February 2011 

NASA/TM-2011-217052 

Conical Shock-Strength Determination on a 

Low-Sonic-Boom Aircraft Model by Doppler 

Global Velocimetry 

Gregory C. Herring and James F. Meyers 

Langley Research Center, Hampton, Virginia 
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February 2011 

NASA/TM-2011-217052 

Conical Shock-Strength Determination on a 

Low-Sonic-Boom Aircraft Model by Doppler 

Global Velocimetry 

Gregory C. Herring and James F. Meyers 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320

443-757-5802 
1

Contents 

Abstract ............................................................................................. 2 

Introduction ......................................................................................... 2 

Methods and Experimental Setup ............................................................... 4 

Results ............................................................................................... 8 

Discussion ..........................................................................................12 

Summary ........................................................................................... 15 

Acknowledgement ................................................................................. 16 

References .......................................................................................... 16 

Table 1 .............................................................................................. 19 

Figures .............................................................................................. 20 
2

Abstract 

A nonintrusive laser technique Doppler global velocimetry (DGV) was used to 

determine conical shock strengths on a supersonic-cruise low-boom aircraft model. The 

work was performed at  Mach 2 in the Unitary Plan Wind Tunnel. Water was added to 

the wind tunnel flow circuit to generate small ice particles used as flow seeding for the 

DGV system. DGV generates two-dimensional (2-D) maps of three components of 

velocity that span the oblique shock. Shock strength (i.e. fractional pressure increase) is 

determined from observation of the flow deflection angle across the shock in combination 

with the standard shock relations. Although DGV had conveniently and accurately 

determined shock strengths from the homogenous velocity fields behind 2-D planar 

shocks [Herring, Meyers, and Hart, Meas. Sci. Tech. 20, 045304-045313 (2009)], the 

inhomogeneous 3-D velocity fields behind the conical shocks presented additional 

challenges. Shock strength measurements for the near-field conical nose shock were

demonstrated and compared with previously-published static pressure probe data for the 

same model in the same wind tunnel. Fair agreement was found between the two sets of 

results. 

1. Introduction

1.1 Background 

Physical probes can sometimes perturb the measured data obtained in complicated 

inhomogeneous flow fields, thus biasing the quality of the measurements. To avoid this 

perturbation, researchers began measuring off-body flow velocities by seeding various 

types of small particles into the flow and measuring the Doppler shift of the Mie 

scattering [1-3] with laser beams. Historically particle-seeded techniques have measured 

only velocity, but not other flow parameters (with rare exception [4]). Hence one area of 

aeronautics research is concentrated on developing optical scattering and fluorescence 

techniques that directly probe the flow molecules to provide pressure and temperature 

information. While avoiding the use of particles, molecular techniques are typically 

expensive and labor-intensive compared to particle methods. If pressure or temperature 

could be derived from simpler particle-based methods, it would enhance the value of the 

particle methods. 
3

Another area of aeronautics research [5] is focused on reducing sonic-boom shock 

strengths and noise to facilitate commercial cross-continental supersonic flights. 

Pressure-signature measurement in wind tunnels has long been an important aspect of 

predicting ground signatures and has traditionally been accomplished with pressure 

probes. However the probe geometry may influence the pressure data through 

interference with the model-induced flow field pressures. Thus optical-based 

noninvasive pressure measurement methods have potential to interface these two related 

research areas and improve the state-of-the art for shock-strength prediction and 

measurement. 

Although shock structures can be visualized directly [6] from velocity maps 

determined from particle-based velocimetry techniques, in certain cases it is possible to 

derive pressure maps from velocity measurements. This is possible because velocity is 

closely related to pressure through the basic fluid dynamical relations. One rare example 

of deriving a pressure map from a velocity map is given in Reference 4. Recently [7] the 

ability of Doppler global velocimetry (DGV) to accurately measure the strengths of twodimensional (2-D) planar shocks from a flat plate was demonstrated. Confidence in the 

accuracy of the measurement of Reference 7 is high because the DGV shock strength 

determinations were compared to another independent optical measurement: laserinduced thermal acoustics (LITA). Good agreement was found between the two optical 

techniques for the shock strength measurements; furthermore, the LITA technique was 

previously compared to and found to agree with static pressure probes in a simple 

homogeneous freestream flow [8].

1.2 Present Work 

With this successful measurement of a planar shock strength in hand, in this 

report the DGV approach of Reference 7 is extended to the measurement of conical shock 

strength in the 3-D flowfield of a lifting body. No physical probe is required for the 

DGV measurement. DGV is a technique [3, 9-13] that uses a single laser beam formed 

into a sheet to measure three components of flow velocity at multiple locations within the 

2-D laser sheet. If the laser sheet is located across a shock and oriented transverse to the 

shock, the measured velocity map can be used to determine the pressure increase across 
4

the shock within the velocity map. The spatially-resolved map of flow velocity is used to 

determine the flow-deflection angle across the shock with about the same spatial 

resolution as that of the velocity map. Then the oblique and normal shock relations can 

be used to determine the pressure rise across the shock. 

