Two-dimensional Turbulent Flow Research Articles

Study on heat transfer and pressure steady-state characteristics of a floating nozzle under a moving wall

This work considers the flow field as two-dimensional turbulent flow and studies the steady-state properties of heat transfer and the pressure of the suspension nozzle. An adiabatic wall parallel to the moving wall and two slit entrances at either end of the adiabatic wall make up the rectangular flow field. The SST k-ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k - \\omega$$\\end{document} turbulence model is used in the turbulence computation. Both qualitative and quantitative analyses are conducted on the distribution of the flow field, temperature field, local Nusselt number, local pressure coefficient, average Nusselt number, and average pressure coefficient under various combination conditions. The findings indicate that when the suspension nozzle's flow field varies greatly, wall-jet velocity ratio is 0.1. A rise in Jet inclination angle is not helpful for the wall's suspension, and it has minimal effect on the flow field. The flow field is greatly influenced by separation space-slit width ratio. Larger separation space-slit width ratio values are advantageous for the wall's heat transmission but unfavorable for the wall's suspension. The flow field is most influenced by wall-jet velocity ratio. The wall's ability to convey heat is stronger the higher the wall-jet velocity ratio, but its ability to support weight falls.

Is a direct numerical simulation (DNS) of Navier-Stokes equations with small enough grid spacing and time-step definitely reliable/correct?

Turbulence is strongly associated with the vast majority of fluid flows in nature and industry. Traditionally, results given by the direct numerical simulation (DNS) of Navier-Stokes (NS) equations that relate to a famous millennium problem are widely regarded as ‘reliable’ benchmark solutions of turbulence, as long as grid spacing is fine enough (i.e. less than the minimum Kolmogorov scale) and time-step is small enough, say, satisfying the Courant-Friedrichs-Lewy condition (Courant number < 1). Is this really true? In this paper a two-dimensional sustained turbulent Kolmogorov flow driven by an external body force governed by the NS equations under an initial condition with a spatial symmetry is investigated numerically by the two numerical methods with detailed comparisons: one is the traditional DNS, the other is the ‘clean numerical simulation’ (CNS). In theory, the exact solution must have a kind of spatial symmetry since its initial condition is spatially symmetric. However, it is found that numerical noises of the DNS are quickly enlarged to the same level as the ‘true’ physical solution, which finally destroy the spatial symmetry of the flow field. In other words, the DNS results of the turbulent Kolmogorov flow governed by the NS equations are badly polluted mostly. On the contrary, the numerical noise of the CNS is much smaller than the ‘true’ physical solution of turbulence in a long enough interval of time so that the CNS result is very close to the ‘true’ physical solution and thus can remain symmetric, which can be used as a benchmark solution for comparison. Besides, it is found that numerical noise as a kind of artificial tiny disturbances can lead to huge deviations at large scale on the two-dimensional Kolmogorov turbulence governed by the NS equations, not only quantitatively (even in statistics) but also qualitatively (such as spatial symmetry of flow). This highly suggests that fine enough spatial grid spacing with small enough time-step alone could not guarantee the validity of the DNS of the NS equations: it is only a necessary condition but not sufficient. All of these findings might challenge some of our general beliefs in turbulence: for example, it might be wrong in physics to neglect the influences of small disturbances to NS equations. Our results suggest that, from physical point of view, it should be better to use the Landau-Lifshitz-Navier-Stokes (LLNS) equations, which consider the influence of unavoidable thermal fluctuations, instead of the NS equations, to model turbulent flows.

The effect of appendages at step on heat transfer in a backward-facing step

The current study analyzes the effect of introduction of an appendage at the step, on forced convection, in a backward-facing step. Two-dimensional turbulent flow over a backward-facing step has been tested numerically. The length and size of the recirculation zone have been observed to vary, by changing the appendage length and Reynolds number. An increase in the average Nusselt number has been observed for an increase in the Reynolds number. In this study, it has been visualized that with the introduction of the appendage, the Nusselt number increases. A vertical fin length of Ya/S = 0.5 has been found to provide the greatest improvement in the heat transfer rate. The average Nusselt number increases by three times for Ya/S = 0.5. However, the Performance Evaluation Criteria (PEC) indicates the appendage of length Ya/S = 0.3 provides consistent values. This arises as the pressure drop increases by 8.75 times for Ya/S = 0.5 when compared with unobstructed case.

