On the development of a subgrid scale clutter model

Abstract

The objective of this research is to develop a subgrid scale model to account for the effects of small objects, or that are difficult to resolve using a CFD mesh. These objects may range in size from being small relative to the grid (i.e. porous media limit) to scales, which are on the same order as the grid size (i.e. a bluff body limit). Transport equations for momentum, turbulent kinetic energy and turbulent kinetic energy dissipation rate are derived based on time and spatial filtering concepts, introducing unknown correlations and surface integral terms that require explicit subgrid scale (SGS) model closures. The SGS models are formulated using a combination of well-established constitutive relations borrowed from the porous media literature and classical relations for drag on bluff bodies. The modeling methodology is exercised for two classes of problems. The first is flow in a porous media for which the obstructions are relatively dense. For this problem, predictions using the SGS clutter model are compared to established correlations taken from the porous media literature. The second class of problems is a cylinder in cross-flow for which both detailed experimental measurements and detailed CFD calculations of velocity deficit and kinetic energy profiles are available. Preliminary results indicate that this modeling approach offers a promising new approach to capture the macroscopic effects of small-scale objects without using excessive mesh resolution. * Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94-AL85000. This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Introduction Capturing object features using a CFD grid that are smaller than one one-hundredth the size of the computational domain requires an excessively large number of grid points and extremely small time steps for numerical integration. Relevant applications include large-scale problems such as forest fires down to smaller scale phenomena such as heat transfer in a porous media. An alternative approach to resolving these features is to use a subgrid scale (SGS) model to represent the macroscopic effects of these small features using reasonably sized CFD grid cells. The focus of this research is to develop engineering SGS models of flow transport in cluttered environments. This work is being carried out in a joint, iterative computational/experimental approach for which detailed experimental and CFD predictions are used to develop macroscopic models of clutter. Subgrid clutter has two main physical effects on the macroscopic flow field. The first is to provide a momentum sink due to viscous and pressure drag forces of gas flowing through clutter. The second is to either increase or decrease the turbulent kinetic energy levels depending on the local clutter size, lc, relative to the characteristic length scale of turbulence, 1T. In a qualitative sense, if lc 1T the turbulent kinetic energy will increase. Previous investigations of turbulent flow in porous media can be found in previous studies of Pedras et a/.'', Nakayama et al.' and Antohe et at.. In these (c)2002 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. studies, time and phase-averaged transport equations of momentum, turbulent kinetic energy and turbulent dissipation rate are derived and constitutive models suggested for accounting for unknown SGS correlations and surface integral terms. These constitutive models are applicable to porous flow for which it is assumed that the characteristic size of the rigid porous matrix is much smaller than the size of the system . This effort extends these approaches to clutter environments for which the length scale of the clutter is not necessarily small relative to the size of the system. Discussion begins with first the mathematical formulation of the two-phase system based on the use of spatial filtering concepts. The filtering of the governing equations results in unknown correlations that require modeling. Constitutive models are formulated based on a linear combination of porous media and bluff body lift and drag relations. Results are then presented in support of the modeling assumptions and predictions of turbulent flow in porous media with comparisons to data from the literature. Experimental measurements of turbulent flow over a single cylinder are presented with comparisons to detailed CFD predictions. The information from the combined experimental and CFD predictions provide further calibration of the clutter model constants as part of future efforts in the clutter model development. Lastly, conclusions are drawn and future efforts are summarized. Mathematical Formulation of Two-Phase System The following mathematical description is limited to a summary of the development of stationary clutter. The section starts with an introduction to the two-phase (Le. spatial) averaging and time-averaging. The reasons for reviewing these mathematical formalities are to familiarize the reader with the concepts of spatial filtering and, more importantly, highlight the restrictions imposed by this approach with respect to the implementation of the clutter model into a CFD code. In the next section, these averaging concepts are applied to conservation equations for momentum, turbulent kinetic energy and dissipation transport. This averaging results in unknown second order correlations and surface integral terms that represent subgrid physics that must be closed with a clutter model. Requirements for the clutter model are provided based on both physical and computational requirements. The end result of this section is to provide a closed set of phase and time averaged transport equations for momentum, turbulent kinetic energy, and turbulent kinetic energy dissipation rate. Two-Phase Averaging The formal averaging for two-phase media was first introduced by Anderson and Jackson through the use of a local filtering function and later refined by Gray et al? and Gough et a/. as presented in text book form by Kuo . Alternatively, Slattery' offers a different derivation based on a spatially dependent volume of integration. The presentation here follows the procedure of Kuo. Figure 1 illustrates a typical phase averaging volume showing the total volume of interest, VT, the volume of the solid clutter, Vc, and the volume of the gas, V.. Figure 1: Illustration of phase averaging volume. Phase averaged properties are obtained by first defining a spatial filtering function, G[(jCf. — jc')/A^ J, with the normalization property: J G[(*,. x')/ A, }dV = I For volume averaging using a cubic volume of length Lj =\yT) on a side then A^ = Lj andG is « i defined as: n-. where His the Heaviside function. Convoluting G with the a gas property of interest, fl , yields a gas phase average quantity, ( * -S (1) which physically represents a spatially averaged property over the volume, VT . It should be emphasized that the volume of integration, V , represents all of the regions occupied by the gas and so does not depend on the location, xt, where the averaging takes place. Of (c)2002 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. more value is the intrinsic average, f , defined as the local average of ft over the gas phase volume, V , for 6 which constitutive and thermodynamic properties are well defined. The intrinsic average is defined as: ( f l ) = J31 fy and is the variable of interest to solve for after phase averaging the transport equations for mass, momentum and energy. The variable, ^, is the void fraction and is defined as the volume of gas divided by the averaging volume, Le. (j> = V IVT = V IA, . ft 5 J Temporal and Spatial Derivatives for Phase Averaging Applying the spatial filtering function to the transport equations requires expressing the phase averaged time and spatial partial derivatives in terms of temporal and spatial derivatives of phase averaged quantities. In the following development, the volume of the gas phase is assumed to change as a function of both time and space to accommodate for the complexities that may be included in future efforts (e.g. decomposing or burning clutter). Simplifications are then imposed to limit the scope to the focus of this work on turbulence modeling on rigid, stationary clutter.