Towards a theoretical model of heat transfer for hot-wire anemometry close to solid walls

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Towards a theoretical model of heat transfer for hot-wire anemometry close to solid walls Yuta Ikeya a,b, Ramis Örlü b,∗, Koji Fukagata a, P. Henrik Alfredsson b a b

Department of Mechanical Engineering, Keio University, Hiyoshi 3-14-1, Yokohama 223-8522, Japan Linné FLOW Centre, KTH Mechanics, Osquars Backe 18, SE-100 44 Stockholm, Sweden

a r t i c l e

i n f o

Article history: Available online xxx Keywords: Hot-wire anemometry Wall turbulence Heat transfer

a b s t r a c t Hot-wire anemometry readings where the sensor is close to a solid wall become erroneous due to additional heat losses to the wall. Here we examine this effect by means of experiments and numerical simulations. Measurements in both quiescent air as well as laminar and turbulent boundary layers confirmed the influences of parameters such as wall conductivity, overheat ratio and probe dimensions on the hot-wire output voltage. Compared to previous studies, the focus lies not only on the streamwise mean velocity, but also on its fluctuations. The accompanying two-dimensional steady numerical simulation allowed a qualitative discussion of the problem and furthermore mapped the temperature field around the wire for different wall materials. Based on these experimental and numerical results, a theoretical model of the heat transfer from a heated wire close to a solid wall is proposed that accounts for the contributions from both convection and conduction. © 2017 Elsevier Inc. All rights reserved.

1. Introduction Hot-wire anemometry (HWA) has been the most widely used laboratory method to measure local fluid velocities in experimental fluid mechanics, which enabled the study of instability waves and turbulent fluctuations quantitatively. Furthermore, it was the only method capable of measuring high frequency and amplitude velocity fluctuations with a high spatial resolution and has been dominant in the experimental field until the development of laserbased techniques such as laser Doppler velocimetry (LDV) and particle image velocimetry (PIV). Nowadays, HWA remains the preferred measurement technique for acquiring data in both laminar and turbulent wall-bounded flows. However, a well-known major drawback in HWA is that a hotwire probe calibrated in the wall-remote region registers a seemingly higher mean velocity in the near-wall region, known as the wall-proximity effect. Additional heat losses from the heated sensor to the cooler wall are erroneously read as an increase in velocity as the wire approaches the wall surface. The wall-proximity effect causes a problem especially for measurements of the velocity derivative close to the wall, or when the absolute wall position needs to be obtained/corrected, through a fit of the data in the linear velocity region existing close to the wall (Örlü et al., 2010).

∗

Corresponding author. E-mail address: [email protected] (R. Örlü).

The wall-proximity effect has been investigated in many studies, and several of them have been concerned with possible correction schemes for the mean velocity and its dependence on operational and geometrical parameters. Generally, it is widely accepted that the wall conductivity, overheat ratio, and sensor dimensions have an influence on the erroneous velocity reading. In particular, it is known that highly conductive materials register larger apparent velocity readings than poorly conductive materials (Polyakov and Shindin, 1978; Bhatia et al., 1982; Durst and Zanoun, 2002). Similarly, larger length-to-diameter ratios l/d of the wire result in a larger apparent velocity reading (Krishnamoorthy et al., 1985; Chew et al., 1995). Also, the larger the overheat ratio is, the larger the apparent velocity reading becomes (Krishnamoorthy et al., 1985; Zanoun et al., 2009). However, despite the aforementioned results the detailed principle of the heat transfer for the hot-wire in the near-wall region including its interaction with the wall material is still not fully understood. Furthermore, most of the previous studies are concerned with errors in the mean velocity and there is little knowledge of the measured turbulence quantities, i.e. turbulence intensity, higher-order moments and in turn probability density distribution. In light of the recent demands for increased accuracies in determining the friction velocity and/or absolute wall-position (Örlü et al., 2010), the interest in higher-order moments in the near-wall region (Örlü et al., 2016) as well as its wall-limiting quantities, e.g. the fluctuating wall-shear stress (Alfredsson et al., 1988; Örlü and Schlatter, 2011; Lenaers et al., 2012; Vinuesa et al. 2017), there is a

http://dx.doi.org/10.1016/j.ijheatfluidflow.2017.09.002 0142-727X/© 2017 Elsevier Inc. All rights reserved.

