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Plate Flow

The flow condition around a plate is composed of three basic parts: turbulent flow, laminar flow, and separated flow.

Fig. 1 : Boundary Layers

The aerodynamicist, Ludwig Prandtl, developed the concept of the boundary layer for fluid flow (Figure 1).  Flow in the region surrounding a body can be divided into the boundary layer and anywhere outside of the boundary layer.  Outside of the boundary layer, the viscosity on the body is negligible, and the flow may be treated as inviscid.  Therefore, only flow inside the boundary layer is important for determining frictional resistance.[2]

The laminar boundary layer can be assumed to have a parabolic velocity profile "U" as shown in Figure 1.  At the body, the velocity of the fluid is zero.  Reynolds number, pressure gradient, and surface roughness determine the length of the laminar boundary layer and disturbed flow.  For a flat, smooth plate in undisturbed flow, the flow typically moves from laminar flow to the transition zone around Rn = 500,000.[1] A description of laminar flow is that it is unmixable, meaning that if the flow were to oscillate between positive and negative flow directions, a fluid particle would always remain on the same streamline. The coefficient of skin friction on a flat plate in laminar flow is shown in Eqn(1).[2]

(1)

Fluctuating velocities and eddying motion characterize the turbulent boundary layer. Compared to the laminar boundary layer, the turbulent boundary layer is thicker, and the velocity profile of the flow increases much faster away from the surface. However, before the flow enters into the turbulent region, it passes through the transition zone (Figure 1). The transition zone contains sudden spots of turbulence generation and a general breakdown of laminar flow. Introduction into the transition zone is primarily a function of Reynolds number, but surface roughness and disturbed flow can also trip the condition from laminar to turbulent.[1] The coefficient of skin friction on a flat plate in turbulent flow is shown in Eqn(2).[2]

(2)

Ship Flow

A special region inside the turbulent boundary layer is called the viscous sublayer (Figure 2). The viscous sublayer is mainly laminar flow, but for a typical sailboat, is only 0.1 mm thick. The importance of the layer's size is that it is affected by surface roughness, and the layer is liable to trip back to the turbulent boundary layer in enough disturbance.[4] When laminar, the viscous sublayer significantly decreases the skin friction on the vessel.

Fig. 2 : Exaggerated flow conditions on a typical sailing vessel

The separated zone is typically found near the stern of a vessel and is a function of the pressure gradient. For a body that is not a flat plate, the pressure gradient will remain positive as the form of the body increases (from the bow to after amidships on a typical sailboat). However, as the form of the body bends inward, the velocity profile will still try to cling to the surface. The resultant is that the flow will invert under this negative pressure gradient. The inverted flow will cause large eddies much larger than the eddies found in normal turbulent flow. Separated flow drag is significantly higher than turbulent flow drag.[4]

For tank testing, the flow condition is a function of Reynolds number, the roughness of the surface of the hull, and the turbulence in the surrounding water. For the full-size ship, the flow condition along most of the hull is turbulent as shown in Figure 2. The flow condition around most of the model would be laminar since the model is small, the tank water is generally stable, and the model surface is very clean. Although a model can be scaled geometrically from a ship, the water and therefore the flow condition cannot. If resistance data with a model in mostly laminar flow were scaled to full-scale, the predicted resistance would be significantly lower than actual.

Flow Scaling

Keeping Reynolds and Froude numbers constant defines dynamic similarity in terms of viscous and wavemaking resistance respectively.[3] Therefore, if both Reynolds and Froude numbers of the model matched those of the ship, then the model's behavior would be scaled identically to the ship's behavior. However, because the acceleration due to gravity and the kinematic viscosity of water cannot be scaled at the same time, perfect modeling conditions cannot exist. In some cases, Reynolds number similarity is used in a wind tunnel for testing submerged appendages. For most hydrodynamic tank testing, however, Reynolds number similarity is neglected because it would require the model to travel at speeds much greater than a tow tank could provide. Froude number similarity is used because the model's speed is slower than the ship's speed.  Furthermore, viscous resistance is fairly well predicted using skin based methods (such as the given ITTC equation) while the wavemaking resistance is generally more sensitive to the details of the hull form and thus must be determined for each new design. Using the Froude hypothesis, the forces on the model can be predicted for the full-scale ship.[5]

The Froude hypothesis states that the wavemaking coefficient is the only variable which can be directly equated between the ship and model. In accordance, for each test performed in the tank, the Froude number of the extrapolated ship's speed is equal to the Froude number of the model. Extrapolation is begun by decomposing the tank test data into the coefficient of wavemaking. Thereafter, the Reynolds number as well as the coefficient of viscous resistance is recalculated for the ship. The new viscous resistance term, the wavemaking term, and a correlation allowance are summed to produce a new coefficient of total resistance for the ship. The correlation allowance mainly takes into effect the discrepancy between the smooth-painted hull of the model and the rough, biologically coated hull of the ship. Typically, the correlation allowance is 0.0004, although this varies in accordance with the assumed roughness of the ship's hull.[5]

Once the coefficient of total resistance for the ship is calculated, the total resistance force is calculated, and the ship prediction is completed. According to dimensional analysis, geometric force scaling can be performed by multiplying the model force by the cubed scaling factor. If this is done for resistance tests, the ship prediction will be over-estimated. The reason for this is that although there is a greater amount of viscous resistance for the ship due to its larger wetted surface area, the coefficient of viscous resistance is smaller. Turbulent flow theory predicts smaller coefficients of viscous resistance for larger Reynolds numbers. Since the ship has a much larger length and travels at larger velocities, the Reynolds number for the ship is larger than for the model. The ITTC resistance equation which is used in this extrapolation reflects the differences in the viscous coefficient.[5]

References