______________________________________________
Proceedings of Combustion Institute – Canadian Section
Spring Technical Meeting
University of Saskatchewan
May 11-14, 2015
Evaluation of Fast RBF Acoustic Tomography Measurement of
Flame Temperature and Velocity Field
Travis Wiens*
Department of Mechanical Engineering, University of Saskatchewan
57 Campus Drive, Saskatoon, SK, S7N 5A9, Canada
Abstract
This paper presents a method of acoustic tomography suitable for real-time measurement and visualization
of temperature and velocity fields within turbulent flows. Acoustic tomography involves surrounding the
measurement volume with a number of acoustic transducers, which are used to determine the sonic time-offlight between each pair of transducers. This sonic data can then be used to reconstruct the temperature and
velocity fields within the measurement volume. The paper presents a method of reconstruction that is highly
computationally efficient and allows for real-time calculation and display of these fields. A simulation study
is conducted to evaluate the feasibility of application of this method to measurement within flames, taking
into consideration features unique to this problem, such as refraction due to thermal gradients and timevarying gas properties of combustion products.
1. Introduction
Tomography refers to the reconstruction of a scalar or vector field from a collection of line integrals through the field.
This is commonly used in medical imaging systems, such as computed tomography (CT) scans. Acoustic tomography
refers to the measurement of fluid velocity and temperature fields from a collection of sonic time-of-flight
measurements through the area of interest. This has been applied to measurement of flame temperature and velocity
in furnaces and burners [1,2,3,4]. However, the methods used in these papers are time consuming, typically requiring
numerical iterative solutions to reconstruct flows and temperatures. In previous papers [5,6], the author has developed
a high-speed non-iterative method of reconstruction of the vector and scalar fields using radial basis function networks.
This paper will investigate the feasibility of applying that method to measurement of flames.
2. Methodology
2.1 Basic Problem Definition
Consider an ethanol pool fire in a large room. Figure 1 shows a temperature and velocity slice at an instant of time.
The flow is surrounded by acoustic transducers, shown as ‘x’s in the Figure.
Figure 1: Slice of temperature and velocity fields for the area above a simulated ethanol pool fire.
*
Assistant Professor, corresponding author, [email protected]
The forward tomography problem is the determination of the time required for a sonic signal to travel from one
transducer to another. The time of flight for path i, between transducers at positions Xi1 and Xi2 is
t i  
Xi 2
Xi 1
dl
c ( X)  U ( X)  Lˆ i
(1)
ˆ(X)i is a
where c(X) is the sonic speed, U(X) is the two-dimensional fluid velocity vector field at position X, L
unit vector parallel to the sound ray, and dl is an integration length along the ray’s path. For ideal gases, the sonic
speed is related to absolute temperature T by
c  T R
(2)
where R is the specific gas constant of the fluid, and  is the fluid’s specific heat ratio.
The inverse problem is to reconstruct the fields c(X) (and thereby T(X)) and U(X) from a number of measurements
ti , and is the basis of tomography. While the Radon Transform may be used to perform this reconstruction, it
requires specific orientation of transducers and only works for a scalar field. Other researchers have developed
reconstruction algorithms to solve this inverse problem but they are typically nonlinear and iterative, and therefore
computationally slow, or only apply to scalar fields [7].
2.2 Simplified Forward Problem
If we assume that there are no large temperature or velocity gradients in the field, the sonic signals will follow straight
paths from transmitter to receiver (i.e. no refraction of sound). If we assume that variations in sonic speed are very
small relative its mean value, we can linearize Equation 1 about the mean sonic velocity c0, and zero fluid speed,
resulting in:
Δ
Δ
∘
d
(3)
where li is the path distance [8].
If we further assume that the fluid has homogenous ideal gas properties of air, we can calculate the sonic speed from
the temperature field. While these assumptions are not controversial for atmospheric flows, flame flows tend to
challenge these three assumptions. The effect of these assumptions will be evaluated in a later section of this paper.
2.3 RBF Tomographic Inversion
In previous work, the author developed a computationally efficient method of solving the inverse problem [5,6]. This
involves using a network of radial basis functions (RBF) to represent the fields. A RBF network is a weighted sum of
nonlinear functions:
∑
(4)
where there are Nr basis functions with centres at positions Xc distributed over the measurement area. The basis
functions, , are nonlinear functions. Commonly used functions are Gaussian (
and polyharmonic
. A polyharmonic spline of 3rd order (n=3) is used for calculations in this paper.
splines, which take the form
The weights, W, are then selected to give the required function output. Notice that, if the parameters Xc are held
constant, finding appropriate weights to approximate some data is a linear inversion problem:
.
