Validation of the discrete element roughness method for predicting heat transfer on rough surfaces
Introduction
Surface roughness relates to many applications such as increased head losses through pipes [1], increased convection in pipes [2], [3], heat transfer on roughened turbine blades [4], [5], [6], and ice accretion [7], [8], [9]. The present efforts aim to improve computational fluid dynamics (CFD) models for heat transfer on large-scale roughness elements in the context of CFD codes that can scale to these various applications.
Strategies to include roughness effects in CFD calculations can be divided into three broad categories, listed here in order of increasing computational cost. One approach aims to modify the no-slip boundary condition at the solid walls [11]. These models work well for small roughness (shorter than the mesh height) for conditions where the equivalent sand-grain roughness is known. The second level of modeling aims to account for the roughness within the flow volume. There tend to be two approaches on this front that include modeling roughness via the turbulence model [12] versus modeling through momentum/energy sources. In terms of the eddy-viscosity models, the models tend to focus on altering the turbulence model to predict the developed roughness boundary profile [12], [13], [14]. These approaches are reasonably well suited for conditions where the flow is fully developed over homogeneous roughness. The alternative forms add momentum and energy source in the roughness regions to emulate roughness effects, typically by a variant of the Discrete Element Roughness Method [15] (DERM). Lastly, the roughness shape can be directly resolved, either in two or three dimensions [16], [17], [18]. In addition to an increase in required computational resources, each of these levels of increasing sophistication demands more knowledge about the details of the roughness geometry. Only the equivalent sand-grain roughness (SGR) height is required in order to use a wall correction. Adding source terms may require, depending on the implementation, knowledge of the roughness statistics and how they vary in space. Directly resolving the roughness geometry requires, in two dimensions, some form of surface tracing and, in three dimensions, a surface scan (via computerized tomography (CT), for example). The present work focuses on DERM as it is both computationally efficient and has the potential to effectively model convective heat transfer on surfaces with large-scale roughness.
The basis of DERM is a decomposition of the total wall skin friction drag and convective heat transfer into components due to the presence of the roughness elements and components due to flow over the remaining smooth surface between the roughness elements if it exists. The components due to the elements are then estimated from empirical correlations as functions of the roughness geometry and the local conditions of the flow at the element location. More sophisticated models also account for the flow accelerations due to the reduction in flow area, or blockage, caused by the presence of roughness. This approach is a natural way to incorporate as many geometric properties of the real roughened surface as are necessary to fully describe the discrete elements being used while still allowing the computational mesh to conform to the underlying smooth wall. DERM is also able to naturally capture the different behaviors of smooth, transitionally rough, and fully rough surfaces without explicitly distinguishing between the three.
DERM as an idea was proposed as early as 1936 by Schlichting [19] and has been applied by several investigators to problems of surface roughness on gas turbine blades [10] and geophysical flows [20], [21], for which DERM is a natural way to capture the effects of very large roughness such as tree canopies and buildings. Pioneering work in DERM includes the research of Lewis [22], Taylor [15], [23], McClain [24], and Aupoix [25], [26], among others. Lewis [22] derived the skin friction for flow over ordered transverse ribs by modeling the flow near the ribs as a lid-driven cavity flow and assuming that, far from the wall, only the statistical properties of the roughness are important. Early investigators, such as Adams and Hodge [27], accounted for roughness effects by combining correlations for the form drag on roughness elements with a modification of the turbulence model. Lin and Bywater [28] and Finson [29], [30] included the element blockage as well as the form drag. Christoph and Pletcher [31], [32] further modified a mixing-length turbulence model to account for roughness effects. Taylor [15] derived the turbulent boundary-layer equations to include blockage effects and form drag acting on roughness elements. These investigations also involved tuning a robust functional form for the drag on the individual roughness elements from experimental data that remains in use. Aupoix [25] rigorously derived the DERM boundary-layer equations from consideration of the spatial-averaging theorem [33] rather than the control-volume approach used by Taylor. McClain in the intervening years has made considerable contributions to the DERM field including developing an understanding of how properly to incorporate knowledge of real gas-turbine-blade roughness geometry into the DERM model [24], [34] and how to deal with the temperature gradients that develop along the height of roughness elements [35]. More recently, Aupoix [26] revisited the volume-averaging approach to DERM and identified two new terms in the volume-averaged RANS equations that arise from the interaction between time-averaging and volume-averaging. To date, the primary disadvantage of DERM methods has been their limitation to boundary-layer solvers. To the author’s knowledge, the only implementation of DERM into a general CFD solver for use in aeronautics is that of K. Fleming in 1992 [36].
