by Richard Garcia, Director of Research, RRS.ORG

---

I would like to set the record straight on a common mistake I see with designing parabolic rocket nozzles.

The simplest and most common nozzle used in amateur rocketry is the simple cone. Most commonly the cone is a 15° half angle. But the simplicity of the cone comes at the cost of reduced performance and increased length. The next step up in complexity is a parabolic bell nozzle.

The parabolic bell nozzle design is a little more complicated and much more difficult to manufacture. Machining a parabolic shape out of metal on a lathe by hand is doable with a digital read-out (DRO) device but the process would likely be very tedious.  In this case, I would recommend looking into some sort of computer-controlled system like CNC or 3D printing. If you’re using an ablative composite like silica-phenolic then tuning a wood mandrel on a lathe is very doable. Beyond the parabolic nozzle the next step up in complexity and efficiency is using the method of characteristics to design a rocket nozzle. This method was pioneered by Gadicharla V.R. Rao. 

https://en.wikipedia.org/wiki/G._V._R._Rao

.These are sometimes called “Rao” nozzles. The parabolic nozzle is a decent approximation of a Rao nozzle, a method also proposed by Rao. Going further to get the last drop of performance would require a detailed computational fluid dynamics (CFD) based optimization.

I’m writing this report to set the record straight on a common mistake I see with determining the proper geometric and dimensional features of specific parabolic rocket nozzles. Almost all the amateur parabolic nozzles I’ve seen up to this point, including several college teams, and even my own early designs, made this mistake. It’s not really their fault though. The parabolic nozzle is commonly mention in rocket literature but rarely is one particular key fact needed to make them correctly explained. The two of the most prolific textbooks on the topic are, “Rocket Propulsion Elements” by Sutton and Biblarz, and “Modern Engineering for Design of Liquid-Propellant Rocket Engines” by Huzel and Huang. Both give their readers charts to help design the parabola and both also neglect to mention this one important fact.

That there is a problem at all will only become apparent when you try to solve for the exact equation of the parabola. If you read through the common literature (a task left to the reader) and go through the process you will see that the parabola is defined by two points and two angles. The two points are the beginning and the ends of the parabola. The beginning point is where the curvature at the throat, typically a circular arc, transitions into the parabola. The end point is the exit of the nozzle. The two angles, θ_n at the beginning of the parabola and θ_e at the exit are the angles between the centerline of the nozzle and lines tangent to the parabola at the two points. These angles are normally found by using graphs given in the literature. These graphs are the angles vs expansion area ratio as shown below. Note that they present the length of the bell nozzle as a percentage, L_f, of the length of an equivalent conical nozzle with the same nozzle expansion area ratio and a 15° half angle.

---

Footnote about the θ_n and θ_e vs expansion area ratio graphs.

I’d like to note that the graph I shared is the in the commonly presented form. However, if you look at the original paper on parabolic nozzles “Approximation of Optimum Thrust Nozzle Contour” by G.V.R. Rao in 1960” you will find the information presented differently. Instead of theta versus epsilon at various percentages of nozzle lengths given in terms of the percentage of an equivalent 15° conical nozzle length of the same expansion ratio (epsilon), the axes of graph are in terms of the ratios r_e/r_t versus L_n/r_t given at various values of θ_n and θ_e. These lines are straight in comparison to the curved lines in the graph above. The more common graphs are supposed to have been derived from the originals. 

However, I have not gone through the math myself, but, I am interested in what the process and results would be. In turn, the originals are supposed to have been derived from full calculations of the Rao method. Again, something I have not tried replicating but it is something I would be interested in seeing the details of. 

Another thing worth mentioning is that I recall reading online that someone had tried to reproduce these results using the method of characteristics. (Unfortunately, I cannot recall exactly where I read this.) The original graphs were based on using a fixed specific heat ratio (gamma) of γ=1.23 and the original paper stated “A different value of γ does not appear to appreciably change the nozzle contour when the area ratio and length are prescribed.” however the testimony I found online stated that they found that the gamma value did significantly affect the angles. I am in doubt over this anonymous internet testimony, but it is something I’m interested in exploring in more detail.  If you ever investigate this yourself, dear reader, be sure to send us what you discover.

