Aerodynamic Model

Here is a list of the main aerodynamic coefficients typically required for building a realistic aircraft flight simulator. These coefficients describe how aerodynamic forces and moments act on the aircraft based on its state (like angle of attack, sideslip, speed) and control inputs.

They are usually represented as functions of variables like Angle of Attack (α), Sideslip Angle (β), Mach number (M), control surface deflections (δe​,δa​,δr​), and sometimes angular rates (p,q,r).

1. Force Coefficients:

• Lift Coefficient (CL​): This is crucial for determining the lift force perpendicular to the relative wind. It primarily depends on:

  • Angle of Attack (α)  

  • Elevator Deflection (δe​)

  • Flap Deflection (δf​)

  • Mach Number (M)

  • Sometimes Pitch Rate (q) and rate of change of alpha (α˙) for dynamic effects (CLq​​,CLα˙​​).

• Drag Coefficient (CD​): Determines the drag force parallel to the relative wind. It's often modelled as:

  • Zero-lift drag (CD0​), which depends on configuration (gear, flaps), Mach number.  

  • Induced drag, which depends strongly on CL2​.  

  • Compressibility drag (significant at high Mach numbers).  

  • Also influenced by flap deflection (δf​), landing gear, speed brakes.

• Sideforce Coefficient (CY​): Determines the force perpendicular to the aircraft's plane of symmetry (sideways force). It primarily depends on:

  • Sideslip Angle (β) (CYβ​​)  

  • Rudder Deflection (δr​) (CYδr​​​)

  • Aileron Deflection (δa​) (CYδa​​​)

  • Sometimes Roll Rate (p) and Yaw Rate (r) (CYp​​,CYr​​).

2. Moment Coefficients:

• Pitching Moment Coefficient (Cm​): Determines the moment tending to pitch the aircraft's nose up or down around the center of gravity. It primarily depends on:

  • Angle of Attack (α) (Cmα​​, related to static stability)

  • Elevator Deflection (δe​) (Cmδe​​​, elevator effectiveness)

  • Pitch Rate (q) (Cmq​​, pitch damping)

  • Rate of change of Angle of Attack (α˙) (Cmα˙​​)

  • Flap Deflection (δf​)

  • Mach Number (M)

• Rolling Moment Coefficient (Cl​): Determines the moment tending to roll the aircraft about its longitudinal axis. It primarily depends on:

  • Sideslip Angle (β) (Clβ​​, dihedral effect)

  • Aileron Deflection (δa​) (Clδa​​​, aileron effectiveness)

  • Rudder Deflection (δr​) (Clδr​​​)

  • Roll Rate (p) (Clp​​, roll damping)

  • Yaw Rate (r) (Clr​​)

• Yawing Moment Coefficient (Cn​): Determines the moment tending to yaw the aircraft's nose left or right about its vertical axis. It primarily depends on:

  • Sideslip Angle (β) (Cnβ​​, weathercock stability)  

  • Rudder Deflection (δr​) (Cnδr​​​, rudder effectiveness)

  • Aileron Deflection (δa​) (Cnδa​​​, adverse/proverse yaw)

  • Yaw Rate (r) (Cnr​​, yaw damping)

  • Roll Rate (p) (Cnp​​)

Summary of Key Dependencies (Stability and Control Derivatives):

Often, these coefficients are represented using stability and control derivatives (the partial derivatives of the coefficients with respect to state variables or controls), especially for simpler models or analysis near a trim condition. The most critical ones include:

• Longitudinal: CLα​​, CD0​​, CLδe​​​, Cmα​​, Cmδe​​​, Cmq​​, Cmα˙​​

• Lateral-Directional: CYβ​​, CYδr​​​, Clβ​​, Clδa​​​, Clp​​, Clr​​, Cnβ​​, Cnδr​​​, Cnδa​​​, Cnr​​, Cnp​​

Implementation:

In a flight simulator, these coefficients are typically stored in multi-dimensional lookup tables derived from wind tunnel data, computational fluid dynamics (CFD), or flight test results. The simulator interpolates these tables based on the current flight state (α,β,M, altitude, configuration, control inputs, angular rates) to calculate the instantaneous aerodynamic forces and moments, which are then used in the equations of motion to update the aircraft's state for the next time step.  

This list covers the fundamental coefficients needed for a reasonably accurate flight dynamics model. More complex simulators might include additional coefficients for effects like ground effect, aeroelasticity, propeller/engine effects, etc.

Representative values or ranges for the main stability and control derivatives for a typical light, single-engine, subsonic aircraft (like a Cessna 172 or Piper PA-28) in a clean configuration (flaps up, gear fixed or up if retractable) during cruise flight.

