CENTRO TECNOLÓGICO
PROGRAMA DE PÓS-GRADUAÇÃO EM ENGENHARIA MECÂNICA
VICTOR WAGNER FREIRE DE AZEVEDO
NEAR-INTERFACE TURBULENCE STRUCTURE IN CO-CURRENT GAS-LIQUID STRATIFIED FLOWS
FLORIANÓPOLIS 2020
Near-interface turbulence structure in co-current gas-liquid stratified flows
Tese submetida ao Programa de Pós-Graduação em Engenharia Mecânica da Universidade Fed-eral de Santa Catarina para a obtenção do título de Doutor em Engenharia Mecânica.
Orientador: Emilio Ernesto Paladino
Florianópolis 2020
Ficha de identificação da obra elaborada pelo autor,
através do Programa de Geração Automática da Biblioteca Universitária da UFSC.
de Azevedo, Victor Wagner Freire
Near-interface turbulence structure in co-current gas liquid stratified flows / Victor Wagner Freire de Azevedo ; orientador, Emilio Ernesto Paladino, 2020.
162 p.
Tese (doutorado) - Universidade Federal de Santa
Catarina, Centro Tecnológico, Programa de Pós-Graduação em Engenharia Mecânica, Florianópolis, 2020.
Inclui referências.
1. Engenharia Mecânica. 2. Turbulência próximo à interface. 3. Simulação numérica direta. 4. Escoamento estratificado. I. Paladino, Emilio Ernesto. II.
Universidade Federal de Santa Catarina. Programa de Pós Graduação em Engenharia Mecânica. III. Título.
Near-interface turbulence structure in co-current gas-liquid stratified
flows
O presente trabalho em nível de doutorado foi avaliado e aprovado por banca examinadora composta pelos seguintes membros:
Prof. Francisco José de Souza, Dr. Universidade Federal de Uberlândia Prof. Jader Riso Barbosa Jr., Ph.D. Universidade Federal de Santa Catarina Prof. Juan Pablo de Lima Costa Salazar, Ph.D.
Universidade Federal de Santa Catarina
Certificamos que esta é a versão original e final do trabalho de conclusão que foi julgado adequado para obtenção do título de Doutor em Engenharia Mecânica.
Prof. Jonny Carlos da Silva, Dr. Eng. Coordenador do Programa
Emilio Ernesto Paladino Orientador
Florianópolis, 05 de Fevereiro de 2020 Documento assinado digitalmente Emilio Ernesto Paladino Data: 06/03/2020 10:22:48-0300 CPF: 006.446.479-22
Assinado de forma digital por Jonny
Carlos da Silva:51451506449
Acknowledgements
First of all, I would like to thank professor Emilio Paladino for all these years of work, friendship and commitment to the challenge that was the present thesis.
To Rafael de Cerqueira for the immeasurable support during most of the steps of the present work.
Special thanks also to Josiane Weise for the many conversations, support and coffees.
To Conrado Zanutto for the friendship and support in Florianópolis and Magdeburg, Germany.
To professor Berend van Wachem for the reception and support at Otto-von-Guericke Universität, in Magdeburg.
To Fabian Denner and Fabien Evrard for the support and beers during my staying in Magdeburg.
To professors Clovis Raimundo Maliska and Antonio Fabio for the reception once again at Sinmec Lab.
To Matthew Stegmeir for support in LDV alignment and operation.
To engineers Thales and Rafael Fank for the support during experimental bench building process and measurements.
To all friends that shared Sinmec Lab during these past years of work, cakes and coffees: Tati, Taísa, Hermínio, Ricieri, Fernando, Gustavo, Gio, Lucas, Aideé, Freitas, Felipe, Adriano, Andreza and Carlos.
To Federal Rural University of the Semiarid Region, in Mossoró, for allowing the development of the present work.
At last, to my wife Luana, my brother Kael and my parents Wagner and Claudia for the immense support in all moments.
“All we have to decide is what to do with the time that is given us”.
Resumo
Escoamentos estratificados estão presentes em diferentes processos na indústria e fenômenos naturais. Embora diferentes modelos de fechamento para este tipo de escoamento estejam disponíveis na literatura, o estudo da estrutura da turbulência na região próxima à interface liquido-gás e suas diferenças e semelhanças com a turbulência nas regiões próximas à parede, mais conhecida, é um tópico ainda aberto na literatura. Poucos estudos na literatura analisaram a estrutura da turbulência nas proximidades de interfaces gás-liquido em escoamentos estratificados e os efeitos de variáveis fundamentais neste tipo de escoamento como altura de filme e tensão superficial não foram analisados. A presente tese estuda a estrutura da turbulência próximo à interface considerando um escoamento estratificado co-corrente limitado por paredes. O estudo numérico é desenvolvido utilizando Simulação Numérica Direta (DNS) baseado no método VOF (do inglês "volume of fluid") para resolver ambas as fases do escoamento levando em conta diferentes tensões superficiais e alturas de filme. A partir destas simulações de grande porte, foi possível obter diferentes estatísticas da turbulência e sua estrutura nas proximidades da interface e paredes. Diferenças na estrutura da turbulência são estudadas com base nos parâmetros descritos, isto é, altura de filme e tensão superficial, e é mostrado que enquanto a tensão superficial tem uma pequena influência no amortecimento da turbulência, a influência da altura do filme é significativa. Os resultados permitiram também observar as diferenças e semelhanças da estrutura da turbulência nas proximidades da interface e da parede. Os principais parâmetros, como velocidade medias e tensões turbulentas, não apresentaram em geral, diferenças significativas, mostrando que o uso de relações de fechamento baseadas em similaridade com o escoamento próximo de paredes é aceitável. No entanto, estatísticas de maior ordem, como dissipação de energia cinética turbulenta, apresentaram diferenças mais significativas. Adicionalmente, um estudo experimental é realizado utilizando Velocimetria por laser Doppler (LDV) para a caracterização da turbulência no filme de líquido baseado em estatísticas de baixa ordem. Resultados experimentais, no escoamento estratificado, mostraram a influência da tensão na interface na velocidade e flutuações no filme de líquido.
Palavras-chave: Turbulência próximo à interface; Simulação Numérica Direta; Escoa-mento estratificado
Resumo Expandido
IntroduçãoEscoamentos multifásicos são encontrados em muitas aplicações industriais e também em fenômenos naturais. A turbulência, neste contexto, é um fenômeno que influencia diretamente a transição entre padrões e o comportamento do escoamento em diferentes regiões do domínio. A interface é a região do domínio onde os mecanismos de troca tanto de massa quanto de quantidade de movimento ocorrem, sendo assim, o estudo do comportamento da turbulência próximo à interface permite a avaliação de como diferentes parâmetros do escoamento influenciam ou são influenciados nessa região. Para esse estudo, o padrão de escoamento estratificado é uma excelente forma de se analisar a interação entre as fases com relação à turbulência. Numericamente, uma das formas mais utilizadas para estudo da turbulência é a simulação numérica direta ("DNS", em inglês). A DNS permite o estudo do fenômeno da turbulência sem a utilização de nenhum modelo de aproximação para modelagem do fenômeno, o que fornece resultados mais confiáveis numericamente. Neste contexto, a influência da parede do escoamento na região próximo à interface, descrita pela variação da altura do filme no escoamento, e a tensão superficial, são objetos de estudo ainda pouco explorados. Uma outra forma de se estudar a turbulência próximo à interface em escoamentos estratificados é através de uma análise experimental. Os métodos não-intrusivos, onde figura-se a velocimetria por Laser Doppler ("LDV", em inglês), são bastante utilizados em diferentes bancadas experimentais para descrever diferentes aspectos do comportamento da turbulência próximo à interface em escoamentos estratificados.