---

In total, you will have six parameters to determine, at point 1 you will have x₁, y₁, and θ_n. At point 2, you will have x₂, y₂, and θ_e. The obvious route from here would be to take the equation of a parabola (more precisely a square root since the x-axis will be our chamber centerline. But hey, a square root is just a parabola on its side), and these six parameters and try to solve for the unknowns in the equation.

This is where you run into the problem. The parabola is over-defined! You can’t solve for a parabola that meets all six criteria. It is impossible. Five of the defining parameters can be correct, but at least one will always be wrong. This is where most people, after considerable effort and eventually the banging of one’s head against a wall, will give up and let one, or more, of the 6 parameters to be wrong, usually by letting the starting angle of the parabola to be wrong.

When plotting a nozzle contour most put the centerline of the nozzle along the x-axis. The mistake everyone makes, including myself, is to make the natural assumption that this parabola is aligned with the nozzle’s same x and y axis in the same way as all of the parabolas in your algebra classes. That is to say the centerline of the parabola is around the x axis, the same axis of axial symmetry as the nozzle. But what everyone misses, including in the common literature, is that the parabola should be canted! That is to say, it should be rotated by some angle in relation to the x and y axis.

The way I found this fact out is by being a prolific reader. I’ve spent a lot of time gathering papers and books on rockets. I’ve only ever found one source that explicitly states the nature of these parabolasand that is in “Liquid Rocket Engine Nozzles NASA SP-8120” It simply refers the them as a “canted-parabola contour” and offers no further explanation. But that clue was enough for me to figure out what was going on. Adding the rotation of parabola adds a degree of freedom in the math. This allows for the equations to be solved with all the required constraints.

---

* Footnote on the NASA SP-8000 publication.

By the way, the NASA SP-8000 series of publications is a fantastic resource for the aspiring rocket scientist. Back in the 1970’s, they tried to write a comprehensive review of the state of the art in rockets and spaceflight. I remember reading somewhere that they fell short of their goal, since there was too much know-how to catalog. But what they did manage to publish is great. Whenever diving into an unfamiliar design topic, I’ll read the relevant SP-8000 article on it first as a good overview.  Some topics may be more outdated than others, but much of it is still great and relevant today.

---

When trying figure this out, I went digging and found the original paper on parabolic nozzles. “Approximation of Optimum Thrust Nozzle Contour, by G.V.R. Rao, 1960” I had hoped it would directly state the parabola should be canted, but it doesn’t. However, it does give enough information to deduce this conclusion.  In the short paper, its one page, Rao describes “a simple geometric construction for the parabolic contour.” It is something that would have been more familiar with the designers and draftsmen of that time. Today, we are more likely use a series of points and connect them with a spline curve, or use mathematical equations in CAD software to draw similar curves. But back then, designers would have to do it by hand on paper. The method suggested by Rao is called the “tangent method” of drawing a parabola. As the name suggests it involves drawing a series of lines that are tangent to the parabola.

Since we are interested in the mathematical equations, I won’t be describing that method any further.  But it is plain to see from figures that the method does not directly depend upon the x, and y axis. The contour is simply plotted on the x-y plane and then the contour is revolved around to x-axis to create the three-dimensional nozzle. Since we cannot fit the desired parabola with the normal parabolic equations, we are left with the question: 

How does one mathematically rotate the equation of a parabola?

Using parametric equations is how we will be able to create aparabola rotated on the x-y plane. Parametric equations also make some of the math easier when solving for circular arcs that are used in the rest of the nozzle contour. 

First let’s start with vertex form a parabola.

To parameterize this, let us set:

And solve for x and y.

