masters thesis: turbulence modeling for heat transfer to …renep/content/msc... · 2016-06-22 ·...

Energy Technology, Process and Energy

Turbulence Modeling for HeatTransfer to Supercritical PipeFlows

Ashish Patel

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Thesis

Turbulence Modeling for HeatTransfer to Supercritical Pipe Flows

Master of Science Thesis

For the degree of Master of Science in Mechanical Engineering at Delft

University of Technology

Ashish Patel

August 8, 2013

Faculty of Mechanical, Maritime and Materials Engineering (3mE) · Delft University ofTechnology

Copyright c© Process and Energy DelftAll rights reserved.

Delft University of Technology

Department of

Process and Energy Delft

The undersigned hereby certify that they have read and recommend to the Faculty ofMechanical, Maritime and Materials Engineering (3mE) for acceptance a thesis

entitled

Turbulence Modeling for Heat Transfer to Supercritical Pipe Flows

by

Ashish Patel

in partial fulfillment of the requirements for the degree of

Master of Science Mechanical Engineering

Dated: August 8, 2013

Supervisor(s):Dr.ir. R. Pecnik

Ir. H. Nemati

Reader(s):Prof.dr.ir. B.J. Boersma

Prof.dr.ir. R.A.W.M. Henkes

Abstract

Over the last few decades there has been an ongoing search for more efficient power plantdesigns. The use of supercritical fluids as a working medium in a power cycle presents apromising new development in this direction. Recently, its use has also been proposed inthe next generation nuclear power plants to have a higher efficiency and reduced size. Safeand efficient design of these systems requires a more fundamental understanding of the tur-bulent heat transfer characteristics of these fluids. However, turbulent flows at supercriticalconditions are extremely complex and their physics and modeling requirements are not wellunderstood. The extreme thermophysical property variations of fluids close to the criticalpoint alter the conventional behavior of turbulence, making the governing equations highlynonlinear. These variations can cause heat transfer deterioration and heat transfer enhance-ment in a very intense and rapid manner. With the increase in computing power it hasbecome possible to accurately compute the flow and thermal structures using Direct Numer-ical Simulation (DNS), applicability of which is still limited to low Reynolds number flow. Inorder to compute flows with high Reynolds numbers, the solution of the Reynolds AveragedNavier Stokes (RANS) equations is needed. The RANS equations contain unknown (closure)terms that need to be closed using turbulence models. Currently, there is no model availablethat can accurately predict heat transfer to supercritical flows, hence analytic and numericalresearch in this emerging field has attracted attention worldwide.

In the present work, first a theoretical framework is established to take into account theextreme fluctuations of thermophysical properties in the RANS equations. As compared toideal gas heat transfer, the number of closure terms for a supercritical heat transfer increases.The effect of flow acceleration due to thermal expansion, buoyancy and property variationson the heat transfer characteristics are also discussed.

Second, conventional turbulence models, namely-the Launder and Sharma (LS), the Myongand Kasagi (MK), the v2 − f (V2F) are implemented in a FORTRAN code and validatedwith canonical cases involving ideal gas (calorically perfect) available in the literature. Theperformance of these turbulence models in predicting heat transfer to supercritical CO2 in avertical tube with different buoyancy effects and flow directions are then assessed by compar-ing their results with the inhouse DNS data. The results of the models are inconsistent fordifferent cases applied. The MK model performs the best among all investigated models, but

Master of Science Thesis Ashish Patel

ii

fails in cases with significant buoyancy effect. The performance of the LS and the V2F modelis in general found to be very poor. However, for downward flows with buoyancy the LSmodel gives a good match with DNS data. Possible deficiencies of these conventional modelsare pointed out.

Finally, the inhouse DNS data for heat transfer to supercritical CO2 is analyzed to evalu-ate the significance of additional closure terms because of large property fluctuations. It isfound that these additional closure terms can significantly affect the enthalpy profile. Theanalysis also points out that the averaged thermophysical properties, especially the specificheat at constant pressure cp, also needs closure due to an averaging artifact called the Jenseninequality.

Ashish Patel Master of Science Thesis

Table of Contents

Acknowledgements ix

1 Introduction 1

1-1 Fluid Properties at Supercritical Pressures . . . . . . . . . . . . . . . . . . . . . 2

1-2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1-3 Motivation and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1-4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Theoretical Framework 9

2-1 Governing Equations: DNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2-2 Governing Equations: RANS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2-3 Turbulence Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2-4 Heat Transfer Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2-4-1 Influence of variation in specific heat, thermal conductivity and viscosity . 22

2-4-2 Influence of flow acceleration . . . . . . . . . . . . . . . . . . . . . . . . 23

2-4-3 Influence of buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2-4-4 Subcritical vs Supercritical . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Flow Domain and Numerical Method 29

3-1 Flow Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3-2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Results and Discussion 33

4-1 Simulation Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4-2 Comparison of DNS and RANS Results . . . . . . . . . . . . . . . . . . . . . . . 35

4-2-1 Wall temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4-2-2 Velocity, enthalpy and turbulent statistics . . . . . . . . . . . . . . . . . 37

4-3 Closure Terms Due to Large Property Variation . . . . . . . . . . . . . . . . . . 57

4-4 Jensen Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67


iv Table of Contents

5 Summary and Conclusions 75

A Transport Equations 77

B Turbulence Model Validation and Mesh Independency 83

B-1 Validation of Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . 83

B-2 Mesh Independency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

C Shear Stress Distribution 91


List of Figures

1-1 Variation of thermophysical properties of carbon-dioxide (CO2) vs. temperatureat P0 = 80 bar and 90 bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2-1 Heat transfer coefficient of supercritical CO2 at P0=80 bar using Dittus-Boeltercorrelation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2-2 Yamagata’s data exhibiting heat transfer enhancement . . . . . . . . . . . . . . 24

2-3 Heat transfer enhancement description . . . . . . . . . . . . . . . . . . . . . . . 24

2-4 Turbulent shear stress for buoyancy opposed and buoyancy aided flow . . . . . . 27

3-1 Geometry of the simulation domain. . . . . . . . . . . . . . . . . . . . . . . . . 29

3-2 Staggered space grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4-1 Streamwise distribution of bulk enthalpy and velocity. . . . . . . . . . . . . . . . 35

4-2 Streamwise distribution of wall temperature from DNS for all cases. . . . . . . . 36

4-3 Comparison of streamwise distribution of wall temperature from RANS with DNSfor all cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4-4 Mean velocity and enthalpy profile for case A . . . . . . . . . . . . . . . . . . . 40

4-5 Reynolds shear stress, ρu′′

ru′′

z/u2τ,0 and radial turbulent heat flux distribution for

case A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4-6 Shear production rate of turbulent kinetic energy for case A. . . . . . . . . . . . 42

4-7 Mean velocity and enthalpy profile for case B . . . . . . . . . . . . . . . . . . . 44


ru′′


case B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4-9 Shear and buoyant production rate of turbulent kinetic energy for case B. . . . . 46

4-10 Mean velocity and enthalpy profile for case C . . . . . . . . . . . . . . . . . . . 49


ru′′


case C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4-12 Shear and buoyant production rate of turbulent kinetic energy for case C. . . . . 51


vi List of Figures

4-13 Comparison of Reynolds shear stress, ρu′′

ru′′

z and ∂uz

∂r/100, based on DNS data for

case C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524-14 Turbulent kinetic energy and total production rate of turbulent kinetic energy based

on DNS data for case C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524-15 Mean velocity and enthalpy profile for case D . . . . . . . . . . . . . . . . . . . 54


ru′′


case D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554-17 Shear and buoyant production rate of turbulent kinetic energy for case D. . . . . 56

4-18 Total production rate of turbulent kinetic energy, based on DNS data for case D. 57

4-19 Turbulent Prandtl number, based on DNS data for all cases. . . . . . . . . . . . 58

4-20 Contribution of fluctuating terms on total viscous shear stress for case A at differentstreamwise locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4-21 Contribution of fluctuating terms on total viscous shear stress for case C at differentstreamwise locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4-22 Contribution of fluctuating terms on total viscous shear stress for case D at differentstreamwise locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4-23 Contribution of fluctuating terms on total heat flux for case A at different stream-wise locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4-24 Contribution of fluctuating terms on total heat flux for case C at different stream-wise locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4-25 Contribution of fluctuating terms on total heat flux for case D at different stream-wise locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4-26 Derivative of heat flux terms for case A at z = 12.8125. . . . . . . . . . . . . . 664-27 One-dimensional analysis of the energy equation to evaluate the significance of

additional closure terms for case A at z = 12.8125. . . . . . . . . . . . . . . . . 664-28 Effect of the Jensen inequality on averaging of thermophysical property at z =

7.8125 for case A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684-29 Streamwise variation of the Jensen inequality on averaging of thermophysical prop-

erty for case A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4-30 Role of fluctuation about a given point on the Jensen inequality. . . . . . . . . . 70

4-31 The Jensen inequality of specific heat cp as function of the enthalpy variance h′′2. 71

4-32 The Jensen inequality of thermal conductivity/specific heat α as function of the

enthalpy variance h′′2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4-33 Production rate of the enthalpy variance h′′2. . . . . . . . . . . . . . . . . . . . 73

4-34 One-dimensional analysis of the energy equation to evaluate the significance of theJensen inequality for case A at z = 12.8125. . . . . . . . . . . . . . . . . . . . . 74

B-1 Validation results for the MK model . . . . . . . . . . . . . . . . . . . . . . . . 84B-2 Validation results for the LS model . . . . . . . . . . . . . . . . . . . . . . . . . 85B-3 Validation results for the LS model applied to ideal gas heat transfer with buoyancy 86

B-4 Validation results for the V2F model . . . . . . . . . . . . . . . . . . . . . . . . 87B-5 Mesh independency study of case A for the MK model . . . . . . . . . . . . . . 88

B-6 Mesh independency study of case A for the LS model . . . . . . . . . . . . . . . 89

B-7 Mesh independency study of case A for the V2F model . . . . . . . . . . . . . . 90

C-1 Shear stress distribution, based on DNS data for case A . . . . . . . . . . . . . . 92

C-2 Shear stress distribution, based on DNS data for case C . . . . . . . . . . . . . . 93

C-3 Shear stress distribution, based on DNS data for case D . . . . . . . . . . . . . . 94


List of Tables

2-1 Constants in turbulence models . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2-2 Functions in turbulence models . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2-3 D and E terms and wall boundary conditions . . . . . . . . . . . . . . . . . . . . 20

4-1 Flow conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4-2 Reynolds number based on friction velocity for different RANS models at the inlet 34


viii List of Tables


Acknowledgements

I would like to thank my supervisor Dr. ir. Rene Pecnik for guiding and supporting methroughout my thesis project. His guidance has not only helped me learn better but has alsoinspired me continuously. Albert Einstein very appropriately said, “It is the supreme art ofthe teacher to awaken joy in creative expression and knowledge.” I believe this quote veryappropriately applies to him. I would also like to thank my daily supervisor, Ir. HassanNemati for his kindness and willingness to help me during any hour of the day. I have grownas a better researcher by working with him. His support gave my thesis the shape and depththat it has now.

I am also thankful to Ir. Jurriaan Peeters and Ir. Enrico Rinaldi for helping me during mytime in need. I am grateful to Prof. Bendiks Boersma and Prof. Piero Colonna for providingtheir valuable support and advice throughout my MSc period. I would also like to expressmy gratitude to the committee members for devoting their time and effort in evaluating mythesis report.

I am thankful to Delft Research Initiative, for providing me with a scholarship to pursuemy MSc studies. The scholarship has helped me focus on my work without being worriedabout the financial issues.

I am thankful to all my friends and colleagues for constantly motivating and helping meduring my stay here in Delft.

Last, but not the least, I express my deepest gratitude to my family for showering theirlove and support on me always. Their faith in me has been a constant source of inspirationfor me.

Delft, University of Technology Ashish PatelAugust 8, 2013


x Acknowledgements


“Our minds are finite, and yet even in these circumstances of finitude we aresurrounded by possibilities that are infinite, and the purpose of life is to grasp asmuch as we can out of that infinitude.”

— Alfred North Whitehead

Chapter 1

Introduction

A supercritical fluid is a substance at a temperature and pressure above its vapor-liquid crit-ical point, where distinct liquid and gas phases do not exist. Close to the critical point thesefluids experience sharp property variations for small changes in temperature. Supercriticalfluids are used for various engineering applications because of their ability to adapt to both-liquid and gas like properties. Some examples from process industry include supercriticalCO2 extraction, supercritical drying, supercritical water oxidation, etc. Supercritical fluidsare also used in the energy industry. The idea of using fluids at supercritical conditions forpower generation was first taken up many decades ago when it was applied to fossil fuel firedsteam generators to increase thermal efficiency of power plants and to avoid the problem ofthermal crisis, which occurs at pressures below the critical value. In the meantime, additionalapplications using fluids at supercritical conditions have been proposed, for example: fuel in-jections in cryogenic rocket nozzles, thermodynamic power cycles operating entirely above thecritical condition, refrigeration systems, carbon capture and storage for enhanced oil recovery,etc. Recently, the concept of a supercritical water reactor (SCWR) has been accepted as oneof the six most promising nuclear reactor concepts recommended by Generation IV Interna-tional Forum (DoE, 2002). The prime reason which makes this class of reactors promising arehigh efficiency, compact size and reduced complexity (Pioro et al., 2004b). This technologycould increase the thermal efficiency of modern nuclear power plants, which have relativelylow efficiency of approximately 33-35 % in comparison with modern supercritical fossil fueledpower units with efficiencies approaching 45-50 % (Oka and Koshizuka, 2000). Due to theincreased popularity of usage of a supercritical fluid in the power industry it has become veryimportant to develop a fundamental understanding of turbulent heat transfer and fluid flowat supercritical state, particularly because heat transfer characteristics of supercritical fluidsare quite different from those of subcritical fluids, due to severe property variations close tocritical point.


2 Introduction

1-1 Fluid Properties at Supercritical Pressures

Supercritical fluids undergo continuous transition from liquid like to a gas like state withincrease of temperature at constant pressure. During this transition all the thermophysicalproperties of fluid vary significantly within a narrow temperature range across the pseudo-critical temperature (Tpc ). Pseudo-critical temperature is defined as the temperature atwhich the specific heat at constant pressure (cp) attains its peak value. Figure 1-1 showsvariation of the thermophysical properties of CO2 at two different supercritical pressures. Itcan be seen that, with increase in temperature, density (ρ), thermal conductivity (λ) anddynamic viscosity (µ) show a drastic decrease around Tpc. λ also shows a small peak valueclose to Tpc. It can also be seen that the extent of property variations decreases as one goesaway from the critical pressure (Pcrit=73.773 bar).

300 310 320 330 340

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c p(k

J/k

g.K

)

ρ(k

g/m

3)

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(a) Density (ρ) and isobaric heat capacity (cp)at P0=80 bar

300 310 320 330 3401E-05

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/m.K

)

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a.s)

Temperature (K)

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300 310 320 330 340

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ρ(k

g/m

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(c) Density (ρ) and isobaric heat capacity (cp)at P0=90 bar

300 310 320 330 3401E-05

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(d) Dynamic viscosity (µ) and thermal conduc-tivity (λ) at P0=90 bar

Figure 1-1: Variation of thermophysical properties based on Span and Wagner (2003a,b) ofcarbon-dioxide (CO2) vs. temperature at P0 = 80 bar and 90 bar. The peak of the heatcapacity at constant pressure indicates the location of the pseudo-critical temperature Tpc. ρ andµ ( ); cp and λ ( ).


1-2 Literature Survey 3

The extreme dependence of fluid properties under supercritical conditions on temperaturemakes the governing equations for flows undergoing heat transfer highly nonlinear and stronglycoupled. The conventional behavior of turbulence is also altered because of this extremeproperty variation, thus making the heat transfer behavior of supercritical fluids peculiarthan most of the subcritical fluids.

1-2 Literature Survey

Research in the field of supercritical heat transfer has been active since the fifties to supportthe thermal design of fossil fueled power plants operating at supercritical pressures. Theinterest in this field regained momentum in the nineties (Pioro et al., 2004b) owing to itspotential to improve the thermal efficiency in modern nuclear plants. Several experimentswere conducted during this period using water or CO2 flowing in heated vertical tubes atsupercritical pressures to collect data for heat transfer distributions and wall temperatures.Most of the experiments were conducted in the turbulent regime with high Reynolds num-bers. Results from these experiments, especially in upward flows, showed peculiar features ofturbulent heat transfer to supercritical fluids such as heat transfer enhancement and deteri-oration (Bae et al., 2005). Peculiar characteristics of heat transfer was often observed whenthe bulk temperature (Tb) was less than the pseudo-critical temperature (Tpc) and the walltemperature (Tw) was higher than the pseudo-critical temperature- Tb < Tpc < Tw. Severalcomprehensive reviews on heat transfer to supercritical fluids have been given by Petukhov(1970); Hall (1971); Jackson and Hall (1979b,a); Polyakov (1991); Pioro et al. (2004b,a);Duffey and Pioro (2005) and more recently by Yoo (2013).

Shitsman (1963) reported the data for heat transfer to supercritical water in upward flowsfor low mass flux flows. He observed localized enhancement of heat transfer for small heatfluxes. With the increase in heat flux, however, the heat transfer progressed towards a dete-riorated condition (highlighted with wall temperature spike). He also reported heat transferrecovery after the deterioration phenomena.

Yamagata et al. (1972) provided data for high mass flux flows with varying heat flux forsupercritical water heat transfer. He observed an enhancement in heat transfer as the bulk en-thalpy approached the pseudo-critical enthalpy. The extent of the enhancement was observedto be reduced with increasing heat flux.

Bishop et al. (1964) reported pressure pulsations when the bulk temperature approachedTpc, after which further investigations were made by several authors using single phase andtwo phase fluid dynamics (Yoo, 2013). The real cause of flow instability at supercriticalpressure is still subject of discussion among researchers and requires further investigation (Liet al., 2009).

Ackerman (1970) investigated heat transfer to water at supercritical pressures flowing insmooth vertical tubes for a wide range of pressure, mass flux, heat flux, and tube diameter. Heobserved that the heat transfer phenomenon was affected by these parameters as well as thebulk temperature. He also reported that sometimes unpredictable heat transfer performancewas observed when Tb < Tpc < Tw. He reported this to be a pseudo boiling process occurringpossibly due to large changes in density across Tpc. The validity of this reasoning is arguedupon by various authors and is not agreed upon (Yoo, 2013).


4 Introduction

The theoretical reasoning that the peculiar heat transfer behavior is indeed because ofbuoyancy and flow acceleration due to thermal expansion, became more acceptable when theexperimental results for upward flows were compared with those for downward flows underotherwise the same conditions by Hall and Jackson (1969); Bourke et al. (1970); Hiroaki et al.(1973); Jackson et al. (2003). It was agreed upon that the buoyancy effect could cause heattransfer deterioration and recovery in upward flow (depending on magnitude of buoyancy),but only enhancement in downward flow. Acceleration (due to thermal expansion) on theother hand could only result in deterioration in both upward and downward flows. Halland Jackson (1969) provided a theoretical explanation on the mechanism of deterioratedheat transfer, suggesting that the dominant factor was the modification of the shear-stressdistribution across the pipe with consequential changes in turbulence production. Hiroakiet al. (1973) studied the effects of buoyancy and flow acceleration owing to thermal expansionon turbulent heat transfer to supercritical fluids in vertical circular tubes. They proved thatthe effects of both buoyancy and acceleration due to thermal expansion in upward flowsoperate quite similarly, resulting in a very rapid decrease of the shear stress near the wall.They also deduced the criteria for the prominent effects of buoyancy and acceleration andproposed a criterion for the reverse transition from turbulent to laminar flow. Jackson and Hall(1979a,b) later provided a general criterion for the onset of the buoyancy effect applicable toboth supercritical and subcritical pressure fluids. After a comparative study of the availableexperimental data, Hall (1971) summarized his findings regarding the peculiarities of heattransfer to supercritical fluids as follows:

• The sharp wall temperature peaks observed in larger pipes (larger diameter) are presentonly in upward flows, whereas the broad peak obtained in the smaller pipe is insensitiveto the flow direction,

• Except for the sharp peaks themselves, the wall temperatures for upward flow are lowerthan those measured in the smaller pipe,

• The wall temperatures for downward flow in larger pipes show no signs of peaks andare considerably lower than those for the smaller pipe.

Later, Kurganov and Kaptilnyi (1993) provided experimental data on flow structure, heattransfer, and hydraulic drag of supercritical (CO2), heated in a vertical tube flowing upwardand downward at high Reynolds numbers. They found that the development of an M-shapedvelocity profile (due to buoyancy) in upward flows favors the recovery of heat transfer dete-rioration.

Recently, several other experimental measurements have been made (for eg: see Kim et al.,2008a; Bae and Kim, 2009; Bae et al., 2010; Song et al., 2008; Kim et al., 2006, 2007; Lichtet al., 2008, 2009), to enable the development of universal heat transfer correlation for super-critical heat transfer.

