GUST GENERATION
4.2 Analysis Techniques
The analysis of these experiments relies heavily on two techniques: the visualization of the flow through ‘unwrapping,’ and the identification of vortices. This section describes these methods and how they were used.
4.2.1 Gust Unwrapping
Many of the flows observed in this thesis consist of flow structures convecting with the constant freestream velocity. This allows the flow to be unwrapped into a larger field of view, converting a time-varying two-dimensional measurement of part of the flow field into a single snapshot of the flow that encompasses the observed time span. This is illustrated in Figure 4.1. Individual frames, at the top of the figure, show only a portion of a structure, which is moving from left to right at speed U. At the bottom, these frames have been combined to show a view of the full structure.
More rigorously, for each velocity field at timet, the ‘initial’ position of that view is computed as x0 = x−tU. The estimated structure is then assembled by averaging together each frame, interpolated onto its ‘initial’ position. With the freestream velocity subtracted, the result is a view of the convecting structure in its own frame.
1Some of the work in this chapter was previously presented in an AIAA conference paper [34].
Figure 4.1: This shows a portion of the uwrapping process. Three snapshots of the vorticity field of a passing gust are shown. With the flow moving at constant speed from left to right, an unwrapped view of the convecting structure can be assembled from the snapshots as in the second plot. In practice, significantly more than three snapshots were used to unwrap the flow.
In this chapter, all of the spatial dimensions have been nondimensionalized by the airfoil’s chord length, ca, independent of the length of the gust-generating device.
The unwrapped gusts present the nondimensionalized vorticity,u-velocity, or kinetic energy. The dimensionless kinetic energy of the unwrapped gusts is defined as (u2+v2)/U2, using the velocities in the frame of the convecting gust.
4.2.2 Vortex Identification
It is not simple to identify a vortex. The theoretical underpinnings of vortex identi- fication are often based on local velocity gradients [9, 11, 37], which is troubling for experimentalists. The presence of noise in experimental measurements greatly re- duces the efficacy of such methods, which motivated Michard and Graftieaux [28,49]
to develop a non-local method of vortex identification: the Γ2 function. This is a Galilean invariant criterion that gives a measure of how circular the mean-subtracted flow is around a point. Equation 4.1 defines this function for two-dimensional flow.
u(x)= 1 A
Z
S
udS, (4.1a)
v(x)= 1 A
Z
S
vdS, (4.1b)
Γ2(x) = 1 A
Z
S
v−v
x− u−u y px2+y2
q
u−u2+ v−v2
dS. (4.1c)
The point around which the flow’s circularity is evaluated isx. The circular region of integration isS, its area is A, and the average velocity components in that region are denoted by an overline. If the mean-subtracted flow is perfectly circular around x, thenΓ2 =±1, depending on the direction of rotation. If the magnitude is greater than 2/π, the region is declared vortical. In a simple flow, this is identical to saying that magnitude of the rotation is greater than that of the shear. Since theΓ2function subtracts the local average velocity, it is invariant to translating frames of reference.
Since theΓ2function integrates over a region, any weak noise is generally averaged out. Unfortunately, this leads to a free parameter: the radius of the integration region. The analyses that follow in the text use a range of integration radii, from 0.5 to 6 cm, which is 5% to 60% of the airfoil’s chord length.
Once a vortical region has been identified, the vorticity within that area can be integrated to estimate the vortex’s circulation. An additional concern with the Γ2 criterion is that these approaches do not entirely capture the vorticity of ideal vortices. For example, when applied to a Lamb-Oseen vortex, the Γ2 criterion estimates its circulation to be 72% of the true value.
TheΓ2function provides an estimate of the location of the vortex’s center: the point with the maximum magnitude of Γ2. The average radial and tangential velocity profiles of the vortex may then be computed by averaging those fields in annuli around the center.
In each set of experiments, theΓ2function was used to determine the circulation of the primary vortex in each frame where it was visible. This led to a distribution of values for each set of experimental parameters. The reported values in this chapter are the 25th, 50th, and 75th percentiles of these values, using the largest radius of integration. This displays both the median and variation in the measurements. This was necessary due to error in the computed velocity fields and variation of the flow over time.
This chapter also compares how well the theoretical values of the circulation match the experimental measurements. In later figures, the plotted values are the ratio be- tween the unsteady thin airfoil theory’s estimate and the median of the experimentally measured circulation, for the different radii of integration. Two fitting methods were
used. The curves labeled “LSQ” were determined by the least-squares division of the array of estimated values by the measured values. The “Mean of Ratio” curves are the average of the ratios for each experiment. If the experimental measurements match the theory, then these curves should be near unity.
They-position of the vortex is reported in a similar fashion, using the distributions of identified vortex center locations. The reported x-velocity of the vortex was computed using a linear least-squares fit of the vortex’s xposition over time.