Thermal signatures of turbulent airflows interacting with evaporating thin porous surfaces

International Journal of Heat and Mass Transfer 87 (2015) 429–446

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Thermal signatures of turbulent airflows interacting with evaporatingthin porous surfaces

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.04.0260017-9310/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Soil and Terrestrial Environmental Physics, Depart-ment of Environmental Systems Science, ETH Zurich, Universitätstrasse 16, CH-8092 Zürich, Switzerland. Tel.: +41 44 633 6157.

E-mail address: [email protected] (E. Haghighi).

Erfan Haghighi ⇑, Dani OrSoil and Terrestrial Environmental Physics, Department of Environmental Systems Science, ETH Zurich, Zürich, Switzerland

a r t i c l e i n f o

Article history:Received 26 September 2014Received in revised form 16 February 2015Accepted 9 April 2015

Keywords:EvaporationPorous surfaceTurbulenceSurface renewalThermal signatures

a b s t r a c t

Evaporative drying of porous surfaces interacting with turbulent airflows is common in various natural(hydrology, climate) and industrial (food, paper, and building materials) applications. Turbulent airflowsinduce spatially-complex and highly intermittent boundary conditions that affect surface evaporation ratesand associated near-surface thermal regimes. Such interactions are particularly significant during stage-Ievaporation (vaporization plane at the surface) where turbulent eddies induce highly localized and inter-mittent variations in evaporative fluxes that leave distinct thermal signatures observable by infrared ther-mography (IRT). A theoretical framework supported by an experimental method was proposed to capitalizeon measured thermal fluctuations of evaporating porous surfaces to determine airflow turbulence charac-teristics and evaporative fluxes. The study focuses on thin surfaces with low thermal capacity to illustrateexperimentally direct links between characteristics of surface thermal fluctuations and momentum-basedturbulent eddy residence times. For most practical applications, the method could be applied using rapid IRmeasurements from a single sensor aimed at the surface. The theoretical links between surface wetness andcharacteristics of surface temperature fluctuations offer opportunities for remote quantification of dryingand energy exchange processes from engineered and natural porous surfaces.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The rates and patterns of vapor transfer from evaporating sur-faces into airflows are determined by interplay between water sup-ply to the surface, energy input, and interacting air boundary layerconductance. The early stages of evaporation from an initially-sat-urated porous medium are often marked by high and relativelyconstant drying rate (the so-called stage-I evaporation) duringwhich the vaporization plane remains at the surface and is sup-plied by capillary liquid flow from a receding drying front [1–4].The porous medium internal transport capacity during this stageis often high and exceeds the diffusive resistance to transportacross an air viscous sublayer adjacent to the surface (i.e., the dry-ing rate is governed by external aerodynamic conditions) [3–9].

Characteristics of airflow play an important role in defining thenature of the viscous sublayer and its influence on surface evapo-ration dynamics. Turbulent flows over surfaces could be repre-sented by populations of fluid parcels (termed eddies) comprisedof a range of sizes and intensities [10] that interact with surfaces

and affect the rates of vapor, heat, and momentum exchange[8,11–14]. The complex and often localized interactions betweenturbulent eddies and surfaces induce distinct surface thermalsignatures [15–22]. Such surface thermal signatures are affectedby surface properties (wetness, thermal capacity), and offer ameans for remote quantification of instantaneous heat and massexchange into turbulent flows [11,16,20,23–27].

Infrared imaging of such interactions between coherent turbu-lence structures (eddies) and surfaces was first studied by Scholset al. [15] and Derksen [28] revealing streaky patterns of surface tem-peratures along airflow directions. Similar patterns in surface tem-peratures were subsequently observed by Nakamura et al. [14],Hetsroni and Rozenblit [16], Gurka et al. [18], Nakamura [20], andHetsroni et al. [29]. According to Katul et al. [30] and Renno et al.[31], rapid surface temperature fluctuations may be attributed toinactive eddy motions due to turbulence in the outer region and con-vective mixed layer processes [32]. Such interactions could bedetected from near-surface pressure fluctuations and from the lowerwave-number region of the longitudinal velocity spectra [30,33].

In contrast with numerous studies focusing on turbulentcoherent structures forming the convective turbulent boundarylayer [34–41], their impacts on surface temperature fluctuationsand the role of these intermittent interactions on surface fluxesis generally less studied. To systematically consider surface

430 E. Haghighi, D. Or / International Journal of Heat and Mass Transfer 87 (2015) 429–446

thermal signatures characteristics and their links with surfacefluxes [42–45], this study aims to use the model of Haghighi andOr [8] and augment it by considering instantaneous energyexchange (due to eddy interactions) to quantify the resulting sur-face fluxes and the corresponding localized temperature dynamics.

Haghighi and Or [8] have used the surface renewal (SR) formal-ism [46–50] to develop a mechanistic model for evaporative fluxesfrom drying porous surfaces into turbulent airflows (Fig. 1). Themodel considers surface-eddy interactions during their finite resi-dence/exposure time over the surface. As eddies are being sweptalong the surface, they become gradually enriched by diffusingvapor (or exchange heat) across a viscous sublayer that forms overthe footprint area (the interacting area with the surface). Eddiesare eventually ejected back to the turbulent flow and subsequentlyreplaced by new eddies. Consequently, an evaporating surface maybe locally cooled down or warmed up relative to air temperature,and the resulting intermittent and spatial surface temperaturedynamics can be resolved using rapid and high resolution infraredthermography (IRT).

Technological advances in IRT methods contributed to its rapidlyexpanding range of applications including remote sensing andhydrology, thermal efficiency of infrastructure, and numerousindustrial and medical applications [25,51–56]. Shahraeeni and Or[53], Qiu and Zhao [57], and Kalma et al. [58] have shown howIRT measurements could be used to remotely estimate spatially

Eddy Residence Time, τ

T0

TeqSurf

ace

Tem

pera

ture

, Ts

Increasing ω

U∞, T∞

Laminar

zx

Fig. 1. Schematic of an idealized surface renewal process by turbulent air parcels. The tparcels (eddies) with different characteristics drawn from a distribution. Vapor transfeindividual pores across a local viscous sublayer forming below an eddy’s footprint (seerenewal eddy, the surface can either cool down or warm up due to the completion betwetemperature (surface response to temperature depression) as a function eddy residencevalues of effective ‘‘thermal thickness’’ for surface temperature fluctuations with the do

variable evaporative fluxes from heterogeneous terrestrial surfaces.Focusing on rapid thermal dynamics, Schimpf et al. [59], Garbe et al.[60], and Asher et al. [61] exploited surface IRT measurements tocharacterize micro turbulence (in water) and estimate air-sea heatand gas fluxes from the statistics of ocean skin temperatures.

Considering the potential offered by these recent developments,the primary objective of this study was to establish quantitativelinks between momentum exchange induced by turbulent airflowsand the resulting temperature fluctuation patterns forming overevaporating porous surfaces and observable by IR imaging. Thespecific objectives of the study were to: (1) establish theoreticallinks between turbulent flow field characteristics and surface tem-perature fluctuations over evaporating surfaces; (2) use the theo-retical framework and reconstruct characteristics of turbulentairflows from rapid measurements of surface temperature fluctua-tions; and (3) use the method and dynamic thermal information toremotely estimate surface evaporation rates (and other scalarexchange). We defer complexities related to surface roughnesson turbulent interactions [62–65] to future studies, and focus hereon interactions with ‘‘smooth’’ evaporating surfaces.

Following this introductory section, we present in Section 2 themodeling framework and discuss the nature of coupling betweensurface temperature dynamics and a turbulent momentum field.Section 3 is devoted to describing experimental considerationsused for exploring similarities between turbulent airflows and

δi

GS

HRn λE

∂T/∂x=0

Ts

Ua, Ta

Turbulent

ω

urbulent airflow interacts with the evaporating surface as a collection of individualr through a unit surface into individual eddies is governed by 3D diffusion fromthe inset). Depending on the temperature difference between the surface and theen inward sensible and outward latent heat fluxes. A typical variation in the surfacetime and surface thermal thickness is shown in the graph. The lines mark differenttted line indicating ‘‘thicker’’ interacting surface layer.

E. Haghighi, D. Or / International Journal of Heat and Mass Transfer 87 (2015) 429–446 431

surface temperature fluctuations. Experimental results addressingthe research objectives are presented in Section 4, followed bysummary and concluding remarks in Section 5.

2. Theoretical analysis of surface temperature dynamics due toturbulent airflows

2.1. Eddy renewal characteristics and air viscous sublayer thickness

The intermittent nature of turbulent airflows gives rise to fluc-tuations in the thickness of viscous sublayers forming beneathinteracting eddies [8,66–68]. Based on own characteristics (sizeand residence time), each eddy forms a well-defined viscous sub-layer that provides the boundary conditions for heat and vaportransfer by thermal conduction and molecular-diffusion, respec-tively [3,7,8,44]. The resulting scalar transfer rates are determinedby temperature and water vapor gradients across the viscous sub-layer. We assume that the eddy properties above this thin bound-ary remain as those of the fully mixed turbulent airflow regime.Consequently, we tacitly assume that scalars in the turbulent floware well mixed, and vertical gradients in scalar values (for purposesof exchange with the surface) occur primarily across the viscoussublayer [8,44]. For brevity, we recap the main results fromHaghighi and Or [8] that are directly relevant for this study.