Please cite this article as: Y. Ikeya et al., Towards a theoretical model of heat transfer for hot-wire anemometry close to solid walls, International Journal of Heat and Fluid Flow (2017), http://dx.doi.org/10.1016/j.ijheatfluidflow.2017.09.002

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Fig. 1. Schematic of the setup for the no flow, i.e. free/natural convection, measurements.

need to revisit the effect of hot-wire measurements close to solid walls. In the present investigation a systematic parameter study on the misreading of hot-wire anemometry in the near-wall region is carried out so that further insight can be provided in order to eventually help researchers to correct for or correctly acknowledge these effects. In particular, measurements under no-flow (i.e. free convection) and flow conditions (i.e. forced convection), in a laminar and a turbulent boundary layer, have been performed by varying the wall material, overheat ratio, and probe dimensions and the details and results are reported in Section 2. The numerical counterpart including a study on the heat conduction inside the wall material is given in Section 3. Based on the results from Sections 2 and 3, a theoretical model to simulate the wall effect is proposed in Section 4 and its capability of estimating the heat transfer from a heated wire close to a solid wall is discussed, upon which the work is summarised and concluded in Section 5. 2. Experimental part 2.1. Natural convection measurements To study the effect of various parameters on the heat transfer from the hot wire in the absence of a forced flow, i.e. under free convection conditions, measurements in an enclosed Plexiglas box were performed. The schematic of the setup is illustrated in Fig. 1. The hot-wire probe can manually be traversed normal to the wall by means of a linear micrometer and to study the influence of the wall distance the output voltage E of the anemometer was acquired at thirty-five heights up to a distance of y = 2 mm from the wall. Additionally, the voltage output at y = 5 mm was recorded as E0 , where the effect of the wall is considered to be negligible. The effect of thermal conductivity of the wall was investigated by changing the wall material between aluminum, brass, steel, Plexiglas, and styrofoam. Besides the wall material, the wire length and resistance overheat ratio

aR =

Rw − R0 , R0

(1)

were also taken as parameters to see their influence on the voltage reading. Here, R0 denotes the sensor electrical resistance at ambient temperature, i.e. the reference state, and Rw denotes the resistance under operation. 2.2. Forced convection wind-tunnel experiment Hot-wire measurements were also carried out inside the Minimum Turbulence Level (MTL) closed-loop wind tunnel located at KTH Royal Institute of Technology in Stockholm, which has a 7 m long test section and a cross-sectional area of 0.8 × 1.2 m2 . Details about the specific wind-tunnel setup can be found in Sanmiguel Vila et al. (2017). A probe was mounted on a computer controlled traversing system above a flat plate as shown in Fig. 2. The flat plate has both

Fig. 2. Schematic of the setup for the wind-tunnel, i.e. forced convection, experiments.

aluminum and Plexiglas surfaces at different spanwise positions at the same streamwise location, which were used to investigate the effect of wall conductivity. Furthermore, both laminar and turbulent boundary layers developing on the plate with zero-pressure gradient were considered with momentum-loss thickness Reynolds numbers (Reθ ) of around 400 and 950, respectively. The established boundary layers adhere to canonical zero-pressure gradient laminar, i.e. Blasius, and turbulent boundary layers as will be shown later on in Fig. 5a) and Figs. 6 and 7, respectively. The sampling frequency in this measurement was 20 kHz and the sampling time was 10 s. The calibration of the probes was carried out in the free-stream and upstream of the flat plate, against a Prandtl tube which was also used to monitor the free-stream velocity in the tunnel. The free-stream velocity is controlled by a computer and the corresponding voltage output from the probe was recorded. The voltage without flow E0 was also recorded and used for the calibration. In the present study, a 4th-order polynomial was used to relate the top-of-the-bridge voltage to the velocity reading (see e.g. George et al., 1989). 2.3. Experimental results Results from the measurements on different wall materials in quiescent air are shown in Figs. 3a) and 4a) and show the expected dependency of the wall conductivity on the hot-wire reading (platinum core wire with 2.5 μm diameter and 0.7 mm nominal length operated at an resistance overheat ratio aR = 0.8). In accordance with Durst et al. (2002), large differences can be observed between poorly conducting walls (Plexiglas and styrofoam with heat conductivities of the order of 10−1 and 10−2 Wm−1 K−1 , respectively), while the results from highly conducting materials (such as aluminum, brass and steel, with heat conductivities of the order of 101 –102 Wm−1 K−1 ) do not vary between each other. Due to the lack of wall materials that show a dependence on the conductivity on the hot-wire reading (in the present study), we are not in a situation to propose a scaling. However, as apparent from Fig. 4a), where the values on the ordinate are scaled with its wall-closest value (where E ∗ = (E − E0 )/E0 ), the curves behave self-similar. The dependency on the overheat ratio for the same probe on the aluminum wall, shown in Fig. 3b), is also in accordance with the main body of previous studies (Krishnamoorthy et al., 1985; Durst and Zanoun, 2002). However, it was found that the three curves for these different overheat ratios can be collapsed into a single curve by utilizing the wall-remote hot-wire output voltage E0 as shown in Fig. 4b), albeit with a noticeable discrepancy for the aR = 0.30 case at the wall proximity. Likewise, the voltage output from probes with different sensor length is plotted in Figs. 3c) and