In addition to the experimental studies mentioned above, computational investigations werealso performed in the past using turbulence models to simulate turbulent heat transfer to su-percritical fluids. Koshizuka et al. (1995) used the Jones and Launder turbulence model(Jones and Launder, 1972) to study heat transfer phenomena in supercritical water. Theresults of this model were compared with experimental data of Yamagata et al. (1972). A


1-2 Literature Survey 5

good qualitative agreement with experimental data was obtained, however quantitative agree-ment was poor specially for deteriorated case. He et al. (2005) compared the performanceof the Launder and Sharma turbulence model (Launder and Sharma, 1974) with their ownexperimental data for mixed convection heat transfer to supercritical CO2; again here thequalitative trends were captured, but for some cases the agreement was poor. Jiang et al.(2008) carried out an experimental and numerical investigation of convective heat transfer forCO2 at supercritical pressures in a 0.27 mm diameter vertical mini-tube. The heating condi-tions were such that buoyancy and flow acceleration had little influence on heat transfer, withno deterioration observed in either of the flow direction. The numerical results correspondedwell with the experimental data using several turbulence models, with the realizable k−ǫ tur-bulence model performing the best. Sharabi et al. (2008) implemented various low Reynoldsnumber models in Fluent for predicting mixed convection heat transfer to supercritical CO2

in non circular flow configurations, such as square and triangular channels. The results werethen compared using the experimental data of Kim et al. (2005). The models were able toreproduce the trend of heat transfer deterioration due to buoyancy, but with a relativelylarge overestimation of measured wall temperatures. Sharabi and Ambrosini (2009) furtheranalyzed the low Reynolds number turbulence model in circular tubes using experimentaldata of Kim et al. (2005) and Yamagata et al. (1972). He et al. (2008b) performed simu-lations for vertical upward and downward flows of supercritical CO2 in a heated tube usingthe v2 − f turbulence model (Behnia et al., 1998) and the AKN model (Abe et al., 1994).The results were compared with experimental data of Fewster (1976). It was found that bothmodels qualitatively showed the general trend of heat transfer deterioration, due to strongbuoyancy influences in upward flow, but significant discrepancies in detailed wall temperaturedistributions were present. Similar studies were done by Wen and Gu (2010) using differentsix low Reynolds number turbulence models and using experimental data of Pis’menny et al.(2006) and Shitsman (1963). Most of the models over-predicted the deterioration and did notreproduce the subsequent recovery of heat transfer.

Bae et al. (2005) investigated turbulent heat transfer to CO2 at supercritical pressure flow-ing in heated vertical tubes using Direct Numerical Simulation (DNS) at a Reynolds numberReb=5400, based on inlet bulk velocity and tube diameter. The temperature range withinthe flow field covered the pseudo-critical region, where significant property variations occur.Both upward and downward flows were considered. The wall temperature distribution showedheat transfer deterioration characterized by the localized peak in upward flows, while no suchanomaly was observed in downward flows. The deterioration occurred at the region whereturbulence was attenuated significantly, and was followed by the enhancement with restora-tion of turbulence with the formation of an M-shaped velocity profile caused by buoyancyeffect. The influence of buoyancy on turbulence statistics was also reported. The results wereconsistent with the observations made by Hall (1971) (also summarized above) and were qual-itatively in good agreement with the experimental data, although direct comparisons werenot made because of the very large difference in the Reynolds numbers. Bae et al. (2008) dida similar DNS study for supercritical CO2 heat transfer in a vertical annulus. DNS studies forheat transfer in air were performed by You et al. (2003), Bae et al. (2006). Bae et al. (2008)also made a comparison with subcritical flows and stated that the deterioration mechanismis not confined to the pseudo-critical region, but is also found at all temperature ranges andeven at subcritical pressures (although the effect is more gradual).

Recently Ničeno and Sharabi (2013) performed Large Eddy Simulation (LES) of turbulent


6 Introduction

heat transfer for supercritical water and compared the results with experimental data fromPis’menny et al. (2006).

He et al. (2008a) assessed the performance of seven low-Reynolds number turbulence modelsin predicting heat transfer to CO2 at supercritical pressures by comparing model predictionswith DNS of Bae et al. (2005). The authors divide the models into two groups based ontheir earlier work for mixed convection for air (Kim et al., 2008b), where they classified themodels as Group 1 and 2. The models whose damping functions are based on variables readilyresponding to buoyancy and flow acceleration were classified as Group 1 and those, whosedamping function do not respond well to buoyancy/flow acceleration effects were classifiedas Group 2. The results showed that the Group 1 models significantly over-predict flowlaminarization and therefore, heat transfer deterioration. However, Group 2 models closelyreproduce the wall temperature variations as exhibited in the DNS for flow laminarization(deterioration), due to modeling error cancellation, but detailed characteristics of the flowand turbulence were not reproduced. However, no model was able to correctly capture therecovery of heat transfer in the strongly buoyancy influenced cases.

From the computational studies reviewed above it is clear that, over quite a range of flowconditions, suitably selected turbulence models can reproduce the general trend of heat trans-fer enhancement and deterioration as found in experiments of supercritical pressure flows.However, the deviation can vary significantly, resulting in over/under prediction of heattransfer deterioration and enhancement. Also, it was seen that a detailed comparison of flowcharacteristics with DNS data was poor and also none of the models were able to correctlycapture the heat transfer recovery.

1-3 Motivation and Scope

Most of the industrial flows involving heat transfer at supercritical conditions are turbulent.Turbulence plays an important role in mixing, thus increasing heat transfer and drag. How-ever the advantage of increased heat transfer surpasses the disadvantage of increased dragin heat exchangers. Turbulent flows at supercritical conditions are extremely complex andtheir physics and modeling requirements are not well understood. The extreme thermophys-ical property variations of fluids close to the critical point alter the conventional behavior ofturbulence, making the governing equations highly nonlinear. In the case of turbulent heattransfer to supercritical fluids these variations can cause heat transfer deterioration and heattransfer enhancement. Although these features are also observed in heat transfer with sub-critical fluids, they are more intense and dynamic for supercritical fluids. With the increasein computing power it has become possible to accurately compute the flow and thermal struc-tures using Direct Numerical Simulation (DNS), however applicability of DNS is still limitedto low Reynolds number flows. In order to compute flows involving high Reynolds numbers,one has to solve the Reynolds Averaged Navier-Stokes (RANS) equations using turbulencemodels. Currently, there is no model available that can accurately predict heat transfer tosupercritical flows, hence analytic and numerical research in this emerging field has attractedattention worldwide. The present work is a step forward to understand the modeling re-quirements for heat transfer in a supercritical fluid. The work can be divided into followingtasks:


1-4 Thesis Outline 7

• Developing a theoretical framework to take into account extreme variations of ther-mophysical properties close to the critical point in the RANS equations and also tounderstand various heat transfer mechanisms.

• Implementing standard turbulence models in a FORTRAN code, which solves the vari-able density low Mach-number approximation of the Navier-Stokes equations on a stag-gered mesh in cylindrical coordinates.

• Validating the turbulence models for canonical cases involving ideal gas (caloricallyperfect) by comparing with data available in literature.

• Testing the performance of various turbulence models for heat transfer to supercriticalCO2, by comparison with the inhouse DNS data. The inhouse DNS was performed byHassan Nemati, PhD candidate at Process and Energy Laboratory of TU Delft.

• Analyzing the data obtained from the inhouse DNS of heat transfer to supercriticalCO2, to highlight additional modeling requirements because of the extreme propertyvariations.

1-4 Thesis Outline

Chapter 2 presents a theoretical background wherein all the governing equations and theirsubsequent simplifications are described. Afterwards the RANS equations are derived us-ing Reynolds/Favre decomposition. The additional terms that are obtained because of highfluctuations of thermophysical properties are highlighted. A short review on the theory ofturbulence modeling is also carried out, after which a brief explanation of various factors thataffect heat transfer behavior is provided.

Chapter 3 presents the flow domain used in the simulation, after which the numerical methodemployed in the implementation of the governing equation is discussed.

Chapter 4 presents the results and discussion using both DNS and RANS data, finallythe report ends with a summary and conclusion in Chapter 5.


8 Introduction


Chapter 2

Theoretical Framework

First, the three dimensional equations (in dimensional and non-dimensional form) governingturbulent flows, which are also implemented in the DNS will be described. Second, theequations will be statistically averaged and simplified to obtain the RANS equations. Third,closure models to account for unknown terms in the RANS equation will be reviewed in theturbulence modeling section. Finally, a discussion on various heat transfer mechanisms andfactors that affect it will be carried out.

2-1 Governing Equations: Direct Numerical Simulation (DNS)

In the DNS study, the low Mach-number approximation (Lele, 1994; Poinsot and Veynante,2005) of the Navier-Stokes equations are solved to simulate a flow undergoing heat transferin a vertical pipe slightly above the supercritical condition. Compared to solving the fullycompressible Navier-Stokes equations, the solution of the low Mach-number Navier-Stokesequations circumvents the severe time step restrictions due to the small time scales of theacoustic waves. Additionally, ignoring compressibility effects and splitting the pressure fieldinto its thermodynamic (P0) and hydrodynamic part (p(x, t)) one can determine all thermo-dynamic state variables, such as density, enthalpy, etc., independently of the hydrodynamicpressure variations as P0 ≫ p(x, t). This means that the heat transfer process considered is athermodynamically constant-pressure process, which is a typical assumption for most of heattransfer problems in low speed incompressible flows (Bae et al., 2005). The viscous dissipationand the gravitational work term are neglected in the energy equation, because their effects aregenerally very small in low speed flows. All the governing equations are written in cylindricalcoordinates, with z the streamwise direction, θ the circumferential direction and r the wallnormal direction. The governing equations in dimensional form can then be written as:

Conservation of mass,∂ρ

∂t+

1

r

∂rρur

∂r+

1

r

∂ρuθ

∂θ+

∂ρuz

∂z= 0. (2-1)


10 Theoretical Framework

Conservation of momentum in r-direction,

∂ρur

∂t+

1

r

∂rρurur

∂r+

1

r

∂ρuruθ

∂θ+

∂ρuruz

∂z− ρuθuθ

r= −∂p

∂r+

1

r

∂rτrr

∂r+

1

r

∂τrθ

∂θ− τθθ

r+

∂τrz

∂z. (2-2)

Conservation of momentum in θ-direction,

∂ρuθ

∂t+

1

r

∂rρuθur

∂r+

1

r

∂ρuθuθ

∂θ+

∂ρuθuz

∂z+

ρuθur

r= −1

r

∂p

∂θ+

1

r2

∂r2τθr

∂r+

1

r

∂τθθ

∂θ+

∂τθz

∂z. (2-3)

Conservation of momentum in z-direction,

∂ρuz

∂t+

1

r

∂rρuzur

∂r+

1

r

∂ρuzuθ

∂θ+

∂ρuzuz

∂z= −∂p

∂z+

1

r

∂rτzr

∂r+

1

r

∂τzθ

∂θ+

∂τzz

∂z+ ρg. (2-4)

Conservation of energy,

∂ρh

∂t+

1

r

∂rρhur

∂r+

1

r

∂ρhuθ

∂θ+

∂ρhuz

∂z=

1

r

∂

∂r(rλ

∂T

∂r) +

1

r

∂

∂θ(λ

r

∂T

∂θ) +

∂

∂z(λ

∂T

∂z), (2-5)

where the diffusion part of the equation can be written in terms of enthalpy using the factthat, dh = cpdT and α = λ

cp. Equation (2-5) then becomes,

∂ρh

∂t+

1

r

∂rρhur

∂r+

1

r

∂ρhuθ

∂θ+

∂ρhuz

∂z=

1

r

∂

∂r(rα

∂h

∂r) +

1

r

∂

∂θ(α

r

∂h

∂θ) +

∂

∂z(α

∂h

∂z). (2-6)

The stress tensors in the momentum equations are given as,

τrr = µ(2∂ur

∂r− 2

3∇ · u), (2-7)

τθθ = µ(2(1

r

∂uθ

∂θ+

ur

r) − 2

3∇ · u), (2-8)

τzz = µ(2∂uz

∂z− 2

3∇ · u), (2-9)

τrθ = τθr = µ(r∂

∂r(uθ

r) +

1

r

∂ur

∂θ), (2-10)

τrz = τzr = µ(∂uz

∂r+

∂ur

∂z), (2-11)

τzθ = τθz = µ(∂uθ

∂z+

1

r

∂uz

∂θ), (2-12)

where,

∇ · u =1

r

∂rur

∂r+

1

r

∂uθ

∂θ+

∂uz

∂z. (2-13)

The temperature and thermophysical properties could be calculated as a function of thermo-dynamic pressure P0 and enthalpy h.


2-1 Governing Equations: DNS 11

Non-dimensional form

The non-dimensional form of the equation is obtained by substituting following dimensionlessvariables in Equation (2-1)–(2-13).

x∗

i =xi

D, t∗ =

t

D/uτ,0, u∗

i =ui

uτ,0, p∗ =

p

ρ0u2τ,0

,

ρ∗ =ρ

ρ0, λ∗ =

λ

λ0, c∗

p =cp

cp0, µ∗ =

µ

µ0,

α∗ =λcp0

λ0cp, h∗ =

h − h0

cp0T0, T ∗ =

T

T0, (2-14)

Q∗ =qwD

λ0T0= q∗Reτ Pr0, q∗ =

qw

ρ0uτ,0cp0T0,

Reτ =ρ0uτ,0D

µ0, P r0 =

µ0cp0

λ0,

1

Fr0=

gD

u2τ,0

=Gr0

β0T0Re2τ Q∗

, Gr0 =ρ2

0gβ0qwD4

µ20λ0

,

where, the subscript 0 refers to the inlet condition, uτ,0 is the friction velocity at the inlet,qw is the constant wall heat flux, Reτ is the friction velocity Reynolds number, Pr0 is thePrandtl number, Fr0 is the Froude number, Gr0 is the Grashof number, and D is the pipediameter. Note, that the non-dimensionalization could also be done using a bulk velocityReynolds number Reb and its implementation is discussed in Section 4-1.

On substituting the non-dimensional variable and dropping the * sign for brevity, the non-dimensional governing equations becomes:


∂t+

1

r

∂rρur

∂r+

1

r

∂ρuθ

∂θ+

∂ρuz

∂z= 0. (2-15)


∂ρur

∂t+

1

r

∂rρurur

∂r+

1

r

∂ρuruθ

∂θ+

∂ρuruz

∂z− ρuθuθ

r= −∂p

∂r+

1

r

∂rτrr

∂r+

1

r

∂τrθ

∂θ− τθθ

r+

∂τrz

∂z. (2-16)

Conservation of momentum in θ-direction,

∂ρuθ

∂t+

1

r

∂rρuθur

∂r+

1

r

∂ρuθuθ

∂θ+

∂ρuθuz

∂z+

ρuθur

r= −1

r

∂p

∂θ+

1

r2

∂r2τθr

∂r+

1

r

∂τθθ

∂θ+

∂τθz

∂z. (2-17)


∂ρuz

∂t+

1

r

∂rρuzur

∂r+

1

r

∂ρuzuθ

∂θ+

∂ρuzuz

∂z= −∂p

∂z+

1

r

∂rτzr

∂r+

1

r

∂τzθ

∂θ+

∂τzz

∂z+

1

Fr0ρ. (2-18)


∂ρh

∂t+

1

r

∂rρhur

∂r+

1

r

∂ρhuθ

∂θ+

∂ρhuz

∂z=

1

Reτ Pr0

(1

r

∂

∂r(rα

∂h

∂r) +

1

r

∂

∂θ(α

r

∂h

∂θ) +

∂

∂z(α

∂h

∂z)

).

(2-19)




τrr =1

Reτµ(2

∂ur

∂r− 2

3∇ · u), (2-20)

τθθ =1

Reτµ(2(

1

r

∂uθ

∂θ+

ur

r) − 2

3∇ · u), (2-21)

τzz =1

Reτµ(2

∂uz

∂z− 2

3∇ · u), (2-22)

τrθ = τθr =1

Reτµ(r

∂

∂r(uθ

r) +

1

r

∂ur

∂θ), (2-23)

τrz = τzr =1

Reτµ(

∂uz

∂r+

∂ur

∂z), (2-24)

τzθ = τθz =1

Reτµ(

∂uθ

∂z+

1

r

∂uz

∂θ), (2-25)

where,

∇ · u =1

r

∂rur

∂r+

1

r

∂uθ

∂θ+

∂uz

∂z. (2-26)

In the momentum equation, 1/Fr0 takes minus sign for upward and plus sign for downwardflow. The Froude number (Fr0), which is a function of Gr0, Reτ , Q∗ and inlet condition, isa representative of buoyancy effect. So given a Reτ , Q∗ and inlet condition, one can vary thetube diameter as a parameter to vary the Grashof number and hence the buoyancy effect.

2-2 Governing Equations: Reynolds Averaged Navier-Stokes (RANS)

Since the pipe is vertically placed and heated symmetrically, the turbulent flow will be sta-tistically homogeneous in the θ direction. The governing equations for the RANS are derivedby taking a statistical average over time and over homogeneous circumferential direction ofEquation (2-15)–(2-26). Two types of averaging, namely Reynolds averaging and Favre aver-aging are applied. Velocity components and enthalpy are Favre averaged, whereas pressureand thermophysical properties are Reynolds averaged.

Reynolds averaging

Under Reynolds averaging a variable γ is decomposed into a mean part γ and a fluctuatingpart γ′, such that,

γ = γ + γ′, (2-27)

where,γ′ = 0, (2-28)

γ =1

NtNθ

Nt∑

1

Nθ∑

1

γ. (2-29)

Nθ is the number of samples in circumferential direction and Nt is the number of samplestaken in time.


2-2 Governing Equations: RANS 13

Favre averaging

In cases of variable density flow, density-weighted averaging is often used to account forstrong density fluctuations, while retaining the relatively simple form of equations obtainedthrough the conventional Reynolds averaging approach. A variable γ is decomposed into itsdensity-weighted average component γ and fluctuating part γ′′, such that,

γ = γ + γ′′, (2-30)

where,

γ =ργ

ρ, (2-31)

γ′′ 6= 0, (2-32)

but,

ργ′′ = 0. (2-33)

Although both Favre and Reynolds decomposition are used, the equation obtained willbe referred to as Reynolds Averaged Navier-Stokes equation. For a statistically steady andhomogeneous (in θ direction) turbulent flow, the averaged governing equations become twodimensional and the averaged velocity and derivative in θ direction disappear, as,

uθ = 0, (2-34)

∂..

∂θ= 0. (2-35)

Decomposing the variables into their respective averaged and fluctuating parts, one gets

ρ = ρ + ρ′, p = p + p′, µ = µ + µ′, α = α + α′, ui = ui + u′′

i , h = h + h′′.(2-36)

Substituting Equation (2-36) in Equation (2-15)–(2-26), applying Reynolds averaging andsimplifying using Equation (2-28),(2-33),(2-34) and (2-35), gives the RANS equations as:


∂t+

1

r

∂rρur

∂r+

∂ρuz

∂z= 0. (2-37)


∂ρur

∂t+

1

r

∂rρurur

∂r+

∂ρuruz

∂z= −∂p

∂r+

1

r

∂rτrr

∂r− τθθ

r+

∂τrz

∂z− 1

r

∂rρu′′

ru′′

r

∂r− ∂ρu′′

r u′′

z

∂z+

ρu′′

θu′′

θ

r.

(2-38)Conservation of momentum in z-direction,

∂ρuz

∂t+

1

r

∂rρuzur

∂r+

∂ρuzuz

∂z= −∂p

∂z+

1

r

∂rτzr

∂r+

∂τzz

∂z+

1

Fr0ρ − 1

r

∂rρu′′

zu′′

r

∂r− ∂ρu′′

zu′′

z

∂z.

(2-39)




∂ρh

∂t+

1

r

∂rρhur

∂r+

∂ρhuz

∂z=

1

Reτ Pr0

(1

r

∂

∂r(rα

∂h

∂r) +

∂

∂z(α

∂h

∂z) +

1

r

∂

∂r(rα

∂h′′

∂r) +

∂

∂z(α

∂h′′

∂z)

+1

r

∂

∂r(rα′

∂h′′

∂r) +

∂

∂z(α′

∂h′′

∂z)

)− 1

r

∂rρh′′u′′

r

∂r− ∂ρh′′u′′

z

∂z.

