Direct numerical simulations with up to 10242 resolution are performed to study statistical properties of the inverse energy cascade in stationary homogeneous two-dimensional turbulence driven by small-scale Gaussian ...
Direct numerical simulations with up to 10242 resolution are performed to study statistical properties of the inverse energy cascade in stationary homogeneous two-dimensional turbulence driven by small-scale Gaussian white-in-time noise. The energy spectra for the inverse energy cascade deviate strongly from the expected k−5/3 law and are close (somewhat flatter) to k−3. The reason for the deviation is traced to the emergence of strong vortices distributed over all scales. Statistical properties of the vortices are explored.
Following the ideas of operator product expansion, the velocity v, kinetic energy K=1/2v2, and dissipation rate ε=ν0(∂vi/∂xj)2 are treated as independent dynamical variables, each obeying its own equation of motion....
Following the ideas of operator product expansion, the velocity v, kinetic energy K=1/2v2, and dissipation rate ε=ν0(∂vi/∂xj)2 are treated as independent dynamical variables, each obeying its own equation of motion. The relations Δu(ΔK)2 ∝ r, Δu(Δε)2 ∝ r0, and (Δu)5≊rΔεΔK are derived. If velocity scales as (Δv)rms∝ r(γ/3)−1, then simple power counting gives (ΔK)rms ∝ r1−(γ/6) and (Δε)rms ∝ 1/√(Δv)rms ∝ r(1/2)−(γ/6). In the Kolmogorov turbulence (γ=4) the intermittency exponent μ=(γ/3)-1=1/3 and (Δε)2=O(Re1/4). The scaling relation for the ε fluctuations is a consequence of cancellation of ultraviolet divergences in the equation of motion for the dissipation rate.
The Kolmogorov relation for the third order structure function is used to derive the energy spectrum in the far dissipation range (k→∞). This contains no unspecified constants. Using methods from matched asymptotic ...
The Kolmogorov relation for the third order structure function is used to derive the energy spectrum in the far dissipation range (k→∞). This contains no unspecified constants. Using methods from matched asymptotic expansions and mild analyticity assumptions, a uniformly valid form for the inertial through the dissipative ranges is obtained. An analogous energy spectrum is presented. This is compared with the results of physical and numerical experiments on the energy spectra E(k). The theoretical predictions are found to deviate by not more than a few percent from the measured data in the entire range of wave numbers where the energy spectrum E(k) varies by more than 30 orders of magnitude.
The Hamiltonian formulation of hydrodynamics in Clebsch variables is used for construction of a statistical theory of turbulence. It is shown that the interaction of the random and large-scale coherent components of t...
The Hamiltonian formulation of hydrodynamics in Clebsch variables is used for construction of a statistical theory of turbulence. It is shown that the interaction of the random and large-scale coherent components of the Clebsch fields is responsible for generation of two energy spectra E(k)∝k−7/3 and E(k)∝k−2 at scales somewhat larger than those corresponding to the -5/3 inertial range. This interaction is also responsible for the experimentally observed Gaussian statistics of the velocity differences at large scales, and the nontrivial scaling behavior of their high-order moments for inertial-range values of the displacement r. The ‘‘anomalous scaling exponents’’ are derived and compared with experimental data.
We consider a family of three-dimensional, volume preserving maps depending on a small parameter epsilon. As epsilon --> 0+ these maps asymptote to flows which attain a heteroclinic connection. We show that for sma...
We consider a family of three-dimensional, volume preserving maps depending on a small parameter epsilon. As epsilon --> 0+ these maps asymptote to flows which attain a heteroclinic connection. We show that for small epsilon the heteroclinic connection breaks up and that the splitting between its components scales with epsilon like epsilon(gamma) exp(-beta/epsilon). We estimate beta using the singularities of the epsilon --> 0+ heteroclinic orbit in the complex plane. We then estimate gamma using linearization about orbits in the complex plane. While these estimates are not proven, they are well supported by our numerical calculations. The work described here is a special case of the theory derived by Amick et al. which applies to q-dimensional volume preserving mappings.
A new rotation symmetry for steady Hele-Shaw flows is reported. In the case when surface tension is neglected, it is shown that if a curve L moving with constant velocity U is a solution to the Hele-Shaw problem, then...
A new rotation symmetry for steady Hele-Shaw flows is reported. In the case when surface tension is neglected, it is shown that if a curve L moving with constant velocity U is a solution to the Hele-Shaw problem, then the curve L obtained from a rotation of L about its center by an arbitrary angle is also a solution with the same velocity U. Similar results hold for the case with surface tension if and only if the Schwarz function of the curve L is regular in the fluid region and at most a linear function at infinity. Several examples in which this principle is used to generate new solutions to the problem are also discussed.
In this letter, a nonlinear deviation from the Navier-Stokes equation is obtained from the recently proposed LBGK models, which are designed as an alternative to lattice gas or lattice Boltzmann equation. The classica...
In this letter, a nonlinear deviation from the Navier-Stokes equation is obtained from the recently proposed LBGK models, which are designed as an alternative to lattice gas or lattice Boltzmann equation. The classical Chapman-Enskog method is extended to derive the nonlinear-deviation term as well as its coefficient. Their analytical expression is derived for the first time, thanks to the simplicity of the LBGK models. A numerical simulation of a shock profile is presented. The influence of the correction on the kinetics of compressible flow is discussed. A complete analysis of the thermodynamics including the temperature will be presented elsewhere.
A new algorithm based on spectral element discretization and non-oscillatory ideas is developed for the solution of hyperbolic partial differential equations. A conservative formulation is proposed based on cell avera...
A new algorithm based on spectral element discretization and non-oscillatory ideas is developed for the solution of hyperbolic partial differential equations. A conservative formulation is proposed based on cell averaging and reconstruction procedures, that employs a staggered grid of Gauss-Chebyshev and Gauss-Lobatto Chebyshev discretizations. The non-oscillatory reconstruction procedure is based on ideas similar to those proposed by Cai et al. (Math. Comput. 52, 389 (1989)) but employs a modified technique which is more robust and simpler in terms of determining the location and strength of a discontinuity. It is demonstrated through model problems of linear advection, inviscid Burgers equation, and one-dimensional Euler system that the proposed algorithm leads to stable, non-oscillatory accurate results. Exponential accuracy away from the discontinuity is realized for the inviscid Burgers equation example.
The short-time behavior of the turbulent viscosity is inferred from the immediate response of the Reynolds stress deduced by Crow [1] for the problem of isotropic turbulence subjected to a mean strain at time t=0. The...
Direct numerical simulations with up to 40962 resolution are performed to address the question of universality of statistical properties of the enstrophy cascade in homogeneous two-dimensional turbulence driven by lar...
Direct numerical simulations with up to 40962 resolution are performed to address the question of universality of statistical properties of the enstrophy cascade in homogeneous two-dimensional turbulence driven by large-scale Gaussian white-in-time noise. Data with different Reynolds numbers are compared with each other. The energy spectrum is found to be very close to 1/k3. It is shown that the primary contribution to the enstrophy transfer function comes from wave-number triads with one small leg and two long ones, corresponding to wave numbers in the inertial range.
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