Two-dimensional Homogeneous Isotropic Turbulence Research Articles

Energy-conserving hyper-reduction and temporal localization for reduced order models of the incompressible Navier-Stokes equations

A novel hyper-reduction method is proposed that conserves kinetic energy and momentum for reduced order models of the incompressible Navier-Stokes equations. The main advantage of conservation of kinetic energy is that it endows the hyper-reduced order model (hROM) with a nonlinear stability property. The new method poses the discrete empirical interpolation method (DEIM) as a minimization problem and subsequently imposes constraints to conserve kinetic energy. Two methods are proposed to improve the robustness of the new method against error accumulation: oversampling and Mahalanobis regularization. Mahalanobis regularization has the benefit of not requiring additional measurement points. Furthermore, a novel method is proposed to perform energy- and momentum-conserving temporal localization with the principle interval decomposition: new interface conditions are derived such that energy and momentum are conserved for a full time-integration instead of only during separate intervals. The performance of the new energy- and momentum-conserving hyper-reduction methods and the energy- and momentum-conserving temporal localization method is analysed using three convection-dominated test cases; a shear-layer roll-up, two-dimensional homogeneous isotropic turbulence and a time-periodic inviscid flow consisting of a vortex in a uniform background flow. Our main finding is that energy conservation in combination with oversampling or regularization leads to a robust method with excellent long time stability properties. When any of these two ingredients is missing, accuracy and/or stability is significantly impaired.

On the impact of dimensionally-consistent and physics-based inner products for POD-Galerkin and least-squares model reduction of compressible flows

Model reduction of the compressible Euler equations based on proper orthogonal decomposition (POD) and Galerkin orthogonality or least-squares residual minimization requires the selection of inner product spaces in which to perform projections and measure norms. The most popular choice is the vector-valued L2(Ω) inner product space. This choice, however, yields dimensionally-inconsistent reduced-order model (ROM) formulations which often lack robustness. In this work, we try to address this weakness by studying a set of dimensionally-consistent inner products with application to the compressible Euler equations. First, we demonstrate that non-dimensional inner products have a positive impact on both POD and Galerkin/least-squares ROMs. Second, we further demonstrate that physics-based inner products based on entropy principles result in drastically more accurate and robust ROM formulations than those based on non-dimensional L2(Ω) inner products. As test cases, we consider the following problems: the one-dimensional Sod shock tube, a two-dimensional Riemann problem, a two-dimensional Kelvin-Helmholtz instability, and two-dimensional homogeneous isotropic turbulence.

Collision statistics of inertial particles in two-dimensional homogeneous isotropic turbulence with an inverse cascade

AbstractThis study investigates the collision statistics of inertial particles in inverse-cascading two-dimensional (2D) homogeneous isotropic turbulence by means of a direct numerical simulation (DNS). A collision kernel model for particles with small Stokes number ($\mathit{St}$) in 2D flows is proposed based on the model of Saffman & Turner (J. Fluid Mech., vol. 1, 1956, pp. 16–30) (ST56 model). The DNS results agree with this 2D version of the ST56 model for $\mathit{St}\lesssim 0.1$. It is then confirmed that our DNS results satisfy the 2D version of the spherical formulation of the collision kernel. The fact that the flatness factor stays around 3 in our 2D flow confirms that the present 2D turbulent flow is nearly intermittency-free. Collision statistics for $\mathit{St}= 0.1$, 0.4 and 0.6, i.e. for $\mathit{St}<1$, are obtained from the present 2D DNS and compared with those obtained from the three-dimensional (3D) DNS of Onishi et al. (J. Comput. Phys., vol. 242, 2013, pp. 809–827). We have observed that the 3D radial distribution function at contact ($g(R)$, the so-called clustering effect) decreases for $\mathit{St}= 0.4$ and 0.6 with increasing Reynolds number, while the 2D $g(R)$ does not show a significant dependence on Reynolds number. This observation supports the view that the Reynolds-number dependence of $g(R)$ observed in three dimensions is due to internal intermittency of the 3D turbulence. We have further investigated the local $\mathit{St}$, which is a function of the local flow strain rates, and proposed a plausible mechanism that can explain the Reynolds-number dependence of $g(R)$. Meanwhile, 2D stochastic simulations based on the Smoluchowski equations for $\mathit{St}\ll 1$ show that the collision growth can be predicted by the 2D ST56 model and that rare but strong events do not play a significant role in such a small-$\mathit{St}$ particle system. However, the probability density function of local $\mathit{St}$ at the sites of colliding particle pairs supports the view that powerful rare events can be important for particle growth even in the absence of internal intermittency when $\mathit{St}$ is not much smaller than unity.

