The intake design is arguably the most critical feature of the ramjet; jet engines can rely upon the compressor to raise the pressure of the air, while pulse-jet engines can utilise the pressure waves from each previous cycle to compress a fresh fuel-air charge. But a ramjet can only rely on the compression produced by the flow of air into the intake as this is all that is available. This pressure recovery can be quite modest at subsonic speeds but as the Mach number increases, very high ram pressure ratios are available for exploitation.

The supersonic intake as used by aircraft are often required to operate over a wide range of airspeeds and flight conditions, providing an engine with a uniform flow and a sufficient quantity of air to prevent the aerodynamic breakdown of the flow into and through the compressor stage. However a ramjet has the benefit that it can be designed to operate only at supersonic speeds, negating the need for by-pass ducts and rounding of the intake lip edges, and other requirements often seen in aircraft using these intakes.

The ideal ram pressure ratio across the intake is related to the Mach number by the following equation. ${\frac{P_i}{P_e}} = {\left(1+\frac{\gamma -1}{2}M^2_i\right)}^{\frac{-\gamma }{\gamma -1}} \qquad\qquad\qquad\qquad\qquad \left(1\right)$

In practice it is not possible to generate isentropic compression in supersonic flow conditions due to the presence of shock waves. The shock presents a sudden discontinuity within the flow where the pressure abruptly changes. Although this rise in pressure is beneficial, the associated turbulence means that full isentropic compression cannot be reached. The stronger the shock wave produced, the greater the losses encountered. By creating multiple weak shock waves, a more reasonably efficient intake can be designed.

The supersonic intake can take one of three configurations characterised by the location of the shock waves produced by the intake, these are: internal compression, external compression, and mixed-flow compression, shown here.

Internal Compression Intake

This intake design achieves compression through production of a series of internal oblique shock waves, with a terminal or ‘normal’ shock wave situated at the throat. This intake does require the use of variable geometry in order to position the shock waves and prevent air spilling around the engine.

At first glance, this intake supports the idea that one could simply design an intake much like a converging-diverging nozzle in reverse, so that compression could be generated  without the use of shock waves, and hence reduce losses. However the use of a fixed intake geometry (needed at the scale of engine considered here) means that during off-design conditions, shock waves will inevitably form at the throat fixing the mass flow rate through the throat, leading to air piling up in the intake and instantaneously leading to formation of a shock across the intake lip; with air spilling around the inlet in the subsonic flow aft of the shock wave.

As such this design has been rejected for this engine, however this may be revisited in future as the conditions within a wind tunnel can be readily fixed to provide adequate design conditions for this type of inlet, however the requirement for variable geometry to swallow the normal shock during the intake starting, may be an issue. Although this may be overcome with bleeding air at strategic locations within the nozzle.

External Compression Intake

This intake relies on one or more oblique shock waves created by a projection ahead of the inlet, these shock waves then intersect with the intake lip where upon a normal shock wave is generated. An external intake that relies only on the normal shock for compression of the airflow is called a ‘pitot’ intake. This represents the simplest form of supersonic intake and is easy to design, manufacture and short in length. Most external compression intakes rely on at least one oblique shock wave as the total pressure recovery increases with the number of shocks generated. However as will be shown later, this advantage may be lost due to the accuracy required of the intake geometry at the extremely small scales this engine will be produced at. Plotting the pressure ratio for a number of ideal intake types show the gains in performance which can be achieved for an increase in complexity. The ideal, isentropic ram pressure ratio (Eqn 1) is also shown to show the losses which occur to due the presence of shock waves.

Mixed Compression Intake

This, as the name suggests, uses both externally generated oblique shocks, reflected internal oblique shocks, and a normal shock wave at the throat region to provide compression. This is a complex design and is generally only utilised for flow regimes where the Mach number is greater than 2.5. Due to the internal shocks, it also requires variable geometries in the form of movable centre bodies, bleeding and by-pass doors to provide adequate control of the shock wave location. For similar reasons as the internal compression intake, this design has been rejected.

