Plane Mixing Layer Research Articles

Effects of Anisotropy, Convection, and Relaxation on Nonlinear Reaction-Diffusion Systems

The effects of relaxation, convection, and anisotropy on a two-dimensional, two-equation system of nonlinearly coupled, second-order hyperbolic, advection–reaction–diffusion equations are studied numerically by means of a three-time-level linearized finite difference method. The formulation utilizes a frame-indifferent constitutive equation for the heat and mass diffusion fluxes, taking into account the tensorial character of the thermal diffusivity of heat and mass diffusion. This approach results in a large system of linear algebraic equations at each time level. It is shown that the effects of relaxation are small although they may be noticeable initially if the relaxation times are smaller than the characteristic residence, diffusion, and reaction times. It is also shown that the anisotropy associated with one of the dependent variables does not have an important role in the reaction wave dynamics, whereas the anisotropy of the other dependent variable results in transitions from spiral waves to either large or small curvature reaction fronts. Convection is found to play an important role in the reaction front dynamics depending on the vortex circulation and radius and the anisotropy of the two dependent variables. For clockwise-rotating vortices of large diameter, patterns similar to those observed in planar mixing layers have been found for anisotropic diffusion tensors.

The Planar Mixing Layer Flame (PMLF): a canonical configuration for studying non-premixed combustion chemistry and soot inception

The study of fuel chemistry and soot inception in non-premixed combustion can be advanced by characterizing flame configurations in which the advection and diffusion transport can be finely controlled, with the ability to decouple pyrolysis from oxidation. Also, the ideal flames to be investigated should be perturbed minimally by probes and thick enough for sampling techniques to yield spatially resolved measurements of their structure. The Planar Mixing Layer Flame (PMLF) configuration introduced herein is established between a fuel and an oxidizer slot jet adjacent to each other and shielded from the ambient air by annularly co-flowing inert nitrogen. The PMLF flow is kept laminar and steady by an impinging flat plate equipped with a rectangular exhaust slit opening which anchors the position of the hot combustion products via buoyancy. The PMLF is accessible to sampling and its flow stability is preserved when using any tested probe. The experiments are complemented with 2D- Computational Fluid Dynamics (CFD) modeling with detailed chemical kinetics. The results demonstrate that the PMLF has a self-similar boundary layer structure whose horizontal cross-sections are equivalent to properly selected and equally thick 1D- Counterflow Flames (CFs). The equivalence allows for excellent predictions of the PMLF thermochemical structure characterized experimentally but at a small fraction of the 2D-CFD computational cost. The 1D-CF equivalence affects even aromatics less than twofold despite their kinetics being known to be very sensitive to the temperature field. Importantly, the PMLF thickness is several millimeters and grows at increasing HABs so that the equivalent 1D-CFs have strain rates as small as 7.0 /s which cannot be studied in CF experiments. As a result, the PMLF emerges as a promising canonical non-premixed flame configuration for studying flame chemistry and soot inception on time scales of tens of milliseconds typical of many combustion applications.

Global dynamics and topology of two-phase mixing layer flow through simultaneous gas and liquid velocity measurements

This study reports the first time-resolved particle image velocimetry characterization of a planar two-phase mixing layer flow, whose velocity field is measured simultaneously in gas and liquid streams. Two parallel air and water flows meet downstream of a splitter plate, giving rise to an initially spanwise invariant configuration. The aim is to elucidate further the mechanisms leading to the flow breakup in gas-assisted atomization. The complete experimental characterization of the velocity field represents a database that could be used in data-driven reduced-order models to investigate the global behaviour of the flow system. After the analysis of a selected reference case, a parametric study of the flow behaviour is performed by varying the liquid ( $Re_l$ ) and gas ( $Re_g$ ) Reynolds numbers, and as a consequence also the gas-to-liquid dynamic pressure ratio ( $M$ ), shedding light on both time-averaged (mean) and unsteady velocity fields. In the reference case, it is shown that the mean flow exhibits a wake region just downstream of the splitter plate, followed by the development of a mixing layer. By increasing both $Re_l$ and $Re_g$ , the streamwise extent of the wake decreases and eventually vanishes, the flow resulting in a pure mixing layer regime. The spectral analysis of the normal-to-flow velocity fluctuations outlines different flow regimes by variation of the governing parameters, giving more insights into the global characteristics of the flow field. As a major result, it is found that at high $Re_g$ and $M$ values, the velocity fluctuations are characterized by low-frequency temporal oscillations synchronized in several locations within the flow field, which suggest the presence of a global mode of instability. The proper orthogonal decomposition of velocity fluctuations, performed in both gas and liquid phases, reveals finally that the synchronized oscillations are associated with a low-frequency dominant flapping mode of the gas–liquid interface. Higher-order modes correspond to interfacial wave structures travelling with the so-called Dimotakis velocity. For lower gas Reynolds numbers, the leading modes describe higher frequency fingers shedding at the interface.