Direct Numerical Simulation in Two Dimensional Homogeneous Isotropic Turbulence Using Spectral Method

Direct numerical simulation (DNS) in two-dimensional homogeneous isotropic turbulence is performed by using the Spectral method at a Reynolds number Re = 1000 on a uniformly distributed grid points. The Reynolds number is low enough that the computational grid is capable of resolving all the possible turbulent scales. The statistical properties in the computed flow field show a good agreement with the qualitative behavior of decaying turbulence. The behavior of the flow structures in the computed flow field also follow the classical idea of the fluid flow in turbulence. Keywords: Direct numerical simulation, Isotropic turbulence, Spectral method. © 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi:http://dx.doi.org/10.3329/jsr.v5i3.12665 J. Sci. Res. 5 (3), 435-445 (2013)

High-order methods for decaying two-dimensional homogeneous isotropic turbulence

Numerical schemes used for the integration of complex flow simulations should provide accurate solutions for the long time integrations these flows require. To this end, the performance of various high-order accurate numerical schemes is investigated for direct numerical simulations (DNS) of homogeneous isotropic two-dimensional decaying turbulent flows. The numerical accuracy of compact difference, explicit central difference, Arakawa, and dispersion-relation-preserving schemes are analyzed and compared with the Fourier–Galerkin pseudospectral scheme. In addition, several explicit Runge–Kutta schemes for time integration are investigated. We demonstrate that the centered schemes suffer from spurious Nyquist signals that are generated almost instantaneously and propagate into much of the field when the numerical resolution is insufficient. We further show that the order of the scheme becomes increasingly important for increasing cell Reynolds number. Surprisingly, the sixth-order schemes are found to be in perfect agreement with the pseudospectral method. Considerable reduction in computational time compared to the pseudospectral method is also reported in favor of the finite difference schemes. Among the fourth-order schemes, the compact scheme provides better accuracy than the others for fully resolved computations. The fourth-order Arakawa scheme provides more accurate results for under-resolved computations, however, due to its conservation properties. Our results show that, contrary to conventional wisdom, difference methods demonstrate superior performance in terms of accuracy and efficiency for fully resolved DNS computations of the complex flows considered here. For under-resolved simulations, however, the choice of difference method should be made with care.