All the intake geometries discussed can be further classified by their form, either two-dimensional or axisymmetric. This relates to whether the intake geometry is rotated about an axis (axisymmetric) or projected along an orthogonal axis (two-dimensional intakes), as shown below. Axisymmetric intakes do have the advantage of a slight increase in weight to pressure recovery ratio, while two-dimensional intakes may prove to provide a better integration with the fuselage and some manufacturing benefits. Two-Dimensional Supersonic Flow Through Intake

Looking at the schematic below, shows an inlet with two shock waves followed by a normal shock wave terminating across the inlet lip, with the relevant stations highlighted. These will be used as the basis for determining the performance and geometry of an intake given some initial operating conditions. What follows will be a discussion of the relevant equations followed by the implementation used to carry this out. This analysis could be done by hand, however it is far easier to solve these equations using something like Python, Matlab, or even Excel, I have made the Matlab code available (link at the bottom).

If it is assumed that the upper limit to the performance of a supersonic intake is due to losses across shock waves, then an estimate of the performance of an intake geometry can be found. Starting with the inputs for a free stream Mach number, specific heat ratio, inlet height and ramp angles, the shock angle ${\vartheta_i}$, formed due to the presence of the wedge angle ${\delta_i}$ in a flow $M_i$, can be found $y =\ \frac{1}{M^2_i}+\left(\frac{1}{M^2_i}+\frac{\gamma +1}{2}-y\right){\left(\frac{y}{1-y}\right)}^{\frac{1}{2}}{\mathrm{tan} \delta \ } \qquad\qquad\qquad \left(2\right)$

This equation cannot be solved explicitly for $y={sin^2}\vartheta$ and requires an iterative approach to find a solution. Assuming ${sin^2}\vartheta=\frac{1}{M^2_i}$ on the right hand side, a new value of ${y}$ can be found, this is then used to update the right hand side until convergence.

Total pressure ratio across shock ${\frac{P_{ti}}{P_{te}}}={\left(\frac{\gamma +1}{2\gamma M^2_iy-\left(\gamma -1\right)}\right)}^{\frac{1}{\gamma -1}}{\left(\frac{(\gamma +1)M^2_iy}{2+(\gamma -1)M^2_iy}\right)}^{\frac{\gamma }{\gamma -1}} \qquad\qquad\qquad \left(3\right)$

Static pressure ratio across shock ${\frac{P_i}{P_e}}= \ \frac{2\gamma M^2_i y-(\gamma -1)}{\gamma +1} \qquad\qquad\qquad\qquad\qquad\qquad \left(4\right)$

Exit Mach number aft of shock wave $M_e= {\left[\frac{4+4\left(\gamma -1\right)M^2_iy+{(\gamma +1)}^2M^4_iy-4\gamma M^4_iy^2}{\left(2\gamma M^2_iy-\gamma +1\right)(2+\left(\gamma -1\right)M^2_iy)}\right]} ^{\frac{1}{2}} \qquad\qquad\qquad \left(5\right)$

Calculation Of Ramp Geometries For Multi-Shock Intake

The overall process I have used to estimate the intake performance and then find the required ramp geometries to ensure that all the shocks converge onto the lip of the intake, is shown in this flowchart.

The process starts by calculating all the shock angles and subsequent flow properties for the given number of ramp angles corresponding to the number shocks required. The next stage begins constructing the geometry of the ramps starting with defining the intersection of the normal shock wave with the ramp as the datum point at x = 0 and y = 0. This datum point is then used to calculate the position of the lip and the location of the preceding changes in ramp angles upstream of the normal shock. Note the signs in the trig equations, as the geometry is constructed from the zero, datum point and in a ‘right to left’ direction, you could of course allow the geometry to be constructed ‘left to right’ and into the positive domain. But I prefer thinking about things with the airflow coming from left to right and it feels more  aesthetically pleasing to me somehow. Diagram showing internal angles for the final shock-normal shock-ramp interaction and relevant equations, inset shows how triangle is formed between ramp and shocks.

