Turbulent radiation interaction in jet flames: Sensitivity to the PDF

International Journal of Heat and Mass Transfer 57 (2013) 250–264

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Turbulent radiation interaction in jet flames: Sensitivity to the PDF Peter S. Cumber ⇑, Owin Onokpe Heriot-Watt University, School of Engineering and Physical Sciences, Riccarton Campus, Edinburgh, United Kingdom

a r t i c l e

i n f o

Article history: Received 19 May 2012 Received in revised form 7 October 2012 Accepted 9 October 2012 Available online 7 November 2012 Keywords: Hydrogen jet fire Turbulence–radiation interaction Stochastic simulation

a b s t r a c t In this paper the modelling of TRI in hydrogen jet flames is investigated. TRI is modelled using a stochastic methodology similar to Faeth’s research group’s methodology. The present work differs from Faeths by investigating the predictive capability of the TRI model to the PDF used to generate realisations of the instantaneous mixture fraction field. Two PDFs are considered, a b PDF and a truncated Gaussian PDF. Surprisingly the predicted heat flux distribution external to the jet flame exhibits sensitivity to the shape of the PDF implemented. This is counter intuitive given previous studies where the predicted flame structure is shown not to be sensitive to the PDF implemented as part of the turbulent combustion model. The sensitivity of the predicted heat flux distribution to the shape of the PDF implemented as part of the TRI model is analysed and the reason for the sensitivity identified. An additional conclusion of this work is the b PDF gives a more accurate prediction of the heat flux distribution compared to a truncated Gaussian PDF. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction

erT 4

Over the last few decades there have been many investigations into the fire dynamics of laboratory scale turbulent jet flames. The fluctuations in temperature and species concentrations leads onto many interesting phenomena to do with turbulence–chemistry interactions [1], pollution emission [2], enhanced heat transfer due to turbulence–radiation interactions (TRI) [3–5], mathematical model development for investigating fire safety and models for the flame structure with a view to study the generation of pollutants such as NOx [6–8]. The focus of the present paper is the modelling of turbulence– radiation interaction in jet flames such that a computational model which can predict accurately the radiation field external to the jet flame on modest computer systems is possible. Hydrogen jet flames are of particular interest as TRI is well known to be important [3,4], and recently there has been much interest in hydrogen as an alternative green fuel.

where e denotes the emissivity of the gas, r is the Stefan– Boltzmann constant and T is the temperature. The over bar indicates the term is time averaged. Due to the nonlinearity in temperature, depending on the magnitude of the fluctuations in temperature and participating species radiation emission is significantly larger than the approximation to the mean emission based on the mean temperature and emissivity evaluated from the mean participating species concentrations.

2. Historical survey 2.1. Turbulence radiation interaction The first investigations of TRI were completed in the 1970s [9]. Cox [9] based on a grey analysis of a turbulent homogeneous volume showed that the mean emission takes the form, ⇑ Corresponding author. Tel.: +44 0131 451 3532; fax: +44 0131 451 3129. E-mail address: [email protected] (P.S. Cumber). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.10.032

erT 4  erT 4 This is a serious problem for most flame structure models where the mean flow fields are available for calculating the radiation source term in the conservation of energy transport equation or similar and when calculating the heat flux distributions external to the flame. In the 1980s and early 1990s a number of papers were published to account for TRI in a relatively simple fashion and demonstrated better agreement with experimental measurements than one based on mean flow fields [10–12]. For example Hall and Vranos [10], starting from the time averaged radiation heat transfer equation showed that using the exponential wide band model of Edwards [13] and Cumber et al. [14] a reasonable approximation for modelling TRI can be found by time averaging band parameters calculated using mean flow field properties and integrating the black body emissive power together with a probability density function (PDF) over instantaneous mixture fraction space. This made it possible to integrate the radiation heat transfer equation once for each band rather than the many integrations required

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251

Nomenclature C1, C2 Cl Cp,i d f ~f ~f 002 g G Ib,k Ik k Ka Kk NR NRay Nsam P Pk pN Q r r, s s Re Rs Rt T Tamb Tadia U0 U V Y H2 O

turbulence model parameters turbulence model parameter specific heat capacity of species i burner diameter instantaneous mixture fraction mean mixture fraction variance of mixture fraction gravitational acceleration Gaussian distribution black body spectral intensity spectral intensity turbulence kinetic energy absorption coefficient spectral absorption coefficient number of control volumes in the radial direction number of rays number of samples probability density function production of turbulence kinetic energy polynomial approximation to the cumulative PDF approximation to the cumulative PDF radial coordinate parameters in the b PDF coordinate parameterising a pencil of radiation Reynolds number spatial correlation temporal correlation temperature bulk air temperature adiabatic temperature source velocity axial velocity component radial velocity component mass fraction of H2O

for the stochastic methodologies discussed below. This shows promising results when simulating a CH4–H2 jet flame, but is restricted to the exponential wide band model. Much research on TRI relates to coupling the radiation fields to the conservation of energy equation or similar to assess the importance of TRI in modifying the flame structure such as the temperature and species fields as well as the density field. When coupling the radiation field to the flame structure and including TRI the starting point is the radiation heat transfer equation in its differential form. To simplify the problem scattering is often ignored, a valid assumption for most jet flames,

dIk ¼ K k Ik þ KkIb;k ds where Ik is the spectral intensity, Kk is the spectral absorption coefficient, Ib,k is the Planck black body distribution. As the flow is turbulent the variables can be decomposed into a mean and fluctuating component.

Ik ¼ Ik þ I0k Ib;k ¼ Ib;k þ I0b;k K k ¼ K k þ K 0k Substituting these into the radiation transfer equation and time averaging,

dIk ¼ K k Ik þ K k Ib;k  K 0k I0k þ K 0k I0b;k ds

x z zL, zU

axial coordinate pseudo random number parameters in the approximation to the cumulative PDF

Greek symbols Dh enthalpy perturbation e dissipation rate of turbulence kinetic energy e emissivity k wavelength l, r2 parameters in the clipped Gaussian and truncated Gaussian PDF leff effective dynamic viscosity q density r Stefan Boltzmann constant rk turbulent Prandtl number for turbulence kinetic energy re turbulent Prandtl number for dissipation rate of turbulence kinetic energy rDh turbulent Prandtl number for enthalpy perturbation Subscripts adia adiabatic property inst instantaneous property RMS root mean squared property St stoichiometric property k spectral property Superscript ‘ fluctuating quantity about a mean value Over bar _ Reynolds average  Favre average

The last two terms in the equation above are due to TRI and these must be closed in some way. Of these terms the first is the most difficult to model as it depends on nonlocal properties. Most researchers into TRI to make progress assume the correlation

K 0k I0k is negligible. Kabasnikov [15] established a criterion for when this assumption is valid based on the optical thickness of the flame, this is called the optically thin fluctuation approximation, (OTFA) or sometimes it is referred to as the optically thin eddy approximation [16]. Song and Viskonta [12] and Kabashnikov and Myasnikova [17], have proposed physical arguments that support the validity of the OTFA. Habibi et al. [18], and Coelho [19] have also simulated a number of jet flames taking a number of different approaches to modelling TRI including the OTFA. One of the conclusions of their investigations being the OTFA is a valid approximation for most jet flames of interest. 2.2. TRI modelling as a stochastic process The first serious attempt to investigate TRI in jet flames, taking account of the spectral nature of flames was Faeth’s group [3,4,20]. Gore et al. [3] use a stochastic methodology to simulate instantaneous realisations of the spectral intensity distribution for pencils of radiation or rays with a horizontal orientation passing through the axis of the jet. The mean spectral intensity is calculated by averaging the instantaneous realisations. In Faeth’s group’s initial