Propagation of wrinkled turbulent flames in the context of hydrodynamic theory

We study the propagation of premixed flames in two-dimensional homogeneous isotropic turbulence using a Navier–Stokes/front-capturing methodology within the context of hydrodynamic theory. The flame is treated as a thin layer separating burnt and unburnt gases, of vanishingly small thickness, smaller than the smallest fluid scales. The method is thus suitable to investigate the flame propagation in the wrinkled flamelet regime of turbulent combustion. A flow-control system regulates the mean position of the flame and the incident turbulence intensity. In this context we study the individual effects of turbulence intensity, turbulence scale, thermal expansion, hydrodynamic strain and hydrodynamic instability on the propagation characteristics of the flame. Results are obtained assuming positive Markstein length, corresponding to lean hydrocarbon–air or rich hydrogen–air mixtures. For stable planar flames we find a quadratic dependence of turbulent speed on turbulence intensity. Upon onset of hydrodynamic instability, corrugated structures replace the planar conformation and we observe a greater resilience to turbulence, the quadratic scaling being replaced by scaling exponents less than one. Such resilience is also confirmed by the observation of a threshold turbulence intensity below which the propagation speed of corrugated flames is indistinguishable from the laminar speed. Turbulent speed is found to increase and later plateau with increasing thermal expansion, this affecting the average flame displacement but not the mean flame curvature. In addition, turbulence integral scale is also observed to affect the propagation of the flame with the existence of an intermediate scale maximizing the turbulent speed. This maximizing scale is smaller for corrugated flames than it is for planar flames, implying that small eddies that will be unable to significantly perturb a planar front could be rather effective in perturbing a corrugated flame. Turbulent planar flames, and more so corrugated flames, were observed to experience a positive mean hydrodynamic strain, which was explained in terms of the overwhelming mean contribution of the normal component of strain. The positive straining causes a decrease in the mean laminar propagation speed which in turn can decrease the turbulent speed. The effect of the flame on the incident turbulent field was examined in terms of loss of isotropy and vorticity destruction by thermal expansion. The latter can be mitigated by a baroclinic vorticity generation which is enhanced for corrugated flames.

Turbulent pair separation due to multiscale stagnation point structure and its time asymmetry in two-dimensional turbulence

The pair separation model of Goto and Vassilicos [New J. Phys. 6, 65 (2004)] is revisited and placed on a sound mathematical foundation. A direct numerical simulation of two-dimensional homogeneous isotropic turbulence with an inverse energy cascade and a k−5/3 power law is used to investigate properties of pair separation in two-dimensional turbulence. A special focus lies on the time asymmetry observed between forward and backward separations. Application of the present model to these data suffers from finite inertial range effects and thus, conditional averaging on scales rather than on time has been employed to obtain values for the Richardson constants and their ratio. The Richardson constants for the forward and backward case are found to be (1.066±0.020) and (0.999±0.007), respectively. The ratio of Richardson constants for the backward and forward cases is therefore gb/gf=(0.92±0.03), and hence exhibits a qualitatively different behavior from pair separation in three-dimensional turbulence, where gb>gf [J. Berg et al., Phys. Rev. E 74, 016304 (2006)]. This indicates that previously proposed explanations for this time asymmetry based on the strain tensor eigenvalues are not sufficient to describe this phenomenon in two-dimensional turbulence. We suggest an alternative qualitative explanation based on the time asymmetry related to the inverse versus forward energy cascade. In two-dimensional turbulence, this asymmetry manifests itself in merging eddies due to the inverse cascade, leading to the observed ratio of Richardson constants.

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.

Viscous Dissipation Model for Fast Vortex Method in Simulation of Decaying Turbulence

For realistic turbulent flow simulation, quantitative representation of turbulent flow dynamics is desired. In the present study, two-dimensional homogeneous isotropic turbulence is simulated by using grid-free vortex method to focus on the viscous dissipation process. The results are compared with those of spectral DNS. Two viscous diffusion models, i.e., core spreading model and Moving Particle Semi-implicit (MPS) Laplacian model, are compared. For the former model, marge and addition process are incorporated to ensure uniform distribution of vortex elements. It is shown that the MPS Laplacian model is superior to the conventional core spreading model in terms of the decay rate of enstrophy and energy spectra. Computational time is remarkably reduced by using Fast Multipole Method (FMM) while retaining accuracy.

Interpolated Differential Operator (IDO) Scheme for Direct Numerical Simulation of Two-Dimensional Homogeneous Isotropic Turbulence

Direct Numerical Simulation (DNS) of two-dimensional homogeneous isotropic turbulence is conducted using the Interpolated Differential Operator (IDO) scheme, which is highly accurate numerical scheme based on spatial discretization. Fifth-order Hermite interpolation and four stage Runge-Kutta time integration methods are employed. The incompressible Navier-Stokes equation is solved by the SMAC algoritim under the same conditions (initial and boundary conditions, comparable grid number, and time step) as the spectral method. It is found that the present IDO scheme has almost comparable accuracy with that of the spectral method in all the statistical quantities and the energy spectrum.

