Part 2: Drag & its Effects

Viscosity of fluids

Thinking of fluids moving in different layers of different velocities, shear stress acts between adjacent layers, just like how friction acts between 2 surfaces in contact at different velocities.

The walls exert such large shear stress onto the liquid which causes liquid at the edges to have 0 velocity, a no-slip condition. Since shear stress is transmitted through the different layers of liquid, the presence of the wall causes a U-shape velocity profile across the cross-section.

Newton’s Law of Viscosity

The dynamic viscosity for most fluids are typically constant, defined as:

The kinematic viscosity is defined as:

Sheer strain Rate

This measures the change in strain of a material with respect to time. The strain rate at some point within the material measures the rate at which the distances of adjacent parcels of the material change with time in the neighbourhood of that point.

Low Re = Laminar Flow High Re = Turbulent Flow Inviscid region: region of low viscous forces (where Bernoulli’s equation holds true)

The Drag Equation

Drag depends on the density of the air, the square of the velocity, the air's viscosity and compressibility, the size and shape of the body, and the body's inclination to the flow. In general, the dependence on body shape, inclination, air viscosity, and compressibility is very complex. Hence, to simplify the model, the shape, inclination and other factors are combined into the drag coefficient, Cd, which is almost always determined experimentally using a wind tunnel. Notice that 0.5rv^2 appears in both the drag equation and Bernoulli’s pressure equation, and is known as the dynamic pressure.

In addition, The Reynolds number (Re), an important dimensionless quantity in fluid mechanics, helps predict flow patterns in different fluid flow situations. At low Reynolds numbers, fluid flow tends to be laminar, while at high Reynolds numbers, flow tends to be turbulent. It is defined as Re=uL, where is the density of the fluid, u is flow speed, L is characteristic linear dimension (distance) and is dynamic viscosity (stickiness) of the fluid.

The Reynolds number decreases as the drag coefficient increases, and the drag coefficient becomes approximately constant (at 0.4) if the Reynolds number is in the range of 1000 and 200000. This means that laminar flow (low Re) is associated with a high drag coefficient turbulent flow is associated with a low drag coefficient, although many other factors will influence the drag coefficient as well.

Pressure vs Frictional Drag

Horizontal forces acting on a moving object within a medium are composed of 2 forces.

• Wall’s shear stress, tangential to the object’s surface, caused by friction. This is known as frictional drag.

• The medium’s pressure distribution around the drag, acting perpendicular to the object’s surface. This is known as pressure drag.

Pressure Drag

Pressure drag occurs when the boundary layer (airflow along an object’s surface) detaches from the object’s surface, creating a wake of recirculating, turbulent flow. The wake is a low pressure area (separation region), generating a

pressure difference in front of and behind an object, thus the medium exerts a backwards force onto the object, slowing it down. Flow separation can also cause vortex shedding, causing vibrations. It has the greatest effect on blunt objects, such as spheres.

Due to Bernoulli’s equation, along the top edges of a body, when flow accelerates, pressure decreases. Once past the biggest cross-section, flow decelerates. If pressure increase is large enough, fluid flows backwards, but as blocked by forward-moving fluid behind, it detaches from the body, causing flow separation.

Surprisingly, introducing turbulent flow decreases separation region and pressure drag. It introduces mixing across layers of flow, causing momentum transfer, helping the gas sustain a larger adverse pressure without separation.

Frictional Drag (Obeys the drag equation)

Frictional drag arises from the friction of the fluid against the object surface. Air in contact with the object will stick to the object's surface, which will tend to stick to the next layer of air and so on, hence the object drags air with it. This layer of air is known as the boundary layer. As such, some of the object’s momentum is imparted into the surrounding air, slowing the object. Frictional drag is evident in viscous mediums, and objects with big surface area along the width (eg. area of top of wing).

Overall drag

Unlike pressure drag, turbulence produces larger shear stresses, increasing frictional drag. Thus, a low Re correlates to low friction drag and high pressure drag.

As seen in the figure on the right, frictional drag mainly affects long and thin objects, while pressure drag mainly affects blunt objects. High angles of attack of airfoils produces high pressure drag, while low angles produces high frictional drag.