Motivated by studies of Meek and Baer [67,68] on the fluctuat-ing nature of the viscous sublayer thickness under turbulentregimes, Haghighi and Or [8] estimated the thickness of the viscoussublayer forming below an eddy’s footprint, di (m), as a function ofeddy residence time, sm;i (s), as

di ¼ cffiffiffiffiffiffiffiffiffiffiffiffimasm;i

pð1Þ

where c ¼ 2:2 is a proportionality constant depending on flowgeometry [8,68–70], ma (m2/s) is the kinematic viscosity of air, andsm;i (s) is the ‘‘momentum’’ residence time of the ith eddy. Studieshave shown that the distribution of eddy residence times followsthe gamma statistical distribution [47,49,50] that is of practicalinterest for various natural applications facilitating parametricand analytical modeling [71–74]. The well-established SR-basedapproaches coupled with the gamma distribution offers a conve-nient representation of eddy residence time distribution /ðsÞ as[25,47,49,50]

/ðsÞ ¼ baþ1

Cðaþ 1Þ sa expð�bsÞ ð2aÞ

b ¼ aþ 1�s ð2bÞ

where CðnÞ ¼ ðn� 1Þ! is the gamma function, and a (�) and b (1/s)are, respectively, the shape and rate parameters that control theshape of the distribution and provide insights on the nature of inter-actions between turbulent airflows and the surface (e.g., a turbulentregime characterized by small or large a (for a constant b) isreferred to as small- or large-scale eddy dominated, respectively[8]), and �s (s) is the mean residence time of the eddy residence timedistribution.

Considering Eqs. (1) and (2), an effective viscous sublayer thick-ness as a function of turbulent flow characteristics that explicitlyconsiders the influence of the eddy spectrum on mass exchangeprocesses is established according to [8]

d ¼ f ðaÞ ma

u�ð3Þ

where u� (m/s) is the friction velocity estimated as [8]

u� ¼0:3

aþ 1U1 ð4Þ

with U1 (m/s) the mean air velocity, and f ðaÞ is obtained from [8]

f ðaÞ ¼ 2:2ffiffiffiffiffiffiffiffiffiffiffiffi112pp

Cðaþ 1Þ1

2ðaþ1Þ ffiffiffiffiffiffiffiffiffiffiffiffiaþ 1p �

1 a ¼ 0Pð2aþ 1Þ a > 0

�ð5aÞ

Pð2aþ 1Þ ¼ ð2aþ 1Þ � ð2ða� 1Þ þ 1Þ � ð2ða� 2Þ þ 1Þ � . . .

� ð2ða� nÞ þ 1Þ; n < a ð5bÞ

with n (-) the largest integer smaller than a. According to Haghighiand Or [8], the occurrence of larger eddies with longer residencetimes in a turbulent airflow regime increases by increasing a andthe mean viscous sublayer thickness becomes thicker. Consideringthe importance of the eddy spectrum shape parameter a in deter-mining the viscous sublayer thickness that, in turn, affects heatand mass exchange rates from the surface [3,8], the study aims toreliably extract values of the eddy spectrum shape parameter a (amomentum field characteristic) from measurements of turbu-lence-induced surface temperature fluctuations (a remotely-sensedsurrogate variable) induced by momentum field fluctuations.Addressing this first research objective requires a better understand-ing of the coupling between fluctuations in turbulence velocity(momentum) field and the resulting surface temperature dynamics.

The Reynolds’ analogy that implies a unique mechanism for thetransport of momentum, heat, and mass suggests the dominance ofa single characteristic time scale for transport across the viscoussublayer. It is the momentum time scale that governs the intermit-tent ejection events of surface interacting eddies back into the fullymixed region [75] and thus defines the residence time for the vis-cous sublayer across which heat and mass exchanges are renewedsimultaneously. Although penetration depths for momentum, heatand mass from the surface may differ due to differences in theirrespective diffusivities, for evaporation, these are of the same order(i.e., the Schmidt and Prandtl numbers are of the order of unity)lending support for a single characteristic time scale for transportacross the viscous sublayer. We thus ignore minute differences inscalar diffusivities, and assume (for simplicity) that momentum,heat, and vapor diffusion rates rapidly attain steady state rates dur-ing the eddy’s sweeping motion over ‘‘smooth’’ surfaces [8,49,75].Hence thermal (dT ) and mass (dv ) boundary layers are assumed tobecome fully developed underneath an eddy’s footprint area (i.e.,dT � dv � d). In subsequent analyses, we parameterize surface tem-perature variations induced by rapid perturbations of heat andmass transfer across the eddy-surface interface during the eddyresidence time as defined by the turbulence regime.

2.2. An instantaneous surface energy balance model

The strongly coupled and intermittent mass and energyexchange during evaporation from porous surfaces into turbulentairflows affects surface temperature dynamics. With the passageof an eddy, an instantaneous and local balance between latent heatand net supply of energy by radiation, convection, and conductionmay form with the eddy. The net radiation Rn (W/m2) is consideredto be the main component of energy supply that is balanced bylatent heat flux kE (W/m2) due to evaporation, sensible heat fluxH (W/m2) from the surface, (vertical) conductive heat flux G (W/m2) resulting from temperature gradients in the porous medium,and by storage in the surface layer S (W/m2) altering its tempera-ture with time [76,77]. This can be expressed by the partitioningof the energy received by the surface layer (Rn � H) between theevaporative flux (kE), the conductive vertical heat flux (G), andthe heat storage in the substrate (S). A one dimensional energy bal-ance for a spatially homogeneous surface underneath the footprintof the ith renewal eddy is accordingly given as

Rn � Hi ¼ kEi þ Gþ S ð6Þ

432 E. Haghighi, D. Or / International Journal of Heat and Mass Transfer 87 (2015) 429–446

where k (J/kg) is the latent heat of vaporization, and

Gþ S ¼ �Ks@T@zþ cvx

@T@t

ð7Þ

in which Ks (W/mK) is the thermal conductivity of a wet porousmedium, Tðz; tÞ (K) represents spatio-temporal distribution of thetemperature profile within the substrate (evaporation zone),cv ¼ qscp (J/m3K) is the volumetric heat capacity of the poroussurface with qs (kg/m3) the apparent density and cp (J/kg K) thespecific heat capacity of the wet surface on mass basis, andx (m) is an effective ‘‘thermal thickness’’ (defined in Eq. (8) below).

The contribution of lateral conductive heat transport across theeddy perimeter is neglected for simplicity, as it would slightlyexpand the lateral extent of an eddy footprint. Moreover, the pathsof rapid and repetitive sweep/ejection of eddies over the interact-ing surface would (statistically) cover the entire surface justifyingthe simplification of assuming no-flux boundary (@T=@x ¼ 0) alongthe eddy lateral boundaries (see Fig. 1). We hence focus on theresulting vertical (1D) temperature fluctuations within the surfacelayer (below an eddy footprint) and associated changes in thermalstorage and short-lived vertical heat flux within the porous med-ium (Eq. (7)).

The extent of 1D transient variations in subsurface temperatureprofile could be estimated to define an effective thermal thicknessx, that senses surface temperature fluctuations. We capitalize onthe known time scale for eddy renewal, and invoke the notion ofthermal damping depth originally derived for sinusoidal tempera-ture fluctuations at the surface [78], yielding an estimate for theeffective thermal thickness as:

x ¼ffiffiffiffiffiffiffiffiffiffiffiDH

p�st

rð8Þ

where DH (m2/s) is the surface thermal diffusivity, and �st (s) is themean residence time of fluctuating ‘‘thermal signatures’’ (termedthermal eddy residence time marked by the index ‘‘t’’ to distinguishit from momentum residence times marked by the index ‘‘m’’; seeSection 4.2.2). Eq. (8) reveals that the effective thermal thicknessvaries with changes in the wetness-dependent thermal diffusivity,and with the time scale of the thermal signatures. This approxima-tion for the temporal effective thermal thickness (Eq. (8)) was eval-uated numerically and found to be a reasonable approximation asshown in Appendix A.

The transient nature of heat (and mass) exchange at the surfaceand the relative importance of various transport processes could becharacterized by the Biot (Bi) and Fourier (Fo) numbers defined as[79]

Bi ¼ hLKs

ð9aÞ

Fo ¼ DHt

L2 ð9bÞ

where h ¼ Ka=d (W/m2K) is the convective sensible heat transfercoefficient with Ka (W/mK) the thermal conductivity of air (seeSection 2.2.1), L (m) is the characteristic length scale through whichthermal conduction occurs (L � x), and t (s) is the characteristictime for a temperature change to occur (� mean residence timeof turbulent momentum field eddies). For a practical range ofparameters in the present study: h ¼ 20� 30 W/mK, Ks ¼ 0:04 W/mK and DH ¼ 2:2� 10�8 m2/s for the cotton cloth used in the exper-iments, L � x ¼ 0:1 mm (see Section 3), and t � 0:1 s (seeSection 4), the Biot and Fourier numbers are of the order of 10�4

to 10�2, and 10�1 to 100, respectively.The Biot number provides a measure of surface temperature

drop relative to the temperature difference between the surface

and the air, with Bi� 1 is indicative of a uniform temperature dis-tribution within the surface layer during the transient process[79,80]. This means that the resistance to heat conduction withinthe substrate is much smaller than the resistance to convectionacross the air boundary layer. The low values of Bi in our setup sup-port the neglect of lateral conductive heat fluxes, and confirm thedominance of surface skin temperature (Ts) as the main parameterthat reflects rapid variations in surface energy balance. The Fouriernumber reflects the ratio of diffusive or conductive transport ratesto changes in storage within the subsurface substrate, with valuesof Fo� 1 suggests that the vertical conductive heat transport isnegligible relative to changes in thermal storage within the effec-tive thermal thickness. Note that for the conditions of this study,the values of Fo are not far from unity (0.1 to 1.0) implying non-negligible contribution of vertical conductive heat transport inthe subsurface layer, as it determines the effective thermalthickness.

