MASTER THESIS, DUAL DIPLOMA PROGRAM ADVANCED LEVEL, 30 ECTS STOCKHOLM - BEIJING, 2018
A Study on Characteristics of a Thin
Liquid Film Flowing down a Uniformly
Heated Plate with Constant Heat Flux
Wang Meng
KTH School of Science Tsinghua University Institute of Nuclear and New Energy Technology TRITA-SCI-GRU 2018:061
www.kth.se www.tsinghua.edu.cn TSINGHUA UNIVERSITY KTH-ROYAL INSTITUTE OF TECHNOLOGY
A Study on Characteristics of a Thin Liquid Film Flowing down a Uniformly Heated Plate with Constant Heat Flux
Wang Meng
Thesis Submitted to
Tsinghua University KTH Royal Institute of Technology In partial fulfilment of the requirement In partial fulfilment of the requirement for the degree of for the degree of Master of Science Master of Science In In Nuclear Science and Technology Engineering Physics
Co-supervisor: Associate Prof. Co-supervisor: Associate Prof. Duan Riqiang Ma Weimin TSINGHUA UNIVERSITY / INSTITUTE OF KTH-ROYAL INSTITUTE OF TECHNOLOGY NUCLEAR AND NEW ENERGY /DEPARTMENT OF PHYSICS TECHNOLOGY
UNDER THE COOPERATION AGREEMENT ON DUAL MASTER’S DEGREE PROGRAM IN NUCLEAR ENERGY RELATED DISCIPLINES
MAY 2018
Abstrakt
Abstrakt
Avhandlingstitel: En studie om egenskaper hos en tunn vätskefilm som flyter ner en enhetligt uppvärmd tallrik med konstant värmeflöde
Fallande vätskefilm används ofta inom teknikområden. Undersökningar av egenskaper hos en tunn flytande film som strömmar ner en likformigt upphettad platta med konstant värmeflöde (hädanefter benämnt HF-tillstånd) utförs i denna avhandling. En tredimensionell andra ordningens modell erhålls för att studera den fallande vätskefilmen för HF fall. Först är de styrande ekvationer och randvillkor för den fallande vätskefilmen bestämdes med flera antaganden. Baserat på den långvågiga egenskaperna hos den fallande vätskefilmen, är en gradient expansionsstrategi sedan utföras genom att införa en vätskefilm parametern ε att uppskatta storleksordningen av varje term i de styrande ekvationerna och motsvarande randvillkor. Förenklingar av de styrande ekvationerna och gränsvillkoren görs genom att försumma termer som är högre än Ο(ε2). Efter idén som i Prandtls gränsskiktsteori är momentumekvationen i cross-stream riktningen integrerad längs filmskiktet för att erhålla ett uttryck av tryck, och därefter ersätts trycket i momentumekvationerna i de andra två riktningarna för att eliminera tryck terms. De flytande filmskikt ekvationerna erhålls sedan. Med hjälp av den vägda restmetoden för att utföra de matematiska approximationerna av filmskiktets ekvationer leder till hela andra ordningsmodellen, vilken dock fortfarande är för komplicerad. Därför utförs ytterligare förenklingar där regularisering proceduren används för att undvika singulariteten. Slutligen erhålls en tredimensionell regulariserad modell (RM) med flytande filmtjocklek, streamwise flödeshastighet, spanwise flödeshastighet och flytande filmytemperatur som variabler. Därefter verifieras validiteten för RM. Att utföra störnings expansion av RM som leder till Benney-ekvation visar att RM är korrekt vid små Reynolds-nummer. God överensstämmelse med linjära stabilitetsegenskaper hos RM med Orr-Sommerfeld visar att RM är korrekt från små till måttliga Reynolds-nummer. En jämförelse med de tredimensionella experimentella resultaten visar att RM kan reflektera de tredimensionella egenskaperna hos den tredimensionella fallande vätskefilmen korrekt. Detaljerad undersökning av den fallande flytande filmen under HF-tillstånd utförs
I Abstrakt genom linjär stabilitetsanalys och tredimensionella simuleringar. Den linjära stabilitets analysen visar att streamwise instabilitet tillväxten hastighet ökar med ökningen av Reynolds-nummer och Marangoni-nummer; spawise instabilitet tillväxten hastighet ökar med ökningen av Marangoni-nummer och minskar med ökningen av Reynolds-nummer. Det finns ett instabilt intervall av störningsvågnummer i både streamwise och spanwise-riktningar. Tredimensionella numeriska simuleringar visar att med den initiala slumpmässiga störningen bildar den flytande filmen som flyter under HF-tillstånd i slutänden tredimensionella rivulet strukturer med ensamma vågor, som liknar de tredimensionella strukturen som bildas under ospecificerat temperatur skick. Värmeöverföringen mellan den flytande filmen och gasen förbättras genom utseende av rivulet strukturer. Om den mest instabila spanwise störningen i dominans introduceras vid den initiala slumpmässiga störningen uppträder rivulet strukturer tidigare, och flytande film brister också tidigare. Om den mest instabila streamwise störningen i dominans införs vid den initiala slumpmässiga störningen, ökar värmeöverföringen, medan den flytande filmen inte brister tidigare. Om båda införs bildas rivulet strukturer i förväg, vilket resulterar i förbättring av värmeöverföring och tidigare sprickning av den flytande filmen. Topparna av rivulets blir bredare, och dalarna mellan rivulets blir smalare såsom Reynolds-nummer ökar. Topparna av rivulets blir bredare, och dalarna mellan rivulets blir smalare såsom Reynolds-nummer ökar. När Reynolds-nummer når en viss nivå visas tredimensionella störningsvågor, som visar liknande egenskaper som vågmönstret i ett isotermiskt tillstånd. Ökande Marangoni-nummer leder till tidigare sprickbildning av den flytande filmen, mer rivulettstrukturer och smalare dalar.
