bayesian inference of a multivariate model in a phase ... · model involving certain significant...

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Bayesian inference of a multivariate model in a phase change situation around a cylinder in staggered arrangement

Ravi Pullepudi1 (), S. K. Maharana2

1. Department of Aeronautical Engineering, MVJ College of Engineering, Bangalore-560067, India 2. Department of Aeronautical Engineering, Acharya Institute of Technology, Bangalore-560107, India Abstract The complexity of problem of fluid flow and heat transfer over an array of circular cylinders is common in industrial applications of fluid dynamics. The complex nature of the problem encountered in industry gives rise to certain significant dimensions in fluid dynamics theory. Some of them are fluid flow interaction, interferences in flow and vortex dynamics which are typically found in compact heat exchangers, cooling of electronic equipment, nuclear reactor fuel rods, cooling towers, chimney stacks, offshore structures, hot-wire anemometry, and flow control. The mentioned structures are subjected to air or water flows and therefore, experience flow induced forces which can lead to their failure over a long period of time. Basically, with respect to the free stream direction, the configuration of two cylinders can be classified as tandem, side-by-side, and staggered arrangements. The Reynolds Averaged Navier–Stokes (RANS) equations have been used to compute the flow and Eulerian model has been used to understand phase change situation in a staggered arrangement of cylinders. In the present study, nucleate boiling has been the cause of heat and mass transfer between the phases for different cylinder configurations in staggered arrangement. The study was carried out by keeping cylinders stationary as well as rotating to understand the difference and impact of these situations on VOF. The profound effects of arrangement of cylinders, the location of cylinder surface, surface temperature of the cylinder were investigated. To have a deeper and meaningful insight into the phase change phenomenon of water into water vapor, Bayesian inference of a multivariate model involving certain significant factors such as Nusselt number (Nu), Prandtl number (Pr), and Stanton number (St) along with volume fraction of vapor phase of water was carried out. The posterior probabilities computed from Bayesian inference for two statistically significant and experimentally verified datasets were obtained during the study. Through Principal Component Analysis (PCA) for the multivariate datasets used in the study, it was observed that some factors are positively and negatively correlated and individually contributing towards a meaningful finding from the study.

Keywords RANS

heat flux

phase change

Eulerian model

volume fraction

Bayesian inference

posterior probability

Article History Received: 5 November 2019

Revised: 12 February 2020

Accepted: 14 February 2020

Research Article © Tsinghua University Press 2020

1 Introduction

Many of the outstanding research activities in the domain of fluid flow and heat transfer have been expanded by concentrating focus on the effect of spacing between the cylinders on the flow characteristics and heat transfer around them. It was keenly noted that the qualitative nature of the flow depends strongly on the arrangement of cylinders (Mittal et al., 1997; Zdravkovich, 1997a; Meneghini et al., 2001). Most of the numerical simulations of flow over

a pair of circular cylinders have utilized finite element methods. The problem numerically investigated by Mittal et al. (1997) is primarily based on finite element method. The finding of their research was reported for Reynolds numbers (Re) of 100 and 1000 for cylinders in staggered and tandem arrangement for various spacings. The conclusion from their study was that the shear layers at Re = 1000 and L/D = 2.5 as compared to the case at Re = 100 caused instability due to the increased velocity of flow. At Re =100 the flow was found to be converged to steady state. Increasing

Vol. 3, No. 2, 2021, 113–123Experimental and Computational Multiphase Flow https://doi.org/10.1007/s42757-020-0060-8

R. Pullepudi, S. K. Maharana

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the gap to L/D = 5.5, the flow at Re = 100 showed unsteady behavior. They indicated that the Strouhal numbers that are associated with the vortex shedding of the twin cylinders could take on the same value.