Please cite this article as: Y. Ikeya et al., Towards a theoretical model of heat transfer for hot-wire anemometry close to solid walls, International Journal of Heat and Fluid Flow (2017), http://dx.doi.org/10.1016/j.ijheatfluidflow.2017.09.002

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Fig. 3. Voltage difference from hot-wire readings in quiescent air as function of wall-distance a) on different wall materials measured with a resistance overheat ratio aR = 0.8, b) at different resistance overheat ratios measured on aluminum, and c) measured by probes with different sensor dimensions on Plexiglas with a resistance overheat ratio aR = 0.8. The diameter of the probes used is consistent throughout the three figures at d = 2.5 μm.

Fig. 4. Voltage difference from hot-wire readings in quiescent air as function of normalised wall-distance scaled with the wall-remote voltage a) on different wall materials measured with a resistance overheat ratio aR = 0.8, b) at different resistance overheat ratios measured on aluminum, and c) measured by probes with different sensor dimensions on Plexiglas with a resistance overheat ratio aR = 0.8. Note that in subplot a), the ordinate (E ∗ = (E − E0 )/E0 ) is additionally scaled by the wall-closest value (E∗ |y → 0 ), thereby forcing the wall-closest value to unity. The diameter of the probes used is consistent throughout the three figures at d = 2.5 μm.

4c). The raw output voltage shows its dependency on the sensor length, namely a higher output is registered with increasing sensor size l/d, as the exposed area of the sensor becomes larger thus more current is supplied. However, the curves do not scale such that the output voltages at different over-heat ratios collapse onto one curve, but appear to reach a converged state for larger l/d ratios. The discrepancy for low l/d might be due to three-dimensional effects. Moving from free to forced convection, Fig. 5 shows the overheat-ratio dependency in a laminar boundary layer. Fig. 5a) depicts the data in outer-scaling using the free-stream velocity and displacement thickness as common for laminar boundary layers and compares it to the Blasius solution (Blasius, 1908). While departures from the linear behaviour are barely visible in this way of scaling, they become more apparent when considering them in a log-log plot in viscous units as done in Fig. 5b). As apparent, the overheat ratio exhibits the same effect as for the results in quiescent air, i.e. the higher the overheat ratio, the stronger the deviation from the linear profile. In both cases, the effect is, however, limited to y+  3, where the superscript ‘+’ denotes scaling in wall units, i.e.

U+ =

U , uτ

y+ =

uτ y

ν

,

(2) (3)

where uτ represents the friction velocity determined as



Fig. 5. Velocity profile in a laminar boundary layer at Reθ ≈ 400 on the aluminum wall at a resistance overheat ratio of aR = 0.3 (thin line) and aR = 0.8 (thick line). a) Velocity and wall distance scaled with the free-stream velocity and displacement thickness, respectively, and compared to the theoretical Blasius solution (Blasius, 1908). b) Velocity and wall distance scaled in wall units and compared to the linear profile U + = y+ . Inset highlights the viscous sublayer.

τw , ρ  ∂U  τw = μ  , ∂ y wall uτ =

(4)

(5)

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Fig. 6. Effect of different wall materials in a turbulent boundary layer at Reθ ≈ 950 measured with a resistance overheat ratio of aR = 0.8: aluminum (red) and Plexiglas (blue). a): Inner-scaled mean and rms profile. b): Diagnostic plot with inset highlighting the viscous sublayer. c): Inner-scaled velocity PDF. Thin lines denote 1, 5, 20, 40, 60 and 90% of the local maximum (thick line) of the PDF. Thin lines (magenta) in a) and b) depict DNS at matched Reθ (Schlatter and Örlü, 2010). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 7. Effect of different resistance overheat ratios aR in a turbulent boundary layer at Reθ ≈ 950 measured on an aluminum wall: aR = 0.3 (black) and aR = 0.8 (red). a): Inner-scaled mean and rms profile. b): Diagnostic plot with inset highlighting the viscous sublayer. c): Inner-scaled velocity PDF. Thin lines denote 1, 5, 20, 40, 60 and 90% of the local maximum (thick line) of the PDF. Thin lines (magenta) in a) and b) depict DNS at matched Reθ (Schlatter and Örlü, 2010). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