This pair of equations is now a parametric form of a parabola. But it is just a regular parabola and we need to rotate it. Recall from your study of trigonometry, that the formulas for rotating a point around the origin by angle θ are:

We can combine equations 3 and 4 by substituting the parametric equations into the place of x₀ and y₀. Doing this, we get:

From this, we can get:

Note that for our use case, because the rotation angle θ is a constant, h∙cosθ-k∙sinθ and h∙sinθ+k∙cosθ are also constants and can be replaced by single terms, c₁ and c₂ respectively. But we can name c₁ and c₂ whatever we like so I shall name them h and k respectively. (Yes, I know this seems like an odd maneuver, but it works fine for us and makes our equations a bit simpler.) With a little rearranging this then gives us the parametric form of a rotated parabola:

Now we are ready to solve for the parabolic nozzle contour. Before we do this for the above equations, we shall discus the rest of the contour as well.  We will use a piecewise set of equations to describe the entire contour. For each piece we will need to solve for parameters and the split points between the different pieces of the function. We will go through each of them in order.  How to size a thrust chamber is beyond the scope of this paper and is an exercise left to the reader. You should already have calculated the radius of the chamber, throat, and nozzle exit, r_c, r_t, and r_e, respectively and the length of the chamber and nozzle, L_c, and L_n respectively.  The figure below shows a typical form of a thrust chamber. You’ll note that I’ve used z and r instead of x and y. This is a notation that I adopted early in my work writing a code for thrust chamber design. I figured z is up and rockets go up, so that’s where I’ll put the axis of symmetry of the chamber. I put the throat at z=0 for convenience. I decided to keep my notation in this report, instead of adopting something more conventional, for my own convenience of not having to rewrite all of my equations.

The first part of our piecewise function f₁ is the chamber section representing by red in the figure. f₁ is a simple horizontal line with the equation. 

f₁ Chamber Section

f₂ Concave Curved Converging Section

The exact converging section shape typically does not strongly affect engine performance. So, the designer has a lot of freedom here butthe following geometric convention is common. Three parts are used, f₂ (blue in figure) is a concave circular arc, followed by f₃(green in the figure) is sloped line and f₄ (orange in the figure) is convex circular arc. I’ve found that using the parametric form of a circle makes the math easier. The parametric form of this circular arcin function 2 is the following. 

Were r_f2 is the radius of curvature for function 2. 

f₃ Linear Converging Section

The third piece is a straight line and we will use the point slope formof a line.  The designer provides the converging angle and a typical rule of thumb value is 30°. Using a bit of trigonometry, we can see that the slope will be tan(-θ_c). For the point we will use (z₄, r₄) which is where f₃ intersects f₄.

f₄ Convex Curved Converging Throat Section

The fourth piece is another circular arc. The radius of curvature is typically given by the designer as a multiple of the throat radius. A typical a rule of thumb value of 1.5 time the throat radius is commonly used. We will call this scalar r_c2. Again, we will use the parametric form.

About solving for the unknowns upstream of the nozzle throat

Solving the converging section is where we will notice a hiccup (minor problem). Say that you have a large throat and a small contraction ratio, A_c/A_t. This could mean the point where f₄ meets the given converging angle, θ_c, of f₃, could be equal to or larger than the chamber radius. A mathematical solution that uses the given parameters would fail to produce real geometry. A second problem could arise even when a valid solution does exist because the result could be an undesirably sharp radius if the point is too close to the chamber radius.

So, we have two solution paths. For the conventional case, we don’t run into the previously mentioned hiccup. Here, functions f₁, f₃, and f₄ are sufficiently defined by the given parameters mentioned so far. But to define where the function f₂ is located, we will need an additional parameter.  In my work, I’ve decided to do this by setting the point r₃ to be a percentage of the vertical distance between the chamber radius and r₄. Where the concave f₂ curve meets the linear converging portion, f₃, is the location of r₃. Where f₃ meets the convex f₄ curve is r₄. My default value is 50% of the way between r₄ and r_c. I’ve named that parameter f_2vp and its value should be between 0 and 1. The letters used mean: function 2, vertical percentage. (You are probably asking yourself “Why not just define r_f2 in the same way as r_f4, with a scaler multiple of r_t?” Well good question. That’s the way I started doing it. It’s been a while but if I remember correctly, I found that it was hard to pick a default value that worked for a large variety of engines and if picked poorly results were either undesirable or didn’t solve. So, I came up with the above method which I found much more satisfactory.)

