Incompressible Mixing Layer Research Articles

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.

Ensemble data assimilation-based mixed subgrid-scale model for large-eddy simulations

An ensemble Kalman filter (EnKF)-based mixed model (EnKF-MM) is proposed for the subgrid-scale (SGS) closure in the large-eddy simulation (LES) of turbulence. The model coefficients are determined through the EnKF-based data assimilation technique. The direct numerical simulation (DNS) results are filtered to obtain the benchmark data for the LES. Reconstructing the correct kinetic energy spectrum of the filtered DNS (fDNS) data has been adopted as the target for the EnKF to optimize the coefficient of the functional part in the mixed model. The proposed EnKF-MM framework is subsequently tested in the LES of both the incompressible homogeneous isotropic turbulence and turbulent mixing layer. The performance of the LES is comprehensively examined through the predictions of the flow statistics including the velocity spectrum, the probability density functions (PDFs) of the SGS stress, the PDF of the strain rate, and the PDF of the SGS energy flux. The structure functions, the evolution of turbulent kinetic energy, the mean flow, the Reynolds stress profile, and the iso-surface of the Q-criterion are also examined to evaluate the spatial–temporal predictions by different SGS models. The results of the EnKF-MM framework are consistently more satisfying compared to the traditional SGS models, including the dynamic Smagorinsky model, the dynamic mixed model, and the velocity gradient model, demonstrating its great potential in the optimization of SGS models for the LES of turbulence.

Sensitivity Analysis of Direct Numerical Simulation of a Spatially Developing Turbulent Mixing Layer to the Domain Dimensions

Abstract Understanding spatial development of a turbulent mixing layer is essential for many engineering applications. However, the flow development is difficult to replicate in physical or numerical experiments. For this reason, the most attractive method for the mixing layer analysis is the direct numerical simulation (DNS), with the most control over the simulation inputs and free from modeling assumptions. On the other hand, the DNS cost often prevents conducting the sensitivity analysis of the simulation results to variations in the numerical procedure and thus, separating numerical and physical effects. In this paper, effects of the computational domain dimensions on statistics collected from DNS of a spatially developing incompressible turbulent mixing layer are analyzed with the focus on determining the domain dimensions suitable for studying the flow asymptotic state. In the simulations, the mixing layer develops between two coflowing laminar boundary layers formed on two sides of a sharp-ended splitter plate of a finite thickness with characteristics close to those of the untripped boundary layers in the experiments by Bell and Mehta, AIAA J., 28(12), 2034 (1990). The simulations were conducted using the spectral-element code Nek5000.

Predominance of pressure transport in spatial energy budget for a mixing layer approaching absolute instability

In this letter, we report the outcome of a spatial energy budget performed for the linear convective instability of the plane incompressible mixing layer within the inviscid framework. We find that, as the critical condition for the onset of absolute instability is approached, the integrated pressure-transport term becomes increasingly more prominent as compared to the integrated production term and it dominates the energy budget completely at the critical condition. This implies that, near the threshold of absolute instability, the growth of disturbances is almost entirely due to the pressure transport mechanism (rather than the production mechanism), which is a striking result. The part of the pressure-transport term that represents the work done by the fluctuating pressure forces is seen to be primarily responsible for the observed shift in the energy balance. These results can help us better understand the physical processes causing absolute instability in a mixing layer. In particular, the redistribution of disturbance energy in streamwise direction by fluctuating pressure, which is "non-local" in character for incompressible flows, seems to play a key role in this respect.

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.