Objetivos
O objetivo geral deste trabalho é estudar numericamente e experimentalmente o com-portamento da turbulência próximo à interface em escoamentos estratificados levando em consideração a influência da variação da altura de filme de líquido no escoamento e da tensão superficial. Isso pode ser descrito em 4 objetivos específicos: a) desenvolver um modelo para DNS em escoamentos estratificados em canais utilizando o Método dos Volumes Finitos e o método "Volume of Fluid" para capturar a interface; b) desenvolver uma simulação numérica direta para escoamentos monofásicos; c) estudar a influência da tensão superficial e altura de filme no amortecimento da turbulência próximo à interface e d) desenvolver uma bancada experimental para medir estatísticas da turbulência próximo à interface utilizando LDV.
Metodologia
Para o desenvolvimento do estudo numérico da turbulência próximo à interface foi utilizado o código computacional Multiflow. O código discretiza as equações governantes seguindo o
Dessa forma, o escoamento estratificado foi implementado no código considerando água e ar como fluidos (fases líquida e gás, respectivamente), separados por uma interface cujo valor da tensão superficial é especificado. O domínio é contornado por paredes sem escorregamento e condições periódicas foram aplicadas nas direções transversal e longitudinal do escoamento, que é movido por um gradiente de pressão constante. As fases foram inicializadas separadamente, como escoamentos monofásicos, e posteriormente juntadas em um único domínio, formando o domínio do escoamento estratificado. As simulações foram rodadas variando-se a tensão superficial e altura de filme no referido domínio. No estudo experimental, uma bancada experimental foi construída considerando uma seção quadrada em acrílico. Água e ar foram injetados na bancada em diferentes posições de forma que o padrão de escoamento estratificado fosse formado. Diferentes vazões de água e ar foram testadas e os resultados foram obtidos através do LDV.
Resultados e discussões
No estudo numérico, os resultados para o escoamento estratificado mostraram que a tensão superficial amortece a turbulência próximo à interface quando considera-se uma maior altura de filme, mas que não possui influência significativa quando considera-se a menor altura de filme. A altura de filme, por sua vez, mostrou ser um parâmetro que influencia diretamente a estrutura da turbulência próximo à interface – mudanças puderam ser notadas quando compararam-se os resultados próximo à parede e próximo à interface. No estudo experimental, pode ser notada a influência da fase gás na fase líquida quando a velocidade do escoamento foi alterada. Notou-se uma maior influência do gás na turbulência do líquido quando aumentou-se a vazão do gás, mesmo para elevadas vazões de líquido.
Conclusões
Os resultados obtidos comprovaram as afirmações presentes na literatura de que a turbulên-cia próximo à interface é amortecida em relação à parede, adicionalmente, o amortecimento foi investigado quanto à variação da tensão superficial e da altura de filme no escoamento estratificado, que apresentaram diferentes formas de influenciar o escoamento: o aumento na tensão superficial provocou um amortecimento maior na maior altura de filme e não apresentou influência na menor altura de filme. Além disso, uma metodologia para solução de escoamentos turbulentos tendo como base o Método dos Volumes Finitos e a utilização de um método de captura da interface foi validada.
Palavras-chave: Turbulência próximo à interface; Simulação Numérica Direta; Escoa-mento estratificado.
Abstract
Stratified flows are present in many different industrial processes and environmental phenomena. Even though some closure models to this kind of flow can be encountered in literature, a detailed understanding of turbulence structure behaviour in the gas-liquid near-interface region compared to well-known near-wall region behaviour is a topic still open in literature. Some studies in literature analysed turbulence structure in the near-interface region of gas-liquid stratified flows, but the effects of fundamental variables in this kind of flow, such as surface tension and film height need further analysis. The present thesis is focused on the study of near-interface turbulence structure considering a co-current stratified flow situation bounded by no-slip walls. Direct Numerical Simulation is applied in a Volume of Fluid framework to solve both phases based on different surface tension coefficients and film heights. These simulations allowed the determination of different turbulence flow statistics and structure in near-wall and -interface regions. Based on the described parameters, i.e., film height and surface tension, differences in turbulence structure are shown; while surface tension has a small influence in turbulence damping, film height influence is expressive. Results also allowed the comparison of turbulence structure between near-wall and -interface regions. Parameters like mean velocity and Reynolds stresses did not show significant differences between the cases, which lead to the conclusion that the application of closure relations based on near-wall-similarity relations is acceptable for the gas phase in stratified flows. Nonetheless, higher-order statistics, such as viscous dissipation rate, showed significant differences. In addition, an experimental study is performed with laser Doppler Velocimetry for the characterization of the turbulent field in the liquid film based on low-order statistics in a stratified flow. The experimental results showed the influence of interfacial shear stress in liquid film velocity and fluctuations.
List of symbols
α Volume fraction a Wave amplitude β Clustering factor Co Courant number cσ Phase velocity df Fringe spacingδ Length scale, interpolation coefficient
δf Film height
η Kolmogorov length scale
Dissipation rate per unit mass
Eii Energy spectrum of autocorrelation functions
F r Froude number
fI Interface friction factor
fG Gas phase friction factor
F Focal length
fσ Surface tension force
g Gravity
Hi,k Interface height
jG Gas phase superficial velocity
jL Liquid phase superficial velocity
kc Critical wavenumber
k Turbulent kinetic energy, Wavenumber
λ Wavelength, Taylor microscale
ν Fluid kinematic viscosity
ω Vorticity
ω0 Vorticity fluctuation
ω0 Angular frequency
p Pressure
QG Gas phase volumetric flow rate
QL Liquid phase volumetric flow rate
Re Reynolds number
Reτ Shear-based Reynolds number
Rλ Taylor-scale Reynolds number
Rii Autocorrelation function of velocity fluctuations
r Curvature radius
ρ Fluid density
σ Surface tension coefficient
t Time
tls Nondimensional time
τη Kolmogorov time scale
τw Wall shear stress
τI Interface shear stress
Ub Bulk mean velocity
ui Velocity
u0i Velocity fluctuation
uτ Friction velocity based on wall shear stress
uI Interface velocity
We Weber number
yI+ Nondimensional normal direction with relation to interface
Abbreviations
CBC Convective Boundedness Criterion
CELESTE Curvature Evaluation with Least-Squares fit for Taylor Expansion CICSAM Compressive Interface Capturing Scheme for
CSF Continuum Surface Force Arbitrary Meshes DNS Direct numerical simulation
FVM Finite volume method
HC Hyper-C
HF Height Function
HRIC High-Resolution Interface Capturing LES Large-eddy simulation
LDV Laser Doppler Velocimetry MPI Message Passing Interface NVD Normalised Variable Diagram PIV Particle Image Velocimetry