Hierarchical structure analysis of forced two-dimensional turbulence simulated by lattice BGK model

Forced two-dimensional homogeneous isotropic turbulence is simulated using lattice BGK model on parallel PC cluster with high resolution of 1024 2. Statistically stationary database over one turnover time is available to do statistic analysis. Based on the database hierarchical structure analysis of vorticity field is done according to She-Leveque (S-L) model. It is found that intermittency exists in this case and the S-L model can depict very well the anomalous scaling law.

509 高強度乱流における水素・空気予混合火炎の構造(熱工学)

Direct numerical simulations of hydrogen-air turbulent premixed flames propagating in two-dimensional homogeneous isotropic turbulence are conducted to investigate the effects of turbulence length scale on the structure of turbulent premixed flames. Detailed kinetic mechanism including 12 reactive species and 27 elementary reactions is used to simulate H_2-O_2-N_2 reaction in turbulence. DNS are conducted for the cases of turbulence length scale of 1.00, 2.42, 4.84 and 7.26 times of the laminar flame thickness under the constant turbulence intensity. With the decrease of the turbulence length scale, particular flame structure can be observed in which distributions of heat release rate and major radicals such as H, O and OH show relatively smooth profiles whereas those of temperature, density and minor radicals show complicated patterns in the flame front. These local flame structures seem to represent the regime of well stirred reaction zone in the turbulent combustion diagram. The local flame elements in turbulence were identified by using local maximum temperature gradients and statistics of turbulent premixed flames are investigated.

Structure of High Intensity Turbulent Premixed Flames.

Direct numerical simulations of hydrogen-air turbulent premixed flames propagating in two-dimensional homogeneous isotropic turbulence are conducted to investigate the effects of turbulence intensity on the structure of turbulent premixed flames. Detailed kinetic mechanism including 12 reactive species and 27 elementary reactions is used to simulate H2-O2-N2 reaction in turbulence. DNS are conducted for the cases of turbulent intensities of 10.0, 20.0 and 30.0 times of the laminar burning velocity. Turbulent burning velocity increases with the increase of turbulence intensity. The flame fronts for the case of lowest turbulence intensity can be classified in the regime of 'wrinkled laminar flame', whereas those for the cases of higher turbulence intensities show the characteristics of 'distributed reaction zone'. The mean flame thickness increases and the mean heat release rate decreases with the increase of turbulence intensity. However, for the higher intensity cases, variances of local flame thickness and heat release rate become large where the minimum local flame thickness decreases and the maximum local heat release rate increases with the increase of turbulence intensity. The local flame thickness and local heat release rate are well correlated with the tangential strain rate at the flame front.

Local Flame Structure of a H2-Air Turbulent Premixed Flame.

Direct numerical simulation of a H2-Air turbulent premixed flame propagating in two-dimensional homogeneous isotropic turbulence was conducted to clarify local flame structure of turbulent premixed flames. Detailed kinetic mechanism including 21 elementary reactions and 8 reactive species was used to simulate H2-Air combustion in turbulence. Local heat releases are well correlated with curvatures of the flame elements. Flames convex toward the burnt gas show large heat release, while those concave toward the burnt side show relatively low heat release. In the range of small curvatures, local heat releases increase linearly with the increase of curvature. Local heat release can be represented as a linear function of strain rate tangential to the flame front. Mean tangential strain rate at the flame fronts shows positive value, which results in the larger mean heat release than that of the laminar flame. The change of local heat release in turbulence can be explained by the modification of local flame structures due to difference of diffusion coefficients of O2 and H2.

Direct Numerical Simulation of Two-Dimensional Homogeneous Isotropic Turbulence by Higher-Order Finite Difference Scheme.(Comparison with the Spectral Method).