Nyckelord: konstant värmeflöde; fallande vätskefilm; linjär stabilitetsanalys; numerisk simulation
II Abstract
Abstract
Falling liquid film is widely used in engineering areas. Investigations on characteristics of a thin liquid film flowing down a uniformly heated plate with constant heat flux (hereinafter referred to as HF condition) are performed in this thesis. A three-dimensional second-order model is obtained to study the falling film for HF case. First, the governing equations and boundary conditions of the falling liquid film are determined with several assumptions. Based on the long-wave characteristics of the falling film, a gradient expansion strategy is then performed by introducing a film parameter ε to estimate the order of magnitude of each term in the governing equations and corresponding boundary conditions. Simplifications of the governing equations and boundary conditions are made by neglecting terms higher than Ο(ε2). Following the idea as in Prandtl’s boundary layer theory, the momentum equation in cross-stream direction is integrated along the film layer to obtain an expression of pressure, and then the pressure is substituted into the momentum equations in the other two directions to eliminate the pressure terms. The film layer equations are then obtained. Subsequently, using the weighted residual method to perform the mathematical approximations of the film layer equations leads to the full second-order model, which, however, is still too complicated. Therefore, further simplifications are carried out where the regularization procedure is adopted to avoid the singularity. Finally, a three-dimensional regularized model (RM) with liquid film thickness, streamwise flow rate, spanwise flow rate, and film surface temperature as variables is obtained. Then the validity of RM is verified. Performing perturbation expansion of RM leading to Benney equation proves that RM is accurate at small Reynolds number. Good agreement of linear stability characteristics of RM with Orr-Sommerfeld proves that RM is accurate from small to moderate Reynolds numbers. A comparison with the three-dimensional experimental results demonstrates that the RM can reflect the nonlinear characteristics of the three-dimensional falling liquid film correctly. Detailed investigations of the falling liquid film under HF condition are performed by linear stability analysis and three-dimensional simulations. The linear stability analysis shows that the streamwise instability growth rates increase with the increase of Reynolds number and Marangoni number; the spanwise
III Abstract instability growth rates increase with the increase of Marangoni number and decrease with the increase of Reynolds number. There is an unstable range of disturbance wave number in both streamwise and spanwise directions. Three-dimensional numerical simulations show that with the initial random disturbance, the liquid film flowing under HF condition eventually forms three- dimensional rivulet structures with solitary waves, similar to the three-dimensional structures formed under specified temperature condition. The heat transfer between the liquid film and the gas is enhanced by the appearance of rivulet structures. If the most unstable spanwise disturbance in dominance is introduced in the initial random disturbance, the rivulet structures arise earlier, and film also ruptures earlier. If the most unstable streamwise disturbance in dominance is introduced in the initial random disturbance, the heat transfer is enhanced, while the film does not rupture earlier. If both are introduced, rivulet structures are formed in advance, resulting in enhancement of heat transfer and earlier rupture of the film. The crests of the rivulets become wider, and the valleys between the rivulets become narrower as Reynolds number increases. When the Reynolds number reaches a certain level, three-dimensional disorder waves appear, showing similar features as the wave patterns in an isothermal condition. Increasing Marangoni number leads to earlier rupture of the film, more rivulet structures, and narrower valleys.
Key Words: constant heat flux; falling liquid film; linear stability analysis; numerical simulation
IV Table of Contents
Table of Contents
Chapter 1 Introduction ...... 1 1.1 Background ...... 1 1.2 Status of the research ...... 4 1.3 Motivation and structure of this thesis ...... 11 1.3.1 Motivation ...... 11 1.3.2 Structure of this thesis ...... 12
Chapter 2 Theory Model ...... 13 2.1 Governing equations and boundary conditions...... 13 2.2 Dimensionless equations, scalings and parameters ...... 16 2.3 Film layer approximation ...... 18 2.4 Three-dimensional Regularized Model ...... 22 2.4.1 Weighted residual methods ...... 23 2.4.2 Full second-order model ...... 23 2.4.3 Regularized model ...... 35 2.5 Summary of This Chapter ...... 37
Chapter 3 Model Validation ...... 39 3.1 Derive the Benney equation from the RM ...... 39 3.2 Linear stability analysis ...... 41 3.2.1 Definition of the flow stability ...... 41 3.2.2 Method of linear stability analysis ...... 42 3.2.3 Linear stability analysis of the falling film for HF case ...... 43 3.3 Three-dimensional verification ...... 48 3.4 Summary of this chapter ...... 50
Chapter 4 Three-dimensional Simulation ...... 51 4.1 Introduction to three-dimensional simulation ...... 52 4.2 Random disturbance ...... 54 4.2 Random disturbance superposed with the most unstable spanwise disturbance ...... 58 4.3 Random disturbance superposed with the most unstable streamwise
V Table of Contents
disturbance ...... 61 4.4 Random disturbance superposed with the most unstable streamwise and spanwise disturbances ...... 64 4.5 Effect of disturbance on heat transfer ...... 66 4.6 Effect of Reynolds number and Marangoni number on the wave patterns ..... 67 4.7 Summary of this chapter ...... 72
Chapter 5 Conclusions and Further Research ...... 73 5.1 Conclusions ...... 73 5.2 Further research ...... 75
References ...... 76
Acknowledgements ...... 79
Resume ...... 80
VI Chapter 1 Introduction
Chapter 1 Introduction
1.1 Background
Falling liquid film is widely used in nuclear engineering, thermal engineering, chemical engineering, and electronics owing to its high rates of heat and mass transfer with small temperature difference, small flow rate and small power consumption, such as cooling towers, AP1000 passive containment cooling system (PCCS), and multiple-effect distillation (MED) for seawater desalination. The AP 1000, a third-generation nuclear power plant designed by Westinghouse, has revolutionized the design of the safety system by using passive methods such as gravity and natural circulation to prevent overheating in the core and containment. Containment is an essential barrier to the radioactive release of nuclear power plants. The AP 1000 uses a passive containment cooling system (PCCS) to ensure the integrity of the containment structure under accident. The function of the PCCS is to keep the temperature of the inner containment below the limit. The PCCS reduces temperature and pressure within the containment under hypothetical design basis accidents, discharges heat from the containment in the event of loss of coolant accident (LOCA) and main steam line break accident (MSLB), and provides a final heat sink under other design basis accidents that cause a significant increase in pressure and temperature within the containment. Its most significant advantage is that it can continuously and reliably cool down the containment by relying solely on the gravity and the natural circulation without any external power or requiring very little external power to prevent the containment from a burst. In addition, it dramatically reduces the influence of external conditions and human factors on safety so as to improve the intrinsic safety of the system. The heat transfer principle of PCCS is shown in Figure 1.1. The heat in the containment is transferred to the condensate film on the inner surface of the steel structure through steam condensation, thermal radiation, and heat convection. Through the heat conduction of the condensate film and the containment wall, the heat is then transmitted to the outer surface of the containment vessel. The heat from the outer surface of the containment is transferred to the water film by heat convection and heat
1 Chapter 1 Introduction conduction, which exchanges heat with the air in the annular channel by convection and evaporation. The water vapor and air density within the inner annular channel are less than the water vapor and air density in the outer annular channel, forming a natural circulation. Hot air and water vapor entrained with hot air are vented from the air diffuser (stack) to the atmosphere to provide passive cooling of the containment to ensure containment integrity without the need for external power. Liquid film flowing on the outer surface of the containment has played a significant role in heat transfer.