A variational data assimilation approach for sparse velocity reference data in coarse RANS simulations through a corrective forcing term

The Reynolds-averaged Navier–Stokes (RANS) equations provide a computationally efficient method for solving fluid flow problems in engineering applications. However, the use of closure models to represent turbulence effects can reduce their accuracy. To address this issue, recent research has explored data-driven techniques such as data assimilation and machine learning.An efficient variational data assimilation (DA) approach is presented to enhance steady-state eddy viscosity based RANS simulations. To account for model deficiencies, a corrective force term is introduced in the momentum equation. In the case of only velocity reference data, this term can be represented by a potential field and is divergence-free. The DA implementation relies on the discrete adjoint method and approximations for efficient gradient evaluation.The implementation is based on a two-dimensional coupled RANS solver in OpenFOAM, which is extended to allow the computation of the adjoint velocity and pressure as well as the adjoint gradient. A gradient-based optimizer is used to minimize the difference between the simulation results and the reference data. To evaluate this approach, it is compared with alternative data assimilation methods for canonical stationary two-dimensional turbulent flow problems. For the data assimilation, sparsely distributed reference data from averaged high-fidelity simulation results are used.The results suggest that the proposed method achieves the optimization goal more efficiently compared to applying data assimilation for obtaining the eddy viscosity, or a field modifying the eddy viscosity, directly. The method works well for different reference data configurations and runs efficiently by leveraging coarse meshes.

Domain Decomposition Strategy for Combining Nonlinear and Linear Reduced-Order Models

Increasingly ubiquitous reliance on expensive, high-fidelity numerical simulations has led to the emergence of reduced-order modeling as an effective method to predict high-dimensional, spatially discretized field outputs in an efficient manner. This study presents a method to construct parametric, nonintrusive reduced-order models (ROMs) by leveraging domain decomposition (DD) to enable using multiple dimension reduction (DR) techniques within different spatial subdomains of the field to construct a single predictive ROM. To enforce smoothness of solutions across subdomains, points at domain interfaces are reconstructed using the data-driven gappy proper orthogonal decomposition method. The method is assessed on three high-dimensional test cases with shocks, including one with two-dimensional turbulent flow around the RAE2822. Results show that fully nonlinear ROMs predict fields more accurately in the vicinity of discontinuous features compared to their linear counterparts, which are more accurate in regions that are far from these discontinuities. Furthermore, irrespective of the DR method, DD-based ROMs outperformed their global counterparts in all test cases. Finally, models that used a combination of DR methods simultaneously showed superior performance near shockwaves and a significant improvement in total error compared to existing approaches that employ a single DR method in all subdomains.

Prediction model based on artificial neural network and bivariate spectral quasi-linearization method for compressible turbulent boundary-layer flow over a smooth flat surface

The compressible two-dimensional turbulent flow solutions at an arbitrary point in time and space by incorporating the mass, momentum, and energy conservation equations over a smooth flat surface and parallel free stream with unfavorable pressure gradient are studied. The Falkner–Skan transformation is applied to the turbulent boundary-layer equations and related boundary conditions, and the resulting nonlinear coupled system of partial differential equations is solved by the bivariate spectral quasi-linearization method. Moreover, to predict the thermal distribution of the flow, an artificial neural network model has been developed with the Nusselt number as target values. Several plots have been depicted, it is evaluated that the mean squared error value is 6.41 × 10−7, the overall coefficient of determination (R) is 0.997 52, and the average error rate is 0.68% for the said model, indicating the attainment of high accuracy for estimation.

Compressible two-dimensional turbulence: cascade reversal and sensitivity to imposed magnetic field

We study the impact of compressibility on two-dimensional turbulent flows, such as those modeling astrophysical disks. We demonstrate that the direction of cascade undergoes continuous transition as the Mach number Ma increases, from inverse at Ma = 0, to direct at . Thus, at comparable amounts of energy flow from the pumping scale to large and small scales, in accord with previous data. For supersonic turbulence with the cascade is direct, as in three dimensions, which results in multifractal density field. For that regime () we derive a Kolmogorov-type law for potential forcing and obtain an explicit expression for the third order correlation tensor of the velocity. We further show that all third order structure functions are zero up to first order in the inertial range scales, which is in sharp contrast with incompressible turbulence where the third order structure function, that describes the energy flux associated with the energy cascade is non-zero. The properties of compressible turbulence have significant implications on the amplification of magnetic fields in conducting fluids. We thus demonstrate that imposing external magnetic field on compressible flows of conducting fluids allows to manipulate the flow producing possibly large changes even at small Mach numbers. Thus Zeldovich’s antidynamo theorem, by which at Ma = 0 the magnetic field is zero in the steady state, must be used with caution. Real flows have finite Ma and, however small it is, for large enough values of I, the magnetic flux through the disk, the magnetic field changes the flow appreciably, or rearranges it completely. This renders the limit Ma → 0 singular for non-zero values of I. Of particular interest is the effect of the density multifractality, at which is relevant for astrophysical disks. We demonstrate that in that regime, in the presence of non-zero I the magnetic field energy is enhanced by a large factor as compared to its estimates based on the mean field. Finally, based on the insights described above, we propose a novel two-dimensional Burgers’ turbulence, whose three-dimensional counterpart is used for studies of the large-scale structure of the Universe, as a model for supersonic two-dimensional magnetohydrodynamic flows.

