Two-dimensional Isotropic Turbulence Research Articles

Low-Order Moments of Velocity Gradient Tensors in Two-Dimensional Isotropic Turbulence

In isotropic turbulence, symmetry of different directions can reduce the number of independent components for velocity gradient tensors. In three-dimensional isotropic turbulence, the independent components under either incompressible or compressible conditions have already been analyzed in the literature. However, for two-dimensional isotropic turbulence, they are still unclear. We derive rigorously the independent components for velocity gradient tensors of two-dimensional isotropic turbulence and give physical explanations. These theoretical results are validated using high-resolution direct numerical simulations (DNSs) of two-dimensional compressible turbulence. Results show that the present DNS setup is still not sufficient to capture the isotropy of third-order moments, suggesting that more investigations on determining the smallest scale and improving the numerical schemes for two-dimensional compressible turbulence are required.

On the application of principal component transport for compression ignition of lean fuel/air mixtures under engine relevant conditions

Principal component transport-based data-driven reduced-order models (PC-transport ROM) are being increasingly adopted as a combustion model of turbulent reactive flows to mitigate the computational cost associated with incorporating detailed chemical kinetics. Previous studies were mainly limited to replicating relatively-simple chemistry in canonical configurations. The objective of the present study, therefore, is to further explore the accuracy of PC-transport ROM on more complex combustion phenomenon where, for example, large hydrocarbon fuel chemistry spanning a broad range of thermochemical space governs sequential multi-stage compression ignition processes. The cumulative error of PC-transport for this problem, and for others that depend upon sequential highly nonlinear physics, has to be minimal as the combustion phasing and heat release rate in internal combustion engines depends upon accurate predictions of minor ignition species whose concentrations start from ashes and grow orders of magnitude over the course of low- and high-temperature autoignition. Specifically, the PC-transport ROM is applied to predict the compression ignition characteristics of lean n-heptane/air and primary reference fuel (PRF)/air mixtures in a two-dimensional (2-D) constant volume computational domain initialized with a two-dimensional isotropic turbulence spectrum and temperature inhomogeneities. PCA is used to define the low-dimensional manifold that represents the original thermochemical state vector, and artificial neural network (ANN) models are adopted to tabulate chemical kinetics, transport, and thermodynamic properties. A series of 2-D pseudo-turbulent simulations are performed at engine pressures by varying the initial mean and r.m.s. of temperature, turbulence intensity, and the composition of fuel/air mixture. The results show that the PC-transport ROM accurately reproduces the instantaneous and statistical ignition characteristics of the fuel/air mixture, aided by pre-processing techniques including species subsetting, data clustering, and data transformation. It is found that PCs are not properly scaled with a power transformer if reactants are included in the species subset, which leads to a decrease in the accuracy of the PC-transport ROM. A separation of the reactants from the species subset ensures that the temporal evolution of the PCs starts from zero and spans orders of magnitude with time, and as such, this approach is found to effectively redistribute both PCs and their source terms with a power transformer. The computational speed-up factor of the PC-transport ROM ranges between 5.1 and 15.0 for the cases with n-heptane/air mixture and PRF/air mixture, respectively. Moreover, a potential further speed-up is anticipated through a combination of reduction in grid resolution requirements and in the stiffness of the chemical system. As an example, many of the pre-processing methods for inhomogeneous compression ignition may also apply to other complex intermittent combustion phenomena.Novelty and significance statement• The PCA-based reduced-order model (PC-transport ROM) has been applied to the multi-stage compression ignition of large hydrocarbon fuels under HCCI-relevant conditions. The present work presents a systematic procedure to accurately capture the two-stage ignition behavior of lean n-heptane/air or PRF50/air mixture.• The present work demonstrates an advantage of the PC-transport ROM in terms of computational speed-up. The computational speed-up factor for the ROM is up to 15, and moreover, a potential additional speed-up is anticipated through the reduction in the spatial and temporal resolution required.• A series of 2-D PC-transport ROMs are conducted to demonstrate the robustness of the ROM. A limitation of the ROM against different operating conditions is also discussed.

