Abstract
Two hotwire flow diagnostics have been developed to measure a variety of turbulence statistics in the buoyancy driven, airhelium Rayleigh–Taylor mixing layer. The first diagnostic uses a multiposition, multioverheat (MPMO) single wire technique that is based on evaluating the wire response function to variations in density, velocity and orientation, and gives timeaveraged statistics inside the mixing layer. The second diagnostic utilizes the concept of temperature as a fluid marker, and employs a simultaneous threewire/coldwire anemometry technique (S3WCA) to measure instantaneous statistics. Both of these diagnostics have been validated in a low Atwood number (A _{ t } ≤ 0.04), small density difference regime, that allowed validation of the diagnostics with similar experiments done in a hotwater/coldwater water channel facility. Good agreement is found for the measured growth parameters for the mixing layer, velocity fluctuation anisotropy, velocity fluctuation p.d.f behavior, and measurements of molecular mixing. We describe in detail the MPMO and S3WCA diagnostics, and the validation measurements in the low Atwood number regime (A _{ t } ≤ 0.04). We also outline the advantages of each technique for measurement of turbulence statistics in fluid mixtures with large density differences.
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Abbreviations
 A _{ t } :

Atwood number \( ( \equiv {{\left( {\rho_{1}  \rho_{2} } \right)} \mathord{\left/ {\vphantom {{\left( {\rho_{1}  \rho_{2} } \right)} {\left( {\rho_{1} + \rho_{2} } \right)}}} \right. \kern\nulldelimiterspace} {\left( {\rho_{1} + \rho_{2} } \right)}}) \)
 α :

Rayleigh–Taylor growth parameter
 α_{CL} :

Rayleigh–Taylor growth parameter determined using centerline v′
 β :

Thermal diffusivity (m^{2}/s)
 c _{ p,1}, c _{ p,2}, c _{ p,mix} :

Specific heat of inlet streams 1 and 2 and the mixing layer (J/kg °C)
 E :

Hotwire anemometer voltage (V)
 E _{cw} :

Coldwire anemometer voltage (V)
 f _{ m,1}, f _{ m,2} :

Mass fraction of streams 1 (top) and 2 (bottom) in the mixing layer
 f _{ v,1}, f _{ v,2} :

Volume fraction of streams 1 and 2 in the mixing layer
 f _{ v,he} :

Volume fraction of helium
 g :

Gravitational acceleration constant (m/s^{2})
 h :

Mixing layer half width (m)
 k :

Wavenumber (m^{−1})
 υ :

Kinematic viscosity (m^{2}/s)
 ρ _{1}, ρ _{2}, ρ _{mix} :

Fluid densities of inlet streams 1 and 2 and the mixing layer (kg/m^{3})
 ρ′:

Density fluctuations inside the mixing layer (kg/m^{3})
 R _{ ρ′v′ } :

Correlation coefficient for ρ′ and v′
 τ :

Nondimensional time
 θ :

Molecular mixing parameter
 t :

Time (s)
 T _{1}, T _{2}, T _{mix} :

Temperature of inlet streams 1 and 2 and the mixing layer (°C)
 U _{eff} :

Hotwire sensor effective (normal) velocity (m/s)
 u′, v′, w′:

Streamwise, vertical, and crossstream velocity fluctuations (m/s)
 \( \bar{U},\bar{V},\bar{W} \) :

Streamwise, vertical, and crossstream mean velocities (m/s)
 X, Y, Z :

