Flow around Submerged Objects

The flow around a submerged object is generally more complicated than the flow in a straight pipe or channel, beeause it is two- or three-dimensional. To understand the delails of the flow around any submerged object, we must first take up the subjects of potential flow and the boundary layer, which we do in Chaps. 10 and 11. 

Frequently we are not interested in the details of the flow but only in the practical problem pf predicting the force on a body due to the flow of fluid around it. For example, the airplane designer wants to know the air resistance of the plane to select the right engine, the submarine designer wants to know the water resistance to determine how fast the submarine can go, and the designer of a chimney wants to know the maximum wind force on it to 

Subsequent workers found that this equation had to be modified by introducing a coefficient, which we call the drag coefficient Cj. This coefficient is not a constant equal to ij, as Newton believed, but varies with varying conditions, as we will see. Introducing it and dividing both sides of Eq. 6.52 by the cross-sectional area of the sphere, we find 

Compare this with the equation for the pressure drop in a long, straight, horizontal pipe j 

From these equations we see that Q plays the same role as /. Equation 6.54 contains the factor Ax/D, which describes the geometry of the system (long, thin pipes have more pressure drop than short, thick ones), but since all spheres have the same shape, there was no need to include such a factor in Eq. 6.53. 

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For flow around submerged objects, a = 1 in the creeping flow regime, and a = 0 for turbulent flow. Since the hydrodynamic drag force exerted by fluid B on solid particle A acts in the opposite direction of va when the fluid is stationary, and the gravitational force acts downward, these two forces are balanced ... 

Fig. 2 shows increased velocity around a submerged object by streamlines that are closer together compared to the region where there is no submerged object. The concept of streamlines can help to define two different flow regimes ... 

The following information is provided for flow around a spherical submerged object with interphase mass transfer into the passing fluid stream ... 

F. 4.18A shows the flow pattern without permeation through the membrane tube but with the no-slip condition imposed. As expected, the gas flows around the cyhnder. Note that inside the cylinder a flow is induced as well this phenomenon is well known (Uhlmarm, 2005). We add force to the cells surrounding the force points on the cylinder surface, but do not enforce any restriction on these cells, thus allowing for internal flow in the submerged object (membrane). Usually, these gas velocities inside the... 

Flow around objects is a somewhat more complicated situation than flow in conduits. The basis for describing flow around a submerged object is the drag force, which is the force in the flow direction exerted by the fluid on the solid surface. 

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