Bayesian comparison of stochastic models of dispersion

Stochastic models of varying complexity have been proposed to describe the dispersion of particles in turbulent flows, from simple Brownian motion to complex temporally and spatially correlated models. A method is needed to compare competing models, accounting for the difficulty in estimating the additional parameters that more complex models typically introduce. We employ a data-driven method, Bayesian model comparison, which assigns probabilities to competing models based on their ability to explain observed data. We focus on the comparison between the Brownian and Langevin dynamics for particles in two-dimensional isotropic turbulence, with data that consist of sequences of particle positions obtained from simulated Lagrangian trajectories. We show that, while on sufficiently large time scales the models are indistinguishable, there is a range of time scales on which the Langevin model outperforms the Brownian model. While our set-up is highly idealised, the methodology developed is applicable to more complex flows and models of particle dynamics.

Shock-induced energy transfers in dense gases

Dense gases are characterised by molecules featuring large numbers of active degrees of freedom (quantified by the cv/R ratio). The isentropes in such gases have the distinct property of following rather closely the isotherms (the two become identical in the limit of cv/R going to infinity). Near the liquid-vapour critical point, this makes the isentropes very shallow and possibly concave (in the pressure-specific volume diagram). Whilst shallow isentropes are desirable when designing expanders (i.e. a large specific-volume increase may be achieved for virtually no pressure drop), could such extreme compressibility effects modify turbulence in a profound manner? This paper discusses two particularly interesting aspects: (i) shock-refraction properties (i.e. the way a shock can redistribute the energy of incoming perturbations), (ii) enstrophy production in homogeneous turbulence. A linear interaction analysis (LIA) is conducted on the shock configuration for which the incoming perturbation is decomposed into linear modes of the compressible Euler equations. The transmission coefficients relative to each eigen modes are solved analytically and results are compared against fully non-linear compressible direct numerical simulation reproducing the weak perturbation of an isolated two-dimensional compression shock wave. The linear analysis is found to be capable of predicting the shock-induced redistribution of the energy of the incoming perturbation between the different eigen modes. Non-ideal gas effects are observed both analytically and numerically with especially an unusual selective response for some particular choice of incoming Mach number. A two-dimensional isotropic turbulence configuration is then numerically investigated for the case of an inviscid compressible dense-gas flow close to the liquid-vapour critical point. Strong non-ideal-gas effects on enstrophy production are observed with the formation of eddy shocklets. In both cases non-convex isentropes close to the liquid-vapour critical point are extremely influential in letting both the shock and the turbulence redistribute any supply of turbulence kinetic energy in ways which are simply not observable in ideal gases. This will hopefully spark enthusiasm amongst turbulence modellers (and their end users?).

Numerical Simulation of Two-dimensional Isotropic Turbulence by the Spectral Lattice Boltzmann Method

We propose the Spectral Lattice Boltzmann Method (SLBM) that is the numerical calculation method of fluids based on the lattice BGK (Bathnagar-Gross-Krook) model discretized with spectral method. Two-dimensional homogeneous isotropic turbulence are simulated by the SLBM. Numerical results agree with another calculation results i.e. the energy spectra proportional to the wave number to -4 and backward cascade of energy is observed. As a result, we obtain the conclusion that homogeneous isotropic turbulence can be simulated by the SLBM, and it is confirmed that this method is useful for numerical simulations of turbulent flows.

On the large-scale structure of homogeneous two-dimensional turbulence

We consider freely decaying two-dimensional isotropic turbulence. It is usually assumed that, in such turbulence, the energy spectrum at small wavenumber, k, takes the form E(k → 0) ∼ Ik3, where I is the two-dimensional version of Loitsyansky's integral. However, a second possibility is E(k → 0) ∼ Lk, where the pre-factor, L, is the two-dimensional analogue of Saffman's integral. We show that, as in three dimensions, L is an invariant and that E ∼ Lk spectra arise whenever the eddies possess a significant amount of linear impulse. The conservation of L is shown to be a direct consequence of the principle of conservation of linear momentum. We also show that isotropic turbulence dominated by a cloud of randomly located monopole vortices has a singular energy spectrum of the form E(k → 0) ∼ Jk−1, where J, like L, is an invariant. However, while E ∼ Jk−1 necessarily implies the existence of a sea of monopoles, the converse need not be true: a sea of monopoles whose spatial locations are not entirely random, but constrained in some way, need not give a E ∼ Jk−1 spectra. The constraint imposed by the conservation of energy is particularly important, ruling out E ∼ Jk−1 spectra for certain classes of initial conditions. Finally, we provide simple explicit examples of random vorticity fields with E ∼ Ik3, E ∼ Lk and E ∼ Jk−1 spectra.