Variable-density effects in incompressible non-buoyant shear-driven turbulent mixing layers

Abstract

Flow structures of a lobed mixer and effects of streamwise vortices on mixing enhancement

Large-scale vortices downstream of a lobed mixer are investigated experimentally using a nanoparticle-based planar laser scattering experimental system. The results are contrasted with additional measurements of a flat plate and a convoluted plate. Three streamwise vortices form downstream of the lobed mixer due to the pressure difference between the peak and trough regions of the trailing edge. This is not the case for the convoluted plate, where the streamwise structures are suppressed and do not appear downstream. Comparison of vortex sizes indicates that streamwise vortices contribute almost 80% of mixing enhancement, which is substantially higher than what could be expected from the increase of the interfacial surface area, especially in the far field of the present compressible mixing layer. It is also quite different from the effects of streamwise vortices on the mixing enhancement in incompressible mixing layers. That no streamwise vortices appear downstream of the convoluted plate indicates that the pressure difference around the corners of the peak and trough regions in the lobed mixer plays an important role for the appearance of the streamwise vortices. Three-dimensional views of the streamwise vortices show that a significant amount of fluid is entrained from the lower stream to the upper stream, resulting in the development of the streamwise vortices for the lobed mixer. The interaction of the streamwise and spanwise vortices then leads to the formation of a large number of small-scale vortices. These two mechanisms enlarge the interfacial surface area of the two streams greatly and substantially improve the turbulent mixing prominently. In addition, the presence of vortex clusters and T-shape vortices in the mixing layer indicate that they are common topological structures in such flows.

The need for prediction in feedback control of a mixing layer

We present the first application of a Smith predictor in the model-based feedback control of a convectively unstable flow. The two-dimensional incompressible mixing layer is chosen as a plant and integrated with a finite-element Navier–Stokes solver. The flow is actuated by a modulation of the inflow mimicking a pulsed jet actuator. The state is monitored by a downstream velocity sensor. The control goal is to command the fluctuation level at the sensor location. As an additional challenge, the incoming shear-flow profile is perturbed by the small noise level. The starting point of this study is a standard low-order control without prediction. A Smith predictor is included to compensate for the convective delay from actuation to sensing. The closed-loop results with a Smith predictor are distinctly better than without it. The design of the linear robust controllers is performed with an open-loop control. We conjecture that predictors may augment feedback control for a large class of convectively unstable flows with significant time-delay from actuation to sensing.

Gradients estimation from random points with volumetric tensor in turbulence

We present an estimation method of fully-resolved/coarse-grained gradients from randomly distributed points in turbulence. The method is based on a linear approximation of spatial gradients expressed with the volumetric tensor, which is a 3×3 matrix determined by a geometric distribution of the points. The coarse grained gradient can be considered as a low pass filtered gradient, whose cutoff is estimated with the eigenvalues of the volumetric tensor. The present method, the volumetric tensor approximation, is tested for velocity and passive scalar gradients in incompressible planar jet and mixing layer. Comparison with a finite difference approximation on a Cartesian grid shows that the volumetric tensor approximation computes the coarse grained gradients fairly well at a moderate computational cost under various conditions of spatial distributions of points. We also show that imposing the solenoidal condition improves the accuracy of the present method for solenoidal vectors, such as a velocity vector in incompressible flows, especially when the number of the points is not large. The volumetric tensor approximation with 4 points poorly estimates the gradient because of anisotropic distribution of the points. Increasing the number of points from 4 significantly improves the accuracy. Although the coarse grained gradient changes with the cutoff length, the volumetric tensor approximation yields the coarse grained gradient whose magnitude is close to the one obtained by the finite difference. We also show that the velocity gradient estimated with the present method well captures the turbulence characteristics such as local flow topology, amplification of enstrophy and strain, and energy transfer across scales.

Linear stability analysis of interactions between mixing layer and boundary layer flows

The linear instabilities of incompressible confluent mixing layer and boundary layer were analyzed. The mixing layers include wake, shear layer and their combination. The mean velocity profile of confluent flow is taken as a superposition of a hyperbolic and exponential function to model a mixing layer and the Blasius similarity solution for a flat plate boundary layer. The stability equation of confluent flow was solved by using the global numerical method. The unstable modes associated with both the mixing and boundary layers were identified. They are the boundary layer mode, mixing layer mode 1 (nearly symmetrical mode) and mode 2 (nearly anti-symmetrical mode). The interactions between the mixing layer stability and the boundary layer stability were examined. As the mixing layer approaches the boundary layer, the neutral curves of the boundary layer mode move to the upper left, the resulting critical Reynolds number decreases, and the growth rate of the most unstable mode increases. The wall tends to stabilize the mixing layer modes at low frequency. In addition, the mode switching behavior of the relative level of the spatial growth rate between the mixing layer mode 1 and mode 2 with the velocity ratio is found to occur at low frequency.