Characterization of the mixing layer self-similarity with multiple parameters

In the paper, results from direct numerical simulation of a planar incompressible mixing layer spatially developing incompressible between two co-flowing laminar boundary layers are used to analyze a possibility for multiple flow parameters to achieve self-similarity within the same flow region. The Reynolds numbers for the boundary layers are 3930 and 2412 based on the free-stream velocities far above and below the splitter plate and the boundary layer thicknesses at the splitter plate trailing edge. The three-dimensional computational domain is sufficiently large for the mixing layer transition to fully turbulent far upstream the domain exit. The mixing layer growth is characterized using various definitions of the mixing layer thickness. It is shown that the proposed mixing layer thickness based on the gradient of the streamwise mean velocity in the transverse direction defines more accurately the area of turbulent mixing. Three regions of the flow linear growth are discovered using a rigorous approach, with only one of them being located within the fully turbulent mixing layer. Other parameters included in the flow self-similarity analysis are the streamwise and transverse mean velocities along with the Reynolds stresses. New normalization is proposed to observe self-similarity of the transverse mean velocity. The flow region where all considered parameters exhibit self-similarity is determined. It is shown that this region is limited by the “pulsating” streamwise distribution of the transverse mean velocity. The computational domain dimension along with the boundary conditions in the transverse direction for all considered parameters are suggested for the Reynolds-averaged Navier–Stokes simulations.

New hyperbolic tangent formula for mixing layer in vegetated flow

Flow velocity profile in the vegetal mixing layer was investigated using data collected in the present experiments and those reported in the literature, which indicates that the inflection point of the velocity profile does not coincide with the middle position of the vegetal mixing layer. Aiming to solve this problem, the hyperbolic tangent formula that is applicable to the plane mixing layer is modified by revising the inflection point and applying different characteristic scales in the free flow and penetration regions. For the problem of predicting flow velocity on top of vegetation, we constructed a new calculation method for flow velocity on top of vegetation based on correlation analysis. The proposed formula yields significant improvements in representing measured velocity profiles in the vegetated open channel flows. In comparison with previous ones, the modified formula performs better in exhibiting the similarity of measured velocity profiles over a wide range of flow and vegetation conditions. The research results provide technical support for the high precision calculation of the vegetated mixing layer.

Scaling patch analysis of planar turbulent mixing layers

Proper scales for the mean flow and Reynolds shear stress in planar turbulent mixing layers are determined from a scaling patch analysis of the mean continuity and momentum equations. By seeking an admissible scaling of the mean continuity equation, a proper scale for the mean transverse flow is determined as Vref=(dδ/dx)Uref, where dδ/dx is the growth rate of the mixing layer width and Uref=Uh−Ul is the difference between the velocity of the high speed stream Uh and the velocity of the low speed stream Ul. By seeking an admissible scaling for the mean momentum equation, a proper scale for the kinematic Reynolds shear stress is determined as Ruv,ref=UavgVref=[12Audδdx]Uref2, where Au=def(Uh−Ul)/(Uh+Ul) is the normalized velocity difference that emerges naturally in the admissible scaling of the mean momentum equation. Self-similar equations for the scaled mean transverse flow V* and Reynolds shear stress Ruv*=Ruv/Ruv,ref are derived from the mean continuity and mean momentum equations. Approximate equations for V* and Ruv* are developed and found to agree well with experimental data.