The available experimental investigations and numerical simulations demonstrated rich hydrodynamic phenomena for the problems of fluid flow over double circular cylinders. Many researchers studied this problem by using numerical simulations. For example the findings of research done by Li et al. (1991) on two flow patterns for the laminar flow over twin tandem circular cylinders were reported for Re = 100. For the large spacing ratio (L/D > 3.0), the vortices are observed to be shed off from both twin cylinders. For L/D < 3.0, it was observed that the vortex shedding happens in the downstream of the cylinder and the vortices, slightly out of phase, from the two cylinders rotate in the same direction in their near wake region

In the experimental work of Bearman and Wadcock (1973), Zhang and Melbourne (1992), Liu et al. (2007), and Ryu et al. (2009) to study wake interaction between two circular cylinders in tandem and side-by-side arrangements, a flow visualization method was used to study the effect of interference between two circular cylinders in side-by-side arrangement at Re = 2.5×104. By applying unstructured spectral element method to investigate the flow pattern of two side-by-side cylinders for different spacings, Liu et al. (2007) carried out their research at low Reynolds numbers.

Ding et al. (2007), Kitagawa and Ohta (2008), Mahír and Altaç (2008), and Singha and Sinhamahapatra (2010) carried out research using numerical method to investigate flow pattern for tandem arrangement of cylinders for both laminar and turbulent regimes. The three-dimensionality effects were studied numerically in the wake of two fixed tandem cylinders at Re = 220 by Deng et al. (2005). In their study virtual boundary method to apply the no-slip condition. From their investigation, like 2-dimensional cases, critical spacing range for which instability occurred fell between 3.5 and 4 of L/D. That means for L/D ≤ 3.5, the flow wake maintained a 2-D state and for L/D ≥ 4, three- dimensionality effects appeared in the wake. In fact, among the numerical works done on flow over a pair of cylinders at high Reynolds numbers, one of the most recent significant studies has been carried out at Re = 2.2×104 by Kitagawa and Ohta (2008) by simulating a three dimensional flow over two tandem cylinders. The researchers have considered a change of gap of 2D to 4D between the two cylinders, analyzed interference effect and vortex interaction of two cylinders and validated their finding with the experimental data for the same Reynolds number.

Of noticeable experimental works at subcritical Reynolds numbers, one can mention the studies of Ljungkrona et al. (1991) at Re = 2×104 and Moriya et al. (2002) at Re = 6.5×104.

They thoroughly investigated the flow characteristics of two tandem cylinders.

The drag forces, pressure distribution, velocity profile, vortex shedding frequency, and flow patterns for the twin circular cylinders in a tandem arrangement have been summarized by Zdravkovich (1997b) in his study.

The intense literature review done above has enormously motivated the authors to understand the phase change phenomenon when water flows over two heated circular cylinders of equal diameters and are in tandem or staggered arrangement kept at different spacing between them and by changing Reynolds number. The earlier work also lacks any attempt to do Bayesian inference on the multivariate data produced from the phase change process. So, first numerical simulation of phase change was attempted and then the dataset was prepared and verified with their counterparts resulted from experiment using staggered arrangement of two, three, and four cylinders, each of equal diameter.

Though the authors’ work on numerical investigation on effect of orientation and rotation on liquid–vapor phase change around a cylinder in staggered arrangement has been reported earlier the current finding of research such as posterior probability of volume of fraction of vapor using Bayesian inference is a new and significant milestone in the current work.

2 Governing equations and methodology

The basic set of governing equations used to solve the multiphase flow problem is given below.

The first step in solving any multiphase problem is to determine which of the regimes provides some broad guidelines for determining appropriate models for each regime, and how to determine the degree of interphase coupling for flows involving bubbles, droplets, or particles, and the appropriate model for different amounts of coupling. Eulerian model is used in the solution of the current problem.

The Eulerian model is the most complex of the multiphase models. It solves a set of momentum and continuity equations for each phase. Coupling is achieved through the pressure and interphase exchange coefficients. The manner in which this coupling is handled depends upon the type of phases involved; granular (fluid-solid) flows are handled differently from non-granular (fluid–fluid) flows.

