Design and validation of a nanoscale cross-wire probe (X-NSTAP)

Abstract

A nano-sized crossed thermal anemometer (X-NSTAP) was developed and validated for measurements of two-components of velocity at high Reynolds numbers. The new sensor design is based on the single-component nanoscale thermal anemometry probe (NSTAP) previously used to acquire streamwise velocity measurements at high Reynolds numbers. The new sensor can, simultaneously, measure two components of velocity with a spatial resolution of $42\times 42\times 50 \, \upmu {\text {m}}$, an order of magnitude smaller in each dimension than conventional cross-wires. The new X-NSTAP design features several structural and manufacturing modifications to improve the aerodynamic performance of the sensor compared to previous nanoscale cross-wire designs. The effects of different manufacturing modifications were evaluated using dye visualizations over scale models of the sensor tip. The pitch sensitivity of the final sensor design was evaluated in an open-loop wind-tunnel and was comparable to the single-component NSTAP design. The X-NSTAP was then deployed in the Princeton Superpipe to acquire axial and radial velocity measurements up to friction Reynolds numbers, $Re_\tau = 24{,}000$ with good agreement to existing studies.

Graphical abstract

Appendix A: “Stress-Calibration” method of Zhao et al. (2004)

In this study, the “stress-calibration” method of Zhao et al. (2004) was to calibrate the angle sensitivity of the two X-NSTAP wires, the major details of which are reproduced below. For a hot-wire inclined with respect to the incoming flow, the “effective” cooling velocity, $u_e$, as determined from the hot-wire output voltage is assumed to be geometrically related to the instantaneous velocity components by an effective cooling angle, $\phi$, by the cosine law of cooling (see Bradshaw 1971).

In a fully developed pipe flow where $\overline{u_r} = 0$, the instantaneous velocities in the coordinate system of the pipe can be geometrically related to the effective cooling velocity by

$$\begin{aligned} u_{e1}&= \overline{u_z} \cos \phi _1 +u_z'\cos \phi _1+u_r'\sin \phi _1 ,\end{aligned}$$

(1)

$$\begin{aligned} u_{e2}&= \overline{u_z} \cos \phi _2 +u_z'\cos \phi _2-u_r'\sin \phi _2, \end{aligned}$$

(2)

where subscripts 1 and 2 denote the cooling velocities and angles for the two different wires in the probe, overlines denote the time-averaged quantity, and $'$ denotes the fluctuating component about the mean value. If one assumes that the local turbulence is spatio-temporally resolved by each wire and the cooling angles are independent of Reynolds number, then Eqs. (1) and (2) can be expressed as

$$\begin{aligned} u_{e1}\cos ^{-1}\phi _1&= \overline{u_z} +u_z'+u_r'\tan \phi _1 = f(E_1,\phi _1) , \end{aligned}$$

(3)

$$\begin{aligned} u_{e2}\cos ^{-1}\phi _2&= \overline{u_z} +u_z'-u_r'\tan \phi _2 = g(E_2,\phi _2), \end{aligned}$$

(4)

where f and g functions that relate the anemometer output voltage, E, for the different wires to the effective cooling velocity. Since $\phi$ is assumed to be a constant for each wire, the calibration functions can be represented as fourth-order polynomials (Bruun (1995)) given by

$$\begin{aligned} f(E_1)&= \sum _{i=0}^{4} a_i E_1^i, \end{aligned}$$

(5)

$$\begin{aligned} g(E_2)&= \sum _{i=0}^{4} b_i E_2^i, \end{aligned}$$

(6)

where the explicit dependence of f and g on $\phi _1$ and $\phi _2$ into absorbed into the calibration constants, $a_i$ and $b_i$. Time averaging Eqs. (5) and (6) yields

$$\begin{aligned} \overline{f}&= \overline{u_z} = \sum _{i=0}^{4} a_i \overline{E_1^i}, \end{aligned}$$

(7)

$$\begin{aligned} \overline{g}&= \overline{u_z} = \sum _{i=0}^{4} b_i \overline{E_2^i}, \end{aligned}$$

(8)

which relates the mean anemometer output to the mean streamwise velocity. Using Eqs. (7) and (8), the calibration coefficients can be determined by varying the mean velocity, $U_z$, and anemometer output, E. Here, calibration was by placing the hot-wire probe and a Pitot tube symmetrically about the pipe centerline where the mean velocity gradient is negligible and turbulence intensity is low and varying the tunnel speed.

