Flight dynamics model for simulating Venturestar style spacecraft

06 Jun 2025

This is an informational post on how to simulate the physics of atmospheric flight of a Venturestar style single-stage-to-orbit space craft. My dad Gerhard Wedekind is an experienced aerodynamics engineer and I asked him to help with making the aerodynamics of the sfsim space flight simulator realistic to some extent. The information in this post is a write-up of relevant formulas and approximate data he obtained from numerical simulation and estimates from aerodynamics knowledge. The information provided in this article is for general informational purposes only and comes without any warranty, express or implied.

Simulation

Here are a few beautiful snapshots from simulation. The first one shows a Mach box for V = 2 Ma and α = 3°.

The next one shows a Mach box for V = 4 Ma and α = 3°.

Finally here is a distribution of the pressure difference between top and bottom of wing.

Coordinate systems

The following drawing shows the body coordinate system (xb, yb, zb) and the wind coordinate system (xw, yw, zw). The wind system is rotated against the body system so that the speed vector (in a stationary atmosphere) points in positive xw.

Note that lift, drag, and side force are defined in the wind system and not in the body system.

• A positive lift force points upwards (negative zw) in the wind system.
• The drag force points backwards (negative xw) in the wind system.
• A positive side force points starboard (positive yw) in the wind system.

Yaw, pitch, and roll moments on the other hand are specified in the body system.

A coordinate system transformation from body system to wind system can be performed using two angles:

• α is the angle of attack
• β is the sideslip angle

When transforming coordinates from body system to wind system, one first rotates by β (sideslip angle) about the body z axis (zb). Then one rotates by α (angle of attack) about the new y axis.

Dynamic pressure

The dynamic pressure q depends on air density ρ and speed V:

Air density (and temperature) as a function of height can be obtained from Hull’s book “Fundamentals of airplane flight mechanics”.

Forces

Drag consists of zero-lift drag and induced drag:

Zero-lift drag is computed using the zero-lift drag coefficient CD0 as well as dynamic pressure q and the reference area Sref: The zero-lift drag coefficient depends on the speed of the aircraft.

Induced drag is determined using the lift coefficient CL, the Oswald factor e, the aspect ratio Λ, as well as q and the reference area Sref. The Oswald factor e depends on the speed of the aircraft. The lift coefficient depends on the angle of attack α.

The aspect ratio Λ depends on wing span b and wing area S:

The lift L is computed using the lift coefficient CL, dynamic pressure q, and the reference area Sref:

The side force Y (and corresponding coefficient) is usually not important but we will look into it later.

Moments

The pitching moment M is computed using the pitching moment coefficient Cm, the dynamic pressure q, the reference area Sref, and the aerodynamic chord cbar: The pitching moment coefficient depends on the lift coefficient CL, the position of the neutral point XN, the centre of gravity xref. and the aerodynamic chord cbar:

The yawing moment N is the product of the yawing moment coefficient Cn, the dynamic pressure q, the reference area Sref, and half the wing span b: The yawing moment coefficient depends on the side slip angle β.

The rolling moment L (using the same symbol as lift for some reason) is the product of the rolling moment coefficient Cl, the dynamic pressure q, the reference area Sref, and half the wing span b: The rolling moment coefficient depends on the angle of attack α and the side slip angle β.

Data Sheet

Here are the parameters for the flight model above:

Note that xref is defined in a coordinate system where x=0 is at the intersection of the inner leading edges (wing apex). The following picture also shows the position of the aerodynamic chord with length cbar. The center of gravity is at 25% of the aerodynamic chord.

Tables

Here is a data table with information for determining the remaining coefficients depending on the airspeed in Mach (Ma). The table shows for each speed:

• a factor to determine the lift coefficient CL
• the position XN of the neutral point relative to the aerodynamic chord (note that the center of gravity xref is at the 25% mark of the aerodynamic chord)
• the Oswald factor e
• a factor to determine the rolling moment coefficient Cl
• a factor to determine the yawing moment coefficient Cn
• the zero-lift drag coefficient CD0

The outlier of Clβα for V = 1.2 Ma (0.5971) should be ignored because the value was changing a lot with mesh resolution.

For small values of α, the lift coefficient increases linearly with α (where α is specified in radians):

For small values of α and β, the rolling moment coefficient increases linearly with the product of α and β (where α and β are specified in radians):

For small values of β, the yawing moment coefficient increases linearly with β (where β is specified in radians):

The following table shows for each speed:

• the value for α at which the linear relation of CL and α breaks down
• the maximum value of CL
• the angle of attack where CL reaches its maximum
• the drag coefficient for 90° angle of attack

Near α=90°, the lift and drag coefficients behave as follows:

At hypersonic speeds (V/Ma=10.0), lift and induced drag coefficients behave as follows:

I.e. the coefficients are stabilising at hypersonic speeds!

Control surfaces

The following table shows parameters to determine different moments generated by control surfaces:

The side force coefficient for a given rudder angle ζ is: The yawing moment coefficient for the rudder is:

The pitching moment coefficient for flaps δF (down is positive) is

The rolling moment coefficient for ailerons with angle ξ (positive: port aileron up, starboard aileron down) is: The yawing moment coefficient is

Angular damping

The formula for roll, pitch, and yaw damping moments (L, M, N) due to roll, pitch, and yaw rates (p, q, r) uses a coefficient matrix: The coefficients for V = 0 Ma are as follows. Note that damping moments are negligible for higher speeds.

Next steps

Using the information, the curves for a full range of angles and speeds need to be fitted and guessed in some places.

Feel free to leave a comment or suggestion below.