Flow past a cylinder: shear layer instability and drag crisis

AbstractFlow past a circular cylinder for Re=100 to 107 is studied numerically by solving the unsteady incompressible two‐dimensional Navier–Stokes equations via a stabilized finite element formulation. It is well known that beyond Re ∼ 200 the flow develops significant three‐dimensional features. Therefore, two‐dimensional computations are expected to fall well short of predicting the flow accurately at high Re. It is fairly well accepted that the shear layer instability is primarily a two‐dimensional phenomenon. The frequency of the shear layer vortices, from the present computations, agree quite well with the Re0.67 variation observed by other researchers from experimental measurements. The main objective of this paper is to investigate a possible relationship between the drag crisis (sudden loss of drag at Re ∼ 2 × 105) and the instability of the separated shear layer. As Re is increased the transition point of shear layer, beyond which it is unstable, moves upstream. At the critical Reynolds number the transition point is located very close to the point of flow separation. As a result, the shear layer eddies cause mixing of the flow in the boundary layer. This energizes the boundary layer and leads to its reattachment. The delay in flow separation is associated with narrowing of wake, increase in Reynolds shear stress near the shoulder of the cylinder and a significant reduction in the drag and base suction coefficients. The spatial and temporal power spectra for the kinetic energy of the Re=106 flow are computed. As in two‐dimensional isotropic turbulence, E(k) varies as k−5/3 for wavenumbers higher than energy injection scale and as k−3 for lower wavenumbers. The present computations suggest that the shear layer vortices play a major role in the transition of boundary layer from laminar to turbulent state. Copyright © 2004 John Wiley & Sons, Ltd.

Stability of lattice Boltzmann methods in hydrodynamic regimes

The Von Neumann linear stability theory as applied by Sterling and Chen [J. Comput. Chem. 123, 196 (1996)] to lattice Boltzmann numerical methods is revisited and extended. A simplifying assumption made by these authors (on the character of the most unstable mode) is abandoned and immediate refinements on their stability results are attained. The inadequacy of uniform background flow, as a general point of expansion, is evident from simulations of simple shear waves. The stability theory is consequently extended to address the destabilizing role of ``background'' shear. To this end, exact time-dependent solutions of the nine-velocity Bhatnagar-Gross-Krook (BGK) lattice Boltzmann model (LB9) are derived and used as expansion points for the stability theory. Calculations reveal both physical and nonphysical instabilities, the former being interpreted via classical inviscid stability theory and the latter forming an empirical instability criterion (fitting better at small values of the viscosity), $N<{R}^{0.56}$, where $N$ is the number of mesh points in the shearing direction and $R$ is the flow Reynolds number. This is interestingly close to the Kolmogorov-Batchelor-Kraichnan inertial range cutoff ${R}^{1/2}$ for two-dimensional isotropic turbulence. In this case, stability seems to require at least the spatial resolution required for accuracy. We also note that the particular class of solutions found above for the LB9 model can be compared directly to corresponding solutions of the Navier-Stokes equations. It is demonstrated that setting $\ensuremath{\tau}=1$, where $\ensuremath{\tau}$ is the relaxation time of the BGK collision operator, provides optimal accuracy in time. This observation may be relevant to current studies as letting $\ensuremath{\tau}\ensuremath{\rightarrow}1/2$ appears to be a common technique aimed at lowering the viscosity and thereby increasing the Reynolds number of LB simulations.

