Navier-Stokes Dynamics Research Articles

Error estimates for finite-element Navier-Stokes solvers without standard Inf-Sup conditions

The authors establish error estimates for recently developed finite-element methods for incompressible viscous flow in domains with no-slip boundary conditions. The methods arise by discretization of a well-posed extended Navier-Stokes dynamics for which pressure is determined from current velocity and force fields. The methods use C1 elements for velocity and C0 elements for pressure. A stability estimate is proved for a related finite-element projection method close to classical time-splitting methods of Orszag, Israeli, DeVille and Karniadakis.

Cascades, thermalization, and eddy viscosity in helical Galerkin truncated Euler flows

The dynamics of the truncated Euler equations with helical initial conditions are studied. Transient energy and helicity cascades leading to Kraichnan helical absolute equilibrium at small scales, including a linear scaling of the relative helicity spectrum are obtained. Strong helicity effects are found using initial data concentrated at high wave numbers. Using low-wave-number initial conditions, the results of Cichowlas et.al. [Phys. Rev. Lett. 95, 264502 (2005)] are extended to helical flows. Similarities between the turbulent transient evolution of the ideal (time-reversible) system and viscous helical flows are found. Using an argument in the manner of Frisch et. al. [Phys. Rev. Lett. 101, 144501 (2008)], the excess of relative helicity found at small scales in the viscous run is related to the thermalization of the ideal flow. The observed differences in the behavior of truncated Euler and (constant viscosity) Navier-Stokes are qualitatively understood using the concept of eddy viscosity. The large scales of truncated Euler equations are then shown to follow quantitatively an effective Navier-Stokes dynamics based on a variable (scale dependent) eddy viscosity.

Calculation of scalar structure functions from a vortex model of turbulent passive scalar transport

A Saffman and Pullin [Phys. Fluids 8, 3072 (1996)] type vortex model for passive scalar structure functions is formulated. The intermittent turbulent fine-scale dynamics in the model is represented by numerical solutions of the advection-diffusion and Navier–Stokes equations in the form of axially strained vortex-scalar structures. The use of these structures is motivated by Pullin and Lundgren’s [Phys. Fluids 13, 2553 (2001)] asymptotic strained spiral vortex model of turbulent passive scalar transport. Ensemble-averaged scalar structure functions, of even orders 2–10, are calculated from a range of vortex-scalar structures using Monte Carlo integration. For axisymmetric strained scalar fields, acceptable agreement of the second-order structure function with experimental data reported by Antonia and Van Atta [J. Fluid Mech. 84, 561 (1978)] is obtained. Structure functions are also calculated for a range of passive scalar spiral structures. These are generated by the winding of single and double scalar patches in single strained vortex patches and in merging strained vortices. Power-law scaling of the second- and higher-order structure functions is obtained from cases involving the winding of single scalar patches in an axisymmetric strained vortex patch. The second-order scaling exponents from these cases are in reasonable agreement with Kolmogorov–Oboukhov–Corrsin scaling and the experimental results of Antonia et al. [Phys. Rev. A 30, 2704 (1984)] and Gylfason and Warhaft [Phys. Fluids 16, 4012 (2004)]. However, the higher-order scaling exponents from these cases fall below theoretical predictions and experimental results. Higher-order moments are sensitive to the composition of the vortex-scalar structures, and various improvements are suggested that could enhance the performance of the model. The present approach is promising, and it is the first demonstration that a vortex model using simplified Navier–Stokes dynamics can produce some scalar structure functions that compare favorably with experimental observations.

Relaxation and self-organization in two-dimensional plasma and neutral fluid flow systems

Extensive numerical studies in the framework of a simplified two-dimensional model for neutral and plasma fluid for a variety of initial configurations and for both decaying and driven cases are carried out to illustrate relaxation toward a self-organized state. The dynamical model equation constitutes a simple choice for this purpose, e.g., the vorticity equation of the Navier–Stokes dynamics for the incompressible neutral fluids and the Hasegawa–Mima equation for plasma fluid flow system. Scatter plots are employed to observe a development of functional relationship, if any, amidst the generalized vorticity and its Laplacian. It is seen that they do not satisfy a linear relationship as the well known variational approach of enstrophy minimization subject to constancy of the energy integral for the two-dimensional (2D) system suggests. The observed nonlinear functional relationship is understood by separating the contribution to the scatter plot from spatial regions with intense vorticity patches and those of the background flow region where the background vorticity is weak or absent altogether. It is shown that such a separation has close connection with the known exact analytical solutions of the system. The analytical solutions are typically obtained by assuming a finite source of vorticity for the inner core of the localized structure, which is then matched with the solution in the outer region where vorticity is chosen to be zero. The work also demonstrates that the seemingly ad hoc choice of the linear vorticity source function for the inner region is in fact consistent with the self-organization paradigm of the 2D systems.

Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics

In this paper, we develop a numerical method to solve Boltzmann like equations of kinetic theory which is able to capture the compressible Navier–Stokes dynamics at small Knudsen numbers. Our approach is based on the micro/macro decomposition technique, which applies to general collision operators. This decomposition is performed in all the phase space and leads to an equivalent formulation of the Boltzmann (or BGK) equation that couples a kinetic equation with macroscopic ones. This new formulation is then discretized with a semi-implicit time scheme combined with a staggered grid space discretization. Finally, several numerical tests are presented in order to illustrate the efficiency of our approach. Incidentally, we also introduce in this paper a modification of a standard splitting method that allows to preserve the compressible Navier–Stokes asymptotics in the case of the simplified BGK model. Up to our knowledge, this property is not known for general collision operators.

Generalized Eulerian-Lagrangian description of Navier-Stokes dynamics

Generalized equations of motion for the Weber-Clebsch potentials that reproduce Navier-Stokes dynamics are derived. These depend on a new parameter, with the dimension of time, and reduce to the Ohkitani and Constantin equations in the singular special case where the new parameter vanishes. Let us recall that Ohkitani and Constantin found that the diffusive Lagrangian map became noninvertible under time evolution and required resetting for its calculation. They proposed that high frequency of resetting was a diagnostic for vortex reconnection. Direct numerical simulations are performed. The Navier-Stokes dynamics is well reproduced at small enough Reynolds number without resetting. Computation at higher Reynolds numbers is achieved by performing resettings. The interval between successive resettings is found to abruptly increase when the new parameter is varied from 0 to a value much smaller than the resetting interval.

Anisotropic developments for homogeneous shear flows

The general decomposition of the spectral correlation tensor Rij(k) by Cambon et al. [J. Fluid Mech. 202, 295 (1989); 337, 303 (1997)] into directional and polarization components is applied to the representation of Rij(k) by spherically averaged quantities. The decomposition splits the deviatoric part Hij(k) of the spherical average of Rij(k) into directional and polarization components Hij(e)(k) and Hij(z)(k). A self-consistent representation of the spectral tensor is constructed in terms of these spherically averaged quantities. The directional and polarization components must be treated independently: representation of the spectral tensor using the spherical average Hij(k) alone proves to be inconsistent with Navier-Stokes dynamics. In particular, a spectral tensor consistent with a prescribed Reynolds stress is not unique. Since spherical averaging entails a loss of information, the description of an anisotropic correlation tensor by spherical averages is limited to weak departures from isotropy. The degree of anisotropy permitted is restricted by realizability requirements. More general descriptions can be given using a higher-order expansion of the spectral tensor. Directionality is described by a conventional expansion in spherical harmonics, but polarization requires an expansion in special tensorial quantities generated by irreducible representations of the rotation group SO3. These expansions are considered in more detail in the special case of axial symmetry.

Coexistence of chaotic and non-chaotic states in the two-dimensional Gauss–Navier–Stokes dynamics

Recently, Gallavotti proposed an Equivalence Conjecture in hydrodynamics, which states that forced-damped fluids can be equally well represented by means of the Navier–Stokes equations (NS) and by means of time reversible modifications of NS called Gauss–Navier–Stokes equations (GNS). This Equivalence Conjecture received numerical support in several recent papers concerning two-dimensional fluid mechanics. The corresponding results rely on the fact that the NS and GNS systems only have one attracting set. Performing similar two-dimensional simulations, we find that there are conditions to be met by the GNS system for this to be the case. In particular, increasing the Reynolds number, while keeping fixed the number of Fourier modes, leads to the coexistence of different attractors. This makes difficult a test of the Equivalence Conjecture, but constitutes a spurious effect due to the insufficient spectral resolution. With sufficiently fine spectral resolution, the steady states are unique and the Equivalence Conjecture can be conveniently established.

Reconnection of colliding vortex rings.