The slightly nonlinear shape of the vertical temperature distri-bution below an evaporating surface could simply be linearizedfrom the surface to the effective thermal thickness (i.e.,@T=@z � DT=x) [81] such that one can assume a relatively constantvertical conductive heat flux within the surface layer over theeddy’s footprint. Thus, the instantaneous energy balance withinthe surface ‘‘skin’’ of known thermal thickness x during renewalof turbulent eddies results in rapid variations in surface tempera-ture dTs at a rate that is inversely proportional to the volumetricheat capacity and the effective thermal thickness, as

dTs

dt

����i

¼ 1cvx

Rn � KsTs � Ta

x� Hi � kEi

� �ð10Þ

with Ts and Ta (K) surface and air temperatures, respectively. Thequantification of surface temperature fluctuations defined in Eq.(10) requires parameterization of variables contributing to the sur-face energy balance. The net long-wave radiation is represented bythe difference between blackbody radiation at the surface tempera-ture (Ts) and that at air temperature (Ta) with the net radiation fluxRn expressed as

Rn ¼ Rswð1�KÞ � reðT4s � T4

aÞ ð11Þ

where Rsw (W/m2) is the incoming short-wave radiation, K (�) is thesurface reflectivity or albedo, r (W/m2K4) is the Stefan–Boltzmannconstant (5:67� 10�8 W/m2K4), and e (�) is the surface emissivity.

2.2.1. The sensible heat fluxThe sensible heat exchange between the surface and the ith

renewal eddy across the viscous sublayer, Hi (W/m2), is formulatedas the product of thermal conductivity and temperature gradientacross the viscous sublayer

Hi ¼ �KadTdz

����i@z¼0

ð12Þ

where Ka (W/mK) is the thermal conductivity of air in the boundarylayer (0.024 W/mK at 25 �C). Evidence suggests that a linear tem-perature profile across a viscous sublayer of thickness 2 mm formswithin a time scale of 0.2 s [44]. We thus consider linear gradientsof temperature and vapor concentration across this layer to forminstantaneously, and Eq. (12) is then rewritten as

Hi ¼ hiðTs � TaÞ ð13Þ

where hi ¼ Ka=di (W/m2K) is the instantaneous convective sensibleheat transfer coefficient assuming that temperature profile at theborder of the viscous sublayer reaches ambient temperature (i.e.,Td ¼ Ta) [44].

E. Haghighi, D. Or / International Journal of Heat and Mass Transfer 87 (2015) 429–446 433

2.2.2. Evaporation and latent heat fluxesEvaporation flux from a wet porous surface into the ith renewal

eddy (across the local viscous sublayer), Ei (kg/m2s), is formulatedas a function of vapor concentration difference [8]

Ei ¼ kiðCs � CaÞ ð14Þwhere Cs and Ca (kg/m3) are the vapor concentrations at the surface(assumed saturated) and in the well-mixed air at the top of the vis-cous sublayer, respectively, and ki (m/s) is the instantaneous masstransfer coefficient that includes surface resistance (due to surfacewetness state) as derived by Haghighi et al. [7]

ki ¼D

di þ P f ðhsurf Þð15Þ

where D (m2/s) is the vapor diffusion coefficient in air, P (m) is theequivalent 1D mean pore size (P ¼ r

ffiffiffiffipp

; with r (m) the pore radius)[7], and the surface wetness-dependent coefficient f ðhsurf Þ isobtained from [6,7]

f ðhsurf Þ ¼1

pffiffiffiffiffiffiffiffiffihsurf

pffiffiffiffiffiffiffiffiffiffiffiffip

4hsurf

s� 1

!ð16Þ

with hsurf (�) the surface water content.The vapor concentration difference in Eq. (14) is a function of

temperature and vapor pressure differences between the surfaceand the free air stream. Assuming that liquid vapor behaves asan ideal gas, one can express the concentration difference as

Cs � Ca ¼Mw

RPsatðTsÞ

Ts� RS

PsatðTaÞTa

� �ð17Þ

where Mw (kg/mol) is the molar mass of evaporating liquid, R(J/mol K) is the universal gas constant (8.314 J/mol K), Psat (Pa) isthe vapor pressure at the liquid surface assumed to be the satura-tion vapor pressure at the surface temperature, and RS (�) is theair relative vapor saturation (air relative humidity, RH (�), for watervapor). In this study, the saturation vapor pressure Psat is computedusing the Antoine equation as [82]

PsatðTÞ ¼ 133:32� 10A� BCþT ð18Þ

where A, B, and C are the Antoine coefficients that vary among liq-uids (see Table 1 for the coefficients and their range of validity forsubstances discussed in this study). Note that for the empirical rela-tions in Eq. (18) the temperature T is expressed in �C and not in K[82].

2.3. Intermittent surface temperature fluctuations

Combining Eqs. (11), (13) and (14) with Eq. (10), we obtain anordinary differential equation (ODE) for the instantaneous varia-tions in surface temperature underneath the ith eddy as

Table 1Coefficients of Antoine equation [82].

Substance A B C Range of validity

Tmin (�C) Tmax (�C)

Water 8.07131 1730.63 233.426 1 100Acetone 7.13270 1219.97 230.653 -64 70

Table 2Physical properties and empirical coefficients.

Substance k (J/kg) cp (J/kg K) D (m2/s) Mw

Water 2450� 103 4180 2:50� 10�5 0.01

Acetone 518� 103 2150 1:24� 10�5 0.05

dTs

dt

����i

¼ 1cvxðRswð1�KÞ þ reðT4

a � T4s Þ þ �hiðTa � TsÞ � kkiðCs � CaÞÞ

ð19Þ

with �hi ¼ hi þ Ks=x. To facilitate an analytical solution for the sur-face temperature Ts, we linearize the long wave radiation and thelatent heat terms. We expand Ts around Ta, and express T4

a � T4s as

4T3aðTa � TsÞ, such that the resulting approximation is very good

for the range of natural surface and air temperatures relevant toevaporation between 0 and 60 �C (see Shahraeeni and Or [81] formore details). We also linearize the saturation vapor concentrationCs as a function of Ts with an excellent accuracy (especially for anoperational range of evaporating surface temperatures of 15 �C to25 �C) as

CsðTsÞ ¼Mw

RPsatðTsÞ

Ts� Mw

RðaTs þ bÞ ð20Þ

where a and b are the linearization constants (see Table 2). The lin-earized form of Eq. (19) subsequently yields a first order ODE thatcould be solved analytically as

Ts; iðtÞ ¼ c4e�ðc2=c1Þt þ c3=c2 ð21Þ

where

c1 ¼ cvxc2 ¼ 4reT3

a þ �hi þ kkiMwa

R

c3 ¼ Rswð1�KÞ þ 4reT4a þ �hiTa þ kkiðCa � Mwb

R Þc4 ¼ T0 � c3

c2

ð22Þ

with T0 (K) the initial surface temperature at the onset of a renewalevent with a time scale of Dt ¼ t � t0 ¼ sm;i (see also Tables 1 and 2for numerical values of the physical properties and empiricalconstants used in Eq. (22)). Note that the exponential form of theanalytical solution in Eq. (21) conforms to other well-establishedsolutions derived for transient heat exchange over a flat plate[79,80]. Eqs. (21) and (22) reflect the instantaneous and fluctuatingnature of surface temperature due to interactions with eddieshaving different characteristic residence times (sm;i) and boundarylayer thicknesses. The transfer coefficients for intermittent sensible(hi) and latent (ki) heat fluxes may also fluctuate due to theirdependency on the instantaneous viscous sublayer thickness,which in turn, is a function of individual eddy residence time sm;i

(see Eq. (1)).Although Eq. (21) was derived for instantaneous surface tem-

perature evolution during a single eddy renewal event, it may alsobe used to describe surface temperature evolution over longerperiods. We use the surface renewal formalism and consider thecumulative thermal effect of a population of eddies with differentresidence times sm;i (drawn from a gamma distribution function)that intermittently interact with the surface in a sequence ofsweep/eject events. The resulting gradual temporal evolution ofsurface temperature (with superimposed transient fluctuations)is computed for any elapsed time from the onset of the turbulent

flow regime given as: t ¼PN

i¼1sm;i, with N the number of eddyreplacements during the time elapsed (the result is similar to a sur-face subjected to periodic thermal transients considered by Glass

(kg/mol) q (kg/m3) e (-) Empirical Coeff.

a (Pa/K2) b (Pa/K)

8 998 0.95 0.467 �128.783

8 327 0.95 3.465 �931.191

434 E. Haghighi, D. Or / International Journal of Heat and Mass Transfer 87 (2015) 429–446

and Özisik [83]). We note that in this calculation the effective ther-mal damping depth x (Eq. (8)) is approximated by the mean eddyresidence time and not by the elapsed time t.