For the second solution path an alternat method for defining the converging section must be used. It will be convenient to set the radius of both of the concave, f₂, and convex, f₄, sections equal. Then we define the curved converging sections, f₂ and f₄, to each take up a percentage of the radial distance between the throat and the chamber radius. The linear section is the remaining percentage. I’ve named this parameter f_{2vp_alt}. The name means: function 2, vertical percentage alternate method. Since there are 3 functions that take up this vertical distance, and two of them are equal the value for f_{2vp_alt} must be between 0 and 0.5. If it is 0 that means that the linear portion takes up all the vertical distance and the curved portions would disappear. If it is 0.5 then the linear portion would disappear. The default vale I use is 1/3. This means that f₂ takes up 1/3 of the distance, and f₄ takes up 1/3 of the distance and the linear section, f₃, takes up the remaining 1/3. It makes for decent looking converging contours.

This will always be solvable with real geometry, but means that theeither the converging angle or the radius of curvature for the converging portions cannot be pre-defined and is instead solved for. At first, I tried to let the converging angle, θ_c be derived. However sometimes the solution would be a very shallow angle creating an overly long converging section. Instead, I found better results by keeping the designer given θ_c and instead solve for the radius of r_f2and r_f4, which would still be kept equal. I was resistant to this at first because it would remove the ability to use the rule of thumb value for r_f4, or any specified value for that matter. This will only work if the assumption that the designer will provide reasonable values for θ_cis met. To summarize, the primary method lets the radius of function 2, r_f2, be an unknown to be solved for, and the converging angle of f₃is given and the radius of f₄ is given. With the alternate method the converging angle is given along with the vertical space the functions occupy. This leaves the radiuses of curvature r_f2 and r_f4, which are equal, to be solved fore.

When I have designed thrust chambers, I use the first method and then only switch to the second method if the first method does not work. The remaining question is when to switch to avoid a sharp corner. My solution was to specify that the concave radius, f₂, take up a minimum percentage of the vertical distance between the throatand chamber radius. My default value for this is 15% but this was somewhat arbitrary. You can also avoid the second method by increasing the thrust chamber contraction area ratio if possible.

Converging section solution, Method 1

The following is a mathematical solution for the converging section when using the regular method for the converging section described above. From f₄ we need to solve for z₄ and its corresponding radial point r₄. For that we need to find when the circular arc is at the givenconverging angle. This is when its derivative is equal to the slope of f₃, the linear section, which is tan(-θ_c). We will take the derivative ofthe parametric equations of function 4. In case you forgot from your calculus class the derivative of a parametric equation is found by taking the derivative of each equation separately with respect to the parametric independent variable, in this case t, which will give us the equations for and . Then you divide the equation for by the equation for and cancels out and you are left with an equation for . Using this method on the equations 11.1 and 11.2 for function 4 we get

We solve this at t₄ by setting the derivative equal to tan(-θ_c) which gives us

Solving for t₄ we get

Note that this solution is periotic so it mathematically solves with any added to it. Through quick trial and error, we find a value n=1will put point (z₄, r₄) in the correct quadrant on the x-y plane. We then plug t₄ into f₄ to get the z₄ and r₄ coordinates. 

Now that we know that value of r₄ we can now check if the criteria needed for switching to the alternate method is met. If not, we can carry on to find r₃. Recalling our assertion that r₃ shall be some given fraction of the vertical distance between r₄ and r_c we see that it would be given by the following equation.

Next, we plug r₃ into the f₃ function [eq 10] and solve to get z₃.

