This document discusses simulating a hot wire anemometer in a turbulent channel flow using computational fluid dynamics to analyze limitations of hot wire measurements. It has successfully simulated a turbulent channel flow and is beginning to simulate a simplified hot wire within the flow. Preliminary results show the hot wire drag fluctuation approximates velocity fluctuations measured without the wire. Future work includes accumulating spectral data and simulating different wire lengths and thermal effects.

1. DNS OF A HOT WIRE ANEMOMETER IN A TURBULENT
CHANNEL FLOW.
J. Herrera-Montojo1,2 and G. Borrell2
1 Mechanical & Aerospace Engineering. University of Strathclyde.
2 School of Aeronautics. Universidad Polit´ecnica de Madrid.
javier.herrera@strath.ac.uk
January 15th, 2014
1 Introduction
Hot wire anemometry is still one of the most suc-
cessful experimental techniques to measure velocity
components in turbulent flows. Hot wires are cheap,
relatively simple to set up, and have excellent spatial
and temporal resolution. Despite hot wire anemome-
try is well established nowadays, and that many details
can be found in handbooks like Tropea et al. (2007),
its limitations are still under discussion. This work in-
tends to address three of them.
The first one involves the usual application of Tay-
lor’s frozen turbulence assumption in hot wire mea-
surements, which supposes that the advection of a field
of turbulence past a fixed point can be taken entirely
due to the mean flow. This hypothesis was proven to
break down in the case of flows with strong shear by
Lin (1953), where turbulent intensities u are compa-
rable to the magnitude of the mean flow U. This as-
sumption may lead to wrong interpretations when wall
bounded flows are measured. A brief explanation can
be found in Moin (2009), and the quantitative analysis
in del ´Alamo and Jim´enez (2009).
The second one is about the effect of the hot wire
length. Many hot wires have a similar slender shape,
with a characteristic diameter many times smaller than
its length. When the length of the wire is larger than
the smallest scales found in the fluid motion, that are
of the order of the Kolmogorov scale η, the anemome-
ter acts as a filter in the direction parallel to the wire.
This behavior is a limiting factor when measuring high
Reynolds number wall-bounded turbulent flows, when
η is smaller than the size of the anemometer. This is a
first order error, given than the signal that is generated
is not able to capture all the energy-containing scales
in the flow, and the turbulent intensities are underval-
ued. This effect was estimated in Ligrani and Brad-
shaw (1987) for the case of single hot wires, and ex-
tended in Sillero and Jim´enez (2013), including multi-
ple wires.
The third limitation, also addressed in Sillero and
Jim´enez (2013), has to do with what the wire is actu-
ally measuring. In the case of a single wire aligned
with a given coordinate, the measurement is a com-
bination of the other two components. In the case of
wall-bounded flows, if u, v, and w are the streamwise,
wall-normal, and spanwise velocities respectively, a
wire aligned with the w component will measure a
combination of u and v. This caveat can be solved by
using a combination of wires, but at the cost of a larger
perturbation on the fluid.
The assessment of the measurement errors intro-
duced by hot wires in previous works is done a poste-
riori, based on how direct numerical simulation (DNS)
data has to be modified to fit anemometer measure-
ments. This allows to estimate each of the previously
mentioned errors one at a time, but it ignores the fact
that those effects are actually combined. Being such a
complex phenomenon, many aspects could be under-
stood by simulating a hot wire in a realistic environ-
ment, like a channel flow. This is, however, a complex
simulation that has never been successfully completed.
The goal of this study is to simulate a schematic
configuration of a hot wire in a turbulent channel flow
at a low but significant Reynolds number with Open-
FOAM (2013), that can be used to analyze the coupled
effects of the described limitations. This would be an a
priori analysis, given that the correct measure can be
obtained just by repeating the simulation without the
wire.
OpenFOAM is a registered trademark of SGI corp.
2 Problem formulation
This problem can be divided in two. First, a DNS
of a turbulent channel has to be simulated with suffi-
cient resolution. Later, once this outer flow has con-
verged, a simplified version of a hot wire is placed in
it, and the output signal of the wire is obtained from
the complete simulation after a short transient.