2.1 Volume fraction equation

The description of multiphase flow as interpenetrating continua incorporates the concept of phasic volume fractions, denoted here by qα . Volume fractions represent the space occupied by each phase, and the laws of conservation of


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mass and momentum are satisfied by each phase individually. The volume of phase q is defined by qV :

dq qV α V= ò (1)

where

1

1n

qq

α=

=å (2)

The effective density of phase q is

ˆqq α qρ ρ=

where qρ is the physical density of phase q. The volume fraction equation may be solved either through implicit or explicit time discretization.

2.2 Conservation equations

The general conservation equations from which the equations are solved are presented below. The three equations are for conservation of mass, conservation of momentum and energy.

2.2.1 Conservation of mass

The continuity equation for phase q is

( ) ( ) ( )1

q

n

q q q qv pq qp qp

α ρ α ρ m m St =

¶+⋅ = - +

¶ å (3)

where qv is the velocity of phase q and pqm characterizes the mass transfer from the pth to qth phase, and qpm characterizes the mass transfer from phase q to phase p. By default, the source term Sq on the right-hand side of the above equation is zero, but we can specify a constant or user-defined mass source for each phase.

2.2.2 Conservation of momentum

The momentum balance for phase q yields

( ) ( )

, ,1( ) ( )

q q q q q q q q q q q

n

pq pq pq qp qp q lift q vm qp

α ρ v α ρ v v α p τ α ρ gt

R m v m v F F F=

¶+⋅ =- +⋅ +

¶

+ + - + + +å

(4)

where

( )T 2 ·3q q q q q q q q qα μ v v Iα λ μτ v

æ ö÷ç= + + - ÷ç ÷çè ø (5)

This is the qth phase stress–train tensor. Here qμ and qλ are the shear and bulk viscosity of phase q, qF

is an external

body force, ,lift qF

is a lift force, ,vm qF

is a virtual mass force,

pqR

is an interaction force between phases, and p is the pressure shared by all phases. pqv is the interphase velocity, defined as follows: if pqm (i.e., phase p mass is being

transferred to phase q), pqv = pv

; if 0pqm < (i.e., phase q mass is being transferred to phase p), pqv = qv . Likewise, if 0qpm > then qpv = qv , if 0qpm < then qpv = pv .

Equation (4) must be closed with appropriate expressions for the interphase force, pqR

. This force depends on the

friction, pressure, cohesion, and other effects, and is subject to the conditions that pq qpR R=-

and 0qqR =

.

The interaction term of the following form:

1 1

( )n n

pq pq p qp p

R K v v= =

= -å å (6)

where ( )pq qpK K= is the interphase momentum exchange coefficient.

2.2.3 Conservation of energy

The current problem uses conservation of energy principle through Eulerian multiphase model given below:

( ) ( )

1

:

( ) (7)

qq q q q q q q q q q q

n

q pq pq pq qp qpp

pα ρ h α ρ u h α τ u q

t t

S Q m h m h=

¶¶+⋅ =- + -⋅

¶ ¶

+ + + -å

where qh is the specific enthalpy of the qth phase, qq is the heat flux, qS is a source term that includes sources of enthalpy (e.g., due to chemical reaction or radiation), pqQ is the intensity of heat exchange between the pth and qth phases, and pqh is the interphase enthalpy (e.g., the enthalpy of the vapor at the temperature of the droplets, in the case of evaporation). The heat exchange between phases must comply with the local balance conditions pq qpQ Q=- and

0pqQ = .

2.3 Turbulence models

In comparison to single-phase flows, the number of terms to be modeled in the momentum equations in multiphase flows is large, and this makes the modeling of turbulence in multiphase simulations extremely complex. One of the three methods for modeling turbulence in multiphase flows within the context of k - models is mixture turbulence models. The description of mixture turbulence model is presented below.

k - mixture turbulence model

The mixture turbulence model is the default multiphase


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turbulence model. It represents the first extension of the single-phase k - model, and it is applicable when phases separate, for stratified (or nearly stratified) multiphase flows, and when the density ratio between phases is close to 1. In these cases, using mixture properties and mixture velocities is sufficient to capture important features of the turbulent flow.

