Turbulent Free Shear Flows Research Articles

Generalized scaling laws for the irrotational motions bordering a turbulent region.

In turbulent free shear flows such as jets and wakes, and also in turbulent boundary layers, the turbulent region is bounded by a region of irrotational flow where the magnitude of the potential velocity fluctuations can be very high. This is particularly true close to the turbulent-nonturbulent interface layer (TNTI) that separates the regions of turbulent (rotational) and nonturbulent (irrotational) fluid motion in these flows. Previous works have shown that for distances from the TNTI x_{2} much bigger than the integral scale L in the nearby turbulent region (x_{2}≫L), the variance of the velocity fluctuations 〈u_{i}^{2}〉 (i=1,2,3) depends on the shape of the kinetic energy spectrum in the infrared region E(k)∼k^{n} [O. M. Phillips, Proc. Camb. Phil. Soc. 51, 220 (1955)10.1017/S0305004100030073; Xavier etal., J. Fluid Mech. 918, A3 (2021)10.1017/jfm.2021.296]. Using rapid distortion theory, we derive the generalized scaling laws for the potential velocity fluctuations, at distances sufficiently far from the TNTI layer, for any value of n. While the cases n=4 (Batchelor turbulence) and n=2 (Saffman turbulence) have been previously derived, with 〈u_{i}^{2}〉∼x_{2}^{-4} and 〈u_{i}^{2}〉∼x_{2}^{-3}, for n=4 and n=2, respectively [O. M. Phillips, Proc. Camb. Phil. Soc. 51, 220 (1955)10.1017/S0305004100030073; Xavier etal., J. Fluid Mech. 918, A3 (2021)10.1017/jfm.2021.296.], we extend these results by including any other value of n. In particular, we obtain 〈u_{i}^{2}〉∼x_{2}^{-2} and 〈u_{i}^{2}〉∼x_{2}^{-4}, for n=1 and n≥5, respectively, while n=3 yields 〈u_{i}^{2}〉∼x_{2}^{-4}ln(x_{2}). These theoretical results are confirmed by direct numerical simulations of turbulent fronts evolving into an irrotational flow region in the absence of mean shear.

Optical temperature measurement in unsteady plasma free jet

An Argon plasma free jet is investigated using spectrally narrow bandpass filtered high-speed imaging. The images were captured at 16 kHz with an exposure time of 3.9 μs and then calibrated for absolute radiance. The free jet exhibited behaviour consistent with turbulent free shear flow and maintains an axisymmetric shape. Significant local fluctuations were observed over time, growing in strength and size as the flow convected downstream. Assuming local thermodynamic equilibrium and self-similar free-jet temperature profiles, the flow radiance is used to determine the local plasma temperature and the jet width. Then, both steady and unsteady flow models were applied to account for the jet fluctuation. In regions of low fluctuations near the nozzle exit, both models show good agreement for centreline temperatures, measuring approximately 11 200 K. In regions of significant fluctuations, the assumption of steady flow leads to an overestimation of 32% for temperature, 18% for jet width, and 41% for total jet power. The unsteady analysis approach results in lower temperatures and smaller jet widths while simultaneously satisfying momentum and energy conservation.

The similarity theory of free turbulent shear flows of viscoelastic fluids

A new theory is formulated for the description of the conformation state of the polymer chains in free turbulent shear flows of viscoelastic fluids. Using self-similarity arguments and new scaling relations for the turbulent flux of conformation tensor we show the existence of minimum and maximum solvent dissipation reduction asymptotes, and four different polymer deformation regimes. The similarities with the maximum drag reduction asymptote of turbulent pipe flow is discussed and new scaling laws are obtained for all components of the mean conformation tensor at each deformation regime. Analytical solutions for the self-similar transverse profiles of the conformation tensor components are also obtained, providing the complete solution for the mean flow problem at the far field. The analysis is developed for both planar jets and wakes and covers the two limits of shear flows, with large and small velocity differences, respectively. Comparisons of the new theoretical results with several direct numerical simulations employing the FENE-P rheological model show excellent agreement.