This report discusses only the DGV measured nose shock strength versus 

downstream location. The entire longitudinal pressure profile across the full shock 

structure (as is traditionally produced by translating a pressure probe) of the model was 

not measured with DGV due to the orientation of the light sheet set to measure the crossflow velocity field below the model. The DGV measured nose shock data are compared 

to traditional static-pressure probe data for the same model in the same wind tunnel with 

approximately the same flow conditions [Mach 2 in the Unitary Plan Wind tunnel 

(UPWT) at NASA’s Langley Research Center (LaRC)]. 

Particle image velocimetry (PIV) and laser Doppler anemometry (LDA) would 

not work as well as DGV with this approach because the larger particles required by 

these techniques would suffer significant velocity lag when traversing the shock. This 

would greatly reduce the spatial resolution of the measurement and offset the physical 

location of the shock from its actual location. In the UPWT, the condensed water seed 

particles are ~ 10 nm [14]. 

2. Methods and Experimental Setup 

2.1 Velocity Measurement 

Descriptions of DGV are found in the literature [3, 9-13]. The frequency of a 

single-longitudinal-mode Ar-ion cw laser is tuned to the peak absorption of an I2

transition near 514 nm. The laser is actively locked to the side of the I2 transition to 

stabilize the frequency drift of the laser. This is accomplished by passing a small fraction 

of the laser beam through a Bragg cell and shifting the wavelength to a position 1⁄2 way 

down the absorption profile. Then the Bragg shifted beam is passed through a vaporlimited I2 vapor cell (the first of two in the setup) and the absorption signal is used to 

control the laser frequency in an active feedback loop. The remaining laser beam power 

is formed into a sheet (~ 200 m thickness) to define a planar interrogation region in the 

test section. In this supersonic flow, Rayleigh scatter from ice particles passing through 
5

the laser sheet is collected by three fiber-optic probes placed to observe the light sheet 

from three different directions (one probe per velocity component). The collected light 

from all three probes is passed to a single receiver system which splits the light into two 

portions, part impinging directly on a cooled-CCD camera (reference) and the remaining 

passing through the second vapor-limited I2 vapor cell to a second CCD camera (signal). 

The three normalized (= signal / reference) amplitude maps are converted to Doppler 

shift maps from the known I2 absorption. Based on the optical geometries, the Doppler 

shifts are then related to the velocity of the ice particles. 

In the present measurement scheme the laser light sheet was located in the vertical 

plane and projected across the test section from a side window. The fiber optic probes 

were located at the far corners of the test section window to maximize geometric 

accuracy when converting the three non-orthogonal velocity measurements to the 

standard orthogonal coordinate system. Thus the technique measures the three 

components of velocity simultaneously with a spatial resolution of 0.2 mm covering a 

region 225 by 150-mm in the center of the test section. 

In the present work,  3 liters of liquid H2O were added to the wind tunnel circuit 

to obtain a flow humidity equivalent to a dew point (DP) of about - 12 C. Water 

condensation in the tunnel nozzle generates a fog of ice crystals in the cold freestream 

flow to produce the needed scattered laser light. The ~ 10-nm particle size insures 

particle-tracking fidelity of the flow. Light scatter from these small particles is in the 

Rayleigh regime rather than the Mie regime. Multiple scattering effects were assumed to 

be minimal, and thus no check or correction for these effects was made. Translating the 

model along the axial direction through the fixed vertical light sheet produced the series 

of measurement planes at various distances downstream of the model nose. All 

measurement ensembles consisted of 50 sequential image pairs, each with an exposure 

integration time of 1 sec, yielding an averaged measurement over 60 sec. Additional 

details of the DGV test were described previously [7, 15]. 

2.2 Shock Strength Determination 

The objective of the present work was to determine the pressure change across a 

conical shock wave based on maps of cross-flow velocity. The velocity measurements 
6

used in this study were previously reported [15]. If a velocity map spans an oblique 

shock, then the flow-deflection angle, , behind the shock can be determined. Once a 

change in the flow velocity vector was measured, the ideal shock relations (found in 

many textbooks and reports, e.g., [16]) were used to calculate the change in pressure 

across the shock. Using the same six-step process reported in [7], both oblique and 

normal shock relations were needed to convert from velocity to pressure information. 

 Velocity vector measurements before and behind the shock provide the angular 

change, , in the flow direction across the shock. The oblique-shock relations, along 

with the incoming freestream Mach number, M1, and  were then used to determine a 

unique shock angle 1. A unique 1 determines the normal component M1N (relative to 

the shock front at that particular location) of the incoming Mach number. Lastly, the 

normal-shock relations determine the fractional pressure increase across the shock using 

this normal component of M1.

Note that the conical shock relations are not used in this approach because the 

model is not an infinitely long cone. Behind a conical shock a 3-D flow exists. As the 

measurement location moves from the shock towards the conical surface (equivalent to 

moving downstream), the flow angle continuously deflects towards the surface as the 

flow is compressed. In the present work on this low-boom aircraft model with its very 

slender form, the shock angle is predominately determined by the incoming Mach 

number and adopts characteristics more similar to a true Mach wave rather than those of 

the ideal semi-infinite cone case of the conical shock tables. The present approach to 

measuring shock strength determines the flow deflection angle as close to the shock as 

possible, where the oblique shock relations hold and give the shock angle. Refer to 

Section 4 for a discussion of a correction to the raw measurement due to the fact that the 

spatial smoothing of the shock location by the modulation transfer function of the small 

lenses used by the DGV optical system prohibited the measurement of the flow deflection 

angle immediately behind the conical shock. 