Now all that remains to be solved is z₂ and the radius of curvature, r_f2, for the concaved converging function f₂. If we take f₂ [equation 9] and plug in the known point z₃, and r_3, which both occur at t₃, we get the following:

We can see that the there are three unknowns z₂, r_f2, and t₃, but there are only two equations. So, we need a third equation to solve this system. We know the derivative of this function is the same as for the function f₄, [equation12] since they are both the equations of a circle. And we know the value of the derivative at this intersection pointfrom the slope of f₃, the linear converging section, which is tan(-θ_c).Setting equation 12 to tan(-θ_c) gives us the third equation we need. 

We then solve this system of three equations, 18.1, 18.2, and 19 for the three unknows z₂, r_f2, and t₃. This gives us:

We now have all the parameters and split points needed for each of the function to fully define the converging section. Equation 20 like in equation 14 also has a periodic portion of the solution, but here n=0.

Converging section solution, Method 2 

With the second method everything is the same as the first method up to calculating t₄ [equation 14] then the methods diverge. In method 2, after calculating t₄ the next things to calculate are r₃ and r₄. This method defines the percentage of the radial distance between the throat and the chamber that the two curved converging functions, f₂and f₄, take up. This definition allows us to create, by reasoning, the equations for r₃ and r₄. They are:

Next the radius r_f4 can be found by taking f_4r [equation 11.2] at t₄ when it will equal r₄,

and then solving for r_f4 and we get 

Then we set r_f2=r_f4. For the remaining unknowns that define the geometry, we can use the same math from the other method. Those remaining unknowns are z₄, z₃, t₃, z₂ and are defined by equations 15.1, 17, 20, and 22, respectively.

f₅ Diverging Throat Section

For the diverging section function 5, magenta in the figure, is again in the parametric form of a circle.

The radius of this function, r_f5, is the diverging radius of the nozzle throat, it is given by the designer and is usually a rule of thumb value of 0.382 times the throat radius. But we still need to find the point z₅, and r₅. To do this we first note that the derivative of f₅ is thesame as for f₂ and f₄ given in equation 12. We know what the angle of the tangent should be at z₅, where f₅ meets the parabolic portion. It should be the θ_n mention above and gotten from the θ_n vs expansion area ratio graph. From the same trigonometry used for the slope of f₃we can set the derivative [eq 12] equal to tan(θ_n) at t₅. Then we solve for the parameter t₅ which gives us:

Like in equation 14 this also has a periodic portion of the solution, but here n=0. We then plug t₅ into f₅ [eq 27] to get z₅ and r₅.

f₆ Parabolic Nozzle Section

The parabolic section of the nozzle, function 6, is yellow in the figure. As derived above in equation 7 we use a pair of parametric equations for a parabola with a rotation applied. I like to use ψinstead of θ here for the sake of symbolic variety.

Solving for the unknowns downstream of the nozzle throat 

All that’s left is to solve for the remaining six unknowns in function 6. These unknowns are, t₅, t₆, m, h k and ψ. Note that the parameter variable t is unique for each section, so it’s technically a different t₅ than for the t₅ for function 5.  We know six key data points on the parabola. We know the starting coordinates z₅, and r₅ from our prior work. We know the end coordinates z₆, r₆ from the exit radius and the length of the nozzle provided by the designer. If we substitute these coordinates into function 6, we would have the following four equations:

But it’s not possible to solve these four equations for six unknowns. We need at least two more equations. We know the slope at the beginning and end of the parabola from θ_n and θ_e which we got from the θ_n and θ_e vs expansion area ratio graphs. This means we can use the derivative of function 6 to give us the last two equations. To find the derivative function first we take the derivative of each parametric equation with respect to the parameter variable, t.

And then we divide by the equation for and we get:

Taking this function at t₅ and t₆ and using trigonometry to find the slope of the tangent lines at beginning and end of the parabola, we see that the derivative should be tan(θ_n) and tan(θ_e) respectively. This gets us last two equations we need for a system of six equations to solve for our six unknowns. 

