Supersonic Wind Tunnel Part 3 – Flow Analysis

I have previously covered some of the analysis used to design the throat section of the wind tunnel, this time I shall cover some of the design used for the rest of the tunnel design. The idea was to use a pressure vessel coupled to a large diameter ball valve, so that large volumes of high pressure air can be dumped through the tunnel. The contours of the wind tunnel must be designed in such a way that air flowing from the pressure vessel flows into the test section free of turbulence and with a constant velocity profile across the section. It is this region which is of interest here.

The wind tunnel design shown here has already gone through several iterations and this is the most up to date version. I can’t foresee many significant changes occurring in the future, although there may be some minor changes to things like the delivery pipe diameter and length, which may vary by a few millimetres during construction.

Wind tunnel profile

Due to the turbulence and high flow speeds formed as air flows from the tank and through the valve, a settling or ‘stilling’ chamber is needed to slow down the expanding air from the tank, remove turbulence, and provide a uniform inlet flow at a constant pressure into the test section. The desired flow speeds can be attained through careful duct design while any turbulence is usually removed via mesh screens, honeycomb panels, or other flow straighteners. The typical flow speed through a settling chamber should be less than 30 m.s-1 to ensure a uniform velocity profile, while a lower limit of 3 m.s-1 is important to negate convection currents forming due to the temperature drop of the expanding air from the tank [1].

A simple cone diffuser was designed to link the delivery air to the settling chamber, the angle of the duct was set to be 10° to prevent flow separating from the diffuser walls for the range of operating pressures to be used (125 – 20 PSI). Separation of the flow from the diffuser walls can lead to both spurious turbulence and an increase in flow speed, both of which are.

The convergent duct which leads to the nozzle throat and accelerates the air in the settling chamber to Mach 1 at the nozzle throat, takes the form of a curve based on the following equation [2]. This curve is also used to transition the circular settling chamber into the rectangular cross-section used for the tunnel. Where x is the axial distance along the duct, y is the height of the duct, and L is the length of the convergent duct required, D_1 is the diameter of the settling chamber, and A_{1,2} are the areas of the delivery pipe and settling chamber respectively.


 y = \frac{D_{1}}{2\left (1+ \frac{x}{L}\left [ \sqrt{\frac{A_{1}}{A_{2}}} -1 \right ] \right )}  \qquad\qquad\qquad\qquad\qquad \left(1\right)


Convergent nozzle profile used based on Equation (1)

In a previous post, an analysis was carried out on the supersonic flow through the nozzle in order to design the contours of the nozzle wall section. However, this was based purely on the assumption that supersonic properties are constant along a line known as a ‘characteristic curve’ and gave no insight into the flow properties, other than Mach number. So another method is required in which to predict the flow properties, both subsonic and supersonic through the wind tunnel.

I have been developing a one-dimensional CFD code over the past few years which can calculate unsteady subsonic and supersonic flows. This has some advantages over a more traditional CFD approach in that it allows me to run less computationally expensive calculations on a desktop computer while retaining some degree of accuracy. Although a two or three dimensional CFD analysis would give much greater accuracy in resolving turbulent flow, the requirement for extremely fine meshes to capture shockwaves and boundary layers, leading to very high cell counts. When coupled with dealing with changes in flow behaviour over time (unsteady), this means that high performance computing is needed, often far exceeding that available on most desktop computers, even my own desktop with an i7-8700K 6 core processor and 64Gb of Ram would take far too long to compute an unsteady CFD solution!

Honeycomb flow straightening panel

Although the main drawback of a one-dimensional flow analysis is the assumption that the flow properties at any point are constant across the plane of the flow; this isn’t a too much of a problem as the settling chamber is designed to remove any significant turbulence that forms. I also plan to include a honeycomb panel aft of the ball valve which should serve to prevent the formation of larger eddies within the delivery pipe as the valve is opened.