In summary, shock strengths are determined from velocity maps as follows: 

(1) Measure (or assume from known wind tunnel conditions) incoming freestream axial 

velocity and Mach number. 

(2) Measure change in transverse velocity across the shock. 
7

(3) Determine flow deflection angle  = ARCTAN (Vtransverse / Vstreamwise). 

(4) Use oblique shock tables to determine shock angle relative to incoming flow. 

(5) Use trigonometry to determine normal component of incoming flow relative to shock. 

(6) Use the normal component of the incoming freestream flow and the normal shock 

tables to determine ratio of pressures and ratio of temperatures across the shock. 

Measurement of the temperature change across the shock is equally feasible with 

the same approach as for pressure. In addition, pressure decreases across expansion fans 

are also obtainable by an analogous method [7]. For expansions, M1 is used with the 

measured flow-deflection angle to determine the Mach number M2 behind the expansion 

with the Prandtl-Meyer flow equations [16]. Then M1 / M2 determines the pressure and 

temperature changes according to the isentropic flow relations [16]. 

2.3 Wind Tunnel and Low-Boom Model 

All measurements were made in test-section 1 of the UPWT at LaRC. A 1/160 

scale model (designed for a low boom at supersonic cruise) was installed in the 1.2 by 

1.2-meter test section and used to generate a near-field sonic boom profile. Additional 

details of the wind tunnel [17] and the low-boom aircraft model [18-20] are available. 

The fuselage was oriented at zero degrees angle-of-attack (AOA), while the wing AOA 

was slightly positive which provided a net lift (i.e., the trailing edge of the wings are 

angles slightly downward relative to the fuselage). The model length, Le, was 9.0 inch, 

and the full wing span, b, was 4.5 inch. Additional details of the model are presented in 

References 19 and 20. 

The present work was performed with a standard Mach number 1.97 airflow 

(velocity = 535 m/sec), a free-steam static pressure of 7.6 kPa, and a freestream static 

temperature of 183 K. However a reduced freestream Mach number = 1.90 (and an 

increased freestream static temperature of 190 K) was used in analyzing the present DGV 

data to account for the addition of significant water to the wind tunnel circuit. 

Condensation of the water vapor into ice crystals in the nozzle slightly heats the flow, 

reducing the Mach number and absolute velocity, while increasing static temperature and 

sound speed [17, 21]. 
8

A schematic of the locations of the DGV and pressure probe measurements is 

shown in the side-view of Figure 1. The vertical thick dashed lines indicate five of the 

many DGV measurement planes at a variety of distances downstream of the nose. The 

single horizontal thick dashed line indicates the line of traverse for the probe 

measurements, which is roughly parallel to the centerlines of both the tunnel and model. 

The thin dotted line indicates the general shock structure. A photograph of a silhouette of 

the model installed in the test section against the green laser light sheet is shown in 

Figure 2. The DGV and pressure probe measurements were made sequentially, separated 

by a few years, and thus the model had been removed and reinstalled between the two 

tests. 

3. Results 

3.1 DGV Velocity Profiles 

Figure 3 shows a typical measured two-dimensional velocity map from Reference 

15 of crossflow velocity (Y component = V), where color gives the magnitude of the 

transverse velocity component over the range -10 to +10 m/sec. The velocity scale is 

given by the reference color bar in the lower right portion of the green shaded area. The 

plane of the map is orthogonal to the freestream direction X and is located 1.39 model 

lengths (12.5 inch) downstream of the nose tip. To determine the pressure change at a 

given downstream location, the transverse velocity profile along the Y direction 

(indicated by arrowed red line) is used to obtain the change in transverse velocity 

component. Moving outward from the model along the red line indicates larger values of 

Y, the transverse distance away from the model (below the model). Note at this far 

downstream location, the nose shock is no longer observable along the red line because 

the nose shock has expanded beyond the limits of the velocity map. However the nose 

shock is observable in the lower-right portion of the velocity map passing through the 

color bar as the curved transition from the freestream green to the faint yellow behind the 

shock. At further upstream locations, the nose shock is easily identifiable along the red 

line and is used to determine the nose shock strengths at these upstream locations. The 

variable Y in this report is equivalent to the variable h in References 19 and 20.
9

The far-downstream location of Figure 3 was not chosen as an example of a map 

used to determine nose shock data, but to illustrate the major features of the flow field 

behind the model. Although these features may be identified with vapor screen 

visualization, Figure 2, only their location can be determined. For example, the nose 

shock cannot be seen in Figure 2, but the wing-tip vortices and shock structures generated 

by the wing can be located. DGV provides quantitative measures of these structures. For 

example the velocity patterns from the two counter rotating wing-tip vortices are found 

by the two blue circles coupled with corresponding red circles (portions of the red circles 

close to the model are lost due to laser flare from the model) showing the typical crossflow patterns of a vortical flow. The red crescent to the right of the model indicates a 

high cross-flow velocity pattern moving away from the model. The corresponding blue 

crescent sections show the second half of the typical sonic “N” velocity pattern. These 

patterns not only show the location of these structures, but their strength. For example, 

the results indicate that the strength of the shock generated by the wing is greater at 45-

degrees than directly below the model. 