Entrainment in a compressible turbulent shear layer

Direct numerical simulations (DNS) of temporally evolving shear layers have been performed to study the entrainment of irrotational flow into the turbulent region across the turbulent/non-turbulent interface (TNTI). Four cases with convective Mach number from 0.2 to 1.8 are used. Entrainment is studied via two mechanisms; nibbling, considered as vorticity diffusion across the TNTI, and engulfment, the drawing of the pockets of the outside irrotational fluid into the turbulent region. The mass flow rate due to nibbling is calculated as the product of the entrained mass flux with the surface area of the TNTI. It is found that increasing the convective Mach number results in a decrease of the average entrained mass flux and the surface area of the TNTI. For the incompressible shear layer the local entrained mass flux is shown to be highly correlated with the viscous terms. However, as the convective Mach number increases, the mass fluxes due to the baroclinic and the dilatation terms play a more important role in the local entrainment process. It is observed that the entrained mass flux is dependent on the local dilatation and geometrical shape of the TNTI. For a compressible shear layer, most of the entrainment of the irrotational flow into the turbulent region due to nibbling is associated with the compressed regions on the TNTI. As the convective Mach number increases, the percentage of the compressed regions on the TNTI decreases, resulting in a reduction of the average entrained mass flux. It is also shown that the local shape of the interface, looking from the turbulent region, is dominated by concave shaped surfaces with radii of curvature of the order of the Taylor length scale. The average entrained mass flux is found to be larger on highly curved concave shaped surfaces regardless of the level of dilatation. The mass fluxes due to vortex stretching, baroclinic torque and the shear stress/density gradient terms are weak functions of the local curvatures on the TNTI, whereas the mass fluxes due to dilatation and viscous diffusion plus the viscous dissipation terms have a stronger dependency on the local curvatures. As the convective Mach number increases, the probability of finding highly curved concave shaped surfaces on the TNTI decreases, whereas the probability of finding flatter concave and convex shaped surfaces increases. This results in a decrease of the average entrained mass flux across the TNTI. Similar to the previous works on jets, the results show that the contribution of the engulfment to the total entrainment is small for both incompressible and compressible mixing layers. It is also shown that increasing the convective Mach number decreases the velocities associated with the entrainment, i.e. induced velocity, boundary entrainment velocity and local entrainment velocity.

Sensitivity of the two-dimensional shearless mixing layer to the initial turbulent kinetic energy and integral length scale.

The shearless mixing layer is generated from the interaction of two homogeneous isotropic turbulence (HIT) fields with different integral scales ℓ_{1} and ℓ_{2} and different turbulent kinetic energies E_{1} and E_{2}. In this study, the sensitivity of temporal evolutions of two-dimensional, incompressible shearless mixing layers to the parametric variations of ℓ_{1}/ℓ_{2} and E_{1}/E_{2} is investigated. The sensitivity methodology is based on the nonintrusive approach; using direct numerical simulation and generalized polynomial chaos expansion. The analysis is carried out at Re_{ℓ_{1}}=90 for the high-energy HIT region and different integral length scale ratios 1/4≤ℓ_{1}/ℓ_{2}≤4 and turbulent kinetic energy ratios 1≤E_{1}/E_{2}≤30. It is found that the most influential parameter on the variability of the mixing layer evolution is the turbulent kinetic energy while variations of the integral length scale show a negligible influence on the flow field variability. A significant level of anisotropy and intermittency is observed in both large and small scales. In particular, it is found that large scales have higher levels of intermittency and sensitivity to the variations of ℓ_{1}/ℓ_{2} and E_{1}/E_{2} compared to the small scales. Reconstructed response surfaces of the flow field intermittency and the turbulent penetration depth show monotonic dependence on ℓ_{1}/ℓ_{2} and E_{1}/E_{2}. The mixing layer growth rate and the mixing efficiency both show sensitive dependence on the initial condition parameters. However, the probability density function of these quantities shows relatively small solution variations in response to the variations of the initial condition parameters.