General Solution for Self-Similar Mixing Layers

Abstract In this note, a longstanding indeterminacy in the solution of self-similar planar mixing layers is resolved. It is shown that the boundary condition that offers the largest set of self-similar solutions is the specification of the vertical velocity at infinity, which also quantifies the value of the stream function and the net inflow/outflow there. This is referred to as the entrainment boundary condition. Other conditions that have been suggested to close the problem yield a subset of solutions and thus represent a loss of generality. The entrainment boundary condition determines the spatial growth rate and the spreading parameter for a given mixing layer, which were previously thought to be determined through experiment and empirical matching to a linearized version of the problem. These statements are validated using existing experimental data.

Evidence for Raupach et al.'s mixing-layer analogy in deep homogeneous urban-canopy flows

The mixing-layer analogy is due to Raupach, Finnigan & Brunet (Boundary-Layer Meteorol., vol. 25, 1996, pp. 351–382). In the analogy, the flow in the roughness sublayer of a homogeneous deep vegetation canopy boundary layer is analogous to a plane mixing layer rather than a surface layer. Evidence for the analogy includes the inflected velocity profile, which resembles the velocity profile in a plane mixing layer, and, most notably, the following estimate as a result of the Kelvin–Helmholtz instability: $\varLambda _x=8.1L_s$ , where $\varLambda _x$ is the spacing of the large-scale eddies, and $L_s$ is the shear length. The mixing-layer analogy has been very successful in vegetation canopy flow research, but has received only limited support in urban-canopy flow research. This work revisits Raupach et al.'s mixing-layer analogy, and we present the evidence for the mixing-layer analogy in urban-canopy flows: the exponential velocity profile in the canopy layer, i.e. $(U-U_c)/(U_h-U_c)=\exp (z/L_m)$ , and $L_m\sim [(U_h/U_c-1)(U_h/U_c+3)]^{-1}$ . Here, $z$ is the vertical coordinate, $L_m$ is the attenuation length and is a measure of the largest eddy in the canopy layer, $U_h$ is the wind speed at the canopy crest and $U_c$ is the velocity in the inactive layer. We conduct direct numerical simulations of various deep homogeneous urban-canopy flows and test the above two scalings. We also discuss why Raupach et al.'s analogy has not seen as many successes in urban-canopy flows as in vegetation canopy flows.

Numerical investigation on fluid characteristics of supersonic mixing layers with splitter plate in a confined space

High-speed airbreathing engines has been going on for more than a century, since French engineer Reina Lauren first developed the concept of the ramjet. Kelvin-Helmholtz (K-H) instability and compressibility effects on the evolution of supersonic mixing layer growing rate have been widely investigated in experimental and computational ways. The present study proposed a quasi-DNS solver which is written in Fortran 90, the mixing process of the supersonic planar mixing layers is analyzed in detail, especially, the process of the vortex rolling up, stretching, pairing and merging are also reported in present study. It is noteworthy that the mechanisms of K-H instability are analysed in detail, and the baroclinic term has a great influence on the growth of the supersonic mixing layer in confined space.

The importance of turbulent equilibrium for Reynolds-stress modeling

Turbulence equilibrium state is analyzed for the modeled Reynolds-stress transport equation, assuming the most general formulation of pressure–strain correlation. In a two-dimensional mean flow at a high-Reynolds number, an algebraic equation system is obtained, providing Reynolds-stress anisotropies as functions of pressure–strain model coefficients. Conversely, the equations provide calibration conditions for the model coefficients to predict specified equilibrium anisotropies. The predicted von-Kármán constant depends on the predicted equilibrium anisotropies and, hence, the pressure–strain model coefficients. Identical equilibrium anisotropies can be obtained with different sets of model coefficients. Identical equilibrium values of invariants of the Reynolds-stress anisotropy tensor can be achieved, despite the differing anisotropy components. Numerical simulations with the Speziale–Sarkar–Gatski (SSG) model, using different sets of model coefficients, confirm the results of the theoretical analysis. They show that the predicted equilibrium value of the Reynolds-shear stress anisotropy determines the predicted skin friction of a boundary layer as well as the predicted spreading rate of a plane mixing layer. However, different values and, hence, different sets of model coefficients are required for achieving good agreement with experimental data for both flows. Therefore, for general improvement of turbulence models, the set of model coefficients probably needs to be adapted to the local type of flow. The required classification is supposed to be suitable for machine learning methods.