Turbulence model form errors in separated flows

Physics-based uncertainty quantification is applied to a boundary layer over a flat plate with a statistically stationary separation bubble to assess sources and dynamics of model error (ME) in two-equation Reynolds-Averaged Navier-Stokes turbulence models in separated flows. Two distinct ME modes are found that correspond to the qualitative behavior of ME in a turbulent wall-bounded flow (upstream of the separation bubble) and turbulent free-shear flow (within the separation bubble), with a superposition of these ME modes in intermediate regions. A complex understanding of ME results as it changes throughout the flow, but is ultimately comprised of the elements of canonical flows.

Structure of turbulence in temporal planar jets

A detailed analysis of the structure of turbulence in a temporal planar turbulent jet is reported. Instantaneous snapshots of the flow and three-dimensional spatial correlation functions are considered. It is found that the flow is characterized by large-scale spanwise vortices whose motion is felt in the entire flow field. Superimposed to this large-scale motion, a hierarchy of turbulent structures is present. The most coherent ones take the form of quasi-streamwise vortices and high and low streamwise velocity streaks. The topology of these interacting structures is analyzed by quantitatively addressing their shape and size in the different flow regions. Such information is recognized to be relevant for a structural description of the otherwise disorganized motion in turbulent free-shear flows and can be used for the assessment of models based on coherent structure assumptions. Finally, the resulting scenario provides a phenomenological description of the elementary processes at the basis of turbulence in free-shear flows.

A parametric and feasibility study for data sampling of the dynamic mode decomposition: range, resolution, and universal convergence states

Scientific research and engineering practice often require the modeling and decomposition of nonlinear systems. The dynamic mode decomposition (DMD) is a novel Koopman-based technique that effectively dissects high-dimensional nonlinear systems into periodically distinct constituents on reduced-order subspaces. As a novel mathematical hatchling, the DMD bears vast potentials yet an equal degree of unknown. This effort investigates the nuances of DMD sampling with an engineering-oriented emphasis. It aimed at elucidating how sampling range and resolution affect the convergence of DMD modes. We employed the most classical nonlinear system in fluid mechanics as the test subject—the turbulent free-shear flow over a prism—for optimal pertinency. We numerically simulated the flow by the dynamic-stress Large-Eddies Simulation with Near-Wall Resolution. With the large-quantity, high-fidelity data, we parametrized and identified four global convergence states: Initialization, Transition, Stabilization, and Divergence with increasing sampling range. Results showed that Stabilization is the optimal state for modal convergence, in which DMD output becomes independent of the sampling range. The Initialization state also yields sufficient accuracy for most system reconstruction tasks. Moreover, defying popular beliefs, over-sampling causes algorithmic instability: as the temporal dimension, n, approaches and transcends the spatial dimension, m (i.e., m < n), the output diverges and becomes meaningless. Additionally, the convergence of the sampling resolution depends on the mode-specific dynamics, such that the resolution of 15 frames per cycle for target activities is suggested for most engineering implementations. Finally, a bi-parametric study revealed that the convergence of the sampling range and resolution are mutually independent.

Exploiting self-organized criticality in strongly stratified turbulence

A multiscale reduced description of turbulent free shear flows in the presence of strong stabilizing density stratification is derived via asymptotic analysis of the Boussinesq equations in the simultaneous limits of small Froude and large Reynolds numbers. The analysis explicitly recognizes the occurrence of dynamics on disparate spatiotemporal scales, yielding simplified partial differential equations governing the coupled evolution of slow large-scale hydrostatic flows and fast small-scale isotropic instabilities and internal waves. The dynamics captured by the coupled reduced equations is illustrated in the context of two-dimensional strongly stratified Kolmogorov flow. A noteworthy feature of the reduced model is that the fluctuations are constrained to satisfy quasilinear (QL) dynamics about the comparably slowly varying large-scale fields. Crucially, this QL reduction is not invoked as an ad hoc closure approximation, but rather is derived in a physically relevant and mathematically consistent distinguished limit. Further analysis of the resulting slow–fast QL system shows how the amplitude of the fast stratified-shear instabilities is slaved to the slowly evolving mean fields to ensure the marginal stability of the latter. Physically, this marginal stability condition appears to be compatible with recent evidence of self-organized criticality in both observations and simulations of stratified turbulence. Algorithmically, the slaving of the fluctuation fields enables numerical simulations to be time-evolved strictly on the slow time scale of the hydrostatic flow. The reduced equations thus provide a solid mathematical foundation for future studies of three-dimensional strongly stratified turbulence in extreme parameter regimes of geophysical relevance and suggest avenues for new sub-grid-scale parametrizations.