PLIC Piecewise Linear Interface Construction SLIC Simple Line Interface Calculation TKE Turbulent kinetic energy
UQ ULTIMATE QUICKEST
List of Figures
Figure 1 – Flow patterns in horizontal flow. . . 30
Figure 2 – Flow patterns in vertical flow. . . 31
Figure 3 – Interface tracking and capturing methods. (a) Interface markers; (b) Interface-fitted mesh; (c) Phase defined by a volume fraction.. . . 38
Figure 4 – Averaged volume compared with bubble volume. . . 38
Figure 5 – LDV measurement principle. . . 45
Figure 6 – Laser Doppler Velocimetry technique. . . 46
Figure 7 – Sketch of the idealized stratified flow. . . 49
Figure 8 – Sketch of the stratified flow simulated problem. The normal boundaries are free slip boundaries. . . 51
Figure 9 – Comparison of mean velocities and shear stress profiles time-averaged in the streamwise direction. . . 53
Figure 10 – Mean streamwise velocity profiles for different air superficial velocities in the liquid side. The column to the right of the legend denotes the mean wave amplitude in mm. . . 57
Figure 11 – Mean vertical velocity profiles for different air superficial velocities. (a) hw = 21 mm; (b) hw = 31 mm. A positive value means an upward movement of the flow . . . 58
Figure 12 – CICSAM scheme applied to unstructured mesh. . . 64
Figure 13 – Stencil for the height function calculation. The index (0, 0) denotes the current cell where the height function is being evaluated based on the presented stencil. . . 66
Figure 14 – Sketch of test case domain with the initialised capillary wave. . . 69
Figure 15 – Normalized wave amplitude in function of the time τ = tω0. The computational meshes have constant spacing in the x- and y-direction of λ/40 and λ/100. . . . 70
Figure 16 – Normalized wave amplitude for the computational mesh with λ/40 spacing and near-interface refinement compared with the mesh with no refinement. . . 70
Figure 17 – Schematic representation of periodic boundary conditions. . . 73
Figure 18 – Relation between mesh size (∆) and Kolmogorov scale for the different meshes used in this study, along the wall normal direction. . . 74
Figure 19 – Non-dimensional velocity profile for different computational meshes. . . 79
Figure 20 – Normalized velocity fluctuations for Reτ = 180. .) denotes Mesh 1, (--) Mesh 2, (:) Mesh 3 and ((--) Mesh 4. The dots represent the results of Moser, Kim e Mansour (1999).. . . 79
Mansour (1999), presented in the first half of the domain y/δ < 1.0. . 80 Figure 22 – Autocorrelation functions in the streamwise direction at center line: (a)
Autocorrelation of u-fluctuations, (b) Autocorrelation of v-fluctuations and (c) Autocorrelation of w-fluctuations. . . 81 Figure 23 – Autocorrelation functions in the streamwise direction near the wall: (a)
Autocorrelation of u-fluctuations, (b) Autocorrelation of v-fluctuations and (c) Autocorrelation of w-fluctuations. . . 82 Figure 24 – Autocorrelation functions in the spanwise direction at center line: (a)
Autocorrelation of u-fluctuations, (b) Autocorrelation of v-fluctuations and (c) Autocorrelation of w-fluctuations. . . 83 Figure 25 – Autocorrelation functions in the spanwise direction near the wall: (a)
Autocorrelation of u-fluctuations, (b) Autocorrelation of v-fluctuations and (c) Autocorrelation of w-fluctuations. . . 84 Figure 26 – One-dimensional energy spectra at center line in the streamwise
direc-tion: (a) Spectra of u-fluctuations, (b) Spectra of v-fluctuations and (c) Spectra of w-fluctuations. . . 85 Figure 27 – One-dimensional energy spectra at the near-wall region in the streamwise
direction: (a) Spectra of u-fluctuations, (b) Spectra of v-fluctuations and (c) Spectra of w-fluctuations. . . 86 Figure 28 – Turbulent kinetic energy budget. (-.) denotes Mesh 1, (- -) Mesh 2, (:)
Mesh 3 and (-) Mesh 4 results. The symbols are the results of Moser et al. (MOSER; KIM; MANSOUR, 1999). . . 89 Figure 29 – Quadrant analysis of the Reynolds shear stress. (-.) denotes Mesh 1
results, (- -) Mesh 2, (:) Mesh 3 and (-) Mesh 4. . . 91 Figure 30 – Ratio between the normalized values of ejections (Quadrant 2) and
sweeps (Quadrant 4). . . 91 Figure 31 – Contour of streamwise fluctuations for y-normal plane at (a) center line
and (b) near-wall region for Mesh 4. . . 92 Figure 32 – Contour of normal fluctuations for y-normal plane at (a) center line and
(b) near-wall region for Mesh 4. . . 92 Figure 33 – Contour of spanwise fluctuations for y-normal plane at (a) center line
and (b) near-wall region for Mesh 4. . . 93 Figure 34 – Sketch of flow initialization process. The phases are run separately and
then merged in one single domain. . . 99 Figure 35 – Initial velocity field for the stratified flow. The gas phase on top and
the liquid phase in the bottom half. The colors indicate the velocity magnitude of each of the phases. . . 101
Figure 36 – Computational mesh with refinement near the walls and interface gen-erated for stratified flow simulations. . . 101 Figure 37 – Non-dimensional velocity profile in the interface reference for the gas
phase. The dashed lines denote the wall-reference values and the contin-uous lines the interface-reference. . . 104 Figure 38 – Velocity-rms fluctuations in the gas phase in the near-interface region.
(-) denote cases with smaller surface tension (Cases "a") and (- -) denote cases with larger surface tension (Cases "b"). (a) δf = 2δ (Case 1a and
1b); (b) δf = δ (Case 2a and 2b). . . . 105
Figure 39 – Reynolds shear stress for δf = 2δ: (a) gas phase and (b) liquid phase.
(--) denotes cases with larger surface tension and ((--) denotes cases with smaller surface tension. . . 106 Figure 40 – Reynolds shear stress for δf = δ. (a) gas phase and (b) liquid phase.
(--) denotes cases with larger surface tension and ((--) denotes cases with smaller surface tension. . . 106 Figure 41 – Normalized vorticity fluctuations compared in the near-wall (- -) and
-interface (-) regions for Cases 1 with F r/We = 1.2 (a) δf = 2δ; (b) δf = δ. . . . 108
Figure 42 – Normalized vorticity fluctuations in the near-interface region for different surface tension coefficients: (a) δf = 2δ; (b) δf = δ. Solid line denotes
cases with smaller surface tension F r/We = 1.2 (Cases "a") and dots denote cases with larger surface tension F r/We = 1.98 (Cases "b"). . . 108 Figure 43 – Autocorrelation functions in the streamwise direction for the
near-interface: (a) streamwise velocity fluctuations; (b) normal velocity fluc-tuations and (c) spanwise velocity flucfluc-tuations. (-) denotes cases with smaller surface tension F r/We = 1.2 (Cases "a") and (- -) denotes cases with larger surface tension F r/We = 1.98 (Cases "b"). The lines were obtained at yI+ ≈ 4 for all cases. . . 110 Figure 44 – One-dimensional energy spectra of autocorrelation functions based on
near-interface (-) and -wall (- -) references for δf = 2δ: (a) streamwise
components; (b) normal components and (c) spanwise components. The black line denotes cases with smaller surface tension F r/We = 1.2 (Cases "a") and red line denotes cases with larger surface tension F r/We = 1.98 (Cases "b"). The lines were obtained at y+i and y+ ≈ 4 in the respective reference. . . 111
near-interface (-) and -wall (- -) references for δf = δ: (a) streamwise
components; (b) normal components and (c) spanwise components. The black line denotes cases with smaller surface tension F r/We = 1.2 (Cases "a") and red line denotes cases with larger surface tension F r/We = 1.98 (Cases "b"). The lines were obtained at y+i and y+ ≈ 4 in the respective reference. . . 112 Figure 46 – Turbulent kinetic energy budget comparison between near-wall and
