the viscosity of bimodal dispersions varies not only with the size ratio of large to small particles, but also depends on the absolute particle size going through a minimum as the size ratio increases. Furthermore, the well-known viscosity minimum for bimodal dispersions with volumetric mixing ratios of around 30/70 of small to large particles is shown to vanish if colloidal interactions contribute significantly. Received: 7 June 2000/Accepted: 12 February 2001     

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.     

Direct Numerical Simulation of Turbulent Flow over a Rectangular Trailing Edge

This paper describes a direct numerical simulation (DNS) study of turbulent flow over a rectangular trailing edge at a Reynolds number of 1000, based on the freestream quantities and the trailing edge thickness h; the incoming boundary layer displacement thickness δ^* is approximately equal to h. The time-dependent inflow boundary condition is provided by a separate turbulent boundary layer simulation which is in good agreement with existing computational and experimental data. The turbulent trailing edge flow simulation is carried out using a parallel multi-block code based on finite difference methods and using a multi-grid Poisson solver. The turbulent flow in the near-wake region of the trailing edge has been studied first for the effects of domain size and grid resolution. Then two simulations with a total of 256 × 512 × 64 (∼ 8.4×10⁶) and 512 × 1024 × 128 (∼ 6.7×10⁷) grid points in the computational domain are carried out to investigate the key flow features. Visualization of the instantaneous flow field is used to investigate the complex fluid dynamics taking place in the near-wake region; of particular importance is the interaction between the large-scale spanwise, or Kármán, vortices and the small-scale quasi-streamwise vortices contained within the inflow boundary layer. Comparisons of turbulence statistics including the mean flow quantities are presented, as well as the pressure distributions over the trailing edge. A spectral analysis applied to the force coefficient in the wall normal direction shows that the main shedding frequency is characterized by a Strouhal number based on h of approximately 0.118. Finally, the turbulence kinetic energy budget is analysed. Received 4 March 1999 and accepted 27 October 2000     

Breakdown in a boundary layer exposed to free-stream turbulence

The natural secondary breakdown modes of a flat plate boundary layer exposed to a free-stream turbulence level of 6.7% are characterised by experimental visualisations. The used experimental set-up is a water channel and no external triggering of the instabilities is used. The visualisations show the presence of two secondary instability modes, a sinuous (antisymmetric) and a varicose (symmetric) mode. The amplitudes of both instabilities increase, according to a non-linear profile, in upstream direction. The amplitudes also experience temporal growth. When a critical amplitude is reached, roll-up structures appear. These structures develop and finally merge, resulting in a turbulent spot. The development of the amplitude is similar in both modes. However, the amplitude of the varicose instability varies between 0.38 and 0.63 [2 π A/ λ] and the sinuous amplitude between 0.11 and 0.23 [2 π A/ λ]. The propagation velocities of the sinuous and varicose instabilities are 0.91 [u/u _blas] and 0.87 [u/u _blas] respectively. This shows that both modes are located in a low speed streak.     

A turbulent boundary layer over a two-dimensional rough wall

Particle image velocimetry (PIV) measurements and planar laser induced fluorescence (PLIF) visualizations have been made in a turbulent boundary layer over a rough wall. The wall roughness consisted of square bars placed transversely to the flow at a pitch to height ratio of λ/k = 11 for the PLIF experiments and λ/k = 8 and 16 for the PIV measurements. The ratio between the boundary layer thickness and the roughness height k/δ was about 20 for the PLIF and 38 for the PIV. Both the PLIF and PIV data showed that the near-wall region of the flow was populated by unstable quasi-coherent structures which could be associated to shear layers originating at the trailing edge of the roughness elements. The streamwise mean velocity profile presented a downward shift which varied marginally between the two cases of λ/k, in agreement with previous measurements and DNS results. The data indicated that the Reynolds stresses normalized by the wall units are higher for the case λ/k = 16 than those for λ/k = 8 in the outer region of the flow, suggesting that the roughness density effects could be felt well beyond the near-wall region of the flow. As expected the roughness disturbed dramatically the sublayer which in turn altered the turbulence production mechanism. The turbulence production is maximum at a distance of about 0.5k above the roughness elements. When normalized by the wall units, the turbulence production is found to be smaller than that of a smooth wall. It is argued that the production of turbulence is correlated with the form drag.