Numerical Simulation of Two-Dimensional Turbulent Dilatable Flows in a Fire

Numerical simulations, based on the theories of computational fluid dynamics and combustion, are currently a powerful tool in the field of fire engineering. Field models allow us to analyze key aspects such as the integrity of buildings and safety. However, it is fundamental to define strategies that allow engineers to obtain a balance between the precision of the results and the computational cost. One of the most relevant sub-models is the turbulence model. This paper presents the research carried out in the field of two-dimensional computational simulations of turbulent dilatable flows to evaluate the behavior of diffusion flames, hot gases, and smoke produced in accidental fires. Several computational simulations have been performed using direct numerical simulations and large eddy simulation turbulence models in the two-dimensional field, analyzing the ability of the models to correctly characterize the transport of hot gases and the behavior of the thermal plumes if the grid resolution is adequate to the physics of the problem. Additionally, several three-dimensional models have been developed to contrast and validate the results obtained in the two-dimensional simulations. In order to validate the capacity to develop a qualitative analysis of two-dimensional models in fire engineering, an evaluation criterion is presented based on the frequency spectral analysis to study the capacity of each type of turbulence model to accurately capture the vorticity of these dilatable flows.

Approximate Method for Calculating the Velocity Profile of a Two-Dimensional Stationary Turbulent Liquid Flow

Approximate Method for Calculating the Velocity Profile of a Two-Dimensional Stationary Turbulent Liquid Flow

REGULARITIES OF HYDROCHEMICAL DYNAMICS IN A TWO-DIMENSIONAL TURBULENT FLOW OF NATURAL WATER

The variability of the composition and properties of natural waters, which creates numerous difficulties in water use, cannot always be explained by the influence of external influencing factors, such as weathering or leaching of rocks, a change in the phases of the water regime, etc. This is especially true for subdiurnal and subhourly quality variability, which can be caused by complex, previously unknown dynamic hydrochemical processes. Such a conclusion follows from the results of the study of turbidity and pH of natural water given in the work, obtained with an increased frequency of measurements. These results indicate the existence of a quasi-cyclic change in the controlled parameters with different periods, from every minute to daily. The study of observational data allows us to make an assumption that in this case the hydrochemical dynamics is due to direct and reverse energy cascades in a two-dimensional turbulent flow of natural water, in which the impurity subsystem is prone to stratification.

Regularities of Hydrochemical Dynamics in a Two-Dimensional Turbulent Flow of Natural Water

Variations in the composition and properties of natural waters, which creates numerous difficulties in water use, cannot always be explained by the influence of external forcing such as weathering or leaching of rocks, changes in the phases of the water regime, and other causes. This is especially true for sub-daily and sub-hourly variations in the water quality, which can be caused by complex, previously unknown dynamic hydrochemical processes. Such a conclusion follows from study of the turbidity and pH of natural water measured with increased frequency, the results of which are presented in this work. These results provide evidence about the existence of quasi-cyclic variations in the controlled parameters with different periods, from minute to daily. Study of the observational data allows us to assume that in this case the hydrochemical dynamics is caused by direct and reverse energy cascades in a two-dimensional turbulent flow of natural water, in which the impurity subsystem may be stratified.

Equivalence of nonequilibrium ensembles: Two-dimensional turbulence with a dual cascade.