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stochastic methodology the non-homogeneous path of a ray is separated into a number of segments that are assumed to be well approximated to be homogeneous and each ray segment is taken to be statistically independent. The instantaneous composition and temperature in each homogeneous ray segment is characterised by the instantaneous mixture fraction via a laminar flamelet library. For each realisation the instantaneous mixture fraction is evaluated from an approximate clipped Gaussian PDF using a pseudo random number generator. The instantaneous realisation of the temperature and participating species are then input into Grosshandler’s narrow band model, RADCAL [21] to produce the instantaneous spectral intensity distribution for a given ray orientation and receiver location. Although the clipped Gaussian PDF can reproduce the intermittency at the boundary of the jet, Faeth et al. [20], also justifies the decision of implementing a clipped Gaussian PDF for the stochastic simulations on the basis of ease of implementation as they found other PDFs difficult to implement. This should be contrasted with a b PDF being commonly used for modelling the flame structure. Thus far to the authors knowledge all stochastic simulations of TRI in jet flames have used a clipped Gaussian PDF for the same reason. In [3,4], the flame structure is calculated using a parabolic Reynolds averaged flow model closed using a k-e-g turbulence model. Turbulent combustion is modelled using a laminar flamelet library, combined with a presumed clipped Gaussian PDF. In all of the simulations presented in [3,4] the primary interest is the radiation fields hence the influence of TRI on the flame structure is not considered and the radiation fields are calculated as a post-process to the flame structure simulation. Radiation heat loss is accounted for by adjusting the temperature flamelet to predict the measured fraction of heat radiated for each flame investigated. Faeth and Co-workers [3,4,22–24] investigated TRI in a number of different jet flames with different fuels, finding significant TRI in hydrogen jet flames [3,4], ethylene jet flames [22], with less significant TRI effects on heat transfer for methane jet flames [23] and carbon monoxide jet flames [24]. In all of Faeth’s groups investigations TRI increased the heat transfer from the flame by a factor of 1.1–4.2 compared to a mean analysis where TRI is not included. Gore et al. [3], measured the spectral intensity distribution using a narrow angle spectrometer and the external radiation heat flux distribution but only modelled TRI when calculating the spectral intensity distribution. Kounalakis et al. [4] extended the stochastic methodology by modelling the temporal correlation,

length and timescales. However the technique requires the measured radiation fields to be available so cannot be considered a predictive model. Chan and Pan [27], investigated the importance of including pre-spatial and temporal correlations in a radiation analysis of a CO/H2 jet flame, using a first order causal stochastic model. They applied the causal model to the spectral intensity at specific wavelengths for horizontal rays passing through the axis of the jet for a number of downstream locations. Instantaneous values of the mixture fraction are evaluated using a clipped Gaussian PDF, similar to Gore et al. [3]. Chan and Pan compared their model predictions with Kounalakis et al. [28] original stochastic methodology where the spatial and temporal correlations are assumed to be negligible and found that the average relative difference between model prediction and measured spectral intensity is worse using the more sophisticated causal stochastic model. Chan et al. [29], extended the original causal stochastic model to include spatial–temporal cross correlation and applied it to the same jet flames as Chan and Pan [27]. Comparing the predicted spectral intensity with the measured value, Chan et al. [29] found that including the cross correlation term improved the predictive capability of the causal model but the results are still marginally inferior to Faeth’s groups original simpler stochastic model. Chan and Pan [30] proposed a final extension to their original approach to produce a general semicausal stochastic model that include both temporal and spatial correlations. The improvement being that the random field includes pre and post spatial correlation. This leads to a linear system to be solved for each realisation. This is a potentially large computational overhead but it can be minimised as the coefficients in the linear system are not time dependent so the inverse matrix or LU decomposition of the matrix for each ray can be calculated once only. This causal model does exhibit better agreement with the measured spectral intensity than Faeth’s group’s original simpler stochastic model for the jet flames considered. This is in contradiction to Faeth et al. [20] as they included both pre and post spatial correlations and found little improvement in accuracy. This is an interesting area of research and looks promising but further validation is required and the requirement for solving a linear system for every realisation makes it unattractive for calculating radiation heat flux distributions or used in coupled simulations. It is also restricted to a clipped Gaussian PDF which ultimately might not be the most appropriate PDF to use. 2.3. Summary

Rt ðDtÞ ¼

f 0 ðtÞf 0 ðt þ DtÞ ðf 0 Þ2

ð1Þ

Including the temporal correlation is useful if knowledge of the temporal properties of the radiation fluctuations is of interest such as in fire detection based on flame flicker, but for evaluating the mean radiation fields is not generally required [4]. Faeth et al. [20] extended the original stochastic methodology to include spatial correlation of the instantaneous mixture fraction field,

Rs ðDsÞ ¼

f 0 ðsÞf 0 ðs þ DsÞ ðf 0 Þ2

ð2Þ

The spatial and temporal correlations are modelled as exponential functions of the local dissipation length scale and a temporal scale estimated using Taylor’s hypothesis [20]. Faeth et al. [20], reported no significant change in the predicted mean spectral intensity distribution with or without including the spatial and temporal correlation, Eqs. (1) and (2). More recently Zheng et al. [25,26] reported excellent results using a stochastic methodology including temporal and spatial correlation of the instantaneous mixture fraction field using a technique based on tomography to estimate the integral

In summary the stochastic methodology is accepted as the most accurate approach to modelling TRI [31], but to date the computational cost has limited its application to calculating spectral intensity distributions [19], and model solutions for comparison with other less accurate approaches. There has been much research into the importance of including temporal and spatial correlations in the stochastic methodology [3,4,20,27,29,30]. The evidence is contradictory but Chan and Pan [30] found improvement in accuracy for a limited validation study if the pre and post spatial correlation were included in the stochastic methodology. The one serious difficulty with this approach is the increase in computational cost compared to less sophisticated methodologies. Further all stochastic simulations completed thus far have used a clipped Gaussian PDF even though many flame structure models based on a presumed PDF use a b PDF [32–34]. This inconsistency is primarily due to difficulties implementing PDFs other than the clipped Gaussian for simulating TRI [20]. The final conclusion that can be drawn from the review of previous investigations [35–37], is the extreme anisotropy of the intensity field external to the jet flame has made it impractical to include TRI in the radiation model and calculate heat flux distributions.