A Compressible Turbulence Model for Pressure—Strain

This work focuses on the performance and validation of compressible turbulence models for the pressure-strain correlation. Considering the Launder Reece and Rodi (LRR) incompressible model for the pressure-strain correlation, Adumitroaie et al., Huang et al., and Marzougui et al., used different modeling approaches to develop turbulence models, taking into account compressibility effects for this term. Two numerical coefficients are dependent on the turbulent Mach number, and all of the remaining coefficients conserve the same values as in the original LRR model. The models do not correctly predict the compressible turbulence at a high-speed shear flow. So, the revision of these models is the major aim of this study. In the present work, the compressible model for the pressure-strain correlation developed by Khlifi−Lili, involving the turbulent Mach number, the gradient, and the convective Mach numbers, is used to modify the linear mean shear strain and the slow terms of the previous models. The models are tested in two compressible turbulent flows: homogeneous shear flow and the newly developed plane mixing layers. The predicted results of the proposed modifications of the Adumitroaie et al., Huang et al., and Marzougui et al., models and of its universal versions are compared with direct numerical simulation (DNS) and experiment data. The results show that the important parameters of compressibility in homogeneous shear flow and in the mixing layers are well predicted by the proposal models.

Non-Equilibrium Time-Relaxation Kinetic Model for Compressible Turbulence Modeling

For the first time, the non-equilibrium time-relaxation kinetic model (NTRKM) is proposed for compressible turbulence modeling on unresolved grids. Within the non-equilibrium time-relaxation framework, NTRKM is extended in the form of modified Bhatnagar-Gross-Krook model. Based on the first-order Chapman-Enskog expansion, NTRKM connects with the six-variable macroscopic governing equations. The first five governing equations correspond to the conservative laws in mass, momentum and total energy, while the sixth equation governs the evolution of unresolved turbulence kinetic energy Kutke. The unknowns in NTRKM, including turbulent relaxation time and source term, are determined by essential gradient-type assumption and standard dynamic modeling approach. Current generalized kinetic model on unresolved grids consequently offers a profound mesoscopic understanding for one-equation subgrid-scale turbulence kinetic energy Ksgs model in compressible large eddy simulation. To solve NTRKM accurately and robustly, a non-equilibrium gas-kinetic scheme is developed, which succeeds the well-established gas-kinetic scheme for simulating Navier-Stokes equations. Three-dimensional decaying compressible isotropic turbulence and temporal compressible plane mixing layer on unresolved grids are simulated to evaluate the generalized kinetic model and non-equilibrium gas-kinetic scheme. The performance of key turbulent quantities up to second-order statistics confirms that NTRKM is comparable with the widely-used eddy-viscosity Smagorinsky model (SM) and dynamic Smagorinsky model (DSM). Specifically, compared with the DNS solution in temporal compressible plane mixing layer, the performance of NTRKM is much closer with DSM and better than SM. This study provides a workable approach for compressible turbulence modeling on unresolved grids.

Production and migration of turbulent kinetic energy in bluff body shear layers

An experimental campaign on a bluff body of rectangular cross section, having a side-length ratio of 5:1 was carried out using both particle image velocimetry and hot wire anemometry at a Reynolds number of 3.04×104. Results show that under a slight angle of attack, the development of unsteadiness in the form of turbulent kinetic energy is significantly amplified on the pressure side of the body. A series of linear corrections fail to fully collapse the behavior of the curved shear onto the growth rates of the canonical planar mixing layer solution as expected. In pursuit of providing a physical mechanism to a non-linear contribution to shear layer transition, inspection of relevant components of the Kármán-Howarth-Monin-Hill equation suggest a re-orientation of energy in scale-space, resulting in a combined forward and reverse energy transfer. It is shown that this reorganization process and energy transfer depend on the orientation of energetic structures within the shear layer.