At locations further upstream, typical transverse velocity profiles and axial 

freestream velocities are shown in Figure 4. Two velocity profiles across the nose shock 

region are shown for four different downstream locations (3.5, 4.0, 4.5, and 5.0 inch from 

the nose tip). The axial freestream velocity (U component along the X axis) is read off 

the left vertical scale, while the transverse velocity (V component along the Y axis) is 

read off the right vertical axis. Each transverse and freestream velocity profile is taken 

from a single line of velocity data (along the equivalent red line illustrated in Figure 3). 

The upper set of four curves in Figure 4 that are approximately flat show (from 3 to 8 

inch) profiles of the freestream velocity U for four downstream locations, while the lower 

set of four curves show profiles of the transverse velocity V. The upper red curve of 

axial velocity corresponds to the lower red curve of transverse velocity. This convention 

applies to each of the four colors. These V components show a clear increase in velocity 

at about Y = 3-4 inch downstream (i.e. shock location) as you move further below the 

model and across the oblique shock. The flow-deflection angle  at the oblique shock at 

each downstream location is determined from these measured velocity profiles according 

to  = ARCTAN (change in transverse velocity across shock / freestream velocity before 
10

shock) = ARCTAN (V / U). The additional increase in velocity at the extreme left hand 

side (Y =1-2 inch region) of the V components is an artificial effect due to the unwanted 

scattered laser light (flare) from the model. 

Shock angles 1 can then be determined from the flow turning angles  and the 

oblique shock tables as in Reference 7. Shock angles determined from the flowdeflection angles are plotted in Figure 5, as a function of distance downstream of the 

model nose. Figure 5 shows 23 separate determinations of shock angle from 

corresponding flow deflection angles, which were derived from velocity maps (each 

averaged over 60 sec). Data is shown from two days (March 16 [yellow triangles] and 17 

[pink squares]), where the model was rotated 180 degrees on March 17 with respect to 

the orientation of March 16. The smooth connection between the two different days of 

data illustrates the strong internal consistency (i.e. precision), of the DGV data, but does 

not make any clear statement about the absolute accuracy. 

The flow-deflection angles from localized spatial regions used to produce these 

localized shock angles 1, i.e. the yellow and pink data of Figure 5, can be checked by 

comparing with a somewhat independent determination of shock angle A by further 

averaging over more extended spatial regions. This second method of determining shock 

angle comes from observing the vertical shock location in each of the DGV transverse 

velocity maps at many downstream locations. These measured shock locations are 

synthesized into the single plot of Figure 6, which effectively shows a 2-D illustration of 

the shock in a single streamwise plane that intersects the model centerline. Several quasiindependent maps of shock position obtained over the four days of testing are shown in 

Figure 6. Each color shows the result of a single test (i.e. a single morning or afternoon 

wind tunnel run) where the DGV sheet position was changed about 7-8 times over the 

approximate two-hour period during that particular test run. As noted before, on some 

days, the model was rotated 180 deg with respect to other days. 

The shock angles A for each individual run (i.e. each color) of Figure 6 are 

determined from A = ARCTAN (slope) = ARCTAN (Y / X) and are also plotted on 

Figure 5 as blue diamonds. The solid black line of Figure 5 is a power-law fit to the blue 

diamonds. Each blue diamond of Figure 5 is from one run (i.e. one color) from Figure 6. 

These angles over extended spatial averages are in fair agreement with the locally-
11

determined angles (yellow triangles and pink squares), but clearly the numbers and trends 

are slightly different. This slight disagreement shows the advantage of the local averages 

in determining spatially-resolved shock angles when the shock is not perfectly straight. 

Both data sets (locally and extended averages) of Figure 5 are essentially showing a slight 

curvature of the conical shock, and the uncertainties for the extended averages are likely 

more affected by this shock curvature. In the discussion below, only the locally averaged 

shock angles (yellow and pink data of Figure 5) are used to determine the DGV-based 

shock strengths. 

3.2 Nose Shock Strength 

Figure 7 shows a summary of previously measured nose shock strength as a 

function of distance downstream of the model. The solid blue diamonds are the raw 

static pressure probe data (read from Figures 14-17 of Reference 19 and from Figures 16-

17 of Reference 20), while the open yellow diamonds show the probe data after the 

original authors have corrected the raw data (read from Figure 9 of Reference 19 and 

Figure 13 of Reference 20). Reference 19 and 20 discuss the method and reasons (e.g. 

smearing of the shock profile due to model vibration) for these corrections. In this report, 

the corrected probe data (open yellow diamonds) are taken as the final probe data for the 

purpose of comparison to DGV. 