Additionally, the analytical solution presented in Eq. (21)enables estimation of the elapsed time for attaining an asymptoticvalue from the onset of a turbulence regime as

teq ¼ �c1

c2ln

Ts � Teq

T0 � Teq

� �ð23Þ

with Teq ¼ c3=c2 (K) the equilibrium surface temperature, andTs ¼ 0:99Teq. For the range of lab conditions studied with:Rsw ¼ 0 W/m2, Ta ¼ 293 to 303 K, U1 ¼ 0:5 to 4 m/s, T0 � Teq ¼10 K, RS ¼ 35 and 0% for water and acetone, respectively, and con-sidering eddies with mean properties (evaluating hi and ki by drather than di) interacting with a saturated cotton cloth(r ¼ 0:03 mm, hsurf ¼ 0:75, and x � 0:1 mm), the surface tempera-ture would attain (near) steady-state value after 100 to 200 s forwater, and within 15 to 40 s for acetone (after which the surfacetemperature would fluctuate about this steady value).

In summary, the theoretical coupling between turbulentmomentum field and surface temperature variations (Eqs. (21)and (22)) combined with rapid (and highly resolved) IR surfacetemperature measurements offer a framework for characterizationof surface-turbulence interactions (parameterized by the eddy res-idence time distribution and the shape parameter a) and a poten-tial method for surface-temperature-based estimation of surfaceevaporative fluxes. In the following, we present the experimental

1

5

6

7

Fig. 2. Experimental setup used to study coherent thermal structures during surfacmeasurement of ambient temperature T1, velocity U1, and relative humidity/vapor sachamber; (4) blowing fans; (5) high-frequency 3D ultrasonic anemometer measuring vel(7) evaporating test surface: a 500� 500� 0:2 mm cotton cloth with pores of size r ¼ 0

setup used to remotely quantify temperature dynamics for anevaporating surface exposed to prescribed turbulent airflowregimes.

3. Experimental considerations

For evaluation of the theoretical framework, we conducted aseries of evaporation experiments (total of 10 runs) under differentturbulent airflow regimes with mean airflow velocities rangingfrom 0.5 to 2.0 m/s in a small wind tunnel (L�W � H:1:2� 1:2� 0:2 m; see Fig. 2). The wind tunnel was equipped withfour blowing fans with built-in straightener grids (TAR/L, LTG Inc.,Germany) providing stable airflows over the evaporating surface(4� 1040 m3/h @ 1050 rpm). A schematic of the experimentalsetup is illustrated in Fig. 2.

We selected a thin cotton cloth (0.2 mm thick) mounted tightlyon a frame of size 500� 500 mm, and initially saturated with ace-tone to provide a thin surface with a high rate of evaporation intoambient air from a surface with low thermal capacity (seeAppendix B). As pointed out by Balick et al. [17], Tiselj et al. [84],and Hunt et al. [85], the magnitude of surface temperature fluctu-ations depends on surface thermal capacity, thickness, and the rel-ative strength of thermal response time for solid and fluid surfaces(see Appendix B). Hence, rapid and dynamically resolved thermalsignatures are expected for thin evaporative surfaces with loweffective heat capacity. Considering Eq. (8) and thermal diffusivityof cotton (DH � 2:2� 10�8 m2/s), the effective thermal thickness of

1 m

0.2

m

2

3

4

e drying: (1) infrared camera; (2) logger for single-point (and low frequency)turation in the middle of duct tunnel above the test surface; (3) wind productionocity of an air layer (Ua) of 50 mm thickness above the test surface; (6) balance; and:03 mm and a porosity of 75%.

Fig. 3. Typical time series of (a) velocity of an air layer (Ua) of 50 mm thicknessabove the test surface measured by the 3D ultrasonic anemometer (along thecenterline), (b) surface temperature obtained from an arbitrary single-point at thecenterline, and (c) acetone mass loss (per area) during drying of a500� 500� 0:2 mm saturated cotton cloth (hsurf � 0:75 and r ¼ 0:03 mm) whichwas exposed to a drying airflow with U1 ¼ 1� 0:25 m/s and T1 ¼ 298 K (measuredat the center of the wind tunnel). The black dashed line shown in (b) is obtainedfrom the analytical model defined in Eqs. (21) and (22) under the given boundaryand initial conditions. The reasonable agreement between model predictions andexperimental data inspires confidence in the proposed mathematical framework.

E. Haghighi, D. Or / International Journal of Heat and Mass Transfer 87 (2015) 429–446 435

the surface for this experiment was simply estimated asx � 0:08 mm for a mean thermal residence time of 1 s. Someexperiments involved multiple thicknesses (from 0.2 to 0.8 mmthick) were also carried out to examine the role of the effectivethickness on the characteristics of surface thermal signatures.The derivation of x (m) uses similar ingredients as the conceptof thermal inertia in remote sensing [86], but it also contains alength scale (effective thickness) for dynamic surface energy bal-ance calculations.

The temperature field at the surface was resolved using an IRimager sampling at 52 Hz. We have used an IR thermal camera(FLIR SC6000, FLIR Systems, MA, USA) with a noise equivalent tem-perature difference (NETD) of 35 mK @ 30 �C. The IR imager isequipped with a quantum well infrared photon detector (QWIP)capable of recording infrared radiation in a narrow range of wave-length of 8.0–9.2 lm, at high resolution (512� 640 pixels). Theintegration time of the detector is 10 ms and it is equipped witha linear Stirling cooler. The thermal images were converted tothe surface temperature Ts assuming a constant surface emissivityof e ¼ 0:95 [87]. The acquired images are transferred to a dedicatedPC with ThermaCAM Researcher software (FLIR Systems, MA, USA)for subsequent analyses.

Details of the velocity of an air layer of 50 mm thickness adjacentto the surface (Ua) was recorded at the edge of the evaporating sur-face using a 3D ultrasonic anemometer with 0.01 m/s accuracy(WindMaster, Gill Instruments Ltd., The Netherland) acquiring mea-surements at 32 Hz. The mean air velocity (U1) was measured at adistance of 100 mm above the test surface (in the middle of the ducttunnel) using a hot-wire anemometer (P600, Dostmann Electronic,Germany), and recorded by a data logger (21X, Campbell ScientificInc., UT, USA). The temperature and relative humidity of the ambi-ent were also monitored at the same location using a temperatureand humidity transmitter (HMT337, Vaisala HUMICAP

�, Finland).

Evaporative mass loss from the surface was measured using elec-tronic balances with �0:01 g repeatability and �0:02 g linearity(ED3202S, Sartorius, Germany).

4. Results and discussion

We first investigated similarities between momentum and ther-mal fluctuation fields using spectral analyses [88] that enabledirect estimates of the extent of coupling between these two pro-cesses. Experimental results for acetone evaporation from the thincotton surface yielded a wide range of highly resolved surface tem-perature fluctuations (discussed in Appendix B). The characteristicsof the surface interacting momentum turbulent field were thendeduced from spatio-temporal IRT measurements, and potentialapplications of such thermal inferences for remote estimation ofsurface evaporative fluxes were explored.

4.1. Surface temperature and turbulent air velocity fluctuations

IRT images and concurrent sonic anemometer airflow measure-ments were sampled at 52 and 32 Hz, respectively. For illustrativepurposes, we present results from an experiment (one out of 10different runs) in Figs. 3(a)–(c). The results demonstrate the highlydynamic evaporative processes into turbulent airflow andassociated surface temperature depression (Figs. 3(a)–(b)). Thefluctuations superimposed onto a mean trend of UaðtÞ andTsðx; y; tÞ are attributed to eddy motions contributing to both U0aand T 0s, where a prime denotes fluctuations about the temporaland tempo-spatial average, respectively. We note that the dashedline shown in Fig. 3(b) is obtained from Eq. (21) for an effectivemean turbulent airflow property under the same experimentalconditions. The agreement between the model prediction and the

experimental data inspires confidence in the proposed couplingapproach.

We further note that the ‘‘deterministic’’ surface temperaturedepression (without instantaneous fluctuations) predicted by themodel (Fig. 3(b)) is consistent with surface temperature adjustmentfollowing exposure to mean turbulent flow. Replacing turbulentmomentum field with ‘‘effective’’ mean properties (hi and ki are eval-uated by the effective viscous sublayer thickness d, rather than di),enables the use of Eq. (21) for parameterizing effective surface tem-perature evolution. Note that here the effective (mean) thermal thick-ness x would simply be estimated from mean thermal eddyresidence time �st (rather than the elapsed time t), and thus is assumedconstant for a stable mean evaporation flux and accordingly does notaffect this alternative approach as t goes to infinity.