Characteristics of Incompressible Free Shear Flows and Implications for Turbulence Modeling

It is shown that in self-preserving incompressible free shear flows governed by the boundary-layer equations, a region must exist in which the Reynolds-stress anisotropies are constant. The theoretical result is confirmed by an analysis of well-established experimental data for the plane jet, the axisymmetric jet, and the plane mixing layer. The values of the corresponding Reynolds-stress anisotropies are determined, revealing differences between the corresponding eigensystems of these flows. While the shape of the corresponding tensor surfaces is similar, the corresponding principal axes have varying inclinations with respect to the flow-aligned coordinate system. Hence, the shear-stress anisotropy differs even in the case of almost identical eigenvalues or invariants of the anisotropy tensor. Numerical predictions can be improved by a calibration of the pressure-strain correlation of a Reynolds-stress model using the observed anisotropies in the constant layer. For an industrial application, this will require an automatic adjustment of model coefficients according to the local flow type, although a self-adaptive model that can switch automatically between the respective sets of tailored coefficients for each local flow type is still being sought.

Towards prediction of wind load on pylons for a neutral atmospheric boundary layer flow over two successive hills

In strong wind situations for a neutral atmospheric boundary layers, wind loads on pylons depend on both the mean velocity and the turbulence fluctuations, predicted by analytical models only for simple topographies. For complex topographies, only experiments or numerical simulations give access to the local flow conditions around the pylon. We present experimental and numerical results using a RANS model for the parametric study of the flow near two successive hills of height H and width L for varying distance λ between the tops. We focus on the modification of flow conditions on the second hill by the presence of the first hill. For λ/H≥19.2, the first hill does not modify the flow structure around the second hill. For decreasing values of λ/H below 19.2, a secondary turbulence intensity peak appears on the frontal part of the second hill. It is associated with the impact of the mixing layer generated at the top of the first hill, whose development is well described by analytical models of the planar mixing layer. These analytical models yields simple estimations of the turbulent kinetic energy k for the pylon’s wind load calculations. Different hill slopes and height ratios are also investigated.

Effect of dielectric barrier discharge plasma actuators on vortical structures in a mixing layer

The influence of spanwise-uniform electro-fluid-dynamic forcing applied by a dielectric barrier discharge (DBD) plasma actuator on the growth of a plane mixing layer and the dynamics of large-scale spanwise vortices are investigated experimentally. A two-dimensional mixing layer formed between two streams of air with different velocities is employed for this study. Quantitative spatio-temporal measurement of the flow field is acquired using high-speed planar particle image velocimetry. The DBD actuator was constructed such that it imparts perturbations into the splitter-plate boundary layer formed by the high-velocity fluid stream, close to the trailing edge. Through this, the fundamental Kelvin–Helmholtz instability of the current mixing layer and its first sub-harmonic are forced. Forcing the fundamental instability results in the inhibition of vortex pairing due to the attenuation of sub-harmonic instabilities, and thus, mixing layer growth is halted in the vicinity of the trailing edge. With sub-harmonic instability forcing, neighboring vortices interact with each other and amalgamate together through mutual induction. This results in a higher growth rate compared to the unforced mixing layer at the streamwise location of this vortex interaction. Eventually, the growth rate of the forced mixing layers becomes similar to that of the unforced case. These results demonstrate the influence of the applied forcing on the spectral signature, growth, and stability characteristics of the plane mixing layer and the dynamics of the coherent vortical structures.

Shock-induced vorticity variation model of supersonic planar mixing layers

Vorticity variation in a supersonic planar mixing layer interacting with an oblique shock wave is investigated analytically and numerically. A model that simplifies the mixing layer to a discontinuous flow is established to solve the post-shock flow parameters, and it is validated through qualitative and quantitative comparisons with the Buttsworth’s model and numerical results. A model to estimate the shock-induced Maximum Vorticity Amplification (MVA) is obtained, which agrees well with the numerical results. The model could estimate the growth rate and maximum vorticity of the shocked mixing layer. The vorticity of the mixing layer is amplified by the shock impingement, even though the vorticity thickness decreases, which can improve the mixing performance for different practical applications.