Local shock angles (yellow and pink data of Figure 5) are used to deduce DGV 

shock strength (fractional pressure change = dP / P) using the oblique and normal shock 

tables as in Reference 7. Raw shock strength determination is shown in Figure 7 from 

two days of testing (March 16 and 17), where the model orientation was rotated 180 

degrees on March 17 with respect to that of March 16. The raw DGV shock strength data 

is about 50% smaller than the corrected pressure probe data. The open black triangles 

show both days of DGV data after a correction (described below in Section 4) because 

the measured flow deflection angles are obtained at a small finite distance downstream of 

the shock (not immediately after the shock). After correction of the DGV data, the DGV 

and the probe data are in modest but not perfect agreement. Near the 20-inch 

downstream location, the corrected DGV results are about 20% smaller than the corrected 

probe results. However the laser technique is nonintrusive and can be used to measure 
12

further upstream and closer to the model nose tip than the probe. The probe 

measurements were terminated at the x  15-inch location to avoid interferences between 

the probe and the model. Table 1 shows the numerical values used in Figure 7.

Figure 8 shows the same data as shown in Figure 7, but plotted as normalized 

pressure change across the shock as a function of span-normalized distance downstream. 

The abscissa of this normalized plot (typically used in the sonic boom community) is

called the far-field parameter and gives a shock strength value as if the pressure signature 

has fully evolved into the N-wave shape, even though a low-sonic-boom signature never 

reaches an N-wave shape. The normalized far field shock strength is of interest in 

minimizing sonic boom strengths on the ground and usually becomes constant as a 

function of distance away from the vehicle or model. This far-field asymptote from 

Reference 20 (based on Whitham theory [22]) is shown with a horizontal dashed line. 

4. Discussion 

4.1 Correction to Raw DGV Determinations of Shock Strength 

In the freestream, the mean free path is ~ 10-5 cm, and the shock thickness is 

expected to be of the order of a few mean free paths. But the velocity profiles in Figure 4 

do not show sharp jumps in velocity over the shock as almost certainly must occur in 

reality. The apparent shock thickness in Figure 4 is ~ 2 cm (i.e. ~ 1 inch) and is artificial. 

This is because the imaging optics of the apparatus consist of 5 mm diameter collection 

lenses (mounted on the ends of optical fibers) located about 3⁄4 of a meter away from the 

laser light sheet. The limited numerical aperture of the light collection optics gives a 

finite modulation transfer function (MTF), i.e. limited spatial resolution. This optical 

arrangement and the MTF are dictated by the limited optical access to the test section as 

restricted by the window supporting structures on the outside of the test section. These 

structures prohibited the use of larger diameter lenses that would yield sharper images, to 

view the measurement plane. Thus the smoothing effect of the small lenses did not allow 

the determination of the transverse velocity immediately behind the shock, but the 

velocity could be determined approximately 1-inch downstream. 

The same problem was encountered in the previous work of Reference 7, where 

the measurement of shock strengths for two-dimensional planar shocks was demonstrated 
13

on a flat plate with the same DGV apparatus. However in the previous work, the flow 

behind the planar shock was uniform (i.e. no compression and change in flow direction 

after the crossing the planar shock) and it did not matter if the finite MTF and artificially 

large shock thickness forced a measurement of the post shock velocity a small finite 

distance downstream from the shock. In fact the uniform flow behind the shock proved 

to be an advantage in that work. 

Figure 9 shows examples of the transverse velocity profiles as a function of 

distance above a flat plate from Reference 7. The flow is effectively from right to left. 

These transverse velocity profiles were used to determine shock strengths from oblique 

planar shocks. The uniform flow (constant velocity) behind the shock is illustrated at 

several downstream locations relative to the leading edge of the plate. There is little 

uncertainty about the velocity values in front of and behind the planar shocks. This 

contributed to the excellent results of shock and expansion strengths derived from DGV 

that compared well to LITA. Although the well-behaved velocity profiles of Figure 9 are 

not used here, they are presented for comparison with the profiles of Figure 4 for the 

present conical shocks. 

In contrast, the shocks in the velocity profiles in Figure 4 are less well defined 

because (1) flare from the model has not been completely removed from the signal and 

(2) the flow behind the conical shocks is 3-dimensional. In the case of an infinitely long 

cone, the flow behind the conical shock would be slowly compressed and turned towards 

the cone as the flow progress downstream. If one measures the transverse velocity 1 inch 

downstream of the shock (which must be done due to the finite MTF), the effect of the 

flow direction change between the shock and the measurement point must be accounted 

for in order to derive the flow direction immediately behind the shock. Thus the raw 

flow angle measurements that are made 1 inch downstream of the conical shocks must be 

corrected to reflect the value immediately behind the shock.

Following the scheme presented in Reference 23, Figure 10 illustrates the 

geometry and physical basis of correction. In order to make this correction, one must 

know the nature of the flow between the measurement point P and the shock (point E on 

the same streamline as point P). If measuring shocks far downstream of the model (by 

many body lengths), then approximating the conical shock as a true Mach wave (i.e. an 
14

extremely weak shock) rather than a typical oblique weak shock would be reasonable. In 

this case, the correction would be zero. If measuring far upstream (i.e. upstream of the 

data presented in Figure 7 where the initial cone shape of the model nose has not yet 

changed to a typical constant-diameter fuselage), then an infinite cone approximation 

would be reasonable and the conical shock tables could be used for the correction. The 

flow under investigation is in-between these two situations, where Figure 7 shows DGV 

data for shock strength from about 0.5 to 2 body lengths downstream of the model nose 

tip. It is assumed that the flow between the measurement point and the shock is 

described by a generic slender 2-D airfoil as in Figure 10. 