We investigate numerically the Navier-Stokes dynamics of reconnecting vortex rings at small Reynolds number for a variety of configurations. We find that reconnections are dissipative due to the smoothing of vorticity gradients at reconnection kinks and to the formation of secondary structures of stretched antiparallel vorticity which transfer kinetic energy to small scales where it is subsequently dissipated efficiently. In addition, the relaxation of the reconnection kinks excites Kelvin waves which due to strong damping are of low wave number and affect directly only large scale properties of the flow.

Exponential mixing of the 2D stochastic Navier-Stokes dynamics

We consider the Navier-Stokes equation on a two dimensional torus with a random force which is white noise in time, and excites only a finite number of modes. The number of excited modes depends on the viscosity v, and grows like v(-3) when v goes to zero. We prove that this Markov process has a unique invariant measure and is exponentially mixing in time.

Large Eddy Simulations Using the Subgrid-Scale Estimation Model and Truncated Navier-Stokes Dynamics

We describe a procedure for large eddy simulations of turbulence which uses the subgrid-scale estimation model and truncated Navier–Stokes dynamics. In the procedure the large eddy simulation equations are advanced in time with the subgrid-scale stress tensor calculated from the parallel solution of the truncated Navier–Stokes equations on a mesh two times smaller in each Cartesian direction than the mesh employed for a discretization of the resolved quantities. The truncated Navier–Stokes equations are solved through a sequence of runs, each initialized using the subgrid-scale estimation model. The modeling procedure is evaluated by comparing results of large eddy simulations for isotropic turbulence and turbulent channel flow with the corresponding results of experiments, theory, direct numerical simulations, and other large eddy simulations. Subsequently, simplifications of the general procedure are discussed and evaluated. In particular, it is possible to formulate the procedure entirely in terms of the truncated Navier–Stokes equation and a periodic processing of the small-scale component of its solution.

Energy dissipation and dynamical complexity in a truncated two-dimensional Navier–Stokes dynamics

The five-mode truncated version of the planar incompressible Navier–Stokes equations developed by Franceschini and co-authors is analyzed from the standpoint of irreversible thermodynamics. The entropy production is expressed in terms of the model variables and its dependence on the control parameter (driving force) is determined. Its variability patterns as the system runs over the periodic and the chaotic attractors encountered at different parameter values are subsequently compared with various indicators of the intrinsic complexity of the dynamics, such as the local Lyapunov exponents and a local generalization of the Kolmogorov entropy. The analysis reveals some unexpected correlations. In particular, it appears that the largest span of values of entropy production occurs during stages dominated by stability, while the maximum value occurs during stages dominated by instability.

Finite-mode spectral model of homogeneous and isotropic Navier-stokes turbulence: a rapidly depleted energy cascade.

An eddy-viscous term is added to Navier-Stokes dynamics at wave numbers k greater than the inflection point kc of the energy flux F(log(k)). The eddy viscosity is fixed so that the energy spectrum satisfies E(k) = E(kc) (k/kc)(-3) for k>kc. This resulting forcing induces a rapid depletion of the energy cascade at k>kc. It is observed numerically that the model reproduces turbulence energetics at k< or =kc and statistics of two-point velocity correlations at scales r>lambda (Taylor microscale). Compared to a direct numerical simulation of R(lambda) = 130 an equivalent run with the present model results in a gain of a factor 20 in CPU time.

Dissipative and dispersive behaviors of lattice-based models for hydrodynamics

Both dissipation and dispersion are present in many complex systems; their interactions through nonlinearity can lead to interesting features. We investigate in this paper the dissipation-dispersion interactions that exist in lattice-based kinetic models for hydrodynamics. The classical Chapman-Enskog expansion is used to derive the dispersion coefficients at third order of Knudsen number. Unlike the dissipation coefficient (viscosity) that is always positive, the dispersion coefficient can be either positive or negative. It would be interesting to know if there is any other physics in these models as compared with the traditional dispersionless Navier-Stokes dynamics. Traveling wave solutions in one dimension are studied and two different solutions have been found: (1) monotonic shock solutions and (2) oscillatory shock solutions, according to different conditions. In two- and three-dimensional systems, whether or not these oscillatory behaviors caused by the interactions between nonlinearity, dissipation, and dispersion have anything to do with vortex cascades (direct or inverse) would be an interesting question and we leave it for future studies.