Streamwise, vertical, and crossstream directions for lab coordinate system (m)
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Acknowledgments
This material is based upon work that is supported by the US Department of Energy under contract number DEFG0302NA00060.
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Appendices
Appendix 1: Uncertainty estimates of measurements
An uncertainty analysis illustrating the propagation of error through the data reduction calculations from hotwires and coldwire voltages to measurements of velocity and density were performed. The analysis for determining uncertainties from data reduction follows the methods described by Kline and McClintock (1953) and Moffat (1988). Additional uncertainties in the statistical turbulence measurements are determined from the methodology for uncertainty estimates in the sampling of random processes described by Benedict and Gould (1996).
1.1 Uncertainty of top (stream 1) and bottom (stream 2) stream fluid densities
The uncertainty of the fluid densities is dependent on the fluid temperatures of each stream and in the case of the bottom stream, additionally the helium metering system. The effect of temperature on the fluid densities from the introduced temperature difference between the fluid streams was obtained from the fits of the equation of states for both helium and air (Kraft 2008). The uncertainties of the fluid temperatures (as the two fluids mix inside the mixing layer) are \( w_{{T_{1} }} = w_{{T_{2} }} = {{\left( {T_{1}  T_{2} } \right)} \mathord{\left/ {\vphantom {{\left( {T_{1}  T_{2} } \right)} 2}} \right. \kern\nulldelimiterspace} 2}, \) where (T _{1} − T _{2})/2 is the maximum change in temperature in each stream fluid properties as it is molecularly mixed. Using these relationships, the uncertainties in the fluid densities following Kline and McClintock (1953) are:
The density of the bottom stream (stream 2) is determined through the relationship (Banerjee and Andrews 2006):
Therefore, the uncertainty of ρ _{2} can then be determined as
The additional uncertainties of \( w_{{A_{c} }} ,\,w_{{U_{m} }} , \) and \( w_{{\dot{m}_{\text{he}} }} \) were evaluated previously in the uncertainty analysis of the helium metering system by Banerjee (2006). The uncertainty of the fluid densities can now be extended to the uncertainty in the Atwood number
1.2 Uncertainty of measuring density using the coldwire anemometer and temperature as a fluid marker
An uncertainty analysis for error propagation through the coldwire measurement of fluid density was also performed. Data reduction of the coldwire voltage proceeds from the measured voltage (E) to fluid temperature (T _{mix}), fluid mass fraction (f _{ m,2}), fluid volume fraction (f _{ v,2}), and finally, fluid density (ρ). The progression of uncertainty through the data reduction is described in the same order. The coldwire anemometer allows temperature to be measured via the anemometer voltage through a linear calibration of temperature versus voltage,
Uncertainty in the measured temperature, \( w_{{T_{\text{mixc}} }} , \) by the coldwire anemometer is determined through the uncertainty of the measured voltage and the uncertainty in the calibration of the coldwire probe. According to Moffat (1988), these uncertainties can be combined in the same manner as which uncertainties have been combined in the standard Kline and McKlintock method,
The measured temperature inside the mixing layer via the coldwire, T _{mix}, combined with the measured freestream temperatures, T _{1} and T _{2}, are then used to determine the mass fraction of fluid 2 given in Eq. 6. The uncertainty in the mass fraction due to the measurement of the fluid temperatures is then found as:
Using the fluid densities of streams 1 and 2, the mass fraction is converted to a volume fraction of fluid 2 by using Eq. 7. The uncertainty in the volume fraction can similarly be determined as:
Finally, the fluid density measured by the coldwire is determined from the densities of stream 1 and 2 and the calculated fluid volume fractions as given in Eq. 8. The uncertainty in the measured fluid density inside the mixing layer is then determined by:
which yields the uncertainty in the measurement of density by the coldwire anemometer and the temperature marker.
1.3 Uncertainty of measuring velocity
A similar uncertainty analysis can be performed to estimate the uncertainty in measuring velocity using the hotwire techniques. Since hotwire voltages are accurately converted to measurements of velocity by accounting for varying temperature and density within the mixing layer, the uncertainty analysis will demonstrate the propagation of the uncertainties in the measured fluid temperature and density through the data reduction for determining velocity. Voltages measured by the hotwire system, E _{ B }, are initially corrected for temperature variations in the passing fluid (Eq. 9). Therefore, the uncertainty in the corrected voltage due to the measured reference temperature for the hotwire, T _{ref}, and the measured fluid temperature, T _{mix}, is
