Drag coefficient

Cauchy momentum equation

In the non-dimensional form of the Cauchy momentum equation, the skin drag coefficient or skin friction coefficient is referred to the transversal area (the area normal to the drag force, so the coefficient is locally defined as:

$${\displaystyle c_{\mathrm{d}}={\displaystyle \frac{\tau }{q}}={\displaystyle \frac{2\tau }{\rho u^2}}}$$

where:

• ${\displaystyle \tau }$ is the local shear stress, which is by definition the stress component in the direction of the local flow velocity;
• ${\displaystyle q}$ is the local dynamic pressure of the fluid
• ${\displaystyle \rho }$ is the local mass density of the fluid;
• ${\displaystyle u}$ is the local flow speed of the fluid

Background

The drag equation

${\displaystyle F_{\mathrm{d}}=\frac{1}{2}\rho u^2{c}_{\mathrm{d}}A}$

is essentially a statement that the drag force on any object is proportional to the density of the fluid and proportional to the square of the relative flow speed between the object and the fluid. The factor of ${\displaystyle 1/2}$ comes from the dynamic pressure of the fluid, which is equal to the kinetic energy density.

The value of ${\displaystyle c_{\mathrm{d}}}$ is not a constant but varies as a function of flow speed, flow direction, object position, object size, fluid density and fluid viscosity. Speed, kinematic viscosity and a characteristic length scale of the object are incorporated into a dimensionless quantity called the Reynolds number ${\displaystyle \mathrm{R}\mathrm{e}}$. ${\displaystyle c_{\mathrm{d}}}$ is thus a function of ${\displaystyle \mathrm{R}\mathrm{e}}$. In a compressible flow, the speed of sound is relevant, and ${\displaystyle c_{\mathrm{d}}}$ is also a function of Mach number ${\displaystyle \mathrm{M}\mathrm{a}}$.

For certain body shapes, the drag coefficient ${\displaystyle c_{\mathrm{d}}}$ only depends on the Reynolds number ${\displaystyle \mathrm{R}\mathrm{e}}$, Mach number ${\displaystyle \mathrm{M}\mathrm{a}}$ and the direction of the flow. For low Mach number ${\displaystyle \mathrm{M}\mathrm{a}}$, the drag coefficient is independent of Mach number. Also, the variation with Reynolds number ${\displaystyle \mathrm{R}\mathrm{e}}$ within a practical range of interest is usually small, while for cars at highway speed and aircraft at cruising speed, the incoming flow direction is also more-or-less the same. Therefore, the drag coefficient ${\displaystyle c_{\mathrm{d}}}$ can often be treated as a constant.

For a streamlined body to achieve a low drag coefficient, the boundary layer around the body must remain attached to the surface of the body for as long as possible, causing the wake to be narrow. A high form drag results in a broad wake. The boundary layer will transition from laminar to turbulent if Reynolds number of the flow around the body is sufficiently great. Larger velocities, larger objects, and lower viscosities contribute to larger Reynolds numbers.

For other objects, such as small particles, one can no longer consider that the drag coefficient ${\displaystyle c_{\mathrm{d}}}$ is constant, but certainly is a function of Reynolds number. At a low Reynolds number, the flow around the object does not transition to turbulent but remains laminar, even up to the point at which it separates from the surface of the object. At very low Reynolds numbers, without flow separation, the drag force ${\displaystyle F_{\mathrm{d}}}$ is proportional to ${\displaystyle v}$ instead of ${\displaystyle v^2}$; for a sphere this is known as Stokes' law. The Reynolds number will be low for small objects, low velocities, and high viscosity fluids.

A ${\displaystyle c_{\mathrm{d}}}$ equal to 1 would be obtained in a case where all of the fluid approaching the object is brought to rest, building up stagnation pressure over the whole front surface. The top figure shows a flat plate with the fluid coming from the right and stopping at the plate. The graph to the left of it shows equal pressure across the surface. In a real flat plate, the fluid must turn around the sides, and full stagnation pressure is found only at the center, dropping off toward the edges as in the lower figure and graph. Only considering the front side, the ${\displaystyle c_{\mathrm{d}}}$ of a real flat plate would be less than 1; except that there will be suction on the backside: a negative pressure (relative to ambient). The overall ${\displaystyle c_{\mathrm{d}}}$ of a real square flat plate perpendicular to the flow is often given as 1.17. Flow patterns and therefore ${\displaystyle c_{\mathrm{d}}}$ for some shapes can change with the Reynolds number and the roughness of the surfaces.