Simulation of a turbulent channel flow
A DNS of a turbulent channel flow at Reτ 600,
the Reynolds number defined as Reτ = uτ h/ν, where
uτ is the friction velocity, h is the half-height of the
channel, and ν is the kinematic viscosity; is simulated

2. in a domain of size 2h×πh×πh/2 in the wall-normal,
streamwise, and spanwise direction respectively. The
discretization is a structured rectangular-prism mesh
based on a 129 × 192 × 192 array of nodes, that will
become unstructured as soon as the wire is placed
within the domain.
A turbulent channel has been chosen because it is
a flow with a strong shear, can be simulated with a
reasonable computational cost, and is a realistic wall
bounded flow. It is also well understood, and many
high quality simulations in the same range of Reynolds
numbers are available to compare with. On the other
hand, this channel is a periodic flow in both stream-
wise and spanwise directions, and any object placed
in it will become an infinite array of identical objects
with the same periodicity.
This relatively cheap simulation can be considered
DNS on its own right, since the resolution is suffi-
cient, and the domain size is able to capture correctly
the smallest scales. A good guide on how to simu-
late channel flows with small boxes can be found in
Lozano-Dur´an and Jim´enez (2014).
This simulation is not straightforward in Open-
FOAM. DNS of channel flows are seldom com-
puted with general purpose codes. In the case of
OpenFOAM, the pressure boundary conditions in the
streamwise direction are periodic. The pressure gra-
dient along the channel had to be enforced by a mo-
mentum source computed each time step. The average
flow-rate is calculated and compared with the desired
value. From this, the needed pressure gradient to ob-
tain the desired average velocity is computed.
Hot wire in a shear flow
Given a channel flow, one can put to test any hot-
wire anemometer in any position and condition. Since
the goal of this project is to understand the possible
limitations of this experimental technique, it is reason-
able to place it where the conditions are more adverse,
where the relative turbulent intensity u /U, is maxi-
mum. This location is at a distance of 15+ (see Kim,
Moin and Moser (1987)), where the + superindex in-
dicates the use of wall units, respect to the wall.
The size of the wire is also important. It is to expect
that, if the effect of a wire too large is analogous to
filter in the spanwise direction, the attenuation of the
signal is proportional to the wire length. According to
the estimations in Ligrani and Bradshaw (1987), and
Sillero and Jim´enez (2013), attenuation becomes an
issue with a wire of length beyond 20+. The chosen
length has been 25+.
Given that this channel flow is periodic in stream-
wise direction, this simulation corresponds to a peri-
odic array of wires. The expectation is that, being such
a small item compared with the rest of the flow, the
feedback of the wake of the wire is not an issue. This
condition may change if the wire is simulated with the
prongs, that are sensibly bigger the wire alone. In con-
sequence, the wire is simulated with no prongs, that is
equivalent to assume that the prongs have no influence
on the measure obtained by the wire.
With this configuration, the relative size of the wire
respect to the complete domain is minuscule, as it can
be seen in figure 1.
Figure 1: Detail of the thickness of the wire at its slim part,
and its relative size compared with the channel.
3 Preliminary results
The channel flow has been sucessfully simulated in
OpenFOAM. A comparison of the present simulation
with some previous cases at similar Reynolds numbers
is shown in figure 2.
(a)
10
0
10
1
10
2
0
0.5
1
1.5
2
2.5
3
y+
u’
+
,v’
+
,w’
+
u’
+
w’
+
v’
+
Reference
Present
(b)
10
0
10
1
10
2
0
5
10
15
20
25
y+
U
+
Reference. Re550
Present
Reference. Re950
Figure 2: Comparison of the present turbulent channel sim-
ulation with some available datasets at similar
Reynolds numbers: (a) turbulent intensities, and
(b) mean velocity profile. The cases used for this
comparison are described in Lozano-Dur´an and
Jim´enez (2014)

3. At this stage of the project, thermal effects have
not been explored yet. The required mesh size to re-
solve the thermal boundary layer around a hot wire is
so fine that the time step becomes a severe limitation.