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In this paper the sensitivity of predicted radiation fields to the PDF implemented is investigated. As well as using measured spectral intensity distributions for validation purposes the measured external radiation heat flux distributions of Gore et al. [3], are also predicted with negligible numerical error using modest computational resources. In the next section the experimental jet flames used for model validation are discussed. Following this the mathematical model for the flame structure and the radiation fields is presented. The main body of the paper is then completed by considering the validation of the flame structure model and the sensitivity of the radiation model predictions to the PDFs implemented are investigated. The main conclusions of the investigation are summarised in the final section of the paper. 3. Experimental jet flames In this investigation we are primarily interested in predicting the radiation fields surrounding jet flames where TRI effects are known to be significant. Therefore based on Faeth and coworkers research [3,4], we will restrict attention to jet flames where the fuel is hydrogen. The mathematical model presented below has been validated using a number of different jet flames, where experimental data characterising the fire structure and the radiation field exists. We will restrict a detailed discussion to just two jet flames, Barlow and Carter [2] and Gore et al. [3] as they are representative of all of the jet flames considered in the investigation. 3.1. Hydrogen jet flame experiments – flame structure Barlow and Carter [2] present measurements characterising three vertical free hydrogen jet flames; an undiluted hydrogen jet flame and two other jet flames with similar source Reynolds numbers but diluted with helium to different degrees. These are particularly attractive experimental flames to simulate as the measured fields are well known to be accurate and have been used for model development and validation by a number of groups [6–8]. Note all three jet flames have been used in the validation study but only the comparisons of predicted vs. measured flame structure will be presented for the undiluted hydrogen jet flame. The burner consists of a straight tube with an inner diameter of 3.75 mm and an outer diameter of 4.8 mm. The average mean source velocity is 296 m/s. There is a coflow of air with a velocity of 1 m/s. Under these source conditions the jet flame is attached to the burner lip. Barlow and Carter [2], use spontaneous Raman scattering to measure the major species and laser induced fluorescence to measure the instantaneous NO and OH fields. The measured temperature is inferred from the major species concentrations. Mean and RMS properties at points in the flame are calculated from 500–800 samples. Further details of the experimental techniques used can be found in [2] and the references cited therein. Measurements available for validation purposes are the mean and RMS temperature and mean and RMS mass fractions and mole fractions of the following species, O2, N2, H2, H2O, NO and OH as both Favre averages and Reynolds averages. We will restrict the validation study to the mean and RMS temperature fields and the mean and RMS H2O mass fraction fields. However the predicted mass fractions of the other major species have also been compared with the equivalent measured quantity and have a similar level of agreement as demonstrated below for the H2O mass fraction field. 3.2. Hydrogen jet flame experiments – radiation fields Gore et al. [3] presented two hydrogen jet flame experiments, with different source Reynolds numbers, Re = 3000 and 5722. For

253

both experiments the source nozzle internal diameter is 5 mm. For the Reynolds numbers Re = 3000 and Re = 5722 the mean source axial velocity components are 66.3 and 108.4 m/s respectively. In most experimental jet flame rigs the source of the flame is designed to give fully developed pipe flow at the nozzle exit i.e., a turbulent velocity of the order of 5% of the mean flow velocity. Gore et al’s. [3] experimental rig is unusual as the source turbulent velocity is significantly above 5% for both jet flames,

pffiffiffiffiffi k0 ¼ 0:184 U0 for the low speed jet, Re = 3000 and

pffiffiffiffiffi k0 U0

¼ 0:23

for the high speed jet, Re = 5722. Gore et al. [3] characterised the flame structure by measuring the chemical species and the temperature along the axis of the jet. The spectral intensity distribution is measured for a number of horizontal paths downstream of the source through the axis of the jet using a narrow angle monochromator with a pyroelectric detector. In addition to the spectral intensity distribution, the radiation heat flux distribution is measured using a medtherm Type 64P-10-22 detector with a field of view of 150° cone angle. The radiation heat flux distribution is measured along a radial line through the jet axis in the plane of the nozzle, x = 0 with the detectors having a vertical orientation and along a vertical line at a distance of r = 0.575 m from the jet axis with the detectors having a horizontal orientation directed towards the jet axis. The heat flux meters are fitted with a sapphire window that restricts the spectral range of the device to 1–6.3 lm. 4. Mathematical model For vertical free jet flames the dominant flow direction is aligned with the buoyancy force such that the mean flame structure is axisymmetric and can be calculated using a system of equations based on the boundary layer equations [38]. The boundary layer equations are parabolic in nature making it possible to solve them using a time marching approach in the axial direction, the dominant flow direction. The solution procedure uses a coordinate transformation together with a normalised stream function. In this way the computational mesh is restricted to the width of the boundary layer, which in this case is the width of the jet flame. Predicted flow fields are calculated on a finite volume mesh that expands in the radial and axial directions. This has the advantages of a CFD simulation technique i.e., the detailed structure of the flame can be calculated accurately [38] with negligible numerical error using a modest computational resource. 4.1. Flame structure The basis of the flow equations is the parabolized Favre averaged Navier Stokes equations in an axisymmetric coordinate system [38]. The system is closed using a variant of the k-e turbulence model [38,39]. The version of the k-e turbulence model implemented is presented in [40] to take account of the round jet/ plane jet anomaly [40,41].

 e ~ leff @k  UkÞ  VkÞ @ðq 1 @ðr q 1 @  ðPk  eÞ þ ¼ r þq @z @r r r @r rk @r e eÞ 1 @ðr q ~ eÞ 1 @  leff @ e U V @ðq e  ðC 1 Pk  C 2 eÞ þq þ ¼ r @z r @r r @r re @r k

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4.2. Modelling the RMS fields

e @U @r

Pk ¼ leff

!2

leff ¼ ll þ

e k dU C 1 ¼ 1:4  3:4 e dz

C l ¼ 0:09;

 k2 Clq

e

!3 cl

C 2 ¼ 1:84;

rk ¼ 1; and re ¼ 1:3

The jet fire model accounts for turbulent combustion using a laminar flamelet combustion model [42]. Combustion is assumed to be infinitely fast with a prescribed probability density function. In previous simulations a b function is used [38,42–44], in this article three PDFs have been considered. This will be discussed further below in the section on modelling RMS fields. The shape of the prescribed PDF at any spatial location is determined by a conserved scalar, the mean mixture fraction and its variance, which are calculated using modelled transport equations [42]. The mean adiabatic temperature and density together with the major chemical species can be evaluated using a flamelet library. The flamelet library is calculated by assuming chemical equilibrium for a range of values of the mixture fraction. For example the mean density and adiabatic temperature are calculated by integrating the appropriate flamelet together with the PDF over instantaneous mixture fraction space.

!1

1

q ¼ R1

qðf ÞPðf Þdf

0

T adia ¼ q

Z

1

0

Tðf Þ

qðf Þ

Pðf Þdf

where the over-bar indicates a Reynolds average. For Favre averaged properties, for example the Favre averaged temperature the expression reads,

Te adia ¼

Z

The model as described above has been applied to a number of different free jet flames with only slight modification [14,34,42– 44,46]. Onokpe and Cumber [46] investigated the calculation of the RMS fields. Two options were considered either use the prescribed PDF in combination with the flamelet or implement a transport equation for the variance of the flow field of interest. Onokpe and Cumber [46], found using a PDF to calculate the RMS field to be accurate for all of the jet flames they considered. Integrating the temperature flamelet together with the PDF over instantaneous mixture fraction space gives the adiabatic mean temperature, similarly using the PDF to calculate the 2nd moment without correcting for radiation heat loss can only give the adiabatic RMS temperature,

Te RMS;adia ¼

0

The RMS temperature field is calculated by assuming the ratio of mean to RMS temperature is conserved,

! Te RMS;adia e T Te adia

Te RMS ¼

For the RMS H2O mass fraction,

e RMS;H O ¼ Y 2

Tðf ÞPðf Þdf

0

with  denoting a Favre average. To account for radiation heat loss a transport equation for a specific enthalpy perturbation is solved, where radiation heat loss is introduced using the optically thin approximation [45].

e DhÞ 1 @ðrq e DhÞ 1 @  leff @ Dh U V @ðq þ 4K a rðT 4  T 4amb Þ þ ¼ r @z r @r r @r rDh @r The mean temperature is then calculated from the adiabatic flame temperature and enthalpy perturbation.