(2-40)


τrr =2

Reτ(µ

∂ur

∂r+ µ′

∂u′′

r

∂r+ µ

∂u′′

r

∂r− 1

3µ∇ · u − 1

3µ∇ · u′′ − 1

3µ′∇ · u′′), (2-41)

τθθ =2

Reτ(µ

ur

r+ µ

u′′

r

r+ µ′

u′′

r

r− 1

3µ∇ · u − 1

3µ∇ · u′′ − 1

3µ′∇ · u′′), (2-42)

τzz =2

Reτ(µ

∂uz

∂z+ µ′

∂u′′

z

∂z+ µ

∂u′′

z

∂z− 1

3µ∇ · u − 1

3µ∇ · u′′ − 1

3µ′∇ · u′′), (2-43)

τrz = τzr =1

Reτ(µ

∂uz

∂r+ µ′

∂u′′

z

∂r+ µ

∂u′′

z

∂r+ µ

∂ur

∂z+ µ′

∂u′′

r

∂z+ µ

∂u′′

r

∂z), (2-44)

where,

∇ · u =1

r

∂rur

∂r+

∂uz

∂z, (2-45)

∇ · u′′ =1

r

∂ru′′

r

∂r+

∂u′′

z

∂z. (2-46)

Note, in most of DNS studies involving variable density, it is common to report turbulentstatistics of diffusion terms (which don’t have density), in form of Reynolds averaged velocityand enthalpy. The Reynolds decomposition, given as,

ui = ui + u′

i, h = h + h′,

is then substituted in the diffusion terms. By doing so, the number of terms reduces signifi-cantly because u′

i, h′, terms drop out (Note: terms of the form µ′u′

i, α′h′ are still present). Thisis not the case when Favre decomposition is used, where u′′

i , h′′ terms are retained because ofthe fact that,

γ′′ 6= 0, ργ′′ = 0.

The RANS equations are solved for ui and h due to which using Reynolds decompositionof ui and h in the diffusion term results in additional unknowns in the form of ui and h.Therefore, from a RANS modeling perspective employing Reynolds decomposition in thediffusion term is not appropriate.


2-3 Turbulence Modeling 15

2-3 Turbulence Modeling

The averaged equations given by Equation (2-37)–(2-46) result in more unknowns (introducedbecause of the nonlinearity of the equations) than equations. In order to account for theseadditional unknowns, a turbulence model is required to close the problem.

15 additional unknowns are generated, which are:

ρu′′

r u′′

r , ρu′′

θu′′

θ , ρu′′

zu′′

z , ρu′′

r u′′

z , ρu′′

r h′′, ρu′′

zh′′,

u′′

r , u′′

z , h′′, µ′∂u′′

r

∂r, µ′

∂u′′

z

∂z, µ′

∂u′′

z

∂r, µ′

∂u′′

r

∂z, α′

∂h′′

∂r, α′

∂h′′

∂z. (2-47)

Note that for subcritical flows with small fluctuations in density and transport propertiesonly the following terms are modeled:

ρu′′

r u′′

r , ρu′′

θu′′

θ , ρu′′

zu′′

z , ρu′′

r u′′

z , ρu′′

r h′′, ρu′′

zh′′. (2-48)

A short discussion on the modeling of this terms is given below, the relevance of the ignoredterms for supercritical heat transfer, will be discussed in Chapter 4. In Chapter 4 it willalso be shown that, apart from the unknown closure terms given in Equation (2-47), theaveraged thermophysical properties specifically cp and α, also become unknown in the verylarge property variation region because for a thermophysical property φ,

φ 6= φ(h, P0). (2-49)

Turbulence models can be categorized into three categories:

• Turbulent/Eddy viscosity models: The turbulent viscosity hypothesis was introduced byBoussinesq in 1877. According to this hypothesis the Reynolds stresses (ρu′′

r u′′

r , ρu′′

θu′′

θ ,ρu′′

zu′′

z , ρu′′

r u′′

z) can be be related to the strain rate of mean flow using a proportionalitycoefficient µt, also known as turbulent viscosity. The different Reynolds stresses thenbecome,

ρu′′

ru′′

r = −µt(2∂ur

∂r− 2

3∇ · u) +

2

3ρk, (2-50)

ρu′′

θu′′

θ = −µt(2ur

r− 2

3∇ · u) +

2

3ρk, (2-51)

ρu′′

zu′′

z = −µt(2∂uz

∂z− 2

3∇ · u) +

2

3ρk, (2-52)

ρu′′

r u′′

z = −µt(∂ur

∂z+

∂uz

∂r), (2-53)

where k (= 12ρ

(ρu′′

r u′′

r + ρu′′

θu′′

θ + ρu′′

zu′′

z)) is the turbulent kinetic energy.

Depending on the number of equations used to calculate µt, the turbulent viscositymodels are further divided into:

Zero equation model/Algebraic models- These models do not require any addi-tional differential equations, and calculate µt directly from an algebraic relation. Thesemodels are too simple and are often not able to properly capture the turbulence physics.



One equation model- They solve one turbulent transport equation with the helpof which µt is represented. The use of these models is very popular in external flowapplications and is used for aerospace applications.

Two equation model- Two of the most commonly used two equation models are thek − ǫ and the k − ω model. Two equation models require the solution of two extratransport equations to evaluate µt, hence providing additional detail of the turbulentflow field. Most often, one of the transported variables is the turbulent kinetic energy(k) . The second transported variable is either the turbulent dissipation (ǫ) (for the k−ǫmodel) or the specific dissipation (ω) (for the k − ω model). These models are furtherclassified based on how they capture wall effects, as High Reynolds Number (HRN)turbulence model and Low Reynolds Number (LRN) turbulence model. HRN modelsmake use of wall functions to bridge the region between the wall and the first meshcell with empirical profiles for velocity and turbulent quantities. It is computationallycheap as it does not require a fine mesh close to the wall in order to resolve the stronggradients. This approach works well for simple flows, however for complex flows itperforms poorly. LRN models on the other hand require to resolve the computationalboundary layer. They make use of damping functions which depend on wall distanceand local flow properties. The term low in LRN models refers to the low value of the

turbulent Reynolds number (Ret = Reρkµǫ

) close to the wall.

v2− f model- This model is an extension of k − ǫ model. It solves two additional

equations, hence making the total number of equations to be solved as four. It doesnot need to make use of wall damping functions because it is valid up to solid walls andis able to correct for turbulence damping due to viscous and pressure effects with thehelp of the additional two equations. The additional variables that are solved for are

v2, which is the wall normal stress (ρu′′

r u′′

r

ρ), and an elliptic relaxation function f . The

anisotropic wall effects are modeled through f , by solving a separate elliptic equationof the Helmholtz type. f along with k enters as a source term in the v2 equation, henceproviding the right scaling for turbulence damping. µt is then calculated as a functionof v2. This model is computationally expensive, but is superior in performance to mostof LRN models.

• Reynolds stress models- These models solve transport equations for the Reynolds stressestogether with a transport equation for the turbulent length scale, which usually is ǫ.The exact transport equations of the Reynolds stresses are given in Appendix A. Theseequations have their own closure problem and require additional modeling.

• Algebraic stress models- These models simplify the stress transport equations to providealgebraic expressions for the Reynolds stresses. These models try to overcome the severecomputational requirement as compared to the Reynolds stress transport models andare in general also more stable to solve.

For a more detailed review on turbulence models see Wilcox (1998) and Davidson (2003).

In the present study, two different LRN k − ǫ model and the v2 − f model (V2F) areused to get the Reynolds stress terms. The use of these class of models is prevalent inengineering community and therefore their applicability in supercritical heat transfer needsfurther investigation. The two LRN models considered here are Launder and Sharma (1974),



(LS) and Myong and Kasagi (1990), (MK). These models represents two different groups interms of their formulation of wall damping functions fµ (Kim et al., 2008b). The dampingfunction fµ in the LS model is based on the turbulent Reynolds number Ret, whereas in theMK model fµ is a strong function of the non-dimensional wall distance y+.

The turbulent heat flux term is commonly modeled using a simple gradient diffusion hy-pothesis, which uses the turbulent viscosity, the turbulent Prandtl number (Prt) and themean enthalpy gradient as,

ρu′′

zh′′ =µt

Prt

∂h

∂z, (2-54)

ρu′′

rh′′ =µt

Prt

∂h

∂r. (2-55)

The above models imply that the turbulent heat flux is directly correlated with turbulentshear stress, which is a valid assumption for ideal gases (Kim et al., 2008b). However, forsupercritical flows this is not necessarily true.

The turbulent heat flux can also be modeled by solving its corresponding transport equation,by using an algebraic form of these transport equations, or by using an analogous eddydiffusivity model of kθ-ǫθ type. These models are not often used and has been investigatedfor ideal gas heat transfer in the past. For more details see Hanjalic (2002), Kenjereš et al.(2005), Kenjereš and Hanjalić (1995), Dehoux et al. (2012), Abe et al. (1995). A preliminaryimplementation of these models in the present work was done, but the results indicated apoor agreement with DNS. Since the focus of this work is to point out additional modelingrequirements, the results obtained from these models are not reported and simple gradientdiffusion hypothesis is used for discussion of the results.

The final RANS equations are obtained by retaining only the conventional terms (ρu′′

ru′′

r ,ρu′′

θu′′

θ , ρu′′

zu′′

z , ρu′′

ru′′

z , ρu′′

rh′′, ρu′′

zh′′) in Equation (2-37)–(2-46) and by using Equation(2-50)–(2-53) for Reynolds stresses and Equation (2-54) and (2-55) for turbulent heat fluxes as,


∂ρur

∂t+

1

r

∂rρurur

∂r+

∂ρuruz

∂z= − ∂pm

∂r+

2

r

∂

∂r[r(

µ

Re+ µt)

∂ur

∂r] +

∂

∂z[(

µ

Re+ µt)(

∂uz

∂r+

∂ur

∂z)]

− 2

3

1

r

∂

∂r[r(

µ

Re+ µt)∇ · u] − 2(

µ

Re+ µt)

ur

r2+

2

3(

µ

Re+ µt)∇ · u.

(2-56)


∂ρuz

∂t+

1

r

∂rρuzur

∂r+

∂ρuzuz

∂z= − ∂pm

∂z+ 2

∂

∂z[(

µ

Re+ µt)

∂uz

∂z]

+1

r

∂

∂r[r(

µ

Re+ µt)(

∂uz

∂r+

∂ur

∂z)] +

1

Fr0ρ − 2

3

∂

∂z(

µ

Re+ µt)∇ · u.

(2-57)


∂ρh

∂t+

1

r

∂rρhur

∂r+

∂ρhuz

∂z=

1

r

∂

∂r[r(

α

RePr0+

µt

Prt)∂h

∂r] +

∂

∂z[(

α

RePr0+

µt

Prt)∂h

∂z]. (2-58)



In above equations, Reynolds number Re, could be Reτ or Reb depending on the choice ofnon-dimensionalizing velocity. In momentum equations, pm is the modified pressure and canbe written as,

pm = p +2

3ρk.

The exact equation for turbulent kinetic energy can be obtained by adding Equation (A-1)–(A-3) (in Appendix A) and dividing by two. Following Lele (1992) and Schilling et al. (2006),the equation obtained can then be written using tensorial notations (for brevity) as:

∂ρk

∂t+ C = Gk + Pk + DF + DS + π, (2-59)

where,

C =∂ujρk

∂xj, Gk = −u′′

j (∂p

∂xj− ∂τij

∂xi), Pk = −ρu′′

i u′′

j

∂ui

∂xj, (2-60)

DF = − ∂

∂xj(ρk′′u′′

j + p′u′′

j − u′′

i τ ′′

ij), DS = −τ ′′

ij

∂u′′

i

∂xj, π = p′

∂u′′

j

∂xj,

with,

τij = (µ + µ′)(∂ui

∂xj+

∂uj

∂xi), τ ′′

ij = (µ + µ′)(∂u′′

i

∂xj+

∂u′′

j

∂xi).

The terms are: C the convective transport, Gk the buoyancy production, Pk the shear pro-duction, DF the turbulent diffusion, DS the dissipation of turbulent kinetic energy and π thepressure strain term.

DF is modeled using a gradient diffusion model, π is 0 for flows with constant density and isneglected in the present study because of no available models; its significance will be studiedin future. The buoyancy production term can be simplified as:

Gk ≈ −u′′

z(∂p

∂z− ∂ ˜τrz

∂r),

−(∂p

∂z− ∂ ˜τrz

∂r) ≈ − 1

Fr0ρ,

Gk ≈ − 1

Fr0ρu′′

z ,

u′′

z = −ρ′u′

z

ρ,

Gk ≈ 1

Fr0ρ′u′

z. (2-61)

Buoyancy production can then be simplified by relating the density fluctuations with tem-perature fluctuations as,

ρ′u′

z ≈ −βρT ′u′

z, (2-62)

where β = −1ρ

∂ρ∂T

. Equation (2-62) then becomes,

Gk ≈ −βρT ′u′

z

1

Fr0. (2-63)



Modeling of Gk is done using generalized gradient diffusion hypothesis, following Ince andLaunder (1995),

ρT ′u′

z = −cTk

ǫ(ρu′′

r u′′

z

∂T

∂r+ ρu′′

zu′′

z

∂T

∂z).

Substitution in Equation (2-63) gives,

Gk ≈ 1

Fr0βcT

k

ǫ(ρu′′

r u′′

z

∂T

∂r+ ρu′′

zu′′

z

∂T

∂z), (2-64)

where, cT = 0.3. The modeled form of k can then be written as,

∂ρk

∂t+

1

r

∂rρkur

∂r+

∂ρkuz

∂z=

1

r

∂

∂r[r(

µ

Re+

µt

σk

)∂k

∂r]+

∂

∂z[(

µ

Re+

µt

σk

)∂k

∂z]+Pk +Gk −ρ(ǫ+D).

(2-65)

The equation for dissipation is modeled on a similar ground as k and is given as,

∂ρǫ

∂t+

1

r

∂rρǫur

∂r+

∂ρǫuz

∂z=

1

r

∂

∂r[r(

µ

Re+

µt

σǫ)∂ǫ

∂r] +

∂

∂z[(

µ

Re+

µt

σǫ)∂ǫ

∂z] + Cǫ1

f11

TtPk

+ Cǫ1f1

1

TtGk − Cǫ2

f2ρǫ

Tt+ ρE. (2-66)

The turbulent viscosity is then calculated as,

µt = ρCµfµk2

ǫ. (2-67)

In Equation (2-65)–(2-67) Cµ, Cǫ1, Cǫ2

, σk, σǫ are empirical constants, which are also usedin high Reynolds number turbulence models. fµ, f1, f2, D and E are parameters used toincorporate wall effects; the values of all these parameters depend on the turbulence modeland are summarized along with the wall boundary condition in Table 2-1–2-3. The turbulenttime scale Tt is given as,

Tt =k

ǫ.

The shear production and the buoyancy production are obtained by substituting Equation(2-50)–(2-53) into Equation (2-60) and (2-64), respectively. The expression obtained in termsof turbulent viscosity then becomes,

Pk = µt[2((∂ur

∂r)2 + (

∂uz

∂z)2 + (

ur

r)2) + (

∂uz

∂r+

∂ur

∂z)2] − 2

3[ρk + µt∇ · u]∇ · u, (2-68)

Gk = −cT β1

Fr0

k

ǫ[µt(

∂uz

∂r+

∂ur

∂z)∂T

∂r+ (2µt

∂uz

∂z− 2

3ρk − 2

3µt∇ · u)

∂T

∂z]. (2-69)

The additional two equations used for the v2 − f model (Behnia et al., 1998) are given as,

∂ρv2

∂t+

1

r

∂rρv2ur

∂r+

∂ρv2uz

∂z=

1

r

∂

∂r[r(

µ

Re+

µt

σk)∂v2

∂r]+

∂

∂z[(

µ

Re+

µt

σk)∂v2

∂z]+ρkf −6ρv2

ǫ

k,

(2-70)



0 =1

r

∂

∂r(r

∂f

∂r) +

∂

∂z(∂f

∂z) − f

L2t

+(C1 − 1)

L2t

(23 − v2

k)

Tt+

C2

L2t

Pk

ρk+

1

L2t

5v2

kTt

, (2-71)

where C1 = 1.4, C2 = 0.3. The turbulent viscosity for the v2 − f model is given as,

µt = ρCµv2Tt. (2-72)

The time and length scale are given as,

Tt = max[k

ǫ, 6

√µ

Reρǫ],

Lt = CLmax[k

3

2

ǫ, Cη(

( µReρ

)3

ǫ)

1

4 ],

where CL = 0.23 and Cη = 70.

Table 2-1: Constants in turbulence models

Model Code Cµ Cǫ1 Cǫ2 σk σǫ Prt

Myoung-Kasagi (1990) MK 0.09 1.40 1.80 1.4 1.3 0.9Launder-Sharma (1974) LS 0.09 1.44 1.92 1.0 1.3 0.9

v2 − f (1998) V2F 0.22 1.40 1.9 1.0 1.3 0.9

Table 2-2: Functions in turbulence models

Code fµ f1 f2

MK [1 − exp(−y+

70 )](1 + 3.45Re0.5

t

) 1.0 [1 − 29exp(−(Ret

6 )2)](1 − exp(−y+

5 ))2

LS exp[ −3.4

(1+Ret50

)2] 1.0 1 − 0.3exp(−Re2

t )

V2F v2Tt/(k2/ǫ) 1 + 0.045

√k

v21.0

Table 2-3: D and E terms and wall boundary conditions

Code D E Wall Boundary Condition

MK 0 0 kw = 0, ǫw = µReρ

∂2k∂y2

LS 2 µReρ

(∂√

k∂y

+ ∂√

k∂z

)2 2 µReρ

µt

ρ[(∂2ur

∂z2 )2 + (∂2uz

∂y2 )2] kw = 0, ǫw = 0

V2F 0 0 kw = 0, ǫw = µReρ

∂2k∂y2 , v2

w = 0, fw = 0

The different parameters in Table 2-1–2-3 are defined as,

Ret = Reρk2

µǫ, y+ = uτ,0 y Re, y = 0.5 − r, uτ,0 =

√−1

Re

(∂uz,0

∂r

)

w

.


2-4 Heat Transfer Mechanisms 21

2-4 Heat Transfer Mechanisms

The objective of this section is to familiarize the reader with different heat transfer termi-nology used in this report and also to provide a brief understanding of various heat transfermechanisms. The common mode of heat transfer in a heat exchanger of a power cycle isconvective heat transfer. Convective heat transfer is a transport mechanism made possiblethrough fluid motion. It involves the combined processes of conduction (heat diffusion) andadvection (heat transfer by bulk fluid flow). Based on the source of fluid motion, convectiveheat transfer can be classified as:

• Forced convection: When the flow is solely driven by external sources (such as a pump, acompressor etc.) and the effect of buoyancy is small, then it is called forced convection.The effect of buoyancy can be considered low for small pipe diameters and heat fluxes.The importance of buoyancy is quantified using non-dimensional numbers such as theGrashof number and the Froude number, as described in Section 2-1.

• Natural/Free Convection: When flow is induced solely by buoyancy forces, resultingfrom the density variations, then it is called natural/free convection. This class ofconvection is not relevant in this work and will therefore not be discussed further.

• Mixed Convection: Mixed (combined) convection is a combination of both forced andfree convection. The flow is governed simultaneously by external pressure gradients andby buoyancy effects arising from density gradients in the flow field.

Because effects of buoyancy depend on flow direction, with respect to gravity, mixed convec-tion can be further classified as:

• Upward flow (buoyancy aided flow)

• Downward flow (buoyancy opposed flow)

• Horizontal flow

Depending on the Reynolds number the flow can be:

• Laminar

• Turbulent

• Transitional

and based on the thermodynamic operating point:

• Subcritical

• Supercritical (present study)



For the sake of brevity, the discussion will be limited to heating processes, i.e. cases whereheat is supplied to the fluid. Also, in case of mixed convection, only vertical flows will beconsidered, as those are the once of practical interest.

In a heated flow the fluid density decreases, whereby the amount of the decrease depends onthe fluid. For instance, in a subcritical state density changes are small and gradual, whereasin a supercritical fluid close to Tpc the changes in density are sharp and large. The decrease indensity causes two effects, namely, buoyancy and flow acceleration due to thermal expansion.Buoyancy occurs because of non uniform gravity forces acting on a fluid element with variabledensities. Flow acceleration due to thermal expansion occurs because of mass conservation,whereby the bulk flow velocity increases with decrease in bulk density. Furthermore, the vari-ation of other thermophysical properties, such as cp, also affect the heat transfer mechanism.These three effects are discussed below. Although, the main focus will be on heat transferto supercritical turbulent flows, attention will also be drawn to similar effects in laminar andsubcritical flows wherever necessary.