Compressibility Effects on Large Structures and Entrainment Length Scales in Mixing Layers

In this work, three-component planar velocity measurements are analyzed to identify the effects of compressibility on mixing layer turbulence, with a focus on the evolution of large-scale structures, dominant spatial eigenmodes, and entrainment length scales. Velocity measurements obtained via stereoscopic particle image velocimetry for five different dual-stream air planar mixing layers with convective Mach numbers of , 0.38, 0.55, 0.69, and 0.88 are analyzed. Results of two-point correlations reveal that length scales of the streamwise velocity fluctuations increase in both the streamwise and transverse directions, whereas the length scales of the transverse velocity fluctuations decrease in the transverse direction. To further investigate these trends, the spatial organization, shape, and dynamics of large-scale turbulent structures are examined via two-dimensional spatial velocity correlations, proper orthogonal decomposition, and linear stochastic estimation. These quantitative techniques support qualitative findings in the literature that report coherent, round roller structures in incompressible mixing layers becoming flattened and elongated longitudinally with increased compressibility. This evolution is likely due to a dominant pulsing motion present for the higher cases and can be linked to compressibility effects on the entrainment mechanisms. Length scales of entrainment are determined using autocorrelations of normal flow velocity components along instantaneously identified turbulent/non-turbulent interfaces and are found to decrease with increasing . This result can heuristically be interpreted as small-scale nibbling being dominant for higher , whereas larger-scale engulfment contributes more in nearly incompressible mixing layers.

On the use of optimization techniques for turbulence model calibration

This article discusses the use of numerical optimization procedures to aid in the calibration of turbulence model coefficients. Such methods would increase the rigor and repeatability of the calibration procedure by requiring clearly defined and objective optimization metrics, and could be used to identify unique combinations of coefficient values for specific flow problems. The approach is applied to the re-calibration of an explicit algebraic Reynolds stress model for the incompressible planar mixing layer using the Nelder–Mead simplex algorithm and a micro-genetic algorithm with minimally imposed constraints. Three composite fitness functions, each based upon the error in the mixing layer growth rate and the normal and shear components of the Reynolds stresses, are investigated. The results demonstrate a significant improvement in the target objectives through the adjustment of three pressure-strain coefficients. Adjustments of additional coefficients provide little further benefit. Issues regarding the effectiveness of the fitness functions and the efficiency of the optimization algorithms are also discussed.

Turbulent Flow in Plant Canopies: Historical Perspective and Overview

Studying the microclimate of plant canopies has long motivated scientists in various research fields such as agronomy, ecology or silviculture, and almost a century has passed since the first measurements of wind speed in a forest stand were published in the scientific literature. The behaviour of wind in canopies is an essential component of their microclimate, which largely conditions the rate of exchange of heat, water vapour, and other gases and particles of interest with the atmosphere. This review examines the evolution of our understanding of turbulent flow in plant canopies, focussing on the period that covers the last fifty years (1970–2020). We first describe how our knowledge and ideas have evolved since canopy flow became a topic of interest, and show how the 1970s was a pivotal decade in this field. Until then, canopy turbulence was considered to result from the superposition of standard surface-layer turbulence and small-scale turbulence generated in the wakes of plant elements. However, it was progressively found that the flow in plant canopies is dominated by large coherent structures, giving canopy turbulence unique characteristics. We thus describe the particular nature and structure of canopy flows, based on experimental observations accumulated over several decades. We show how canopy turbulence was reconsidered on the basis of a now widely-accepted analogy with a plane mixing layer, and we examine the significance of a key parameter, the “canopy-shear length scale”. Investigating the effects of canopy density and atmospheric stability, we then discuss the extent of the mixing-layer analogy and the limits of our current understanding of canopy turbulence. Finally, we review the modelling tools used in this field and show how their development has evolved to date to meet our needs. In conclusion, we present a historical summary of the evolution of this research field and suggest future directions.