To carry out the flow simulation, a set of xy coordinates of the upper and lower wind tunnel wall profile was needed. This was done using Excel see figure 5, the coordinates can then be saved as a text file and imported into Matlab easily. I shall cover the details of the flow solver I developed in another post as it would be far too complicated to go through here. However, the program divides the wind tunnel profile into a large number of ‘slices’, it then calculates the flow properties at each slice based on a set of initial conditions before advancing in time and calculating the new flow state at each slice. This allows to see how things like density, pressure, temperatures etc, propagate through things like rocket nozzles, ducts, and even things like pulsejets, which rely on high frequency pressure changes at high temperatures and flow speeds. Being a one dimensional simulation, it does not show things like shockwaves.

A screen grab of the spreadsheet used to quickly change and generate profile coordinates during the design process

The simulation was started with the air in the tunnel at standard atmospheric conditions and the face of the delivery pipe set to 125 PSI to simulate a sudden opening of the ball valve. I was interested in how long it would take for supersonic flow to develop through the nozzle and to see how long it took to reach steady-state operating conditions.

The initial outputs from the simulation look to confirm the nozzle design I carried out in this post, with a predicted Mach number of just over 2.5 in the test section (see the graph below). Although the solver does not account for any losses due to boundary layers, it does allow for an estimate for the effect of wall friction on the flow, but this was not included here due to having no empirical data. I expect that actual wall effects would slowly reduce the Mach number at an almost constant rate along the test section. However an estimate of the theoretical boundary layer expected at the mid point of the test section[3],  gives a thickness of only ~0.13 mm at the mid-point along the test section, as such I expect the Mach number to be closer to 2.5 during operation but there shouldn’t be any significant drop in the flow speed.

Steady State Mach Number Through Tunnel

As the unsteady flow through the settling chamber was my primary concern, the velocity of the flow at entrance to the diffuser, settling chamber mid-point, and the start of the convergent section (shown on plot), was plotted over time to highlight the changing flow conditions during tunnel operation. There appears to be an initial, very large pressure rise into the chamber followed by some damped periodic oscillations, as the flow changes direction, before settling out to give a flow speed into the nozzle of ~12 m.s-1 into the convergent nozzle, matching the design specifications.

Unsteady velocity of the air flow into, through, and out of the settling chamber. (Excel’s new graphing tools look very shiny, not sure if I like them or not, this dark scheme seemed to highlight the three lines the best)

To understand the reasons why it is best to look at the unsteady pressures as the valve is opened. The following video shows contours of pressure, normalised by atmospheric pressure, as the compressed air leaves the tank and begins expanding down the wind tunnel. We can now begin to understand the reasons for the dynamic velocity behaviour, where the flow slows down, stops, and flows back towards the pressure vessel.

As the high pressure air in the tank begins to expand along the delivery pipe and into the settling chamber, there is a significant pressure drop and the flow quickly accelerates reaching sonic conditions. As the flow ‘chokes’ at the diffuser the mass flow rate becomes fixed, however the flow is also choked in the nozzle throat upstream and as this has a much smaller cross-sectional area where the mass flow rate is much lower. Therefore, there is a significant pressure rise inside the settling chamber, reducing the pressure ratio between the tank and the diffuser leading to flow reversal, as the pressure momentarily exceeds the tank pressure and the air begins to flow backwards. When the pressure inside the settling chamber has dropped below the tank pressure, the cycle starts again giving rise to the velocity fluctuations seen in the graph above and the video blow.

The pressure fluctuations have little effect on the flow upstream of the nozzle, as once choked flow occurs throughout the test section the Mach number does not alter significantly, see video below.

The overall predicated performance appears to be promising with the desired Mach number in the test section being achieved together with an adequate flow speed through the settling chamber. The flow appears to reach steady-state after 0.1 seconds, given my run time will be around 2.5 seconds in total, I think that’s acceptable. Although the large transitory pressure rise in the settling chamber means that some careful consideration will be required, as I was going to originally use fibreglass and epoxy to fabricate the tunnel sections. In all likelihood I probably will still use composites to build this section albeit with some extra reinforcement using carbon fibre tows.



[1] Pope, A. and Goin K., “High-Speed Wind Tunnel Testing“, Wiley, New York, 1965

[2] Seddon, J and Goldsmith, E, “Intake Aerodynamics, Second Edition“, AIAA, 1999

[3] R Ladenburg, “On Laminar and Turbulent Boundary Layer in Supersonic Flow“, 1949


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