with ρ , μ and τ w denoting the fluid density, dynamic viscosity and wall shear stress, respectively. The mean streamwise velocity and root-mean-square profiles for a turbulent boundary layer are shown in Fig. 6a) for two different wall materials (aluminum and Plexiglas) and display no differences in the inner layer of the boundary layer. As a reference also the direct numerical simulation (DNS) of a zero-pressure gradient turbulent boundary layer by Schlatter and Örlü (2010) at matched Reθ is shown. The marginal differences in the outer layer are due to slightly different conditions of the boundary layers (i.e. slight differences in the Reynolds number as well as probable inhomogeneities in the spanwise direction, which can easily be caused by tripping effects at these low Reynolds numbers, see e.g. Schlatter and Örlü, 2012 and Sanmiguel Vila et al., 2017). One should, however, recall that the accurate determination of the absolute wall position in wall-bounded flows is by no means trivial (Örlü et al., 2010), and that small differences can easily be obscured (due to inaccuracies in the absolute wall position and/or the determined friction velocity by shifting the profiles by less than one inner unit) in a semi-logarithmic plot. If one instead considers the profiles in the so-called diagnostic plot (Alfredsson et al., 2011) shown in Fig. 6b), which is independent of the wall position and the friction velocity, differences do appear in the region U/U∞ < 0.25; which here corresponds to the viscous sublayer (Alfredsson and Örlü, 2010). As apparent, the measured turbulence intensity (and in turn the related rms value of the fluctuating wall shear stress, i.e. τw,rms /τw = lim urms /U) is y→0

reduced for highly conducting materials. To illuminate this effect further Fig. 6c) depicts the probability density distribution (PDF) for the streamwise velocity fluctuations in inner scaling. In accordance with Alfredsson et al. (2011), the PDF contour lines should in

such a plot be parallel to each other in the viscous sublayer, which is observed for the contour lines at higher velocities. The deviation at lower velocities is more apparent for the highly conducting wall material. The aforementioned observations can also be made when considering the effect of the overheat ratio as demonstrated in Fig. 7. 3. Numerical part 3.1. Physical model and boundary conditions In the present study, the hot-wire sensor is modelled as an infinitely long cylinder parallel to a wall and normal to the flow as shown in Fig. 8. The entire computational domain is divided into a fluid and solid region. The wire center is located at (x/d, y/d ) = (0, yw /d ) and the domain is within −30 0 0 < x/d < 60 0 0 in the streamwise direction. The fluid region is extended from 0 < y/d < yw /d + 50 0 0 and the solid region is within −50 0 0 < y/d < 0 in the wall-normal direction. The two-dimensional steady numerical simulations use OpenFOAM (version 2.2.2) and the unstructured mesh is created with ANSYS ICEM. The domain contains 339,040 points in the fluid region (180 nodes on the circumference of the hot-wire surface) and 197,600 points in the solid region. A Couette flow is reproduced to simulate the phenomenon of a hot wire located in the viscous sublayer. In the present calculation, the inner-scaled distance between the wire center and the wall surface is changed by varying the velocity gradient S = dU/dy|inlet . The temperatures at the inflow and the top moving wall are set to T∞ = 20◦ C. For the calculation, two temperature overheat ratios aT = 0.27 and 0.96 were investigated, where the temperature overheat ratio is defined with the surface temperature of the wire Tw

Please cite this article as: Y. Ikeya et al., Towards a theoretical model of heat transfer for hot-wire anemometry close to solid walls, International Journal of Heat and Fluid Flow (2017), http://dx.doi.org/10.1016/j.ijheatfluidflow.2017.09.002

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The inlet velocity at the height of the wire center Uw and the wire diameter d are employed to normalize the velocity components and coordinates, respectively, while the temperature is scaled as T ∗ = (T − T∞ )/(Tw − T∞ ). The thermophysical properties ρ ∗ , μ∗ , k∗ , and c∗p (density, dynamic viscosity, thermal conductivity and specific heat at constant pressure, respectively) in the equations are chosen as 7th-order polynomial functions of temperature and normalized by the corresponding values at inflow temperature T∞ . The other non-dimensional parameters are the Eckert number, the Prandtl number and the Reynolds number, which are defined as follows:

Eckert number: Prandtl number:

Ec =

Uw 2 , c p∞ (Tw − T∞ )

Pr =

μ∞ c p∞ k∞

Fig. 8. The computational domain with the boundary conditions.

