Flows Over Smooth and Rough Flat Plates
We consider the boundary layer development of incompressible uniform 2D water flow over a flat plate of length L (see Figure 1). The boundary layer develops from the plate leading edge lying at the upstream computational domain boundary. The boundary layer at the leading edge is considered laminar. Then, at some distance from the plate leading edge the boundary layer automatically becomes turbulent (if this distance does not exceed L).
The geometry model is shown in Figure 2. The problem is solved as internal in order to avoid a conflict situation in the corner mesh cell where the external flow boundary and the model wall intersect. In the internal flow problem statement, to avoid any influence of the upper model boundary or wall on the flow near the flat plate, the ideal wall boundary condition has been specified on the upper wall. The plate length is equal to 10 m, the channel height is equal to 2 m, the walls’ thickness is equal to 0.5 m.
To solve the problem, an incoming uniform water flow of a certain velocity (see below), temperature of 293.2 K, turbulence intensity of 1%, and turbulence length of 0.01 m is specified at the channel inlet, whereas the water static pressure of 1 atm is specified at the channel outlet.
The flow computation is aimed at predicting the flat plate drag coefficient, defined as (see Refs. 21 and 25):
where F is the plate drag force, A is the plate surface area, ρ is fluid density, and V is the fluid velocity.
According to Refs. 21 and 25, the plate drag coefficient value is governed by the Reynolds number, based on the distance L from the plate leading edge (ReL = ρ·V·L/μ, where ρ is the fluid density, V is the incoming uniform flow velocity, and μ is fluid dynamic viscosity), as well as by the relative wall roughness L/k, where k is the sand roughness. As a result, Refs. 21 and 25 give us the semi-empirical flat plate CD(ReL) curves obtained for different L/k from the generalized tubular friction factor curves and presented in Figure 3 (here, ε ≡ k). If the boundary layer is laminar at the plate leading edge, then the wall roughness does not affect CD until the transition from the laminar boundary layer to the turbulent one, i.e., the CD(ReL) curve is the same as for a hydraulically smooth flat plate. The transition region’s boundaries depend on various factors, the wall roughness among them. Here is shown the theoretical transition region for a hydraulically smooth flat plate. The transition region's boundary corresponding to fully turbulent flows (i.e., at the higher ReL) is marked in Figure 3 by a dashed line. At the higher ReL, the semi-empirical theoretical curves have flat parts along which ReL does not affect CD at a fixed wall roughness. These flat parts of the semi-empirical theoretical curves have been obtained by a theoretical scaling of the generalized tubular friction factor curves to the flat plate conditions under the assumption of a turbulent boundary layer beginning from the flat plate leading edge.
To validate the Flow Simulation flat plate CD predictions within a wide ReL range, we have varied the incoming uniform flow velocity at the model inlet to obtain the ReL values of 105, 3∙105, 106, 3∙106, 107, 3∙107, 108, 3∙108, 109. To validate the wall roughness influence on CD, the wall roughness k values of 0, 50, 200, 103, 5∙103, 104 mm have been considered. The Flow Simulation calculation results obtained at result resolution level 5 and compared with the semi-empirical curves (Refs. 21 and 25) are presented in Figure 3.
As you can see from Figure 3, CD(ReL) of rough plates is somewhat underpredicted by Flow Simulation in the turbulent region, at L/k = 1000 the CD(ReL) prediction error does not exceed about 12%.