Time evolution of the eddy viscosity in two-dimensional navier-stokes flow

The time evolution of the eddy viscosity associated with an unforced two-dimensional incompressible Navier-Stokes flow is analyzed by direct numerical simulation. The initial condition is such that the eddy viscosity is isotropic and negative. It is shown by concrete examples that the Navier-Stokes dynamics stabilizes negative eddy viscosity effects. In other words, this dynamics moves monotonically the initial negative eddy viscosity to positive values before relaxation due to viscous term occurs.

The subgrid-scale estimation model for high Reynolds number turbulence

We propose a formulation of the subgrid-scale estimation model in which the effects of the estimated subgrid scales on the resolved scales are obtained through the truncated Navier–Stokes dynamics and the calculation of the subgrid-scale stress tensor is not required. For high Reynolds number isotropic turbulence the model predicts the k−5/3 spectrum with the correct value of the Kolmogoroff constant.

Statistical-mechanical predictions and Navier-Stokes dynamics of two-dimensional flows on a bounded domain.

In this paper the applicability of a statistical-mechanical theory to freely decaying two-dimensional (2D) turbulence on a bounded domain is investigated. We consider an ensemble of direct numerical simulations in a square box with stress-free boundaries, with a Reynolds number that is of the same order as in experiments on 2D decaying Navier-Stokes turbulence. The results of these simulations are compared with the corresponding statistical equilibria, calculated from different stages of the evolution. It is shown that the statistical equilibria calculated from early times of the Navier-Stokes evolution do not correspond to the dynamical quasistationary states. At best, the global topological structure is correctly predicted from a relatively late time in the Navier-Stokes evolution, when the quasistationary state has almost been reached. This failure of the (basically inviscid) statistical-mechanical theory is related to viscous dissipation and net leakage of vorticity in the Navier-Stokes dynamics at moderate values of the Reynolds number.

Taylor expansions in powers of time of Lagrangian and Eulerian two-point two-time velocity correlations in turbulence

A method is developed for generating the Taylor expansions in powers of the time difference of the Lagrangian and Eulerian two-point two-time velocity correlations in turbulence. The expansions are based on the Taylor series of the Eulerian and Lagrangian velocity fields subject to given dynamics along with initial and boundary conditions. The lowest few coefficients in the expansions enable us to construct approximations to the correlations. An application of the method to turbulence obeying the Navier-Stokes dynamics yields approximations, particularly Padé approximations that agree well with direct numerical simulations of homogeneous isotropic turbulence at moderate Reynolds numbers. The ratios of the second-order to the zeroth-order coefficients of the Taylor series of the Lagrangian and Eulerian correlations give, respectively, the estimates for the Lagrangian and Eulerian micro time scales τL and τE. An analysis of a high resolution (5123 grid points) direct numerical simulation database at large Reynolds number suggests the scalings τL∝k−2/3 and τE∝k−1 for wave numbers k in the inertial subrange. The role of flow structures in turbulence in determining the time scales is also discussed.

Predictive Turbulence Modeling by Variational Closure

We show that a variational implementation of probability density function (PDF) closures has the potential to make predictions of general turbulence mean statistics for which a priori knowledge of the incorrectness is possible. This possibility exists because of realizability conditions on “effective potential” functions for general turbulence statistics. These potentials measure the cost for fluctuations to occur away from the ensemble-mean value in empirical time-averages of the given variable, and their existence is a consequence of a refined ergodic hypothesis for the governing dynamical system (Navier–Stokes dynamics). Approximations of the effective potentials can be calculated within PDF closures by an efficient Rayleigh–Ritz algorithm. The failure of realizability within a closure for the approximate potential of any chosen statistic implies a priori that the closure prediction for that statistic is not converged. The systematic use of these novel realizability conditions within PDF closures is shown in a simple 3-mode system of Lorenz to result in a statistically improved predictive ability. In certain cases the variational method allows an a priori optimum choice of free parameters in the closure to be made.

Ultrametric Structure of Multiscale Energy Correlations in Turbulent Models

The ultrametric structure of the energy cascade process in a dynamical model of turbulence is studied. The tree model we use can be viewed as an approximated one-dimensional truncation of the wavelets-resolved Navier-Stokes dynamics. Varying the tree connectiveness, the appearance of a scaling transition in the two-point moments of energy dissipation is detected, in agreement with experimental turbulent data.