2-4-1 Influence of variation in specific heat (cp), thermal conductivity (λ) andviscosity (µ)

Strong variations of thermophysical properties, especially the peak in specific heat (cp) closeto the pseudo-critical point, have a strong affect on heat transfer mechanisms. In fact, theenhancement in heat transfer observed by Yamagata et al. (1972) is a result of a sharp increasein cp. This effect can only be observed when buoyancy and flow acceleration effects are small,i.e for low heat heat flux and high mass flux cases. The effect of property variations hasbeen included in heat transfer correlations. Using the empirically obtained Dittus-Boeltercorrelation (Bergman et al., 2011), which is commonly used for turbulent forced convectionapplication, the Nusselt number (Nu) is given as,

Nu = 0.023Re0.8b Pr0.4

b . (2-73)

The heat transfer coefficient (htc) can then be calculated as a function of thermophysicalproperties as,

htc = (0.028m0.8

D1.8)(λ0.6

b µ−0.4b )(c0.4

p,b), (2-74)

where htc = Nuλb

D, m = ρbubπ

D2

4 and the subscript b represents the bulk quantities. Note,htc can also be written in terms of wall heat flux (qw), wall temperature (Tw) and bulktemperature (Tb) as,

htc =qw

Tw − Tb. (2-75)

For a constant heat and mass flux, the downstream increase in Tb is fixed in order to satisfy theglobal energy balance. Hence, the wall temperature gives a good indication of the heat transfercharacteristic. A higher wall temperature corresponds to a poor heat transfer coefficient,whereas a lower wall temperature indicates good heat transfer.

Equation (2-74) can be used to analyze heat transfer to CO2 at P0 = 80 bar. In Figure 1-1it can be seen that cp,b increases towards its peak value and that λb and µb decrease, as theenthalpy approaches the pseudo-critical enthalpy (hpc = 345.87kJ/kg). Therefore, Equation



300 310 320 330 340 350 360 370 380 390 4005

10

15

20

25

htc

[kW

/m2K

]

Hb

Figure 2-1: Heat transfer coefficient of supercritical CO2 at P0=80 bar using Dittus-Boeltercorrelation.

(2-74) states that an increase in cp,b and decrease in µb causes htc to increase, whereas adecrease in λb causes htc to reduce. Figure 2-1 shows the dependence of htc on these threeparameters, where the effect of cp,b seems dominant. htc reaches a peak when the bulkenthalpy (Hb) approaches the pseudo-critical enthalpy.

The values of htc obtained in Yamagata’s experiment using supercritical water were actuallyhigher than those given by the Dittus-Boelter correlation for very low heat fluxes. However,the trend in heat transfer behavior was similar. A summary of Yamagata’s results are providedby Licht et al. (2008) and is shown in Figure 2-2. It can be seen that the heat transferenhancement reduces with increasing heat flux.

The reason for the enhanced heat transfer can now be explained physically using Figure 2-3.At low heat fluxes the energy input is not large enough to overcome the large values of specificheat close to the wall. Thus, a low wall to bulk temperature gradient is obtained. As theheat flux increases the large values of specific heat at the wall cannot sustain the energyinput, therefore the high cp region shifts inward to enable a higher gradient of temperatureto support the incoming flux. This results in a higher wall to bulk temperature gradientthus impairing the heat transfer (Licht et al., 2008)). On further increasing the heat flux,effects due to buoyancy and flow acceleration become important and the influence of propertyvariations cannot be carried out on a standalone basis.

The dependence of heat transfer on λb is straightforward, as a lower conducting fluid willhave bad heat transfer property. A decrease in µb will reduce the viscous action on turbulence,hence promoting better heat transfer. Heat transfer enhancement effects due to increase incp,b can also be observed for laminar flows (Peeters et al., 2013).

2-4-2 Influence of flow acceleration

Flow acceleration effects occur due to thermal expansion, and in upward flows also due tobuoyancy. However, the net acceleration of buoyancy on the total cross-section is zero, whereas



Figure 2-2: Yamagata’s data exhibiting heat transfer enhancement, mass flux = 1400 kg/m2s(Adapted from Licht et al. (2008)).

Figure 2-3: Heat transfer enhancement description (Adapted and modified from Licht et al.(2008)).

thermal expansion causes net acceleration as the the bulk velocity along the axial directionincreases. In general, buoyancy is a result of radial gradients in density, and flow acceleration,due to thermal expansion, is a result of axial gradients in density (Li et al., 2009). In upwardflows, buoyancy aids the flow in the sense that due to density decrease the gravity forceis lower close to the wall. This increases the velocity close to the wall, causing local flowacceleration. In downward flows, buoyancy opposes the flow, because the lower gravity forceis in flow direction, due to which the flow close to the wall decelerates as compared to thecore. Buoyancy causing local flow acceleration/deceleration (depending on flow direction) isalso sometimes referred to as external or indirect effect, however it also has another role calledstructural or direct effect (Petukhov et al., 1988); it is discussed in detail in Section 2-4-3.

In laminar convection, flow acceleration close to the wall increases the energy transport,due to advection, and thus enhances heat transfer. Similarly, flow deceleration results in lowerenergy transport thus deteriorating the heat transfer. Therefore, in forced convection flows theheat transfer is enhanced. For upward mixed convection, heat transfer is further enhancedbecause of additional acceleration due to buoyancy. In downward flows, heat transfer isdeteriorated when deceleration due to buoyancy is larger than acceleration due to thermalexpansion.



In turbulent convection the effects are opposite. Although, flow acceleration increase thevelocity close to the wall, it also reduces turbulence production. Since, turbulent mixing dom-inates the heat transfer mechanism, the heat transfer effectiveness decreases. In downwardflows with buoyancy, turbulence production increases, when deceleration due to buoyancy islarger than acceleration due to thermal expansion, thereby enhancing heat transfer.

This decrease in turbulence production with flow acceleration can be explained by lookingat the production term Pk of turbulent kinetic energy k,

Pk = −ρu′′

ru′′

r

∂ur

∂r− ρu′′

r u′′

z

∂ur

∂z− urρu′′

θu′′

θ

r− ρu′′

ru′′

z

∂uz

∂r− ρu′′

zu′′

z

∂uz

∂z. (2-76)

For flows without flow acceleration (constant density flows), the only significant term inEquation (2-76) is −ρu′′

ru′′

z∂uz

∂r. Similarly, the production for ρu′′

r u′′

z is given as,

Pρu′′

r u′′

z= −ρu′′

ru′′

r

∂uz

∂r− ρu′′

zu′′

z

∂ur

∂z− ρu′′

zu′′

r

∂ur

∂r− ρu′′

r u′′

z

∂uz

∂z. (2-77)

Here, the most significant term is −ρu′′

ru′′

r∂uz

∂r.

However, for flows with flow acceleration, the term −ρu′′

zu′′

z∂uz

∂zin Equation (2-76) and

term −ρu′′

ru′′

z∂uz

∂zin Equation (2-77) also become important. Both the terms give a negative

contribution to the production term (since ∂uz

∂z> 0 ), hence reducing the overall turbulence

level. In addition, the significant terms −ρu′′

ru′′

z∂uz

∂rand −ρu′′

ru′′

r∂uz

∂rreduce. The velocity

gradient ∂uz

∂rin the viscous dominant region increases, where it has a small influence on the

turbulence production. On the other hand, the velocity gradient decreases further away fromthe wall, thus decreasing the turbulence production. Because of the coupled nature of theReynolds stress equations, the decrease in one of the Reynolds stress affect the productionof other Reynolds stress further, until an equilibrium is obtained. Similar observations weremade by Kline et al. (1967), where for accelerating turbulent boundary layers, they observedreduction in bursting frequency which is the primary mechanism for turbulence production.

Note that the turbulent kinetic energy production term in an eddy viscosity turbulencemodel is given as,

Pk = µt[2((∂ur

∂r)2 + (

∂uz

∂z)2 + (

ur

r)2) + (

∂uz

∂r+

∂ur

∂z)2] − 2

3[ρk + µt∇ · u]∇ · u. (2-78)

The acceleration effect, ∂uz

∂z> 0 in Equation (2-78) actually results in a positive contribution

to the turbulent kinetic energy production; however, the turbulence models still manage tocapture the decrease in turbulence with flow acceleration. The reason for this is that thevelocity gradient ∂uz

∂rreduces further away from the wall, where µt is high, hence decreasing

the turbulent production.

Note, that an important distinction for heat transfer in an ideal gas and a supercritical fluidcould be made, using flow acceleration due to thermal expansion. In forced convection withideal gases flow acceleration due to thermal expansion is ignored (Boussinesq approximation(Kim et al., 2008b)). This decouples the momentum equations with the energy equation. Inother words the turbulence is not modified. In supercritical flows, however, large and sharpchanges in density make flow acceleration (due to thermal expansion) important, and resultin a decrease of turbulence.



2-4-3 Influence of buoyancy

The effect of buoyancy plays an interesting role in terms of heat transfer recovery (afterdeterioration) for turbulent flows and usually dominates the flow behavior. As described inthe previous section buoyancy has two effects (Petukhov et al., 1988), namely:

• External or indirect effect: External effect is essentially the response of turbulencedue to distortions of the mean velocity profile; the local flow acceleration/decelerationdiscussed in the previous section falls under this category.

• Structural or direct effect: Structural effect refers to turbulence production/destructiondue to density fluctuations. The buoyancy production term in the turbulent kineticenergy equation falls under this category.

External effect

As mentioned previously, buoyancy is a result of a radial density gradient and it does notchange the bulk velocity. Therefore, local flow acceleration close to wall must be compensatedin the core to satisfy the integral mass flux balance. This results in a velocity reduction inthe core. The amount of the reduction depends on the buoyancy magnitude; the higher thebuoyancy, the higher the velocity reduction in the core. This ability of the buoyancy to tune itseffect based on its amount, makes its effect peculiar. In contrast to this, the flow accelerationdue to thermal expansion is limited and constrained by mass conservation; also the velocityreduction in the core is small, as to some extent it is compensated by the increase in thebulk velocity. The effect of buoyancy not only results in a very fast reduction of turbulence(for high buoyancy cases), but also in a generation of an ’M’ shaped velocity profile (∂uz

∂r> 0

away from the wall). This creates additional shear, which results in turbulence productionthus recovering the turbulence and the heat transfer.

The buoyancy effect can be explained schematically using Figure 2-4. The figure shows aqualitative representation of the buoyancy effect on the turbulent shear stress. For upwardflows with buoyancy, the decrease in turbulence progresses as the flow proceeds downstreamand the extent of reduction is determined by the amount of buoyancy. For high buoyancy casesthe turbulent shear stress at some downstream location may become so small that the flowcan be considered to be laminarized (re-laminarization). On further increasing the buoyancy,the turbulent shear stress becomes negative in the core (because of M shaped velocity profile)and turbulence is regenerated in that region, thus recovering the heat transfer effectiveness.In case of downward flows, the turbulent shear stress shows an increase due to buoyancy.

Note, that the figure only represents a qualitative picture of the buoyancy effect. As willbe shown in Chapter 4, the Reynolds shear stress is not negative over the complete radialdomain. It exhibits positive values close to the wall where ∂uz

∂r< 0, and negative further away

from the wall where ∂uz

∂r> 0. This can be explained physically by the fact that in the region

where ∂uz

∂r< 0, any fluid element with outward radial (positive) fluctuation u′′

r , will introducepositive values of u′′

z , as it comes from a location where uz is higher than the new location.Therefore, the correlation between u′′

r and u′′

z will be positive. Similarly, in the region where∂uz

∂r> 0, positive fluctuation of u′′

r will introduce negative values of u′′

z , hence the correlationamong them will be negative.



Figure 2-4: Turbulent shear stress for buoyancy opposed and buoyancy aided flow (Adapted fromKim et al. (2008b)).

Structural effect

Any destabilizing phenomena will act as a source of turbulence. For example, when a fluidinside a horizontal channel is heated from below and cooled from above, an unstable stratifi-cation is obtained. The lighter fluid from the bottom starts moving upwards and the denserfluid starts moving downwards. A stable stratification is obtained when the heating andcooling walls are interchanged.

Depending on the stability of the stratification, buoyant turbulent production can act as asource (positive) or as a sink (negative). The buoyant turbulent production term is given as,(see Equation 2-61)

Gk ≈ ρ′uz′

1

Fr0.

If in a stably stratified flow (∂ρ∂z

< 0) a fluid particle ’A’ is displaced from a lower positionto a higher position (due to velocity fluctuations), where the density of the surrounding fluidis lower, then the buoyancy force will push the higher density particle back to its originalposition; thus, damping the vertical turbulent fluctuations. Buoyancy production term actsas sink for stably stratified flows. In an unstable stratified flow (∂ρ

∂z> 0), the opposite occurs,

thus enhancing the vertical turbulent fluctuations. The buoyancy production term acts assource for unstable stratified flows.

Note, that it is not necessary that upward flows are always stably stratified. However,downward flows are always unstably stratified. In case of upward flows, where flow undergoes



recovery after deterioration, the radial density distribution also changes in streamwise direc-tion. This makes the flow locally unstably stratified at some locations, generating positivebuoyancy production of turbulence. Therefore, both the external and the structural effect ofbuoyancy contribute to the recovery process.

The buoyant turbulent kinetic energy production, arises from the buoyant production ofρu′′

zu′′

z (ρu′′

zu′′

z being one of the contributing elements of turbulent kinetic energy). On similargrounds, other turbulent statistics that are influenced by structural effects of buoyancy areρu′′

ru′′

z and ρu′′

zh′′, the buoyant production of which are given as,

Gρu′′

r u′′

z≈ ρ′ur

′1

Fr0,

Gρh′′u′′

z≈ ρ′h′

1

Fr0.

It is now worthwhile to explain the peculiar observations for supercritical heat transfermade by Hall (1971) using the theoretical background described above:

• The sharp wall temperature peaks observed in larger pipes are present only in upwardflows, whereas the broad peak obtained in the smaller pipe is insensitive to the flowdirection. – The sharp wall temperature peak in upward flows in larger pipes occursbecause of initial turbulence reduction due to buoyancy. In a small pipe the buoyancyeffect is negligible and the only dominant effect is flow acceleration due to thermalexpansion. It is present even in forced convection and hence independent of direction.

• Except for the sharp peaks themselves, the wall temperatures for upward flow are lowerthan those measured in the smaller pipe. – Buoyancy in upward flows, which firstdeteriorates turbulence, later also enhances turbulence resulting in better heat transferthan in forced convection.

• The wall temperatures for downward flow in larger pipes show no signs of peaks andare considerably lower than those for the smaller pipe. – In downward flows, buoyancyalways aids in turbulence production and hence results in better heat transfer.

2-4-4 Subcritical vs Supercritical

Having discussed the mechanisms that govern the heat transfer process, a short summary ofheat transfer differences between subcritical and supercritical state can be made. In case ofsubcritical flows, the variation of cp, µ and k are small, hence the effects of property variationsare completely absent. Although, flow acceleration and buoyancy effects are present also insubcritical flows, their influence on heat transfer is gradual, based on the fact that densitychanges are small and uniform.

Another major difference that actually makes supercritical fluids superior in terms of sub-critical fluids is the absence of boiling crisis. Boiling crisis is encountered in a subcritical heattransfer when the fluid reaches its saturated state and starts its phase transition. Boilingcrisis is marked with a very sharp increase in wall temperature. This problem is not encoun-tered for a supercritical heat transfer. Although, heat transfer deterioration is observed insupercritical flows, the increase in wall temperature is rather smooth and gradual.


Chapter 3

Flow Domain and Numerical Method

A short description on the flow domain and the numerical method employed will be presentedhere. The discussion will be focused only on the RANS implementation. The RANS code hasbeen developed by modifying an inhouse FORTRAN code used for DNS simulations.

The inhouse DNS was performed by Hassan Nemati, PhD candidate at Process and EnergyLaboratory of TU Delft. The description of the DNS code is provided in Nemati et al. (2013).

3-1 Flow Domain

The RANS simulation consist of two parts, namely the inflow generator and the simulationof the developing pipe flow (Figure 3-1). A periodic adiabatic pipe flow simulation (withpipe length over pipe diameter, L/D = 5) is used to generate the inflow conditions for thedeveloping pipe with constant wall heat flux and L/D = 20.

Figure 3-1: Geometry of the simulation domain.


30 Flow Domain and Numerical Method

The radial mesh distribution for all simulations is given by a hyperbolic function definedby Equation (3-1), with a=2.4, b=0.12, R=0.5, nr,max the number of grid points to ensurea high resolution close to the wall. A constant mesh spacing in the streamwise direction isused.

r

R=

[1 −tanh{a(b − nr

nr,max)}

tanh(ab)]

[1 − tanh{a(b − 1)}tanh(ab)

]

(3-1)

The inflow generator has a mesh resolution of 96x8 along the radial and axial direction,respectively, whereas for the developing pipe the mesh is 96x128.

3-2 Numerical Method

The RANS code solves the two dimensional momentum and energy equations given by Equa-tion (2-56)-(2-58), along with Equation (2-65) and (2-66) for k and ǫ. In case of the v2 − fmodel, Equation (2-70) and (2-71) are also solved. The equations are solved using a staggeredspace grid as shown in Figure 3-2. The scalar equations (h, k, ǫ, v2, f) are discretized at thecell center, whereas the momentum equations are discretized at the velocity points, which arelocated at the cell face.

For the momentum equation, an unsteady solver is used. A second order central differencescheme is used to discretize the spatial derivatives. An Euler time integration scheme is usedfor the unsteady term. The diffusive terms in r-direction are treated implicitly, while all theother terms are discretized explicitly. The momentum equation is initially solved for ρui,as it facilitates the pressure correction scheme (based on projection method). The pressurecorrection scheme, using tensor notation, is shown below:

(ρui)n+1 = (ρui)

n + ∆t

[(−∂(ρuiuj)

∂xj+

∂

∂xj

[(

µ

Re+ µt)

∂ui

∂xj

])n

− ∂pn+1

∂xi

]. (3-2)

For a steady state flow using the continuity equation,

∂(ρuj)n+1

∂xj= 0. (3-3)

Figure 3-2: Staggered space grid.


3-2 Numerical Method 31

A guess of the velocity is made by neglecting the pressure,

(ρu)k = (ρui)n + ∆t

[(−∂(ρuiuj)

∂xj+

∂

∂xj

[(

µ

Re+ µt)

∂ui

∂xj

])n], (3-4)

which gives,

(ρui)n+1 − (ρui)

k = −∆t∂pn+1

∂xi. (3-5)

By taking the divergence of Equation (3-5) a Poisson equation for the pressure is obtainedas,

∂(ρui)n+1

∂xi− ∂(ρui)

k

∂xi= −∆t

∂2pn+1

∂x2i

, (3-6)

using Equation (3-3) the Poisson equation simplifies to,

∂(ρui)k

∂xi= ∆t

∂2pn+1

∂x2i

. (3-7)

The Poisson equation for the pressure is solved using a Fast Fourier Transform (FFT) algo-rithm.

Substituting the value of pressure in Equation (3-2) gives (ρu)n+1 from which un+1 can becalculated as,

un+1i =

(ρu)n+1

ρn+1, (3-8)

where ρn+1 is obtained by using the enthalpy at new iteration level.

For the scalar equations, steady state solvers are used because of a faster convergence rateas compared to an unsteady solver. The solution obtained using both steady and unsteadysolvers were found to be the same. The steady state solver is obtained by keeping the radialdiffusive terms at a new iteration level and obtaining other terms from the previous iteration.Under-relaxation factors are used to increase the stability of the solution. Here, a second ordercentral difference scheme is used to discretize the spatial derivatives in the diffusion terms.The Koren slope limiter (Koren, 1993) is used to discretize the advection term to reduceoscillations occurring due to sharp gradients. The elliptic equation f in the v2 − f model issolved using a Helmholtz solver. This solver is derived from the existing Poisson solver byaccounting for the additional term in the Helmholtz equation and setting the correspondingboundary conditions.

The thermophysical properties are tabulated for a thermodynamic pressure of P0 = 80 barand a spline interpolation scheme is used to get the properties from the table as a functionof enthalpy. The value of the thermophysical property α, which is required at the cell faces,is calculated by first interpolating the enthalpy value at cell faces and then calculating thecorresponding value of α from the table.

A no slip boundary condition is used for the velocity at the wall. The wall conditions fork, ǫ, v2 and f are summarized in Table 2-3. The inflow generator uses a zero wall heat fluxboundary condition for the energy equation and a periodic boundary condition in streamwisedirection for all velocity components and scalars. For the developing pipe a constant wallheat flux of Q∗ = 2.4 is used. The inlet velocity and scalar conditions for developing pipe


32 Flow Domain and Numerical Method

are obtained from the fully developed solution of the inflow generator. The outlet boundaryconditions for both velocity and scalars are such that their gradient at the outlet is equalto the gradient at the upstream node. Message passing interface (MPI) routines are used toenable communications between cores for parallel computing.

The Poisson solver for the pressure uses the von Neumann boundary condition at the wall.For the inflow generator a periodic pressure boundary condition in streamwise direction isused. For developing case, the boundary condition at the inlet and the outlet is von Neumann.The boundary condition for the Helmholtz solver is the same as that of the Poisson solver,except at the wall, where the Dirichlet boundary condition of f = 0 is applied.