Large eddy simulation of two-dimensional isotropic turbulence

Large eddy simulation (LES) of forced, homogeneous, isotropic two-dimensional (2D) turbulence in the energy transfer subrange is the subject of this paper. A difficulty specific to this LES and its subgrid scale (SGS) representation is in that the energy source resides in high wave number modes excluded in simulations. Therefore, the SGS scheme in this case should assume the function of the energy source. In addition, the controversial requirements to ensure direct enstrophy transfer and inverse energy transfer make the conventional scheme of positive and dissipative eddy viscosity inapplicable to 2D turbulence. It is shown that these requirements can be reconciled by utilizing a two-parametric viscosity introduced by Kraichnan (1976) that accounts for the energy and enstrophy exchange between the resolved and subgrid scale modes in a way consistent with the dynamics of 2D turbulence; it is negative on large scales, positive on small scales and complies with the basic conservation laws for energy and enstrophy. Different implementations of the two-parametric viscosity for LES of 2D turbulence were considered. It was found that if kept constant, this viscosity results in unstable numerical scheme. Therefore, another scheme was advanced in which the two-parametric viscosity depends on the flow field. In addition, to extend simulations beyond the limits imposed by the finiteness of computational domain, a large scale drag was introduced. The resulting LES exhibited remarkable and fast convergence to the solution obtained in the preceding direct numerical simulations (DNS) by Chekhlovet al. (1994) while the flow parameters were in good agreement with their DNS counterparts. Also, good agreement with the Kolmogorov theory was found. This LES could be continued virtually indefinitely. Then, a simplified SGS representation was designed, referred to as the stabilized negative viscosity (SNV) representation, which was based on two algebraic terms only, negative Laplacian and positive biharmonic ones. It was found that the SNV scheme performed in a fashion very similar to the full equation and it was argued that this scheme and its derivatives should be applied for SGS representation in LES of quasi-2D flows.

Numerical simulation of plane Poiseuille turbulence

Statistical properties of plane Poiseuille flow are studied by a two-dimensional numerical simulation based on the Fourier-Chebyshev spectral method with 256 × 257 modes. The dynamic evolution of energy, enstrophy and palinstrophy is revealed as well as coherent structures representing sweep and ejection of high vorticity regions close to the wall. The streamwise power spectra of the vorticity correlation and probability distribution functions of vorticity ω and its streamwise derivative ωx are examined by dividing the fluid volume into two regions, i.e., inside and outside the viscous sublayer. The powers of the spectra near and far from the wall are −4 and −2, respectively, and the latter corresponds to the energy spectra of power −4 in two-dimensional isotropic turbulence. The distributions of ω and ωx are shown to be Gaussian in the viscous sublayer and exponential in the outer region. The difference of the distributions is discussed between two-dimensional decaying and plane Poiseuille turbulence.

Inverse energy cascade in a nearly two-dimensional turbulence

Spectral structures of homogeneous axisymmetric turbulence subjected to a uniform magnetic field B0 and a randomly stirred two-dimensional force fed in a narrow band of wavenumbers is studied, using the energy spectrum equation that is closed by the eddy-damped quasinormal Markovian (EDQNM) approximation. When the interaction parameter N=σB20L/(ρ0u) is large (where σ is the electric conductivity, ρ0 the uniform fluid density, L the integral scale, and u the root mean square velocity), the existence of the inverse energy cascade toward large scales with a k−5/3 energy spectrum and the enstrophy cascade toward small scales with k−3 energy spectrum are observed in the evolution of the energy of velocity components transverse to B0. The energy of the velocity components parallel to B0 shows a k−1 energy spectrum, which agrees with the k−1 variance spectrum predicted by Lesieur and Herring [J. Fluid Mech. 161, 77 (1985)] for a passive scalar convected by two-dimensional isotropic turbulence. This is of great interest because it is found that in the limit of two-dimensional turbulence the equation for the parallel velocity components is reduced to the same equation as that for a passive scalar convected by two-dimensional isotropic turbulence.

The return of strongly anisotropic turbulence to isotropy

Energy spectrum equations for homogeneous axisymmetric turbulence are formulated using the eddy-damped quasinormal Markovian (EDQNM) approximation. The equations, which consist of two spectrum functions of the two variables k (the magnitude of the wavevector k) and μ (the cosine of the angle between k and the direction of axisymmetry), are computed directly, without any reduction, under initial conditions for different degrees of anisotropy. In this paper, the results for cases of medium anisotropic and nearly two-dimensional initial conditions are reported. A remarkable result was that in the nearly two-dimensional case, some characteristic properties of two-dimensional isotropic turbulence, such as inverse energy cascades, are observed. Also, particular attention is paid to the effects of pressure-strain correlation and inertial energy transfer on the return of anisotropic turbulence to isotropy.