The present research is focused on improving the heat transfer modeling in CFD as it occurs on accreted ice, which is relevant to aircraft icing. Surface roughness caused by ice accretion is known to cause a significant increase in skin-friction drag [37], [38] due to increased boundary-layer mixing, increased surface area, and the form drag of the individual roughness elements. With respect to glaze icing, the presence of ice roughness impedes the flow of unfrozen surface water [39], potentially modifying the resulting ice shape. Because iced surface roughness tends to occur near the airfoil leading edge, the associated aerodynamics shows several complications beyond those seen in flat-plate studies. The statistical properties of the roughness change in the chord-wise, and possibly also in the span-wise, direction. In addition, the roughness is typically in a region of favorable pressure gradient and accelerating flow [40]. Further, since the boundary layer is very thin near the leading edge, the dimensionless roughness height tends to be correspondingly large. This complicates the application of wall-correction terms to account for the roughness. Although not directly evaluated in this effort, these are the underlying issues that the present work is aiming to address.
The present work aims to extend the application of DERM from boundary-layer codes, to implementation in a general-purpose CFD code relevant to aircraft ice accretion. The work is outlined as follows. The DERM theory is developed and its implementation in the context of a general CFD code is described. Specifically, the full formulation is presented and discussed in the context of its implementation into a CFD code. The resulting implemented model is then validated with respect to a series of canonical validation cases. Using these results, we discuss sensitivities to the model as well as where DERM can be improved in the future.
Section snippets
DERM background
The fundamental principle of DERM is the representation of the roughness in a flow using a sub-grid-scale model, i.e., accounting for roughness without explicitly resolving the roughness features. A sketch depicting the approach is provided in Fig. 1. The real rough surface (left) is approximated using volumetric source terms in the same region, shown in the box at the right. An intermediate step, shown in the middle panel, involves approximating the potentially very complicated real roughness
Flow solver
In this work, the flow solver is based on the commercial CFD software, Star-CCM+ [41]. Star-CCM+ is a comprehensive finite-volume fluid-dynamics simulation package. The spatial discretization schemes used in the present work are formally second-order. While Star-CCM+ is capable of using numerous discretization schemes, flow solvers, and turbulence models, only those used in the present work are mentioned in this section. The present work focused on solving the compressible flow equations in the
Results
Upon implementation of the aforementioned model, validation is required for its application. In the scope of this effort, the method is validated using a series of canonical cases for which there are ample validation experimental data and/or correlations. The goal of these efforts is to establish that the DERM method and implementation are well suited to predicting the impact of roughness on the boundary layer and the resulting heat transfer. In Section 4.1, the CHT-based thermal model of the
DERM accuracy
The validation presented for a variety of cases in Section 4 gives strong evidence that the DERM method can capture the momentum and energy losses due to roughness for a variety of roughness regimes and shapes. The predictions for SGR cases typically lie within of the correlations in skin friction and in heat transfer coefficient. Results in skin friction and heat transfer for large, ordered roughness elements were typically within . Hence, the fundamental DERM model is considered to
Conclusions
In this work, the DERM equations are implemented into a general-purpose CFD flow solver to enable the ability to handle large-scale, complicated roughness. The overall process of implementing DERM is possible but leaves challenges with obtaining the geometric inputs relevant to a DERM model. For this reason, this work also developed the DERM-model inputs relevant to SGR and expanded the understanding of the DERM-model inputs. The DERM implementation was validated using a variety of experimental
Conflict of interest
This work has no conflict of interest.
Acknowledgments
The authors thank Richard E. Kreeger of the NASA Glenn Research Center for his technical guidance and support. This research is partially funded by the Government under Agreement No. W911W6-11-2-0011. The U.S. Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation thereon. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of
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