Chapter 4

Results and Discussion

The simulations conditions for both the DNS and the RANS simulations will be described,after which the performance of RANS models for different cases will be compared with DNSresults. Thereafter, the significance of additional closure terms that are generated because oflarge property variations will be discussed using the inhouse DNS data. Finally, a discussionwill be carried out on the Jensen inequality and its applicability on Reynolds/Favre averagingfor supercritical flows.

The work of Bae et al. (2005) has been taken as a reference for the development of theinhouse DNS study. The reason for choosing this reference work was to have a validationcase for DNS of supercritical heat transfer and also to look into some additional details thatwere unanswered or unclear. The work of He et al. (2008a), who assessed the performanceof different RANS models, by comparison with DNS data of Bae et al. (2005) provided acomparable reference for the inhouse RANS study.

4-1 Simulation Conditions

The simulation domain for both the DNS and the RANS study, as shown in Figure 3-1consists of two parts: inflow generator and developing pipe. In order to study the developingheat transfer, the inlet profile generated from the inflow generator is advected through a pipeheated with constant wall heat flux. To allow for upstream diffusion and avoid inconsistenciesat the inlet boundary, heating in the developing pipe started few grid points downstream ofthe inlet. For all the simulations the inflow conditions correspond to P0 = 80 bar and T0 =301.15 K; Table 4-1 summarizes further flow conditions providing details on buoyancy andflow direction. Note that in all the simulations, the condition of Tb < Tpc < Tw is maintained,except for few initial grid points downstream of the inlet, to investigate the peculiar behaviorof the heat transfer.

For the DNS studies, the equations are non-dimensionalized using the friction velocityReynolds number Reτ . The pressure gradient in the periodic pipe is set to -4. For a fully


34 Results and Discussion

Table 4-1: Flow conditions

Case Type Dir. Reτ (DNS) Reb Pr0 Gr∗ Q∗

A Forced - 360 5300 3.19 0 2.4B Mixed Up 360 5300 3.19 21e6 2.4C Mixed Up 360 5300 3.19 566.7e6 2.4D Mixed Down 360 5300 3.19 167.9e6 2.4

developed flow, the relation between pressure gradient and wall shear stress is given as:

Sτw = −Adp

dzL,

where S = πDL is the surface area of the pipe and A = πD2/4 is the cross section area ofthe pipe. Hence, the pressure gradient becomes,

dp

dz= −4

τw

D, (4-1)

with τw and D being 1. Reτ = 360 which corresponds to a bulk velocity Reynolds numberReb of 5300 is used for the DNS study. The DNS simulations were validated with Eggels(1994), for the periodic pipe, and with Bae et al. (2005) for the developing pipe. For moredetails on DNS simulation refer Nemati et al. (2013).

In contrast to the DNS, all the RANS simulations have been performed using a bulk velocityReynolds number Reb of 5300 as the non-dimensionalizing parameter. With this, the bulkvelocities are set, instead of the friction velocity, ensuring the same mass to heat flux ratios asin the DNS. In fact, by setting the friction velocity in the RANS, different turbulence modelswould result in different mass fluxes due to their individual wall damping functions. Thepressure gradient in the periodic pipe (inflow generator) is iterated to obtain Reb=5300. Thecorresponding Reτ values obtained by the different turbulence models are given in Table 4-2.The validation and mesh independency results of different turbulence models are presentedin Appendix B-1 and Appendix B-2, respectively.

Since the non-dimensional heat flux Q∗ is the same (=2.4) for all cases, the streamwisedistributions of bulk parameters also remain the same. The bulk mass flux Gb, bulk velocityUb and bulk enthalpy Hb are defined (based on mass and energy conservation) as follows:

Gb =1

A

∫ρuzdA, Hb =

1

GbA

∫ρuzhdA,

ρb = ρ(P0, Hb), Ub =Gb

ρb

. (4-2)

Table 4-2: Reynolds number based on friction velocity for different RANS models at the inlet

Turbulence model Reτ

MK 373.3LS 344.7

V2F 351.9


4-2 Comparison of DNS and RANS Results 35

5 10 15 200

0.002

0.004

0.006

0.008

0.01

0.012

Hb

z

(a) Bulk enthalpy

5 10 15 201

1.02

1.04

1.06

1.08

1.1

Ub

z

(b) Bulk velocity

Figure 4-1: Streamwise distribution of bulk enthalpy and velocity. All cases( ); Analyticsolution for bulk enthalpy(◦).

Figure 4-1 shows the streamwise bulk enthalpy Hb and bulk velocity Ub obtained for all DNSand RANS cases. Hb is also compared with the analytic solution obtained from the globalenergy balance, given as:

Hb =4Q∗

RebPr0

L

D. (4-3)

4-2 Comparison of DNS and RANS Results

The objective of this section is to investigate the performance of different conventional tur-bulence models and to highlight their deficiencies in terms of damping functions, buoyancyproduction, eddy viscosity hypothesis and turbulent Prandtl number. The mechanism of heattransfer (discussed in Chapter 2) will also be revisited wherever necessary with the help ofDNS results.

Before comparing inhouse RANS with inhouse DNS results, the RANS results have beencompared with the results of He et al. (2008a) first. Although the comparison is not shown,a few comments are given. The results with the LS and the MK are found to be consistentwith He et al. (2008a), however the V2F model gave very different results. First, the V2Finlet profiles (periodic pipe simulation) in He et al. (2008a) did not match with Eggels (1994)and our own RANS simulations. Second, the V2F model for the developing pipe showedthe best agreement with DNS, while our results showed the opposite. In order to validateour V2F implementation in the RANS solver, we first performed a channel flow simulationswith Reτ = 590 and compared the results with Pecnik and Iaccarino (2007). The results arefound to be in perfect agreement (see Appendix B-1), ensuring our correct implementation.Furthermore in He et al. (2008a), the shear and buoyancy production term in the turbulentkinetic energy equations are reported ambiguously, the only information given for the MK



model is the wall temperature distribution, and no downward heated flows are analyzed. Forthese reasons a detailed comparison with He et al. (2008a) is not possible.

4-2-1 Wall temperature

Figure 4-2 shows DNS results for wall temperature for the four cases analyzed. For smallbuoyancy effects (case B), the heat transfer deterioration is higher (as seen by the higher walltemperature) than that of forced convection (case A). By increasing the buoyancy effect (caseC) the wall temperature remains lower than for case A at all downstream locations (even forthe deteriorated region until z approximately 9), hence showing enhanced heat transfer ascompared to case A. It should be noted that the buoyancy influence in case C is significantlyhigher than that of case B; for intermediate buoyancy cases the peak in wall temperaturebefore recovery are usually higher than that of forced convection cases (see case C of Baeet al., 2005). For downward flows (case D), the wall temperatures are significantly lower thanthose of case A showing enhanced heat transfer always.

Figure 4-3 shows a comparison of DNS wall temperatures with results obtained using differ-ent turbulence models. The MK model shows a good agreement for case A and B, whereas forcase C and D it performs poorly. The V2F and the LS model give similar wall temperaturedistributions and are significantly higher for case A and B. For case C, both the LS and theV2F model again overpredict the wall temperature and also the onset of recovery (markedby decrease in wall temperature after peak) are computed further downstream as comparedwith DNS data. For case D, the LS model closely represents the DNS wall temperature whilethe V2F model gives an oscillatory behavior.

0 5 10 15 201

1.05

1.1

1.15

1.2

Tw

z

Figure 4-2: Streamwise distribution of wall temperature from DNS for all cases. case A( );case B( ); case C( ); case D( ).



0 5 10 15 201

1.05

1.1

1.15

1.2

1.25

1.3

Tw

z

(a) case A

0 5 10 15 201

1.05

1.1

1.15

1.2

1.25

1.3

Tw

z

(b) case B

0 5 10 15 201

1.04

1.08

1.12

1.16

1.2

Tw

z

(c) case C

0 5 10 15 201

1.02

1.04

1.06

1.08

1.1

1.12

Tw

z

(d) case D

Figure 4-3: Comparison of streamwise distribution of wall temperature from RANS with DNSfor all cases. MK( ); LS( ); V2F( ); DNS(◦).

4-2-2 Velocity, enthalpy and turbulent statistics

A RANS and DNS comparison is done for the following averaged profiles: streamwise velocity,enthalpy, Reynolds shear stress (ρu′′

r u′′

z), radial turbulent heat flux (ρu′′

rh′′), and productionrate of turbulent kinetic energy (both shear and buoyancy), to highlight few of the deficienciesof conventional turbulence models. A case by case comparison is made for the four simulatedconditions. First, for each case observations made using DNS data will be discussed. Second,a brief comment on the performance of the RANS model will be made and finally DNS datawill be analyzed further to highlight the deficiencies of conventional turbulence models. Fourstreamwise locations are used to study the turbulent statistics, whereby the first streamwiselocation corresponds to the inlet condition. The inlet condition is used as a reference state



to study the downstream development of the flow.

Case A

Case A represents forced convection. Figure 4-4 shows the RANS and the DNS comparisonfor velocity and enthalpy profile, Figure 4-5 shows the comparison for Reynolds shear stressand radial turbulent heat flux, which are the closure terms that are modeled using the eddyviscosity hypothesis. Figure 4-6 shows the comparison for the shear production rate of tur-bulent kinetic energy. As discussed in Chapter 2, flow acceleration due to thermal expansioncauses a decrease in turbulence.

DNS observation: On proceeding downstream, Figure 4-4 shows that the velocity profileflattens out towards the core and the enthalpy increases close to the wall while remainingnearly constant in the core. The decrease in turbulence is evident as the Reynolds shearstress in Figure 4-5 and the shear production rate in Figure 4-6 both decrease as one proceedsdownstream. Interestingly, the DNS data for the turbulent heat flux show a constant peak atall locations, except at the inlet (where the heat flux is not active yet). This can be explainedby means of the production of the radial turbulent heat flux P

ρu′′

r h′′ ∝ −ρu′′

r u′′

r∂h/∂r. As

ρu′′

ru′′

r decreases due to deterioration, ∂h/∂r increases, maintaining Pρu′′

r h′′ nearly constant.

RANS performance: At the inlet of the pipe, it can be seen that the MK and the LSmodel show a good agreement with DNS for the velocity (Figure 4-4), Reynolds shear stress(Figure 4-5) and the peak of shear production (Figure 4-6). A less good agreement is seen forthe V2F model. Proceeding downstream, the MK model continues to agree well, while theLS and the V2F perform very poorly. Although, the MK model does not provide an exactmatch over the complete radial domain, it provides a close match of both Reynolds shearstress and turbulent heat flux close to the wall. On the other hand, the LS and the V2Fperform poorly over the complete domain, overpredicting the deterioration. Also, none of themodels correctly predict the peak magnitude of shear production, however the MK model isat least able to capture the location of peak.

RANS modeling deficiency: Figure 4-5 and 4-6 provide details as to why the MK modelgives better results than the V2F and the LS model. Downstream of the pipe inlet, the DNSresults show that the values for the Reynolds shear stress close to the wall are relatively large(even though the overall magnitude decreases) and that the peak of the shear productionslightly moves, first towards the wall and then away. The inference from this is the growthof the viscous dominant region. An indication of the growth of the viscous dominant regioncan also be made by plotting the viscous and turbulent shear stress distributions, to revealwhere the turbulent shear stress starts dominating the viscous shear stress (see AppendixC). The results obtained with the LS and the V2F show a severely deteriorated Reynoldsshear stress profile close to the wall. The peak of the turbulent kinetic energy productionfor the LS model initially shows an inward shift towards the wall, after which the peaksignificantly moves outwards. For the V2F model the peak immediately moves outwards. Forthe MK model, the location of production peak remains the same. This leads to the firstconclusion of the turbulence model deficiency, i.e the inability of the damping functions tomodel the viscous effects close to the wall. The damping function formulations for the LSand the V2F model are based on local flow quantities, while the damping function in the MKmodel is strongly correlated with the wall distance, therefore mitigating the modeling error



for this case. Cases where DNS viscous dominant region shows more movement will highlightthe damping function deficiency for the MK model (see case C and D). A constant turbulentPrandtl number (Prt) is another topic of discussion in RANS modeling. In this case, however,the use of constant Prt is found to be satisfactory (see Figure 4-19). Prt shown in Figure4-19 is calculated using DNS data, as:

Prt =ρu′′

ru′′

z

ρu′′

r h′′

∂h∂r

∂uz

∂r

. (4-4)



0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

uz/U

b,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.04

0.08

0.12

0.16

0.2

h

r

(a) MK

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

uz/U

b,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.04

0.08

0.12

0.16

0.2

h

r

(b) LS

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

uz/U

b,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.04

0.08

0.12

0.16

0.2

h

r

(c) V2F

Figure 4-4: Mean velocity (left column) and enthalpy profile (right column) for case A. RANS:z = 0.3125( ); 6.5625( ); 14.6875( ); 19.0625( ). DNS: z = 0.3125(�);6.5625(△); 14.6875(⋄); 19.0625(◦).



0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

ρu

′′ ru

′′ z/u

2 τ,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

−ρu

′′ rh

′′/q

∗

r

(a) MK

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

ρu

′′ ru

′′ z/u

2 τ,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

−ρu

′′ rh

′′/q

∗

r

(b) LS

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

ρu

′′ ru

′′ z/u

2 τ,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

−ρu

′′ rh

′′/q

∗

r

(c) V2F

Figure 4-5: Reynolds shear stress, ρu′′

r u′′

z /u2τ,0 (left column) and radial turbulent heat flux (right

column) distribution for case A. RANS: z = 0.3125( ); 6.5625( ); 14.6875( );19.0625( ). DNS: z = 0.3125(�); 6.5625(△); 14.6875(⋄); 19.0625(◦).



10-3 10-2 10-1 1000

10

20

30

40

50

60

70

80

90

−ρu

′′ ru

′′ z∂

uz

∂r

/u3 τ,0

1 − 2r

(a) MK

10-3 10-2 10-1 1000

10

20

30

40

50

60

70

80

90

−ρu

′′ ru

′′ z∂

uz

∂r

/u3 τ,0

1 − 2r

(b) LS

10-3 10-2 10-1 1000

10

20

30

40

50

60

70

80

90

−ρu

′′ ru

′′ z∂

uz

∂r

/u3 τ,0

1 − 2r

(c) V2F

Figure 4-6: Shear production rate of turbulent kinetic energy for case A. RANS: z =0.3125( ); 6.5625( ); 14.6875( ); 19.0625( ). DNS: z = 0.3125(�);6.5625(△); 14.6875(⋄); 19.0625(◦).

Case B

Case B represents an upward flow case with small buoyancy influence. As discussed in Chapter2, buoyancy in upward flows causes local flow acceleration thereby decreasing the turbulencefurther with respect to forced convection. The recovery in heat transfer due to buoyancy isnot observed within the simulated domain of L/D=20. The comparison of DNS with RANSfor case B are shown in Figure 4-7 (velocity and enthalpy profile), Figure 4-8 (Reynolds shearstress and radial turbulent heat flux) and Figure 4-9 (shear and buoyant production rate ofturbulent kinetic energy).

DNS observation: The results presented in Figure 4-7, 4-8, 4-9 show the same trend as thosein case A, but with a larger decrease in turbulence and hence higher wall temperature (higherheat transfer deterioration). Because of the structural effects of buoyancy (see Chapter 2)



there is an additional source of turbulent kinetic energy production, namely buoyant turbulentkinetic energy production which is shown in Figure 4-9. Because of a stably stratified densityfield, the buoyancy production is negative (see Chapter 2), similar to the dissipation, but herethe transfer of turbulent energy is back to the mean flow, instead of dissipation to heat byviscous action. Its value is very small as compared to the shear production rate in this case.

RANS performance: The RANS performance is similar to that in case A, with the MKmodel closely reproducing the DNS data, whereas the LS and the V2F performing poorly.None of the turbulence models are able to capture the buoyant production term, but its effectcan be considered negligible as its value compared to shear production rate is small (see DNSresults).

RANS modeling deficiency: The superior performance of the MK model, as compared tothe LS and the V2F, is primarily because of better capturing of the viscous effect close towall (as explained for case A). Looking at Figure 4-19, using a constant value of Prt seemsincorrect in this case, as Prt decreases to 0.6 close to the wall at downstream locations.Therefore, the constant Prt assumption can be regarded as invalid for modeling supercriticalheat transfer.



0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

uz/U

b,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.04

0.08

0.12

0.16

0.2

h

r

(a) MK

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

uz/U

b,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.04

0.08

0.12

0.16

0.2

h

r

(b) LS

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

uz/U

b,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.04

0.08

0.12

0.16

0.2

h

r

(c) V2F

Figure 4-7: Mean velocity (left column) and enthalpy profile (right column) for case B. RANS:z = 0.3125( ); 6.5625( ); 14.6875( ); 19.0625( ). DNS: z = 0.3125(�);6.5625(△); 14.6875(⋄); 19.0625(◦).



0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

ρu

′′ ru

′′ z/u

2 τ,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

−ρu

′′ rh

′′/q

∗

r

(a) MK

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

ρu

′′ ru

′′ z/u

2 τ,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

−ρu

′′ rh

′′/q

∗

r

(b) LS

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

ρu

′′ ru

′′ z/u

2 τ,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

−ρu

′′ rh

′′/q

∗

r

(c) V2F


r u′′


column) distribution for case B. RANS: z = 0.3125( ); 6.5625( ); 14.6875( );19.0625( ). DNS: z = 0.3125(�); 6.5625(△); 14.6875(⋄); 19.0625(◦).



10-3 10-2 10-1 1000

10

20

30

40

50

60

70

80

90

−ρu

′′ ru

′′ z∂

uz

∂r

/u3 τ,0

1 − 2r10-3 10-2 10-1 100-4

-3

-2

-1

0

1

ρ′u

′ z/F

r 0

1 − 2r

(a) MK

10-3 10-2 10-1 1000

10

20

30

40

50

60

70

80

90

−ρu

′′ ru

′′ z∂

uz

∂r

/u3 τ,0

1 − 2r10-3 10-2 10-1 100-4

-3

-2

-1

0

1

ρ′u

′ z/F

r 0

1 − 2r

(b) LS

10-3 10-2 10-1 1000

10

20

30

40

50

60

70

80

90

−ρu

′′ ru

′′ z∂

uz

∂r

/u3 τ,0

1 − 2r10-3 10-2 10-1 100-4

-3

-2

-1

0

1

ρ′u

′ z/F

r 0

1 − 2r

(c) V2F

Figure 4-9: Shear (left column) and buoyant (right column) production rate of turbulent kineticenergy for case B. RANS: z = 0.3125( ); 6.5625( ); 14.6875( ); 19.0625( ).DNS: z = 0.3125(�); 6.5625(△); 14.6875(⋄); 19.0625(◦).



Case C

Case C represents a high buoyancy case for upward flows. The flow undergoes both, deteriora-tion and recovery. The comparison of DNS and RANS results for case C are shown in Figure4-10 (velocity and enthalpy profile), Figure 4-11 (Reynolds shear stress and radial turbulentheat flux) and Figure 4-12 (shear and buoyant production rate of turbulent kinetic energy).

DNS observation: In Figure 4-10 it can be seen that the streamwise velocity has a typical M-shaped profile in the downstream region. The enthalpy profile shows a nearly constant valueclose to the wall and increases in the core region proceeding downstream (note: previous casesshowed a lower increase in the core). A deterioration followed by recovery in turbulence canbe observed by means of the Reynolds shear stress in Figure 4-11. In the deterioration region(until z ≈ 9), the Reynolds shear stress decreases becoming negative at locations slightly awayfrom the wall. As z increases the negative portion of the Reynolds shear stress moves towardsthe center of the pipe. At some downstream locations the Reynolds shear stress is negative inmost of the radial domain, except close to the wall where it remains positive. This is wherethe recovery starts. The recovery onset closely follows the velocity profile, which deforms intoan M-shape. The turbulent heat flux also increases after the recovery onset. If one comparesthe Reynolds shear stress and turbulent heat flux for case A and C at z=6.5625, it can be seenthat the values for case A are higher for both the Reynolds shear stress and the turbulentheat flux; yet the wall temperature for case A is higher than that for case C. The reason forthis is the very high local flow acceleration due to buoyancy close to the wall, which makes thelaminar effect, i.e. energy transport by advection (see Chapter 2) also important. As statedearlier, for intermediate buoyancies (Gr0,caseB < Gr0 < Gr0,caseC) the wall temperature forupward flows is usually higher than the one for forced convection in the deterioration regimeand may become smaller after the onset of recovery. Figure 4-12 shows a negative value ofthe shear production rate in between two positive peaks in the recovery regime. The reasonfor this is that the velocity gradient and the Reynolds shear stress change sign at differentlocations. So, the location at which ∂uz

∂r= 0, ρu′′

r u′′

z 6= 0. This can easily be observed in Figure

4-13, which shows both ∂uz

∂rand ρu′′

ru′′

z ; it can be seen that where ρu′′

r u′′

z = 0, ∂uz

∂ris very large

and vice versa. Another interesting point to note is the buoyancy production term (Figure4-12), which in this case has the same order of magnitude and the same sign as the shearproduction rate. The positive value of buoyant production reveals the presence of a locallyunstable stratified density fields (see Chapter 2). When the shear production rate decreases tozero (z=6.5625), because of the deteriorating Reynolds shear stress, the buoyancy productionrate is the highest, therefore providing a continuous source of turbulence production. As theshear stress production rate recovers, the buoyancy production rate decreases. This results ina considerable magnitude of turbulent kinetic energy close to the wall, even if the Reynoldsshear stress deteriorates, as can be seen in Figure 4-14. The figure shows the turbulent kineticenergy and its total production rate (shear+buoyancy). This phenomenon is rarely observedin flows undergoing small changes in density (ideal gas), where the buoyancy production termis not as dominant. There the turbulent kinetic energy also shows a decrease along with theReynolds shear stress in the deterioration regime (see Bae et al., 2006).

