## 2

In equation (11), y = 0 is the "no-slip" location. The two terms in the numerator of equation (11) represent the contributions of skin friction on the smooth part of the wall and roughness-element drag. The method used for this study treats any point below the mean hydraulic elevation as flat surface at the mean hydraulic elevation and adds the skin friction associated with the flat surface of the mean hydraulic elevation. The blockage fraction, a, at the mean hydraulic elevation is used in equation (11) to calculate the drag associated with the skin friction.

Since there is little difference between the flat surface of a sparsely-spaced cone or spherical segment rough surface and its mean hydraulic elevation, Taylor essentially used this option to validate the cone and spherical segment model used in BLACOMP. The model worked very well but was mainly validated for sparsely-spaced roughness elements. Most of the rough surfaces validated by Taylor (1983) had spacings of more than twice the base diameters of the blockages.

The drawback to the above option concerns the "no-slip" condition at the mean hydraulic elevation. The "no slip" condition implies that the fluid velocity at the surface or reference elevation must be equal to the surface velocity and the temperature of the fluid at the surface must equal the temperature of the surface. While solution of the boundary-layer equations on a surface requires applying the "no slip" condition at a reference surface, there is flow beneath the mean hydraulic elevation for randomly-rough surfaces. While there is some type of flow beneath the mean elevation, the fact that the discrete-element model is a spacially-averaged model means that these secondary flows cannot be resolved with the discrete-element model.

As shown in Figure 2 and discussed above, the open regions below the mean hydraulic elevation are small and isolated. The flow in these open regions is likely local cavity flow or likely resembles "skimming" flow as described by Morris (1972). If the flow below the mean hydraulic elevation is cavity flow or "skimming" flow, then specifying the correct boundary condition at the mean hydraulic elevation other than the "no slip" condition would be difficult. The skin friction contribution from the flow in the randomly-sized-and-spaced valleys would be difficult to analyze, but is different from the skin friction and heat transfer from a flat surface. Until more information is obtained on the flow in the valleys, the effects of the flow below mean hydraulic elevation are best treated as skin friction on a "flat surface" placed at the mean hydraulic elevation.

Like the skin friction calculations, applying the "no-slip" condition at the mean hydraulic elevation was also used in the heat transfer predictions. The velocity and temperature profiles were found by applying the "no-slip" condition at the mean hydraulic elevation. The Stanton number, as presented in equation (12), was then evaluated by adding the heat convected from the roughness elements and the heat convected from the flat surface area on the mean hydraulic elevation.

f dy

LtLp 0

Surface Descriptions

Two randomly-rough surfaces, the Deposit surface and the Erosion 2 surface, were used in the modeling study. The Deposit surface contains relatively large, anisotropic roughness elements aligned in the direction of the flow. The Deposit surface has a centerline average roughness, Ra, of 1.183 mm. A bitmap image of a section of the Deposit surface is shown in Figure 4. The Erosion 2 surface contains smaller, isotropic roughness elements. The Erosion 2 surface has a centerline average roughness of 0.504 mm. A bitmap image of a section of the Erosion 2 surface is shown in Figure 5.

Figure 4. Bitmap Image of the Deposit Surface (Flow Direction is Top to Bottom)

Figure 5. Bitmap Image of the Erosion 2 Surface (Flow Direction is Left to Right) Along with the randomly-rough surfaces, two layered-analog surfaces were studied. The surfaces were derived from the Deposit surface and the Erosion 2 surface. The spacings and heights of the roughness elements are random as measured from the original randomly-rough surface, but the roughness elements were generated by placing ellipsoidal blockages of equivalent area and eccentricity at the height of the original random blockage element. Because the blockage elements were created at distinct heights between the mean hydraulic elevation and the maximum elevation of the surface, the layered-analog surfaces have a "layered" appearance. The layered representation of the Deposit surface, the Deposit Layered surface, is shown in Figure 6. The layered representation of the Erosion 2 surface, the Erosion 2 Layered surface is shown in Figure 7.

Figure 5. Bitmap Image of the Erosion 2 Surface (Flow Direction is Left to Right) Along with the randomly-rough surfaces, two layered-analog surfaces were studied. The surfaces were derived from the Deposit surface and the Erosion 2 surface. The spacings and heights of the roughness elements are random as measured from the original randomly-rough surface, but the roughness elements were generated by placing ellipsoidal blockages of equivalent area and eccentricity at the height of the original random blockage element. Because the blockage elements were created at distinct heights between the mean hydraulic elevation and the maximum elevation of the surface, the layered-analog surfaces have a "layered" appearance. The layered representation of the Deposit surface, the Deposit Layered surface, is shown in Figure 6. The layered representation of the Erosion 2 surface, the Erosion 2 Layered surface is shown in Figure 7.

Figure 7. Bitmap Image of the Erosion 2 Layered Surface

Two cone surfaces, the Deposit Cone surface and the Erosion Cone surface, were also studied. These surfaces were used to validate the model developed by Taylor for non-sparse cone surfaces. The Deposit Cone surface consisted of 4.420-mm tall cones with base diameters of 5.969 mm. The cones were evenly spaced 12.065-mm apart. The spacing-to-base-diameter ratio was 2.730. The Erosion Cone surface consisted of 1.905mm tall cones with base diameters of 4.445 mm. The cones were evenly spaced 5.207mm apart. The spacing-to-base-diameter ratio was 1.171. The cone surfaces were also created based on the randomly-rough surfaces. The cone surfaces were created with

equivalent centerline averaged roughness, Ra = |, values as their respective randomly-rough surfaces. Cone Surface Results

The method presented concerning the reference surface and flow characteristics below the datum was made concerning a randomly-rough surface, but if the method presented can be used for randomly-rough surfaces, the method should also hold for surfaces with cone roughness-elements. The skin friction coefficients of the two cone surfaces were evaluated using the entire height of the roughness element from the solid surface following the model developed by Taylor (1983) and using the mean hydraulic height of the roughness elements and adding wall friction on the unblocked area at the mean hydraulic elevation. Figure 8 demonstrates the variables involved in the determining the characteristics of cone surfaces above the mean hydraulic elevation. The height of the cone above the mean hydraulic elevation and the diameter of the cone at the mean hydraulic elevation are calculated using equations (13) and (14), respectively.

The heights and diameters of the Deposit Cone surface cones above the mean hydraulic elevation are 4.137 mm and 5.587 mm, respectively. The heights and diameters of the Erosion Cone surface cones above the mean hydraulic elevation are 1.541 mm and 3.596 mm, respectively. The discrete-element model was run for the cone surfaces using each of the following options:

1) the full cone placed on the surface floor with shear acting on the surface floor,

2) the partial cone referenced to the mean hydraulic elevation with shear acting on the area not blocked at the mean hydraulic elevation,

The results are presented in Table 1.

Figure 8. Mean Hydraulic Elevation of the Cone Surface Table 1. Cone Surface Results
 Surface Erosion Cone % Difference Deposit Cone % Difference Mean Reynolds Number
0 0