-interface values for δf = 2δ. (a) F r/We = 1.2 (Case 1a) and (b) F r/We = 1.98 (Case 1b). (- -) denotes near-wall values and (-) denotes
near-interface values. . . 113 Figure 47 – Turbulent kinetic energy budget comparison between near-wall and
-interface values for δf = δ. (a) F r/We = 1.2 (Case 2a) and (b) F r/We
= 1.98 (Case 2b). (- -) denotes wall values and (-) denotes near-interface values. . . 114 Figure 48 – Turbulent kinetic energy budget comparison between near-interface
values for (a) δf = 2δ and (b) δf = δ. Solid line denotes cases with
smaller surface tension (Cases "a") and dots denote cases with larger surface tension (Cases "b"). . . 114 Figure 49 – Total turbulent kinetic energy comparison between cases with different
surface tension coefficient and film heights. (a) Cases with δf = 2δ
(Cases 1) and (b) Cases with δf = δ (Cases 2). (-) denotes Cases "a"
(smaller surface tension) and (- -) denotes Cases "b" (larger surface tension). . . 115 Figure 50 – Planes presentation scheme for the flow visualization results. . . 116 Figure 51 – Streamwise velocity-rms fluctuations for (a) F r/We = 1.2 (Case 1a)
and (b) F r/We = 1.98 (Case 1b) in the near-interface region (yi+≈ 4) for the gas phase and δf = 2δ. . . . 117
Figure 52 – Streamwise velocity-rms fluctuations for (a) F r/We = 1.2 (Case 1a) and (b) F r/We = 1.98 (Case 1b) in the near-interface region (yi+≈ 4) for the liquid phase and δf = 2δ. . . . 117
Figure 53 – Streamwise velocity-rms fluctuations for (a) F r/We = 1.2 (Case 2a) and (b) F r/We = 1.98 (Case 2b) in the near-interface region (yi+≈ 4) for the gas phase and δf = δ. . . . 118
Figure 54 – Streamwise velocity-rms fluctuations for (a) F r/We = 1.2 (Case 2a) and (b) F r/We = 1.98 (Case 2b) in the near-interface region (yi+≈ 4) for the liquid phase and δf = δ. . . . 118
Figure 55 – Spanwise vorticity-rms fluctuations hω+
zrmsi for (a) F r/We = 1.2 (Case
1a) and (b) F r/We = 1.98 (Case 1b) at x-z plane with y+i ≈ 4 for
δf = 2δ for the gas phase. . . . 119
Figure 56 – Spanwise vorticity-rms fluctuations hω+
zrmsi for (a) F r/We = 1.2 (Case
1a) and (b) F r/We = 1.98 (Case 1b) at x-z plane with y+i ≈ 4 for
δf = δ for the gas phase. . . . 120
Figure 57 – Spanwise vorticity-rms fluctuations hω+
zrmsi for (a) F r/We = 1.2 (Case
1a) and (b) F r/We = 1.98 (Case 1b) at x-z plane with y+i ≈ 4 for
δf = 2δ for the liquid phase. . . . 120
Figure 58 – Spanwise vorticity-rms fluctuations hω+
zrmsi for (a) F r/We = 1.2 (Case
1a) and (b) F r/We = 1.98 (Case 1b) at x-z plane with y+i ≈ 4 for
δf = δ for the liquid phase.. . . 121
Figure 59 – Spanwise vorticity-rms fluctuations hωzrmsi for (a) F r/We = 1.2 (Case
1a) and (b) F r/We = 1.98 (Case 1b) at center x-y plane for δf = 2δ. . 121
Figure 60 – Spanwise vorticity-rms fluctuations hωzrmsi for (a) F r/We = 1.2 (Case
2a) and (b) F r/We = 1.98 (Case 2b) at center x-y plane for δf = δ. . . 122
Figure 61 – Reynolds stress Du0iu0jE for the (a) gas and (b) liquid phases for δf = 2δ
cases. (-) denotes √F r/We = 1.2 and (- -) denotes √F r/We = 1.98
results. The dots denote DNS results, included for comparison in the gas phase. . . 124 Figure 62 – Velocity-rms fluctuations components in the wall and interface references:
(a) streamwise, (b) normal and (c) spanwise components. The dots represent DNS results for the refined mesh. . . 126 Figure 63 – Autocorrelation functions of the velocity fluctuations in the streamwise
direction at y+ ≈ 7 x-z plane: (a) streamwise, (b) normal and (c) spanwise components. The dots represent DNS results for the refined mesh. . . 127 Figure 64 – Turbulent kinetic energy budget for near-interface region compared with
near-wall region for (a) √F r/We = 1.2 and (b) √F r/We = 1.98. The
dots represent DNS results for the refined mesh. . . 128 Figure 65 – Flow loop scheme with the test section and devices used in the
mea-surements. . . 130 Figure 66 – Schematic view of the experimental bench structure. The gray parts are
the aluminium profiles that support the channel structure. The acrylic test section is indicated and presented in scale.. . . 131 Figure 67 – LDV system components presentation. . . 135 Figure 68 – Measurement of one point with LDV. The picture show the probe with
the laser beams forming the measurement volume in the test section. . 136 Figure 69 – Schematic view of the flat window geometry and probe lens. . . 137
velocities: (a) mean values and (b) non-dimensionalized values compared with the ideal log-law (dashed line). . . 140 Figure 71 – Mean streamwise velocity for jL ≈ 0.04 m/s for different air superficial
velocities: (a) mean values and (b) non-dimensionalized values compared with the ideal log-law (dashed line). . . 141 Figure 72 – Mean streamwise velocity for jL ≈ 0.055 m/s for different air superficial
velocities: (a) mean values and (b) non-dimensionalized values compared with the ideal log-law (dashed line). . . 141 Figure 73 – LDV non-dimentionalized mean streamwise velocity compared with
DNS results for the liquid phase. The solid line (-) represents the liquid phase velocity non-dimentionalized by uτliq obtained in the simulation
for F r/We = 1.2 and δf = 2δ: (a) Reτ ≈ 100 and (b) Reτ ≈ 180.. . . . 142
Figure 74 – Mean vertical velocity for (a) jL ≈ 0.03 m/s, (b) jL ≈ 0.04 m/s, (c) jL ≈ 0.055 m/s and (d) jL ≈ 0.07 m/s for different air superficial
velocities. . . 143 Figure 75 – Turbulent intensities profiles for jL ≈ 0.03 m/s for different air superficial
velocities: (a) streamwise fluctuations urms =
√
< u0u0 > and (b) vertical fluctuations vrms=
√
< v0v0 >.. . . 144 Figure 76 – Turbulent intensities profiles for jL ≈ 0.04 m/s for different air superficial
velocities: (a) streamwise fluctuations urms =
√
< u0u0 > and (b) vertical fluctuations vrms=
√
< v0v0 >.. . . 145 Figure 77 – Turbulent intensities profiles for jL ≈ 0.055 m/s for different air
super-ficial velocities: (a) streamwise fluctuations urms=
√
< u0u0 > and (b) vertical fluctuations vrms =
√
List of Tables
Table 1 – Fluids properties for capillary wave test case. . . 68 Table 2 – Computational meshes parameters. ∆y+
c denotes the non-dimensional
mesh spacing at domain center.. . . 75 Table 3 – Computational meshes parameters with relation to the Kolmogorov and
Taylor length scale. . . 75 Table 4 – A posteriori calculated values for Reτ and Kolmogorov length and time
scales: η and τη. . . 77
Table 5 – Fluid properties. . . 96 Table 6 – Phases initialization parameters. The pressure gradients are different
for δf = 2δ and δf = δ cases. The pressure gradient values specified are
applied during initialization process for each phase.. . . 99 Table 7 – Computational model parameters for ReτL ≈ 100 cases. δL denotes the
liquid film height of the respective simulation.. . . 102 Table 8 – Computational model parameters for initialization cases. The average
∆y+ value for the phases is shown. . . . 102 Table 9 – A posteriori calculated values for Reτ in each phase and Kolmogorov
length and time scales for stratified flow: η and τη. . . 103
Table 10 – Phases initialization parameters for Reτ ≈ 180 cases. The pressure
gradient values specified are applied during initialization process for each phase. . . 123 Table 11 – Computational model parameters for initialization cases. The averaged
∆y+ value for the phases is shown. . . 123 Table 12 – A posteriori calculated values for Reτ and Kolmogorov length and time
scales: η and τη. . . 124
Table 13 – Experimental parameters followed in the present study. The liquid veloc-ity was kept constant and the gas velocveloc-ity varied in a limited range of Reynolds numbers ReG. The denoted Cases 1 are obtained considering Re∗L ≈ 100, Cases 2 considering Re∗
L ≈ 130 and Cases 3 considering Re∗L≈ 171. j denotes the superficial velocity of the respective fluid and
u the estimated mean velocity. . . . 132 Table 14 – Volumetric flow rate of the different points. The values needed to be
rounded in order adequate to the rotameter resolutions. . . 133 Table 15 – Specifications for 2-Beam Probe with TLN06-350 lens. The wavelength
of 514.5 nm is for the green beam (used in the vertical channel) and the value of 488 nm is for the blue beam (used in the streamwise channel). . 138
Contents
1 INTRODUCTION . . . 29 1.1 Two-phase flows patterns . . . 29 1.2 Near-interface turbulence . . . 31 1.3 Motivation . . . 34 1.4 Objectives . . . 34 1.4.1 Specific objectives . . . 34 1.5 Thesis structure . . . 35
2 BACKGROUND AND LITERATURE REVIEW . . . 37
2.1 The Volume of Fluid Method . . . 37 2.2 Surface Tension and Interface curvature calculation . . . 41 2.2.1 Direct Differentiation Methods . . . 42 2.2.2 CELESTE . . . 43 2.2.3 Height function method . . . 44 2.3 Laser Doppler Velocimetry . . . 44 2.4 Length and time scales in turbulent flows . . . 46 2.5 Review of recent literature . . . 50 2.5.1 Numerical study of the near-interface turbulence . . . 50 2.5.2 Experimental study of near-interface turbulence in stratified flow . . . 56 3 NUMERICAL IMPLEMENTATION . . . 61 3.1 Overview of the numerical code . . . 62 3.1.1 CICSAM . . . 63 3.1.2 Height function method . . . 65 3.2 Solitary capillary wave test case . . . 67 3.3 Single-phase channel flow . . . 71 3.3.1 Model set-up . . . 72 3.4 Results for single-phase channel flow . . . 78 4 STRATIFIED CHANNEL FLOW . . . 95 4.1 Model set-up . . . 95 4.1.1 Time-step constraint . . . 98 4.1.2 Flow initialization . . . 98 4.1.3 Simulations cases . . . 100 4.2 Results . . . 103 4.2.1 Mean velocity and fluctuations . . . 103
4.2.3 Vorticity fluctuations . . . 107 4.2.4 Autocorrelation functions and spectra . . . 109 4.2.5 Turbulent kinetic energy budget . . . 113 4.2.6 Flow structure visualization . . . 116 4.3 Results for Reτ ≈ 180 cases . . . 122
5 EXPERIMENTAL RESULTS . . . 129 5.1 Experimental bench design . . . 129 5.1.1 Flow data specification . . . 130 5.2 LDV data processing . . . 132 5.3 Experimental campaign . . . 134 5.4 Results . . . 138 6 SUMMARY AND CONCLUSIONS . . . 147
Bibliography . . . 151
A – TURBULENT STATISTICS CALCULATION . . . 161 A.1 Velocity mean and fluctuations . . . 161 A.2 Autocorrelation functions and spectra . . . 162 A.3 Turbulent kinetic energy budget . . . 162
29
1 Introduction
Multiphase flows can be encountered in several industries, in particular, oil, gas and energy transformation; and in nature. The turbulence in multiphase flows has been extensively studied in the literature. However, it is an extremely complex phenomenon as it combines the well-known complexity of turbulence with the presence of interfaces, which can affect flow structure. For that reason, different ways to solve such flows have been proposed lately. The turbulence phenomena near interfaces are important in stratified flows as it will influence the interfacial transfer mechanisms and flow pattern transition.
Even though many researchers have dedicated time to study turbulence, it is always a challenge to study this phenomenon due to the complexity of the flow and modeling, even for single-phase flows. Besides that, the detailed study of a turbulent flow demands a large computational power because of the many length scales involved in flows. With the advancements in technology, higher Reynolds numbers can be simulated with more refined computational meshes, which are contributing to the understanding of turbulence structures in different types of flow.
Regarding multiphase flows, besides the development of the velocity field, the challenge consists mainly of maintaining the interface sharp, avoiding its breakup, and properly solving the volume fraction field. For this reason, the scientific community yet not reached a high Reynolds number taking into account interface deformations in stratified flows, compared to single-phase flow research.
1.1
Two-phase flows patterns
Multiphase flows appear in several industrial processes, from pump flows to nuclear reactors. During the past years, many models were implemented to predict the different flow patterns behaviours in many different situations (COLLIER; THOME,1994;ISHII; HIBIKI,2010). The variation in fluid superficial velocity generates different multiphase flow patterns – from the bubbly flow (characterized by the presence of dispersed bubbles in a continuous core) to the annular flow (characterized by the presence of an annular film in a continuous core).
The way that phases arrange in horizontal and vertical flows is different owing to the influence of gravity. In general, the stratified flow pattern is characterized by the presence of the liquid phase in the lower part of the domain and the gas phase in the upper part. In stratified flows, the interface can be smooth or wavy, depending on the flow conditions. The transfer mechanisms of heat and mass through the interface are directly
Figure 1 – Flow patterns in horizontal flow.
Source: Azzopardi(2006).
related to the near-interface turbulence structure.
The multiphase flows patterns were formalized in order to describe the configurations observed in gas-liquid flows. Initially, the patterns were established by visual observation in transparent flow sections – for horizontal and vertical flows – and this remains the primary definition. Varying the fluid superficial velocity the pattern changes from bubbly to annular flow, as shown in Figure1.
When compared to the vertical flow patterns, the main change in horizontal patterns is due to the gravity influence, which makes the patterns more complex. The gravity, in horizontal flows, imposes a non-symmetry in the flow.
Looking at Figure 1, the bubbly flow is characterized by the presence of dispersed gas bubbles in a continuous liquid core. Gravity tends to make bubbles accumulate in the upper part of the domain. Increasing the fluid superficial velocity, the stratified flow pattern appears. In this pattern, the liquid flows in the lower part of the domain with the gas above it. The interface is initially smooth, however, increasing the gas superficial velocity waves are formed on the interface – stratified wavy pattern. The transition from smooth to the wavy pattern – that will develop to slug – is initially governed by the capillary waves. The plug flow pattern is characterized by the presence of a Taylor bubble in the domain surrounded by a thin film of liquid. The bubble travels through the top of the domain. The slug flow is like the plug flow with bigger liquid bubbles. The "slugs" have a diameter in the size of the domain and contain many smaller gas bubbles inside it. If the "slug" does not fill the domain completely, the pattern can be called semi-slug
flow, although this pattern can be considered as part of the wavy flow. The last observed
1.2. Near-interface turbulence 31
near the walls of the domain with a continuous gas core. Some liquid can be entrained as drops in the gas core. Because of the gravity effect, the film is thicker on the bottom of the domain, but with an increase in gas superficial velocity, the film becomes more circumferentially uniform (AZZOPARDI, 2006).
As commented before, the vertical flow patterns are almost similar to the horizontal ones, as presented in Figure 2. Patterns vary from bubble flow to annular flow, passing through the churn flow, according to fluid superficial velocity. The main difference from the horizontal pattern is the symmetry presented. The film thickness, presented in the annular pattern, is the same in all domains in the vertical pattern, which does not happen in the horizontal pattern – the gravity causes the film to be more concentrated at the bottom of the domain.
Figure 2 – Flow patterns in vertical flow.
Source: Azzopardi(2006).
1.2
Near-interface turbulence
The presence of deformable interfaces between fluids affects directly turbulence structure as interfacial shear and buoyancy influence the many turbulence length scales in flows. The presence of waves also introduces some fluctuations in the velocity field as interface deformation results in an increase of form drag, which affects directly turbulence behaviour. According to Fulgosi et al.(2003), the main interest in turbulence studies is how to relate the transfer mechanisms with the surface tension, and how they are connected with the turbulence structures.
Researches in turbulence look to reveal the turbulence aspect through the solution of the Navier-Stokes equations. The many orders of magnitude length scales that are
present in the flow make the prediction of the turbulence behaviour very difficult and hard to predict.
During the past decades, different approaches were developed for turbulent flow study. Models like RANS (Reynolds-averaged Navier-Stokes equations) and LES (Large-eddy Scales) have been improved in previous works and are demonstrating to reliably predict turbulent flow behaviour in different situations (HAMBA,2003;ARGYROPOULOS; MARKATOS, 2015). Nonetheless, such models apply mathematical models to approach the flow behaviour, leaving for the Direct Numerical Simulations (DNS) the task of solving the turbulent flow without the application of any numerical approach. DNS solves the Navier-Stokes equations in complete form (without using any averaging or filtering procedure), giving full details of the flow in time and space (MOIN; MAHESH, 1998). Hence, to simulate a turbulent flow, a small time-step and an extremely refined mesh are needed in order to capture the many turbulent scales. The direct numerical simulations provide high fidelity results, promoting a priori or a posteriori validations for theories or/and experiments. However, DNS is restricted to simple geometries and low Reynolds numbers because of the high computational cost.
Direct Numerical Simulations play an increasingly important role in the development of closure models for turbulence. The main goal is to provide a better understanding of the basic behaviour of multiphase flows as well as to provide data for the generation of closure models for engineering simulations of averaged flow fields (TRYGGVASON et al., 2006). Since stratified flows are present in many applications, this kind of flow is the best way to study the flow in the near-interface region. Lombardi, de Angelis e Banerjee (1996) developed the first model to study the near-interface turbulence using DNS in a gas-liquid stratified flow. The coupling at the interface region was made by the velocity and shear stress. It was concluded that the near-interface turbulent structures at the gas side are similar to the near-wall structures. This model was further worked by de Angelis, Lombardi e Banerjee(1997), which developed a DNS for a wavy non-deformable interface, against the plain interface of the previously cited work.
The study of deformable interfaces was first developed byFulgosi et al. (2001) and complemented later byFulgosi et al.(2003) using the pseudospectral method developed by de Angelis, Lombardi e Banerjee (1997). The flow was modeled at the gas side, where the analogy interface-wall can be applied. The results confirmed that analogy, but the simulations were performed for a low Reynolds number. The shear stress tensor fluctuations analysis showed that the flow is dampened near the interface, which resulted in the proposition of a damping function by Lakehal, Reboux e Liovic(2005) and will be described later on this work.
The great majority of the works in literature solve the turbulent flow using spectral methods, even in multiphase flow situations, because of its high order interpolation
1.2. Near-interface turbulence 33
functions, which provide precision in the calculation of high-order flow statistics. However, the finite-difference-based methods (like finite volume method – FVM or finite element method – FEM) are starting to take their place in recent works owing to the suitability of these methods to parallelization techniques and applicability to complex geometries. The main issue, according to Sengupta, Mashayek e Jacobs (2008), of using FVM is the solution of dissipative scales, where the low-order schemes adopted by this method when compared against the high-order of spectral methods, present poor results with a large presence of truncation errors (KRAVCHENKO; MOIN,1997). This problem is solved with the application of a much more refined computational mesh in simulations that apply FVM to discretize the governing equations when compared to models that use spectral methods. Besides that, in multiphase flow situations, the need for the application of an interface capture method along with interface curvature calculation techniques demand the solution of the system of equations in the space-time domain, rather than the frequency domain (as in spectral methods), when the solution of the entire domain is of interest (coupled solution between the phases).
The computational study previously developed with DNS, regarding stratified flows, was later developed using large eddy simulation (LES). The main objective was the decrease in simulation time providing results close to DNS results, as in the work of Liovic e Lakehal (2007), where a model based on LES was introduced to model the heat and mass transfer across the interface. The results obtained for the interfacial flow, using LES and a sub-scale model (SGS) with the damping function obtained by Lakehal, Reboux e Liovic (2005), were compared to the DNS results, satisfactorily reproducing them.
Besides the numerical studies, stratified flow can be also analysed based on an experimental approach. The main goal of the experimental research in turbulence is to characterize the interaction of the different parameters with the flow by real flow measurements, once the full reproduction of the flow in both phases is not straightforward in numerical studies, as will be described later in the present thesis. The experimental results can be used to validate a numerical model or as a basis for the development of theoretical models.
The experimental works involving the near-interface turbulence analysis were mainly developed using Laser Doppler Velocimetry (LDV) and Particle Image Velocimetry (PIV). Both methods are non-intrusive and are highly recommended for interfacial flow studies. Among the published works found in the literature, most of them used PIV (AYATI et al., 2014;KUMARA; HALVORSEN; MELAAEN,2010), however, this technique is not precise at interface capturing, although interface capturing can be possible with it (BIRVALSKI et al., 2014).
The studies involving near-interface turbulence structure, although improved in the past years, still leave some unclear topics that need better analysis. Despite its high
computational cost, the numerical experiments using DNS still are the best way to study the turbulence structure near the interface.
1.3
Motivation
The previously mentioned works in Section 1.2 modeled computationally and experimentally the near-interface behaviour of the stratified flow. However, some topics still need attention to complement the understanding of this phenomenon.
The influence of variables like the surface tension in the near-interface flow statistics in a coupled gas-liquid solution still needs more studies. Another parameter of interest is the real influence of the wall in the interface region; in other words, how the distance of the interface from the wall affects flow statistics? This topic was first mentioned in the work published by Fernandino e Ytrehus(2008), but its relation with the surface tension coefficient is still unknown, as far as this research went. Additionally, it is interesting to check the real limitation of the flow DNS solution based on the Finite Volume and VOF methods, which is closer to what can be found in modern commercial software, for instance.
1.4
Objectives
The main objective of this work is the study of the near-interface turbulence behaviour in a stratified gas-liquid flow numerically and experimentally. The character-ization will be done through the flow development using DNS with advanced interface capture methods and curvature calculation. The experimental study will be developed on a stratified flow using Laser Doppler Velocimetry (LDV) for the characterization of the flow statistics. The influence of surface tension and film height in the near-interface turbulence will be analysed and how these variables influence the turbulence damping in this region.
1.4.1
Specific objectives
• Develop a model for DNS in stratified co-current flow in a channel, using finite volume method in volume of fluid (VOF) framework;
• Develop a direct numerical simulation for channel single phase flows using finite-volume-method;
• Study the influence of the surface tension and film height in near-interface turbulence damping;
1.5. Thesis structure 35
• Develop an experimental bench for LDV measurements and measure flow statistics in stratified flow situation for different flow velocities.
1.5
Thesis structure
The present work briefly reviews the theoretical fundamentals of the present study in Chapter 2, which include a literature review of the main published works in near-interface turbulence research. In Chapter 3 the basis of the numerical code adopted for the simulations is explained along with the description of the contributions made to it in the present work. Still in the referred chapter, DNS of a single-phase channel flow is performed to serve as a basis for the multiphase flow DNS formulation in Chapter 4, the main contribution of the present work. At last, the developed experimental bench for the study of stratified flows is described and the experimental procedure commented in Chapter 5. The conclusions and suggestions for future works regarding this topic and the methodology developed in the present thesis are present in Chapter 6.
37
2 Background and literature review
This chapter presents a theoretical background and literature review that supports the proposed study. First, the theoretical fundamentals of the flow modeling that will be used in the numerical part of the study, including an overview of the fundamentals of the Volume of Fluid method and interface curvature calculation, are presented in Sections 2.1 and 2.2. The working principle of the LDV, that was used in the experimental approach in the present work, is described in Sec. 2.3, and comments about turbulent scales are presented in Sec. 2.4. The chapter is concluded with the review of recent literature that served as a basis for the development of the present work in Sec. 2.5.
2.1
The Volume of Fluid Method
This section presents an overview of multiphase flow models and provides a basic understanding of the models adopted in this work.
The solution of multiphase flows involves the solution of the governing equations for both phases coupled by the interface. The interface motion can be modelled with different methods, which include the interface capturing and interface tracking methods, represented in Fig. 3. In the interface tracking methods the exact position of the interface is known and tracked from an initial position through a Lagrangian tracking, and, in the interface capturing methods, the interface is defined implicitly from the volume fraction field, which must be accurately resolved.
Figure 3presents different interface tracking and capturing methods. The method known as Lagrangian interface tracking is presented in Fig. 3a and defines the interface by the motion of markers over a fixed mesh. The interface fitting method is presented in Fig.3b and "fit" computational mesh to the interface. This method allows an easy application of jump conditions in the interface region but is not capable of working with high interface deformations. These approximations belong to the interface tracking methods, which also include the front-tracking (TRYGGVASON et al., 2001) method.
The interface capturing method is a volume-based method. Thus, the interface is defined in the volumes by a volume fraction, as presented in Fig 3c. Unlike the interface tracking method, the position of the interface in this method is extracted implicitly from the volume fraction field. The Volume of Fluid method (VOF), which will be used in the present work, is an interface capturing method.
The VOF method, developed byHirt e Nichols (1981), is used to distinguish two fluids and is among the most used methods in two-phase flows modeling with large scale
Figure 3 – Interface tracking and capturing methods. (a) Interface markers; (b) Interface-fitted mesh; (c) Phase defined by a volume fraction.
(a) (b) (c)
Source: Adapted from Ubbink(1997).
Figure 4 – Averaged volume compared with bubble volume.
Source: Adapted from Gopala e van Wachem (2008)
interfaces. The method assigns a volume fraction α to each phase, in every mesh cell, representing the local value, defined as
α(x, t) = 1, fluid A 0, fluid B .
Hence, a mesh cell with volume fraction value between 0 and 1 characterizes an interface region. Following the description of Gopala e van Wachem (2008) of the VOF method, the volume fraction can be defined as
• α = 1: control volume is filled with phase A • α = 0: control volume is filled with phase B • 0 < α < 1: interface region.
The governing equations are volume-averaged, following the cases as presented in Fig.4. In the present case, the average was done compared with the bubble size.
2.1. The Volume of Fluid Method 39 • Case 1: – Mass conservation: ∂hρAi ∂t + ∇ · hρAuAi = 0 (2.1) – Momentum: ∂hρAuAi ∂t + ∇ · hρAuAuAi = ∇ · hTAi + hρAgi (2.2) • Case 2: – Mass conservation: ∂hρBi ∂t + ∇ · hρBuBi = 0 (2.3) – Momentum: ∂hρBuBi ∂t + ∇ · hρBuBuBi = ∇ · hTBi + hρBgi (2.4) • Case 3: – Mass conservation: ∂hαρAi ∂t + ∇ · hαρAuAi = 0 (2.5) ∂h(1 − α)ρBi ∂t + ∇ · h(1 − α)ρBuBi = 0 (2.6) – Momentum: ∂hαρAuAi ∂t + ∇ · hαρAuAuAi = ∇ · hαTAi + hαρAgi + 1 V Z SI TA· nIABdS (2.7) ∂h(1 − α)ρBuBi ∂t + ∇ · h(1 − α)ρBuBuBi = ∇ · h(1 − α)TBi + h(1 − α)ρBgi + 1 V Z SI TB· nIABdS (2.8) – Jump condition: 1 V Z SI [−TA· nIAB− TB· nIAB] dS = − 1 V Z SI mσABdS (2.9)
The fluid properties are discontinuous at the interface and are based on the volume fractions
ρ = hαρA+ (1 − α)ρBi (2.10)
and
Thus, the equations solved with the VOF method are the mass conservation, momentum conservation and volume fraction advection,
∂ρu ∂t + ∇ · ρu = 0, (2.12) ∂ρu ∂t + ∇ · (ρuu) = −∇p + ∇ · T + ρg + fσ (2.13) and ∂α ∂t + ∇ · (uα) = 0. (2.14)
It is important to use an interface reconstruction method to properly capture the interface position. Two approaches can be applied for this: compressive methods and geometric methods. The compressive methods discretize the VOF equation (Eq.2.14) using standard numerical differencing schemes. These methods use spatial advection approaches based on the donor-acceptor approach, which basic formulation idea is that the volume fraction in the downwind cell (called the "acceptor") is used to predict the volume fraction transported through it in a time-step. However, this approach may lead to non-physical results, which makes necessary the application of a filter in the solution of the equations to avoid non-realistic values for the volume fraction (values greater than 1 and lesser than 0).
Another compressive method used to capture interfaces are the High Resolution differencing schemes (HR schemes). The convective transport equations are discretized with higher order differencing schemes in this method, which includes CICSAM (Compressive Interface Capturing Scheme for Arbitrary Meshes), developed by Ubbink e Issa (1999).
Otherwise, the interface can be transported using a geometric method, which means that a representation of the interface is reconstructed from the volume fraction field. The most notable geometric methods are the SLIC (Simple Line Interface Calculation) and PLIC (Piecewise Linear Interface Construction) methods (NOH; WOODWARD, 1976; YOUNGS,1982). These methods give a good approximation of the shape of the interface and allow proper calculation of the fluxes through faces of the control volume, however, they are better applied to structured orthogonal grids. The implementation in non-orthogonal or unstructured meshes become extremely complex.
The major advantage of compressive VOF methods compared to geometric methods are the straightforward implementation, the computational efficiency and the applicability to arbitrary meshes (DENNER, 2013). Hence, in the present work, it was opted for the application of CICSAM to model the interface in the stratified flow. This method will be detailed later in this work in Sec.3.1.1.
2.2. Surface Tension and Interface curvature calculation 41
2.2
Surface Tension and Interface curvature calculation
Surface tension is a property of the fluid which permits the interface to resist external forces. From a molecular point of view, forces act uniformly in all directions in the bulk fluid region. Considering the presence of a fluid-fluid interface, this force is non-uniform, which produces a resultant force – surface tension force fσ. This resultant
force at interface region opposes any variation in interface area, i.e. a certain amount of work is generated proportional to the area increase (YEOH; TU, 2009; DENNER, 2013). The amount of work generated is characterized by a surface tension coefficient σ, which represents the energy per unit area to increase the interface area. Considering an interface without external forces such as gravity, surface tension is defined by the Young-Laplace equation (FAGHRI; ZHANG, 2006)
∆p = pi− po = σ 1 r1 + 1 r2 = σκ, (2.15)
where κ denotes the mean curvature of the interface, r1 and r2 denotes the principal
curvature radii of a three-dimensional interface and pi and po denotes the pressure inside
and outside the interface, respectively. Brackbill, Kothe e Zemach (1992) proposed the
Continuum Surface Force (CSF) model to numerically describe the effect of surface tension.
The CSF model transforms the surface tension1 into a volumetric source term, distributing the surface force2 over a small finite transition region. The model is constructed from Young-Laplace equation (Eq.2.15). Thus, the surface force acting at the interface is
fσ = σκ∇α, (2.16)
where the interface curvature is defined as
κ = ∇ · n, (2.17)
and n denotes the interface normal vector, which is given by n = ∇α
|∇α|. (2.18)
The volumetric surface force (Eq. 2.16) is included in the source term of the momentum equation (Eq. 2.13), here repeated,
∂ρu
∂t + ∇ · (ρuu) = −∇p + ∇ · T + ρg + fσ. (2.19)
1 the surface tension is physically defined as existing only over a surface which represents the interface. 2 force acting at the interface due to surface tension.
When using CSF, a problem related to it is the appearance of "unphysical" flow around the interface, also called parasitic currents or spurious currents. In general, they are caused by the discretization of the molecular surface force at the macroscopic scale (DENNER, 2013). Parasitic currents are also generated due to a local imbalance between the pressure gradient at the interface and the surface force or due to an inaccurate estimation of the interface curvature, as described byFrancois et al.(2006).
When simulating interface flows, particularly when using a VOF method, the implementation of interface normal vectors and curvature is of extreme importance. In order to determine the interface curvature, the volume fraction has to be twice differentiable – the first derivative determines the surface force and the second derivative of the interface curvature. Because of the application of compressible and geometric reconstruction methods, the volume fraction field α is a continuum, however, its variation is not smooth, which leads to problems in the evaluation of the gradients. Thus, smoothing techniques, as convolution, need to be applied to volume fraction fields (DENNER,2013).
As commented early, the interface in the VOF method is represented by a volume fraction value between 0 and 1. Hence, only the cells that contain an interface are known, but not the exact location of the interface. This represents the difficulty in the numerical evaluation of the interface curvature, which in turn makes difficult the application of the surface tension and fluid properties to this particular region of the flow. As also commented before, the surface tension force depends directly on the curvature. Thus, this variable must be modeled correctly when using VOF.
There are some methods well established in literature to treat the interface using VOF, such as the direct differentiation methods (BRACKBILL; KOTHE; ZEMACH,1992) and the height function techniques (CUMMINS; FRANCOIS; KOTHE, 2005; LÓPEZ et al.,2009;GLITZ, 2012). The direct differentiation methods evaluate the curvature by differentiating the volume fraction field directly, using finite volume and least-squares methods. The height function technique evaluates the curvature by a height function approximation y = h(x). As the volume fraction is not a smooth function, a convolution method can be applied to diminish the curvature estimate errors. Brackbill, Kothe e Zemach (1992) and Francois et al. (2006) applied the convolution in interface treatment as a method to interpolate the curvature value at the control volume centre.
2.2.1
Direct Differentiation Methods
Direct differentiation methods act differentiating the volume fraction directly. The interface curvature is obtained by differentiating the volume fraction twice through a basic finite difference, finite volume or least-squares methods. In all methods, the curvature is
2.2. Surface Tension and Interface curvature calculation 43
defined by Eq. 2.16, repeated here,
κ = ∇ · ∇α |∇α|
!
. (2.20)
Finite difference and volume methods do not require significant computational resources, however, they suffer from inaccurate curvature estimates, because of the dis-continuous nature of the volume fraction (DENNER, 2013). A reformulation of Eq.2.20 was proposed by Brackbill, Kothe e Zemach(1992), applying a standard finite difference method to the volume fraction to shift the major contribution of the curvature from the edges to the center of the differentiation stencil
κ = 1 ∇α " ∇α |∇α| · ∇ ! |∇α| − (∇ · ∇α) # . (2.21)
The interface can be evaluated, using a finite volume approach, using the Gauss theorem (KOTHE et al.,1996)
κP = − 1 VP Z VP ∇ · ndV ≈ − 1 VP X f nfAf = − 1 VP X f ∇α |∇α| f Af. (2.22)
2.2.2
CELESTE
Another way to calculate the curvature was obtained by Denner (2013), the CELESTE method (Curvature Evaluation with Least-Squares fit for Taylor Expansion). The method is based on the least-squares method with a higher-order Taylor series approach for the interface normal vector calculation. In three-dimensions, the second-order Taylor series expansion of the volume fraction α from cell P to its neighbour cells Q is defined as
αQ = αP + 3 X i=1 ∂α ∂xi P (xi,Q− xi,P) + 3 X i=1 3 X j=1 ∂2α ∂xi∂xj P (xi,Q− xi,P)(xj,Q− xj,P) 2 + O(∆x 3, ∆y3, ∆z3). (2.23)
As denoted by Eq.2.18, the interface normal vector is obtained by normalizing the first derivatives of the volume fraction. As commented by Denner (2013), the inclusion of the higher-order derivatives of the volume fraction field in the linear equation system is essential for the accuracy of curvature evaluation in the following step, as the interface curvature depends on the second derivative of volume fraction field.
After the solution of the previously mentioned linear system, the interface normal vector n is determined from the unitary vector of ∇α. Once determined the interface normal vector field in the entire domain, its values are approximated in a least-squares fit
through the following Taylor series expansion, n X Q=1 nkQ = nkP + n X Q=1 " ∂nk ∂x P (xQ− xP) # + n X Q=1 " ∂nk ∂y P (yQ− yP) # + n X Q=1 " ∂nk ∂z P (zQ− zP) # , (2.24)
where k is a component of the n = [nx, ny, nz] vector. After obtaining the solution vector,
the curvature is obtained in a control volume P as
κP = − ∂ni ∂xi P . (2.25)
The CELESTE method is capable of dealing with structured and unstructured meshes when estimating the curvature and vanishing parasitic currents (DENNER et al., 2014). Denner e van Wachem (2014) applied the CELESTE with the CICSAM scheme (Sec.3.1.1) and obtained satisfactory results that accurately predicted the two-phase flow
dynamics in complex surface-tension-dominated situations.
2.2.3
Height function method
One straightforward approach to interface curvature calculation mainly in stratified flows is the height functions. The method is easy to implement and provides precise results for the interface curvature in flows with capillary waves. This method was an alternative to the development of the present model, which demonstrated good results, as will be detailed in Sections 3.1.2 and 3.2.
2.3
Laser Doppler Velocimetry
The Laser Doppler velocimetry (LDV) is an optical method that allows the determi-nation of the fluid velocity with high resolution in time. When compared to other traditional methods, as hot-wire anemometry, LDV has many advantages. As a non-invasive method, the flow is not perturbed when the measurements are made (FELDMANN; MAYINGER, 2012). This technique is adopted in the experimental approach of the present work.
The most used LDV technique is the two beams system, presented in Fig. 5. In this technique, one single laser beam with wavelength λ is equally split into two beams with different wavelengths. The resulting beams are converged through a probe to a point, creating a measurement volume. Seed particles are added to the flow and scatter the light of the beams when passing through the measurement volume. The photodetector (or light
