Good morning, and happy Thesis Tuesday! Today we'll be continuing our look at external flow with a brief discussion on drag.
Drag is a net force in the direction of the flow due to the pressure and shear forces on the surface of the object that is moving through a fluid. If the pressure distribution and wall shear stress are known, drag can be determined from:
D = int(p*cos(theta))dA + int(tauw*sin(theta))dA
however, there are very few cases for which this data can be found analytically. Typically, drag is determined from a given drag coefficient, CD:
CD = D/(0.5*rho*U*U*A)
As we will see, CD = f(shape, Re, Ma, Fr, epsilon/l), where epsilon is surface roughness.
Friction drag, Df, is the portion of drag due to the viscous shear stress on the object (the second term in the drag equation above). Typically, the surface of an object rotated in any fashion has parts parallel and perpendicular to the flow, but if we examine the case of a very thin flat plate, it can be seen that there is no friction drag force on the plate rotated perpendicularly to the flow. The friction drag on a flat plate can be determined by:
Df = 0.5*rho*U*U*b*l*CDf
Pressure drag, Dp, is the portion of drag due to the pressure (or normal stresses) on an object (the first term in the drag equation above). The pressure drag can be found by:
Dp = 0.5*rho*U*U*A*CDp
where: CDp = int(Cp*cos(theta))dA/A
Here, Cp is a pressure coefficient:
Cp = (p - po)/(0.5*rho*U*U)
where po is a reference pressure, which does not influence the drag (the difference is important).
In most cases, the net effects of friction drag and pressure drag are considered instead of examining each type of drag individually as in the second equation above. As mentioned before, drag is influenced by many different aspects of the flow.
One of the most important contributors to the drag coefficient is the shape of the submerged body. Obviously, objects range in size from streamlined to blunt; more blunt objects (l/D -> 0, D >> l) result in larger drag coefficients. For extremely thin, streamlined bodies (l/D -> infinity, D << l) such as thin airfoils, the bodies are essentially tread as flat plates.
Reynolds number also affects the drag coefficient. At very low Re (<1), inertial effects are very small and CD = 2*C/Re, where C is a constant dependent upon size. For moderate Re, CD = Re^-0.5. For very high Re, CD increases for streamlined bodies and decreases for blunt bodies. For extremely blunt bodies, there CD depends little upon Reynolds number.
For sufficiently large object velocities (Ma > 0.5), compressibility effects become important and the drag coefficient becomes a function of Mach number (Ma = U/c). Sharp-pointed bodies develop their maximum drag coefficient around Ma = 1 (sonic flow), while that for blunt bodies increases with Ma far above Ma = 1.
For streamlined bodies, drag increases with increased surface roughness; however, for blunt bodies the opposite is true.
The Froude number (U/sqrt(g*l)) is the ratio of free-stream speed to a typical wave speed on the interface of two fluids, such as the surface of the ocean. Wave drag, Dw can be a complex function of the Froude number and body shape:
CDw = Dw/(0.5*rho*U*U*l*l)
Often approximate drag calculations for a complex body can be obtained by examining the body as a collection of various parts. The drag on each simpler part can be calculated and added together to determine the overall drag on the body.
So that's drag! Until next week, happy studying!
