Session FO: Turbulence Modeling I*

Lagrangian dynamics and statistical geometric structure of turbulence

The local statistical and geometric structure of three-dimensional turbulent flow can be described by properties of the velocity gradient tensor. A stochastic model is developed for the Lagrangian time evolution of this tensor, in which the exact nonlinear self-stretching term accounts for the development of well-known non-Gaussian statistics and geometric alignment trends. The non-local pressure and viscous effects are accounted for by a closure that models the material deformation history of fluid elements. The system is forced with a simple, white in time, Gaussian noise. The resulting stochastic system reproduces many statistical and geometric trends observed in numerical and experimental 3D turbulent flows. Examples include the non-Gaussian statistics of velocity gradient components, the preferential alignment of vorticity, nearly log-normal statistics of the dissipation, the tear-drop shape of the so-called R-Q joint probability density and anomalous scaling of velocity derivatives.   

Search for local low dimension in transitional flow behind a bed form

We study experimentally the dynamics of vortices formed behind a bed form in an open channel flow. The Reynolds number is 453 which corresponds to transition flow. On the lee side of the bed form, vortical structures are generated, deformed and shed with no apparent identifiable regularity or predictable shape. This phenomenon is interpreted as a spatiotemporal chaotic system. We determine regions of low dimensionality in this flow by adapting the method Local Bred Vectors originally proposed by Patil et al. (Phys Rev Lett 2001) to analyze the atmosphere. In contrast to the original proposal, with our method no theoretical model is required. Then, we correlate the local dimension field with the predictability of specific regions of the flow. To this end, we use a forecast strategy similar to that described in Kantz and Ragwitz (Int. J. Bif. Chaos 2005). The methodology of the analysis described is general and not restricted to the flow behind a bed form.   

Dynamics of Episodic POD

In this talk we will discuss the derivation of evolution equations, which take the form of discrete maps, for the modes of episodic POD. The method of episodic POD is an extension of standard POD that leads to the construction of spatio-temporal POD eigenfunctions called ``episodic modes.'' In contrast to standard POD, this method produces dynamical structures that evolve in space and time. Classical methods of low-dimensional modeling rely on decomposing flow-fields into a set of spatial functions and temporal coefficients. By substituting the expansion into the governing equations, one obtains a finite-dimensional system of ODEs. However, for the case of the Navier-Stokes equations, this method requires finesse, not only because of the nonlinearity of the equations, but also because of the intricacies of the pressure term. Conventional schemes suggest eliminating the pressure term using a vorticity formulation, or a model for the pressure term. In the method of episodic POD, the need for pressure modeling can be eliminated by using the pressure term as a constraint on the expansion coefficients of two consecutive episodes, ensuring spatio-temporal continuity. This method has the advantage that the formulation leads to a non-linear system of algebraic equations. The formulation also enables us to progress over contiguous chunks of time rather than computing the velocity state at every instance in time. Fluid dynamical examples are presented that validate this method.   

Evolution of Recurrent Coherent Structures in a Minimal Channel

Many classes of turbulent flows, including wall-bounded turbulence, are known to exhibit organized behavior in the form of coherent structures underneath their chaos and nonlinearity. One of the current methods to detect organized flow phenomena is the method of Proper Orthogonal Decomposition (POD). POD enables us to decompose flow fields into a set of basis functions and coefficients such that they provide a compact and finite-dimensional representation of an infinite- dimensional system. In this talk, we look at a relatively new method for identifying the evolution of coherent structures that exhibit temporal periodicity. The method of episodic POD is an extension of standard POD, wherein flow realizations are grouped together based on a specified time scale to create a modified ensemble such that each member of the ensemble consists of a group of realizations. Application of POD analysis to this new ensemble leads to the construction of spatio-temporal POD eigenfunctions called ``episodic modes.'' By appropriately selecting the time scale, the first episodic mode represents the most dominant recurrent behavior in the flow. It is shown that the most dominant spatio-temporal coherent structure arises from the spanwise variation of the streamwise component of the velocity. Qualitative analysis on the turbulent statistics seems to suggest the presence of some form of self-sustaining process.   

Turbulent Flow past a Blunt Nosed Body with Spinning Base

Turbulent flow over blunt-nosed cylinders that are spinning about their axis is analyzed with applications to the development of projectiles. In this study, computations are performed using an anisotropic two-equation Reynolds-stress model to study the flow past spinning projectiles of circular cross-section at zero angle of attack. The model utilizes a phenomenological treatment of the energy spectrum to include the effects of rotation and compressibility. The resulting set of modeled form of transport equations for the turbulence kinetic energy and the scalar form of turbulence dissipation are solved along with the time-averaged equations of motion using an efficient finite-volume algorithm. The model is applied for several test cases to validate its predictive capabilities for capturing the effects of curvature, swirl and compressibility. Computations for the flow past axially rotating cylinders are performed and the results are shown to be in agreement with the experimental results of Carlucci {\&} Thangam (2001). Both the cases of axial flow past single rigid cylinder as well as that of flow past cylinders with a free-spinning and finned base are analyzed. The model performance and its potential for applications involving the design of projectiles are discussed.   