T ¼ T adia  P

Dh i Y i C p;i

The absorption coefficient, Ka is adjusted to give good agreement for the measured mean temperature in the near field of a turbulent jet flame. This approach is more sophisticated than accounting for the radiative heat loss by adjusting the temperature flamelet. Strictly speaking a grey absorption coefficient is not valid as emission from gases is spectrally banded, however this approach has been used successfully in other computational studies [34,42–44]. A further consideration is this aspect of the model should be viewed as a convenient way of accounting for radiation heat loss and allowing the investigation to focus on the calculation of the radiation fields external to the flame.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 2 e H O  Y H O ðf Þ Pðf Þdf Y 2 2 0

In Onokpe and Cumber [46], three PDFs are considered, a b PDF, a clipped Gaussian PDF and a truncated Gaussian PDF. All three PDFs give similar predictions of the mean and RMS fields but the predictions based on the b PDF and the truncated Gaussian PDF are marginally closer to the measured flow fields. Therefore in this study we will only consider the b PDF and the truncated Gaussian PDF. The functional form of the b PDF is given below,

Pðf Þ ¼ R 1 0

1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 2 Te adia  Tðf Þ Pðf Þdf

f r1 ð1  f Þs1 f r1 ð1  f Þs1 df

where

 0  1 ~f 1  ~f r ¼ ~f @  1A ~f 002 and

! 1  ~f s¼r ~f The shape of the truncated Gaussian is,

Gðf Þ ¼ exp 

Pðf Þ ¼ R 1 0

ðf  lÞ2 2r 2

!

Gðf Þ Gðf Þdf

2

r and l are adjusted to give the desired 1st and 2nd moments ~f ¼

Z

1

f Pðf Þdf

0

~f 002 ¼

Z

1

 2 ~f  f Pðf Þdf

ð3Þ

0

similar to Lockwood and Naguib [47]. This is a simpler relationship than the Clipped Gaussian PDF and has a similar shape for values of

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the mixture fraction remote from the end points, f = 0 and f = 1, and for small values of the mixture fraction variance. As part of a turbulent combustion model the parameters r2 and l in the PDF can be viewed as functions of the mean mixture fraction and mixture fraction variance and the system of equations, Eq. (3) implicitly define the functional relationships.



l ~f ; ~f 002



and



r2 ~f ; ~f 002



The parameters r2 and l can be evaluated as a ‘once and for all’ calculation by solving Eq. (3) numerically using Newton’s method [48], for a range of values of mean mixture fraction and mixture fraction variance. This information can then be encoded in a turbulent combustion model as a look up table as discussed by Lockwood and Naguib [47]. In Onokpe and Cumber [46] this was done for the jet flame simulations considered for the clipped Gaussian PDF and the truncated Gaussian PDF but there was no improvement in predictive capability for the jet fires [2,49,50]. Therefore for improved efficiency and ease of implementation the approximations,

l ¼ ~f r2 ¼ ~f 002 are used. As a consequence of this the truncated Gaussian PDF satisfies the 0th moment exactly and the 1st and 2nd moment approximately, with good agreement for the range of the mean mixture fraction and mixture fraction variance discussed above. This approximation for the truncated Gaussian PDF is similar to the approximation to the clipped Gaussian PDF implemented in all of Faeth and co-workers investigations into TRI in jet flames [3,4,20,22–24], in that the 0th moment is satisfied exactly but the 1st and 2nd moments are only satisfied approximately.

Fig. 1. Predicted mean and RMS temperature for x/d = 22.5, for Barlow and Carter [2] hydrogen jet fire using a number of different finite volume grids.

predicted flow fields significantly. Therefore for all simulations in this article NR = 160 control volumes in the radial co-ordinate direction spanning the jet radius, are used to calculate the flame structure.

4.3. Boundary conditions and finite volume grid 4.4. Radiation model In the jet flame simulations presented below the bulk inlet conditions are given by the nozzle diameter and the average source velocity. The mean stream-wise velocity distribution and radial velocity distribution are taken to be consistent with fully developed pipe flow. Source turbulence profiles are calculated by assuming the turbulence is isotropic and in equilibrium. A uniform turbulence velocity consistent with the measured source turbulence velocity where available is used or a value of 5% of the source velocity is specified. A grid sensitivity study for all jet flames simulated was completed. Fig. 1 shows the predicted radial mean temperature and RMS temperature distribution at x/d = 22.5 for the undiluted hydrogen jet flame [2], calculated using a b PDF. Three predictions are shown with 40, 80 and 160 control volumes in the radial coordinate direction. For each simulation following the first with every doubling of the number of control volumes in the radial direction the fractional step in the axial direction is halved, doubling the number of control volumes in the axial direction. From Fig. 1 it is clear that the mean temperature distribution using NR = 80 and NR = 160 control volumes in the radial coordinate direction are similar. In addition simulations with NR = 320 control volumes in the radial coordinate direction confirmed that the predicted flow fields would not change significantly if a finer computational grid is used. In addition previous studies [46], have shown this level of resolution is sufficient to calculate the flame structure such that further refinement of the finite volume grid does not change the

To calculate the spectral emission from the flame at the end of a ray the spectral intensity is modelled using a statistical narrow band model, RADCAL [11,51]. This is a radiation model developed by Grosshandler and has been well validated in the open literature for fire applications [11,51,52]. In RADCAL the equation of radiative transfer ignoring scattering is solved in its integral form [11].

Ik ðLÞ ¼ Ik;w ejk ðLÞ þ

jk ðLÞ ¼

Z

Z 0

L



K k ðl Þdl

jk ðLÞ







Ib;k ðl Þexpððjk ðLÞ  jk ðl ÞÞÞdjk ðl Þ ð4Þ



0

Ik,w is the intensity entering the flame, taken to be a black body at ambient temperature and L is the length of the ray from the flame boundary through the body of the flame to the receiver. The major complexity of evaluating the spectral intensity using Eq. (4) is the evaluation of the spectral absorption coefficient, Kk which varies in a complex way with species concentration and temperature, further details of RADCAL can be found in [21,51]. To evaluate the total intensity Eq. (4) is integrated with approximately 200–250 narrow bands spanning the spectral range of the narrow angle monochromator detector and the heat flux meters. In the simulations presented below scattering has been ignored. This is a reasonable approximation given the clean combustion characteristics of hydrogen.

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CL …

Ray intersecting a spectrometer

Portion of finite volume mesh

tions including TRI presented below use 32 samples to calculate the mean spectral intensity distribution. For a ray segment a realisation of the instantaneous mixture fraction to calculate Reynolds averaged properties is calculated using a pseudo random number generator satisfying the cumulative PDF.