The k and equations describing this model are as follows:

( ) ( )m m mρ kt

kρ v¶=+⋅

¶ ,t m

k

μk

σæ ö÷ç⋅ +÷ç ÷÷çè ø , –k m mG ρ (8)

and

( ) ,

1 , 2

( )

( )

t mm m m

ε

k m m

μρ ρ v

t σ

C G C ρk

æ ö¶ ÷ç+⋅ = ⋅ ÷ç ÷÷ç¶ è ø

+ -

(9)

where the mixture density mρ and velocity mv are computed from

1

N

m i ii

ρ α ρ=

=å (10)

and

1

1

Ni ii

m Ni ii

α ρ viv

α ρ=

=

=åå

(11)

The turbulent viscosity, ,t mμ is computed from

,t m mμ ρ= Cμ

2k

(12)

and the production of turbulence kinetic energy, Gk,m is computed from

,, (t mk mm vG μ= + ( mv

)T): mv (13)

The constants in these equations are given below for the single-phase k– model.

1 20.09, 1.44, 1.92μ CC C= == , σ 1.0k = , 1.3σ =

2.4 Bayesian inference

Bayesian viewpoint is taken to draw statistical inference from a sample. In Bayesian paradigm, which is fundamentally a statistical/probabilistic paradigm wherein the uncertainty of a new observation is updated by the models of probability of another distributed. To understand it better the concept of Bayes theorem needs to be understood where relation between the updated knowledge (the “posterior”), the prior knowledge (the “prior”) and the knowledge coming from

the observation (the “likelihood”) exist (Rocca, 2019). The concept has been outlined below.

Let us assume a model where data are generated from a probability distribution depending on an unknown parameter θ. Models are the mathematical formulation of the observed events. Parameters are the factors in the models affecting the observed data. If we have the prior knowledge about θ, then the probability of θ is P(θ). As soon as data are observed, the prior knowledge about θ is updated by using Bayes theorem whose basic understanding is outlined below. The foundation of Bayes theorem is based upon conditional probability. An important part of Bayesian inference is the establishment of parameters and models. This could be understood by simplifying the above definition of conditional probability.

The Bayes theorem is P(θ︱data) =(P(data︱θ)P(θ))/ P(data). Here P(θ) is the “prior” that is the strength of our belief in θ without the data. The “posterior”, P(θ︱data) is the strength of our belief in θ when the data have been taken into account. The “likelihood”, P(data︱θ) is the probability that the data could be generated by the model with parameter values θ. The “evidence”, P(data), is the probability of the data according to the model, determined by summing across all possible parameter values weighted by the strength of belief in those parameter values. For the computation of posterior there is a need to have three terms such as a prior, a likelihood, and evidence. In most of the situations the first two things (the prior and the likelihood) are known. If they are not known then they could be expressed easily as part of the assumed model. The computation of the third term, that is the evidence, is computed using the following way:

( ) ( )data (data ) dP P θ P θ θ= ò ︱

For a dataset with lower dimension this integral is easy to compute; however the difficulty arises for higher dimensions and hence certain approximation technique is adopted to know the posterior of such cases. Some of the well-known techniques are Markov Chain Monte Carlo (MCMC) and Variational Inference methods. In this report MCMC has been followed.

The basic approach of Bayesian inference as described above was used to get the essential information from the dataset obtained from the numerical simulations as mentioned in the next section.