Once the hotwire voltages are corrected for variations in the fluid temperature, voltages are converted to effective velocities measured by the wire sensor. This was accomplished through the calibrated wire response due to velocity and concentrations of helium (fluid density). A series of simplifications and assumptions is used to simplify the uncertainty analysis. For the purpose of this analysis, the mean velocity vector for the buoyancydriven flow will be considered, \( \left\langle {U,V,W} \right\rangle = \left\langle {U_{m} ,0,0} \right\rangle . \) Considering the statistical velocity vector for the flow under consideration allows for further simplification in the effective velocities of each wire sensor. In the situation of a onedimensional velocity vector, the effective velocities of each sensor are approximately equal, \( U_{{{\text{eff}},1}} \cong U_{{{\text{eff}},2}} \cong U_{{{\text{eff}},3}} . \) In addition, the responses of each of the three wires are very similar such that the response from wire 1 will be used to approximate the behavior of all the hotwires to both helium and velocity.
To account for the effects of helium and velocity on the hotwire voltage response, the calibration data points are curve fit using a threedimensional surface fit using Table Curve 3D. This allows the sensitivities of voltage to velocity and concentrations of helium to be directly evaluated. As previously mentioned, the fit which is utilized is of the form
This allows the uncertainty in determining the effective velocities of the wire sensors to be calculated according to
The final steps in the data reduction of hotwire voltages to measurements of velocity converts the wire effective velocities, U _{eff}, to a global velocity component through the coordinate transformations described in Sect. 3. Therefore, the uncertainty in measuring the velocity in the global coordinate system is a function of the uncertainty of the wire effective velocity. The general documented uncertainty in measuring velocity via a threewire probe due to its design and geometry can also be included in the uncertainty analysis as an additional uncertainty
The uncertainty in measuring velocity due to the geometry of the probe is ~3% (Frota and Moffat 1983). Combining this uncertainty in the general threewire hotwire probe performance with the additional uncertainties associated with the introduced temperature and density variations leads to a final estimate of uncertainty of ~7% for A _{ t } = 0.03.
1.4 Uncertainty in turbulent statistics
The method of Kline and McClintock (1953) is ideal for determining the uncertainty in a single sample measurement from a variety of sources such as instrumentation and human error. However, the method is not intended to be applied to multiple sample measurements of random processes, where higher order statistics are desired such as in fluid turbulence (Benedict and Gould 1996). Fortunately, the analysis of Benedict and Gould (1996) addresses this necessity for the measurement of uncertainty in higher order statistics as found in turbulence measurements. Benedict and Gould use general relationships for the sample variance of central moments obtained from literature for a univariate (autocorrelation) or bivariate (crosscorrelation) central moment. These relationships are derived for any distribution, without assuming a normal distribution. Not restricting the estimates of uncertainties to assumptions of normal distributions is important, since many aspects of turbulence are nonGaussian. The general relationships for a sample variance of any central moment utilize the measured sample to estimate the variance in a desired turbulent statistic. Examples of estimates in the variance of some common turbulent statistics are shown in Table 6 where N is the number of independent samples. The calculated variance in the desired turbulent statistic is then used to determine the measurement uncertainty by defining the 5–95% confidence interval. Benedict and Gould applied these concepts to commonly measured turbulent statistics, and confirmed uncertainty estimates with their own experimental turbulence measurements. Their methodology and statistical analysis will be used to estimate the uncertainty in all turbulent statistics presented in this dissertation.
1.5 Uncertainty in the growth parameter α
Uncertainty in the measured growth parameter, α, is determined using the Kline and McKlintock method with individual parameter uncertainties estimated from previous estimates through the Kline and McClintock and Benedict and Gould procedures. The growth parameter is found from the selfsimilar relationship of the growth of the mixing layer half width as given in Eq. 11. The uncertainty is found through the sensitivities of the parameters used to calculate the growth parameter:
1.6 Uncertainty in the molecular mixing parameter θ
The uncertainty in determining the degree of molecular mixing, θ, is determined in a similar manner to the uncertainty of the measured growth parameter. The definition for molecular mixing is used to determine the uncertainty of the measurement,
By combining the sensitivities of the parameters used to calculate molecular mixing, the uncertainty is
1.7 Uncertainty due to propagation of measurement error
The previous sections describe the statistical error for the measurement of variance of velocity statistics using the methods of Benedict and Gould (1996), and assume the statistical limitations of the measurement to be the dominant source of uncertainty. A test for the added uncertainty due to propagation of the measurement error was also performed, as described below. Since the individual measured velocities are known from the captured velocity traces, we extend the use of Kline and McClintock (1953) analysis to estimate how the measurement error of each individual velocity sample contributes to the uncertainty in the calculation of the variance (see Eq. 34). In this procedure, each velocity measurement at a particular instant in time represents an independent source of error for the calculation of variance.