 1 P ðf Þ ¼ P ~f ; ~f 002 ; f R finst

z ¼ R0 1

…

0

Ray intersecting a heat flux meter

x r CL

Fig. 2. A schematic diagram showing the flame structure finite volume mesh, a ray traced through the flame and the points the flame variables are evaluated for input to the narrow band radiation model.

RADCAL requires the temperature and participating species along the ray. The nonhomogeneous path is represented as a sequence of ray segments that are taken to be homogeneous, see Fig. 2. The finite volume mesh used in the flame structure model is not appropriate as it is too fine a representation of the nonhomogeneous path for calculating the spectral intensity distribution. A number of numerical experiments were completed where the number of segments was systematically increased and the relative difference in spectral intensity noted. In all of the simulations presented below 10 ray segments internal to the flame per ray are used to calculate the spectral intensity. It is estimated that the relative difference between the spectral intensity calculated using 10 ray segments and the fully converged value is approximately 2%. The same approach was adopted for calculating the radiative heat flux at a receiver, see Fig. 2. Similar to Faeth and most other researchers interested in calculating the external radiation field for jet flames the discrete transfer method developed by Lockwood and Shah [53] is used to calculate the radiative heat flux external to the jet flame. To improve ray convergence a staggered ray mesh is used [35]. When the spectral intensity and radiative heat flux is calculated without including TRI the mean temperature and participating species concentrations are used as an input to the narrow band model, RADCAL. To incorporate TRI effects into the simulation a stochastic methodology is used where a number of samples of the instantaneous spectral intensity at the receiver are calculated and the average taken.

q



P ðf Þdf

ð5Þ

P ðf Þdf

This is an implicit equation that is difficult to solve in a fast and efficient manner and would make the methodology too slow to calculate the spectral intensity or worse still the radiative heat flux distribution on conventional computer platforms. If we consider the radiative heat flux distribution, Eq. (5) would require solution for the number of receivers multiplied by the number of rays, multiplied by the number of ray segments, multiplied by the number of samples. A conservative estimate requires the solution of Eq. (5) of the order of 10 million times per simulation. To avoid this Eq. (5) is approximately inverted for each ray segment leading to an expression for the cumulative probability density function expressed in the form,

finst ¼ Q ðzÞ where Q takes the form,

Q ðzÞ ¼ pN ðzÞ zL 6 z 6 zU pN ðzÞ ¼

N X ai zi i¼0

For 0 6 z 6 zL and zU 6 z 6 1 a linear function is imposed such that

Qð0Þ ¼ 0 and

Qð1Þ ¼ 1 together with the conditions that Q is continuous at the points zL and zU. The values of zL and zU are determined by the specification of pN(z). pN(z) is a polynomial that is determined by a least squares best fit to points,

ðz; finst Þi

i ¼ 1; 2;    ; Nfit

ð6Þ

The pairs (z, finst)i are generated as solutions to Eq. (5) on a uniform mesh spacing in the z coordinate. Away from the boundaries of mixture fraction a polynomial fits the data, Eq. (6), well. We only include pairs, Eq. (6) in the curve fitting operation where,

0:005 6 z 6 0:995 From this it follows that

zL ¼ zi ;

i ¼ minj f0:005 6 zj g

and

zU ¼ zi ;

i ¼ maxj fzj 6 0:995g

The RMS error for the approximation is given as,

ERMS

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Nfit uX 2 ¼t Qðzi Þ  finst;i i¼1

Nsample

Ik ðLÞ ¼

X 1 ðIk ðLÞÞi Nsample i¼1

After systematically varying the number of samples used and noting how the mean spectral intensity distribution changed all simula-

It was found that the RMS error did not decrease significantly if the order of the polynomial was increased beyond third order. Hence a cubic polynomial is used in the approximation of the cumulative probability distribution in the simulations presented below. See Fig. 3 for a comparison of the cumulative probability density

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Fig. 3. Cumulative PDF distribution calculated with a truncated Gaussian PDF, (a)     ~f ; ~f 002 ¼ ð0:5; 0:025Þ and (b) ~f ; ~f 002 ¼ ð0:03; 0:0015Þ.

function using the truncated Gaussian PDF and the cubic approximation. Two comparisons are made,

ð~f ; ~f 002 Þ ¼ ð0:5; 0:025Þ and

  ~f ; ~f 002 ¼ ð0:03; 0:0015Þ: the first of relevance in the fuel rich region of the jet flame, near the source, and the second with a much lower value of mixture fraction more representative of the PDF close to the edge of the jet. In both cases the agreement between the analytic solution and the cubic approximation is excellent. Having described all aspects of the mathematical model we will now discuss the predictive capability of the model. 5. Flame structure validation Before the models capability with regard to predicting the radiation fields can be considered it must be confirmed that the flame structure is reproduced adequately. The first jet flame considered is Barlow and Carter’s hydrogen jet flame [2]. This is a jet flame used previously in a number of other model validation studies [6–8] and as such is interesting to simulate for model comparison purposes. Fig. 4 shows the predicted and measured Reynolds averaged mean temperature and RMS temperature as a function of radial co-ordinate at four axial locations, x/d = 22.5, 45, 90 and 180. Two predictions are shown, calculated using a b PDF and a truncated Gaussian PDF. Considering the mean temperature field, there is a clear difference between the two predicted curves, calculated with a b PDF and a truncated Gaussian PDF. The largest difference between the two predictions occurs in the shear layer of the jet. However both predictions are in good agreement with the measured mean temperature distribution, with the b PDF prediction being in closer agreement with the measured data, but the

Fig. 4. The mean and RMS radial temperature distributions for Barlow and Carter’s [2] hydrogen jet flame, (a) x/d = 22.5, (b) x/d = 45, (c) x/d = 90 and (d) x/d = 180.

improvement is marginal. Comparing the predicted mean temperature distribution simulated here with the equivalent prediction using a PDF Transport model and the PEUL model [8] and the

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models. The relevant figures in [6,8] are Figs. 4 and 9 respectively as these figures show a comparison of the respective models with the measured mean temperature distribution for Barlow and Carter jet flame [2]. However it should be realised that these more sophisticated models are demonstrably more accurate at modelling intermediate species such as NO and OH than the laminar flamelet combustion model with an equilibrium flamelet. The predicted Favre RMS temperature field calculated with a b PDF and the truncated Gaussian PDF is also shown in Fig. 4. The two predictions are similar in the near field. Further downstream the predicted RMS temperature calculated with the truncated Gaussian PDF is marginally closer to the measured distribution than the prediction calculated using a b PDF. Fig. 5 shows the predicted and measured Favre averaged mean and RMS H2O mass fraction at x/d = 22.5, 45, 90 and 180. The predictions calculated with a b PDF and a truncated Gaussian PDF are similar and in close agreement with the measured distributions for x/d = 22.5 and 45. Further downstream the predicted mean H2O mass fraction over predicts the measured distribution, but the agreement is still acceptable. The maximum relative difference between the measured and predicted H2O mass fraction distribution occurs on the jet axis and is of the order of 25%, but in the shear layer the agreement is much better. An inspection of Fig. 5 suggests the prediction calculated with the b PDF is marginally closer to the measured distributions than the truncated Gaussian PDF, but both are acceptable. Comparing the predicted mean H2O mass fraction distributions in Fig. 5 with the equivalent in [8], the present model is as accurate as or slightly more accurate than the model predictions calculated using a PDF transport model and PEUL model, see Fig. 8 in [8]. 6. Radiation field predictions In this section the models capabilities with regard to the radiation fields is considered. Firstly the ability of the model to predict the spectral intensity distribution for a given receiver orientation and location is presented. The second part of the validation study the model is applied to the prediction of the external heat flux distribution. For each of the spectral intensity distributions and heat flux distributions calculated the sensitivity of the predicted fields to the PDF implemented will be considered. The model is validated using the hydrogen jet flames presented in [3]. Gore et al. [3] took measurements of mean flow properties and radiation properties in two hydrogen jet flames, with different source velocities, characterised by the source Reynolds number, Re = 3000 and Re = 5722. In this section we will present comparisons of predicted and measured radiation properties for both of Gore et al’s. [3] jet flames. 6.1. Spectral intensity distributions

Fig. 5. The mean and RMS radial H2O mass fraction distributions for Barlow and Carter’s [2] hydrogen jet flame, (a) x/d = 22.5, (b) x/d = 45, (c) x/d = 90 and (d) x/ d = 180.

