Three regularization models of the Navier–Stokes equations (original) (raw)

A modified-Leray-α subgrid scale model of turbulence

Nonlinearity, 2006

Inspired by the remarkable performance of the Leray-α (and the Navier-Stokes alpha (NS-α), also known as the viscous Camassa-Holm) subgrid scale model of turbulence as a closure model to Reynolds averaged equations (RANS) for flows in turbulent channels and pipes, we introduce in this paper another subgrid scale model of turbulence, the modified Leray-α (ML-α) subgrid scale model of turbulence. The application of the ML-α to infinite channels and pipes gives, due to symmetry, similar reduced equations as Leray-α and NS-α. As a result the reduced ML-α model in infinite channels and pipes is equally impressive as a closure model to RANS equations as NS-α and all the other alpha subgrid scale models of turbulence (Leray-α and Clark-α). Motivated by this, we present an analytical study of the ML-α model in this paper. Specifically, we will show the global well-posedness of the ML-α equation and establish an upper bound for the dimension of its global attractor. Similarly to the analytical study of the NS-α and Leray-α subgrid scale models of turbulence we show that the ML-α model will follow the usual k −5/3 Kolmogorov power law for the energy spectrum for wavenumbers in the inertial range that are smaller than 1/α and then have a steeper power law for wavenumbers greater than 1/α (where α > 0 is the length scale associated with the width of the filter). This result essentially shows that there is some sort of parametrization of the large wavenumbers (larger than 1/α) in terms of the smaller wavenumbers. Therefore, the ML-α

Highly turbulent solutions of the Lagrangian-averaged Navier-Stokes α model and their large-eddy-simulation potential

Physical Review E, 2007

We compute solutions of the Lagrangian-Averaged Navier-Stokes α−model (LANS−α) for significantly higher Reynolds numbers (up to Re ≈ 8300) than have previously been accomplished. This allows sufficient separation of scales to observe a Navier-Stokes inertial range followed by a second inertial range specific to LANS−α. Both fully helical and non-helical flows are examined, up to Reynolds numbers of ∼ 1300. The analysis of the third-order structure function scaling supports the predicted l 3 scaling; it corresponds to a k −1 scaling of the energy spectrum for scales smaller than α. The energy spectrum itself shows a different scaling which goes as k 1 . This latter spectrum is consistent with the absence of stretching in the sub-filter scales due to the Taylor frozen-in hypothesis employed as a closure in the derivation of LANS−α. These two scalings are conjectured to coexist in different spatial portions of the flow. The l 3 (E(k) ∼ k −1 ) scaling is subdominant to k 1 in the energy spectrum, but the l 3 scaling is responsible for the direct energy cascade, as no cascade can result from motions with no internal degrees of freedom. We demonstrate verification of the prediction for the size of the LANS−α attractor resulting from this scaling.

On a Leray- model of turbulence

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2005

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Dynamical modelling of sub-grid scales in 2D turbulence

We develop a new numerical method which treats resolved and sub-grid scales as two different fluid components evolving according to their own dynamical equations. These two fluids are nonlinearly interacting and can be transformed one into another when their scale becomes comparable to the grid size. Equations describing the two-fluid dynamics were rigorously derived from Euler equations [B. Dubrulle, S. Nazarenko, Physica D 110 (1997) 123-138] and they do not involve any adjustable parameters. The main assumption of such a derivation is that the large-scale vortices are so strong that they advect the sub-grid scales as a passive scalar, and the interactions of small scales with small and intermediate scales can be neglected. As a test for our numerical method, we performed numerical simulations of 2D turbulence with a spectral gap, and we found a good agreement with analytical results obtained for this case by Nazarenko and Laval [Non-local 2D turbulence and passive scalars in Batchelor's regime, J. Fluid Mech., in press]. We used the two-fluid method to study three typical problems in 2D dynamics of incompressible fluids: decaying turbulence, vortex merger and forced turbulence. The two-fluid simulations performed on at 128 2 and 256 2 resolution were compared with pseudo-spectral simulations using hyperviscosity performed at the same and at much higher resolution. This comparison shows that performance of the two-fluid method is much better than one of the pseudo-spectral method at the same resolution and comparable computational cost. The most significant improvement is observed in modeling of the small-scale component, so that effective inertial interval increases by about two decades compared to the high-resolution pseudo-spectral method. Using the two-fluid method, we demonstrated that the k −3 tail always exists for the energy spectrum, although its amplitude is slowly decreasing in decaying turbulence.

Dynamical modeling of sub-grid scales in 2D turbulence

Physica D: Nonlinear Phenomena, 2000

On the properties of similarity subgrid-scale models as deduced from measurements in a turbulent jet

1994

The properties of turbulence subgrid-scale stresses are studied using experimental data in the far field of a round jet, at a Reynolds number of R, z 310. Measurements are performed using two-dimensional particle displacement velocimetry. Three elements of the subgrid-scale stress tensor are calculated using planar filtering of the data. Using a priori testing, eddy-viscosity closures are shown to display very little correlation with the real stresses, in accord with earlier findings based on direct numerical simulations at lower Reynolds numbers. Detailed analysis of subgrid energy fluxes and of the velocity field decomposed into logarithmic bands leads to a new similarity subgridscale model. It is based on the 'resolved stress' tensor L,,, which is obtained by filtering products of resolved velocities at a scale equal to twice the grid scale. The correlation coefficient of this model with the real stress is shown to be substantially higher than that of the eddy-viscosity closures. It is shown that mixed models display similar levels of correlation. During the a priori test, care is taken to only employ resolved data in a fashion that is consistent with the information that would be available during largeeddy simulation. The influence of the filter shape on the correlation is documented in detail, and the model is compared to the original similarity model of Bardina et al. (1980). A relationship between L,, and a nonlinear subgrid-scale model is established. In order to control the amount of kinetic energy backscatter, which could potentially lead to numerical instability, an ad hoc weighting function that depends on the alignment between Lii and the strain-rate tensor, is introduced. A 'dynamic' version of the model is shown, based on the data, to allow a self-consistent determination of the coefficient. In addition, all tensor elements of the model are shown to display the correct scaling with normal distance near a solid boundary.

Three regularization models of the Navier–Stokes equations (original) (raw)

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