Turbulence In Incompressible Fluid Research Articles

Vorticity spectra in high Reynolds number anisotropic turbulence

Assuming a turbulent flow to be homogeneous and isotropic allows for significant theoretical simplification in the description of its motions. The validity of these assumptions, first put forth by Kolmogorov [A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluids for very large Reynolds numbers,” C. R. Acad. Sci. URSS 30, 301 (1941)], has been the subject of considerable analytical development and extensive research as they are applied to actual flows. The present investigation describes the one-dimensional vorticity spectra of two flow fields: a single stream shear layer and the near surface region of an atmospheric boundary layer. Both flow fields exhibit a power-law region with a slope of −1 in the one-dimensional spectra of the spanwise component of vorticity in the same wave-number range for which the velocity spectra indicated the isotropic behavior. This is in strong disagreement with the isotropic prediction, which does not exhibit a power-law behavior.

Performance analysis of a parallel finite element solution to the direct numerical simulation of fluid turbulence on Linux PC clusters

The parallel performance of a direct numerical simulation (DNS) computer code was tested on three different sets of Linux PC clusters including Intel^(R) Pentium^(R) (P) III 800MHz, P III 1GHz, and Xeon(TM) 2.8GHz. The DNS code is used to simulate incompressible fluid turbulence and is developed by the Galerkin finite element method (FEM) with equal-order interpolating polynomials. It employs domain decomposition for data structure together with data parallelism for matrix assemblies and operations. This paper discusses the scalability of the code and contrasts the code performance on the Linux PC clusters.

A high‐order finite difference method for incompressible fluid turbulence simulations

AbstractA Hermitian–Fourier numerical method for solving the Navier–Stokes equations with one non‐homogeneous direction had been presented by Schiestel and Viazzo (Internat. J. Comput. Fluids 1995; 24(6):739). In the present paper, an extension of the method is devised for solving problems with two non‐homogeneous directions. This extension is indeed not trivial since new algorithms will be necessary, in particular for pressure calculation. The method uses Hermitian finite differences in the non‐periodic directions whereas Fourier pseudo‐spectral developments are used in the remaining periodic direction. Pressure–velocity coupling is solved by a simplified Poisson equation for the pressure correction using direct method of solution that preserves Hermitian accuracy for pressure. The turbulent flow after a backward facing step has been used as a test case to show the capabilities of the method. The applications in view are mainly concerning the numerical simulation of turbulent and transitional flows. Copyright © 2003 John Wiley & Sons, Ltd.

Effect of the external magnetic field on the MHD turbulence spectra

The turbulent properties of conducting fluids in an external constant magnetic field are known to change with increasing field strength. A study is made of the behavior of the second-order structural function of the velocity field in a homogeneous incompressible turbulent fluid in the presence of an external uniform magnetic field. It is shown that, depending on the magnetic field strength, there may be different governing parameters of the system in both the inertial and dissipative intervals of turbulence. This leads to new spectral scalings that are consistent with experimental ones.

A probabilistic method for Navier-Stokes vortices

Consider a Navier-Stokes incompressible turbulent fluid in R2. Let x(t) denote the position coordinate of a moving vortex with initial circulation Γ0 > 0 in the fluid, subject to a force F. Define x(t) as a stochastic process with continuous sample paths described by a stochastic differential equation. Assuming a suitable notion of weak rotationality, it is shown that the stochastic equation is equivalent to a linear partial differential equation for the complex function ψ, i∂ψ/∂t = [-Γ0Δ + F] ψ, where |ψ|2 = ρ(x,t), ρ being the probability density function of finding the vortex centre in position x at time t.

A probabilistic method for Navier-Stokes vortices

Consider a Navier-Stokes incompressible turbulent fluid in R 2. Let x(t) denote the position coordinate of a moving vortex with initial circulation Γ0 > 0 in the fluid, subject to a force F. Define x(t) as a stochastic process with continuous sample paths described by a stochastic differential equation. Assuming a suitable notion of weak rotationality, it is shown that the stochastic equation is equivalent to a linear partial differential equation for the complex function ψ, i∂ψ/∂t = [-Γ0Δ + F] ψ, where |ψ|2 = ρ(x,t), ρ being the probability density function of finding the vortex centre in position x at time t.

The Navier–Stokes-alpha model of fluid turbulence

We review the properties of the nonlinearly dispersive Navier–Stokes-alpha (NS- α) model of incompressible fluid turbulence — also called the viscous Camassa–Holm equations in the literature. We first re-derive the NS- α model by filtering the velocity of the fluid loop in Kelvin’s circulation theorem for the Navier–Stokes equations. Then we show that this filtering causes the wavenumber spectrum of the translational kinetic energy for the NS- α model to roll off as k −3 for kα>1 in three dimensions, instead of continuing along the slower Kolmogorov scaling law, k −5/3, that it follows for kα<1. This roll off at higher wavenumbers shortens the inertial range for the NS- α model and thereby makes it more computable. We also explain how the NS- α model is related to large eddy simulation (LES) turbulence modeling and to the stress tensor for second-grade fluids. We close by surveying recent results in the literature for the NS- α model and its inviscid limit (the Euler- α model).