Reynolds number: as

aT =

Tw − T∞ , T∞

(6)

i.e. the wire temperature is set to Tw = 10 0 and 30 0 °C. No-slip conditions are applied at the solid walls and a zero-gradient Neumann boundary condition for velocity and temperature are applied at the outlet. In the solid region, the Dirichlet boundary condition T = T∞ was applied at the upstream wall, and adiabatic Neumann conditions were set at the bottom and the downstream boundaries. These two regions were coupled by means of the temperature continuity and heat-flux conservation at the interfaces, namely,



Tfluid = Tsolid

and

k

∂T ∂y





= fluid

k

∂T ∂y



.

(7)

Re =

,

ρ∞Uw d . μ∞

(12) (13)

(14)

The heat loss from the wire was evaluated as the mean Nusselt number Nu on the wire surface, which is calculated from the local Nusselt number Nu(θ ). The heat flux at a certain point on the surface q˙ (θ ) is calculated as

q˙ (θ ) = −k(Tw )

 ∂ T (r, θ )  , ∂ r r=d/2

(15)

where r and θ are the polar coordinates originated at the wire centre. By normalizing q˙ (θ ) with a reference heat flux q˙ c = k(Tf )(Tw − T∞ )/d the local Nusselt number is obtained as:

Nu(θ ) =

solid

k(Tw ) q˙ (θ ) =− q˙ c k(Tf )

 ∂ T ∗ (r∗ , θ ∗ )  ∗ , ∂ r∗ r =0.5

(16)

The thermal conductivity of the solid region was set to ksolid = 205 and 0.19 Wm−1 K−1 , corresponding to the properties of aluminum and Plexiglas, respectively.

where θ ∗ = θ /(2π ) and Tf is the film temperature: Tf = (Tw + T∞ )/2. By taking the average of Nu over the wire surface, the mean Nusselt number is obtained through:

3.2. Mathematical model

Nu =



1 0

Nu(θ ∗ )dθ ∗ .

A built-in solver chtMultiRegionSimpleFoam capable of calculating conjugate heat transfer in fluid and solid zones is used for the simulation. The governing equations in the fluid region are the conservation of mass, momentum and energy for compressible flow and are shown here in their non-dimensional, steady, form (superscript ∗ denotes a non-dimensional quantity).

3.3. Numerical results

∂ ρ ∂

Nu



∗

Ui∗ x∗i



= 0,



(8)

 ∗

∂ ρ ∗Ui∗U j ∂ P∗ = − ∗ + ρ ∗ g∗j ∗ ∂ xi ∂xj   ∗  ∂ U j ∂ Ui∗ 2 ∂ Uk∗ 1 ∂ ∗ + + − δ , μ Re ∂ x∗i ∂ x∗i ∂ x∗j 3 ∂ x∗k i j  ∗ ∗ ∗ ∗   ∂ ρ ∗ h∗Ui∗ Ec ∂ ρ Ui U j U j + ∂ x∗i 2 ∂ x∗  ∗ ∗ i 1 ∂ k ∂h = + Ecρ ∗Ui∗ g∗i . RePr ∂ x∗i c∗p ∂ x∗i

(9)

(10)

k ∂ T = 0. ρ ∗ c∗p ∂ x∗i ∂ x∗i ∗

2

Collis and Williams (1959) suggested that the Nu-Re relationship needed a correction in order to take the effect of the overheat ratio into account. The proposed relation is:

∗

(11)

 T −0.17 f T∞

= f (Ref ),

(18)

where Ref = ρfUw d/μf and the subscript f indicates the corresponding value at the film temperature. The calculated surfaceaveraged Nusselt number of the wire at aT = 0.27 as a function of Reynolds number is shown in Fig. 9 where aT is a constant. The result of the present cases on an aluminum wall is plotted together with the numerical data of Shi et al. (2003) and shows good agreement for the same wire height even though the case of Shi et al. (2003) has a flow beneath the aluminum wall. Furthermore, the heat transfer from the wire in the free stream, i.e. wallremote region, is plotted as calibration curves on the same figures. These calibration curves are expressed in the form

Nu

In the solid region, the heat-conduction equation is solved:

(17)

 T −0.17 f T∞

= n1 Ref n2 + n3 ,

(19)

where the coefficients were found to be n1 ≈ 0.60, n2 ≈ 0.40 and n3 ≈ 0.22 for both temperature overheat ratios. The results of the cases with the wire close to walls deviate further from the calibration curves as the Reynolds number decreases, and also as the