Previously to study the complete problem, the goal is
to understand the flow around the hot wire as a solid
body in a channel flow.
This simplified problem can give an approximation
of the effect of the three phenomena described in the
introduction. Instead of using the total heat transfer
of the wire, drag is used. Despite these two quantities
are loosely related, at this low Reynolds number, they
are roughly proportional (see White (1946)). Drag can
give, therefore, a good approximation of the frequency
response of the actual hot wire, and how the measure
is influenced by the different components of velocity.
An important condition for the results is that the
average wall-normal velocity is zero. This means that
the average value of the streamwise component of drag
Dx (capital letters denote average values), could be ap-
proximated as function of only the average streamwise
velocity U.
The interesting quantity is the fluctuation of the to-
tal drag d, that can be used to measure the velocity
fluctuation u. However, drag has to be a function of
both components of the velocity fluctuations. The val-
ues of d, u, and v for a short simulation time is pre-
sented in figure 3.
Figure 3: Time evolution of the fluctuations of d (right axis),
u and v (left axis). The drag is measured over the
wire, while velocities are obtained repeating the
experiment without the wire.
4 Conclusions and future work
The most important conclusion of this work, con-
sidering that this project is on a very preliminary stage,
is that simulating hot wires is possible, and it can be
useful to understand the limitations of its use in wall
bounded flows. The immediate goal is to accumulate
enough data to obtain a converged spectrum for d .
Once this first experiment is finished, other wire sizes
have to be introduced to explore if different lengths
can be modelled by using different filter widths in the
spanwise direction.
The next step after the solution without thermal ef-
fects has been explored is to mesh the thermal bound-
ary layer around the wire and to solve the coupled
problem. This simulation does not require a huge
amount of memory, but it needs to run for a very long
time due to the enormous scale separation between the
different lengths involved in the phenomenon.
Acknowledgments
The authors woud like to thank Javier Jim´enez,
Julio Soria, and Juan Antonio Sillero. This project
would not have been possible whithout their excellent
advice. The work of G. Borrell is funded by the the
European Research Council grant ERC-2010.AdG-
201000224.
References
del ´Alamo, J. C. and Jim´enez, J. (2009) Estimation of tur-
bulent convection velocities and corrections to Taylor’s
approximation. J. Fluid Mech. 640, 5-26.
Kim, J. Moin, P. and Moser, R. D. (1987) Turbulence statis-
tics in fully developed channel flow at low Reynolds num-
ber. J. Fluid Mech. 177, 133-166.
Ligrani, P. M. and Bradshaw, P. (1987) Spatial resolution and
measurement of turbulence in the viscous sublayer using
subminiature hot-wire probes. Exp. in Fluids. 5, 407-417.
Lin, C. C. (1953) On Taylor’s hypothesis and the accelera-
tion terms in the Navier-Stokes equations. Q. Appl. Math.
10 (4), 295306.
Lozano-Dur´an, A. and Jim´enez, J. (2014) Effect of the com-
putational domain on direct simulations of turbulent chan-
nels up to Reτ = 4200. Physics of Fluids (in press).
Moin, P. (2003) Revisiting Taylor’s hypothe-
sis. Journal of Fluid Mechanics, 640, pp 1-4
doi:10.1017/S0022112009992126
OpenFOAM Foundation. (2011-2013) The OpenFOAM
documentation, version 2.2.2.
Sillero, J.A. and Jim´enez, J. (2013) Effects Of Hot-wire
Measurement In Wall-bounded Flows Studied Via Direct
Numerical Simulation. European Turbulence Conference
14, Lyon.
Tropea, C. Yarin, A. L. and Foss, J. F. Eds. (2007) Springer
Handbook of Experimental Fluid Mechanics. Chapter 5,
Velocity, Vorticity and Mach Number.
White, C. M. (1946) The drag of cylinders in fluids at slow
speeds. Proc. of The Royal Soc. A. 186, 472-479.