Figure 1.1 Heat transfer principle of PCCS
Vertical evaporating falling film flow is the core of multi-effect distillation seawater desalination process. Because of the excellent heat transfer and flow characteristics of falling film such as high heat transfer efficiency, small heat loss and short contact with hot wall, multiphase distillation desalination process systems for evaporators show significant advantages over other thermal desalination processes, such as efficiency, input-output ratio and scale control, which are considered as the one of the most promising technologies for desalination. Thin film evaporation technology has also become the major technology of chemical evaporation equipment. Thin film absorbers and thin film reactors are also an important class of chemical equipment; in addition, film production, polymer processing, and enhanced fueling of the slurry fuel also involve the heat and mass transfer of the liquid film. The application of the characteristics of liquid film flow to solve the problem of high efficient and safe heat transfer under the high heat flux encountered in the high-tech field has drawn more and more attention.
2 Chapter 1 Introduction
To give full play to the advantages of liquid film to enhance the heat and mass transfer, a crucial issue is to understand its inherent flow process and heat and mass transfer mechanism. However, the nature of the falling liquid film flow shows considerably complex phenomena. The formation and evolution of disturbing wave on the liquid film lead to complex liquid film flow process. As Chang, H. C pointed[1] out, falling liquid film flow shows very rich wave mechanics. The disturbing wave action on the liquid film is twofold. On the one hand, the disturbing wave enhances the transfer of heat and mass between the gas-liquid interface. On the other hand, the increase of disturbance wave leads to the instability of the liquid film flow, which causes the liquid film to rupture and fall off, so that the wall cannot be covered entirely with the liquid film. This not only fails to achieve the expected purpose but also exacerbates the deterioration of the heat transfer process. Once the liquid film breaks, dry spots or dry areas on the heat transfer surface can cause a variety of serious consequences such as degeneration of the heat-sensitive material, coking of the non-heat-sensitive material, clogging of the heat transfer tube. In some cases, the heat transfer surface will be overheated or burnt due to a sharp increase of temperature in the dry area. To enhance the ability of heat transfer of the liquid film, one must have a deep understanding of the influencing factors and the mechanism of the instability of the liquid film in heating process. Therefore, it is imperative for engineering applications to study the characteristics of falling liquid film with heating. Falling liquid film flow is an open flow system. Under external perturbation, the film flow shows rich flow phenomena, such as interface instability, free surface waves, three-dimensional spatial-temporal evolution and transition from laminar flow to turbulent flow. Numerous studies have shown that the spatial evolution of the falling film flow field is very similar to other classical hydrodynamic instabilities such as the free-convection boundary layer and the shear instability transition in the channel, but the instabilities of falling film are more accessible to be analyzed. It is hoped that inspiring conclusions can be drawn from the flow of thin film to understand the physical mechanism of the transition from laminar to turbulent flow. Turbulence is still a challenging and strategic issue for fluid mechanics and even for the whole physics. Therefore, the study of film flow has great theoretical significance.
3 Chapter 1 Introduction
1.2 Status of the research
The investigations of falling liquid film are mainly performed by experimental research and numerical research. The numerical researches of liquid film flow are divided into two kinds: linear analysis and nonlinear analysis. The linear analysis assumes that the disturbance is infinitesimal, so the changes caused by the disturbance are linear, which helps to understand the stability and flow behavior of the liquid film easily. The nonlinear analysis is used to study the specific evolution after disturbance occurs. The linear analysis method is mainly based on solving the Orr-Sommerfeld equation (O-S equation). Such method can predict the instability growth rates caused by a given small disturbance, but the non-linear behavior such as large amplitude solitary waves and three-dimensional evolution cannot be well described by such method. Nonlinear analysis methods mainly include direct numerical simulation and investigation with nonlinear model equations derived from the governing equations and boundary conditions by reasonable approximations. The study of liquid film flow can be traced back to Nusselt's theoretical study[2] in 1916. He assumed that the liquid film is steady flow along the plate with a smooth surface, without taking into account the interaction between the gas-liquid interfaces, and obtained the expression of the film thickness (Nusselt's solution).
Figure 1.2 Two waveforms observed by Kapitaz and his son
A pioneering experimental study of liquid film flowing along an inclined plate was carried out by Kapitza and his son in the middle of the twentieth century[3]. Experiments
4 Chapter 1 Introduction showed a wealth of wave dynamic phenomena, and two typical waves were formed: periodic sine waves and solitary waves. It is a simple open flow system showing very complex and plentiful dynamic phenomena. From then on, the research on the dynamics of the film falling has attracted the interest of many researchers. Based on the Orr-Sommerfeld equation, Benjiamin[4] and Yih[5] conducted a linear stability analysis of two-dimensional liquid film flowing down an inclined plane with the assumption of long-wave perturbation, small surface tension, and small Reynolds number. They first proposed the concept of critical Reynolds number. The flow is long-wave unstable when the Reynolds number exceeds a specific critical value, obtained as Re=5/6cotβ (Re is Reynolds number, and β is the angle of inclination), which means that the vertical falling liquid film is always long-wave unstable. Linear theory cannot predict the various nonlinear flow phenomena away from the entrance. Benney[6] derived a nonlinear single evolution equation, referred to as the Benney equation (BE), under the assumption of low Reynolds number and long waves. The Benney equation gives same critical Reynolds number as the value got by Benjiamin[4] and Yih[5]. However, Pumir et al.[7] found that the Benney equation had a finite-time blow-up behavior at large Reynolds number. Scheid et al.[8] studied the validity domain of the Benney equation, showing that Benney equation is exact in the limit of small Reynolds number. Thus, the BE can be used as a benchmark to examine the accuracy of other models at small Reynolds number. The Benney equation preserved only the first-order approximation when long-wave perturbations were performed. Panga and Balakotaiah[9] have derived the evolution equation that considered second-order diffusion terms, but still did not eliminate the singularity. Ooshida[10] used a Padé-like regularization method to eliminate the singularity formation in the Benney equation. The model equations derived by Ooshida did remove the singularity, but the wave velocity and amplitude were underestimated at moderate Reynolds number. Shkadov[11] derived an integral-boundary-layer model (hereinafter referred to as the Kapitza-Shkadov model) based on the boundary layer theory and Karman-Pohlhausen averaging method. The Kapitza-Shkadov model is suitable for moderate Reynolds number[12], but does not predict accurate linear stability threshold at small Reynolds number, except for the vertical case. To remedy this problem, Ruyer-Quil and Manneville[13-15] used the weighted residuals combined with the gradient expansion and derived weighted residuals model and regularized model[16]. The regularized model
5 Chapter 1 Introduction
(hereinafter referred to as the RM) gives accurate critical condition and is applicable from small to moderate Reynolds number[17]. Compared to weighted residuals model (hereinafter referred to as the WRM), the RM is more straightforward than the full second-order model, which is one of the WRM; and when it comes to the three-dimensional situation, the RM is more accurate than the simplified second-order model, which is also one of the WRM[18]. Considering the accuracy, simplicity, and validity of the models mentioned above, the RM is most suitable to study the falling film. The three-dimensional numerical simulation results with RM coincided well with the experimental results[19].
