How it Works

Ideal Nozzle

The Ideal Nozzle contour is calculated using the Method of Characteristics (MoC). The main contour defining parameter for the Ideal Nozzle method is the exit Mach number. An initial parabola is applied at the Nozzle Throat according to Sauer's method, where the initial flow Mach number is set to 1.1. From here on the Characteristic lines, emitted from the supersonic flow, are traced and a linearized approximation to the real flow properties is calculated with a Euler-method error correction for accuracy.
The initial expansion contour is chosen to be a circular arc with a user-set radius. Calculating the Characteristic points downstream of the initial parabola, once the user-set exit Mach Number at the Nozzle main Axis (y=0) is reached, the circular arc has reached its maximum wall angle.
For the Ideal Nozzle Contour, the flow at the Nozzle exit is shock-free, parallel and uniform. Therefore, the turning contour of the nozzle is calculated via mass flow balance across each characteristic line.
The Method of Characteristics not only yields the ideal De-Laval Nozzle Contour, but also offers a solution of the supersonic flow at each Characteristic point (marked as the red points in the graphic below). The corresponding flow properties throughout the nozzle: Velocity, Pressure, Temperature, Density can be investigated via the Plot settings.

Required Parameters:

• Resolution of Characteristic Mesh
• Desired Exit Mach Number
• Initial Mach Number (= 1.1)
• Combustion Chamber Pressure
• Ambient Pressure
• Combustion Chamber Temperature
• Isentropic Exponent of Exhaust Gas
• Specific Gas Constant
• Nozzle Throat Radius

Truncated Ideal Nozzle (TIC)

The Truncate Ideal Contour, or TIC is derived from the Ideal Nozzle solution. The full-length ideal contour is truncated at a user-defined geometric Boundary. The truncation parameters can be set to: (Max Relative Nozzle Length, Max absolute Nozzle Length, Max absolute Nozzle Width, Expansion Ratio or Nozzle exit Angle). In addition, by selecting both the Max absolute Nozzle Length and Max absolute Nozzle Width, a Geometric Envelope can be set, which defines the maximum Nozzle Size.
In addition, the TIC Method offers the inclusion of boundary layer calculations. These base on the Laminar Blasius' Boundary Layer solution, which uses the Local Reynolds number, Flow Velocity and Density to calculate the Flow Shear Force at the Nozzle Wall. The Reynolds number is calculated via local distance from Nozzle exit, local flow velocity and local kinematic viscosity. The Viscosity on the other hand is calculated through the three parameters Sutherland Constant , Reference Dynamic Viscosity and Reference Temperature. Typical values for these parameters can be obtained from the following table:

Gas	Reference Dynamic Viscosity μ₀ [Pa·s]	Reference Temperature T₀ [K]	Sutherland Constant Sμ [K]
Air	1.716 × 10⁻⁵	273	111
Argon	2.125 × 10⁻⁵	273	114
CO₂	1.370 × 10⁻⁵	273	222
CO	1.657 × 10⁻⁵	273	136
N₂	1.663 × 10⁻⁵	273	107
O₂	1.919 × 10⁻⁵	273	139
H₂	8.411 × 10⁻⁶	273	97
Steam	1.12 × 10⁻⁵	350	1064

The Boundary Layer Calculation additionally yields the displacement thickness, which is used to calculate the final Nozzle Contour Geometry.

Required Parameters:

• All parameters from Ideal Nozzle
• Truncation Criterion
• Reference Dynamic Viscosity
• Sutherland Constant

Thrust Optimized Parabola (TOP)

The Thrust Optimized Parabola (TOP) contour is generated by appending a quadratic Bezier curve to an initial circular arc. The defining Parameters for the contour are the maximum wall angle, which defines its positional vector through the circular arc, the exit wall angle and the exit Mach number of the flow. The positional vector of the exit point is calculated via isentropic calculations for the Expansion Ratio, as well as the corresponding length of a 80% Conical Nozzle, which determines the TOP Nozzle Length.
The results of the standard TOP method do not yield any information about the flow field and therefore the Nozzle efficiency. By selecting the -Include Flowfield Calculation- Button, it is possible to run a complete characteristic mesh calculation with the pre-calculated TOP contour. Instead of the massflow comparision used for the Ideal Nozzle, the parabola is used as a boundary condition. SINCE THE TOP CONTOUR EXPANDS THE FLOW NON-IDEALLY, THE OCCURING SHOCK WAVES (which can be seen in the characteristic mesh) CAN DISRUPT THE LINEARIZATION PROCESS OF THE MOC. UNCERTAINTIES MAY OCCUR DUE TO THIS PROBLEM.
The TOP Contour Generation also includes a Boundary Layer Calculation, similar to the TIC Nozzle.

• All parameters from Ideal Nozzle
• Max. Wall Angle
• Exit Wall Angle
• Reference Dynamic Viscosity
• Sutherland Constant