2.5 Methodology

A sketch definition of numerical set-up (shown in x–y plane) for two circular cylinders in tandem arrangement with boundary conditions is shown in Fig. 1. The necessary


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dimensions of the fluid domain are expressed in terms of the diameter (D) of a cylinder. The diameters of both the cylinders are the same. The two cylinders are arranged in tandem inside the flow domain. The mesh density is higher near the walls of cylinders. The total number of nodes of the mesh generated is 5117 and that of elements is 3984. The flow enters at inlet where the inlet velocity is specified (u = U∞) and leaves the domain at outlet. The temperature at the cylinder surface is maintained at 573 K. All the walls are stationary and have no slip boundary conditions.

The commercially available computational fluid dynamics (CFD) solver Ansys® Fluent was used for the simulation of fluid flow and heat transfer over the cylinders. The working fluid is water (liquid phase), upon phase change, becoming steam (vapour phase).

3 Results and discussion

The effect of orientation of a cylinder in an arrangement on the volume fraction of each phase has been the focus. Hence two different arrangements such as tandem and staggered are considered during the computation to understand the phase change of liquid to vapor phase over a cylinder whose surface is heated to a certain temperature (above the saturation temperature of vapor).

There has been a validation of a few quantities such as the mean drag coefficient, mean lift coefficient, and Strouhal numbers of the isolated circular cylinder with that of the other researchers and the details are mentioned in Table 1. In Fig. 2 is plotted the mean drag coefficient versus Re of the flow.

The comparison of the present values of mean drag coefficient with the data published earlier is fairly good. The comparison of the non-dimensional quantities such as mean drag coefficient, M

DC , mean lift coefficient MLC , and

Strouhal number St was done with those published by leading researchers and the authors, due to good comparison and validation of the above-mentioned quantities, have carried

Table 1 Comparison of the mean drag coefficient MDC , M

LC , and St

MDC M

LC St

Present study 1.41 0.692 0.1902

Lui et al. (2014) 1.337 0.685 0.1955

Rajani et al. (2009) 1.3380 0.4276 0.1936

Wang et al. (2009) — 0.71 0.1950

Zhang et al. (2008) 1.34 0.66 0.1970

Linnick and Fasel (2005) 1.34–1.37 0.71 —

Farrani et al. (2001) 1.36–1.39 0.71 —

He et al. (2000) 1.36 — 0.1978

Zdravkovich (1997a, 1997b) 1.43 — —

Henderson (1995) 1.34–1.37 — 0.1971

Fig. 2 Comparison of mean drag coefficients for various Reynolds numbers for a single circular cylinder.

out their study to investigate the flow over two heated circular cylinders.

Phase change of liquid to vapor

In Fig. 3(a) is shown the contours of liquid–vapor phase for two non-rotating cylinders in tandem arrangement and Fig. 3(b) shows the variation of volume fraction (VOF) of vapor around the same two cylinders shown in Fig. 3(a). The range of variation of VOF is between 0.5 and 0.9 around both the cylinders and it is noted that cylinder-1 (extreme left one) experiences more vapor (volume fraction is close

Fig. 1 Physical domain (shown in x–y plane) for two circular cylinders with boundary conditions.


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to 0.9) at a location of 120° on its surface and cylinder-2 faces more vapor (volume fraction is almost 0.85) at a location around 350° on its surface.

In Fig. 4(a) is shown the contours of liquid–vapor phase for two rotating cylinders in tandem arrangement and Fig. 4(b) shows the variation of volume fraction of vapor around the same two cylinders shown in Fig. 4(a). The range of variation of VOF varies between 0.51 and 0.95 around cylinder-1 and the variation of VOF is between 0.67 and 0.89 for cylinder-2. It is noted that cylinder-1 (extreme left one) experiences more vapor at a location of 50° on its surface and cylinder-2 faces more vapor at a location around 350° on its surface.