Drag coefficient examples

General

In general, ${\displaystyle c_{\mathrm{d}}}$ is not an absolute constant for a given body shape. It varies with the speed of airflow (or more generally with Reynolds number ${\displaystyle \mathrm{R}\mathrm{e}}$). A smooth sphere, for example, has a ${\displaystyle c_{\mathrm{d}}}$ that varies from high values for laminar flow to 0.47 for turbulent flow. Although the drag coefficient decreases with increasing ${\displaystyle \mathrm{R}\mathrm{e}}$, the drag force increases.

Aircraft

As noted above, aircraft use their wing area as the reference area when computing ${\displaystyle c_{\mathrm{d}}}$, while automobiles (and many other objects) use projected frontal area; thus, coefficients are not directly comparable between these classes of vehicles. In the aerospace industry, the drag coefficient is sometimes expressed in drag counts where 1 drag count = 0.0001 of a ${\displaystyle c_{\mathrm{d}}}$.

Automobile

Blunt and streamlined body flows

Concept

The force between a fluid and a body, when there is relative motion, can only be transmitted by normal pressure and tangential friction stresses. So, for the whole body, the drag part of the force, which is in-line with the approaching fluid motion, is composed of frictional drag (viscous drag) and pressure drag (form drag). The total drag and component drag forces can be related as follows:

$${\displaystyle \begin{array}{cc}{c}_{\mathrm{d}}& ={\displaystyle \frac{2F_{\mathrm{d}}}{\rho v^{2}A}}\\ & =c_{\mathrm{p}}+c_{\mathrm{f}}\\ & =\underset{c_{\mathrm{p}}}{\underbrace{{\displaystyle \frac{2}{\rho v^{2}A}}{\displaystyle {\int }_S\mathrm{d}S(p-p_o)\left(\hat{\mathbf{n}}\cdot \hat{\mathbf{i}}\right)}}}+\underset{c_{\mathrm{f}}}{\underbrace{{\displaystyle \frac{2}{\rho v^{2}A}}{\displaystyle {\int }_S\mathrm{d}S\left(\hat{\mathbf{t}}\cdot \hat{\mathbf{i}}\right)T_{\mathrm{w}}}}}\end{array}}$$

where:

• ${\displaystyle A}$ is the planform area of the body,
• ${\displaystyle S}$ is the wetted area of the body,
• ${\displaystyle c_{\mathrm{p}}}$ is the pressure drag coefficient,
• ${\displaystyle c_{\mathrm{f}}}$ is the friction drag coefficient,
• ${\displaystyle p_{\mathrm{o}}}$ is the pressure far away from the body (note that this constant does not affect the final result),
• ${\displaystyle p}$ is pressure at surface dS,
• ${\displaystyle \hat{\mathbf{i}}}$ is the unit vector in direction of free stream flow
• ${\displaystyle \hat{\mathbf{n}}}$ is the unit vector in the direction perpendicular to the body surface dS, pointing from the fluid to the solid,
• ${\displaystyle \hat{\mathbf{t}}}$ is the unit vector in the direction of the shear stress acting on the body surface dS,
• ${\displaystyle T_{\mathrm{w}}}$ is the magnitude of the shear stress acting on the body surface dS,

Therefore, when the drag is dominated by a frictional component, the body is called a streamlined body, while in the case of dominant pressure drag, the body is called a blunt or bluff body. Thus, the shape of the body and the angle of attack determine the type of drag. For example, an airfoil is considered a body with a small angle of attack by the fluid flowing across it. This means that it has attached boundary layers, which produce much less pressure drag. The wake produced is very small and drag is dominated by the friction component. Therefore, such a body (here an airfoil) is described as streamlined, whereas for bodies with fluid flow at high angles of attack, boundary layer separation takes place. This mainly occurs due to adverse pressure gradients at the top and rear parts of an airfoil.

As a result, wake formation occurs, leading to eddy formation and pressure loss due to pressure drag. In such situations, the airfoil is stalled and has higher pressure drag than friction drag. In this case, the body is described as a blunt body.

A streamlined body looks like a fish (tuna), Oropesa, etc. or an airfoil with small angle of attack, whereas a blunt body looks like a brick, a cylinder or an airfoil with high angle of attack. For a given frontal area and velocity, a streamlined body will have lower resistance than a blunt body. Cylinders and spheres are taken as blunt bodies because the drag is dominated by the pressure component in the wake region at high Reynolds number.

To reduce this drag, either the flow separation could be reduced or the surface area in contact with the fluid could be reduced (to reduce friction drag). This reduction is necessary in devices like cars, bicycle, etc. to avoid vibration and noise production.