Nonequilibrium turbulent dissipation in buoyant axisymmetric plume

This paper established the existence of non-Kolmogorov turbulence in free shear flows. The authors propose a set of dissipation laws and link them with the spreading rates as well as the entrainment coefficient, which varies with the rise of the plume in contrast to the Kolmogorov case. The work also show how the parameters of two stretching transformations determine both Kolmogorov and non-Kolmogorov turbulence.

Universality of small-scale motions within the turbulent/non-turbulent interface layer

Abstract

Internal layers in turbulent free-shear flows

In turbulent free-shear flows, layers of intense shear bound regions of nearly uniform momentum. The thickness of these layers scales as the Kolmogorov length scale and the velocity across these layers presents a jump of approximately 10% of the characteristic large-scale velocity of the flow. In layers of intense shear, rotation dominates, whereas in layers of intense scalar gradient, strain is prevalent.

Prandtl’s extended mixing length model applied to the two-dimensional turbulent classical far wake

Despite its limitations, Prandtl’s mixing length model is widely applied in modelling turbulent free shear flows. Prandtl’s extended model addresses many of the shortfalls of the original model, but is not so widely used, in part due to additional mathematical complexities that arise in its derivation and implementation. Furthermore, in both models, Prandtl neglects the kinematic viscosity on the basis that it is much smaller in magnitude than the turbulent viscosity. Recent work has shown that including the kinematic viscosity in the original model has both mathematical and physical advantages. In the present work, a novel derivation of the extended model is provided, and it is demonstrated that similar advantages are again obtained when the kinematic viscosity is included. Additionally, through the use of scaling techniques, similarity mean velocity profiles of the extended model are derived, resulting in a single nonlinear ordinary differential equation that is solved numerically with a Hermite spectral method. The computed profiles for the normalized similarity mean velocity and shear stress are compared with experimental observations and shown to be in excellent agreement.

Spatially evolving cascades in temporal planar jets

Abstract

Length scales in turbulent free shear flows

ABSTRACTWe address the important point of the proportionality between the longitudinal integral lengthscale (L) and the characteristic mean flow width (δ) using experimental data of an axisymmetric wake and a turbulent planar jet. This is a fundamental hypothesis when deriving the self-similar scaling laws in free shear flow. We show that L/δ is indeed constant, at least in a range of streamwise distances between 15 and 50 times the characteristic inlet dimension. We revisit turbulence closure models such as the Prandtl mixing length and the eddy viscosity in the light of the non-equilibrium dissipation scaling. We show that the mixing length model does not comply with the scalings stemming from the non-equilibrium version of the theory even if it does comply with the theory’s equilibrium version. Similarly, the eddy viscosity model holds in the case of the non-equilibrium version of the theory provided that the eddy viscosity is constant everywhere. We conclude by comparing the results of the different models with each other and with experimental data and with an improved model (following Townsend) that corrects for the eddy viscosity by considering the intermittency of the flow.