Now all we have to do is simultaneously solve the system. Here are the six equations fully written out.

As far as I can tell this can’t be solved symbolically. I’ve tried several methods including a few computer algebra systems, but always failed. (Give it a try yourself and let us know what you come up with.) 

Instead, I solve the system of equations 41 through 46 numerically for m, h, k, phi, t₅ and t₆. This can easily be done with a calculator like the Texas Instruments, TI-89. But the way I solve this and the rest of the calculations is with a piece of custom software I wrote. My programing language of choice for doing this has been Octave, but it is possible to do with any Turing-complete programming language.  Once you’ve done this and solved for those unknowns you now have a fully mathematically defined parabolic nozzle contour.  One final note worth mentioning is that I typically use these equations to find the value r at a particular z location. If this z is in one of the parametric sections, then I first solve the equation f_z(t)=z for t. Then I take the value for t and use it to solve f_r(t)=r for the value of r.

Hopefully, that clears up any confusion and set the record … canted.

---

List of symbols

A_c​ = Cross sectional area of the thrust chamber

A_c/A_t​ = Thrust chamber contraction area ratio

A_e​ = Cross sectional area of the nozzle exit

A_t​ = Cross sectional area of the nozzle throat chamber

c₁-c₂​ = Numerical constants

f₁-f₆​ = Parts of the piecewise function that defines the nozzle contour

f_2vp​ = Fraction of the vertical distance between r₄ and r_c occupied by f₂

f_{2vp_alt}​ = Fraction of the vertical distance between r_t and r_c occupied each by f₂ and f₄

f_r1-f_r6​ = Radial portion of the of the piecewise functions that defines the nozzle contour

f_z1-f_z6​ = Axial portion of the of the piecewise functions that defines the nozzle contour

h​ = Horizontal translation of parabola

k​ = Vertical, or radial, translation of parabola

L_c​ = Length of the thrust chamber, measured from the injector to the nozzle throat

L_f​ = Length of a bell nozzle given as a percentage of an equivalent 15° half angle conical nozzle

L_n​ = Length of the nozzle, measured from the nozzle throat to the nozzle exit

m​ = Scalar that is the stretch factor of a parabola 

n​ = scalar for periodic solution, n can mean any whole number

r​ = Radial coordinate

r₁ – r₆​ = Radial coordinate of points 1 through 6, see figure for location of points

r_c​ = Radius of the cylindrical portion of the thrust chamber

r_c1​ = Scalier for the concave converging radius r_f2 = r_c1 ∙ r_t

r_c2​ = Scalier for the convex converging radius r_f4 = r_c2 ∙ r_t rule of thumb value of 1.5

r_d​ = Scalier for the diverging arc radius r_f5 = r_d ∙ r_t rule of thumb value is 0.382

r_e​ = Radius of the nozzle exit

r_f2​ = Radius of curvature for function 2 

r_f4​ = Radius of curvature for function 4

r_f5​ = Radius of curvature for function 5

r_t​ = Radius of the nozzle throat

t​ = Parametric independent variable, it is unique for each pair of parametric equations

t₁ – t₆​ = Parametric independent variable at points 1 through 6, and unique for each parametric function

x​ = Horizontal coordinate

y​ = Vertical coordinate

z​ = Coordinate perpendicular to the x-y plane

z₁ – z₆​ = Axial coordinate of points 1 through 6, see figure for location of points

ε​ = Nozzle expansion area ratio, A_t/A_e

θ​ = Rotation angle of a parametric parabola

θ_c​ = Converging angle of the linear converging section, function 3

θ_e​ = Angle between the nozzle centerline and parabola tangent line at the the nozzle exit

θ_n​ = Angle between the nozzle centerline and parabola tangent line at the beginning of the parabola

ψ​ = Rotation angle of the parabola of the nozzle contour

---

For questions and comments:

research @ rrs.org