## Modeling Progress Report

Stephen T. McClain and B. K. Hodge Department of Mechanical Engineering Mississippi State University

### Introduction to the Discrete-Element Model

The discrete-element model is formulated for roughness elements with three-dimensional shapes for which the element cross-section can be defined at every height, y. This form of the discrete-element roughness model has its origin in the work of Finson (1976). The differential equations including roughness effects are derived by applying the basic conservation statements for mass, momentum, and energy transfer to a control volume such as that shown in Figure 1. Basic to this approach is the idea that the two-dimensional, time-averaged turbulent boundary-layer equations can be applied in the flow region below the crests of the roughness elements. The flow variables have been spatially averaged over the transverse (z) direction and the streamwise (x) direction. The physical effects of the roughness elements on the fluid in the control volume are modeled by considering the flow blockage, the local element heat transfer, and the local element form drag. The blockage factors, a, are defined as the fraction of the area open to flow. The form drag force on the control volume is due to the portion of a roughness element penetrating the control volume and is expressed using a local drag coefficient as

Likewise, the rate of heat transfer between the portion of the element penetrating the control volume and the fluid is expressed using a local Nusselt number as

Using the above ideas, the continuity, momentum, and energy equations for a steady, Reynolds averaged, two-dimensional turbulent boundary layer with uniform roughness become and

ox oy

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