To study the relationship between momentum fluctuations in tur-bulent air velocity and surface temperature, we computed the powerspectral densities for the fluctuations in U0a and T 0s using the fastFourier transform [88] depicted in Figs. 4(a) and (b), respectively.The results (averaged over 10 runs) describe the distribution of thepower of a signal (or time series) across different spatial frequencies(wavenumbers). The one-dimensional wavenumber is computedusing Taylor’s [89] hypothesis and the velocity measurements.Katul et al. [30] have argued that eddy motions responsible for surfacetemperature fluctuations may be decomposed into an active compo-nent inducing shear at the surface and an inactive component whichis produced by turbulence in the outer region away from the surface.These two contributions are characterized by power spectra with -1and -5/3 power laws for the inactive and shear-inducing eddymotions, respectively [30]. Evidence of these contributions are shownin Fig. 4(a) for the U0a spectrum; whereas in Fig. 4(b) only the �5/3power law appears to be reflected in the spectrum of T 0s. These resultsare similar to Katul et al.’s [30] findings, where T 0s spectra follow�5/3power law for the entire range of wavenumbers (Fig. 4(b)).

The results in Fig. 4(a) where large-scale fluctuations (smallwavenumbers) in the U0a spectrum follow �1 power law (attribu-ted to inactive eddy motions), yet, thermally, this class of eddiesare expected to exert stronger influence on the thermal signaturesthrough shear affecting the viscous sublayer near the surface(hence inducing �5/3 power law for thermal fluctuations). Thus,

Fig. 4. Power spectral density for (a) air velocity fluctuations U0a (measured at afrequency of 32 Hz) and (b) surface temperature fluctuations T 0s (52 Hz) for all 10runs of the acetone evaporation test. The �5/3 and �1 power laws are also shown.

436 E. Haghighi, D. Or / International Journal of Heat and Mass Transfer 87 (2015) 429–446

the similarity between the two spectra in the inertial range(wavenumbers larger than 10 1/m) supports the proposedapproach that considers eddies contributing to heat and massexchange from the surface through their impact on the viscoussublayer thickness. We also performed a more complete waveletspectral analysis to investigate similarities between the two sig-nals in space and time with the results presented in Appendix C.

Such inherent coupling between airflow turbulence characteris-tics and surface temperature fluctuations offers opportunities forusing a resolved thermal regime to estimate (invert for) the momen-tum field fluctuations (turbulent structures), as discussed next.

4.2. Surface-temperature-based characterization of turbulent airflows

4.2.1. Thermal manifestation of eddy renewal eventsWe capitalized on the spatio-temporal IRT information to make

inferences on eddy-surface interactions (renewal events). Asequence of spatially-resolved snapshots of T 0sðx; yÞ (the meanvalue was spatially averaged over the instantaneous image) duringthe thin cotton surface drying is shown in Fig. 5 demonstratingspatio-temporal evolution of thermal fluctuations over a footprintof size 200 � 200 mm in the center of the 500 � 500 mm test

surface (to avoid edge effects). We note that a time stamp ofDt ¼ 5 s was used to visualize ‘‘macroscopic’’ fluctuations (includ-ing a population of small-scale fluctuations; see Fig. 3(a)) whosethermal spatial distribution can be more easily captured anddetected over a relatively large footprint. The IR images show thatair parcels with different characteristics in contact with the surfaceinduce different surface temperature patterns. An initially positiveT 0s (within the solid white box) can be attributed to a sweep eventby a fast and warm air parcel (relative to surface temperature).Shortly thereafter, several small cold spots appear due to evapora-tive cooling; these then grow, merge, and move along the airflowdirection culminating in an ejection event. As an eddy rises fromthe surface (renewed by another eddy) the surface may warmagain and the cycle is repeated. The same story (sequential hotand cold thermal signatures) is realized from the dashed whitebox, however, with a phase difference.

To further illustrate the spatio-temporal evolution of struc-tures related to the momentum field through their thermalimprints over the surface, we draw a streamwise line throughthe center of the image (the dashed line in Fig. 6(a)) and plotthe time evolution of T 0s along this line for the entire drying pro-cess (Fig. 6(b)). We then zoom-in on to examine a narrow win-dow of a few seconds (3–4 s) from the onset of drying(Fig. 6(c)). The results clearly illustrate the advective nature ofthermal signatures with thermal structures lasting a few secondsas seen from the downward slope of the hot and cold stripes of T 0sin Fig. 6(c). The estimated horizontal velocity from the slope ofthe stripes is 1.5 m/s, close to the directly measured mean windvelocity in the middle of the duct tunnel that advects large-scalestructures along the surface.

Next, we evaluate how surface thermal signatures could beused to characterize the turbulent momentum field (parameter-ized by the eddy spectrum shape parameter a) and yield estimatesfor mean evaporation flux from the surface.

4.2.2. Thermally-deduced eddy residence time distributionAs discussed in Section 2.1 above, the quantification of the vis-

cous sublayer thickness (and evaporative fluxes) requires prescrib-ing the shape parameter of the eddy residence time distribution (a)that is parametrically expressed by the gamma statistical distribu-tion [47,50]. Prominent in Eq. (2) are the shape a and rate b param-eters that control the shape of the distribution. Considering aconstant shape parameter a, the distribution may shift towards lar-ger residence times (low frequency) by decreasing the rate param-eter b (i.e., the mean residence time �s in Eq. (2b) increases). Thus,different distributions characterized by a similar shape parametera can be distinguished through differences in the rate parameterb affected by surface thermal properties deduced from IRTmeasurements.

Invoking the analytical description of surface temperature vari-ations defined in Eq. (21), we express the residence time of ‘‘ther-mal signatures’’ of eddies (the index ‘‘t’’ marks thermal residencetimes, distinguished from momentum residence times marked bythe index ‘‘m’’) as a function of surface thermal properties, as

st; i ¼ �cvxc2

ln 1� DTsjiDTmax

� �ð24Þ

in which DTmax ¼ Teq � T0 is the maximum temperature variationachievable on the top surface due to maximum elapsed time of Neddy replacement with Teq ¼ c3=c2 the equilibrium surface temper-ature. Fig. 7 shows typical parameters used in Eq. (24) which can beextracted from given surface temperature time series.

For prescribed airflow conditions (i.e., c2 and DTmax are given),Eq. (24) shows that the thermal residence time of an eddy, thatresults in a well-defined temperature depression DTs, is directly

Fig. 5. Snapshots of surface temperature fluctuations (averaged spatially over the footprint) T 0s (K) at Dt ¼ (a) 0, (b) 5, (c) 10, and (d) 15 s measured at 52 Hz during theacetone evaporation (wind direction is from top to bottom). The sequence of the snapshots reveals spatio-temporal evolution of thermal fluctuations over the footprint.Considering the thermal field within the solid white box, an initially positive T 0s can be attributed to a sweep event, when a population of fast moving air parcels (eddies)comes in contact with the surface. Shortly thereafter, several small cold spots appear; these grow, combine, and move in the wind direction initiating an ejection event (b). Asthe population of eddies rises from the surface (renewed by another family of eddies), the surface begins to warm again and the cycle repeats (c and d). A similar sequence ofevents (sequential hot and cold thermal signatures) can be realized from the dashed white box but with a phase difference.

E. Haghighi, D. Or / International Journal of Heat and Mass Transfer 87 (2015) 429–446 437

proportional to surface volumetric heat capacity and effective ther-mal thickness. This implies that typical thermal properties of nat-ural surfaces are likely to dampen observable thermal signaturesand filter out rapid perturbations (as discussed in Section 3.2 andAppendix B). On the other hand, such proportionality betweenthermally-deduced eddy residence time (i.e., from thermalresponse time of the surface) and surface properties could beexploited as it imparts no effects on the momentum dominatedshape parameter a, and only eddies with sufficiently long mean res-idence time will imprint their effect on ‘‘surface thermal fluctua-tions’’. This predictable shift could be of great importance forfield-scale applications where evaporation takes place from surfaceswith relatively large effective heat capacity (and thermal inertia).

4.2.2.1. Extracting. a and b Parameters from Surface TemperatureFluctuations

As shown theoretically, surface temperature fluctuations pro-vide a means for characterizing the surface interacting momentumfield intermittency (renewal eddies) through the correspondinggamma density function parameters (a and b). The eddy residencetime distributions depicted in Fig. 8 were deduced from direct airvelocity measurements and from IR imagery (symbols) shown inFigs. 3(a) and (b), respectively. The eddy residence time distribu-tions were obtained from the time difference between two consec-utive peak and valley points in the wind velocity and surface

temperature time series (see Fig. 7), as described in Seo and Lee[50] and Watanabe and Mori [90]. In practice, there is a lowerthreshold for the time difference between consecutive peaks andvalleys longer than the measurement’s ‘‘sampling time’’ (1/Hz),otherwise, these are considered as noise and filtered out, accord-ingly. Note that time differences belonging to a surface tempera-ture difference of smaller than the IR camera’s NETD are alsotreated as noise and filtered.

