While it is relatively simple and commonplace to leverage CFD to determine static stability characteristics of an aircraft such as *C _{Lα }* and

*C*, it is more challenging to determine dynamic stability characteristics. Here we will demonstrate how CFD can be used to estimate the pitch damping derivative (

_{Mα }*C*of an aircraft. For this study, we will be analyzing the eFlyer 2 which is an all-electric aircraft being developed by Bye Aerospace.

_{mq}Challenges:

- Modeling an aircraft in a steady maneuver using CFD
- Predicting aerodynamics in non-inertial reference frames

Values:

- Determining dynamic stability characteristics of an aircraft
- Higher fidelity aerodynamic database modeling

**What is Pitch Damping?**

Physically, pitch damping is the tendency for the aircraft to produce a restoring pitching moment when subjected to a pitch rate *q** in rad/s. It plays a major role in longitudinal dynamic stability of an aircraft through damping of the short period and phugoid modes. Mathematically, this is represented by *C _{mq}* which is the change in pitching moment due to a

*non-dimensional*pitch rate

*q*defined by

While in a constant pitch rate maneuver it is the aircraft that is actually moving, the perceived effect in the aircraft body frame is that of a curved freestream flow as depicted below.

The result is a change in the effective angle of attack of the lifting surfaces due to the pitch rate. Inspection of the flowfield geometry quickly reveals that the tail will see the highest change in α due to the pitch rate.

**Estimating Pitch Damping**

The change in effective angle of attack can be estimated using simple geometric relations to be

where* l*_{T} is the distance from the aerodynamic center of the tail to the CG. With the change in effective angle of attack known, the lift and ultimately the pitching moment due to pitch rate can be estimated simply from the aircraft geometry.

The engineer should be aware that large values of *q* can produce values of Δα that exceed the linear region of the lift curve. For this reason, it is advisable to limit the non-dimensional pitch rate such that the change in effective angle of attack remains small – in this case we use *q* = 0.01.

The lift curve slope of the tail is estimated by adjusting the 2D lift curve slope with finite span effects.

Finally, the pitch damping can be estimated with the following:

**Modeling a Pitch Maneuver Using CFD**

Modeling any dynamic maneuver in CFD is typically a non-trivial task. However, for a constant rate maneuver such as a constant pitch rate pull-up, clever use of non-inertial reference frames greatly simplifies this problem. Solvers such as Star-CCM+ and FUN3D – for example, possess this capability. By solving the Navier-Stokes equations in a coordinate system that is rotating at a rate of *q** rad/s centered at the center of rotation of the pull-up – that is, a distance R = V⁄q* from the CG, the curved freestream flow from the maneuver can be simulated.

Starting with a mesh of the eFlyer 2 aircraft,

the analysis is performed in 2 steps: 1) perform a simulation at 0 pitch rate and angle of attack to determine C_{Lo} and C_{mo}, 2) perform a simulation at a known pitch rate (in this case using a non-inertial reference frame to determine C_{L} and C_{m} during the maneuver.

Inspecting the flowfield for the pitch rate simulation it is apparent that using the non-inertial reference frame does indeed curve the flowfield as desired.

Finally, we take the forces and moments from both simulations and arrive at the CFD prediction for *C _{Lq}* and

*C*.

_{mq}Recall that the preliminary estimation of the aerodynamic effects due to pitch rate only accounted for the effect on the tail and not on the fuselage or wing which were captured in the CFD which explains the differences in *C _{Lq}*. However, since the lift on the tail will dominate the pitching moment, the pitch damping predicted by the hand calculation and CFD match very closely.

**Summary**

Pitch damping plays a critical role in the longitudinal dynamics of an aircraft. Estimation of pitch damping using CFD can be performed by leveraging non-inertial reference frames within a CFD solution and is easily verified by hand calculation.

**Written by Eddie Szwabowski**

Eddie Szwabowski is an Aerospace Engineer with a background in Aerodynamics, CFD, and programming. Outside of work he enjoys playing video games, shooting guns, and eating steak.