Please cite this article as: Y. Ikeya et al., Towards a theoretical model of heat transfer for hot-wire anemometry close to solid walls, International Journal of Heat and Fluid Flow (2017), http://dx.doi.org/10.1016/j.ijheatfluidflow.2017.09.002

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Fig. 9. Heat transfer from the heated cylinder at aT = 0.27 on an a) aluminum and b) Plexiglas wall.

wire-wall distance becomes smaller. In addition, employing an aluminum wall is found to result in larger heat transfer than the case with Plexiglas as it is observed in the experiment. Figs. 10 and 11 present velocity profiles which were converted from the Nusselt numbers of the present results with the obtained calibration curve in order to express the results such that they correspond to the “acquired” velocity readings from a hot-wire probe. The “measured velocity” increases with decreasing innerscaled height in the viscous sublayer: the typical wall-effect is demonstrated. Also, the effect of the overheat ratio and the wall conductivity is found to be consistent with the experimental result (Fig. 11). The present results on an aluminum wall agree as well with the result of Zanoun et al. (2009), when compared at the same sensor height of yw /d = 100. However, it is observed that the various curves do not necessarily collapse onto a single line with respect to different wire locations yw . This indicates that innerscaling is not sufficient when wire-wall heat transfer is analyzed. Fig. 12 shows the representative result of the temperature distribution around the wire. It is apparent that, with the presence of a solid wall, there is a steep temperature drop beneath the wire. Especially for the aluminum wall cases, the temperature at the interface is always approximately identical to the atmospheric value T∞ , while for the Plexiglas wall a high-temperature region is apparent, which shifts downstream as the flow velocity increases. 4. Wire-wall heat-transfer model

Fig. 10. The “measured velocity” of the hot-wire at aT = 0.27 on an a) aluminum and b) Plexiglas wall.

Fig. 11. Comparison of the measured velocity of the hot-wire at different overheat ratios and on different walls.

from a wire to supporting prongs is also neglected, i.e. the wire is assumed to be long enough compared to its diameter. The total heat transfer on the wire surface is assumed to be a superposition of the two contributions, i.e. convection and conduction:

Nutotal = f1 (Nuconv ) + f2 (Nucond ),

(20)

where the first term on the right hand side denotes the heat transfer due to convection, and the second term on the right hand side, i.e. the conduction part, is interpreted as how much the presence of the wall distorts the temperature field from its unaffected state.

4.1. Assumptions 4.2. Convection part of the model In this section, a theoretical, yet occasionally empirical model on the heat transfer between a hot-wire, a solid wall and surrounding air is proposed by combining the knowledge obtained through the experiment and the numerical analysis. To establish a model in a form which is as simple as possible, radiation is assumed to be small enough to be negligible, and heat conduction

In the numerical part of the present study, the calibration curve coupling the heat transfer Nu to the Reynolds number Ref was derived. Although this calibration curve was found to agree with correlations by several previous researches in the range of 6 × 10−3 < Ref < 1, those reference curves are not able to predict the heat

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Fig. 12. Local temperature distribution around the wire at aT = 0.27 at the location of y∗w = 100. Velocity gradients are S = 10, S = 100 and S = 10 0 0, respectively from left to right and the corresponding Reynolds numbers are Ref = 1.3 × 10−3 , Ref = 1.3 × 10−2 and Ref = 1.3 × 10−1 .

transfer in the lower Reynolds number range. This is due to the difficulty to simulate free convection effects, since the results in common simulation domains are dependent on the simulation domain itself. In fact, the calibration curves in the previous numerical works often neglect the gravitational effect (Lange and Durst, 1998; Shi et al., 2003) and are not adequate for the discussion of mixed convection. The experimental work of Collis and Williams (1959) provides data in the region Re∞ < Gr1∞/3 , where natural convection starts to dominate the total heat transfer. Here the Grashofs number is defined as

Grf =

g(Tw − T∞ )d3 . Tf νf2

(21)

They observed that the Nusselt number deviates from their correlation in the low Reynolds number region, however, no correlation for this behavior was proposed. It is stated in their work that this mixed convection registers lower Nusselt numbers than the value of pure free convection, i.e. without flow applied. On the other hand, heat transfer from a heated cylinder under pure free convection has been studied both in experiments and numerically, although estimates deviate from each other, possibly due to the difference of the experimental and numerical setup. Collis and Williams (1954) employed relatively long sensors (l/d ≥ 2 × 104 ) suitable to compare with the present twodimensional calculation, and their measurements were conducted in the Grashof number range of 10−10 ≤ Grf Prf ≤ 10−3 , in which the present case is included. Morgan (1975) gave a correlation to their result:

Nunc = 0.675(Grf Prf )0.058 .