Anomalous scalings and dynamics of magnetic helicity

It is demonstrated that the two-point correlation function of the magnetic helicity in the case of zero mean magnetic field has anomalous scalings for both compressible and incompressible turbulent helical fluid flow. The magnetic helicity in the limit of very high electrical conductivity is conserved. This implies that the two-point correlation function of the conserved property does not necessarily have normal scaling. The reason for the anomalous scalings of the magnetic helicity correlation function is that the magnetic field in the equation for the two-point correlation function of the magnetic helicity plays a role of a pumping with anomalous scalings. It is shown also that when magnetic fluctuations with zero mean magnetic field are generated the magnetic helicity is very small even if the hydrodynamic helicity is large.

Dynamics of Particles Advected by Fast Rotating Turbulent Fluid Flow: Fluctuations and Large-Scale Structures

Dynamics of particles advected by fast rotating incompressible turbulent fluid flow is studied. Fast rotation and particle inertia imply the divergent particle velocity field and result in both intermittency in spatial distribution of particles and formation of the large-scale inhomogeneous structures. A nonzero mean helicity of fluid flow causes an additional mean nondiffusive turbulent flux of inertial particles. Intermittency in the systems with and without external pumping is studied. Fast rotation causes anomalous scaling already in the second moment of inertial particle number density and may result in excitation of a small-scale instability of inertial particle distribution, which leads to the formation of small-scale particle clusters. We discuss the relevance of our results for atmospheric, astrophysical, and industrial turbulent rotating flows. [S0031-9007(98)07241-X]

Turbulence in incompressible fluids and magnetic fields

The Hamiltonian dynamics of a two-degree-of-freedom system is shown to mimic the structure of an incompressible fluid if the fluid streamlines are considered instead of the pathlines used by many authors. This approach is formally equivalent to the determination of magnetic field lines, which satisfy the same mathematical equations. Turbulence in this case is seen as Hamiltonian chaotic dynamics perturbed by time, which plays the role of the codimension parameter of such chaos. Any spatial Lie symmetry leads to a laminar flow, whereas stationary flow could be considered chaotic but not turbulent. The transition from laminar to turbulent flow is associated with the destruction of KAM trajectories produced by time evolution.

Interaction of turbulence and large-scale vortices in incompressible 2D fluids

It is commonplace in 2D fluid dynamcis that intense large-scale vortices arise during turbulence decay. Both the large-scale vortex and the turbulent components will be important in the further evolution: the vortices will modify the turbulent dynamics and in turn, will be modified by the turbulence. Using Wigner functions, we derive a new two-component model to describe interaction of turbulence and intense large-scale vortices in incompressible inviscid 2D fluids. We apply this model to study the dynamics of a vortex dipole propagating through turbulence, a problem that allows an elegant analytic solution.

A path integral for turbulence in incompressible fluids

In this paper a classical path integral is formulated for incompressible fluids that evolve according to the Navier–Stokes equation. The path integral propagates probability distributions deterministically on the space ℊvol of solenoidal velocity fields. We construct a set of ISp(2) charges associated with the geometry of ℊvol and its Poisson structure, and a pair of supersymmetry charges connected with the Hamiltonian. These charges generate exact symmetries of the classical path integral when the viscosity is set equal to zero. When the effect of dissipation is included, the charges associated with the Poisson structure and the Hamiltonian are no longer conserved. Charges that generate Kolmogorov scaling and Galilean transformations are also constructed. The classical path integral is formulated in terms of vorticity as well.

Self-Excitation of Fluctuations of Inertial Particle Concentration in Turbulent Fluid Flow.

A mechanism for intermittency in spatial distribution of small inertial particles advected by a turbulent incompressible fluid flow is discussed. The mechanism is related to self-excitation (i.e., exponential growth without an external source) of fluctuations of concentration of small particles in a turbulent fluid flow. The effect is caused by the inertia of particles which results in a divergent velocity field of particles. An equation for the high-order correlation functions of concentration of small inertial particles is derived. It is shown that the growth rates of the higher moments of particle concentration are higher than those of the lower moments, i.e., particle spatial distribution is intermittent. Similar phenomena occur for noninertial admixtures advected by divergent turbulent velocity field.

Inverse cascades in incompressible fluid and magnetofluid turbulence

Absolute equilibrium statistical theory and numerical simulations are reviewed in the context of inverse cascades in two- and three-dimensional incompressible fluid and magnetofluid turbulence. Turbulent fluctuations of physically interesting quantities undergo inverse cascade to larger spatial scales, leading to self-organization under certain circumstances. In particular, most systems with more than one quadratic ideal invariant, or, having some kind of imposed anisotropy, exhibit inverse cascades. Anisotropic fluid turbulence in the presence of a uniform rotation and magnetofluid turbulence in the presence of a uniform magnetic field are considered.

