Hydrodynamics
Hydrodynamics is a branch of fluid mechanics that involves mathematical analysis of the forces occurring at a fluid-object interface, e.g., between a submarine and the surrounding water or between a pipe wall and the flowing water. In general, it does not matter mathematically whether the object or the fluid is moving, rather it is the relative motion between the two that is important. The fluid is considered to be composed of particles that flow in layers. These layers become distorted and slide over one another (like playing cards) when the fluid intercepts an object.
Historically, hydrodynamics focused on theoretical equations for the flow of a fictitious (ideal) fluid that was assumed to be inviscid (i.e., totally without viscosity). In other words, this fluid would experience no frictional effects between its own moving layers or between fluid layers and object surfaces. These equations, though mathematically correct, indicated that a submerged object would experience zero drag (i.e., no resistance from the fluid to motion), a phenomenon that was known to be false from practical experience. This contradiction was called d'Alembert's paradox, after Jean le Rond d'Alembert, an eighteenth century scientist.
In 1904, Ludwig Prandtl published the boundary layer theory, which showed that flow around a submerged solid object contained two distinct regions with very different flow characteristics. Flow in a thin boundary layer around the object was affected by the surface of the object and by effects of internal friction due to the fluid's viscosity. This is called the viscous effect. Outside of the thin boundary layer, the viscous effect was negligible, and the free-flowing fluid behaved like an ideal fluid.. Prandtl's boundary layer concept solved d'Alembert's paradox and laid the groundwork for modern hydrodynamics (and aerodynamics, in which air is the "fluid"). Boundary layer formation also occurs in flow over spillways and through pipes, channels, nozzles, and orifices.
The velocity of flow at the very bottom of the boundary layer is zero, because flow is retarded by viscous forces (which are also called shear stresses) and by attractive forces that pull water particles to the object surface (i.e., surface tension). These forces lessen with increasing distance from the object surface, so that velocity increases within the boundary layer until it is the same as the free-flow velocity. Thus, a velocity field or velocity gradient exists within the boundary layer.
Fluid-object interactions are often illustrated using flow lines to represent flow behavior. Prior to the interface, the approaching flow is a pattern of smooth parallel flow lines. This is called rectilinear flow. Velocity along each line is constant (although different lines may have different velocities). At the front surface of the object, the flow lines part to move around it. For a smooth streamlined object with its axis aligned with flow (e.g., a thin plate situated parallel to flow), the disturbance in the flow lines is minor, and they quickly retake a rectilinear pattern after leaving the trailing edge of the object. This flow is considered one-dimensional, because all of the velocity vectors remain virtually parallel. Mathematical analysis becomes considerably more complex when two- and three-dimensional flow fields are considered.
Consider the flow lines for a flat plate situated perpendicular to flow. They approach the plate in rectilinear flow and then are forced widely apart upon impact. They move across the plate surface (where the boundary layer forms), but are unable to make the sharp turn following the plate edge. The boundary layer separates from the plate surface and plunges into the free-flow area. An area of extreme turbulence occurs directly behind the plate, characterized by swirling eddies and vortices (i.e., a wake). The eddies dissipate their energy with increasing distance from the plate and eventually flatten out into uniform flow lines again.
Flow within the boundary layer along the plate surface can be laminar (uniform) or turbulent. This is important because the shear forces (and thus the drag forces) are higher in areas of turbulent flow. For the parallel-flow plate, flow is usually laminar near the plate's leading edge, and then becomes turbulent at some point along the surface. Quantitatively, this is represented by the dimensionless Reynolds Number (NRe). For any distance x along the plate, NRe,x= xv/, where v is the bulk fluid velocity, is the fluid density, and is the fluid viscosity. The NRe,x value at which flow becomes turbulent varies between 10,000 and 3,000,000, depending on the smoothness of the plate and the turbulence of the approaching flow. If the transition from laminar to turbulent flow occurs nearer the plate's leading edge, then the total drag is higher than it would be for a transition located farther downstream.
Besides shear, drag forces can also be due to pressure, gravity, and compressibility effects. Although determination of drag forces is very complex for most flow situations, drag coefficients can be calculated for certain regular-shaped objects immersed in low-velocity flows. For example, the drag coefficient for a smooth sphere subjected to flow at very low Reynolds number is CD=24/NRe, which is a derivation of Stokes' law. The drag force is then calculated from D=CDAv2/2g, where A is the area of the sphere projected onto a plane perpendicular to flow, is the specific weight of the fluid, v is velocity, and g is gravitational acceleration. Drag coefficients for plates and cylinders can be similarly calculated or approximated from tables if the Reynolds number is known.
Other hydrodynamic forces acting at the fluid-object interface can be determined using quantitative parameters including the Froude Number(which is a ratio of inertial forces to gravitational forces), the Weber Number (which is a ratio of inertial forces to surface tension forces), the pressure coefficient (which is a ratio of pressure forces to inertial forces), and the Mach Number (which is a ratio of inertial forces to compressibility forces).
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