The fitted gamma density functions to the experimental data(lines) support the potential for using thermal fluctuations todeduce the turbulent airflow characteristics (typically measuredby 3D anemometers). The time distribution spectra of both themomentum- and surface-temperature fluctuations share the sameshape parameter a ¼ 2, which is also in agreement with the previ-ously reported experimental values in the range of a ¼ 0 to 2 [8].Consequently, this unique shape parameter for T 0s and U0a enablesquantification of an effective viscous sublayer thickness (linkedto the momentum field characteristic) from rapid thermal mea-surements (see Eqs. (1)–(4)). An interesting feature of this experi-ment was the ability to better resolve the dynamics of thermalfluctuations due to the higher sampling frequency of the IR camera(52 Hz for IR imagery relative to 32 Hz by the sonic anemometer),while in typical terrestrial applications we expect the reverse toapply (due to higher thermal capacity of terrestrial surfaces rela-tive to the thin evaporative surface used in this exploratory study).

Fig. 6. (a) Dashed line through the image oriented in the streamwise direction indicates centerline of the drying surface, (b) Temporal evolution of T 0s recorded (at 52 Hz)during acetone evaporation along the centerline for the whole period of the drying process, and (c) for a few seconds after the onset of drying (marked by the red rectangle in(b)). The white squares shown in (a) indicate data sampling locations analyzed in Fig. 8. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

Fig. 7. Typical surface thermal information (Eq. (24)) extracted from a given surfacetemperature time series.

Fig. 8. Comparison of eddy residence time distributions deduced from air velocity(3D sonic anemometer, 32 Hz) and temperature (IR imager, 52 Hz) fluctuationsmeasurements during acetone evaporation. Wind data were measured at differentheights above the surface (z indicates the height of the bottom of the sonicanemometer) revealing variations in turbulent structures (parameterized by a)with the measurement height. The error bars shown over the temperature-baseddata represent confidence interval (standard deviation) of the data taken from fivedifferent locations at the centerline (white squares shown in Fig. 6(a)).

438 E. Haghighi, D. Or / International Journal of Heat and Mass Transfer 87 (2015) 429–446

The robustness of the newly proposed surface-temperature-based approach was further evaluated by monitoring wind velocityat different heights above the surface using the 3D sonicanemometer. The results depicted in Fig. 8 reveal systematic vari-ations in the turbulent regimes (parameterized by a) with mea-surement height such that the turbulent flow gradually becomesdominated by larger eddies (with longer residence time) with

increasing the measurement height (z). We note that z indicatesthe height of the bottom of the 3D sonic anemometer and the

E. Haghighi, D. Or / International Journal of Heat and Mass Transfer 87 (2015) 429–446 439

measurements volume contains an air layer of 150 mm thickness(the gap between sonic transducers). Such systematic variationsin the eddy spectrum parameter with height are indicative of vari-ations in the nature of the flow field from shear flow near the sur-face to free shear flow away from the surface [8,91]. These nuancedvariations may not be captured by classical flux gradient methodslimiting the quantification of mean viscous sublayer thicknessusing Eq. (3). Additionally, air velocity measurements recordedby the sonic anemometer even at the surface may contain fluctua-tions at scales comparable with the sensor’s physical dimensions(i.e., a 150 mm gap between sonic transducers relative to a viscoussublayer thickness of a few millimeters). This inherent measure-ment bias may shift the measured eddy residence time distributiontowards lower frequencies or longer eddy residence times (withlarger a) [92] than the population of surface-interacting eddies thatsuggests a potential advantage of inferences made at the surfacewhere renewal eddies ‘‘directly’’ leave their imprints.

The results in Fig. 8 demonstrate that surface temperature fluc-tuations spectra are consistent over the surface and independent ofany particular sampling location. The distributions of thermal fluc-tuations obtained from five different positions along the centerline(white squares in Fig. 6(a)) collapse into a unique eddy residencetime spectrum (the standard deviations are shown by the errorbars in Fig. 8). The spatially-consistent results are of practicalimportance suggesting the possibility of replacing costly IRT bylow cost and simpler single-point IR measurements (with a spatialresolution of larger than the representative elementary area of thesurface) useful for widespread field and industrial applications.

4.2.2.2. Effects of surface thermal properties. The magnitude of tem-perature fluctuations over typical natural surfaces will be damp-ened by their relatively high surface heat capacity cv , and theirlarge effective thermal thickness x (see Appendix B). Eq. (24)shows that thermally-deduced eddy residence time is proportionalto surface thermal properties. Hence, while the shape of the eddyresidence time distribution characterized by the shape parametera may be constant, the rate parameter b is expected to vary withthe mean response time of measured thermal signatures due tosurface thermal properties.

The experimental results depicted in Fig. 9 illustrate how ther-mally-deduced eddy residence time distribution varies with effec-tive surface thermal thickness for acetone evaporation from acotton cloth of different thicknesses. These experimental resultsconfirm (1) the independence of the shape parameter a from

ωωω

Fig. 9. Variation of surface-temperature-based eddy residence time distributionwith surface thickness measured during acetone evaporation.

surface thermal properties, and (2) the sensitivity of the rateparameter b to surface thermal properties. Hence, informationgleaned from surface temperature fluctuations provide direct linksbetween the rate parameter b and surface wetness through effectson surface effective thermal thickness and heat capacity (Eq. (24)).This link is of potential importance for various remote sensing ofhydrological and meteorological near surface processes that willbe explored in future studies.

4.3. Turbulent evaporative fluxes quantified by surface thermalfluctuations

A potential application of thermally-deduced eddy residencetime distributions discussed in the previous subsection is the esti-mation of average heat and mass fluxes induced by eddy-surfaceinteractions. These exchange processes are dominated by the vis-cous sublayer thickness, shown to be a function of the eddy spec-trum shape parameter a (Eqs. (3)–(5)). Substituting a obtainedfrom surface temperature measurements discussed in the previoussubsection into Eqs. (4) and (5) yields the effective (mean) viscoussublayer thickness defined in Eq. (3). This, in turn, enables estima-tion of mean evaporation flux from drying surfaces by substitutingthe mean mass transfer coefficient k (m/s) evaluated by d into Eq.(14) as (see Haghighi and Or [8] for detailed discussions)

E ¼ Ddþ Pf ðhsurf Þ

ðCs � CaÞ ð25Þ

where all terms have been defined previously.

4.3.1. Acetone evaporation from the thin cotton clothEvaluating Eq. (25) by airflow and surface properties given in

Fig. 3 and the thermally-deduced eddy spectrum shape parameterobtained in Fig. 8 (a ¼ 2), the mean evaporation flux is estimatedas Etheory ¼ 0:0011 kg/m2s, a value in good agreement with the fluxdetermined by direct measurement of mass loss, Eexperiment ¼ 0:0013kg/m2s (see Fig. 3(c)). Fig. 10 depicts a comparison between modelpredictions (using the surface-temperature-based a) and experi-mental results obtained under similar conditions as described inFig. 3 for a range of mean air velocities (as marked). The agreementbetween model predictions and measured fluxes lends support tothe proposed surface-temperature-based estimates of parametersthat control the viscous sublayer thickness (Eqs. (3)–(5)). More

Fig. 10. Comparison between model estimations using the thermally-deduced eddyspectrum parameter a ¼ 2 (used for calculation of the viscous sublayer thicknessvarying from 4 to 1.5 mm while air velocity changes from 0.5 to 2 m/s) andexperimental data. The square symbol corresponds to the sample measurementsshown in Fig. 3 and the solid line represents 1:1 line.

440 E. Haghighi, D. Or / International Journal of Heat and Mass Transfer 87 (2015) 429–446

detailed discussion of the sensitivity of estimated evaporativefluxes to the eddy shape parameter a that affects the viscous sub-layer thickness is provided in Haghighi and Or [8].

4.3.2. Plant leaves and canopy interactionsPlant leaves represent natural evaporating surfaces with inher-

ently thin thermal thickness (and low thermal inertia). Studies byPaw U et al. [93], Braaten et al. [94], and Snyder et al. [95] haveexplored the behavior of coherent turbulent structures withinplant canopies that periodically sweep through the canopy (SRevents). Extracted from high-frequency air temperature measure-ments above canopies, characteristics of these coherent structureswere subsequently used by many investigators [96–99] to estimatesurface sensible and latent heat fluxes using the SR formalism.

The link between turbulence and canopy temperature perturba-tions (observable by IRT [100]) could be used to characterize tur-bulent coherent structures that contribute to surface fluxes[30,101]. Invoking a similar line of reasoning, we hypothesize thatthe use of surface thermal information proposed in this studycould be used to quantify turbulence-induced heat and massexchange from assemblies of leaf surfaces. We note that concur-rent stomatal thermal regulation via transpiration cooling maymediate turbulence induced variations in leaf temperatures

Fig. 11. (a) Air velocity and surface temperature data reproduced from Katul et al.[30] which were recorded over a grass-covered field (small thermal inertia); (b)Comparison of eddy residence time distribution deduced from air velocity andtemperature fluctuations shown in (a).

[102–104], however, such adjustments occur at longer time scales(102 to 103 s [104,105]) relative to characteristic turbulent resi-dence times (a few seconds) and are neglected here.