Figure 1.3 Falling film evolution excited by natural noise in intermediate Reynolds number region. For high Reynolds numbers (Re > 50), region (III) may not appear
As theoretical research continues to develop, experimental research is underway, and more phenomena are observed. Four distinct wave regions of falling film evolution by natural excitation in intermediate Reynolds number region were summarized by Chang[1] as shown in Figure 1.3. In region I, the infinitesimal disturbances at the entrance are amplified downstream, and a monochromatic wave is formed at the end of this area, indicating that the instability is convective instability, not absolute instability. The amplitude of the monochromatic wave is amplified exponentially downstream. In region II, due to the weakly nonlinear effect, the monochromatic wave exponential growth is suppressed and tends to a saturation value dependent on the wavenumber, Reynolds number, Weber number. The finite-amplitude waves travel about 10 wavelengths before they go to region III. In region III, subharmonic instability[20] and sideband instability[21] are two dominant instabilities of these waves. These instabilities lead to increase of wave speed, wavelength, and amplitude. The waves evolve into teardrop humps, which have steep fronts, and there is a series of bow waves in front of them. These humps are identical, and their characteristic lengths are much shorter than the length between two humps. Thus, these humps are referred as solitary waves or
6 Chapter 1 Introduction solitary humps. In region IV, these solitary waves undergo nonstationary transverse variations. These transverse variations progress slowly in the direction of the parallel walls and eventually cause the adjacent crests to merge and separate at different locations. However, in the streamwise direction, the waveform remains in solitary shape, except the merging position. Krantz & Goren[22], Alekseenko[23], Nosoko et al.[24], Liu & Gollub[25] and Liu et al.[26] performed some representative experiments on the wave characteristics of falling liquid film. Alekseenko[23] pointed out that under sufficient level of flow disturbance, the shape of the steady-state wave is only affected by the initial disturbance frequency and has no relation to the amplitude. When the disturbance frequency is relatively low, a solitary wave is generated; when the disturbance frequency is relatively large, a periodic sine wave is generated; and when the disturbance frequency is the same as the natural frequency, the observed waveforms are similar in both cases, showing that natural frequency disturbance is only a special case of forced perturbation experiments.
Figure 1.4 Schematic description of the spatial evolution of film flows
Due to the difficulty of generating regular waves under natural disturbance, it is inconvenient to analyze its characteristics. Most experiments use periodically forced method to introduce disturbance. Liu and his colleagues[25,26] conducted extensive study on the unstable frequency range of several typical waveforms, the transition from two-dimensional wave to three-dimensional wave, and the evolution of the small perturbation of liquid film at the inlet. They found that synchronous instability showed up under low frequency while subharmonic instability showed up under high frequency
7 Chapter 1 Introduction for 3D flow as shown in Figure 1.4. In the experiment, the disturbance of liquid film was introduced by natural or periodic forcing method. Takamasa and Hazuku[27] conducted an experimental study on the flow behavior of vertical plate falling film using a laser focus displacement meter (LFD). The transient film thickness and film velocity were measured experimentally. The influence of Reynolds number and inlet length on the falling film was studied. The experiment also obtained the empirical relationship between wave frequency, wave velocity, and inlet length. Moran et al.[28] measured the film thickness, the velocity distribution of the mainstream and the wall shear stress of the laminar falling film on the inclined plate. The Nusselt theory was found to underestimate the film thickness and overestimate the velocity profile of the water film at a Reynolds number from 11 to 220. In addition, the peak value of the wall shear stress often precedes the peak of the film thickness, indicating that the liquid film undergoes acceleration or deceleration in the wavy region. With the gradual improvement of computing abilities, some scholars began to use direct numerical simulation to simulate the flow of thin films. Malamataris et al.[29] performed two-dimensional numerical simulations using the Lagrangian finite element method and obtained numerical results very close to experimental results by Liu and Gollub[25]. They first verified the validity of the numerical method by comparing the linear evolution stage results with the linear stability theory. The non-linear evolution observed in the experiment is reproduced. That is, as the perturbation frequency decreases, two-dimensional thin film flows develop into saturated periodic wave, quasi-periodic wave, multi-peak wave structure, and solitary wave, respectively. They also investigated the interaction of solitary waves and the flow structures within the liquid film at various interface waveforms, including velocity distribution, pressure distribution at the interface and wall. The two-dimensional flow simulation of the vertical falling liquid film performed by Nave et al.[30] were in good agreement with the experiment by Nosoko et al.[24], and the wave structure in the transition from two-dimensional to three-dimensional flow was given. There are two major difficulties in direct numerical simulation, one is the huge computational cost and the other one is the need to deal with the motion interface. This poses a high challenge to the numerical solution. At present, the results of direct numerical simulation do not show much advantage over the weighted residual model.
8 Chapter 1 Introduction
Most of the researches above focus on the isothermal case. For the heated falling film, the situation is different. Apart from the long-wave hydrodynamic instability, referred to as H-mode, there exists other instability caused by heating. As we know, the surface tension varies with the temperature. For most liquids, the surface tension decreases as the temperature increases. Consider a falling liquid film heated from below; when a long-wave small disturbance is applied to the film, the temperature at trough is larger than the temperature at crest; thus, surface tension at trough is smaller than at crest, the surface tension gradient driving the liquid away from the trough to the crest. The disturbance on the liquid film is further amplified. This instability is called as long-wave Marangoni instability or thermocapillary instability, which is classified by Goussis and Kelly[31] as the S-mode. There also exists a P-mode[31] motion caused by heating, which yields the Bénard cells whose size is of the same order of magnitude with film thickness. The study of falling liquid films focuses on thin film, whose thickness is much smaller than the characteristic length scale. So the P-mode motion does not appear in our study. Lin[32] and Sreenivasan[33] performed linear stability analysis on films flowing down a uniformly heated plate. They found not only the surface wave instability, but also the Marangoni instability caused by the Marangoni effect exist. Kelley et al.[34] studied the stability characteristics of uniformly heated falling films at small dip angles. Goussis and Kelly[31,35] conducted a detailed study of the interaction between surface wave instability and Marangoni instability, and confirmed the existence of two modes of thermocapillary instability, the heating of the film is an unstable effect, while cooling has a stabilizing effect. The S-mode Marangoni instability is dominant in the case of small flow rates[31]. At large flow rates, the hydrodynamic H-mode is dominant, and the unstable effect of inertia becomes predominant. However, S and H modes can coexist and reinforce each other for a reasonably wide range of parameters[31,36]. They also noted the presence of a stable region where increasing or decreasing the Reynolds number made the stable liquid film unstable. Kabov et al.[37,38] performed experimental studies on liquid film flowing down a vertical plane with a local heater. Regular horseshoe-like structures were formed at the upper edge of the heater and rivulet structures were formed along the heater due to the interaction of hydrodynamic instability and Marangoni instability as shown in Figure 1.5.