The flow over the airfoil, immediately after the conical shock and above the 

model, begins by being directed maximally upward, but slowly turns downwards as one 

moves downstream until midway between the nose shock and the tail shock where the 

flow direction regains its original horizontal direction. This downward deflection of the 

flow, due to the slight and gradual compression behind the shock, is a reasonable 

approximation until reaching the general location of the tail shock. 

The transverse velocity change cannot be determined upstream of point P in 

Figure 10 because of the finite MTF in the DGV optical system. This is about 38% of the 

distance downstream (i.e. 1 inch downstream of the shock) between the nose shock and 

the midpoint of the nose and tail shocks. First, approximating a flow path streamline as a 

parabola (dashed line and bold arrows in Figure 10) in the region between the nose and 

tail shocks, the flow direction is expected to regain its horizontal attitude at the midpoint 

between the nose and tail shocks. Second the flow angle change between the shock and 

the midpoint is approximated to be linear. With these approximations, the flow direction 

has changed downward by about 1 / (1-0.38) = 1.62  60% between the shock and the 

measurement point. Adjusting the measured flow directions by this 60% and then 

determining the shock strengths, produces the corrected data points (black color) of 

Figures 7 and 8. These corrected data show better agreement with the corrected probe 

measurements, but may be starting to diverge from the probe results near the 20-inch 

downstream location. 
15

4.2 Uncertainties for DGV-Based Shock Strength 

The spread in the DGV data of Figures 7 and 8 (relative to an imaginary smooth 

curve fit through the data) suggests that the statistical uncertainty (precision) is about +

10% for a single time-averaged (60 sec) measurement of shock strength. Estimating the 

systematic uncertainty (accuracy) is more difficult due to the 60% correction that has to 

be made to the current raw measurements. In addition, the lack of confident comparison 

benchmarks also contributes to the lack of information necessary to access the accuracy. 

In other words, how accurate are the probes? Assuming that the corrected pressure probe 

measurements of Figures 7 and 8 are truly correct, one might estimate that the DGV 

results are trending approximately 20% low for the last three points near the 20-inch 

downstream location. Thus the systematic uncertainty in the DGV method is estimated to 

be < 20%.

Although unlikely (partially because the probe results are corrected by only ~ 

20% and the DGV results are corrected by ~ 60%), it is possible that the DGV results are 

more accurate than the probe results. See References 18-20 for a discussion on the 

uncertainties of the pressure probe measurements. Note that in Reference 7, the DGV 

results agreed with another optical method (LITA) to within a few per cent (normalized 

to the absolute pressure) for shock strength measurements. However in that work, the 

DGV raw data did not require the 60% correction that was needed in the present work. 

5. Summary 

DGV was used to study conical shock strengths on a supersonic-cruise low-boom 

aircraft model by measuring and mapping off-body flow velocity. The non-uniform 

nature of the velocity field and the limited MTF effectively forced a velocity 

measurement slightly downstream of the shock position. This presented a challenge for 

accurately converting the DGV velocity map to pressure change, requiring a correction to 

the raw shock determinations. 

Near-field determinations of the shock strength across the nose shock (in the 

region where the shock is still ~ axially symmetric) were made from the velocity maps 

and compared favorably with static pressure probe data. At  2 body lengths downstream 

the DGV values are trending  20% lower than the probe values. This comparison 
16

provides confidence in the ability of the laser technique to quantitatively measure shock 

strengths where the flow between the shock and the measurement point is well known (or 

alternatively for cases where one can measure the flow angle immediately behind the 

shock). Flows more complicated than the one studied here will present additional 

challenges for the successful use of this method. 

Two advantages of the laser-based method are its nonintrusive nature and its 

ability to measure further upstream towards the apex of the conical shock in a region 

where the probe is not used because of probe-model interferences. Each of the two 

techniques has to correct its respective raw measurement (pressure probe by  20% and 

DGV by  60%) for a smearing of the sharp shock feature. However the physical basis 

of the smearing and corresponding correction is different for each technique, arguing for 

the reality of the general agreement of the two methods. 

6. Acknowledgements 

The authors thank A. A. Cavone, M. T. Fletcher, and J. W. Lee for assembly and 

operation of the laser apparatus and R. J. Mack and L. M. Weinstein for helpful 

discussions. 

