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Video 1. Video Snippet of CFD Simulation

Within this post we will examine a basic design approach for the Kiel probe sizing. This probe is intended for total pressure measurement, necessary for airspeed calculations. From now on we will assume that airspeeds are well below 0.3 times the speed of sound, so it's safe to say that we are not facing any remarkable compressibility effects.  Air density is denoted with \(\rho\) and relative airspeed is \(V\).

Let's start with a review of the total pressure probe described in NACA-TN2530, where you can find the section view visible in the following figure. After CFD simulation with the NACA design we will propose and analyze a custom design.

Total pressure \(P_t=P_s+q+\rho gz\) is a sum of three terms. The first term corresponds to the static pressure, the second is the dynamic pressure \(q=\frac{1}{2}\rho V^2\) and the third accounts for gravity effects. If we assume that our flow is adiabatic and incompressible, then in our case the Bernoulli's principle holds quite well and along the streamlines \(V^2/2+gz+P_s/ \rho\ =constant\).

Figure 1. NACA-TN2530 Total Pressure Probe 

If we add, only to drop it immediately afterwards, the inviscid flow hypothesis, the total pressure does not depend on the geometry of the probe; every point inside the outer tube has the same total pressure. The probe is composed by two main parts: the external shield and the enclosed total pressure port. In the next figures, in the direction from upstream to downstream, you can see the inlet convergent conical nozzle, the cylindrical throat, the divergent nozzle section and the outlet holes. On the centerline of the inlet nozzle the total pressure tap is situated . The outlet is composed by 24 holes of 6.35mm diameter, with a total outlet area of 7.6e-4 \(m^2\). Inlet area is 5.1e-4 \(m^2\). The outlet section has 50% more area that inlet section, which helps minimize the downstream blockage effects.

Flow is axial, along the major tube axis. The mass is conserved across any section \(i\) of the probe of area \(A_i\) and mass flow rate \(\dot{m}=\rho VA_i\) is constant. Air enters the shield and accelerates until it reaches the maximum speed at the throat, where the minimum cross-section area is found, then decelerates through the divergent exit cone and finally exits the probe radially. Pressure is maximum at the inlet, then decreases to its minimum value at the throat and then the rises progressively up to the divergent section. To evaluate the design performance a CFD approach will be used. In Figure 2 you can find the 3D model of the NACA probe ready to be processed by Salome mesher. The cylindrical section and the trailing cone were added to have a smooth flow stream.

All the simulations are conducted in a 3D domain, but for better visual representation many results are presented as 2D slices. The intention behind these simulations is to better understand the behavior of the probe under different angles of attack, not to verify the NACA results. In case of doubt, it is better to run the simulation at the same conditions using two or more meshes. Usually a good solution doesn't change with the use of two well-designed meshes.

We simulated the probe with zero angle of attack and we obtained a total pressure value of 101340.1Pa. The actual free-stream total pressure is 101340.6Pa, so at 0 degrees the probe is performing well. Simulation related problems apart, since we don't know some dimensions or details of the NACA probe there will be a certain baseline difference between our results and those of NACA report. It is also worth noting that the magnitude of relative wind for all our simulations is 5 m/s.

The simulated probe is expected to work well up to about 40 degrees and this behaviour was verified with CFD. Probe measurements started to diverge from their ideal values at 35 degrees. You can find a limited subset of the results depicted in Figure 3.

Now that we are familiar with the probe design, let's address some issues. The position of the pressure tap is not arbitrary and it will have an impact on the probe overall performance. Referring to Figure 3, at high angles of attack a significant wake will appear, which also interacts wit the head of the probe. With that kind of effect in mind, we may prefer placing the pressure tap near the probe throat, in order to mitigate issues at the probe inlet. However, by inspection of table I of reference, "Tube 2-a" and "Tube 2-b" in particular, we see a contradicting indication. The table highlights the fact that, in terms of maximum angle of attack, the probe with the pressure port placed nearest the probe entry is performing better. The stated Mach number for the lab test is 0.26. In such a regime, our initial assumptions do not hold: the flow is compressible (Air flow accelerates inside the convergent section), hence we are reading a table that is not addressing our design approach. The behavior of the Kiel probe at low airspeed will be different, so we should validate by ourselves the pressure tap position impact at low speeds. Consider also the boundary layer, the more you retract the total pressure port inside the shield the more that tap is near the walls.

Another important aspect is the shape of the inlet. The simpler conical geometry seems perform worse, compared to curved inlets. The shape of the inlet modifies the velocity profile and a curved inlet will produce a smoother airspeed transition. With an elliptic nozzle, the area reduction gradient is greater at the inlet and progressively decreases to zero at the throat. In this way, the velocity gradient is kept low when the velocity magnitude is at maximum. For very low airspeed the probe can manifest measurement issues related to the boundary layer. That effect needs a dedicated analysis.

Other aspect to consider is the divergent section length. In this section the pressure should grow from its minimum value to the exit value. The smoother the transition is, the better is the pressure recovery. At the exit cone the air is flowing from a low pressure zone to a high pressure zone, hence separation of flow from the internal wall should be expected. Despite the fact we cannot use the Venturi tubes formulas for prediction of the pressure loss as a function of diverging cone angle, it's possible to observe that proposed angles for diverging cones in various design standards are between 5° and 15°. That range provides a valid initial estimate value for the area gradient in our probe design.

Regarding the exhaust holes, they should be placed in a position that minimizes the internal pressure variation at high angles of attack. It is expected that at high AoA, an airflow will be established between the holes placed upstream and the holes placed downstream and the impact of this flow on probe performance should be investigated.

In this article, we have familiarized with the total pressure probe and highlighted some key design points. In the next article we will proceed to present some results regarding the proposed BAD probe.