In Fig. 5(a) is shown the contours of liquid–vapor phase for three non-rotating cylinders in staggered arrangement and Fig. 5(b) shows the variation of volume fraction (VOF) of vapor around the two cylinders (extreme left and right) shown in Fig. 5(a). By introducing another cylinder in the arrangement described in Fig. 4(a) a new staggered arrangement was made which is shown in Fig. 5(a). The arrangement has significantly affected the phase interaction between the liquid and vapor phases. There is a lowering of VOF (~0.2) in cylinder-2 and the VOF for cylinder-1 has

also been affected (increased) by 10%–15%. It is also observed that more time is needed for a better VOF to be seen around the cylinders at hand. The range of variation of VOF varies between 0.4 and 0.9 around cylinder-1 and the variation of VOF is between 0.2 and 0.85 for cylinder-2. It is noted that cylinder-1 (extreme left one) experiences more vapor at a location of 0° and 350° on its surface and cylinder-2 faces more vapor at a location around 200° on its surface.

In Fig. 6(a) is shown the contours of liquid–vapor phase for four non-rotating cylinders in staggered arrangement and Fig. 6(b) shows the variation of volume fraction of vapor around the two cylinders (extreme left and right) shown in Fig. 6(a). By introducing another cylinder in the arrangement described in Fig. 6(a) a new staggered arrangement was made which is shown in Fig. 6(a). The arrangement has significantly affected the phase interaction between the liquid and vapor phases. There is an enhancement of VOF (~0.4 in the lower side and ~1.0 in the upper side) in cylinder-2 and the VOF for cylinder-1 has also been affected by 5%. The range of variation of VOF varies between 0.45 and 0.9 around cylinder-1 and the variation of VOF is between 0.41 and 1.0 for cylinder-2. It

Fig. 3 Liquid–vapor phase for two non-rotating cylinders in tandem.

Fig. 4 Liquid–vapor phase for two rotating cylinders (104 rad/s) in tandem.


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is noted that cylinder-2 (extreme right one) experiences more vapor at a location of 0° on its surface and cylinder-1 faces more vapor at a location around 130° on its surface.

The volume fraction of water vapor not only changes with surface temperature and location of cylinder surface but it also has profound interaction and connection with Nusselt number (hD/k), Prandtl number (μCp/k), and Stanton number (h/ρCpU∞) of the flow. Hence it is necessary to understand their deeper correlation and inter-variable interaction. One of the techniques to do it is Bayesian inference. This allows one to observe the posterior probability of a variable. In this study it was achieved by the BEST (Bayesian Estimation Supersedes the t-test), a package in Rstudio®. Here, VOF of water vapor is a variable (or a factor) that has been predicted around two cylinders: extreme left one (cylinder-1) and extreme right one (cylinder-2) in the staggered arrangement described above by using BEST with Markov chain Monte Carlo (MCMC) method. The function BEST with MCMC was used to predict the probability of the statistical information of the multivariate dataset obtained from CFD solver. The intent was to compare the prior belief (before the model was used, shown in histogram in red line

of Fig. 9) and posterior belief (after the model was used, shown in blue line in Fig. 9 just behind the histogram) of VOF around the two cylinders and the behavior of the other computed variables such as Nu, St, Pr, and curve length.

In the present study, it is also noted that very high Nusselt number (Nu = 1000–8000 for both the cylinders) is an indication of extremely high convective heat transfer near each cylinder’s boundary. So, the volume fraction (VOF) of vapour attains a very high value (~1.0) near the same boundary and the amount of vapour being formed is quite good. The Prandtl number (Pr) noted during the study (for both the cylinders) is varying within 4.0 and 6.0 for both the cylinders. The momentum diffusivity of the fluid is better than its thermal diffusivity and its velocity boundary layer over the surface of cylinder is thicker than the thermal boundary layer at the same location. It allows more vigorous momentum exchange during phase change between the two phases. For a higher Pr, there is a decrease of VOF. The thicker boundary layer (Pr > 4.0) deters heat exchange between two phases and hence lower VOF. Stanton number (Nu/(Re×Pr)), which is an indication of skin friction drag (due to a viscous flow moving over the

Fig. 5 Liquid–vapor phase for three non-rotating cylinders in staggered arrangement.