Important Considerations:

• Units: These derivatives are typically given per radian when used in equations of motion. Values per degree would be smaller by a factor of π/180≈0.01745.  

• Reference Geometry: Values depend on the reference wing area (S), wingspan (b), and mean aerodynamic chord (cˉ) used for non-dimensionalization.

• Flight Condition: Values change with angle of attack (α), Mach number (M), altitude, and configuration (flaps, gear, power setting). These are typical values for low α, low subsonic flight.

• CG Location: Longitudinal static stability derivatives (Cmα​​) are particularly sensitive to the center of gravity (CG) location.  

• Approximation: These are ballpark figures to give you a sense of magnitude and relative importance. Real aircraft data will vary.

Here are some representative values (mostly per radian):

Longitudinal Coefficients & Derivatives:

Lateral-Directional Coefficients & Derivatives:

Clβ (the change in non‑dimensional rolling‑moment coefficient for a small sideslip angle β).

A negative value gives the familiar dihedral effect that tends to roll the aeroplane back toward level flight.

Open‑class sailplane / long‑span UAV:   ‑0.02 → ‑0.05,  Very slender wing, usually mild dihedral; lateral stability kept small to minimise roll‑drag.

Light GA, high‑wing (e.g. C‑172):  ‑0.04 → ‑0.09,  High wing + 3–5° dihedral and sizeable fin give a clearly stabilising dihedral effect.

Commuter turboprop, high‑wing (DHC‑6 Twin Otter) ≈ ‑0.08, Value taken from the published Twin Otter JSBSim data set FlightGear

Narrow‑body jet transport (B737, A320): ‑0.07 → ‑0.15, 25–35° wing sweep adds to geometric dihedral; large vertical tail adds further negative contribution.

Wide‑body / military transport (C‑130, B777): ‑0.10 → ‑0.20, High wing (C‑130) or large vertical tail (wide‑body) produce stronger restoring roll moment.

Modern relaxed‑static‑stability fighter (F‑16, F/A‑18): ‑0.02 → ‑0.06, Low or even negative geometric dihedral; designers keep (

Delta / canard fighters (Mirage III, Eurofighter) ≈ 0 → ‑0.03, Delta wings give little dihedral effect; value may cross zero at high α.

Flying‑wing / blended‑wing ≈ 0 (design target) Artificial roll control replaces natural dihedral stability.

Conversion: per‑degree values are simply Clβ (/deg)=Clβ/57.3C_{l_\beta}^{\,(\text{/deg})}=C_{l_\beta}/57.3Clβ​(/deg)​=Clβ​​/57.3.
Thus −0.10 rad‑¹ ≈ −0.0018 deg‑¹.

Why the numbers sit where they do

• Wing geometry – The classic DATCOM first‑order estimate is;

Cl_β_wing ≈ −(AR / 2) Γ 

(AR = aspect ratio, Γ = dihedral in radians). Higher wings on the fuselage behave as if they carried extra dihedral.

• Wing sweep – Each 10° of sweep is roughly worth −0.01 to −0.02 rad‑¹ because the windward wing sees more lift.

• Vertical tail – A weathercocking fin placed above the roll axis adds a negative ClβC_{l_\beta}Clβ​​ roughly proportional to its side‑force derivative CYβC_{Y_\beta}CYβ​​.

• Propellers & jets – Slipstream on the up‑wind wing of a multi‑engine turboprop can increase the magnitude (see discussion of prop‑induced dihedral effect aeroscience).

• Anhedral – High‑g fighters often use −3° to −5° anhedral to cancel the vertical‑tail contribution and keep the derivative small. Anecdotally, designers try to keep the overall value around −0.04 rad‑¹ for highly agile jets.

• Sign convention – Most flight‑mechanics literature (and simulation formats such as FS .air files) take negative ClβC_{l_\beta}Clβ​​ as “stable”. See the practical note for modellers fsdeveloper.com.

Using these numbers in preliminary design

1. Set a target band for your mission:

   • Trainer / utility aircraft ⇒ aim for −0.05 ± 0.02 rad‑¹.

   • High‑maneuver fighter  ⇒ aim for −0.03 ± 0.01 rad‑¹.

2. Allocate the budget between wing geometry (dihedral or anhedral), wing sweep and the vertical tail.

3. Check coupling – A large negative ClβC_{l_\beta}Clβ​​ damps dutch‑roll but can slow aileron response; very small magnitudes require a roll‑rate damper.

4. Iterate with a higher‑fidelity method (VLM, CFD, or wind‑tunnel) once geometry freezes.

These ranges are meant only as first‑cut reference numbers; detail design, Reynolds‑number effects and non‑linear aerodynamics will shift the precise value.