Optimal prediction for moment models: Crescendo diffusion and reordered equations

A direct numerical solution of the radiative transfer equation or any kinetic equation is typically expensive, since the radiative intensity depends on time, space and direction. An expansion in the direction variables yields an equivalent system of infinitely many moments. A fundamental problem is how to truncate the system. Various closures have been presented in the literature. We want to study moment closure generally within the framework of optimal prediction, a strategy to approximate the mean solution of a large system by a smaller system, for radiation moment systems. We apply this strategy to radiative transfer and show that several closures can be re-derived within this framework, e.g. $P_N$, diffusion, and diffusion correction closures. In addition, the formalism gives rise to new parabolic systems, the reordered $P_N$ equations, that are similar to the simplified $P_N$ equations. Furthermore, we propose a modification to existing closures. Although simple and with no extra cost, this newly derived crescendo diffusion yields better approximations in numerical tests.Comment: Revised version: 17 pages, 6 figures, presented at Workshop on Moment Methods in Kinetic Gas Theory, ETH Zurich, 2008 2 figures added, minor correction

The Characteristic Length Scale of the Intermediate Structure in Zero-Pressure-Gradient Boundary Layer Flow

In a turbulent boundary layer over a smooth flat plate with zero pressure gradient, the intermediate structure between the viscous sublayer and the free stream consists of two layers: one adjacent to the viscous sublayer and one adjacent to the free stream. When the level of turbulence in the free stream is low, the boundary between the two layers is sharp and both have a self-similar structure described by Reynolds-number-dependent scaling (power) laws. This structure introduces two length scales: one --- the wall region thickness --- determined by the sharp boundary between the two intermediate layers, the second determined by the condition that the velocity distribution in the first intermediate layer be the one common to all wall-bounded flows, and in particular coincide with the scaling law previously determined for pipe flows. Using recent experimental data we determine both these length scales and show that they are close. Our results disagree with the classical model of the "wake region".Comment: 11 pages, includes 2 tables and 3 figure

A Model of a Turbulent Boundary Layer With a Non-Zero Pressure Gradient

According to a model of the turbulent boundary layer proposed by the authors, in the absence of external turbulence the intermediate region between the viscous sublayer and the external flow consists of two sharply separated self-similar structures. The velocity distribution in these structures is described by two different scaling laws. The mean velocity u in the region adjacent to the viscous sublayer is described by the previously obtained Reynolds-number-dependent scaling law $\phi = u/u_*=A\eta^{\alpha}$, $A=\frac{1}{\sqrt{3}} \ln Re_{\Lambda}+ \frac 52$, $\alpha=\frac{3}{2\ln Re_{\Lambda}}$, $\eta = u_* y/\nu$. (Here $u_*$ is the dynamic or friction velocity, y is the distance from the wall, $\nu$ the kinematic viscosity of the fluid, and the Reynolds number $Re_{\Lambda}$ is well defined by the data) In the region adjacent to the external flow the scaling law is different: $\phi= B\eta^{\beta}$. The power $\beta$ for zero-pressure-gradient boundary layers was found by processing various experimental data and is close (with some scatter) to 0.2. We show here that for non-zero-pressure-gradient boundary layers, the power $\beta$ is larger than 0.2 in the case of adverse pressure gradient and less than 0.2 for favourable pressure gradient. Similarity analysis suggests that both the coefficient B and the power $\beta$ depend on $Re_{\Lambda}$ and on a new dimensionless parameter P proportional to the pressure gradient. Recent experimental data of Perry, Maru\v{s}i\'c and Jones (1)-(4) were analyzed and the results are in agreement with the model we propose.Comment: 10 pages, 9 figure

A Note on the Intermediate Region in Turbulent Boundary Layers

We demonstrate that the processing of the experimental data for the average velocity profiles obtained by J. M. \"Osterlund (www.mesh.kth.se/$\sim$jens/zpg/) presented in [1] was incorrect. Properly processed these data lead to the opposite conclusion: they confirm the Reynolds-number-dependent scaling law and disprove the conclusion that the flow in the intermediate (`overlap') region is Reynolds-number-independent.Comment: 8 pages, includes 1 table and 3 figures, broken web link in abstract remove

Renormalization group and perfect operators for stochastic differential equations

We develop renormalization group methods for solving partial and stochastic differential equations on coarse meshes. Renormalization group transformations are used to calculate the precise effect of small scale dynamics on the dynamics at the mesh size. The fixed point of these transformations yields a perfect operator: an exact representation of physical observables on the mesh scale with minimal lattice artifacts. We apply the formalism to simple nonlinear models of critical dynamics, and show how the method leads to an improvement in the computational performance of Monte Carlo methods.Comment: 35 pages, 16 figure

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

Processing the data from a large variety of zero-pressure-gradient boundary layer flows shows that the Reynolds-number-dependent scaling law, which the present authors obtained earlier for pipes, gives an accurate description of the velocity distribution in a self-similar intermediate region of distances from the wall adjacent to the viscous sublayer. The appropriate length scale that enters the definition of the boundary layer Reynolds number is found for all the flows under investigation. Another intermediate self-similar region between the free stream and the first intermediate region is found under conditions of weak free stream turbulence. The effects of turbulence in the free stream and of wall roughness are assessed, and conclusions are drawn.Comment: 77 pages, (includes 61 charts and graphs

Fluid--Gravity Correspondence under the presence of viscosity

The present work addresses the analogy between the speed of sound of a viscous, barotropic, and irrotational fluid and the equation of motion for a non--massive field in a curved manifold. It will be shown that the presence of viscosity implies the introduction, into the equation of motion of the gravitational analogue, of a source term which entails the flow of energy from the non--massive field to the curvature of the spacetime manifold. The stress-energy tensor is also computed and it is found not to be constant, which is consistent with such energy interchange

On the structure of the energy distribution function in the hopping regime

The impact of the dispersion of the transport coefficients on the structure of the energy distribution function for charge carriers far from equilibrium has been investigated in effective-medium approximation for model densities of states. The investigations show that two regimes can be observed in energy relaxation processes. Below a characteristic temperature the structure of the energy distribution function is determined by the dispersion of the transport coefficients. Thermal energy diffusion is irrelevant in this regime. Above the characteristic temperature the structure of the energy distribution function is determined by energy diffusion. The characteristic temperature depends on the degree of disorder and increases with increasing disorder. Explicit expressions for the energy distribution function in both regimes are derived for a constant and an exponential density of states.Comment: 16 page