Acceleration feature points of unsteady shear flows

A framework for extracting features in 2D transient flows, based on the acceleration field to ensure Galilean invariance is proposed in this paper. The minima of the acceleration magnitude (a superset of acceleration zeros) are extracted and discriminated into vortices and saddle points, based on the spectral properties of the velocity Jacobian. The extraction of topological features is performed with purely combinatorial algorithms from discrete computational topology. The feature points are prioritized with persistence, as a physically meaningful importance measure. These feature points are tracked in time with a robust algorithm for tracking features. Thus, a space-time hierarchy of the minima is built and vortex merging events are detected. We apply the acceleration feature extraction strategy to three two-dimensional shear flows: (1) an incompressible periodic cylinder wake, (2) an incompressible planar mixing layer and (3) a weakly compressible planar jet. The vortex-like acceleration feature points are shown to be well aligned with acceleration zeros, maxima of the vorticity magnitude, minima of the pressure field and minima of λ 2 .

Optimal nonlinear eddy viscosity in Galerkin models of turbulent flows

AbstractWe propose a variational approach to the identification of an optimal nonlinear eddy viscosity as a subscale turbulence representation for proper orthogonal decomposition (POD) models. The ansatz for the eddy viscosity is given in terms of an arbitrary function of the resolved fluctuation energy. This function is found as a minimizer of a cost functional measuring the difference between the target data coming from a resolved direct or large-eddy simulation of the flow and its reconstruction based on the POD model. The optimization is performed with a data-assimilation approach generalizing the 4D-VAR method. POD models with optimal eddy viscosities are presented for a 2D incompressible mixing layer at $\mathit{Re}=500$ (based on the initial vorticity thickness and the velocity of the high-speed stream) and a 3D Ahmed body wake at $\mathit{Re}=300\,000$ (based on the body height and the free-stream velocity). The variational optimization formulation elucidates a number of interesting physical insights concerning the eddy-viscosity ansatz used. The 20-dimensional model of the mixing-layer reveals a negative eddy-viscosity regime at low fluctuation levels which improves the transient times towards the attractor. The 100-dimensional wake model yields more accurate energy distributions as compared to the nonlinear modal eddy-viscosity benchmark proposed recently by Östh et al. (J. Fluid Mech., vol. 747, 2014, pp. 518–544). Our methodology can be applied to construct quite arbitrary closure relations and, more generally, constitutive relations optimizing statistical properties of a broad class of reduced-order models.

Cluster-based reduced-order modelling of a mixing layer

AbstractWe propose a novel cluster-based reduced-order modelling (CROM) strategy for unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger’s group (Burkardt, Gunzburger & Lee,Comput. Meth. Appl. Mech. Engng, vol. 196, 2006a, pp. 337–355) and transition matrix models introduced in fluid dynamics in Eckhardt’s group (Schneider, Eckhardt & Vollmer,Phys. Rev. E, vol. 75, 2007, art. 066313). CROM constitutes a potential alternative to POD models and generalises the Ulam–Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron–Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the snapshot data are clustered into a small number of representative states, called centroids, in the state space. These centroids partition the state space in complementary non-overlapping regions (centroidal Voronoi cells). Departing from the standard algorithm, the probabilities of the clusters are determined, and the states are sorted by analysis of the transition matrix. Second, the transitions between the states are dynamically modelled using a Markov process. Physical mechanisms are then distilled by a refined analysis of the Markov process, e.g. using finite-time Lyapunov exponent (FTLE) and entropic methods. This CROM framework is applied to the Lorenz attractor (as illustrative example), to velocity fields of the spatially evolving incompressible mixing layer and the three-dimensional turbulent wake of a bluff body. For these examples, CROM is shown to identify non-trivial quasi-attractors and transition processes in an unsupervised manner. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, for the identification of precursors to desirable and undesirable events, and for flow control applications exploiting nonlinear actuation dynamics.