9 Chapter 1 Introduction
Figure 1.5 Final regular structure on a local heater
To study the nonlinear behavior of a uniformly heated falling film, Joo et al.[36] deduced the Benney-type long wave evolution equation that considered the evaporation of the film surface: they found that the inertia effects became more pronounced as the dip angle increased, while thermocapillary forces amplified the disturbance caused by inertia effect. Because the Benney-type equations are only suitable for flows with small Reynolds numbers, Kalliadasis et al.[39] derived the integral boundary layer equations for moderate Reynolds numbers. After their linear stability analysis, they obtained the stability results consistent with Goussis and Kelly[31]. In addition, they constructed a solitary wave solution by integrating the boundary layer equations, predicting the existence of solitary waves in a uniformly heated film. Scheid et al.[40] extended the weighted residual model[16,17] to consider the three-dimensional case of uniform heating and carried out linear stability analysis and numerical simulation. They reproduced the rivulet structure in experiment[41] and discussed quite a lot about the formation mechanism of rivulet structure. All the models mentioned above mainly focus on the isothermal falling film or falling film being heated at specified temperature (ST) boundary condition. Only a few investigations on the heated falling film with uniform heat flux (HF) boundary condition were performed[42,43]. Scheid[43] first established the two-dimensional regularized model under HF condition, but no further research is performed with this model. A detailed summary of the derivation method and model equations can be seen in Kalliadasis et al.[18]. Based on a new set of test functions satisfying all the boundary conditions, a first-order two-dimensional model was built by Trevelyan et al.[42] and comparisons between HF and ST cases were conducted. The linear stability analysis and two-dimensional spatio-temporal dynamics showed that falling film exhibited similar properties under HF and ST condition. Here we can anticipate that three-dimensional evolutions of the falling film still have the similar features between HF and ST
10 Chapter 1 Introduction condition. However, this model developed by Trevelyan et al.[42] is a first-order two-dimensional model. It cannot be used to study the three-dimensional evolution of the falling film, neither can the model developed by Scheid[43]. A three-dimensional model is eagerly needed to study the three-dimensional evolutions of the falling film under HF condition.
1.3 Motivation and structure of this thesis
1.3.1 Motivation
Based on the experimental and numerical studies on falling film flow, many wavy characteristics and intrinsic mechanisms of falling film flow are not fully understood, which greaty limits the role of falling film flow in engineering application. Therefore, it is still necessary to further study the falling film flow mechanism to obtain a more thorough scientific understanding, which will be an essential basis for improving the passive containment cooling system. In experimental studies, the thickness of the liquid film is only 1 to 2 mm, and the time it takes for the liquid film to pass through the measurement area is also very short. Therefore, the experimental study of the wavy characteristics of the falling liquid film is limited to indirectly measuring the values of wall shear stress, film thickness and wave velocity with time, so the heat and mass transfer mechanism, pressure and velocity distribution in the liquid film cannot be studied. Therefore, it is necessary to carry out the corresponding research through numerical simulation and theoretical analysis. Direct numerical simulation requires tremendous computational efforts, especially in the case of three-dimensional liquid films. Compared to the direct numerical simulation and experimental studies, the benefits of the model equations are apparent. Model equations are based on physical approximation. They can catch the flow characteristics and are simpler than N-S equations, easy to perform numerical calculation, and more convenient for theoretical analysis. Most of the current model equations focus on isothermal falling film or nonisothermal falling film heated at specified temperature (ST) boundary condition. There are only a few researches on falling film with uniform heat flux (HF) boundary condition, which is closer to practical engineering applications. The reason is that the
11 Chapter 1 Introduction boundary condition in ST case is simpler than HF case, and only the heat transfer between the plate and the liquid film needs to be considered. The corresponding model equations are much simpler. But in HF condition, both the heat transfer from the heated plate to the film and the heat loss from the plate to ambient gas need to be taken into consideration. The complexity of model equations is apparent. However, the HF condition is more realistic than the ST one. It is more close to the actual situation. Hence, investigations carried out under HF condition are needed.
1.3.2 Structure of this thesis
In chapter 1, the phenomenon, basic principle and research status of falling liquid film are introduced, and direction of this research is determined based upon literature review. In chapter 2, the governing equations and the boundary conditions are presented and a three-dimensional regularized model under HF condition is built. In chapter 3, the verification of the regularized model is performed to lay the foundation for subsequent numerical simulation works, and linear stability analysis is conducted to study the influence of the flow rate, heat flux, and disturbance on the stability of the film. In chapter 4, nonlinear numerical simulations with the RM is carried out to investigate the three-dimensional evolutions of the falling film and influence of flow rate, heat flux, and disturbance on the evolution of the falling film. In chapter 5, the conclusions are summarized, and further research is proposed.
12 Chapter 2 Theory Model
Chapter 2 Theory Model
In this chapter, the physical model of the falling liquid film is determined, and then the corresponding mathematical model is established. Finally, several methods are used to simplify the mathematical model.
2.1 Governing equations and boundary conditions
Figure 2.1 Sketch of the thin liquid film flowing down an infinite vertical plate
An incompressible Newtonian viscous thin liquid film driven by gravity g flows down an infinite vertical plate, as shown in Figure 2.1. In the coordinate system of Figure 2.1, x is the streamwise direction, y is the cross-stream direction normal to the plate, z is the spanwise direction and the origin is at the plate. The plate is rigid and solid, and infinite in x and z directions. The angle of inclination between a supporting plate and the horizontal plane is β. Film flow is heated by a heat source uniformly embedded in the plate with constant heat flux Qw. Considering practical applications, it is assumed that some heat Qloss is leaked into atmosphere pre-wall area on the opposite side of the supporting plate, in which the heat transfer coefficient between the plate wall and atmosphere is αw. The other
Qw−Qloss is all absorbed by film and eventually released into the atmosphere as well, in which the heat transfer coefficient on the free surface of film flow is α.
h N is the mean flat film thickness. The free surface temperature of the film is Ts. The physical properties of the film such as density ρ, viscosity µ, thermal conductivity λ, the heat capacity cp and the thermal diffusivity χ = λ/(ρcp) are assumed to be constant. The assumption that the density ρ is constant holds for very thin film, as the one
13 Chapter 2 Theory Model considered in this work, where the buoyancy effect can be neglected[44,45]. The surface tension of the film is assumed to decrease linearly with the temperature, σ =σa−γ (Ts−Ta), where σa is film surface tension at temperature Ta, and γ = −(dσ/dTs)|Ta measures the sensitivity of surface tension to temperature variations and is positive for most liquids. The ambient gas is assumed to have infinite heat capacity and to keep at the constant temperature Ta and constant pressure Pa. The gas is also considered as mechanically ‘passive’ in a sense that the viscous stress from the gas on the film surface is neglected. The liquid is considered as nonvolatile so that evaporation effects can be neglected. In the energy equation, the contribution of viscous dissipation is neglected. The governing equations for uniformly heated film flow include the continuity equation, Navier–Stokes equations and the energy equation with appropriate boundary conditions. u 0 , (2-1) du P 2u F , (2-2) dt dT 2T , (2-3) dt where ∇ = (∂x, ∂y, ∂z) is the gradient operator, d / dt t u is the “material derivative”, u = (u, v, w) and F ()gsin , g cos , 0 are the fluid velocity and body force, respectively, while T and P are respectively the temperature and pressure. These governing equations need to be completed by the boundary conditions. At the plate y = 0, there is a no-slip and no-penetration boundary condition, so the corresponding boundary condition is u 0 . (2-4)
And also, a constant heat flux Qw is uniformly distributed on the plate. The corresponding boundary condition is
yTQTT w w() a , (2-5) which is gained by solving the steady state energy equation in the plate. The derivation process is shown as follows.