7. References 

[1] Hunter, Jr. W W and Foughner, Jr. J T, editors 1982 Flow visualization and laser 

velocimetry for wind tunnels, NASA Report CP-2243

[2] Yeh Y and Cummins H Z 1964 Localized fluid flow measurements with an He-Ne 

laser spectrometer, Appl. Phys. Lett. 4 176-178

[3] Samimy M and Wernet M P 2000 Review of planar multiple-component 

velocimetry in high-speed flows, AIAA Journal 38 553-574

[4] van Oudheusden B W, Scarano F, Roosenboom E W M, Casimiri E W F, and 

Souverein L J 2007 Evaluation of integral forces and pressure fields from planar 

velocity data for incompressible and compressible flows, Exp. Fluids 43 153-162

[5] Aronstein D C and Schueler K L 2005 Two supersonic business aircraft conceptual 

designs, with and without sonic boom constraint, J. of Aircraft 42 775-786
17

[6] Murphy M. J. and Adrian R. J. 2007 Particle response to shock waves in solids: 

dynamic witness plate/PIV method for detonations, Exp. Fluids 43 163-171

[7] Herring G. C., Meyers J. F., and Hart, R. C., 2009 Shock-strength determination with 

seeded and seedless laser methods, Meas. Sci. Tech. 20 045304-045-313 

[8] Hart R C, Herring G C, & Balla R J 2007 Pressure measurement in supersonic 

airflow by differential absorptive laser-induced thermal acoustics, Opt. Lett. 32

1689-1691

[9] Elliot G S and Beutner T J 1999 Molecular filter based planar Doppler velocimetry, 

Prog. Aerosp. Sci. 35 799-845

[10] Willert C, Stockhausen G, Klinner J, Lempereur C, Barricau P, Loiret P, and Raynal 

J C 2007 Performance and accuracy investigations of two Doppler global 

velocimetry systems applied in parallel, Meas. Sci. Tech. 18 2504-2512 

[11] Meyers J F 1995 Development of Doppler Global Velocimetry as a Flow 

Diagnostics Tool. Measurement in Fluids and Combustion Systems, Special 

Issue, Meas. Sci. Tech. 6 769-783

[12] Meyers J F, Lee J W, and Cavone, A A 2007 Doppler Global Velocimetry Springer 

Handbook of Experimental Fluid Mechanics, Ed: C. Tropea, A. L. Yarin, and J. F. 

Foss (Springer-Verlag, Berlin) Chapter 5.3.4. 

[13] Meyers J F 1992 Doppler Global Velocimetry - The Next Generation? AIAA 17th 

Aerospace Ground Testing Conference, Nashville, TN (July 6-8) Paper 92-3897

[14] Benedict L H, Griffith W C, Yanta W J, Spring III W C, and Boyd C F 1991 

Quantification of vapor screen analysis, AIAA 22nd Fluid Dynamics, Plasma 

Dynamics, and Lasers Conference, Honolulu, Hawaii (June 24-26) Paper 91-1691

[15] Meyers J F, Lee J W, Fletcher M T, Cavone A, Viramontes J A G 2006 Supersonic 

flow field investigations using a fiber-optic based Doppler global velocimeter, 

13th International Symposium on Applications of Laser Techniques to Fluid 

Mechanics Lisbon, Portugal, (June 26-29) Paper 1019

[16] AMES Research Staff 1953 Equations, tables, and charts for compressible flow,

NASA NACA Report-1135

[17] Jackson C M, Jr., Corlett W A, and Monta W J 1981 Description and calibration of 

the Langley unitary plan wind tunnel, NASA Report TP-1905
18

[18] Mack R J 2003 A supersonic-cruise business jet concept designed for low sonic 

boom, NASA Report TM-2003-21443

[19] Mack R J and Kuhn N 2004 Determination of extrapolation distance with measured 

pressure signatures from two low-boom models, NASA Report TM-2004-213264

[20] Mack R J and Kuhn N 2006 Determination of extrapolation distance with pressure 

signatures measured from two to twenty span lengths from two low-boom models, 

NASA Report TM-2006-214524 

[21] Herring G C 2008 Mach-number measurement with laser and pressure probes in 

humid supersonic flow, AIAA Journal, 46, 2107-2109 

[22] Whitham G B 1952 The flow pattern of a supersonic projectile, Communications on 

Pure and Applied Mathematics, 5, 301-348

[23] Thompson P A 1972 Compressible-Fluid Dynamics, 1

st ed., (New York, McGrawHill) pp. 262 Chapter 5 
19

Table 1

Raw Raw Corrected Corrected Raw Corrected

Method DGV DGV DGV DGV Probe Probe

and Date Distance Flow Turn Shock Flow Turn Shock Shock Shock

or Reference Downstream Angle Strength Angle Strength Strength Strength

(inch) (deg) (deg)