LES Study of 3D Incompressible Temporal Mixing Layer Using Different Well-Known Subgrid Scale (SGS) Models

In this paper, 3D incompressible temporal mixing layer at three different Reynolds numbers has been investigated using large eddy simulation (LES) turbulence model. To consider the effects of the subgrid scales, six different well-known SGS models, Smagorinsky, dynamic Smagorinsky, Yoshizava one equation, Spalart–Allmaras one equation, Lagrangian dynamic Smagorinsky two equation (LDS) and six-equations Reynolds stress (RSM) were applied. The computed LES results for vorticity thickness, average total kinetic energy, and Reynolds shear stress components of stress tensor revealed that the LDS and RSM models agree well with DNS data. Different steps of forming of coherent flow structures at different times were also discussed.

Redefining Mixing Length in Turbulent Mixing Layer in Terms of Shear-Induced Vorticity

An incompressible turbulent planar mixing layer is composed of two different flow types in its flow field, namely a shear layer in the central region and two free streams in each outer high- and low-speed sides. Shear layer is formed right after the trailing edge of the splitter plate and develops stream-wisely through successively distinct regions, namely the near field region and the self-preserving region. A new definition of the mixing length (lω) is proposed on the basis of an effectively pure shear-induced vorticity component (ΩSH) by means of a triple decomposition method, that is, lω = yH - yL where yH and yL are the two transverse positions, at which |ΩSH| normalized with the maximum ∂U/∂y at the virtual origin is equal to 0.05, in the high- and low-speed free stream sides, respectively. It is shown that the linear growth rate of lω along stream-wise distance can be, then, used as one of the necessary and sufficient conditions for identifying the achievement of the self-preserving state in turbulent mixing layer.

The critical Reynolds number of a laminar incompressible mixing layer from minimal composite theory

According to parallel-flow theory based on the Orr–Sommerfeld equation, a mixing layer is unstable at all Reynolds numbers. However this is untenable from energy considerations, which demand that there exist a non-zero Reynolds number below which disturbances cannot extract net energy from the mean flow. It is shown here that a linear stability analysis of similarity solutions of the plane mixing layer, including the effects of flow non-parallelism using the minimal composite theory and the properties of adjoints, following Govindarajan & Narasimha (Theor. Comput. Fluid Dyn. vol. 19, 2005, p. 229) resolves the issue by yielding a non-zero critical Reynolds number for co-flowing streams of any velocity ratio. The critical Reynolds number for the total disturbance kinetic energy is found to be close to 30 for all velocity ratios in the range from zero to unity.

Similarity-based constitutive relations for local mass fluxes in incompressible mixing layers

The local concentrations of the two fluids within a mixing layer produced by an interfacial instability are determined by their individual continuity equations. Solution of these equations requires constitutive relations for the local mass fluxes of the two fluids. We derive explicit analytical expressions for these fluxes in planar incompressible mixing layers characterized by a single integral length scale h(t), which is presumed to be provided by a suitable mix or turbulence model. Elementary scaling arguments imply that in mixing layers of this type, the mean volume fraction profile alpha(x,t) depends on x and t only through the similarity variable s=x/h(t). The continuity equations determine alpha(s) in terms of the mass fluxes. We invert this relationship to obtain constitutive relations for the mass fluxes in terms of an arbitrary given alpha(s) obtained from experiment or direct numerical simulation. These relations can then be employed in more complicated situations (e.g., problems lacking spatial symmetry) in which the volume fraction is not known a priori. The resulting mass fluxes are not inherently diffusional in character, but can be transformed into an alternative equivalent diffusional form that may be advantageous for computational purposes. These results are further generalized to the case in which the light and heavy regions of the mixing layer exhibit different piecewise self-similar scaling behavior characterized by two independent integral length scales h1(t) h2(t) and that depend differently on time. The validity and practical utility of the formulation are confirmed by a test calculation.