The LagRST Model Applied to a q-Vortex

A turbulence model is needed which is less complicated than existing Reynolds stress models but which captures non-equilibrium effects in flows exhibiting streamline curvature or rotation. In this investigation, we study a non-equilibrium turbulence modeling idea proposed independently by Knight and Saffman and later by Olsen and Coakley. The equilibrium Reynolds stress tensor is computed with a standard two-equation model and the Boussinesq approximation. The actual Reynolds stresses are then solved from a lag equation. In order to study the lag Reynolds stress transport model, an idealized quasi-steady q-vortex problem is considered in which the mean velocity, turbulent kinetic energy and dissipation rate are taken from a DNS solution. The lag equation is solved analytically to obtain the actual Reynolds stresses. The Reynolds stresses predicted with the lag equation exhibit magnitudes comparable to and contours in better alignment with those of the DNS data. In contrast, the Reynolds stresses obtained with the Boussinesq approximation are too great and have contours aligned with the contours of the corresponding components of strain rate tensor, which is inaccurate. The lag model more realistically solves for the non-equilibrium q-vortex Reynolds stresses.   

A velocity estimation model constrained by subgrid-scale dissipation

We propose a subgrid model that estimates the subgrid velocity, while constraining the overall subgrid--scale dissipation. The motivation for the model is (i) it is not always practical to estimate filtered variables from experiment for comparison, (ii) eddy viscosity models do not account for backscatter, while models that account for backscatter can suffer from inadequate dissipation, and (iii) to explore the possibility of using LES to also predict instantaneous velocities in reasonable agreement with experiment. The modeling procedure consists of two distinct steps: in the first (predictor) step, a purely dissipative SGS model is computed. As the second (corrector) step, a similarity-type model is considered with an adjustable constant. The constant is determined by minimizing the difference of SGS dissipation between the predictor and corrector steps in the least square sense. The dynamic Smagorinsky model is used in the predictor step, and a new SGS velocity estimation model is applied to the corrector step in which the 'projected' SGS velocity is estimated on the resolved scale without extending the grid resolution. When applied to isotropic turbulence, the proposed model predicts good statistics while exhibiting realistic backscatter. Simulations of turbulent channel flow are in progress, and will be discussed.   

A Conditionally Cubic-Gaussian Stochastic Lagrangian Model for Acceleration in Isotropic Turbulence

The modeling of fluid-particle acceleration in isotropic turbulence via stochastic models for the Lagrangian velocity, acceleration and a dissipation rate variable is considered. We use data from direct numerical simulations (DNS) up to Taylor-scale Reynolds number 650 to construct a stochastic model that is exactly consistent with Gaussian velocity and conditionally cubic-Gaussian acceleration statistics. This model captures both the effects of intermittency of dissipation on acceleration and the conditional dependence of acceleration on pseudo-dissipation (which differs from the Kolmogorov 1962 prediction). The large-time behavior of the new model is that of a velocity-dissipation model that can be matched with DNS data for conditional second-order Lagrangian velocity structure functions. As a result, the diffusion coefficient for the new model incorporates two-time information and their Reynolds-number dependence from DNS. Model predictions for conditional and unconditional velocity statistics are shown to agree well with DNS.   

Structure Functions in Isotropic Turbulence: DNS and Stochastic Models

Lagrangian velocity structure functions (up to tenth order) are extracted from direct numerical simulations (DNS) of isotropic turbulence and are compared to predictions from several stochastic models. The DNS are of statistically-stationary homogeneous isotropic turbulence for a range of Taylor-scale Reynolds numbers up to 650. As expected, the Lagrangian velocity structure functions extracted from the DNS reveal strong intermittency with the most pronounced non-Gaussian behavior in the dissipation range. The Langevin model for velocity and Sawford's 1991 model for acceleration are both Gaussian, and hence do not represent such behavior. The recently-developed conditionally cubic Gaussian (CCG) model for acceleration accounts (to some extent) for intermittency and non-Gaussianity. In this model, the pseudo-dissipation is taken to be log-normal, and the acceleration conditional on pseudo-dissipation is taken to be cubic Gaussian. Because this model accurately describes the one-time velocity and acceleration statistics, its predictions of the structure functions are accurate at small and large times. At intermediate times they exhibit strongly non-Gaussian behavior and anomalous scaling, in moderate agreement with the DNS data.