The sensitivities for each independent parameter, ith point in the trace, must be evaluated and combined to yield an overall measurement uncertainty (Eqs. 35, 36).
It was observed from Eq. 36 that each point will contribute a unique uncertainty to the calculation of variance which is summed for the entire trace. This allowed for a direct estimate of the uncertainty propagation of velocity measurement error for the calculation of variance. This calculation was performed using the Utrace at τ = 1.33 with the S3WCA technique. The estimated measurement uncertainty for \( \overline{{u^{\prime 2} }} \) using the described procedure is found to be 3.8%. This is much smaller than the previously determined statistical uncertainty for \( \overline{{u^{\prime 2} }} , \) which using Benedict and Gould (1996) were estimated to be 16%. Following Moffat (1988), we can combine the uncertainties from both the measurement and statistical sources to determine a total uncertainty in \( \overline{{u^{\prime 2} }} \) (Eq A.22).
The total uncertainty in \( \overline{{u^{\prime 2} }} \) is found to be 16.4%, which demonstrates the dominance of the statistical uncertainty for these highly dynamic, turbulent experiments. Thus, we believe that the estimation of uncertainty based on the statistical uncertainty is an acceptable procedure for the current set of measurements.
1.8 Summary
Using the combined methodology of Kline and McClintock (1953) and Benedict and Gould (1996), the procedure for estimates of uncertainties in statistical measurements using both the hotwire techniques has been described. A summary of the individual uncertainties using the outlined analysis is shown in Table 7 for both the S3WCA and the MPMO techniques. Uncertainties in the measured velocities and their variances are found to be larger for the S3WCA technique. However, this is expected as the MPMO technique does not attempt to determine instantaneous measurements, but determines timeaverage statistics directly from timeaverages of the hotwire voltage fluctuations. The complexity and instantaneous nature of the S3WCA diagnostic results in larger measurement uncertainties, however, it is a more powerful measurement tool. The uncertainties of ~16% for the variance of measured velocity fluctuations, however, is reasonable for a hotwire diagnostic (Bruun 1995), although not as accurate as the MPMO diagnostic. The primary sources of error in the measurements are in the determination of the sensed fluid density, large hotwire sensitivities to helium, uncertainty in the inlet stream properties from the helium metering system, and the applied inlet stream temperature difference. Uncertainties in the measured statistics can be improved by using large sample sizes to include more independent measurements of the largescale structures.
Appendix 2: Validation of S3WCA technique in aironly shear layer
To gain experience with the threewire diagnostic and to verify the data reduction technique, shear layer experiments in air were performed. Two velocity profiles were measured across a shear layer following the experiments of Bell and Mehta (1990). The conditions for the two velocity profiles measured are shown in the Table 8. The amount of shear between the top and bottom streams of the shear layer is defined as
and the mean velocity at the centerline of the shear layer is
A Reynolds number for the shear layer is defined as
where w is the shear layer width defined by the 10 and 90% velocity differences. A nondimensional width is defined as
The conditions were to provide a comparison to the data of Bell and Mehta (1990), while attempting to achieve large Re _{ w }. The mean velocity profiles of Fig. 16 collapse well and show the expected linear behavior within the shear layer. This is expected as the mean velocity profiles of Bell and Mehta begin to collapse in advance of the selfsimilar flow. As shown by Bell and Mehta, the asymmetry of the velocity fluctuation profiles is dependent on the initial boundary layers of the two air streams. In many shear layer experiments, care is taken to manipulate these boundary layers to achieve the desired behavior.
The measured velocity fluctuation profiles are shown in Fig. 17. Similar asymmetry between the lowspeed and highspeed sides of the shear layer are seen in Bell and Mehta (1990). As selfsimilarity is approached, the velocity fluctuation profiles should become symmetric. Although the downstream data (x = 1.9 m) is not yet symmetric, it approaches symmetry when compared with the upstream data (x = 1.6 m). Selfsimilarity should be found at Re ≫ 10^{4}. Therefore, similarity is neither expected nor found for the two cases. Most shear layer experiments are performed at velocities an order of magnitude larger than can be achieved in the lowspeed wind tunnel, thus allowing shear layer researchers to easily achieve Re ≫ 10^{4}. The peak magnitudes of the velocity fluctuations are reported in Table 9; even though not yet selfsimilar, the peak magnitudes appear to be approaching the selfsimilar values reported by Bell and Mehta (1990). Overall good agreement is found between the measured shear layer velocity statistics and those obtained by Bell and Mehta.
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Kraft, W.N., Banerjee, A. & Andrews, M.J. On hotwire diagnostics in Rayleigh–Taylor mixing layers. Exp Fluids 47, 49–68 (2009). https://doi.org/10.1007/s0034800906363
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DOI: https://doi.org/10.1007/s0034800906363