(22)

The relation below is proposed in the present study to relate the Nusselt number Nunc at Ref = 0 calculated with relation

(22) and the original calibration points Nucal, orig for Ref > Gr1f /3 :

Nuconv = Nucal,new



= Nunc exp −

Ref nconv





+ Nucal,orig 1 − exp −

Ref nconv



. (23)

Here, the original calibration curve Nucal, orig is simply expressed as in Eq. (19). In accordance with the finding by Collis and Williams (1959), this relation converges to Nunc in the limit of Ref → 0, while it asymptotes to the original calibration curve for larger Reynolds numbers. Furthermore, the undershoot of the Nusselt number in the mixed convection region 0 < Ref < Grf 1/3 is simulated. By choosing the coefficient such that the improved calibration curve deviates by 1% from pure forced convection curve at Ref = Grf 1/3 , the heat transfer due to convection is determined with nconv = 2.9 × 10−3 and 2.3 × 10−3 for overheat ratios aT = 0.27 and 0.96, respectively, where pure forced convection was calculated by eliminating the gravitational acceleration in the original calibration cases. The calculated Nusselt number at aT = 0.27 through Eq. (23) is presented in Fig. 13. 4.3. Conduction part of the model Heat conduction between primitive geometries is determined uniquely. For a long cylinder with the diameter d, the length l and the temperature T1 located at yw apart from an unbounded isothermal wall with the temperature T2 as shown in Fig. 14, the conduction shape factor Hcond, isoth is calculated as (see e.g. Incropera et al., 2006, Chapter 4.3)

Hcond,isoth =

2π l , acosh(2yw /d )

(24)

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Fig. 13. Modeled heat transfer due to heat convection at aT = 0.27 in the freestream.

Fig. 15. The modeled heat transfer from the wire at aT = 0.27 on an aluminum wall at y∗w = 100.

high Ref , conduction effects are expected to become smaller so any functional description needs to take this into account. With this in mind we propose the following functional form for the heat transfer due to conduction:

f2 (Nucond ) = f2 (Nuconv , yw ) = g(Nuconv )Nucond (yw ),

(27)

where

ncond1



g(Nuconv ) = Fig. 14. A long cylinder with the diameter d, the length l and the surface temperature T1 located yw away from an unbounded isothermal wall with the temperature T2 .

and the heat transfer from this cylinder is calculated consequently as

Q˙ = q˙ π dl = kmed Hcond,isoth (T1 − T2 ),

Nucond,isoth =

q˙ , q˙ ref

(25)

1 + acosh 1 +

where kmed is the thermal conductivity of the medium where the cylinder is placed, and the reference heat flux is calculated as q˙ ref = k(T1 − T2 )/d. The conductivity k here needs to be taken at the film temperature Tf = (T1 + T2 )/2. Heat conduction becomes less intense with increasing wire-wall distance, which means that the temperature gradient around the wire surface becomes more gentle as the sensor moves away from the interface. In other words, the wall presence distorts the temperature distribution from the unaffected state, i.e. the state without a wall, and the distortion becomes more significant as the sensor height becomes smaller. Furthermore, it has been seen from Fig. 12 that the temperature gradient around the wire becomes gentler and the Nusselt number becomes smaller as the flow velocity decreases if there is no wall nearby, i.e. heat diffuses farther around the wire at lower velocities, although the influenced area reduces when a solid wall is placed nearby. Hence, the temperature field should be altered more significantly with decreasing flow velocity when compared at the same wire height. Lowering the wire height and decreasing the flow velocity have similar effects on the alteration of the temperature field, whereby the conduction part in the total heat transfer is proposed following the form of Eq. (24). It is clear that the amount heat conducted to the wall depends on the temperature field around the sensor, which depends on the height of the sensor above the wall, the wall material but also the convective heat transfer from the wire. At large Nuconv , i.e.