Detailed surface IR images are not required for inferences of tur-bulent interactions (see Section 4.2.2.1), and a single-point rapidthermal measurement over an evaporating surface (or canopy) suf-fices for extraction of surface-temperature-based eddy residencetime distributions. We re-analyzed such (rapid) single-point IR-measured surface temperature and air velocity fluctuations over agrass-covered surface reported by Katul et al. [30] as depicted inFig. 11(a). The results in Fig. 11b represent the eddy residence timedistributions obtained from direct air velocity fluctuations and indi-rectly from the surface-temperature-based approach. As for the lab-oratory-scale results depicted in Fig. 8, the results in Fig. 11(b)confirm the similarity between the distributions obtained by veloc-ity and thermal measurements, both yield a unique eddy spectrumshape parameter a for the U0a and T 0s distributions. For the data ofKatul et al. [30], the surface temperature and wind velocity fluctua-tions were both recorded at the same sampling frequency (5 Hz),hence information concerning faster thermal fluctuations was notcaptured due to the surface thermal inertia. Regardless, the thermalproperties of the surface were not expected to affect the shapeparameter a (see Section 4.2.2). A potential application of the pro-posed approach would be the estimation of evaporative fluxes fromcanopies using Eq. (25) based on rapid thermal measurements.Successful determination of such fluxes would require supplemen-tal information such as mean stomatal size and density.

5. Summary and conclusions

The study develops a theoretical framework for linking rapidthermal fluctuations from an evaporating surface with characteris-tics of turbulent airflows interacting with the surface. The intermit-tent regime associated with turbulent eddy-surface interactionsgives rise to structured temperature fluctuations; these are linkedvia surface renewal formalism and parametrically by the gammadistribution parameters. We extended a recently proposed modelof Haghighi and Or [8] for turbulent exchange from evaporatingsurfaces to explicitly consider instantaneous and local variationsin surface energy balance induced by evaporative cooling beneatheddy footprints.

Results of acetone evaporation from thin cotton cloths con-firmed that the temperature-based eddy spectrum shape parame-ter a was constant for momentum and surface thermal fluctuationsand values were consistent with previously-reported (momentum-based) measurements of Haghighi and Or [8]. Surface thermalinformation offers direct access to surface-turbulence interactionswhere exchange processes take place (instead of measured at someheights above the surface). The results also showed that theinferred statistical properties of surface thermal fluctuations (andturbulence) are independent of the measured location over homo-geneously wet surfaces such that a single measurement over thesurface could provide necessary information for deducing theshape and rate parameters, and facilitate estimates of surface evap-oration. This, in turn, makes the proposed approach a good candi-date for (large-scale) hydrological applications potentiallycomplementing momentum-based approaches.

As a preliminary test of the approach for practical field applica-tions, the model was evaluated using reported values of thermalfluctuations from short grass leaves (surfaces with relatively lowthermal inertia) to deduce characteristics of the turbulent airflow(that was measured and reported independently). The agreementbetween surface-temperature- and momentum-based eddy resi-dence time distributions obtained thermally (from a point mea-surement) and by direct wind speed measurements, respectively,

E. Haghighi, D. Or / International Journal of Heat and Mass Transfer 87 (2015) 429–446 441

inspires confidence in the utility of the method. We also discussedhow such thermally-deduced eddy residence time distributionscould be used to quantify evaporative fluxes from thin porous sur-faces (including plant leaves).

Additional tests would be required to evaluate model applicabil-ity for more general scenarios of evaporation from natural surfaces;nevertheless, the study provides new insights on the role of airflowturbulence in surface-turbulence interactions (irrespective ofmomentum field measurements) at this critical interface wheremost of the exchange processes originate. We envision that theapproach could potentially provide a building block for upscalingto field- and landscape-scale applications with relatively high ther-mal inertia (e.g., water evaporation from bare soil surfaces) throughfurther evaluations of likely effects of surface hydro-thermal prop-erties (water content, volumetric heat capacity, and effective ther-mal thickness) on the attenuation of surface temperaturefluctuations. Although the primary focus of the study was on turbu-lence interactions with an evaporating porous surface, the analysiscould be extended easily to infer sensible heat flux exchanges overdry surfaces using similar thermal fluctuation concepts.

Conflicts of interest

None declared.

Acknowledgements

The constructive comments from anonymous reviewers led to animproved paper. The authors gratefully acknowledge funding by theSwiss National Science Foundation of the project ‘‘Evaporation fromterrestrial surfaces – linking pore scale phenomena with landscapeprocesses’’ (200021-113442) and the financial support of theGerman Research Foundation DFG of the project ‘‘Multi-ScaleInterfaces in Unsaturated Soil’’ (MUSIS; FOR 1083). The assistanceof Mr. Fabian Rüdy with the illustration shown in Fig. 2 and techni-cal assistance of Daniel Breitenstein and Hans Wunderli (ETHZurich) are greatly appreciated.

Appendix A. Evaluation of the analytical approximation for theeffective thermal thickness

To evaluate the performance of the analytical approximation forthe effective thermal thickness defined in Eq. (8); temperature

sTx

z

50 m

m

1 mm

∂T/∂

x=0

T∞

Solu

tion

dom

ain

∂T/∂

x=0

Fig. A1. Schematic of an evaporating (saturated) soil column below an eddyfootprint (dimensions are not to scale). Solution domain along with initial andboundary conditions used for numerical solution of the temperature field beneaththe surface are presented.

ig. A2. Comparison of the analytical approximation for the effective thermalickness (Eq. (8)) with numerical simulations under saturated conditionsurf ¼ 0:38 and DH ¼ 4� 10�7 m2/s): (a) probability density of momentum andermal eddy residence times extracted from measured wind velocity and surfacemperature time series, (b) temperature profile beneath the top soil obtained from

umerical simulations for different eddies characterized by sm (the red profiledicates the result belonging to the eddy with mean properties), and (c)

omparison between numerical estimates of the thermal thickness (blue circles)btained from (b) with the analytical approximation (solid red line) which isvaluated by �st obtained from (a). (For interpretation of the references to color inis figure legend, the reader is referred to the web version of this article.)

Fth(hthtenincoeth

442 E. Haghighi, D. Or / International Journal of Heat and Mass Transfer 87 (2015) 429–446

distribution (2D unsteady heat condition) beneath an evaporatingsaturated soil column (hsurf ¼ 0:38 and DH ¼ 4� 10�7 m2/s) duringmomentum eddy residence times was resolved numerically usingan explicit finite difference method. The governing equation usedis a 2D unsteady Laplace equation for heat conduction below theevaporating surface as

@T@t¼ DH

@2T@x2 þ

@2T@z2

!ðA1Þ

with initial and boundary conditions as

Tðt ¼ 0Þ ¼ T1Tðz ¼ 0Þ ¼ Ts

Tðz! �1Þ ¼ T1@T=@xjx¼0 and 1 mm ¼ 0

ðA2Þ

A schematic of the evaporating soil column and solutiondomain along with the initial and boundary conditions are shownin Fig. A1. We note that even grid spacing of Dx ¼ Dz ¼ 0:02 mm inx- and z-directions was used and that the time interval was deter-mined by Dt ¼ 0:5� CFL (with CFL ¼ Dz2=4DH the Courant–Friedrichs–Lewy condition) to ensure stability of the numericalsolution.

Considering measured surface temperature time series for a sat-urated soil sample (not reported here) as the top boundary condi-tion (T ¼ Ts) during momentum-based eddy residence time ofindividual eddies extracted from wind velocity measurements sm

(Fig. A2a), Fig. A2b illustrates a numerically resolved temperatureprofile along the saturated soil column and its variation withmomentum-based eddy residence times (sm) drawn from the spec-trum shown in Fig. A2a. The 2D temperature field shown in theinset was obtained for an eddy with mean properties (s ¼ �sm)and a corresponding 1D profile is indicated by the thick red line.The numerical results show variation of the thermal thickness(the depth at which the effects of the temperature variationinduced at the surface is dampened and the temperature profilebelow the surface reaches its initial status) with the eddy residencetime whose mean value tends to a depth of 1 mm below the sur-face for the whole spectrum of eddies interacting with the satu-rated surface.

Comparing the numerically obtained thermal thicknesses withthe analytical approximation obtained from Eq. (8), which is eval-uated by the mean thermal eddy residence time �st (Fig. A2a),Fig. A2c reveals that there is a fair agreement between the twoapproaches suggesting confidence in the analytical approximationused in this study. We note that surface thermal fluctuations con-tain a more integrated measure of the energy partitioning interac-tions as they implicitly consider changes in energy partitioningbetween latent, sensible, and conductive soil heat fluxes duringeddy residence times; these in turn are reflected in the resultingst and then in the estimated effective thermal thickness. For allpractical purposes the estimated x based on the value of �st is ofthe same order as the one that would have been obtained fromthe complete solution of the energy balance over the same surface.

Appendix B. Detection limits for infrared thermography ofevaporating surfaces

B.1. Temperature resolution

The use of IRT measurements for the proposed inferenceshinges on the presence of surface temperature fluctuation ampli-tudes, DTs ¼ Ts � T0, larger than the temperature resolution ofthe IRT device, DTIR. The temperature resolution of an IR camerais specified by the so-called noise-equivalent temperature

difference (NETD) for a blackbody, DTIR0. Since the noise intensityis independent of surface emissivity [106], the values of the noiseamplitudes for a blackbody and a non-blackbody emissivity aresimilar yielding the following relation (invoking the binomial the-orem and assuming Ts � DTIR0 and Ts � DTIR) [106]

DTIR ¼DTIR0

eðB1Þ

Eq. (B1) indicates that the temperature resolution for a non-blackbody is proportional to the reciprocal of the surface emissiv-ity (e).