Computations of 2-dimensional homogeneous isotropic turbulence by the spectral method and the higher-order finite difference scheme (FDS) have been conducted with the grids of 128×128 and 64×64. The energy spectrum obtained by the FDS is compared with that obtained by the spectral method to investigate the effect of discretization. From these studies, the following conclusions are obtained. (1) As for the discretization of the nonlinear term, with the 128×128 grids, the energy spectrum obtained by the 5th-order upwind FDS coincides with that obtained by the spectral method up to the cutoff wave number, while the 3rd-order upwind FDS shows deflection. (2) With the 64×64 grids, the 5th-order upwind FDS is still valid for DNS, while the 6th-order central FDS requires the use of dealiasing operation or energy-conserving form for the nonlinear term. (3) With the nonlinear term discretized by the 5th-order upwind FDS, the 4th-order FDS is adequate for the diffusion term.

Global spectra of energy and enstrophy and their fluxes during July 1979

Transient and stationary spectra of kinetic energy (KE), available potential energy (APE) and enstrophy (EN), and their spectral fluxes as a function of the two-dimensional wavenumbern were computed for July 1979. Triangular truncation at zonal wavenumber 42 was used for computation. The slopes of various spectra in the wavenumber range 14≤n≤25 were obtained by fitting a straight line in log-log scale by the least square method. The transientKE, APE andEN spectra in the lower (upper) troposphere had slopes −2·21 (−2·30), −2·65 (−2·64) and −0·36 (−0·46), respectively. The effect of stationary and divergent motion on the slope values was investigated. The possible correlation between the slope and percentage of transient component in the combined energy and enstrophy was examined to identify the transient motion of the atmosphere with the two-dimensional homogeneous isotropic turbulence. The vertically averaged slope of kinetic energy and enstrophy in the lower (upper) troposphere was close to the value at 700 (200) hPa level.

Large-scale direct simulation of two-dimensional homogeneous isotropic turbulence using a higher-order method of lines.

A new higher order method is devised for the numerical simulation of two-dimensional homogeneous isotropic turbulence. The spatial derivatives of the Navier-Stokes equations are discretized by means of the modified differential quadrature (MDQ) method. The resulting system of ordinary differential equations in time is integrated by the fourth-order Runge-Kutta-Gill (RKG) schemes. The elliptic (Poisson) equation in solved by a new variable-order multigrid method. The direct simulations are presented at resolutions up to 1024×1024. The results suggest that the present higher order method is more efficient, and that the history of the vorticity field consists of three stages, i. e., development, peak, and decay. The corresponding power laws of inertial energy spectrum are k-4, k-3, and k-4, respectively.

Diffusion of a passive scalar in two-dimensional turbulence

We study the spectral statistics of the fluctuations of a passive scalar convected by a two-dimensional homogeneous isotropic turbulence using the eddy-damped quasinormal Markovian (EDQNM) theory, and – in certain cases – the near-equivalent test-field model (TFM). For zero correlation between scalar and vorticity fields it is known that these closures lead to inertial–convective ranges following respectively a k−1 law in the enstrophy-cascade range and a k−5/3 law in the inverse-energy-cascade range. We show that the scalar cascade in the latter range is direct, and is characterized by a positive eddy diffusivity. For forced flows in which correlation between vorticity and scalar forcing is prescribed, the k−5/3 range is replaced by a k1/3 range if the correlation is perfect, and for imperfect correlation we describe an analysis that bridges the ranges k1/3, k−1.We also examine the infrared (k → 0) behaviour of the energy and scalar spectra. Statistically steady injection of energy and scalar variance at a wavenumber kI produces an energy spectrum E(k) ∼ k3, and a scalar spectrum E0(k) ∼ k. In the unforced case, for any initial conditions, the energy spectrum develops a k3 range for k less than the wavenumber characteristic of the energy-containing eddies and a k−3 range for larger k. This allows a demonstration – via closure – that this energy-bearing wavenumber decreases as t−1 and the enstrophy as t−2 (modulo logarithmic corrections), as predicted by Batchelor (1969). Finally we show the scalar-fluctuation variance decays as the enstrophy, if the enstrophy spectrum is considered as a passive scalar. If not, the decay exponent is proportional to the ratio of the characteristic eddy-damping rates of the velocity and scalar third-order moments.