CMC model of Fairweather and Woolley [6] the present model gives a more accurate prediction than these more sophisticated

For the high speed jet flame, Gore et al. [3], measured the spectral intensity for receivers with a horizontal orientation with a line of sight intersecting the axis of the jet for three downstream locations, x/d = 50, 90 and 130. Fig. 6 shows a comparison of the measured and predicted spectral intensity calculated using a b PDF. For each downstream location two predictions are shown, one based on the mean flow fields and a second where the stochastic methodology is used to simulate TRI. As expected the TRI predictions of spectral intensity overpredict the spectral intensity calculated using the mean flow fields although the difference between the two predictions is relatively small. One conclusion of Gore et al’s. [3] investigation is TRI for the two jet flames, Re = 3000 and 5722 is significant with their stochastic predictions being as much as twice the mean property prediction. It should be noted however

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Fig. 6. Comparison of the measured and predicted spectral intensity using a b PDF for horizontal chords through the axis of Gore et al. [3], hydrogen jet flame, Re = 5722, (a) x/d = 50, (b) x/d = 90 and (c) x/d = 130.

that the similarity between the predicted spectral intensity calculated when TRI effects are included or not in the present investigation is consistent with the work of Zheng et al. [25]. Zheng et al. [25] found that for a 50% N2 diluted hydrogen jet flame TRI effects are relatively small for horizontal rays passing through the jet axis compared to horizontal rays that do not intersect the jet axis. The increase in TRI effects for off axis rays is believed to be due to TRI being promoted primarily by fluctuations in the instantaneous fields in the vicinity of the stoichiometric conditions, [3,4] and a ray passing through a jet flame that does not intersect the jet axis

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will have larger portions of the ray in the shear layer where the mean concentration fields are at or close to the stoichiometric condition. Comparing the TRI prediction of spectral intensity with the measured spectral intensity distribution in Fig. 6, overall the agreement is acceptable given the uncertainties in the experimental measurements [3], the accuracy of the flame structure model and the narrow band radiation model, RADCAL. For x/d = 50 the agreement is very good, with the agreement deteriorating further downstream with a tendency to underpredict the measured spectral intensity distribution. Considering the predicted spectral intensity distribution calculated with the stochastic methodology, thus including TRI, these predictions can be compared with the equivalent model predictions in Gore et al. [3], see Fig. 4 in [3], overall the predicted spectral intensity distributions using the present model are in closer agreement with the measured spectral intensity distributions than Gore et al. [3], model but the improvement is marginal at best for x/ d = 90 and 130. The basis of the two models is similar. There are two main differences; firstly the way radiation heat loss in the flame is accounted for is different and the second difference is the PDF implemented in the stochastic methodology. In the present model radiation heat loss is accounted for by solving an enthalpy perturbation transport equation with a sink term for radiation heat loss. Gore et al. [3] introduce radiation heat loss into their model by adjusting the temperature flamelet such that the model globally reproduces the measured fraction of heat radiated. The use of an enthalpy perturbation transport equation is more sophisticated than adjusting the temperature flamelet. In Fig. 6 the predicted spectral intensity distributions are calculated using a b PDF, whereas Gore et al. [3], implement a clipped Gaussian PDF to simulate the instantaneous mixture fraction field. This is not just a question of a different shape for the PDF, the b PDF satisfies the 0th, 1st and 2nd moments exactly, whereas the way Gore et al. [3], have implemented the clipped Gaussian PDF it only satisfies the 0th moment exactly. Gore et al. [3], implementation of the clipped Gaussian PDF satisfies the 1st and 2nd moments approximately for values of the mean mixture fraction remote from the boundaries of the mixture fraction space, f = 0 and f = 1. Fig. 7 shows a comparison of the predicted spectral intensity distribution calculated using a truncated Gaussian PDF with the measured spectral intensity distribution for the same ray orientations and receiver locations as considered in Fig. 6, that is x/ d = 50, 90 and 130 with all three rays having a horizontal orientation and the rays intersect the jet axis. As for the simulations calculated using a b PDF, for the truncated Gaussian PDF two predicted spectral intensity distributions are shown for each downstream location, with and without TRI accounted for in the model. Comparing Figs. 6 and 7 the predictions using the two different PDFs are very similar. It is not observable from a visual comparison but in general the b PDF predictions are in slightly better agreement with the measured spectral intensity distribution than the truncated Gaussian PDF. Typically the b PDF yields a predicted spectral intensity distribution that is 2% higher than the predictions based on a truncated Gaussian PDF. Overall it can be concluded that for horizontal rays that intersect the jet axis the spectral intensity distribution is insensitive to the PDF implemented. A further conclusion is satisfying the 0th, 1st and 2nd moments is not critical to the models capability to predict the spectral intensity distribution for horizontal rays that intersect the jet axis. Therefore the differences in predicted spectral intensity distribution for the present model and Gore et al’s model are likely to be predominantly due to the way radiation heat loss is modelled. With these conclusions in mind we take the investigation further by considering the external radiation heat flux distributions surrounding the hydrogen jet flames.

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Fig. 8. Ray refinement study of the heat flux distributions for Gore et al. [3] hydrogen jet flame, Re = 5722, incorporating TRI calculated using the b PDF, (a) x = 0, horizontal distribution and (b) r = 0.575 m vertical distribution.

(a)

(b) Fig. 7. Comparison of the measured and predicted spectral intensity using a truncated Gaussian PDF for horizontal chords through the axis of Gore et al. [3], hydrogen jet flame, Re = 5722, (a) x/d = 50, (b) x/d = 90 and (c) x/d = 130.

6.2. Heat flux distributions Before any evaluation of the respective models and PDFs can be presented a ray sensitivity study must be completed to ensure all predictions are free from significant numerical error. Fig. 8 shows three predictions of the two heat flux distributions calculated using the discrete transfer method, each using a different number of rays per receiver. The predicted heat flux distributions are calculated using the stochastic methodology implemented with a b PDF

Fig. 9. The horizontal heat flux distribution of Gore et al. [3] hydrogen jet flame, Re = 3000, at x = 0, (a) b PDF and (b) truncated Gaussian PDF.