Fig. 6 Liquid–vapor phase for four non-rotating cylinders in staggered arrangement.


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circular cylinder), indicates that cylinder-2 has a higher (20%) value of friction than that over cylinder-1. Cylinder-2 might be facing a higher viscous drag. In both the cases VOF is falling with increasing St which is absolutely true because the viscous sublayer over the cylinder surface might be obstructing the heat exchange between the wall and fluid and hence the phase change.

Reference is made to Fig. 7(a). In this is shown the difference of means of VOF of water vapour formed around each cylinder. This is posterior probability of the difference in means. It says that there is merely 1% variance between the two means of VOF distribution around each cylinder. The 95% high density intensity (HDI) of the data (collected on volume of fraction of water vapor around cylinders when they are immersed in water and heated) is visible around means, in other terms 65.6% to 34.4% of VOF is around the mean. As per the expectation the heat has been properly utilized to generate sufficient amount of water vapor (close to 1).

From Fig. 7(b), drawn with the region of practical equivalence (ROPE), it was noticed that the VOF has clustered within ±1% of ROPE where the same mean of the data set is maintained (spread is the minimum). The probability that the difference in reaction time is precisely zero is zero. More interesting is the probability that the difference may be too small to matter. We can define a region of practical equivalence (ROPE) around zero, and obtain the probability that the true value lies therein.

Fig. 7 Posterior probability of the difference in means.

Figure 8 shows the posterior plot for difference in standard deviations for the VOF data around the same two cylinders. The deviation is just close to 0.0106 which confirms that the means are quite good to predict a very good VOF around the cylinders.

Table 2 is a summary of the information received through BEST from the dataset of two cylinders (upstream cylinder is denoted as cylinder-1 and downstream cylinder is denoted as cylinder-2).

In Table 3, Rhat is the Brooks–Gelman–Rubin scale reduction factor, which is 1 on convergence and n.eff is the effective sample size, which is less than the number of simulations because of autocorrelation between successive values in the sample. The values of n.eff around 10,000 are needed for stable estimates of 95% credible intervals.

In Fig. 9 is presented a comparison of two data groups: Group 1 data (VOF of vapour) is for cylinder-1 and Group 2 data (VOF of vapour) is for cylinder-2, with their posterior predictions. The histograms in red colour are the data for each cylinder. Some are outliers out of 13 observations. The posterior prediction is drawn from the Bayesian algorithm and the belief is a normal distribution and is clustered around their individual means (0.7137 for cylinder-1 and 0.7333 for cylinder-2).

Figure 10 represents the PCA analysis which signifies that variance in the dataset is most significant. Here in the VOF dataset for the two cylinders, there are 5 dimensions (all are not principal components). The PCA shows the variance in the direction of the component which drives the data distribution and spread. The directions with the largest variances are the most significant. PCA fundamentally reduces the dimensions of the dataset into new dimensions (Dim-1 and Dim-2) without changing the essence of the variation of original dataset. This dimensionality reduction algorithm operates on the five variables and outputs two new variables (Dim-1 and Dim-2) that represent the original variables, a projection or “shadow” of the original data set. Each dimension represents a certain amount of the variation (i.e., information) contained in the original data set. In Fig. 10, Dim-1 and Dim-2 represent 59.5% and 40.5% respectively. Similarly, in Fig. 12, Dim-1 and Dim-2 represent

Fig. 8 Posterior plot for the difference in standard deviation.


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Fig. 9 Comparison of two data groups with their posterior predictions.

Fig. 10 Principal component analysis (PCA) for two cylinders.

69.7% and 21.4% where another data set was used. Out of the two components (dimensions: Dim-1 and Dim-2) in

the data set the significant information is driven by the principal component and other dimensions are not significant.