DGV 3/17 1.8 0.618 0.036 1.001 0.059

DGV 3/17 2.3 0.528 0.031 0.855 0.050

DGV 3/17 2.8 0.449 0.026 0.728 0.043

DGV 3/17 3.3 0.506 0.030 0.819 0.048

DGV 3/17 3.8 0.392 0.023 0.634 0.037

DGV 3/17 4.3 0.415 0.024 0.672 0.040

DGV 3/17 4.8 0.392 0.023 0.634 0.037

DGV 3/17 5.3 0.414 0.024 0.671 0.039

DGV 3/17 5.8 0.346 0.020 0.561 0.033

DGV 3/17 6.3 0.336 0.020 0.544 0.032

DGV 3/17 6.8 0.336 0.020 0.545 0.032

DGV 3/17 7.3 0.282 0.017 0.457 0.027

DGV 3/17 7.8 0.338 0.020 0.547 0.032

DGV 3/17 8.3 0.293 0.017 0.474 0.028

DGV 3/16 9.3 0.281 0.017 0.455 0.027

DGV 3/16 10.3 0.226 0.013 0.366 0.022

DGV 3/16 11.3 0.281 0.017 0.455 0.027

DGV 3/16 12.3 0.225 0.013 0.364 0.021

DGV 3/16 13.3 0.203 0.012 0.328 0.019

DGV 3/16 14.3 0.225 0.013 0.365 0.021

DGV 3/16 15.3 0.169 0.010 0.274 0.016

DGV 3/16 16.3 0.170 0.010 0.276 0.016

DGV 3/16 17.3 0.148 0.009 0.240 0.014

Probe [19, 20] 14.1 0.0210 0.0240

Probe [19, 20] 21.2 0.0140 0.0199

Probe [19, 20] 28.3 0.0140 0.0172

Probe [19, 20] 35.3 0.0110 0.0141

Probe [19, 20] 70.6 0.0060 0.0069

Probe [19, 20] 106 0.0047 0.0054
20 Sting

Nose

Shock

Tail

Shock





Multiple 

Transverse

DGV Sheets

Longitudinal Line

Of Static Pressure

Probe Stations



Wing

Mach 2

Freestream

Airflow

Full Nose

Angle  27 deg

Wing AOA =  = 2 deg

Fuselage AOA = 0 deg

Fuselage

Y

X

Figure 1 Side-view schematic of test setup, showing model, nose and tail shocks 

(thin black dashed lines), five representative light sheet locations (vertical 

blue heavy dashed lines indicate laser sheets transverse to freestream 

direction) and pressure probe locations (heavy red horizontal dashed line 

parallel to tunnel centerline). Fuselage AOA = zero. 
21

Figure 2 Top-view photograph of low-boom model installed in the UPWT in front 

of green laser light sheet. 
22

Figure 3 2-D transverse map of measured transverse velocity component V at 1.39 

times the model length (= 2.78 span widths) downstream from the nose. 

Color indicates transverse velocity component defined by the calibration 

bar at lower right (units = m/sec). 
23

Figure 4 Axial velocity U (upper set of four approximately flat curves read on left 

vertical scale) and transverse velocity V (lower set of four rising curves 

read on right vertical scale) at four different downstream locations. 
24

Figure 5 Shock angles 1 determined from flow-deflection angles from single 60-

sec temporally averaged measurements from two different days, March 16 

(yellow triangles) and 17 (pink squares). A second method for 

determining shock angles A (blue diamonds) is from extended spatially 

averaged 2-D shock location maps from Figure 6. The solid line is a 

power law fit to only the blue diamonds. 
25

Figure 6 2-D maps of shock location used to determine spatially averaged shock 

angles A (blue diamonds in Figure 5). Each blue diamond in Figure 5

comes from the arctangent of the slope of one colored data set here, where 

Y is the vertical shock location as a function of downstream distance X. 
26

Figure 7 Comparison of nose shock strength determined from DGV on two days 

(March 16 and 17) and from static pressure probes [19, 20] at various 

distances downstream of the model nose. 
27

Figure 8 Same data as shown in Figure 7. Both the shock strength (ordinate) and 

distance downstream (abscissa) have been normalized to reference values 

to illustrate a constant far field value. 
28

Figure 9 Example of velocity profiles from Reference 7 that were used to determine 

shock strength across 2-dimensional planar shocks from a flat plate in

Reference 7. 
29

2-D Airfoil

Mach 2

Nose

Shock

Tail

Shock

P

DGV

Light

Sheet

Midpoint

X = L / 2

E

S



L

x

Y

Figure 10 Geometry of typical supersonic flow over a generic lifting slender body. 
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Conical Shock-Strength Determination on a Low-Sonic-Boom Aircraft 

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Herring, Gregory C.; Meyers, James F.

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14. ABSTRACT

A nonintrusive technique Doppler global velocimetry (DGV) was used to determine conical shock strengths on a supersonic-cruise 

low-boom aircraft model. The work was performed at approximately Mach 2 in the Unitary Plan Wind Tunnel. Water is added to the wind 

tunnel flow circuit, generating small ice particles used as seed particles for the laser-based velocimetry. DGV generates two-dimensional 

(2-D) maps of three components of velocity that span the oblique shock. Shock strength (i.e. fractional pressure increase) is determined from 

observation of the flow deflection angle across the shock in combination with the standard shock relations. Although DGV had conveniently 

and accurately determined shock strengths from the homogenous velocity fields behind 2-D planar shocks, the inhomogeneous 3-D velocity 

fields behind the conical shocks presented additional challenges. Shock strength measurements for the near-field conical nose shock were 

demonstrated and compared with previously-published static pressure probe data for the same model in the same wind tunnel. Fair

agreement was found between the two sets of results.

15. SUBJECT TERMS

Velocity measurement; Supersonic speed; Flow velocity; Doppler global velocimetry; Sonic boom

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