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to simulate TRI. To produce Fig. 8 the heat flux at a receiver is initially calculated using a ray mesh of 3  1 rays, the incident unit hemisphere being discretised in local to the receiver 2D spherical co-ordinates, the rotational direction and azimuthal direction. Following this a finer mesh is generated by doubling the number of rays in each spherical co-ordinate direction. This is done until the change in the predicted incident heat flux distribution is sufficiently small to conclude the predicted heat flux distribution is insensitive to further ray refinement. It is possible to implement a more efficient method for calculating the heat flux than the discrete transfer method, such as an adaptive discrete transfer method [36,54], or the reverse Monte–Carlo method implemented with Sobol sequences [55], or even an adaptive Monte–Carlo method such as hot sampling [37,56]. Ultimately this was not done as the original discrete transfer method [53,57] is commonly used to calculate heat flux distributions without modification and it is possible to model TRI and calculate heat flux distributions using the discrete transfer method using modest computational resources. Fig. 8a shows the predicted incident heat flux for a radial line of vertically orientated receivers in the plane of the jet flame source, x = 0 m, calculated using NRay = 192, 768 and 3072 rays. All three predictions are similar the relative difference between the 768 ray and 3072 ray prediction is less than 2%. In Fig. 8b the receivers are organised into a vertical line at r = 0.575 m with each receiver having a horizontal orientation directed towards the axis of the jet. Again three curves are shown, the predicted heat flux distribution using NRay = 192, 768 and 3072 rays per receiver. Comparing Fig. 8a and b we see that the vertical heat flux distribution demonstrates more sensitivity to the number of rays used in the integration of the total intensity distribution than the horizontal heat flux distribution. For the vertical heat flux distribution ray independence using 3072 rays was confirmed by calculating the vertical heat flux distribution using 12,288 rays. An interesting point is the ray independence study was also completed for the predicted heat flux distribution calculated using a truncated Gaussian PDF and it was found that a finer ray mesh is required to achieve ray independence in the heat flux distributions compared to when a b PDF is implemented as part of the stochastic methodology. In conclusion all predicted heat flux distributions calculated using the b PDF in subsequent figures use 3072 rays per receiver, whereas 12,288 rays per receiver are used for predicted heat flux distributions calculated using the truncated Gaussian PDF. Figs. 9 and 10 show predicted and measured heat flux distributions for the Re = 3000 jet flame. Fig. 9 shows a comparison of the predicted and measured horizontal heat flux distribution in the plane x = 0, using a b PDF and a truncated Gaussian PDF and with and without including TRI, four predictions in total. The first point of note is the predicted heat flux distributions calculated using the stochastic methodology to model TRI are in good agreement with the measured distributions. The predictions based on the mean flow fields, so no TRI is included as expected underpredict the measured heat flux distribution. For the remainder of the analysis the predicted heat flux distributions calculated without including TRI will not be discussed, the curves are included in the figures for reference only. There are minor differences between the two predictions modelling TRI using a b PDF and a truncated Gaussian but the level of agreement with the measured heat flux is similar. Although the two predictions including TRI are very similar it should be noted that the b PDF model requires 25% of the number of rays to calculate the heat flux compared to the TRI model based on the truncated Gaussian PDF. Fig. 10 shows a similar comparison between the measured and predicted vertical heat flux distribution at r = 0.575 m. The trends in accuracy are similar to the comparison of model predictions and measured heat fluxes shown in Fig. 9. A careful inspection of Fig. 10 shows that unlike Fig. 9 the two predictions including TRI

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Fig. 10. The vertical heat flux distribution for Gore et al. [3] hydrogen jet flame, Re = 3000, at r = 0.575 m, (a) b PDF and (b) truncated Gaussian PDF.

in the model are significantly different, with the relative difference increasing with increasing axial coordinate. For example the predicted heat flux calculated with a b PDF over predicts the measured data downstream of x = 1 m and the equivalent calculated with a truncated Gaussian PDF under predicts the measured data. The reason for the sensitivity to the PDF increasing with receiver height is the flame profile in the field of view of the receiver increases with height. Therefore for receivers located near the source the majority of the incident intensity distribution is made up of ambient background intensity and the generic shape of the PDF has little or no impact on the predicted heat flux. The predicted heat flux distribution including TRI using a b PDF is in closer agreement with the measured heat flux distribution than the prediction calculated using a truncated Gaussian PDF but the improvement is marginal. Figs. 11 and 12 show predicted and measured heat flux distributions for the Re = 5722 jet flame. Considering Fig. 11 similar to the low Reynolds number jet flame, Re = 3000 the predicted heat flux distributions calculated using the stochastic methodology to model TRI are in good agreement with the measured distributions. For r = 59  103 m the relative error between the measured and predicted heat flux is 6.7% when TRI is modelled with a b PDF, whereas the predicted heat flux calculated using the mean flow fields under predict the measured heat flux by 25%. The other point that is clear is the model including TRI using a b PDF, see Fig. 11a, is in closer agreement with the measured heat flux distribution compared to the model including TRI using a truncated Gaussian PDF, see Fig. 11b. The relative error between the measured and predicted heat flux including TRI and the truncated Gaussian PDF at r = 59  103 m is 14%. Fig. 12 shows a similar comparison between the measured and predicted vertical heat flux distribution at r = 0.575 m. The trends in accuracy are similar to the comparison of model predictions and measured heat fluxes shown in Fig. 11. The prediction

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Fig. 11. The horizontal heat flux distribution for Gore et al. [3] hydrogen jet flame, Re = 5722, at x = 0, (a) b PDF and (b) truncated Gaussian PDF.

Fig. 12. The vertical heat flux distribution for Gore et al. [3] hydrogen jet flame, Re = 5722, at r = 0.575 m, (a) b PDF and (b) truncated Gaussian PDF.