,

(28)

with (ncond1 , ncond2 , ncond3 ) being fitted to numerical data. With this expression for g the contribution of the conduction to the total heat transfer goes to zero at high Nu. Nucond (yw ) is approximately equal to Nucond, isoth (yw ) for highly conductive walls and can be written as

Nucond (yw ) = (26)

Nuconv ncond2 ncond3

kmed 2 . kf acosh(2yw /d )

(29)

As can be expected the value of Nucond (yw ) decreases with increasing yw /d. 4.4. Final form of the model on the wire-wall heat transfer By interpreting the total heat transfer as the sum of the pureconvection state without a wall and the distortion from it due to the wall, relation (20) is written as

Nutotal = Nuconv + f2 (Nucond ),

(30)

where the convection part and the conduction part are denoted as Eqs. (23) and (27), respectively. The conduction is likely to be independent of the temperature loading aT considering the results in the natural convection measurement in the experimental part. By choosing the coefficients empirically as (ncond1 , ncond2 , ncond3 ) ≈ (0.22, 20, 3.9 × 10−10 ) for aT = 0.27 the modeled function of the total heat transfer yields the curves presented in Fig. 15, and it captures the behavior of the simulation results for the wire on an aluminum wall at y∗w = 100 for both overheat ratios. Nearly identical results are obtained for aT = 0.96 and are here omitted for the sake of brevity. The discussion so far is oriented for a certain parameter configuration; however, it is possible to generalize it for a wider range of parameters including the overheat ratio and wall conductivity. The overheat ratio is likely to be related to the convection part, while the wall conductivity is related to the conduction part. By collecting more numerical data, it should be possible to improve the dependencies of model coefficients on these parameters, which were determined empirically in the present paper.

Please cite this article as: Y. Ikeya et al., Towards a theoretical model of heat transfer for hot-wire anemometry close to solid walls, International Journal of Heat and Fluid Flow (2017), http://dx.doi.org/10.1016/j.ijheatfluidflow.2017.09.002

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However, one should note that more consideration on each parameters should be taken into account for the practical use of the present model, e.g. the temperature distribution inside poorly conducting walls should also be dependent on flow velocity, and the wire length and consequently three-dimensional effects of convection and conduction should be important.

9

good agreement with the present numerical results, and the possibility of the generalization for the various heat conductivities and overheat ratios is suggested. However, it is still necessary to widen the parameter range to comprehend the parameter dependency of the model coefficients, also to improve its applicability. Acknowledgement

5. Summary and conclusions In the present paper, an experimental and numerical investigations of HWA measurement close to solid walls were carried out. Through the experimental part, the parameter dependency of the HWA output on several parameters under no-flow and flow conditions were studied. The thermal conductivity of the wall material affects the HWA reading, viz., walls with higher thermal conductivity lead to higher output voltages, i.e. larger overestimation of the velocity. It should be noted that highly conductive materials, such as aluminum, brass and steel show similar results despite the fact that the conductivity differs a factor of ten between them. For poorly conductive materials, Plexiglas shows a much larger effect than styrofoam, despite the fact that Plexiglas has a thermal conductivity three orders of magnitude less than the metals. In addition, employing higher overheat ratios or longer sensors contribute to larger velocity overestimations as these factors contribute to the additional heat loss from the sensor, however, the no-flow measurement illuminated that the voltage output at different overheat ratios can be scaled by a single curve. The present study focused not only on the mean output but also on the fluctuations. The measured turbulence intensity and the velocity PDF are also affected by the wall conductivity and the overheat ratio: employing higher conductivity and higher temperature loading of the wire suppresses the reading of the turbulence intensity, and results in a narrower PDF in the low speed region within the viscous sublayer. The difference of the output voltage by varying the parameters can be seen only in the viscous sublayer, and one should note that the effect within the sublayer can easily be “hidden” when measured velocity profiles are employed to determine the absolute wall position and friction velocity as is common. In the numerical part, a two-dimensional steady simulation of the flow and temperature field around the heated wire in air coupled with a solid wall was conducted. The calculated results showed a consistent parameter dependency with previous numerical and experimental studies. Furthermore, the present study pointed out that the inner-scaled velocity as a function of the inner-scaled wire height does not necessarily yield a single curve when various non-scaled wire heights are concerned. This implies that the inner scale is not adequate to discuss the present problem of heat transfer nevertheless it has been used in numerous publications. By investigating the temperature distribution around the wire and inside the wall, aluminum was found to act nearly isothermal, while Plexiglas accumulates heat inside, thus creating a high temperature region. It was also found that the temperature distribution resembles that of the freestream state with the increasing flow velocity or the wire-wall distance. By considering the findings from the experimental part and the numerical analysis, a theoretical model on the wire-wall heat transfer was proposed. The model consists of the superposition of the contributions from convection and conduction. For the convection part, the improved calibration curve capable of simulating the natural convection effect is considered. The proposed model shows

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Please cite this article as: Y. Ikeya et al., Towards a theoretical model of heat transfer for hot-wire anemometry close to solid walls, International Journal of Heat and Fluid Flow (2017), http://dx.doi.org/10.1016/j.ijheatfluidflow.2017.09.002