Finding the required ramp geometry to ensure intersection of each shock with the cowl is simpler than it first appears and just requires some simple trigonometry to find angles and lengths of triangles constructed using the ramp angles and shock waves. Looking at the previous schematic showing the shock and ramp geometries, a triangle between the lip, ramp and the last oblique shock wave can be formed. The angles between the horizontal axis of the intake and the ramp (side s) together with the shock angle (side c) are all known, while the length of side a defined as the required inlet height and equivalent to the length of the normal shock, which forms a right angle with the ramp.

For the preceding shocks upstream the same process is used, but this time the new side a of the traingle formed between the ramp and shock waves is now defined using side from the previous shockwave. Diagram showing internal angles for the final shock-shock-ramp interaction and relevant equations, inset shows how triangle is formed between ramp and shocks.

This approach was transferred into a Matlab script and used to evaluate various ramp angles and number of shock waves for their performance. It should be noted that for any given Mach number there will be a maximum shock wave angle, beyond this no solution exists for a straight, oblique shock wave. Instead the shock wave detaches from the ramp and becomes curved, forming a bow shock, this manifests itself as a solution in the complex plane using the script I’ve linked to in this post. The script can produce solutions for multiple shock waves, I’ve tried it up to ten shocks at Mach 4.5 (see below), but this required very small changes in the ramp angles on the order of half a degree (note it will only display text data on the graph for five or fewer shock inlet designs). Ten shock, Mach 4 intake profile, with 0.25 degree changes in ramp angles! Not quite sure how practical such an inlet would be though.

Looking at something more sensible, two different ramp profiles are shown in the figures below, showing a two-shock and three-shock solution for a freestream Mach number of 2.5. It may not be immediately obvious at first sight, but the inlet height sets the scale for the rest of the intake geometry. Due to the small scales involved, small angle changes over just a few millimetres for an inlet height of 15 mm will be tricky to build given the equipment I have to hand, therefore the use of three shock intakes has been neglected for now and the intake modelling will go ahead based on a two shock geometry.  One thing that became apparent was choosing the ramp angles for a two-shock intake, it seemed as though I was blindly entering values and seeing what happened while struggling to see the bigger picture in the data I was getting. At first I started plugging values into the script and noting down the outputs but I soon started noticing a pattern in the results. So the script was modified to loop through a range of ramp angles for the second ramp while holding the first ramp angle fixed, with the loop terminated when the shock was predicted to become detached (in other words the solution gave a complex number result). Plotting the total pressure ratio across both shock waves for a number of fixed first ramp angles, then varying the angle of the second ramp, shows that as the shock becomes stronger the pressure ratio increases as expected, heading towards a locus point. However the gradient of the lines also increases with an increase in initial ramp angle, meaning that a large initial ramp angle together with a smaller increase in secondary ramp angle gives a higher overall pressure ratio than the opposite case of a shallower initial ramp angle followed by a larger increase in the secondary ramp. I still need to digest this graph and plot some more data to fully understand the processes at work here, but for now I chose a geometry that was slightly conservative as an initial starting point. Variation in static pressure ratio for increasing fixed first ramp angle (a – h) and varied secondary ramp angles. Red cross indicates the predicted design performance of the intake geometry shown below.

After some deliberation, and an effort to find a design point that was sufficiently far enough from generating a strong shock and subsequent detachment, while having a reasonable pressure recovery. This ramp geometry chosen is shown below, and the predicted pressure ratio is highlighted by the red cross on the graph previously. I haven’t included the pressure rise across the normal shock wave here, nor the pressure rise in the subsonic diffuser. I’m still playing about with the idea of integrating this script into a more comprehensive tool to run a full steady state stage analysis for the entire engine as it would be nice to tie this all together. More on this next time.

## One thought on “Ramjet Part 3 – Intake Geometry & Shock Waves”

1. JaeSan says: