SYMPOSIUM ON UNDERGRADUATE RESEARCH EXPERIENCES

Investigation of the Modified Cosine Equation for Angle Calibrations of X-array Hot Wires in Transonic Air Flow

Sephir Hamilton¹ and Jack Seiner²

Mechanical and Aeronautical Engineering

Abstract (a complete copy of the experiment may be found on the web at http://www.clarkson.edu/~hamiltsd/larsstek.html

Experiments were conducted to determine if the modified cosine law suggested by Hinze (1959) [5] is accurate for angle calibrations of an X-array hot wire probe in transonic air jets. Hinze's equation relates the angle of a probe sensor (with respect to the air flow) and the effective velocity (as measured by the sensor) to the free stream velocity of the jet. Once calibrated, the cross wire probes measure transient flow speed and direction with very high frequency response. However, the accuracy of all measurements rests on the validity of the calibration. Several tests show data that fits Hinze's cosine equation well for a limited range of sensor angles even at high subsonic Mach numbers (~0.85). The tests also identify regions of the angle calibration curve that may warrant further investigation.

Hot wire anemometry literally means a process to measure air flow rate with a heated wire. As pointed out by Lomas (1986) [8], measuring the heat transfer of a probe to extrapolate fluid behavior such as flow direction, temperature, and turbulence intensity as well as flow rate are common. Thus, the name given to this process in the early 1900s does not fully describe its potential as a measurement tool.

The experiments focused on a hot wire technique that measures fluid flow and direction by combining two hot wires in a plane to form an 'x.' It is called a cross wire (see figure 1). Single wire probes only require a velocity-voltage calibration. Cross wires are calibrated differently than single wire probes because they have an added angle variable.

One may calibrate a cross wire in two steps. Performed first is a velocity-voltage calibration for each of the two sensor wires using an appropriate technique. Then, one must find the angle response of the combined sensors. If the wires lost heat in only one dimension (radially from the cylindrical wire), simple trigonometry would show that the velocity measured by the sensor is equal to the free stream velocity times the cosine of theta:

Since the heat loss of the wire is not zero at theta equals ninety degrees, the cosine relationship does not hold. Hinze suggested a modified cosine equation in 1959 that added a term for the heat loss in the direction parallel to the wire (sine of theta times a constant k). This equation:

Figure 2 - (top) angle calibration of probe in air at Mach 0.80, (bottom) same calibration at Mach 0.159. Basic shape of curve held for all Mach #s in between.

This investigation looked primarily at the accuracy of Hinze's modified cosine equation as a curve fit for the angle calibration of an X-array hot wire probe. The curve fit equally well at all Mach numbers in the range 0.16 < M < 0.86 (see figure 2 above). Shocks that appeared at subsonic rates (see figure 3) were exclusive to the particular testing apparatus used, and were further determined to not affect average voltage or pressure measurements. A separation region that formed around the probe at certain angles is not directly related to data shifts observed over certain angle ranges.

Despite the verification that Mach number does not affect the cosine, there were some noticeable changes in data as Mach number increased. At higher Mach numbers, the data still held the cosine curve shape but the lower leg continuously declined leaving large deviations at high angles (theta > 65). Further investigation may reveal that certain applications that rely on measurements at high angles should not use the modified cosine equation. The data shift at low angles may also warrant continued investigation if one applies the modified cosine equation to make measurements at very low sensor angles (theta < 20). However, for the range of sensor angles 25 < theta < 60 the modified cosine equation is accurate for probe calibration in transonic flow as high as Mach 0.85.

References

1. Batchelor, B. (1995) "Lighting Advisor". Internet:
   http://www.cs.cf.ac.uk/Lad/text95.html, University of Wales, UK.
2. Cole, D and Glauser, M.N. (1996) "Utilizing a Flying Hot-Wire System to Study the Flow in an Axisymetric Sudden Expansion". Report Number MAE-315, Clarkson University.
3. Friehe, C.A. and Schwarz, W.H. (1968) "Deviations from the Cosine Law for Yawed Cylindrical Anemometer Sensors". Journal of Applied Mechanics, December, pp.655 - 662.
4. George, W.K., Beuther, P.D. and Shabbir, A. (1987) "Polynomial Calibrations for Hot Wires in Thermally-Varying Flows". ASME Symposium on Thermal Anemometry, Cincinnati, 53, pp.1-6.
5. Hinze, J.O. (1959) Turbulence, McGraw-Hill, New York.
6. Horstman, C.C. and Rose, W.C. (1977) "Hot-Wire Anemometry in Transonic Flow". AIAA Journal, Vol. 15, No. 3 , pp. 395 - 401.
7. Jones, G., Stainback, P. and Nagabushana, K. (1992) "A Comparison of Calibration Techniques for Hot-Wires Operated in Subsonic Compressible Slip Flows". AIAA 17th Ground Testing Conference, AIAA-92-4007 , Huntsville, AL.
8. Lomas, C.G. (1986) Fundamentals of Hot Wire Anemometry, Cambridge University Press, New York.
9. Ukeiley, L.S. (1998) Private Communications.

1. Class of 1999, NASA LARSS Fellow, Oral Presentation
2. NASA Langley Research Center Scientist at Aeroacoustics Branch

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