Hydrodynamic Drag

resistance to the motion of a body by the fluid around it or resistance to the motion of a fluid caused by the action of pipe walls, channels, and the like. If a liquid (or gas) flows around a stationary body or, vice versa, if a body moves through a stationary medium, the hydrodynamic drag is the projection of the vector sum of all forces acting on the body onto the direction of motion. The hydrodynamic drag is X = c_x(ρv²/2)S, where ρ is the density of the medium, v is the velocity, and S is the characteristic area for the given body. The dimensionless coefficient of hydrodynamic drag c_x depends on the shape of the body, its position with respect to the direction of motion, and the numbers of similarity. The force with which a fluid acts on each element of a moving body’s surface can be resolved into normal and tangential components, that is, into the pressure force and the friction force. Pressure drag is the resultant of all pressure forces, projected on the direction of motion. Friction drag is the result of all friction forces, likewise projected on the direction of motion. If a body’s pressure drag is small in comparison with its friction drag, the body is considered well streamlined. The hydrodynamic drag of poorly streamlined bodies is almost wholly due to pressure drag. If a body moves near the surface of water, waves can form and a wave drag arises.

When dealing with the flow of fluids through pipes, channels, and the like, hydraulics distinguishes two types of hydrodynamic drag: longitudinal drag and local drag. Longitudinal drag is directly proportional to the length of the stream section. Local drags are associated with changes in the structure of the flow in a short interval on flowing past different obstructions (such as valves and gates). Local drags also occur where the stream abruptly widens or narrows or where the flow direction changes. In hydraulic calculations the hydrodynamic drag is given by the value of the “lost” head h_v. This value represents that part of the specific energy of the stream which is irreversibly lost overcoming the resisting forces.

For the motion of the fluid caused by pressure drop, the value of h_v for a given length of pipe is computed using Darcy’s formula: h_v = λ(l/d)(v²/2g), where λ is the drag coefficient, l and d are the pipe length and pipe diameter, v is the average velocity, and g is the acceleration due to gravity. The coefficient λ is determined by the type of flow. For laminar flow this coefficient depends only on the Reynolds number (linear resistance dependence). For turbulent flow it also depends on the roughness of the pipe walls. For very large Reynolds numbers (on the order of 10⁶ or more), λ depends only on roughness (quadratic resistance dependence). Local hydrodynamic drags are evaluated using the general formula h_v= ξv²/2 g, where ξ is the coefficient of local drag, which varies with the type of obstruction and depends on the value of the Reynolds number.

Numerical values of the coefficients λ and ξ are obtained by formulas given in handbooks. The value h_v for open streams is also found using special formulas. Hydrodynamic drags in open streams and in penstocks are due to the same physical causes.

A correct computation of the value of hydrodynamic drag is of great importance in the design and construction of a wide range of structures, installations, and equipment (such as hydraulic engineering structures, turbine installations, air-and gas-purifying equipment, gas mains, oil mains, water mains, engines, compressors, and pumps).