The Effect of the Finite Rossby Radius on Two-Dimensional Isotropic Turbulence

The effect of the finite outer Rossby radius on two-dimensional isotropic turbulence in a single-layered fluid is investigated using the modified cumulant expansion method. This type of turbulence may be considered as a model of geostrophic turbulence in the atmosphere. The beta effect is neglected in order to avoid extra complexity. It is found that turbulent motions with scale larger than the Rossby radius are strongly suppressed and consequently the inverse cascade of the energy is remarkably reduced. At large wavenumbers, the similarity law of the energy spectrum is identical to that of two-dimensional turbulence without rotation. The kinetic energy decreases in time, while the potential energy increases.

The modified cuinulant expansion for two-dimensional isotropic turbulence

The two-dimensional isotropic turbulence in an incompressible fluid is investigated using the modified zero fourth-order cumulant approximation. The dynamical equation for the energy spectrum obtained under this approximation is solved numerically and the similarity laws governing the solution in the energy-containing and enstrophy-dissipation ranges are derived analytically. At large Reynolds numbers the numerical solutions yield the k−3 inertial subrange spectrum which was predicted by Kraichnan (1967), Leith (1968) and Batchelor (1969) assuming a finite enstrophy dissipation in the inviscid limit. The energy-containing range is found to satisfy an inviscid similarity while the enstrophy-dissipation range is governed by the quasi-equilibrium similarity with respect to the enstrophy dissipation as proposed by Batchelor (1969). There exists a critical time tc which separates the initial period (t < tc) and the similarity period (t > tc) in which the enstrophy dissipation vanishes and remains non-zero respectively in the inviscid limit. Unlike the case of three-dimensional turbulence, tc is not fixed but increases indefinitely as the viscosity tends to zero.

The Integral Invariants of Two-Dimensional Inviscid Flow and Their Role in the Theory of Two-Dimensional Turbulence

It is shown that the invariance of the vorticity, kinetic energy and enstrophy integrals determines the form of the partial differential equation that governs the detailed evolution of two-dimensional nondivergent flows of homogeneous inviscid fluids, to within an arbitrary time scale. As a result, the invariance of integrals of other functions of vorticity is not independent of the invariance of the three basic integrals. An even more important consequence is that the nonlinear interactions between different scales of motion in two-dimensional isotropic turbulence are completely characterized by the invariance of the kinetic energy and enstrophy integrals in inviscid flow.

Numerical simulation of developing and decaying two-dimensional turbulence

Two-dimensional isotropic turbulence is investigated in its development from an arbitrarily specified initial flow through its transformation into a statistically self-preserving decaying flow. Numerical simulation is the principal method of investigation. The early development is characterized by rapid growth of the mean squared vorticity gradient, and this growth is found to be predicted satisfactorily by the quasi-normal hypothesis. During the later states of decay the numerical results are found to be generally consistent with the predictions by Kraichnan, Leith and Batchelor of a k−3 inertial range spectrum. The dimensionless constant of the spectrum is found to be near 2, about half the value found earlier for turbulence maintained by a constant forcing amplitude. The results are also consistent with Batchelor's predictions of the time-dependent behaviour of certain quadratic moments: An inconsistency in those predictions is pointed out, however, which can be resolved by altering the inertial range spectrum by a logarithmic term, as suggested by Kraichnan. The most important two-point Eulerian correlation functions are exhibited. An investigation is made of the Gaussianity of the flow with results indicating a strong tendency toward intermittency in the enstrophy dissipation.

On the large-scale atmospheric turbulence

The dynamical equations of the velocity correlation and spectrum functions for three-dimensional anisotropic turbulence in a stratified, rotating atmosphere are derived and discussed. The characteristics of the horizontal two-dimensional isotropic turbulence are analyzed.

THE CORRELATION BETWEEN WIND AND HEIGHT ON AN ISOBARIC SURFACE

Abstract Since the two point correlations of winds on an isobaric surface approximate those expected from a two-dimensional isotropic turbulence, it follows that in a flow that is geostrophic the two point correlations of wind and height must have a specific functional form. The form of the two point correlation function for wind and height is predicted from relations between this function and the correlation functions for the two point correlations of wind and the two point correlations of height. The theoretical predictions are compared with observational data and found to be reasonably well realized. The two point correlation function for wind and height is applied to height and wind observations at a point for the estimation of these quantities at a distance of several hundred nautical miles.