The thickness and temperature of the plate are noted as hw and Tw, respectively.
The wall thermal conductivity is noted as λw. So the energy equation becomes
Qw wyyT w 0, (2-6) hw
14 Chapter 2 Theory Model where the Qw/hw is the heat flux per unit plate thickness. Considering that the plate is just a heat source and the gas and liquid are just heat sinks, the plated thickness will be shrunk to zero once we get the formula of the plate temperature Tw for the sake of simplicity. So the heat conduction term in streamwise direction λw∂xxTw is negligible in (2-6). The solution of equation (2-6) is
Qw 2 Tw y AB y , (2-7) 2wh w where A and B are determined by two boundary conditions. On the solid-liquid interface, the continuity of temperature is imposed:
y0 : Tw T , (2-8) where T is the film temperature on the wall. On the solid-gas interface, the Newton’s law of cooling is applied:
y hw:() w y T w w T w T a . (2-9)
With (2-8) and (2-9), A and B are obtained as h QTT(1 w w ) ( ) w2 w a A w , (2-10) w wh w B T . (2-11) Using the continuity of heat fluxes at the solid-liquid interface gives
y0 : y T w y T w w A . (2-12)
Taking the limit of hw→0 leads to
QTTw w() a wA() w QTT w w a (2-13) w which gives the formula of (2-5). At the free surface y = h (x, z, t), kinematic condition, continuity of stress, and the Newton’s law of cooling are applied:
th u x h w z h v , (2-14)
[]()Τ Τa n s n n , (2-15)
15 Chapter 2 Theory Model
TTT n () a , (2-16) where T = −PI + P is the stress tensor for the liquid with I the identity matrix, P = 2μE is the deviatoric stress tensor and E(1/ 2)( u ( u )t ) is the rate-of-strain tensor. For the gas, the stress tensor is Ta = −PaI. s ()I nn is the surface gradient operator andn ( xh , 1, z h ) / n is the outward-pointing unit vector normal to the boundary 2 2 1/2 with n(1 ( x h ) ( z h ) ) . As mentioned above, surface tension σ =σa−γ (Ts−Ta), where σa is film surface tension at temperature Ta, and γ = −(dσ/dTs)|Ta.
2.2 Dimensionless equations, scalings and parameters
In the following, the Buckingham pi-theorem is used to make similarity analysis. Film flow is featured with film thickness h, velocity distribution U and surface temperature, which are determined by the physical properties of fluid, configuration of plate, hydro- and thermo- qualities, and the disturbance wave on free surface. Their dependence relation can be written as follows
(,,)(,,,,,,,,,,,,)U T h f a w g Qw q N k . (2-17)
Considering that ρ, μ, λ, γ, g are dimensional independent parameters, the Nusselt [18] scaling based on these independent parameters and the Nusselt flat film thickness h N is used here to nondimensionalize the governing equations and boundary conditions,
t l (,,)xyz (',',') xyzhNN , hhh ' , tt ' , (2-18) hN
h2 h NN (,,)(',',')uvw uvw , PPPl a ' 2 , TTTT a Δ N , (2-19) t l t
2 1/3 2 1/3 where l ( / g sin ) is the viscous-gravity length scale, t ( / ( g sin ) ) is the viscous-gravity time scale, u l / t is the viscous-gravity velocity scale, 2 P l hN / t is the pressure scale and ΔTN Q w h N/ Δ Th N is the temperature scale with ΔT= Qw l / , hN hN / l . The primes indicate that variables are dimensionless. For simplicity, the primes will be dropped in the following part. With dimensionless variables, the governing equations become, u 0 , (2-20) du 3Re 1 P 2 u , (2-21) dt x
16 Chapter 2 Theory Model
dv 3Re Ct P 2 v , (2-22) dt y dw 3Re P 2 w , (2-23) dt z dT 3PrRe 2 T , (2-24) dt and the boundary conditions become: at the plate y = 0, u v w 0 , (2-25)
yTBT 1 w , (2-26)
at the film surface y = h(x, z, t),
v t h u x h w z h , (2-27)
2 P[( huhwhhuwhuv )2 ( ) 2 ( ) ( ) n2 xx zzxzzx xyx 1 h ( v w ) v ] ( We MT )[ h (1 ( h )2 ) (2-28) z z y yn3 xx z 2 zzh (1 ( x h ) ) 2 x h z h xz h ],
1 M( T h T ) [2( h v u )(1())( h2 u v ) x x yn x y x x y x (2-29)
zh ( z u x w ) x h z h ( z v y w )],
1 M( T h T ) [2( h v w )(1())( h2 w v ) z z yn z y z z y z (2-30)
xh ( z u x w ) x h z h ( y u x v )], 1 BT() h T h T T . (2-31) n x x z z y The concrete formulas and detail physical meanings of these dimensionless number is explained here. The Prandtl number, Pr / , compares the momentum 2 diffusivity to the thermal diffusivity; the Kapitza number, Γ al / , compares the surface tension force to the force of inertia; the free-surface Biot number, Bi l / , and the plate Biot number, Biw w l / , are the dimensionless heat transfer coefficients and have the same physical meaning as α and αw, respectively; 2 Ma T l / , where T Qw l / , is the Marangoni number comparing the force induced by the surface tension gradient to the force of inertia; the Reynolds number,
17 Chapter 2 Theory Model
Re qNNN// u h ( u N is the average velocity), is the dimensionless flow rate; 2 MΔ TNNN/ gh sin Ma / h is the modified Marangoni number; 2 2 Wea/ gh N sin / h N is the Weber number, comparing the surface tension pressure to the viscous normal stress generated by gravity at the film surface;
B hN / hN Bi is the modified free surface Biot number and Bw w h N /
hN Bi w is the modified wall Biot number.