including TRI using a b PDF is in very close agreement with the measured distribution. The relative error for the predicted heat flux at x = 0.484 m is 2.6% compared to 40% for the relative error for the predicted heat flux based on the mean flow fields. Similar to Fig. 11 the model prediction including TRI using the truncated Gaussian PDF has a larger relative error of 10% for x = 0.484 m than the model prediction including TRI with a b PDF. Considering the two jet flames, Re = 3000 and Re = 5722 together, superficially it initially looks like there is more sensitivity to the PDF used to calculate the vertical heat flux distributions in the high Reynolds number jet flame. A moment’s reflection though reveals that this is possibly what would be expected if there was sensitivity to the PDF in the predicted distributions as the heat flux meters are in the same locations for both jet flames. Therefore the profile of the jet flame in the field of view for any given heat flux meter for the low speed jet flame is smaller than the high speed jet flame meaning the contribution to the heat flux from the background ambient radiation is higher implying the way TRI is modelled has less of an impact on the predicted heat flux. Gore et al. [3], calculated the heat flux distribution using the mean flow fields only i.e., TRI is neglected. Considering the predicted heat flux distributions based on the mean flow fields, as expected both models tend to underpredict the measured heat flux distributions. The level of agreement between the model and measured heat fluxes when TRI is neglected in this investigation is similar to Gore et al. [3], see Fig. 7 in [3]. From Fig. 7 in [3], it is clear that the peak in the predicted vertical heat flux distribution occurs closer to the jet source than in the measured heat flux distribution, this would indicate a deficiency in the flame structure model as the location of the hot emitting region of the flame is too close to the jet source in the simulation. These results as expected confirm that including TRI in the model significantly improves the model predictive capability with regards to calculating heat flux distributions in hydrogen jet flames. It can also be inferred that the b PDF yields more accurate predictions of heat flux than the truncated Gaussian PDF. A caveat to this is the predicted heat flux distribution demonstrates significant sensitivity to the shape of the PDF used in the stochastic methodology to simulate TRI. This is a surprising result as it is counter most other findings relating to jet flame modelling and PDFs. For example Lockwood and Naguib [47], examined the sensitivity of the mean flow fields to the PDF implemented for a number of natural gas jet fires and found the predicted flow fields insensitive to the PDF implemented. Similarly Onokpe and Cumber [46], examined the sensitivity of the mean and RMS fields of a number of hydrogen jet flames [2,49,50], and found little difference in the predictions using a b PDF, a clipped Gaussian PDF and a truncated Gaussian PDF, although perhaps crucially what differences there are, primarily in the shear layer where the mean mixture fraction is relatively small near the stoichiometric value. Also the predicted spectral intensity distributions shown in Figs. 6 and 7 demonstrate little sensitivity to the PDF implemented. The sensitivity of the predicted heat flux distribution to the PDF implemented seems to be due to the difference in shape of the PDFs for mean mixture fractions close to the lower boundary of mixture fraction space. The heat flux to a receiver is calculated as a numerical integration on the incident hemisphere of the total incident intensity distribution. The total intensity is evaluated at the centroid of elements defined on the unit hemisphere by the ray mesh specified, by solving the equation of radiative heat transfer, Eq. (4) along rays traversing the computational domain. Of the rays that intersect the flame, some pass through or near the axis of the flame, but the majority define chords where large parts of the ray are in the shear layer where the differences in the cumulative PDFs of instantaneous mixture fraction are largest. Therefore the relatively small differences in the predicted mean and RMS flow

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sensitivity to the PDF implemented in the stochastic TRI model and when the radiation heat flux distribution is calculated these differences accumulate. The final issue to consider is the computational run-time of simulating the radiative heat flux distribution with and without modelling TRI. All simulations discussed in this article were completed on a standard PC with a Dual Intel(R) Core TM 2 Processor with a clock speed of 1.8 GHz. The flame structure model takes approximately 10 min to produce the predicted flow fields required as inputs to the radiation model. To calculate the total intensity for a receiver when neglecting TRI takes approximately 2  103 s. This translates into approximately 6 s to calculate the heat flux at a receiver based on 3072 rays. The equivalent timing data when including TRI in the radiation model are 5.9 s to evaluate the total intensity for a given ray and approximately 9 min to evaluate the heat flux for a receiver evaluated using 3072 rays. Therefore the TRI implementation requires approximately 100 times the runtime of a radiation simulation based on the mean flow properties.

7. Conclusion

Fig. 13. Projections of the mean intensity distribution onto a plane through the jet axis normal to the receiver located at x = 0.484 m for Gore et al. [3] hydrogen jet flame, Re = 5722, (a) 32 samples to calculate the mean and (b) 512 samples.

fields due to the shape of the PDF are amplified when calculating the heat flux. Why the b PDF is the more accurate PDF for simulating TRI is likely to be due to it satisfying the 0th, 1st and 2nd moments exactly for all values of the mean mixture fraction but critically for low values of the mean mixture fraction close to the stoichiometric ratio. For hydrogen burning in air the stoichiometric mean mixture fraction is fst = 0.0286 where the truncated Gaussian PDF and the clipped Gaussian PDF as implemented in [3], will only satisfy the 0th moment exactly. The above is a logical reason for the increased sensitivity of the heat flux distribution to the PDF implemented than exhibited in previous investigations [46,47], but more evidence is required before it can be considered a credible reason for the sensitivity. Fig. 13 shows the predicted total intensity for a horizontal line of rays incident on a receiver with a horizontal orientation located at (r, x) = (0.575, 0.484 m). The orientations of the rays are expressed in local spherical coordinates on the unit hemisphere and then projected onto a vertical plane normal to the receiver intersecting the jet axis. From this it is possible to identify the jet radius to be approximately 0.075 m at a height of x = 0.484 m. Fig. 13a shows a comparison of the predicted total intensity calculated using a b PDF and a truncated Gaussian PDF to model TRI. For each value of the total intensity plotted 32 samples are used to calculate the mean total intensity, hence the non-smooth behaviour of the two curves. Fig. 13b gives much smoother predictions of the total intensity curves as 512 samples are used to calculate the mean total intensity. Fig. 13 is consistent with the explanation of why the heat flux distribution is sensitive to the PDF used in the stochastic model formulation as the largest difference between the two mean total intensity curves occurs in the shear layer of the jet near the jet boundary, exactly where the differences in shape of the PDF is the most significant. It follows that although little sensitivity to the PDF is exhibited in the turbulent combustion model or the spectral intensity distribution for the ray orientations considered above, for rays that pass through the shear layer there is some

In this paper TRI in hydrogen jet flames is simulated. In a previous paper [46], the flame structure model was extensively validated using the free jet flame measurements for a hydrogen jet flame and two hydrogen jet flames diluted with helium [2,49], two additional hydrogen jet flames [3] and a nitrogen diluted jet flame [50]. The specific objective of the study is to investigate the sensitivity of radiation fields to the PDF implemented to stochastically simulate the instantaneous mixture fraction field. All previous stochastic simulations of TRI in jet flames use a clipped Gaussian PDF [3,4,20,25–30], as it is relatively straight forward to implement. Predictions of the spectral intensity distribution and external heat flux distribution are compared with the equivalent measured distributions in Gore et al. [3]. It is found that the predicted spectral intensity distribution including TRI in the model give reasonable agreement with the measurements of Gore et al. [3]. The predicted spectral intensity distributions are marginally more accurate than Gore et al’s model, probably due to the way radiation heat loss is incorporated into the respective models. The predicted spectral intensity distributions using a b PDF and a truncated Gaussian PDF, see Figs. 6 and 7 are very similar suggesting the exact shape of the PDF is not important when simulating the spectral intensity distribution for horizontal rays passing through the axis of the jet flame. Turning to the prediction of the heat flux distribution, TRI is found to significantly enhance the received external heat flux distribution. Indeed it is possible to accurately predict the external radiation heat flux distribution provided TRI is included in the computational model. Surprisingly the predicted heat flux does demonstrate significant sensitivity to the PDF used to simulate the instantaneous mixture fraction field. This is a surprising result as all other investigations of the mean and RMS flow fields of a jet flame suggest the exact shape of the PDF is not critical to accurately predict the flame structure of a jet fire [46,47]. The reason for the sensitivity to the PDF implemented is they differ crucially for a region of mixture fraction space near the boundary f = 0, which includes the stoichiometric value. It is the rapid variation of instantaneous properties around the instantaneous stoichiometric mixture fraction that promotes TRI in hydrogen jet flames, [3]. One final conclusion is the model including TRI using the b PDF to simulate the instantaneous mixture fraction field is the most accurate of the models for simulating the external radiation heat flux distribution. It is likely that the b PDF is the more accurate PDF as it satisfies the 0th, 1st and 2nd moments.

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The next step in the model development is to improve the computational efficiency of the TRI model by incorporating techniques such as Sobol sequences [55], hot sampling [37,56], and adaptive quadrature [36,54].

[28]

[29]

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