Radiographic Evidence fork−5/3Scaling of Density Power Spectra

Experiments on the explosively driven compression and reexpansion of metal cylinders are reported. Analysis of radiography data taken with a pulsed x-ray source reveals scale invariant density fluctuations. The marginal density spectra are consistent with a {ital k}{sup {minus}8/3} power law, where {ital k} is wave number, corresponding to scalar density spectra varying as {ital k}{sup {minus}5/3}. Remarkably, the same scaling behavior is observed for a variety of metals with widely different mechanical properties. The relation of these results to fully developed turbulence in both incompressible fluids and compressible fluids with large density variations is discussed. {copyright} {ital 1996 The American Physical Society.}

Probability distribution functions for small scale turbulence

The exact expression for the probability distribution function (pdf),P(Δur), of a velocity difference Δur, over a distancer, in incompressible fluid turbulence, obtained from the Navier-Stokes equations, is used as a basis for deriving approximate profiles forP(Δur). These approximate forms are deduced from an approximate factorisation of the underlying functional probability distribution of the flow field, in which the individual factors capture different physical effects.P(Δur) is represented as the integral, with respect to the spatially averaged dissipation rateer, of the product of the conditionalpdf of Δur givener, and thepdf ofer. The approximation yields the latter as a log-Poissonpdf, while the conditionalpdf is found to be a Gaussian for a transverse increment, and the product of a Gaussian and a cubic polynomial for a longitudinal increment. This approximation is equivalent to the refined similarity hypothesis coupled with the log-Poisson distribution, and it possesses the characteristic features ofP(Δur), including symmetric profiles for transverse increments, asymmetric profiles for longitudinal increments, and the development of pronounced non-Gaussian features at small separations. The associated scaling exponents for longitudinal and transverse structure functions are shown to be identical, in this approximation, and to assume the log-Poisson form.

Turbulent thermal diffusion of small inertial particles.

A new physical effect, turbulent thermal diffusion, is discussed. This phenomenon is related to the dynamics of small inertial particles in incompressible turbulent fluid flow. At large Reynolds and Peclet numbers the turbulent thermal diffusion is much stronger than the molecular thermal diffusion. It is shown that inertia of particles under certain conditions can cause a large-scale instability of spatial distribution of particles. Inertial particles are concentrated in the vicinity of the minimum (or maximum) of the mean temperature of the surrounding fluid depending on the ratio of material particle density to that of the surrounding fluid.

Probability distribution functions for Navier–Stokes turbulence

The probability distribution function (PDF), P(Δur), of a velocity difference, Δur, over a distance r in incompressible fluid turbulence is derived from the Navier–Stokes equations using the underlying functional probability distribution. Two different types of approximation are used to evaluate the resulting functional integral for P(Δur). The first is based on the saddle-point technique. It is used to examine the non-Gaussian features of P(Δur) and to demonstrate, in particular, that its tail has the characteristic exponential form associated with intermittency. The second approximation is developed for the purpose of deriving the anomalous scaling exponents of the structure functions from P(Δur). It represents P(Δur) as the integral with respect to the spatially averaged dissipation rate, εr, of the product of the PDF of Δur conditioned to a particular εr, and the PDF of εr. These are coarse-grained PDFs, obtained using the renormalization group. The former is approximately Gaussian, whereas the latter is given by a constrained Gaussian functional integral, which is evaluated approximately using a transformation that enables it to be represented in terms of a Poisson process. This approach yields the scaling exponents in the form discussed by She and Lévêque [Phys. Rev. Lett. 72, 336 (1994)], and also gives the complete third-order structure function exactly.

A control‐volume based finite element method for three‐dimensional incompressible turbulent fluid flow, heat transfer, and related phenomena

AbstractA control‐volume based finite element method of equal‐order type for three‐dimensional incompressible turbulent fluid flow, heat transfer, and related phenomena is presented. The discretization equations are based mainly on the physics of the phenomena under consideration, more than on mathematical arguments. Special emphasis is devoted to the discretization of the convective terms and the continuity equation, and to the treatment of the boundary conditions imposed by the use of a high Reynolds k‐ϵ, type turbulence model. The pressure‐velocity coupling in the fluid flow calculation is made from a derivative of the original SIMPLER method, without pressure correction. The discretized equations are solved in a sequential, rather than a coupled, form with significant advantage in the required computer time and storage. The method is an extension of a former version proposed by us for two‐dimensional, laminar problems, and is here successfully applied to the following situations: three‐dimensional deflected turbulent jet, and flows in 90° and 45° junctions of ducts with rectangular cross sections. The calculated results are in very good agreement with the experimental and numerical (obtained with the well established finite difference method) data available in the literature.

Isotropic and anisotropic turbulence in Clebsch variables

Three-dimensional turbulence of incompressible fluid is described by using Clebsch canonical variables. This reveals the families of new local integrals of motion so that there are additional cascade spectra besides the energy cascade. A weakly anisotropic spectrum of developed turbulence is shown to be as universal as isotropic Kolmogorov spectrum. The correlation functions of three-dimensional incompressible turbulence approach their isotropic values in the inertial interval so that the share taken by the anisotropic parts of velocity correlators decrease with the wavenumber as k − 2 3 , which satisfactorily fits the experimental data. The complementarity of the turbulence description in Clebsch and velocity variables is demonstrated.