From the definition of these dimensionless number, Pr, Γ, Bi, Biw and β are determined once a gas-liquid-solid system is fixed, because they depend on the physical parameters of the system. Re, Ma and disturbance wavenumber k are free variables and can be varied with different flow rate, heat flux and disturbance frequency. In this paper, water at 20 C and 1.0 bar is chosen as fluid media of film flow, the inclination angle of the supporting plate is taken as / 2. the thermodynamic boundary conditions on both the opposite wall of the supporting plate and free surface are assumed as natural convection. So the following parameters are taken as the following values, Γ =3375,
Pr=7, Bi=0.12, Biw=0.6 and / 2 .
2.3 Film layer approximation
Until now, the equations formed by the governing equations and boundary conditions are still difficult to solve. Therefore, there is still a need to simplify these equations further. In our studies, the falling film is long-wave unstable, so that the film thickness is very small in comparison with the typical length of the waves, also meaning that the slope of the film interface is small. However, the dimensionless equations above do not consider the difference between the characteristic lengths in different directions. Accounting for the small ratio of film thickness to the streamwise and spanwise characteristic length scales, the mathematical formulation is simplified by the gradient expansion strategy[18], in which it is postulated that all hydro- and thermo- qualities are slowly varied in the streamwise and spanwise and time as done in the long-wave expansion. An ordering parameter is introduced to estimate the amplitude order of each term in the governing equations and corresponding boundary conditions through 2 the transformation (,,)(,,)t x z t x z , and (,)(,)xx zz xx zz . The governing equations and their corresponding boundary conditions are nondimensionalized based on the Shkadov scaling proposed by Shkadov[46], in which ε was introduced to redefine the scaling scales in different directions. ε is also called as
18 Chapter 2 Theory Model
‘film parameter’, which is a small number (=We-1/3 ) and compares the film thickness to the streamwise characteristic length scale. The Nusselt film thickness h N is naturally selected as the scale for the y. The streamwise and spanwise coordinates are compressed by taking its scale as 1/ε times the scale for the cross-wise coordinate y. ε appears as the coefficients of each term in the equations and thus can be used to estimate the magnitude orders of each term. So the whole equations can be simplified by neglecting certain orders. Now the nondimensionalization becomes
(,)(',')/,xz xzhNNNN yyhhhhtttl ', ', ' / h , (2-32)
(,)(',')/uw uwh2 tlv , vh '/ 2 tlPP , Plht ' /, 2 N N a N (2-33) TTTTa Δ N .
Based on the “film parameter” ε, a new set of dimensionless parameters are introduced:3 Reis the reduced Reynolds number; M is the reduced film Marangoni number; Ct is the reduced inclination number; 2 is the viscous dispersion number. In order to make the derivation process clearer, these new dimensionless numbers are not used in the derivation process. The dispersion effects induced by streamwise viscous terms play a significant role [15] 2 in the falling film . Thus the terms up toΟ() need to be retained to capture the dispersion effects. Then the governing equations become u 0 , (2-34) du 3Re 1 P 2 u u 2 u , (2-35) dt x xx yy zz dv 32 Re Ct P v , (2-36) dt y yy dw 3Re P 2 w w 2 w , (2-37) dt z xx yy zz dT 3PrRe 2 T T 2 T , (2-38) dt xx yy zz where terms of Ο() 3 in y component of the momentum equation have been neglected. The boundary conditions at plate y=0 are u v w 0 , (2-39)
19 Chapter 2 Theory Model
yTBT 1 w , (2-40) and at the free surface y = h(x, z, t) are
v t h u x h w z h , (2-41)
2 P2 ( y vhuhw x y z y ) ( WeMT )( xx h zz h ) , (2-42)
2 yu M x [ z h ( z u x w ) x w 2 x h (2 x u z w ) x v ], (2-43)
2 yw M z [ x h ( z u x w ) x w 2 z h (2 z w x u ) z v ] , (2-44)
2 2 2 yTBT [[()())]/2 x h z hBT x hThT x z z ], (2-45) where terms of Ο() 3 and higher in the boundary conditions have been neglected. θ is the film surface temperature θ = T |y = h, and [(i ih y )T]| h i with i = x, z. Following the idea as in Prandtl’s boundary layer theory, the y-direction momentum equation is integrated along the film layer to obtain an expression of pressure P, and then substitute this pressure expression into the momentum equation in the other two directions to eliminate the pressure term.
Since the pressure contributions in (2-35) and (2-37) occur through the term x, z P , terms ofΟ() 2 and higher in (2-36) need to be ignored to maintain the consistency of the equations atΟ() 2 . From boundary conditions (2-43) and (2-44), we can see that at the free surface, yu and y w are of Ο() , so that 2xh y u and 2zh y w are of Ο() 2 in equation (2-42). Then equation (2-42) becomes
2 2 P|h 2 y v | h We ( xx h zz h ) O ( ) , (2-46) where 2We has been kept because of the assumption We is a large number (We 3 ). Integrating equation (2-36) across the film and using the boundary conditions (2-46) and (2-39), we get
2 2 PCthy( ) Weh ( xx zz h ) ( y v y v | h ) O ( ) , (2-47) where the first term on the right side is hydrostatic pressure, the second is corresponding to surface tension effects, and the third item is higher-order viscous effects. Substituting pressure expression into the streamwise momentum equation (2-35) and spanwise momentum equation (2-37), we get
20 Chapter 2 Theory Model
du 3Re 1 u Ct h 2 [2 u u w ( v | )] dt yy x xx zz xz xyh (2-48)
xxxh xzz h ,
dw 3Re w Ct h 2 [2 w w u ( v | )] dt yy z zz xx xz z y h (2-49)
xxzh zzz h , where the cross-wise velocity component v is calculated through the continuity equation y by integrating in the cross direction v 0 ()xu z w dy . Governing equations (2-34), (2-48)-(2-49), (2-38) and boundary conditions (2-39) -(2-41), (2-43)-(2-45) are called as the second-order film layer equations since the assumptions leading to these equations are essentially the same with those in the derivation of the Prandtl equations of the boundary layer theory in aerodynamics. Here we rewrite the second-order film layer equations in the Shkadov scaling, including the dimensionless parameters. u 0 , (2-50) du 1u h [2 u u w ( v | )] dt yy x xx zz xz x y h (2-51) 3 We ( xxx h xzz h ),
dw w h [2 w w u ( v | )] dt yy z zz xx xz z y h (2-52) 3 We ( xxz h zzz h ), dT Pr () T T T , (2-53) dt xx zz yy u v w 0 , (2-54)
yTBT 1 w , (2-55)
v t h u x h w z h , (2-56)
yu x [ z h ( z u x w ) x w 2 x h (2 x u z w ) x v ], (2-57)
yw z [ x h ( z u x w ) x w 2 z h (2 z w x u ) z v ] , (2-58)
2 2 yTBT[(( x h ) ( z hBT ) ) / 2 x hThT x z z ] . (2-59)
21 Chapter 2 Theory Model
Comparing with the Nusselt scaling and viscous-gravity scaling, the advantage of the application of the Shkadov scaling is that it locates the transition between the drag-gravity and drag-inertial regimes at δ≃1 and makes the balance among the gravity, viscosity, and surface tension apparently, which is necessary to sustain the strongly nonlinear waves. Moreover, a technical advantage of this scaling is that its parameters retain values close to unity in the region of moderate Reynolds numbers and for large Weber number, which is rather useful to the convergence of numerical schemes. Thus the Shkadov scaling is adopted in the following part.
