An Introduction to Shock Waves

When you think of shock waves, I would wager that you picture a supersonic jet zooming past overhead. Or maybe you have experienced the famous (or infamous) “sonic boom” that accompanies shock waves attached to airplane engines. The engineering challenges associated with the often-troublesome behavior of shock waves is present in all scales, from carefully designing the bodywork of the aforementioned fighter jets, to the equally intricate details of flow passages and blade design in turbomachinery. The first step in taking into account the effect of shock waves is to understand what they are. In this post we will be reviewing a short introduction into what shock waves are and a few applications where they might be relevant.

Figure 1: Schlieren image showing the shock waves of a supersonic jet
Figure 1: Schlieren image showing the shock waves of a supersonic jet. Source

What are shock waves?

Shockwaves are non-isentropic pressure perturbations of finite amplitude and from the second law of thermodynamics we can say that shockwaves only form when the Mach number of the flow is larger than 1. We can distinguish between normal shocks and oblique shocks. In normal shocks, total temperature is constant across the shock, total pressure decreases and static temperature and pressure both increase. Across oblique shocks, flow direction changes in addition to pressure rise and velocity decrease.

Weak shocks are nearly isentropic, therefore there are smaller overall losses when flow goes through multiple oblique shocks rather than a single normal shock. Engineers can use this property to their advantage. Since oblique shocks turn supersonic flow, multiple oblique shocks are used to achieve specified diffusion with lower losses than normal shocks. This is especially apparent in the example of intake geometries of aircraft engines, where often local supersonic flow must be slowed down before entering the compression stages.

Let’s take a closer look at shock waves in blades. As the tangential blade Mach number increases, the stage total temperature and pressure ratio increase. The appearance of strong shocks at the blade tip as well as vibratory stresses that may reach structural limitations pose a limit to the increase of the tangential blade Mach number. In order to avoid these strong bow shocks, engineers design blade tip section to be made thin with typical relative tip Mach numbers of 1.2 to 1.7. This helps reduce the drag from the blade and keep a cleaner flow for the downstream stages while also ensuring the extended life of the engine by reducing potential destructive vibrations that can result from strong bow shocks.

Figure 2 Interferogram of the shock wave boundary layer interaction in the mid-section of a turbine rotor blade
Figure 2: Interferogram of the shock wave/ boundary layer interaction in the mid-section of a turbine rotor blade. Source

In the design stage it is important to understand where one might have supersonic flow in the flow passage. Turbomachinery design and analysis tools such as AxSTREAM can help with this. For example, during preliminary design stages of turbomachinery, from radial inflow turbines to axial turbines and everything in between, users can filter solutions by selecting their preferred range of Mach numbers. This allows avoiding supersonic flow and potential shock waves in flow passages which would generally result in additional losses. After this, blade profiling and further analysis can help to avoid shocks in unwanted locations. Shown below is the user interface for AxSTREAM that allows users to visualize the distribution of the Mach number of the flow along the flow passage and therefore assess areas where shockwaves might appear.

Figure 3 Meridional view of axial turbine showing coloured distribution of Mach number
Figure 3: Meridional view of axial turbine showing colored distribution of Mach number

Where else can we find shock waves?

Shock waves are an important phenomenon to take into account when designing the combustor of a gas turbine engine. Shock waves can increase viscous losses through boundary layer interaction. As M increases from low values, viscous effects will diminish as 1/M2, but once high Mach numbers are reached the viscous losses tend to increase drastically. In designing a combustor, there is a need to slow down the fuel and air mixture in order to achieve full mixing over a shorter distance. In typical applications, the Mach number of the flow leaving the compressor is kept approximately between 0.1 to 0.5. This usually necessitates the use of a diffusor between the compressor and combustor in order to achieve the required flow speed of 10-20 m/s.

Let’s take a quick look at shock waves in nozzles of aircraft engines. We’ll specifically look at the most likely scenario for shock waves to occur, which is for supersonic flow in convergent-divergent (CD) nozzles. When the exit pressure is small enough, a normal shock will occur in the divergent section of the nozzle. An isentropic supersonic flow exists between the throat and shock while downstream of the shock there is an isentropic subsonic flow. There is therefore an increase in entropy across the normal shock, as we would expect. This shock occurs because the exit pressure of the nozzle is too high to correspond to the isentropic solution for the given area. The shock position is determined by the increase in static pressure across the wave as well as the fact that the static pressure in the divergent part of the subsonic flow behind the shock must be just right to give the correct exit pressure. The shockwave moves further downstream towards the exit of the nozzle as the pressure is further reduced, right up until the shock stands directly at the exit.

Figure 4: Shock wave arrangements in CD nozzles
Figure 4: Shock wave arrangements in CD nozzles. Source

Reducing the exit pressure further results in supersonic external flow. There can exist three cases. In the overexpanded case, the back pressure is still higher than the isentropic pressure at the exit. This means that the flow must be compressed to reach the back pressure, resulting in oblique shock waves. In the second case, where back pressure and exit pressure are equal, the exit flow would be straight and smooth since there is no pressure mismatch. In the third case, the flow is underexpanded. The flow must expand to reach back pressure resulting in expansion waves. There exists only one allowable back pressure for supersonic flow for any given nozzle geometry. For this reason, variable geometry nozzles are ideal but most often in the interest of simplicity and reduced costs, fixed geometries are used. In either case, a lot of careful thought needs to go into designing not only the nozzle but all aspects of an engine where shock waves might occur. Predicting their appearance will allow to design to avoid any destructive behavior they might exhibit.

Figure 5: Mach disks or "shock diamonds" formed at nozzle exit
Figure 5: Mach disks or “shock diamonds” formed at nozzle exit. Source

We have only taken a very quick look into what shock waves are and where they can appear. As you can see, shock waves appear in all aspects of turbomachinery and are a very important aspect to consider. With design tools such as AxSTREAM, their effects can be predicted and designed for far easier than in the past. Shock waves are an ever-present design challenge for engineers as aircraft chase faster speeds and turbomachines chase higher efficiency, so let’s use the tools at our disposal to reach all our design goals!

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