We examine the conjecture of equivalence of nonequilibrium ensembles for turbulent flows in two dimensions in a dual-cascade setup. We construct a formally time-reversible Navier-Stokes equationin two dimensions by imposing global constraints of energy and enstrophy conservation. A comparative study of the statistical properties of its solutions with those obtained from the standard Navier-Stokes equationsclearly shows that a formally time-reversible system is able to reproduce the features of a two-dimensional turbulent flow. Statistical quantities based on one- and two-point measurements show an excellent agreement between the two systems for the inverse- and direct-cascade regions. Moreover, we find that the conjecture holds very well for two-dimensional turbulent flows with both conserved energy and enstrophy at finite Reynolds number.

Numerical study of transient convective turbulent boundary layer flow along a vertical plate: analysis of kinetic energy and its dissipation rate

ABSTRACT The present article numerically investigates the turbulent buoyancy-driven (natural convection) flow along a vertical plate with a low Reynolds turbulence two-equation k-ϵ model. The deployed turbulence model is appropriate for low Reynolds number (LRN) turbulent flow adjacent to a solid boundary, and the turbulence kinetic energy (TKE) and dissipation rate of TKE are estimated using the momentum equations and are solved simultaneously with the mean flow conservation equations. Two-dimensional time-dependent viscous incompressible turbulent flow is simulated. This flow domain is governed by a highly non-linear group of partial differential equations, namely the time-averaged continuity, momentum, and energy, and also the flow property is determined through TKE, and dissipation rate of TKE equations. Since these equations are not solvable using analytical methods, an implicit second-order finite difference method is employed to solve the governing turbulent flow equations numerically. The simulated time-averaged velocity, temperature, TKE, and dissipation rate of TKE profiles along with friction factor and heat transfer rate are computed for different values of turbulent Reynolds and Prandtl numbers. Average velocity, temperature, turbulence energy, and dissipation rate under both transient and steady-state conditions are decreased with increment in . There is a decrement in average transient velocity and turbulence energy rates with increasing values, whereas the average temperature profile increases. Average steady temperature and turbulence energy profiles are also enhanced with rising values of . The majority of researchers focus on laminar flow and thus far turbulent flow has mainly been addressed through experiments only. To better elucidate the flow in this condition, the novelty of the present study is to utilize a numerical FDM procedure to probe more deeply into the turbulence characteristics in buoyancy-driven (natural convection) flow along a vertical plate with a low Reynolds turbulence two-equation k-ϵ model, as this topic is fundamental to many applications in science and engineering. Furthermore, the results of the turbulent flow simulations obtained from the current model are corroborated with existing results for the special case of laminar flow and a good correlation is achieved.

Inducing intermittency in the inverse cascade of two-dimensional turbulence by a fractal forcing

Can the inverse cascade of two-dimensional turbulence display intermittency? This work shows that if forced in a fractal set of dimension smaller than two, a two-dimensional turbulent flow develops clusters of different size vortices not uniformly spread in space as the ones shown in the figure, that display intermittent properties.

Irreversible energy extraction from negative-temperature two-dimensional turbulence.

The formation and transition of patterns of two-dimensional turbulent flows observed in various geophysical systems are commonly explained in terms of statistical mechanics. Different from ordinary systems, for a two-dimensional flow, the absolute temperature defined for a statistical equilibrium can take negative values. In a state of negative temperature, the second law of thermodynamics predicts that energy in microscopic fluctuations is irreversibly converted to a macroscopic form. This study explores the possibility of this one-way energy conversion in a two-dimensional flow using a basic conceptual model. We consider an inviscid incompressible fluid contained in a bounded domain, the shape of which is distorted by an externally imposed force. Unlike the usual fixed boundary cases, the flow energy within the domain is exchanged with the external system via pressure work through the moving lateral boundary. Concurrently, the flow field remains constrained by vorticity conservation. Beginning from a state of Kraichnan's grand-canonical ensemble, when the domain shape is distorted from one shape to another in a finite time, the Jarzynski equality is established. This equality states that, on average, the direction of a net energy flow through the boundary during a cycle of domain distortion changes with the sign of the initial temperature of the system. Numerical experiments are carried out to verify this theoretical argument and to investigate the parameter dependence of the energy exchange rate.

Stochastic model for the alignment and tumbling of rigid fibers in two-dimensional turbulent shear flow

Non-spherical particles transported by an anisotropic turbulent flow preferentially align with the mean shear and intermittently tumble when the local strain fluctuates. Such an intricate behaviour is here studied for inertialess, rod-shaped particles embedded in a two-dimensional turbulent flow with homogeneous shear. A Lagrangian stochastic model for the rods angular dynamics is introduced and compared to the results of direct numerical simulations. The model consists in superposing a short-correlated random component to the steady large-scale mean shear, and can thereby be integrated analytically. To reproduce the single-time orientation statistics obtained numerically, it is found that one has to properly account for the combined effect of the mean shear, for anisotropic velocity gradient fluctuations, and for the presence of persistent rotating structures in the flow that bias Lagrangian statistics. The model is then used to address two-time statistics. The notion of tumbling rate is extended to diffusive dynamics by introducing the stationary probability flux of the rods unfolded angle. The model is found to reproduce the long-term effects of an average shear on the mean and the variance of the fibres angular increment. Still, it does not reproduce an intricate behaviour observed in numerics for intermediate times: the unfolded angle is there very similar to a L\'evy walk with distributions of increments displaying intermediate power-law tails.

Statistics of clustered vortices in the inverse energy cascade of two-dimensional turbulence

Strong, long-lived vortices play a prominent role in two-dimensional (2D) turbulent flows, and understanding their statistics is key to unravelling 2D turbulence. Here it is shown that clustered same-sign vortices in forced 2D turbulence have a scale-invariant distribution of areas and are described by a mean-field theory. Larger clusters come apart more quickly, and the flow surrounding the remaining clusters becomes progressively more randomized and neutral, while vortex merger concentrates coherent enstrophy into a smaller and smaller fraction of the domain.

Anisotropic Lagrangian dispersion in zonostrophic turbulence in a closed basin

This article studies the anisotropic particle dispersion in a continuously forced, two-dimensional turbulent flow on a β-plane. The flow is immersed in a large-scale closed basin with free-slip walls. The anisotropy is analyzed in two sets of numerical experiments characterized by the magnitude of the imposed, time-dependent forcing (weak and strong). Both experiments exhibit typical features of zonostrophic turbulence: eddy motions that, on average, form alternating east–west circulation bands due to the β-effect. The dispersion anisotropy is investigated through three Lagrangian statistics calculated by zonal and meridional components: (i) relative dispersion between pairs of particles; (ii) dispersion ellipses; and (iii) finite-scale Lyapunov exponents (FSLE), also measured with particle pairs. In the experiment with weak forcing, the relative dispersion and dispersion ellipses show anisotropy with a zonal preference toward the west; however, the FSLE did not reveal significant anisotropy. In the experiment with strong forcing, the relative dispersion and dispersion ellipses show zonal anisotropy toward the west when the particles are far from the boundaries. As the particles reach the western wall and are redistributed to fill the domain, the anisotropy ceases. The FSLE show zonal anisotropy for a wide range of particle separations. The results are examined further by using no-slip boundary conditions and a rectangular domain geometry.

New numerical methods for calculating statistical equilibria of two-dimensional turbulent flows, strictly based on the Miller–Robert–Sommeria theory

New numerical methods are proposed for the mixing entropy maximization problem in the context of Miller–Robert–Sommeria’s (MRS) statistical mechanics theory of two-dimensional turbulence, particularly in the case of spherical geometry. Two of the methods are for the canonical problem; the other is for the microcanonical problem. The methods are based on the original MRS theory and thus take into account all Casimir invariants. Compared to the methods proposed in previous studies, our new methods make it easier to detect multiple statistical equilibria and to search for solutions with broken zonal symmetry. The methods are applied to a zonally symmetric initial vorticity distribution which is barotropically unstable. Two statistical equilibria are obtained, one of which has a wave-like structure with zonal wavenumber 1, and the other has a wave-like structure with zonal wavenumber 2. While the former is the maximum point of the mixing entropy, the wavenumber 2 structure of the latter is nearly the same as the structure that appears in the end state of the time integration of the vorticity equation. The new methods allow for efficient computation of statistical equilibria for initial vorticity distributions consisting of many levels of vorticity patches without losing information about all the conserved quantities. This means that the statistical equilibria can be obtained from an arbitrary initial vorticity distribution, which allows for the application of statistical mechanics to interpret a wide variety of flow patterns appearing in geophysical fluids.

Preferential transport of swimmers in heterogeneous two-dimensional turbulent flow

We investigate the performance of active swimmers in a strongly heterogeneous two-dimensional weakly turbulent flow. We demonstrate that there are three regimes of preferential transport for rod-like swimmers as the swimmers' intrinsic speed increases. We further reveal that the three regimes are due to the relative strengths of three different effects: the intrinsic speed of the swimmers, the reorientation ability of the shear layer at the interface of two flow regions, and the attracting Lagrangian Coherent Structures of the flow field.