RANS performance: In this case, the turbulence dynamics are so strong, that none of theturbulence models are able to capture the correct flow behavior (this is also reflected on thewall temperature profiles as discussed earlier). From Figure 4-10, it can be seen that theRANS models, especially the LS and the V2F, tend to overpredict the M-shaped velocity



profile, and all models overpredict the wall enthalpy, while the core enthalpy remains nearlyconstant proceeding downstream. The ability of the models to capture the Reynolds shearstress and the turbulent heat flux (Figure 4-11) is very poor. Not only a delay of the recovery,but also the extent of the recovery is poorly captured. None of the models are able to correctlyreproduce the turbulent heat flux. The shear production rate (Figure 4-12) is also very poorlycaptured, and only the MK model is able to show the two peaks, but both the location andmagnitude of the peaks are wrong. Furthermore, no model is able to capture the buoyancyproduction rate of turbulent kinetic energy (Figure 4-12), which in this case is very significant.

RANS modeling deficiency: The turbulence model deficiencies, as observed in the pre-vious cases, amplify the failure to predict the heat transfer and additionally highlight otherweaknesses. The negative shear production rate observed in the DNS data cannot be capturedby the eddy viscosity turbulence models at all. The eddy viscosity is related to the Reynoldsstresses and the mean deformation rate, as:

ρu′′

ru′′

z = −µt(∂ur

∂z+

∂uz

∂r), (4-5)

for which ρu′′

r u′′

z ≈ 0 for ∂uz

∂r= 0, leading to a break down of the eddy viscosity hypothesis.

The analysis of the viscous dominated region (as done for case A) is performed using the sumof the shear and buoyant production rate from DNS in Figure 4-14. The peak of the totalproduction rate moves towards the wall, and a second smaller peak appears further awayfrom the wall in the recovery region. Here again, the damping functions for the LS and theV2F model wrongly reproduce the viscous dominated region as shown in Figure 4-12. Alsothe damping function formulation for the MK model fails and the peak locations are wronglypredicted. The modeling error is further amplified because of the very poor prediction of thebuoyant turbulent production rate (see Figure 4-12), which results in an overprediction ofdeterioration in turbulent kinetic energy. As the model for turbulent heat flux is based on theReynolds shear stress, its value is also underpredicted. In fact, the turbulent Prandtl numberconcept fails as the turbulent heat flux and the Reynolds shear stress are weakly correlated.The value of Prt, calculated using DNS data, can be seen in Figure 4-19; it not only becomesnegative at many locations, but also infinity at few places as ∂uz

∂ris 0 at those places.



0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

uz/U

b,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.04

0.08

0.12

0.16

0.2

h

r

(a) MK

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

uz/U

b,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.04

0.08

0.12

0.16

0.2

h

r

(b) LS

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

uz/U

b,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.04

0.08

0.12

0.16

0.2

h

r

(c) V2F

Figure 4-10: Mean velocity (left column) and enthalpy profile (right column) for case C. RANS:z = 0.3125( ); 6.5625( ); 14.6875( ); 19.0625( ). DNS: z = 0.3125(�);6.5625(△); 14.6875(⋄); 19.0625(◦).



0 0.1 0.2 0.3 0.4 0.5-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

ρu

′′ ru

′′ z/u

2 τ,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

1.2

−ρu

′′ rh

′′/q

∗

r

(a) MK

0 0.1 0.2 0.3 0.4 0.5-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

ρu

′′ ru

′′ z/u

2 τ,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

−ρu

′′ rh

′′/q

∗

r

(b) LS

0 0.1 0.2 0.3 0.4 0.5-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

ρu

′′ ru

′′ z/u

2 τ,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

1.2

−ρu

′′ rh

′′/q

∗

r

(c) V2F


r u′′


column) distribution for case C. RANS: z = 0.3125( ); 6.5625( ); 14.6875( );19.0625( ). DNS: z = 0.3125(�); 6.5625(△); 14.6875(⋄); 19.0625(◦).



10-3 10-2 10-1 100-40

-20

0

20

40

60

80

100

−ρu

′′ ru

′′ z∂

uz

∂r

/u3 τ,0

1 − 2r10-3 10-2 10-1 100-10

0

10

20

30

40

50

60

70

ρ′u

′ z/F

r 0

1 − 2r

(a) MK

10-3 10-2 10-1 100-40

-20

0

20

40

60

80

100

120

140

160

180

200

220

−ρu

′′ ru

′′ z∂

uz

∂r

/u3 τ,0

1 − 2r10-3 10-2 10-1 100-10

0

10

20

30

40

50

60

70

ρ′u

′ z/F

r 0

1 − 2r

(b) LS

10-3 10-2 10-1 100-40

-20

0

20

40

60

80

100

120

140

−ρu

′′ ru

′′ z∂

uz

∂r

/u3 τ,0

1 − 2r10-3 10-2 10-1 100-10

0

10

20

30

40

50

60

70

ρ′u

′ z/F

r 0

1 − 2r

(c) V2F

Figure 4-12: Shear (left column) and buoyant (right column) production rate of turbulent kineticenergy for case C. RANS: z = 0.3125( ); 6.5625( ); 14.6875( ); 19.0625( ).DNS: z = 0.3125(�); 6.5625(△); 14.6875(⋄); 19.0625(◦).



10-3 10-2 10-1 100-1.5

-1.2

-0.9

-0.6

-0.3

0

0.3

ρu

′′ ru

′′ z,

∂u

z

∂r

/100

1 − 2r

Figure 4-13: Comparison of Reynolds shear stress, ρu′′

r u′′

z and ∂uz

∂r/100, based on DNS data for

case C. ∂uz

∂r/100 ( ); ρu′′

r u′′

z ( ). z = 9.6875(�); 14.6875(△); 19.0625(◦).

10-3 10-2 10-1 1000

1

2

3

4

5

ρu

′′ iu

′′ i/2

1 − 2r10-3 10-2 10-1 100-10

0

10

20

30

40

50

60

70

80

−ρu

′′ ru

′′ z∂

uz

∂r

+ρ

′u

′ z/F

r 0

1 − 2r

Figure 4-14: Turbulent kinetic energy (left column) and total production rate of turbulent kineticenergy (right column) based on DNS data for case C. z = 0.3125(�); 6.5625(△); 14.6875(⋄);19.0625(◦).

Case D

Case D represents a downward flow with significant buoyancy. In downward flows buoyancycauses the flow to locally decelerate close to the wall. If the deceleration is higher than the flowacceleration due thermal expansion, turbulence increases (see Chapter 2). The comparison ofDNS and RANS results for case D are shown in Figure 4-15 (velocity and enthalpy profile),Figure 4-16 (Reynolds shear stress and radial turbulent heat flux) and Figure 4-17 (shear andbuoyant production rate of turbulent kinetic energy).

DNS observation: From Figure 4-15 it can be seen that the streamwise velocity increasestowards the core and decreases close to the wall because of the buoyancy force. On proceedingdownstream, the enthalpy profile shows an increase in the core, whereas the profile close



to the wall remains the same. Enhancement in turbulence can be seen from Figure 4-16,which shows a downstream increase in the Reynolds shear stress. The turbulent heat fluxremains nearly constant. The buoyant turbulent production rate is small compared to theshear production (Figure 4-17). However, its value is considerable and is positive indicatingan unstably stratified density field (see Chapter 2). The total production rate of turbulentkinetic energy is shown in Figure 4-18.

RANS performance: For this case, the LS model closely matches the DNS wall tempera-ture, whereas the MK and the V2F model perform poorly. Velocity profiles shown in Figure4-15 indicate that the MK and the LS model perform well, while the V2F agreement is poor.In Figure 4-16, it can be seen that the LS model gives the best match for the Reynolds shearstress and the turbulent heat flux. The V2F model shows turbulence deterioration (afterinitial increase) towards the end of the simulation domain, where the Reynolds shear stressand turbulent heat flux decreases. The LS model gives the best agreement for the shearproduction rate (Figure 4-17) with a very close match for the peak magnitude and the peaklocation. The peak location and magnitude of the MK model remains almost fixed, whereasthe V2F model shows a decrease in the shear production after the initial increase. None ofthe turbulence models are able to reproduce the buoyant production rate.

RANS modeling deficiency: The superior performance of the LS model can be attributedto its ability to better resolve wall damping effects. The MK model, which performs goodfor case A and B with a nearly constant viscous dominated region, fails in this case. Theviscous region shows a significant movement towards the wall, as can be seen by the peakof the turbulent shear production (Figure 4-17) and the total production rate (Figure 4-18)from the DNS results. The peak location for the MK remains at the same location, due tothe wall distance dependence of the damping function. The LS and the V2F model capturethis inward movement of the production peak fairly well, however the strong decrease inthe production rate for the V2F deteriorates its predictive capability towards the end of thesimulation domain. Note, that since none of the turbulence models are able to predict thebuoyant turbulent production rate, the total turbulence production rate is lower as comparedto the DNS; still the LS model is able to capture the Reynolds shear stress profile fairly well,which might indicate a wrong scaling of the dissipation rate ǫ. In Figure 4-19, it can be seenthat the effect of a constant Prt is insignificant for this case. In case of downward flows, thedeficiency of turbulence models is not well reflected on the wall temperature, as the marginof wall temperature reduction is lower for enhanced heat transfer.



0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

uz/U

b,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.04

0.08

0.12

0.16

h

r

(a) MK

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

uz/U

b,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.04

0.08

0.12

0.16

h

r

(b) LS

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

uz/U

b,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.04

0.08

0.12

0.16

h

r

(c) V2F

Figure 4-15: Mean velocity (left column) and enthalpy profile (right column) for case D. RANS:z = 0.3125( ); 6.5625( ); 14.6875( ); 19.0625( ). DNS: z = 0.3125(�);6.5625(△); 14.6875(⋄); 19.0625(◦).



0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

ρu

′′ ru

′′ z/u

2 τ,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

−ρu

′′ rh

′′/q

∗

r

(a) MK

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

ρu

′′ ru

′′ z/u

2 τ,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

−ρu

′′ rh

′′/q

∗

r

(b) LS

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

ρu

′′ ru

′′ z/u

2 τ,0

r0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

−ρu

′′ rh

′′/q

∗

r

(c) V2F


r u′′


column) distribution for case D. RANS: z = 0.3125( ); 6.5625( ); 14.6875( );19.0625( ). DNS: z = 0.3125(�); 6.5625(△); 14.6875(⋄); 19.0625(◦).



10-3 10-2 10-1 1000

10

20

30

40

50

60

70

80

90

100

110

120

130

140

−ρu

′′ ru

′′ z∂

uz

∂r

/u3 τ,0

1 − 2r10-3 10-2 10-1 100-5

0

5

10

15

20

25

30

35

ρ′u

′ z/F

r 0

1 − 2r

(a) MK

10-3 10-2 10-1 1000

10

20

30

40

50

60

70

80

90

100

110

120

130

140

−ρu

′′ ru

′′ z∂

uz

∂r

/u3 τ,0

1 − 2r10-3 10-2 10-1 100-5

0

5

10

15

20

25

30

35

ρ′u

′ z/F

r 0

1 − 2r

(b) LS

10-3 10-2 10-1 1000

10

20

30

40

50

60

70

80

90

100

110

120

130

140

−ρu

′′ ru

′′ z∂

uz

∂r

/u3 τ,0

1 − 2r10-3 10-2 10-1 100-5

0

5

10

15

20

25

30

35

ρ′u

′ z/F

r 0

1 − 2r

(c) V2F

Figure 4-17: Shear (left column) and buoyant (right column) production rate of turbulent kineticenergy for case D. RANS: z = 0.3125( ); 6.5625( ); 14.6875( ); 19.0625( ).DNS: z = 0.3125(�); 6.5625(△); 14.6875(⋄); 19.0625(◦).


4-3 Closure Terms Due to Large Property Variation 57

10-3 10-2 10-1 1000

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

−ρu

′′ ru

′′ z∂

uz

∂r

+ρ

′u

′ z/F

r 01 − 2r

Figure 4-18: Total production rate of turbulent kinetic energy, based on DNS data for case D.z = 0.3125(�); 6.5625(△); 14.6875(⋄); 19.0625(◦).

4-3 Closure Terms Due to Large Property Variation

In this section, the significance of additional closure terms in the RANS equations, due to largeproperty variations (see Chapter 2), are discussed using DNS averaged data. The significanceof these additional closure terms are found to be minor in the momentum equation, howeverin the energy equation they are relevant.

Momentum equation

The averaged momentum conservation equations for r- and z-directions are given in Equations(2-38) and (2-39), which contain the stress tensors, rewritten below for clarity:

τrr =2

Reτ(µ

∂ur

∂r+ µ′

∂u′′

r

∂r+ µ

∂u′′

r

∂r− 1

3µ∇ · u − 1

3µ∇ · u′′ − 1

3µ′∇ · u′′), (4-6)

τθθ =2

Reτ(µ

ur

r+ µ

u′′

r

r+ µ′

u′′

r

r− 1

3µ∇ · u − 1

3µ∇ · u′′ − 1

3µ′∇ · u′′), (4-7)

τzz =2

Reτ(µ

∂uz

∂z+ µ′

∂u′′

z

∂z+ µ

∂u′′

z

∂z− 1

3µ∇ · u − 1

3µ∇ · u′′ − 1

3µ′∇ · u′′), (4-8)

τrz = τzr =1

Reτ(µ

∂uz

∂r+ µ′

∂u′′

z

∂r+ µ

∂u′′

z

∂r+ µ

∂ur

∂z+ µ′

∂u′′

r

∂z+ µ

∂u′′

r

∂z), (4-9)

∇ · u =1

r

∂rur

∂r+

∂uz

∂z, (4-10)

∇ · u′′ =1

r

∂ru′′

r

∂r+

∂u′′

z

∂z. (4-11)



0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

Pr t

y+

(a) case A

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

Pr t

y+

(b) case B

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Pr t

y+

(c) case C

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

Pr t

y+

(d) case D

Figure 4-19: Turbulent Prandtl number, based on DNS data for all cases. z = 3.4375( );9.6875( ); 14.6875( ); 19.0625( ).

In the r-momentum equation, the unknown closure terms due to fluctuations in viscosity µand velocity u′′

i are all found to be negligible and are therefore not shown.

In the z-momentum equation, significant closure terms are found in τrz and are (µ′ ∂u′′

z

∂r+

µ′ ∂u′′

r

∂z) and µ(∂u′′

z

∂r+ ∂u′′

r

∂z). The z-momentum equation can therefore be simplified by using the

following stress tensors,

τzz =2

Reτ(µ

∂uz

∂z− 1

3µ∇ · u), (4-12)

τrz = τzr =1

Reτ(µ

∂uz

∂r+ µ′

∂u′′

z

∂r+ µ

∂u′′

z

∂r+ µ

∂ur

∂z+ µ′

∂u′′

r

∂z+ µ

∂u′′

r

∂z), (4-13)

For brevity τrz can be written as,

τrz = τzr =1

Reτ(µtz,r + µ′t′′

z,r + µt′′

z,r), with tz,r =∂uz

∂r+

∂ur

∂z, (4-14)



The last two terms (µ′t′′

z,r, µt′′

z,r) in the equation above, appear only due to the fluctuationsin properties and will be zero for constant property flows.

A comparison of µtz,r, µ′t′′

z,r, µt′′

z,r and τzr,tot(= µtz,r + µ′t′′

z,r + µt′′

z,r) is made for cases A,C and D in Figure 4-20, 4-21 and 4-22, respectively. It can be seen that the additional termsare negligible compared to the high values of the averaged viscous shear, µtz,r, in case C. Forcase A and D, where the averaged viscous shear stress has a lower value close to the wall, theadditional terms have a slightly higher influence on the total viscous shear stress, τzr,tot, ascan be seen by comparing µtz,r and τzr,tot. However, in general, the importance of additionalclosure terms in the momentum equation could be considered minor for the cases investigated.From a physical point of view, this can be attributed to the fact that the gradient of the meanflow (uz) close to the wall is very high compared to that of the fluctuating component (u′′

z).

0 0.1 0.2 0.3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

µt z

,r,µ

t′′ z,r

,µ′t′

′ z,r

,τzr,

tot

1 − 2r

(a) z=2.8125

0 0.1 0.2 0.3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

µt z

,r,µ

t′′ z,r

,µ′t′

′ z,r

,τzr,

tot

1 − 2r

(b) z=9.6875

0 0.1 0.2 0.3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

µt z

,r,µ

t′′ z,r

,µ′t′

′ z,r

,τzr,

tot

1 − 2r

(c) z=14.0625

0 0.1 0.2 0.3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

µt z

,r,µ

t′′ z,r

,µ′t′

′ z,r

,τzr,

tot

1 − 2r

(d) z=19.0625

Figure 4-20: Contribution of fluctuating terms on total viscous shear stress for case A at differentstreamwise locations. τzr,tot ( ); µtz,r ( );µt′′

z,r ( ); µ′t′′

z,r ( ).



0 0.1 0.2 0.3-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

µt z

,r,µ

t′′ z,r

,µ′t′

′ z,r

,τzr,

tot

1 − 2r

(a) z=2.8125

0 0.1 0.2 0.3-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

µt z

,r,µ

t′′ z,r

,µ′t′

′ z,r

,τzr,

tot

1 − 2r

(b) z=9.6875

0 0.1 0.2 0.3-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

µt z

,r,µ

t′′ z,r

,µ′t′

′ z,r

,τzr,

tot

1 − 2r

(c) z=14.0625

0 0.1 0.2 0.3-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

µt z

,r,µ

t′′ z,r

,µ′t′

′ z,r

,τzr,

tot

1 − 2r

(d) z=19.0625

Figure 4-21: Contribution of fluctuating terms on total viscous shear stress for case C at differentstreamwise locations. τzr,tot ( ); µtz,r ( );µt′′

z,r ( ); µ′t′′

z,r ( ).



0 0.1 0.2 0.3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

µt z

,r,µ

t′′ z,r

,µ′t′

′ z,r

,τzr,

tot

1 − 2r

(a) z=2.8125

0 0.1 0.2 0.3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

µt z

,r,µ

t′′ z,r

,µ′t′

′ z,r

,τzr,

tot

1 − 2r

(b) z=9.6875

0 0.1 0.2 0.3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

µt z

,r,µ

t′′ z,r

,µ′t′

′ z,r

,τzr,

tot

1 − 2r

(c) z=14.0625

0 0.1 0.2 0.3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

µt z

,r,µ

t′′ z,r

,µ′t′

′ z,r

,τzr,

tot

1 − 2r

(d) z=19.0625

Figure 4-22: Contribution of fluctuating terms on total viscous shear stress for case D at differentstreamwise locations. τzr,tot ( ); µtz,r ( );µt′′

z,r ( ); µ′t′′

z,r ( ).



Energy equation

The averaged energy equation is rewritten for clarity as,

∂ρh

∂t+

1

r

∂rρhur

∂r+

∂ρhuz

∂z=

1

Reτ Pr0

(1

r

∂

∂r(rα

∂h

∂r) +

∂

∂z(α

∂h

∂z) +

1

r

∂

∂r(rα

∂h′′

∂r) +

∂

∂z(α

∂h′′

∂z)

+1

r

∂

∂r(rα′

∂h′′

∂r) +

∂

∂z(α′

∂h′′

∂z)

)− 1

r


r

∂r− ∂ρh′′u′′

z

∂z.

(4-15)

Among the additional closure terms generated, namely: 1r

∂∂r

(rα∂h′′

∂r), ∂

∂z(α∂h′′

∂z), 1

r∂∂r

(rα′ ∂h′′

∂r),

∂∂z

(α′ ∂h′′

∂z) only the radial gradients, i.e. 1

r∂∂r

(rα∂h′′

∂r) and 1

r∂∂r

(rα′ ∂h′′

∂r), play a significant role.

The total radial heat flux can then be written as,

qr,tot = αh,r + αh′′

,r + α′h′′

,r, with h,r =∂h

∂r(4-16)

Again, in the above equation terms αh′′

,r and α′h′′

,r are zero for constant property flows.

Figure 4-23, 4-24 and 4-25 show the comparison of different radial flux terms at differentdownstream locations for cases A, C and D respectively. It is important to note that thetotal heat flux at the wall, qr,tot, for all the cases is equal to 2.4, which is the non-dimensionalconstant wall heat flux used in all the simulations. The averaged heat flux, αh,r, is modifiednot only close to the wall but also at the wall, where its value is higher than 2.4. This can bethought of as a quenching effect due to αh′′

,r and α′h′′

,r, which have negative flux values at thewall. This will result in a higher average enthalpy at the wall, hence affecting also the heattransfer coefficient.

In fact, consider the following thought experiment. The wall temperature distributionsas shown in Figure 4-2 are obtained using a constant wall heat flux. If this temperaturedistribution is used as an isothermal boundary condition in another DNS, the added heat inthe pipe might be higher. The reason for this is the higher heat transfer coefficient obtainedbecause α′h′′

,r will be zero (as α′=0), however αh′′

,r will not vanish (as both, ∂h′′

∂r6= 0 and

α 6= 0 ) and its magnitude might change. Therefore, additional simulations will need to becarried out in the future to verify this effect.

Figure 4-26 shows the derivatives of αh,r, (αh′′

,r + α′h′′

,r) and ρu′′

r h′′, as they appear in theenergy equation, and it can be seen that the magnitude of the additional terms is significantand dominating close to the wall. A one-dimensional analysis is carried out to investigate theinfluence on the enthalpy profile by solving h from,

1

r

∂

∂r(rα

∂h

∂r) = −1

r

∂

∂r(rα

∂h′′

∂r) − 1

r

∂

∂r(rα′

∂h′′

∂r) + Reτ Pr0

1

r


r

∂r, (4-17)

once without, and once with the terms containing αh′′

,r and α′h′′

,r. The values for ρh′′u′′

r ,

(αh′′

,r + α′h′′

,r) and α are obtained using DNS averaging and are frozen during the iterations.

The solution of Equation (4-17) with (αh′′

,r + α′h′′

,r) resembles the DNS enthalpy profile,whereas the solution without the additional term results in a lower wall enthalpy, as theaverage heat flux is higher in the DNS because of the quenching effect (Note: the enthalpyprofiles are pivoted about r=0). Although the one-dimensional analysis does not take intoaccount the coupling of all the terms, it provides a qualitative description on the effect of theadditional terms.



0 0.1 0.2 0.3-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

αh

,r,α

h′′ ,r,α

′h

′′ ,r,q

r,to

t

1 − 2r

(a) z=2.8125

0 0.1 0.2 0.3-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

αh

,r,α

h′′ ,r,α

′h

′′ ,r,q

r,to

t1 − 2r

(b) z=9.6875

0 0.1 0.2 0.3-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

αh

,r,α

h′′ ,r,α

′h

′′ ,r,q

r,to

t

1 − 2r

(c) z=14.0625

0 0.1 0.2 0.3-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

αh

,r,α

h′′ ,r,α

′h

′′ ,r,q

r,to

t

1 − 2r

(d) z=19.0625

Figure 4-23: Contribution of fluctuating terms on total heat flux for case A at different streamwiselocations. qr,tot ( ); αh,r ( );αh′′

,r ( ); α′h′′

,r ( ).



0 0.1 0.2 0.3-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

αh

,r,α

h′′ ,r,α

′h

′′ ,r,q

r,to

t

1 − 2r

(a) z=2.8125

0 0.1 0.2 0.3-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

αh

,r,α

h′′ ,r,α

′h

′′ ,r,q

r,to

t1 − 2r

(b) z=9.6875

0 0.1 0.2 0.3-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

αh

,r,α

h′′ ,r,α

′h

′′ ,r,q

r,to

t

1 − 2r

(c) z=14.0625

0 0.1 0.2 0.3-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

αh

,r,α

h′′ ,r,α

′h

′′ ,r,q

r,to

t

1 − 2r

(d) z=19.0625

Figure 4-24: Contribution of fluctuating terms on total heat flux for case C at different streamwiselocations. qr,tot ( ); αh,r ( );αh′′

,r ( ); α′h′′

,r ( ).



0 0.1 0.2 0.3-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

αh

,r,α

h′′ ,r,α

′h

′′ ,r,q

r,to

t

1 − 2r

(a) z=2.8125

0 0.1 0.2 0.3-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

αh

,r,α

h′′ ,r,α

′h

′′ ,r,q

r,to

t1 − 2r

(b) z=9.6875

0 0.1 0.2 0.3-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

αh

,r,α

h′′ ,r,α

′h

′′ ,r,q

r,to

t

1 − 2r

(c) z=14.0625

0 0.1 0.2 0.3-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

αh

,r,α

h′′ ,r,α

′h

′′ ,r,q

r,to

t

1 − 2r

(d) z=19.0625

Figure 4-25: Contribution of fluctuating terms on total heat flux for case D at different stream-wise locations. qr,tot ( ); αh,r ( );αh′′

,r ( ); α′h′′

,r ( ).



0.45 0.46 0.47 0.48 0.49 0.5-0.2

-0.1

0

0.1

0.2

0.3

∂q

i

∂r

r

Figure 4-26: Derivative of heat flux terms for case A at z = 12.8125. 1r

∂rαh∂r

( ); − 1r

∂rρu′′

rh′′

∂r

( ); 1r

∂r(αh′′+α′h′′))∂r

( ).

0 0.02 0.04 0.06 0.08 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

h

1 − 2r

Figure 4-27: One-dimensional analysis of the energy equation to evaluate the significance ofadditional closure terms for case A at z = 12.8125. DNS ( ); no αh′′

,r + α′h′′

,r and α fromDNS ( ).


4-4 Jensen Inequality 67

4-4 Jensen Inequality

Description

As mentioned earlier, averaged thermophysical properties, specially cp and α, also form aclosure problem, since for a thermophysical property in a turbulent flow

φ 6= φ(h, P0). (4-18)

This is due to the Jensen inequality. This inequality states that the convex transformation ofa mean is less than or equal to the mean after a convex transformation (Jensen, 1906). Therole of Jensen inequality applied to Reynolds/Favre averaging has not been reported in theliterature before and the present work forms a starting point to direct the attention towardsthis finding.

According to the Jensen inequality,∑

pif(xi) ≥ f(∑

pixi), (4-19)

where p1, ....., pn are positive numbers, which sum to 1, and f is a real continuous function

that is convex (∂2f(x)∂x2 > 0). If f is concave (∂2f(x)

∂x2 < 0) the inequality becomes,∑

pif(xi) ≤ f(∑

pixi). (4-20)

Translated into RANS averaging with pi = 1N

(N = NtNθ, which is the averaging operator),for a convex function f we get,

1

N

∑f(xi) ≥ f(

1

N

∑xi). (4-21)

If f is the thermophysical property φ and xi is the enthalpy hi, then Equation (4-21) becomes,

1

N

∑φ ≥ φ(

1

N

∑hi), (4-22)

which can be written asφ ≥ φ(h). (4-23)

Similarly, for a concave φ(h) the inequality becomes,

φ ≤ φ(h). (4-24)

In other words, it is not correct to calculate the averaged thermophysical properties usingan averaged enthalpy or temperature. This inequality, hence generates additional closureproblems as φ is not known.

Figure 4-28 demonstrates the Jensen inequality for all the thermophysical properties usingcase A. The DNS averaged property is compared with the properties calculated using Favreaveraged enthalpy h and Reynolds averaged temperature T . It can be seen that deviationsfor cp and α are large with respect to both h and T . It can also be seen that cp < cp(h) inthe concave part and cp > cp(h) in the convex part. For ρ and µ the inequality is negligiblewith respect to h, however using T shows differences. This is caused by the larger curvatureof the functional relation between temperature and the thermophysical properties. Since theJensen inequality is most significant for cp and α, further discussion will be focused only onthem.



0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

ρ

r

(a) ρ

0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

µ

r

(b) µ

0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2

λ

r

(c) λ

0.25 0.3 0.35 0.4 0.45 0.50

2

4

6

8

10

c p

r

(d) cp

0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

α=

λ cp

r

(e) α = λcp

Figure 4-28: Effect of the Jensen inequality on averaging of thermophysical property (φ) at z =7.8125 for case A. φ ( ); φ(h) ( ); φ(T ) ( )



0.4 0.42 0.44 0.46 0.48 0.50

2

4

6

8

10

~

c p

r

(a) cp

0.4 0.42 0.44 0.46 0.48 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

~

α

r

(b) α

Figure 4-29: Streamwise variation of the Jensen inequality on averaging of thermophysicalproperty (φ) for case A. φ ( ); φ(h) ( ). z = 3.4375(�); 12.8125(△); 19.0625(◦).

Role of the enthalpy variance

Figure 4-29 shows the results of the inequality for cp and α, for case A at different streamwiselocations. At all streamwise locations the values for φ(h) show the same peak value. This isbecause cp(h) and α(h) are interpolated on a continuous distribution of h and at the pseudo-critical point, cp(h) and α(h) have their maximum and minimum, respectively. However, thepeaks of cp and α change their magnitude in streamwise direction. The different peak valuesare due to the turbulent enthalpy fluctuations only, as the functional relation (convexity/con-cavity) between φ and h does not change. The Jensen inequality is an averaging artifact thatdepends on the curvature of the function and on the fluctuations of the averaging quantity.The higher the curvature and the fluctuations, the higher the differences between φ and φ(h).This can be explained by taking a simple example of a convex function f(x) = 4 − (x − 4)2.Considering different fluctuation magnitudes about a point whose average is 4, i.e. x = 4, wecan analyze the Jensen inequality (Figure 4-30). In Figure 4-30(a) x fluctuates in the rangeof 3 to 5, such that x = 4; whereas in Figure 4-30(b) the fluctuation range is small (3.5 to4.5). It can be seen that the Jensen inequality is higher for case with larger fluctuations.

In order to estimate the extent of the Jensen inequality it is suggested to use the enthalpy

variance (h′′2) as a parameter. Figure 4-31 and 4-32 shows the Jensen inequality for cp andα, along with the enthalpy variance for case C and D at two streamwise positions. It can beseen that, the lower the enthalpy variance the lower the difference between φ and φ(h), with

both the values approaching each other as h′′2 goes to zero (towards laminar). This providesa basis for turbulence modelers to account for the extent of Jensen inequality.

The dependence of the Jensen inequality on the enthalpy variance is shown in the previousparagraph, however it is interesting to point out that a sharp curvature of thermophysicalproperties also plays a dominant role in the production of the enthalpy variance. The produc-tion term in the enthalpy variance transport equation (given in Appendix A) can be written



1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

f(x

),f

(x)

x

(a)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

f(x

),f

(x)

x

(b)

Figure 4-30: Role of fluctuation about a given point on the Jensen inequality. f(x) fluctuationrange ( ); f(x) ( ). f(x) = 4 − (x − 4)2. f(x) = 4; f(x) = 3.65 (◦); f(x) = 3.9083(⋄).

as:

Ph′′2

= − 2ρu′′

r h′′∂h

∂r− 2ρu′′

zh′′∂h

∂z+ 2h′′

1

r

∂

∂r(rα

∂h

∂r) + 2h′′

1

r

∂

∂r(rα′

∂h

∂r) + 2h′′

∂

∂z(α

∂h

∂z)

+ 2h′′∂

∂z(α′

∂h

∂z). (4-25)

The two dominant terms are −2ρu′′

r h′′ ∂h∂r

and 2h′′ 1r

∂∂r

(rα∂h∂r

), where the second term canfurther be decomposed as:

2h′′1

r

∂

∂r(rα

∂h

∂r) = 2h′′

∂h

∂r

1

r

∂

∂r(rα) + 2h′′α

∂2h

∂r2. (4-26)

The first term on the right hand side is zero for fluids for which α is constant, howeverfor fluids where α shows large changes, its value becomes significant and contributes to theproduction of the enthalpy variance. Figure 4-33 shows the enthalpy variance together with

the two significant terms of its production rate. The two subcomponents of 2h′′ 1r

∂∂r

(rα∂h∂r

) (as

given in Equation (4-26)) are also shown in Figure 4-33. It shows that 2h′′ ∂h∂r

1r

∂∂r

(rα) has asignificant role in the production of enthalpy invariance. Hence, the large and sharp changesin α are not only subjected to the Jensen inequality, because of the enthalpy variance, butare also one of the contributing factors in the production of the latter.

The significance of the Jensen inequality is evaluated using a similar one-dimensional anal-ysis, as it has been done for evaluating the significance of additional closure terms. Equation(4-17) is solved for h by first using all the terms including α from DNS and then the valueof α is replaced with α(h) using the property table. The results of this analysis are shown inFigure 4-34. It can be seen that the enthalpy profile changes considerably. The comparisonis again made by pivoting the two profiles at r=0.



0.4 0.42 0.44 0.46 0.48 0.50

2

4

6

8

10

0.00E+00

3.00E-04

6.00E-04

9.00E-04

1.20E-03

1.50E-03

c p h′′2

r0.4 0.42 0.44 0.46 0.48 0.50

2

4

6

8

10

0.00E+00

3.00E-04

6.00E-04

9.00E-04

1.20E-03

1.50E-03

c p h′′2

r

(a) case C

0.4 0.42 0.44 0.46 0.48 0.50

2

4

6

8

10

0.00E+00

3.00E-04

6.00E-04

9.00E-04

1.20E-03

1.50E-03

c p h′′2

r0.4 0.42 0.44 0.46 0.48 0.50

2

4

6

8

10

0.00E+00

3.00E-04

6.00E-04

9.00E-04

1.20E-03

1.50E-03

c p h′′2

r

(b) case D

Figure 4-31: The Jensen inequality of specific heat cp as function of the enthalpy variance h′′2.

z = 2.8125 (left column); z = 19.0625 (right column). cp ( ); cp(h) ( ); h′′2 ( ◦ ).



0.4 0.42 0.44 0.46 0.48 0.50

0.2

0.4

0.6

0.8

1

0.00E+00

3.00E-04

6.00E-04

9.00E-04

1.20E-03

1.50E-03

α h′′2

r0.4 0.42 0.44 0.46 0.48 0.50

0.2

0.4

0.6

0.8

1

0.00E+00

3.00E-04

6.00E-04

9.00E-04

1.20E-03

1.50E-03

α h′′2

r

(a) case C

0.4 0.42 0.44 0.46 0.48 0.50

0.2

0.4

0.6

0.8

1

0.00E+00

3.00E-04

6.00E-04

9.00E-04

1.20E-03

1.50E-03

α h′′2

r0.4 0.42 0.44 0.46 0.48 0.50

0.2

0.4

0.6

0.8

1

0.00E+00

3.00E-04

6.00E-04

9.00E-04

1.20E-03

1.50E-03

α h′′2

r

(b) case D

Figure 4-32: The Jensen inequality of α as function of the enthalpy variance h′′2. z = 2.8125

(left column); z = 19.0625 (right column). α ( ); α(h) ( ); h′′2 ( ◦ ).



0.45 0.46 0.47 0.48 0.49 0.5-8.0E-03

-4.0E-03

0.0E+00

4.0E-03

8.0E-03

1.2E-02

0.0E+00

3.0E-04

6.0E-04

9.0E-04

1.2E-03

1.5E-03

Ph

′′2

h′′2

r0.45 0.46 0.47 0.48 0.49 0.5

-8.0E-03

-4.0E-03

0.0E+00

4.0E-03

8.0E-03

1.2E-02

0.0E+00

3.0E-04

6.0E-04

9.0E-04

1.2E-03

1.5E-03

Ph

′′2

h′′2

r

(a) case B

0.45 0.46 0.47 0.48 0.49 0.5-8.0E-03

-4.0E-03

0.0E+00

4.0E-03

8.0E-03

1.2E-02

0.0E+00

3.0E-04

6.0E-04

9.0E-04

1.2E-03

1.5E-03

Ph

′′2

h′′2

r0.45 0.46 0.47 0.48 0.49 0.5

-8.0E-03

-4.0E-03

0.0E+00

4.0E-03

8.0E-03

1.2E-02

0.0E+00

3.0E-04

6.0E-04

9.0E-04

1.2E-03

1.5E-03

Ph

′′2

h′′2

r

(b) case C

0.45 0.46 0.47 0.48 0.49 0.5-8.0E-03

-4.0E-03

0.0E+00

4.0E-03

8.0E-03

1.2E-02

0.0E+00

3.0E-04

6.0E-04

9.0E-04

1.2E-03

1.5E-03

Ph

′′2

h′′2

r0.45 0.46 0.47 0.48 0.49 0.5

-8.0E-03

-4.0E-03

0.0E+00

4.0E-03

8.0E-03

1.2E-02

0.0E+00

3.0E-04

6.0E-04

9.0E-04

1.2E-03

1.5E-03

Ph

′′2

h′′2

r

(c) case D

Figure 4-33: Production rate of the enthalpy variance h′′2. z = 2.8125 (left column); z =

19.0625 (right column). 2h′′ 1r

∂∂r

(rα ∂h∂r

) ( ); 2h′′ ∂h∂r

1r

∂∂r

(rα) ( ); 2h′′α∂2h∂r2 ( );

−2ρu′′

r h′′ ∂h∂r

( ); h′′2 ( ◦ ).



0 0.02 0.04 0.06 0.08 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

h

1 − 2r

Figure 4-34: One-dimensional analysis of the energy equation to evaluate the significance ofthe Jensen inequality for case A at z = 12.8125. DNS ( ); α(h) from the property table( ).


Chapter 5

Summary and Conclusions

In the present work, a comprehensive review on different mechanisms affecting heat transfercharacteristics of supercritical flows is carried out. The physical mechanism of heat transferis then further investigated using inhouse DNS data. In addition, RANS simulations of aheated vertical pipe with supercritical CO2 are carried out using conventional low-Reynoldsnumber eddy viscosity turbulence models. The turbulence models are implemented in aninhouse FORTRAN code and validated using canonical cases available in the literature. Theperformance of the turbulence models is assessed by comparison with the inhouse DNS data.Furthermore, a theoretical framework has been established to take into account the large ther-mophysical property variations close to pseudo-critical point in the RANS averaged governingequations.

The investigation of the heat transfer mechanisms using the inhouse DNS data has beencarried out for four cases with different buoyancy conditions and flow directions (upwardand downward). It is observed that flow acceleration caused by thermal expansion reducesturbulence. In upward flows with small buoyancy, the turbulence is further deterioratedbecause of additional local acceleration. For the case with high buoyancy in an upward flow,a deterioration followed by recovery in turbulence is observed. The recovery onset closelyfollows the velocity profile, which deforms into an M-shape. The buoyancy production ofturbulent kinetic energy is found to be of major significance in this case. It was observedthat when ρu′′

ru′′

z becomes very small, the turbulent kinetic energy still remains considerablebecause of high buoyant turbulent production. This phenomena is rarely observed in flowsundergoing small changes in density (ideal gas), where the buoyancy production term is notas dominant. For downward flows with intermediate buoyancy, an enhancement in turbulenceis observed due to local flow deceleration close to the wall.

The implemented conventional turbulence models are: the MK (Myong and Kasagi, 1990),the LS (Launder and Sharma, 1974) and the V2F model (Behnia et al., 1998). The resultsof the models are inconsistent for different cases applied. The MK model performs the bestamong all investigated models, but fails in cases with significant buoyancy effect. The per-formance of the LS and the V2F model is in general found to be very poor. However, for


76 Summary and Conclusions

downward flows with buoyancy the LS model gives a good match with DNS data. A com-parison with DNS data highlights some of the weaknesses of these turbulence models. It isobserved that the correct formulation of damping functions is very important to capture theheat transfer behavior; it is recommended to create a new formulation of damping functionsusing the DNS data in the future. The buoyant turbulent production model based on ageneralized gradient diffusion hypothesis is found to be completely flawed for all turbulencemodels and needs significant improvement. It is also found that the use of a constant tur-bulent Prandtl number in the energy equation is wrong, specially in buoyancy affected caseswhere it is observed that the relation between Reynolds shear stress and turbulent heat fluxdoes not hold at several places. It is therefore important to solve a separate equation forthe turbulent heat flux. A breakdown of the eddy viscosity hypothesis is also observed forupward flows with high buoyancy, where ρu′′

ru′′

z 6= 0 at locations with the strain rate beingapproximately zero. The ǫ equation is also found to be deficient in some cases.

The RANS equations obtained by taking into account the extreme thermophysical propertyfluctuations resulted in some additional closure terms. These additional closure terms arefound to be of major significance in the energy equation. A one-dimensional analysis iscarried out to evaluate their effect on the enthalpy profile. The analysis resulted in a lowerwall enthalpy if the additional terms are ignored.

The analysis of DNS data also pointed out that cp and α become unknown because of anaveraging artifact called the Jensen inequality. Because of this inequality, the average of athermodynamic property is not equal to the value obtained from the property data using anaveraged state value; φ 6= φ(h, P0). The enthalpy fluctuation is pointed out to be the source

of the Jensen inequality. Hence, it is recommended to use the enthalpy variance h′′2 as one ofthe modeling parameters to close cp and α. The significance of the Jensen inequality on theenthalpy profile is evaluated using a one-dimensional analysis. The enthalpy profile obtainedusing the DNS averaged α (α) is compared with the one obtained using α from the propertytable α(h, P0). A significant difference between the two profiles is observed.


Appendix A

Transport Equations

The transport equations for Reynolds stresses (u′′

r u′′

r , u′′

θu′′

θ , u′′

zu′′

z , u′′

r u′′

z) and enthalpy variance

(h′′h′′) are presented for a statistically homogeneous (in θ direction) turbulent flow.

Transport equation for u′′

r u′′

r

∂ρu′′

r u′′

r

∂t+

1

r

∂rurρu′′

ru′′

r

∂r+

∂uzρu′′

ru′′

r

∂z= − 2ρu′′

r u′′

r

∂ur

∂r− 2ρu′′

r u′′

z

∂ur

∂z

− 1

r

∂rρu′′

ru′′

r u′′

r

∂r− ∂ρu′′

r u′′

r u′′

z

∂z+ 2

ρu′′

r u′′

θu′′

θ

r

− 21

r

∂rp′u′′

r

∂r+ 2

1

rp′

∂ru′′

r

∂r− 2u′′

r

∂p

∂r

+ 2u′′

r

r

∂rτrr

∂r+ 2

u′′

r

r

∂τrθ

∂θ− 2

u′′

r τθθ

r+ 2u′′

r

∂τrz

∂z(A-1)


θ u′′

θ

∂ρu′′

θu′′

θ

∂t+

1

r

∂rurρu′′

θu′′

θ

∂r+

∂uzρu′′

θu′′

θ

∂z= − 2

urρu′′

θu′′

θ

r

− 1

r

∂rρu′′

ru′′

θu′′

θ

∂r− ∂ρu′′

θu′′

θu′′

z

∂z− 2

ρu′′

r u′′

θu′′

θ

r

+ 21

rp′

∂u′′

θ

∂θ

+ 2u′′

θ

r2

∂r2τθr

∂r+ 2

u′′

θ

r

∂τθθ

∂θ+ 2u′′

θ

∂τθz

∂z(A-2)


78 Transport Equations


z u′′

z

∂ρu′′

zu′′

z

∂t+

1

r

∂rurρu′′

zu′′

z

∂r+

∂uzρu′′

zu′′

z

∂z= − 2ρu′′

r u′′

z

∂uz

∂r− 2ρu′′

zu′′

z

∂uz

∂z

− 1

r

∂rρu′′

ru′′

zu′′

z

∂r− ∂ρu′′

zu′′

zu′′

z

∂z

− 2∂u′′

z p′

∂z+ 2p′

∂u′′

z

∂z− 2u′′

z

∂p

∂z

+ 2u′′

z

r

∂rτzr

∂r+ 2

u′′

z

r

∂τzθ

∂θ+ 2u′′

z

∂τzz

∂z(A-3)


r u′′

z

∂ρu′′

r u′′

z

∂t+

1

r

∂rurρu′′

r u′′

z

∂r+

∂uzρu′′

r u′′

z

∂z= − ρu′′

r u′′

r

∂uz

∂r− ρu′′

zu′′

z

∂ur

∂z− ρu′′

zu′′

r

∂ur

∂r− ρu′′

r u′′

z

∂uz

∂z

− 1

r

∂rρu′′

ru′′

r u′′

z

∂r− ∂ρu′′

r u′′

zu′′

z

∂z+

ρu′′

θu′′

θu′′

z

r

− ∂u′′

zp′

∂r− ∂u′′

r p′

∂z

+ p′∂u′′

z

∂r+ p′

∂u′′

r

∂z− u′′

z

∂p

∂r− u′′

r

∂p

∂z

+u′′

z

r

∂rτrr

∂r− u′′

zτθθ

r+ u′′

z

∂τrz

∂z+

u′′

r

r

∂rτzr

∂r

+u′′

r

r

∂τzθ

∂θ+ u′′

r

∂τzz

∂z(A-4)

where,

u′′

r

r

∂rτrr

∂r=

u′′

r

r

∂r(µ + µ′)(trr + t′′

rr)

∂r=

u′′

r

r

∂rµtrr

∂r+

u′′

r

r

∂rµ′trr

∂r+

1

r

∂ru′′

r µt′′

rr

∂r+

1

r

∂ru′′

r µ′t′′

rr

∂r

− µt′′

rr

∂u′′

r

∂r− µ′t′′

rr

∂u′′

r

∂r

u′′

r

r

∂τrθ

∂θ=

u′′

r

r

∂(µ + µ′)(trθ + t′′

rθ)

∂θ=

u′′

r

r

∂µtrθ

∂θ+

u′′

r

r

∂µ′trθ

∂θ+

1

r

∂u′′

r µt′′

rθ

∂θ+

1

r

∂u′′

r µ′t′′

rθ

∂θ

− µ

rt′′

θr

∂u′′

r

∂θ− µ′t′′

θr

r

∂u′′

r

∂θ

u′′

r τθθ

r=

u′′

r(µ + µ′)(tθθ + t′′

θθ)

r=

u′′

r µtθθ

r+

u′′

r µ′tθθ

r+

u′′

r t′′

θθµ

r+

u′′

rµ′t′′

θθ

r


79

u′′

r

∂τrz

∂z= u′′

r

∂(µ + µ′)(trz + t′′

rz)

∂z=u′′

r

∂µtrz

∂z+ u′′

r

∂µ′trz

∂z+

∂u′′

rµt′′

rz

∂z+

∂u′′

rµ′t′′

rz

∂z

− µt′′

rz

∂u′′

r

∂z− µ′t′′

rz

∂u′′

r

∂z

u′′

θ

r2

∂r2τθr

∂r=

u′′

θ

r2

∂r2(µ + µ′)(tθr + t′′

θr)

∂r=

u′′

θ

r2

∂r2µtθr

∂r+

u′′

θ

r2

∂r2µ′tθr

∂r+

1

r2

∂r2u′′

θµt′′

θr

∂r

+1

r2

∂r2u′′

θµ′t′′

θr

∂r− µt′′

θr

∂u′′

θ

∂r− µ′t′′

θr

∂u′′

θ

∂r

u′′

θ

r

∂τθθ

∂θ=

u′′

θ

r

∂(µ + µ′)(tθθ + t′′

θθ)

∂θ=

u′′

θ

r

∂µtθθ

∂θ+

u′′

θ

r

∂µ′tθθ

∂θ+

1

r

∂u′′

θµt′′

θθ

∂θ+

1

r

∂u′′

θµ′t′′

θθ

∂θ

− µ

rt′′

θθ

∂u′′

θ

∂θ− µ′t′′

θθ

r

∂u′′

θ

∂θ

u′′

θ

∂τθz

∂z= u′′

θ

∂(µ + µ′)(tθz + t′′

θz)

∂z=u′′

θ

∂µtθz

∂z+ u′′

θ

∂µ′tθz

∂z+

∂u′′

θµt′′

θz

∂z+

∂u′′

θµ′t′′

θz

∂z

− µt′′

θz

∂u′′

θ

∂z− µ′t′′

θz

∂u′′

θ

∂z

u′′

z

r

∂rτzr

∂r=

u′′

z

r

∂r(µ + µ′)(tzr + t′′

zr)

∂r=

u′′

z

r

∂rµtzr

∂r+

u′′

z

r

∂rµ′tzr

∂r+

1

r

∂ru′′

zµt′′

zr

∂r+

1

r

∂ru′′

zµ′t′′

zr

∂r

− µt′′

zr

∂u′′

z

∂r− µ′t′′

zr

∂u′′

z

∂r

u′′

z

r

∂τzθ

∂θ=

u′′

z

r

∂(µ + µ′)(tzθ + t′′

zθ)

∂θ=

u′′

z

r

∂µtzθ

∂θ+

u′′

z

r

∂µ′tzθ

∂θ+

1

r

∂u′′

zµt′′

zθ

∂θ+

1

r

∂u′′

zµ′t′′

zθ

∂θ

− µ

rt′′

zθ

∂u′′

z

∂θ− µ′t′′

zθ

r

∂u′′

z

∂θ

u′′

z

∂τzz

∂z= u′′

z

∂(µ + µ′)(tzz + t′′

zz)

∂z=u′′

z

∂µtzz

∂z+ u′′

z

∂µ′tzz

∂z+

∂u′′

zµt′′

zz

∂z+

∂u′′

zµ′t′′

zz

∂z

− µt′′

zz

∂u′′

z

∂z− µ′t′′

zz

∂u′′

z

∂z



where,

trr = (2∂ur

∂r− 2

3(1

r

∂rur

∂r+

1

r

∂uθ

∂θ+

∂uz

∂z))

tθθ = (2(1

r

∂uθ

∂θ+

ur

r) − 2

3(1

r

∂rur

∂r+

1

r

∂uθ

∂θ+

∂uz

∂z))

tzz = (2∂uz

∂z− 2

3(1

r

∂rur

∂r+

1

r

∂uθ

∂θ+

∂uz

∂z))

trθ = tθr = (∂

∂r(uθ

r) +

1

r

∂ur

∂θ)

trz = tzr = (∂uz

∂r+

∂ur

∂z)

tzθ = tθz = (∂uθ

∂z+

1

r

∂uz

∂θ)

t′′

rr = (2∂u′′

r

∂r− 2

3(1

r

∂ru′′

r

∂r+

1

r

∂u′′

θ

∂θ+

∂u′′

z

∂z))

t′′

θθ = (2(1

r

∂u′′

θ

∂θ+

u′′

r

r) − 2

3(1

r

∂ru′′

r

∂r+

1

r

∂u′′

θ

∂θ+

∂u′′

z

∂z))

t′′

zz = (2∂u′′

z

∂z− 2

3(1

r

∂ru′′

r

∂r+

1

r

∂u′′

θ

∂θ+

∂u′′

z

∂z))

t′′

rθ = t′′

θr = (∂

∂r(u′′

θ

r) +

1

r

∂u′′

r

∂θ)

t′′

rz = t′′

zr = (∂u′′

z

∂r+

∂u′′

r

∂z)

t′′

zθ = t′′

θz = (∂u′′

θ

∂z+

1

r

∂u′′

z

∂θ)


81

Transport equation for h′′h′′

∂ρh′′h′′

∂t+

1

r

∂rurρh′′h′′

∂r+

∂uzρh′′h′′

∂z= − 2ρu′′

r h′′∂h

∂r− 2ρu′′

zh′′∂h

∂z+ 2h′′

1

r

∂

∂r(rα

∂h

∂r)

+ 2h′′1

r

∂

∂r(rα′

∂h

∂r) + 2h′′

∂

∂z(α

∂h

∂z) + 2h′′

∂

∂z(α′

∂h

∂z)

− 1

r

∂rρu′′

r h′′h′′

∂r− ∂ρu′′

zh′′h′′

∂z

+ 21

r

∂

∂r(rαh′′

∂h′′

∂r) + 2

1

r

∂

∂r(rα′h′′

∂h′′

∂r)

+ 2∂

∂z(αh′′

∂h′′

∂z) + 2

∂

∂z(α′h′′

∂h′′

∂z)

− 2α∂h′′

∂r

∂h′′

∂r− 2α′

∂h′′

∂r

∂h′′

∂r

− 2α∂h′′

∂z

∂h′′

∂z− 2α′

∂h′′

∂z

∂h′′

∂z(A-5)


Appendix B

Turbulence Model Validation andMesh Independency

B-1 Validation of Turbulence Models

The validation of turbulence models shown in this section are generated from the inflow gen-erator using either Reτ or Reb as the non-dimensional parameter. Inflow generator representsa case where all thermophysical properties are constant i.e an isothermal case. After thevalidation of code for an isothermal case, the LS model of the code is validated for ideal gasheat transfer wherein density is variable and buoyancy effect also comes into action.

Myong and Kasagi model (MK)

The MK model is validated using data from Hrenya et al. (1995). The test case is a turbulentpipe flow at Reb = 5300 and uses Reb as non-dimensional parameter. Figure B-1 showsthe validation result for normalized turbulent kinetic energy k/u2

τ,0 and normalized turbulentdissipation rate ǫ/u4

τ,0.

Launder and Sharma model (LS)

The LS model is validated using data from Keshmiri et al. (2012). The first test case forthe LS model is a constant property turbulent flow in a pipe with Reb = 5300, here againReb is used as a non-dimensional parameter. Figure B-2 shows the validation result for thefirst test case, it shows the comparison of streamwise velocity (uz), Reynolds stress (ρu′′

r u′′

z),turbulent kinetic energy (k) and damping function (fµ). The second test case consists ofan ideal gas flow undergoing heat transfer under buoyancy effect. The buoyancy effect isaccounted by using Boussinesq approximation. Figure B-3 shows the validation result forsecond case. The validation results in this case correspond to fully developed profiles for caseC of Keshmiri et al. (2012), which represents moderate amount of buoyancy and results inflow laminarization.


84 Turbulence Model Validation and Mesh Independency

v2-f model (V2F)

The V2F model is validated using data from Pecnik and Iaccarino (2007). The test case is aturbulent channel flow with Reτ = 590 (based on half channel width), the non-dimensionalparameter used here is Reτ . Figure B-4 shows the validation results for the V2F model, itshows the streamwise velocity (uz), turbulent kinetic energy (k), turbulent dissipation rate(ǫ), wall normal fluctuation (v2) and elliptic relaxation parameter (f).

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

3.5

4

k/u

2 τ,0

r

(a) k

0.4 0.42 0.44 0.46 0.48 0.50

0.05

0.1

0.15

0.2

ǫ/u

4 τ,0

r

(b) ǫ

Figure B-1: Validation results for the MK model. present: ( ). literature: (◦).


B-1 Validation of Turbulence Models 85

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

uz

r

(a) uz

0 0.1 0.2 0.3 0.4 0.50

0.0005

0.001

0.0015

0.002

0.0025

0.003

ρu

′′ ru

′′ z

r

(b) ρu′′

r u′′

z

0 0.1 0.2 0.3 0.4 0.50

0.002

0.004

0.006

0.008

0.01

0.012

0.014

k

r

(c) k

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

f µ

r

(d) fµ

Figure B-2: Validation results for the LS model. present: ( ). literature: (◦).



0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

uz

r

(a) uz

0 0.1 0.2 0.3 0.4 0.5-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

ρu

′′ ru

′′ z

r

(b) ρu′′

r u′′

z

0 0.1 0.2 0.3 0.4 0.50

0.0005

0.001

0.0015

0.002

k

r

(c) k

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

f µ

r

(d) fµ

Figure B-3: Validation results for the LS model applied to ideal gas heat transfer with buoyancy,validation result correspond to case C of Keshmiri et al. (2012). present: ( ). literature: (◦).


B-1 Validation of Turbulence Models 87

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14

16

18

20

22

uz

r

(a) uz

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

k

r

(b) k

0 0.1 0.2 0.3 0.4 0.50

20

40

60

80

100

120

ǫ

r

(c) ǫ

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

v2

r

(d) v2

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

7

8

9

10

f

r

(e) f

Figure B-4: Validation results for the V2F model. present: ( ). literature: (◦).



B-2 Mesh Independency

Mesh is refined in the radial direction by increasing the number of mesh in radial directionfrom 96 to 120, number of mesh along axial direction is 128. Mesh refinement did not changethe solution for any of the turbulence models. Figure B-5, B-6 and B-7 shows the meshindependency results applied to case A for the MK, the LS and the V2F model respectively.The figure compares the streamwise velocity (uz), enthalpy profile (h), Reynolds shear stress(ρu′′

r u′′

z) and radial turbulent heat flux (ρu′′

rh′′). Similar study done for axial direction (figurenot shown) also didn’t result in any change in solution. Hence, the chosen mesh could beconsidered as adequate.

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

uz

r

(a) uz

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

h

r

(b) h

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

ρu

′′ ru

′′ z/u

2 τ,0

r

(c) ρu′′

r u′′

z

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

−ρu

′′ rh

′′/q

∗

r

(d) ρu′′

r h′′

Figure B-5: Mesh independency study of case A for the MK model. nr,max = 96: z =0.3125( ); 6.5625( ); 14.6875( ); 19.0625( ). nr,max = 120: z = 0.3125(�);6.5625(△); 14.6875(⋄); 19.0625(◦).


B-2 Mesh Independency 89

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

uz

r

(a) uz

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

hr

(b) h

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

ρu

′′ ru

′′ z/u

2 τ,0

r

(c) ρu′′

r u′′

z

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

−ρu

′′ rh

′′/q

∗

r

(d) ρu′′

r h′′

Figure B-6: Mesh independency study of case A for the LS model. nr,max = 96: z =0.3125( ); 6.5625( ); 14.6875( ); 19.0625( ). nr,max = 120: z = 0.3125(�);6.5625(△); 14.6875(⋄); 19.0625(◦).



0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

uz

r

(a) uz

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

hr

(b) h

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

ρu

′′ ru

′′ z/u

2 τ,0

r

(c) ρu′′

r u′′

z

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

−ρu

′′ rh

′′/q

∗

r

(d) ρu′′

r h′′

Figure B-7: Mesh independency study of case A for the V2F model. nr,max = 96: z =0.3125( ); 6.5625( ); 14.6875( ); 19.0625( ). nr,max = 120: z = 0.3125(�);6.5625(△); 14.6875(⋄); 19.0625(◦).


Appendix C

Shear Stress Distribution

In order to investigate the development of viscous dominant region, viscous shear stress andturbulent shear stress are plotted (using DNS data) as a function of y+ at different down-stream positions for cases A, C and D. The intersecting point of these two stresses gives ainterpretation of the viscous dominant region. This analogy of determining viscous dominantregion is not appropriate for case C where Reynolds shear stress approaches zero whereasturbulent kinetic energy still exhibits considerable values. Figure C-1, C-2 and C-3 shows theshear stress distribution for case A, C and D respectively. For case A, it can be seen that theviscous dominant region moves slightly, first towards the wall and then away. For case D, theregion moves significantly towards the wall. Note that in Figure C-2 sign of Reynolds shearstress is reversed at locations z=14.6875 and 19.0625 owing to negative values of Reynoldsshear stress there.


92 Shear Stress Distribution

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

ρu

′′ ru

′′ z,−

µR

eτ(∂

uz

∂r

+∂

ur

∂z

)

y+

(a) z=0.3125

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

ρu

′′ ru

′′ z,−

µR

eτ(∂

uz

∂r

+∂

ur

∂z

)y+

(b) z=6.5625

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

ρu

′′ ru

′′ z,−

µR

eτ(∂

uz

∂r

+∂

ur

∂z

)

y+

(c) z=14.6875

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

ρu

′′ ru

′′ z,−

µR

eτ(∂

uz

∂r

+∂

ur

∂z

)

y+

(d) z=19.0625

Figure C-1: Shear stress distribution, based on DNS data for case A. ρu′′

r u′′

z ( ); µReτ

(∂uz

∂r+

∂ur

∂z)( ).


93

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

ρu

′′ ru

′′ z,−

µR

eτ(∂

uz

∂r

+∂

ur

∂z

)

y+

(a) z=0.3125

0 4 8 12 16 20 24 28 32 36 40

0

0.4

0.8

1.2

1.6

2

2.4

ρu

′′ ru

′′ z,−

µR

eτ(∂

uz

∂r

+∂

ur

∂z

)y+

(b) z=6.5625

0 4 8 12 16 20 24 28 32 36 40-2

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

2

2.4

−ρu

′′ ru

′′ z,−

µR

eτ(∂

uz

∂r

+∂

ur

∂z

)

y+

(c) z=14.6875

0 4 8 12 16 20 24 28 32 36 40-2

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

2

2.4

−ρu

′′ ru

′′ z,−

µR

eτ(∂

uz

∂r

+∂

ur

∂z

)

y+

(d) z=19.0625

Figure C-2: Shear stress distribution, based on DNS data for case C. ρu′′

r u′′

z ( ); µReτ

(∂uz

∂r+

∂ur

∂z)( ).


94 Shear Stress Distribution

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

ρu

′′ ru

′′ z,−

µR

eτ(∂

uz

∂r

+∂

ur

∂z

)

y+

(a) z=0.3125

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

ρu

′′ ru

′′ z,−

µR

eτ(∂

uz

∂r

+∂

ur

∂z

)y+

(b) z=6.5625

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

ρu

′′ ru

′′ z,−

µR

eτ(∂

uz

∂r

+∂

ur

∂z

)

y+

(c) z=14.6875

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

ρu

′′ ru

′′ z,−

µR

eτ(∂

uz

∂r

+∂

ur

∂z

)

y+

(d) z=19.0625

Figure C-3: Shear stress distribution, based on DNS data for case D. ρu′′

r u′′

z ( ); µReτ

(∂uz

∂r+

∂ur

∂z)( ).


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