Introduction

Waverider Generator is a python package which can be used to generate hypersonic waveriders based on an efficient parametrisation described in [1]. The method makes use of the oscultating cone inverse design method and four design parameters X1, X2, X3 and X4. Please consult the paper referenced for any further clarification on the geometrical meaning of these inputs.

Example of generated waverider

Coordinate System

The coordinate system is defined such that:

• $x$ is the streamwise direction
• $y$ is the transverse direction
• $z$ is the spanwise direction

with the origin at the tip of the waverider.

Coordinate System

Flow Field

Oblique Shock

In the flat region of the shockwave, the lower surface is determined via the $\theta$- $\beta$ - $M_{\infty}$ equation, which relates the deflection angle $\theta$ to the shock angle $\beta$ in an oblique shock.

$$ \tan(\theta) = \frac{2 \cot\left(\beta\right) \left(M_{\infty}^2 \sin^2\left(\beta \right) - 1\right)}{M_{\infty}^2 (\gamma + \cos\left(2 \beta\right)) + 2} $$

Where $\gamma=1.4$ is the ratio of specific heats for air and $M_{\infty}$ is the freestream Mach number.

Osculating Cone Theory

In the curved region of the shockwave, the osculating cone theory is used whereby conical flow is locally applied at each osculating plane. The Taylor-Maccoll ODE, which describes conical flow, is solved and the resulting streamlines are propagated until the back of the waverider is reached to produce the lower surface. See [2] for more information on the Osculating Cone Theory.

Required Inputs

• Design parameters X1, X2, X3 and X4. Note this is entered as a list dp of four elements where the parameters are organised in the order listed here. Refer to the examples.
• Freestream Mach number M_inf.
• Shock angle in degrees beta.
• Height of the waverider at the base plane height. Note that the length of the waverider is determined as $height/\tan(\beta)$ so a user may choose to determine the height required for a desired length.
• Width of the waverider width. Note that this refers to half of the total width of the waverider due to the symmetry.
• Number of points to be used for interpolating the shockwave and the upper surface curve (n_shockwave and n_upper_surface respectively). Both are integers greater than 10 to preserve quality.

Optional Inputs

• Number of osculating planes used in the generation of the geometry n_planes. Note that this doesn't include the symmetry plane and the tip. A minimum number of planes is set at 10 to preserve quality and this is the default value.
• Number of points in the streamwise direction for the generation of the upper surface as well as the flat part of the shockwave on the lower surface n_streamwise. A minimum is set at 10 to preserve quality and this is the default value.
• Maximum step to be used in the tracing of the streamlines for the lower surface delta_streamwise. Note this is entered as a percentage of the length of the waverider and a maximum is set at 20% to preserve quality. The default value is 5%.

Summary of inputs

Input	Type	Conditions
X1, X2, X3 and X4	list	$0\leq X2,X3,X4 \leq 1$ $0 \leq X1 < 1$ $\frac{X2}{\left(1-X1\right)^4}\leq\ \frac{7}{64}\left(\frac{width}{height}\right)^4$ (see [1] for more information on this design condition)
M_inf	float, int	M_inf>0
beta	float, int	0<beta<90
height	float, int	height>0
width	float, int	width>0
n_shockwave and n_upper_surface	int	n_shockwave,n_upper_surface>=10
n_planes	int	n_planes>=10
n_streamise	int	n_streamwise>=10
delta_streamise	float	0<delta_streamwise<=0.2

Usage and functionality

A user can create a waverider instance by importing the waverider class from waverider_generator.generator. Everything takes place inside the class constructor so the user only needs to initialise the instance to create the geometry. An example is shown below:

from waverider_generator.generator import waverider as wr
M_inf=5
beta=15
height=1.34
width=3
dp=[0.11,0.63,0,0.46]
n_planes=20
n_streamwise=10
delta_streamwise=0.1
waverider=wr(M_inf=M_inf,beta=beta,height=height,width=width,dp=dp,n_upper_surface=10000,n_shockwave=10000,n_planes=n_planes,n_streamwise=n_streamwise,delta_streamise=delta_streamwise)

CAD Export

To export the geometry into a file CAD, the user can import the to_CAD function from waverider_generator.cad_export. An example is shown below:

#%%
# assuming a waverider instance has been created with the code above
from waverider_generator.cad_export import to_CAD

waverider_cad=to_CAD(waverider=waverider,sides='both',export=True,filename='waverider.step',scale=1000)
waverider_cad

The to_CAD function is detailed below:

Input	Type	Conditions	Description
waverider	waverider	NA	waverider instance
sides	str	"left", "right" or "both"	Side(s) of the waverider to generate in the CAD
export	bool	True or False	Setting this to True exports the CAD to the current directory, False does not
filename	str	The extension must be one which cadquery can generate a CAD file in	Name of the CAD file to be created
scale (optional)	float,int	scale>0	Scale factor by which the final geometry is scaled. By default, cadquery exports geometries in mm so the default value for scale is 1000 to obtain dimensions in meters. Setting this to 1 keeps the dimensions in mm. It is recommended to keep the default value of 1000 and work in meters from the start.

Output	Type	Conditions	Description
waverider_cad	cq.Solid (cadquery solid)	NA	A cadquery solid corresponding to the waverider generated. This can be previewed in a Jupyter Notebook, therefore avoiding the step of exporting and importing into a seperate CAD software each time.

Jupyter Notebook CAD preview example

Plotting tools

The package also allows the user to plot basic figures to analyse the geometry generated before a CAD export. This is done by importing Plot_Base_Plane and Plot_Leading_Edge from waverider_generator.plotting_tools. An example, also included as an example file, is shown below

Note that the latex input into the plotting functions is a boolean and determines whether or not to plot the figure with the default LaTex font. The user is required to install a LaTex distribution on their system for this feature to work.

#%%
from waverider_generator.generator import waverider as wr
from waverider_generator.plotting_tools import Plot_Base_Plane,Plot_Leading_Edge
import matplotlib.pyplot as plt

M_inf=5
beta=15
height=1.34
width=3
dp=[0.11,0.63,0,0.46]
n_planes=20
n_streamwise=10
delta_streamwise=0.1
waverider=wr(M_inf=M_inf,beta=beta,height=height,width=width,dp=dp,n_upper_surface=10000,n_shockwave=10000,n_planes=n_planes,n_streamwise=n_streamwise,delta_streamise=delta_streamwise)
#%%
'''
PLOT BASE PLANE
'''
base_plane=Plot_Base_Plane(waverider=waverider,latex=True)
#%%
'''
PLOT LEADING EDGE
'''
leading_edge=Plot_Leading_Edge(waverider=waverider=True,latex=True)
plt.show()

Base Plane

Leading Edge Plot

Dependencies

The package requires the following libraries to be installed:

• numpy
• cadquery
• scipy
• matplotlib

Optional:

• a LaTex distribution such as MiKTex (refer to previous section)

License

MIT License

Copyright (c) 2024 jade5638

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Citing

Please use the reference.bib file included to reference this work.

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References

[1] Jiwon Son, Chankyu Son, and Kwanjung Yee. 'A Novel Direct Optimization Framework for Hypersonic Waverider Inverse Design Methods'. In: Aerospace 9.7 (June 2022), p. 348. issn: 2226-4310. doi: 10.3390/aerospace9070348.

[2] Helmut Sobieczky, F Dougherty, and Kevin Jones. “Hypersonic Waverider Design from Given Shock Waves”. In: May 1990