Effective forcing for direct numerical simulations of the shear layer of turbulent free shear flows

A numerically efficient configuration to simulate turbulent flows is to use triply periodic domains, with numerical forcing techniques to sustain turbulence. Previous homogeneous shear turbulence simulations considered only idealized homogeneous shear flows and not the statistically stationary shear turbulence observed in practical free shear flows. In contrast, the current study mathematically derives the complete forcing technique from the large scales of the turbulent free shear flows. Different statistically stationary free shear flows are considered in this study, namely, a nearly homogeneous shear turbulent flow, turbulent mixing layer, a turbulent planar jet, and a turbulent round jet. The simulations are performed on triply periodic, statistically homogeneous cubic domains in the vicinity of the shear layer in the self-similar region. An a priori analysis is performed to calculate the effects of the different forcing terms and to predict the expected turbulence quantities. The forcing technique is then used to perform direct numerical simulations at different Reynolds numbers. Numerical results for the different cases are discussed and compared with results from experiments and other simulations of free shear turbulent flows. Anisotropy is observed both in the components of velocity and vorticity, with stronger Reynolds number dependence in the anisotropy of vorticity. Energy spectra obtained from the present homogeneous shear turbulence agree well with the spectra from temporally evolving shear layers. The results also highlight the effects of the additional forcing terms that were neglected in previous studies and the role of shear convection and the associated splitting errors in the unbounded evolution of previous numerical simulations.

Adjoint-based sensitivity and ignition threshold mapping in a turbulent mixing layer

Successful ignition in non-premixed turbulent flows remains a fundamental challenge in combustion systems. Current design strategies typically rely on iterative testing to map the spatial distribution of ignition probability. We propose to accelerate this by formulating the adjoint of the perturbed and linearised governing equations in such a way that sensitivity of an ignition indicator can be obtained with a cost comparable to the flow solution. A space–time discrete adjoint method for multi-component chemically reacting flows is developed, and the gradient formed via the corresponding adjoint solution is used to identify regions favourable to ignition in a direct numerical simulation of non-premixed turbulent free shear flow. This approach requires a specific definition of an ignition metric, although this can be problematic because ignition either succeeds or fails after some period and thus gradients for some metrics become ill-defined near the ignition threshold. To this end, a quantity of interest is designed to provide short-time sensitivity in conjunction with an indicator function over a long-time period that informs whether successful ignition occurred. The gradients are used in a line-search algorithm to map the ignition boundary under specific constraints. Finally, parametric sensitivity is evaluated at different flow realisations to analyse factors governing local sensitivity in unsteady chemically reacting flows.

Multi-particle dispersion during entrainment in turbulent free-shear flows

Multi-particle dispersion is studied using direct numerical simulations of temporally evolving mixing layers and planar jets for tetrahedra consisting of four fluid particles which are seeded in the turbulent regions or in the non-turbulent regions near the turbulent/non-turbulent interface (TNTI). The modified Richardson law for decaying turbulence is observed for particle pairs. The size dependence of the mean and relative motions of the entrained tetrahedra indicates that the characteristic length scale of the entrained lumps of fluid is approximately 10 times the Kolmogorov microscale. When the tetrahedra move within the TNTI layer they are flattened and elongated by vortex stretching at a deformation rate that is characterized by the Kolmogorov time scale. The shape evolutions of the tetrahedra show that in free-shear flows, thin-slab structures of advected scalars are generated within the TNTI layers.

Prediction of transitional and fully turbulent flow using an alternative to the laminar kinetic energy approach

ABSTRACTThis paper presents a new eddy-viscosity turbulence model for prediction of laminar, transitional and turbulent flow, as an alternative to the popular laminar kinetic energy framework. The model represents a conceptually different description of the transition process, in which the pressure-strain terms of the Reynolds stress budget are suppressed in pretransitional flow regions, and transition is viewed as activation of the pressure-strain terms. This concept has been implemented using an existing linear-eddy viscosity Reynolds-averaged Navier–Stokes transitional model as a baseline, which has been modified to more accurately capture the physics of fully turbulent free shear flows. The model is tested for several boundary layer and free shear flow test cases. The simulations show engineering accurate results, qualitatively equivalent to the baseline transition-sensitive model for boundary layer test cases, and substantially improved over the baseline model for fully turbulent free shear flows. The results suggest that the new model formulation may have a wider range of application than the baseline model for complex transitional and turbulent flows.