(5)
Two RBF networks are used to model the temperature and velocity fields. The estimated sonic speed field is given by
Δ ̂
.
The velocity field is represented by the stream function, with its estimate given by the RBF network
(6)
ˆ ( X)  Φ ( X) W
.
(7)
With some algebraic manipulation, Equation 3 can be written as
Δ
Δ d
∘ d.
(8)
By substituting Equation 6 into the first integral, we get
Δ d =
d
∑
d
(9)
As the line integral of the basis function is independent of the flow (assuming no refraction) and can be calculated
. A similar procedure can be
from the problem geometry in advance, this can be simplified to the linear form
applied to the velocity’s stream function and Equation 9 can be written in the form
W
(10)
where d is a Np x 1 row vector of data,  is an Np x 2 Nr matrix including the integral in Equation 9, and W is an 2 Nr
x 1 parameter vector to be solved for using standard linear matrix methods. Notice that the computationally intensive
matrix inversion need only be performed when the problem geometry changes, not for every time-of-flight
measurement. This allows the temperature and velocity fields to be quickly determined in real time at useful sample
rates for turbulent flows (i.e. at rates in the kHz range).
2.4 Simulation Study
A simulation study was conducted in order to evaluate the effects of the assumptions in Section 2.1 (namely: no
refraction of sound paths, homogenous ideal gas properties, and linearized sonic speed). Target temperature, flow
fields, and combustion product concentrations were calculated using the NIST Fire Dynamics Simulator (FDS).
This program’s ethanol pool fire example was selected as a test case. A representative frame of data, occurring 2.86
seconds after ignition was used as the reference flow and temperature field. Temperatures, velocities, and gas
species concentrations were calculated on a 50 mm grid on a plane through the centre of the 500 mm by 500 mm
pool, as shown in Figure 1.
This flow field was used to evaluate the accuracy of the inversion algorithm. 16 transducers were located in a circle
of 1 m diameter around the measurement area, located 2 m above the pool, as shown in Figure 1. Times of flight for
each path were calculated by numerically integrating across the calculated target field for the following cases:
1) Straight sonic paths between transducers with time of flight calculated using the linearized Equation 3,
ignoring changes in ideal gas properties due to combustion products. (This matches the assumptions in the
reconstruction scheme, and the reconstruction of fields should be at its best).
2) Same as case 1, the sonic paths are curved to follow refraction due to temperature gradients.
3) Same as case 1, except the travel time is calculated using the nonlinear Equation 1, rather than its
linearization in Equation 2. Ideal gas properties for air are used.
4) Same as case 1, except ideal gas properties vary across the measurement area according to properties of
calculated combustion products.
5) Includes effects of refraction, nonlinear time of flight and combustion products.
For each case, the root mean squared error, ET, was calculated between the target temperature and the reconstructed
temperature, averaging over the measurement area. The RMS error in velocity vector, EU, was also calculated. This
procedure was repeated 10 times using different randomly generated RBF positions in order to calculate the variance
in the mean error values.
3. Results and Discussion
The results of the analysis are summarized in Table 1, showing the RMS error in temperature and velocity that can be
attributed to refraction, nonlinear sonic speed, and the effect of combustion products. Figures 2 and 3 show the target
and reconstructed temperature and velocity fields for cases 1 (best) and 5 (worst). Notice that the curved sonic paths
shown in white are refracted by the temperature gradient in the case 5 data.
Table 1. RMS reconstruction error in temperature and velocity for each of the five cases. Case 1 has no refraction
effects, nonlinear sonic speed or combustion product effects. Case 2 includes refraction, case 3 includes nonlinear
travel time calculation, and case 4 includes combustion gas properties. Case 5 includes all three effects.
Case
1
2
3
4
5
50.99 +/- 1.11
84.91 +/- 0.45
73.00 +/- 0.61
61.30 +/- 1.73
100.15 +/- 0.57
0.928 +/- 0.014
2.334 +/- 0.035
2.028 +/- 0.009
0.897 +/- 0.049
2.479 +/- 0.0487
Target
2.5
2.4
2.3
2.2
y (m)
2.1
2
1.9
1.8
1.7
1.6
1.5
-0.5
0
x (m)
Reconstruction: Case 5
2.5
2.5
1200
2.4
2.4
1100
2.3
2.3
1000
2.2
2.2
900
2.1
2.1
800
2
2
1.9
1.9
1.8
1.8
1.7
1.7
1.6
1.6
700
600
500
400
300
1.5
-0.5
0
x (m)
1.5
0.5 -0.5
0
x (m)
0.5
Figure 2. Target and reconstructed temperature fields for case 1, which does not include effects of refraction,
nonlinear sonic speed or combustion products, and case 5, which includes all effects.