While the streamwise sensitivity of the individual wires has been established, one still needs to determine the cooling angles, $\phi _1$ and $\phi _2$, to fully relate the anemometer outputs to the instantaneous velocity components. Here, we will use the known distribution of the total stress, $\tau _T$, in fully-developed, incompressible, turbulent pipe flow to determine, $\phi _1$ and $\phi _2$. For a smooth pipe, the total stress distribution is given by

$$\begin{aligned} \frac{\tau _T}{\rho } = u_\tau ^2 = \overline{u_z u_r} - \nu \frac{{\text {d}} U_z}{{\text {d}}r} =-\frac{1}{2\rho }\frac{{\text {d}}P}{{\text {d}}z}(r), \end{aligned}$$

(9)

where ${\text {d}}P/{\text {d}}z$ is the streamwise pressure-gradient. Considering the fluctuation components of Eqs. (3) and (4) gives

$$\begin{aligned}&u_z'+u_r'\tan \phi _1 = f-\overline{f}=f', \\&u_z'-u_r'\tan \phi _2 = g-\overline{g}=g', \end{aligned}$$

which can be expressed as

$$\begin{aligned}&u_z'+ \zeta u_r' = \frac{1}{2}(f'+g'), \end{aligned}$$

(10)

$$\begin{aligned}&\xi u_r' = \frac{1}{2}(f'-g'), \end{aligned}$$

(11)

where

$$\begin{aligned}&\zeta =\frac{1}{2}(\tan \phi _1-\tan \phi _2) , \end{aligned}$$

(12)

$$\begin{aligned}&\xi = \frac{1}{2}(\tan \phi _1+\tan \phi _2) . \end{aligned}$$

(13)

Multiplying both Eqs. (10) and (11) by (11) and averaging gives

$$\begin{aligned}&\xi \overline{u_z'u_r'}+ \zeta \xi \overline{u_r'^2} = \frac{1}{4}(\overline{f'^2}-\overline{g'^2}) ,\end{aligned}$$

(14)

$$\begin{aligned}&\xi ^2 \overline{u_r'^2} = \frac{1}{4}\overline{(f'-g')^2}. \end{aligned}$$

(15)

Through simple rearranging, Eq. (10) can be substituted into Eq. (11) giving

$$\begin{aligned} \xi ^2\overline{u_z' u_r'} = \frac{\xi }{4}(\overline{f'^2}-\overline{g'^2}) - \frac{\zeta }{4}(\overline{f'^2-g'^2}). \end{aligned}$$

(16)

To determine $\zeta$ and $\chi$, one simply gathers and computes the fluctuating statistics corresponding to $\overline{f'^2}$, $\overline{g'^2}$, and $\overline{f'^2-g'^2}$ at different radial locations in the pipe where the viscous stress contribution is negligible. Conveniently this can be accomplished during the course of a typical experimental measurement, making angular calibration and data acquisition one and the same. Since the Reynolds shear stress component can be determined from the pressure-gradient and radial location in this region, the only unknowns left in Eq. (16) are $\zeta$ and $\chi$. Substituting the measurements from the different radial locations into Eq. (16) and conducting a nonlinear curve fit allows us to determine the error minimizing values for $\zeta$ and $\chi$ and by extension, $\phi _1$ and $\phi _2$. Once the cooling angles are known, the fluctuating velocity components, $u_z'$ and $u_r'$, can be related to the $f'$ and $g'$ through

$$\begin{aligned}&u_z'=\frac{f'+g'}{2}-\frac{\zeta (f'-g')}{2\xi } , \end{aligned}$$

(17)

$$\begin{aligned}&u_r' = \frac{(f'-g')}{2\xi }. \end{aligned}$$

(18)

If the wires suffer from low Péclet number effects, the individual wire signals will be correlated to each other as one wire heats the other, and vice versa. Compared to the radial velocity measurements, the thermal correlation is not expected to significantly influence the axial velocity measurements. Since the probe is calibrated using varying streamwise velocities, the thermal correlation between the wires is explicitly accounted for in the mean velocity measurements. From Eq. (18), one can see that the radial measurements will be strongly impacted by the thermal correlation between the wires as the difference between the wire signals will be materially diminished. However, Eq. (17) shows that the fluctuating component of the axial velocity is computed using both the sum and difference between the two fluctuating wire signals, $f'$ and $g'$, the former of which is larger in magnitude and proportionally less affected by this consideration.