The kinematics of the reduced velocity gradient tensor in a fully developed turbulent free shear flow

AbstractThis paper examines the kinematic behaviour of the reduced velocity gradient tensor (VGT),$\tilde{\unicode[STIX]{x1D608}}_{ij}$, which is defined as a$2\times 2$block, from a single interrogation plane, of the full VGT$\unicode[STIX]{x1D608}_{ij}=\partial u_{i}/\partial x_{j}$. Direct numerical simulation data from the fully developed turbulent region of a nominally two-dimensional mixing layer are used in order to examine the extent to which information on the full VGT can be derived from the reduced VGT. It is shown that the reduced VGT is able to reveal significantly more information about regions of the flow in which strain rate is dominant over rotation. It is thus possible to use the assumptions of homogeneity and isotropy to place bounds on the first two statistical moments (and their covariance) of the eigenvalues of the reduced strain-rate tensor (the symmetric part of the reduced VGT) which in turn relate to the turbulent strain rates. These bounds are shown to be dependent upon the kurtosis of$\partial u_{1}/\partial x_{1}$and another variable defined from the constituents of the reduced VGT. The kurtosis is observed to be minimised on the centreline of the mixing layer and thus tighter bounds are possible at the centre of the mixing layer than at the periphery. Nevertheless, these bounds are observed to hold for the entirety of the mixing layer, despite departures from local isotropy. The interrogation plane from which the reduced VGT is formed is observed not to affect the joint probability density functions (p.d.f.s) between the strain-rate eigenvalues and the reduced strain-rate eigenvalues despite the fact that this shear flow has a significant mean shear in the cross-stream direction. Further, it is found that the projection of the eigenframe of the strain-rate tensor onto the interrogation plane of the reduced VGT is also independent of the plane that is chosen, validating the approach of bounding the full VGT using the assumption of local isotropy.

DNS of Differential Thermal and Mass Diffusions in Free Turbulent Shear Flows

The diffusion of scalars in turbulent flows is the combined effect of the molecular diffusion and the turbulent diffusion, and the latter usually dominates. We investigated the diffusion of scalars with different diffusivities through two canonical free turbulent shear flows, i.e., the mixing layer and the homogeneous isotropic flow, by means of direct numerical simulation (DNS). In the mixing layer, the molecular diffusion has a big impact on the diffusion of scalars in the laminar region where the scalar gradient is high. In the turbulent region, the molecular diffusion has a negligible effect on the diffusion. In the homogeneous isotropic flow, the higher molecular diffusivity strengths the dispersion of scalars of relatively smaller values. However, the molecular diffusion exhibits a counter effect on the scalar diffusion for the relatively larger value. This work is intended to figure out the feasibility of the assumption that the thermal and mass diffusions of a contaminated content are equal, in other words, unit Lewis number assumption, which is often adopted.

Entrainment in plane turbulent pure plumes

AbstractTurbulent jets and plumes are commonly encountered in industrial and natural environments; they are, for example, key processes during explosive eruptions. They have been the objects of seminal works on turbulent free shear flows. Their dynamics is often described with the concept of the so-called entrainment coefficient, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\alpha $, which quantifies entrainment of ambient fluid into the turbulent flow. This key parameter is well characterized for axisymmetric jets and plumes, but data are scarcer for turbulent planar plumes and jets. The data tend to show that the Gaussian entrainment coefficient in plane pure plumes is about twice the value for plane pure jets. In order to confirm and to explain this difference, we develop a model of entrainment in turbulent plane jets and plumes taking into account the effect of buoyancy on entrainment, as a function of the shape of the velocity, buoyancy and turbulent shear stress profiles. We perform new experiments to better characterize the rate of entrainment in plane pure plumes and to constrain the values of the model parameters. Comparison between theory and experiments shows that the enhancement of entrainment in plane turbulent pure plumes relative to plane turbulent pure jets is well explained by the contribution of buoyancy.