That (K)
Reconstruction: Case 1
0.5
2.5
Target
Reconstruction
y (m)
2.4
2.4
2.3
2.3
2.2
2.2
2.1
2.1
2
2
1.9
1.9
1.8
1.8
1.7
1.7
1.6
1.6
1.5
-0.5
0
x (m)
6
Target
Reconstruction
0.5
1.5
-0.5
5
4
3
EU (m/s)
2.5
2
1
0
x (m)
0.5
0
Figure 3. Target and reconstructed velocity fields for cases 1 (left) and 5 (right), with shading denoting error.
As the largest errors are due to refraction and the nonlinear time-of-flight relationship, further study was performed to
quantify the effect of these errors as the maximum temperature changes. The maximum temperature in the field was
varied by rescaling the reference temperature field; the RMS errors were then calculated for maximum temperatures
varying between 168 and 1325 oC, shown in Figure 4. As expected, the errors due to refraction and nonlinearity
increase as the temperature variation within the field increase.
RMS Temperature Error/Temperature Range (%)
7
6.5
Case 1: Base
Case 2: Refraction
Case 3: Nonlinear c
6
5.5
5
4.5
4
3.5
0
200
400
600
800
1000
Field Temperature Range (K)
1200
1400
Figure 4: Effect of field temperature range on temperature accuracy when including effects of refraction and
nonlinear sonic speed.
4.
Conclusion
This paper evaluates the feasibility of measuring flame temperatures by reconstructing the temperature and velocity
fields of an ethanol pool fire under a number of assumptions. It was found that the reconstruction was not significantly
degraded by the assumption of constant gas properties, but that there are significant errors due to refraction of sound
waves by the temperature gradient and also due to the linearization of the sonic travel time. These errors together
combined to cause an RMS error of 100.1 K in temperature (over a field with a temperature range of 1286 K) and an
RMS error of 2.48 m/s in velocity (with a maximum field velocity of 9.09 m/s). Based on these results, the highspeed tomography method may be used (which requires the simplifying assumptions tested here), but significant errors
should be expected. Further work will continue on a high-speed tomography method that takes refraction and
nonlinear flight times into account.
References
[1] Sielschott, Helmut. "Measurement of horizontal flow in a large scale furnace using acoustic vector tomography."
Flow measurement and instrumentation 8.3 (1998): 191-197.
[2] Li, Yan-Qin, and Huai-Chun Zhou. "Experimental study on acoustic vector tomography of 2-D flow field in an
experiment-scale furnace." Flow Measurement and Instrumentation 17.2 (2006): 113-122.
[3] Lu, J., et al. "Acoustic computer tomographic pyrometry for two-dimensional measurement of gases taking into
account the effect of refraction of sound wave paths." Measurement Science and Technology 11.6 (2000): 692.
[4]Gan, Tat Hean, and David A. Hutchins. "Air-coupled ultrasonic tomographic imaging of high-temperature flames."
Ultrasonics, Ferroelectrics, and Frequency Control, IEEE Transactions on 50.9 (2003): 1214-1218.
[5] Wiens, Travis, and Paul Behrens. "Turbulent flow sensing using acoustic tomography." Proceedings of Inter-noise
2009 (2009).
[6] Wiens, Travis. “Sensing of Turbulent Flows Using Real-Time Acoustic Tomography,” Proceedings of the XIXth
Biennial Conference of the New Zealand Acoustical Society, Auckland, 2008.
[7]Tabuchi, H., et al. "Computerized Tomography with radial basis functions network: a neuro-fuzzy approach."
Neural Networks, 1995. Proceedings., IEEE International Conference on. Vol. 5. IEEE, 1995.
[8] V. Ostashev, S. Vecherin, D. Wilson, A. Ziemann, and G. Goedecke, “Recent Progress in Acoustic Tomography
of the Atmosphere,” IOP Conference Series: Earth and Environmental Science, vol. 1, no. 1, 2008.