# Can drag coefficients and forces be broken up in to orthoganol components

I am trying to understand how a rigid body traveling through air (and the drag) changes with cross wind. Without crosswind, I can determine the drag by looking up the drag coefficient (in a table) and knowing it's velocity. If I add some cross wind, can I just break up the apparent wind in to x,y,z components and determine the drag coefficient by looking up the projected shape of the rigid body in the three planes and treat each of three problems separately to determine the forces/accelerations separately? Does aerodynamics work that way? Are there published/common tables or formulas that I can use for common shapes that are traveling at an angle to the wind?

• Commented Oct 19, 2022 at 16:27

## 3 Answers

The best way to do this is to write the aerodynamic force as

$$\mathbf{F} = \dfrac{1}{2}{\overline{\rho}} \overline{U}^2 \overline{S} \mathbf{c_F}(Re, M, \alpha, \beta)$$.

This expression is motivated by dimensional analysis, where the physical dimension of the force is the product of a meaningful reference dynamical pressure $$\overline{q} = \frac{1}{2} \overline{\rho}\overline{U}^2$$ and a meaningful surface $$\overline{S}$$ of the body (usually wing area for an aircraft, frontal area for bluff bodies like cars, buildings, ...).

Beside these quantities, the aerodynamic coefficient vectors $$\mathbf{c_F}$$ is a function of non-dimensional number only, like Reynolds number $$Re$$, Mach number $$M$$ or the 2 angles that define the relative direction of the body w.r.t the free stream velocity. Then you can project this vector of aerodynamic coefficients in the set of directions you prefer, like:

• body axes: usually in a reference frame used for the design of the body, with:
• two axis in the symmetry plane (if there is a symmetry plane): coefficients $$c_x(Re, M, \alpha, \beta)$$, $$c_z(Re, M, \alpha, \beta)$$;
• on orthogonal to it: coefficient $$c_y(Re, M, \alpha, \beta)$$
• wind axes:
1. drag in the direction of the wind, $$c_D(Re, M, \alpha, \beta)$$;
2. lift orthogonal to the wind in the plane of symmetry of the body, $$c_L(Re, M, \alpha, \beta)$$;
3. side force orthogonal to these two directions, $$c_S(Re, M, \alpha, \beta)$$

You can take a look at frames in aircraft aerodynamics here https://jsbsim-team.github.io/jsbsim-reference-manual/mypages/user-manual-frames-of-reference/, it's similar for the

No, you cannot split the velocity into components and use the components to calculate aerodynamic forces.

### Theory

First consider the concept of the angle of attack $$\alpha$$, which describes the orientation of the object relative to the airstream. The aerodynamic forces $$F_{\rm air}$$ that develop are not only a function of the airstream speed $$v_{\rm air}$$ and the geometry, but the angle of attack also. Not only the magnitude and direction of the force vary but also the location at which this force is felt, called the center of pressure.

In order to make sense of all the different variability what is done typically is loading on the body is decomposed into two components, drag and lift along the direction of the body, and include a torque applied to the body to account for the offset between the center of pressure and the center of mass.

See Aerodynamic Lift, Drag and Moment Coefficients for more details

The loading is described by three coefficients, $$c_L$$ coefficient of lift, $$c_D$$ coefficient of drag, and $$c_M$$ coefficient of moment defined as follows

\begin{aligned} F_{\rm lift} & = (\tfrac{1}{2} \rho_{\rm air} S_{\rm total} v_{\rm air}^2) c_L \\ F_{\rm drag} & = (\tfrac{1}{2} \rho_{\rm air} S_{\rm total} v_{\rm air}^2) c_D \\ M_{\rm turn} & = (\tfrac{1}{2} \rho_{\rm air} S_{\rm total} \ell v_{\rm air}^2) c_M \\ \end{aligned}

where $$\rho_{\rm air}$$ is the density of air, $$S_{\rm total}$$ is some reference (fixed) area value, like the wing area, or the frontal area, $$v_{\rm air}$$ is the airstream speed, and $$\ell$$ is some reference length like the chord length or the total length of the object.

Now the aerodynamic loading is described by three dimensionless coefficients $$c_L$$, $$c_D$$ and $$c_M$$, and their values are plotted against the angle of attack from various experiments run in wind tunnels, or CFD software.

Some examples are shown below

For specific geometries and conditions, you might be able to make some additional assumptions. For example sometimes drag is described as a quadratic function of lift $$c_D \approx c_{D_0} + \beta\, c_L^2$$, or note that the ratio of lift to drag varies linearly with attack angle $$c_L/c_D \propto \alpha$$. But all that depends on the situation.

Here is an example a common relationship between $$c_D$$ and $$c_L$$

### Answer

For your case having $$c_D$$ and $$c_L$$ for $$\alpha=0$$ only isn't sufficient information to describe what will happen when the angle of attack is more than zero (cross-wind situation).

Note that you are proposing using the components of velocity $$v \cos \alpha$$ and $$v \sin \alpha$$ for the aero forces.

For example, if you tried to calculate lift this way

$$F_{\rm lift} = \tfrac{1}{2} \rho S c_L \left( v \sin \alpha \right)^2$$

you end up with $$F_{\rm lift} \propto \alpha^2$$ which is incorrect behavior. The simplest models should have $$F_{\rm lift} \propto \alpha$$ for small values of $$\alpha$$, which allows lift to switch signs when the angle of attack switches signs.

In the simplest idealizations (flows without viscosity), velocity fields can be found using Laplace's equation, which is a linear partial-differential equation for which superpositions of solutions at different angles are meaningful. Such flows have no drag however (apart from trailing-vortex induced drag on wings, for example).

In cases of streamlined shapes with viscous effects, Laplace's equation can often be used in combination with boundary-layer results to give good approximations to the real flow at specific angles, but the ability to superimpose solutions (while possible) is no longer rigorously valid.

In the most general case, the Navier-Stokes equations must be used. These are nonlinear and it is not possible to combine the results of the individual cases to get the total drag at arbitrary angles.