Using Eq. (21), we define a dimensionless temperature variation(surface time constant to response to an induced temperaturedepression) during the residence time of a single eddy of averageproperties as

DT�s ¼DTs

DTmax¼ 1� e�ðc2=c1Þ�sm ðB2Þ

where DTmax ¼ Teq � T0 is the maximum temperature variationachievable on the top surface due to maximum elapsed time of Neddy replacement and �sm � 112va=u2

� is the (momentum-based)eddy mean residence time [8].

Eq. (B2) contains a time constant that characterizes the surfaceresponse to temperature depression induced by a renewal eddyduring its residence time. For known airflow conditions (with con-stant DTmax), a cursory inspection of Eqs. (B2) and (21) reveals thatthe amplitude of surface temperature fluctuations is enhanced bydecreasing the volumetric heat capacity cv and the surface effec-tive thermal thickness x (Fig. B1a). The comparison depicted inFig. B1a reveals that the attenuation of thermal fluctuationsdecreases with increasing cv and x, hence thermal fluctuationsof natural surfaces would require the use of sensitive IR imagers.In addition to surface thermal properties, airflow properties suchas air temperature and air velocity also contribute to the surfacetemperature fluctuations DTs, through effects on both DTmax andDT�s .

For standard laboratory conditions (e.g., Rsw ¼ 0 W/m2,T0 ¼ 293 K, and RS ¼ 35 and 0% for water and acetone, respec-tively), Fig. B1b represents the expected variations of DTs over athin layer of liquid saturated cotton cloth (x ¼ 0:08 mm) exposedto various mean wind velocities and air temperatures. The resultsclearly show that thermal fluctuations are enhanced by diminish-ing the air velocity (increasing eddy residence times); however,the effect of the air temperature is also a function of the surface ini-tial temperature T0 (especially for water). Even though the evapo-rating surface temperature may increase under certaincircumstances (such as evaporation into very hot airflows), gener-ally, evaporation results in a surface temperature depression(evaporative cooling, marked by DTs < 0) due to phase changefrom liquid to vapor [53,107]. Thus, for many natural conditions,where the surface temperature decreases by evaporation, theamplitude of surface thermal fluctuations may decrease withincreasing the air temperature (counteracting the evaporativecooling) as shown in Fig. B1b. Fig. B1b also reveals that tempera-ture fluctuations are considerably enhanced for evaporating sur-faces saturated with acetone whose large vapor pressure gradientis less dependent on the air temperature.

In summary, based on the analyses in Figs. B1a and 1b, acetoneappears to offer certain advantages as the evaporation liquid espe-cially in the context of accentuating surface thermal fluctuations(practically independent of air temperature). Furthermore, ourexperimental setup was equipped with a high sensitivity IR camerawith a specified DTIR0 ¼ 20 to 30 mK that could resolve tempera-ture fluctuations over thin surfaces under commonly experiencedrange of air velocities (1 to 5 m/s) and air temperatures (273 to323 K).

Fig. B1. Theoretical analysis of temperature resolution detectable using IRmeasures under prescribed conditions: (a) Variation of the dimensionless surfacetemperature fluctuations, DT� , with surface thermal properties (effective thermalthickness and volumetric heat capacity) over water and acetone surfaces exposed toa prescribed airflow condition (Rs ¼ 0:0 W/m2; Ta ¼ 298 K; Ua ¼ 2 m/s; and RS ¼ 35and 0% for water and acetone, respectively); (b) Variation of surface temperaturedepression over a thin layer of water and acetone (with a damping depth ofx ¼ 0:08 mm) exposed to various mean wind velocities and air temperatures,under prescribed conditions (Rs ¼ 0:0 W/m2; To ¼ 293; and RS ¼ 35 and 0% forwater and acetone, respectively).

Fig. B2. Theoretical analysis of upper limit of fluctuating frequency detectableusing IR measures under the prescribed conditions: (a) Variation of the maximumresolvable frequency, f max, with surface thermal properties (effective thermalthickness and volumetric heat capacity) over a layer of water and acetone exposedto a drying airflow (Rs ¼ 0:0 W/m2; Ta ¼ 298 K; and U1 ¼ 2 m/s); (b) Variation off max over a thin layer of water and acetone (with a damping depth of x ¼ 0:08 mm)exposed to various mean wind velocities and air temperatures, under prescribedconditions (Rs ¼ 0:0 W/m2; To ¼ 293; and RS ¼ 35 and 0% for water and acetone,respectively).

E. Haghighi, D. Or / International Journal of Heat and Mass Transfer 87 (2015) 429–446 443

B.2. Upper limit of thermal fluctuations frequency

We may consider the maximum frequency of eddy-inducedthermal fluctuations with respect to limits of detectability. Basedon Eq. (21) this could be expressed simply as DTs ¼ DTIR (the upperlimit of detectability) that corresponds to a critical thermal fluctu-ations frequency, f max (1/s). The value of f max can be determined as afunction of DTIR (the minimum resolvable temperature fluctuation)for known thermo-physical properties of the surface and the air

f max ¼ �c2

c1

1

ln 1� DTIRDTmax

ðB3Þ

For given airflow conditions, Eq. (B3) suggests that the maxi-mum resolvable frequency by an IR camera (with a specifiedDTIR0) increases with decreasing the surface effective thermalthickness and with decreasing the volumetric heat capacity(Fig. B2a). The results shown in Fig. B2a reveal that the phase delayof surface temperature fluctuations decreases with decreasing the

volumetric heat capacity and the surface effective thermal thick-ness (less dampened frequency due to the surface thermal inertia).In other words, retaining high-frequency thermal fluctuationsrequires a reduction in surface thermal properties (i.e., low volu-metric heat capacity and thin effective thermal thickness).

Simulation results under the prescribed conditions depicted inFig. B2b show the maximum detectable frequency of thermal fluc-tuations f max by a high sensitive IR camera with an NETD ofDTIR0 ¼ 20 to 30 mK. For the normal conditions where DTs < 0,the range of f max is increasing by increasing the air velocity andby decreasing the air temperature. The results also reveal thatresolving fluctuations for an air velocity of 1 m/s for a wide rangeof air temperatures (273 6 Ta 6 323) would require a rapid IRcamera with a frame rate of 300 Hz.

Appendix C. Wavelet spectral analysis

To quantitatively assess similarities between turbulent air char-acteristics and the resulting surface temperature fluctuations, we

444 E. Haghighi, D. Or / International Journal of Heat and Mass Transfer 87 (2015) 429–446

have used wavelet spectral analysis [108,109]. A wavelet is a math-ematical function used to distinguish the various contributions ofdifferent space and time scales to observed function or continu-ous-time signal. The effect of each scale component can then bestudied at a resolution that matches its scale.

A wavelet transform is the representation of a function bywavelets which are scaled and translated copies (‘‘baby’’ wavelets)of a finite-length or fast-decaying oscillating waveform (‘‘mother’’wavelet) [108,109]. The wavelet transform is often compared withthe traditional Fourier transforms, in which signals are representedas a sum of sinusoids. The main difference is that wavelets arelocalized in both time and frequency whereas the standardFourier transform is only localized in frequency. Wavelets oftenyield a better signal representation using multiresolution analysis,with balanced resolution at any time and frequency. Wavelettransforms have several advantages over traditional Fourier trans-forms for representing functions that include discontinuities andsharp peaks, and enable accurate deconstruction and reconstruc-tion of finite, non-periodic and/or non-stationary signals [110].

For a time series F, its wavelet function WF is given by [108]

WFðs; sÞ ¼Z

FðtÞwsðt � sÞdt ðC1Þ

where s, s, and ws are respectively scale, time, and mother waveletfunction (Morlet wavelet in this study). Thus, wavelet analysisyields not only the spectral measure of variance, but also the timeinstant when it appears. One can use wavelets to detect surfacerenewal events and analyze wind speeds and surface temperatures[30,108,111].

Figs. C1a and 1b depict the wavelet scalogram of the air velocityand surface temperature fluctuations in which colors represent thepercentage of energy for each continuous wavelet coefficient of thesignal vectors Ua and Ts at real positive scales using the motherwavelet. A scalogram is a visual method of displaying the time-scale (frequency) localization property of a continuous wavelettransform using a mother wavelet. As clearly seen, there is a gen-eral similarity between U0a and T 0s scalograms especially for higherfrequencies (reflected by small scales) where the fluctuations aremore apparent and the T 0s time series lags behind U0a by about 10s. We note that the scale parameter shown on the y-axis is definedas the stretching factor such that the mother wavelet stretches

Fig. C1. Scaled wavelet scalogram of (a) air velocity and (b) surface temperaturefluctuations recorded during acetone evaporation until the surface dries out (45 s).In this plot, coefficient values are shown by colors (z-axis) and x and y axesrepresent time interval and scale range related to the coefficients, respectively.

(lower frequency) by increasing the scale parameter. Consistentwith the time plot of Ua and Ts in Figs. 3(a) and(b), and with theenergy spectra shown in Figs. 4(a) and (b), the U0a scalogramincludes larger scale fluctuations detected at larger scales (com-pared to T 0s scalogram). This is attributed mainly to differences inmeasurement frequency (52 Hz and 32 Hz for IR imager and sonicanemometer, respectively).

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