Figure 11 shows a scree plot wherein the eigenvalues are on the y-axis and the number of factors on the x-axis. The factors with the highest percentage of explained variances are considered out of all the 5 factors. In this case two factors (or 2 dimensions) are having 69.7% and 21.4% variability as shown in Fig. 12. Basically a scree plot displays how much variation each principal component captures from the data. Principal components are created in order of the amount of variation they cover: PC1 captures the most variation, PC2 the second most, and so on. Each of them contributes some information of the data, and in a PCA, there are as many principal components as there are characteristics.

Fig. 11 Scree plot (eigenvalues).

Table 2 Summary of the information received through BEST

Mean Median Mode HDI% HDIlo HDIup compVal%>compVal

mu1 0.7137 0.7132 0.7117 95 0.6473 0.7817 —

mu2 0.7332 0.7333 0.7325 95 0.6579 0.8070 —

muDiff -0.0195 -0.0199 -0.0246 95 -0.1187 0.0827 34.4

sigma1 0.1138 0.1092 0.1028 95 0.0651 0.1742 —

sigma2 0.1259 0.1208 0.1131 95 0.0739 0.1886 —

sigmaDiff -0.0122 -0.0114 -0.0106 95 -0.0987 0.0712 37.9

Nu 35.7715 27.0997 11.1142 95 1.1747 96.0244 —

log10nu 1.4039 1.4330 1.4998 95 0.6478 2.1085 —

effSz -0.1714 -0.1682 -0.1921 95 -1.0082 0.6547 34.4

Table 3 MCMC (Markov chain Monte Carlo) fit results for BEST analysis

mean sd median HDIlo HDIup Rhat n.eff

mu1 0.7137 0.03397 0.7132 0.64731 0.7817 1 53969

mu2 0.7332 0.03759 0.7333 0.65793 0.8070 1 58789

nu 35.7715 30.20913 27.0997 1.17473 96.0244 1 20442

sigma1 0.1138 0.02904 0.1092 0.06506 0.1742 1 31951

sigma2 0.1259 0.03148 0.1208 0.07391 0.1886 1 32700

“HDIlo” and “HDIup” are the limits of a 95% HDI credible interval, “Rhat” is the potential scale reduction factor (at convergence, Rhat = 1), and “n.eff” is a crude measure of effective sample size.


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Fig. 12 Variable PCA.

From Fig. 12 that shows the graphs of variable PCA, it is noted that positive correlated variables point to the same side of the plot. Negative correlated variables point to opposite sides of the graph. That means VOF, Nu, and curve length are towards one side and St and Pr are to another side. So, Nu, curve length, and VOF are positively correlated. Similarly Pr and St are positively correlated.

Fig. 13 Contribution of each variable towards Dim-1 and Dim-2 for principal component analysis.

Contribution of each variable towards Dim-1 and Dim-2 for principal component analysis is presented in Fig. 13. It is noted from Fig. 13(a) that Pr, VOF, and Nu have higher contributions (above 20% each) for Dim-1 and for Dim-2, there is a significant contribution (above 20%) from St compared to other four variables.

4 Conclusions

It was observed during the study that the nucleate boiling phenomenon has been the cause of heat and mass transfer between the phases for three different configurations such as two, three, and four cylinder arrangements. The focus of the study was on staggered arrangement of cylinders. The study was done when cylinders were stationary; however, a case of rotation of a cylinder was also presented to see its impact on VOF. The profound effects of arrangement of cylinders, the location of cylinder surface, and surface temperature of the cylinder were investigated and presented. Bayesian inference of the multivariate model involving significant factors such as Nusselt number (Nu), Prandtl number (Pr), and Stanton number (St) along with volume fraction of vapor phase of water was carried out by BESTMCMC function of Rstudio. The posterior probabilities, computed from Bayesian inference, for two statistically significant and experimentally verified datasets were obtained and presented. Principal component analysis (PCA) for the multivariate datasets was done and it was observed how some factors are positively and negatively correlated and individually contributing towards a meaningful finding from the study.

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