2.4 Three-dimensional Regularized Model
Although the second-order film layer equations have eliminated the pressure, there is no strong simplification and they are still complicated. The complexity of these equations makes calculation difficult to carry out and time consuming. Then reduced dimensional models, while keeping the essential dynamic characteristics of the full equations, are eagerly needed. In general, there are two typical ways to simplify the film layer equations. One is the long-wave perturbation expansion of all relevant variables in the region of small Reynolds number, and the other is integrating the film layer equations across the film layer following the Kármán–Pohlhausen averaging method in boundary layer theory in aerodynamics, which was introduced by Kapitza and Shkadov[11,47], as mentioned in Chapter 1. Then the averaging method was improved with a gradient expansion combined with “weighted residuals” technique using polynomials as test functions[14,15]. Scheid et al.[40] reduced the boundary layer model of a three-dimensional falling film with specified temperature (ST) boundary condition to a regularized model with four evolution equations by weighted residual method with regularization procedure. In the regularized model, film thickness h, the streamwise flow rate q and spanwise flow rate q across the film, and the surface temperature θ, depend only on time t and coordinates x, z. Scheid[43] derived a regularized model for a two-dimensional falling film with heat flux (HF) boundary condition using the same method. However, there is not any three-dimensional regularized model for HF case yet due to the complexity of the equations. Thus, here we will follow the weighted residual method and regularization procedure that Scheid[43] used and expand the two-dimensional model to three-dimensional by considering the variables in both the streamwise and spanwise
22 Chapter 2 Theory Model directions.
2.4.1 Weighted residual methods
Weighted residual methods are general terms of methods for solving differential equations in applied mathematics. The starting point is to assume that the solutions of these differential equations can be well approximated by a finite sum of trial functions. For example,
N Y(,)()() x t cj t j x , (2-60) j1 where φj are the trial functions, and the coefficients cj are determined by a chosen way. Weighted residual methods represent a particular set of methods where an integral error between the approximate solution and the exact solution is minimized in a certain way and thereby defining the specific method. The error produces the residual R. To choose the coefficients cj such that the residual R becomes small (in fact 0) over a chosen domain is the aim of weighted residual methods. A group of weight functions Wj are used to achieve this purpose in the integral form W( x ) Rdx 0 . (2-61) j Based on the choice of trial functions, there are four main categories, which are finite volume methods, finite element methods, finite difference methods, spectral methods, and so on. According to different weight functions, weighted residual methods can also be divided into subdomain method, collocation method, least square method, Galerkin method, and so on. The Galerkin method, in which weight functions are chosen to be identical to the trial functions, is the most commonly employed version of weighted residual methods. It has been proved that Galerkin method is the most efficient one in comparison with other methods when applied in the falling film problem[18]. Hence, the Galerkin method is used in our following work.
2.4.2 Full second-order model
The streamwise and spanwise velocity components are expanded as
jmax uxyzt(,,,) axztfj (,,)(/(,,)) j yhxzt , (2-62) j0
23 Chapter 2 Theory Model
jmax wxyzt(,,,) bxztfj (,,)(/(,,)) j yhxzt . (2-63) j0
The set of the trial functions fj(y/h(x,z,t)) for the streamwise and spanwise velocity component are taken as the following orthogonal polynomials with the help of a Gram-Schmidt orthogonalization procedure. aj(y / h(x, z, t)) and bj(y / h(x, z, t)) are respectively the associated coefficients. 1 f() y y y 2 , (2-64) 0 2 17 7 7 f() y y y2 y 3 y 4 , (2-65) 1 6 3 12 13 57 111 99 33 f() y y y2 y 3 y4 y5 y 6 , (2-66) 2 2 4 8 16 32 531 2871 6369 29601 9867 f() y y y2 y3 y 4 y5 y 6 , (2-67) 3 62 124 248 2480 4960 where the cross-stream variable y = y/h(x, z, t) , f0( y ) is semi-parabolic function, corresponding to the Nusselt flat film velocity profile. f1(y ), f2(y )and f3(y ) correct the streamwise and spanwise velocity component up to first-order approximation for free surface deformation. However, considering the convenience working with amplitudes
y y homogeneous in streamwise flow rate q 0 u dy and spanwise flow rate q 0 w dy . The streamwise and spanwise velocity components are finally expanded as 3 45 210 434 u() qsssfy sfy sfy sfy , (2-68) h 1 2 3 0 h 1 1 h 2 2 h 3 3 3 45 210 434 w() qrrrfy rfy rfy rfy , (2-69) h 1 2 3 0 h 1 1 h 2 2 h 3 3 where s1,, s 2 s 3 and r1,, r 2 r 3 are at most first-order inertia corrections to the Nusselt flat semi-parabolic velocity distribution (they can also contain terms of second-order). The cross-wise velocity component v is calculated through the continuity equation by integrating in the cross-stream direction
y v() u w dy . (2-70) 0 x z As well, considering the convenience working with surface temperature homogeneous in temperature field, the temperature field across film is expanded in the following form
4 1 j T Fy ()()() Fh t1 t 2 t 3 t 4 g 0 y ( 1) t jg j y . (2-71) 2 j1
24 Chapter 2 Theory Model
The first term on the right side is linear zeroth-order temperature profile across the film in terms of the surface temperature θ. And F is effective heat flux as follows 1 B F w . (2-72) 1 Bw h
The set of the trial functions for temperature field gi(y ) are taken as the following orthogonal polynomials, which can also correct the temperature distribution up to first order approximation.
g0 ( y ) 1, (2-73)
2 g1 ( y ) 1 3 y , (2-74)
2 3 g2 ( y ) 1 15 y 16 y , (2-75)
2 3 4 g3 ( y ) 1 45 y 112 y 70 y , (2-76)
23 4 5 g4 () y 1105 y 448 y 630 y 288 y . (2-77)
And t1, t2, t3, t4 are at most first-order corrections to the Nusselt flat linear temperature distribution (they can also contain terms of second-order). The spectral Galerkin method, which is a particular case of the weighted residual method, obtained when the set of weight functions Wj(y ) are chosen the same as the set of the trial functions separately for streamwise momentum equation, spanwise momentum equation and temperature equation, is used to solve numerically the problem. After substituting the expansions of the streamwise and span-wise and temperature field into the governing equations and their corresponding boundary conditions, projection of them on the set of the trial functions engenders the equations on q , q , θ, sj, rj, tj. Here for clarity, we show the detail process of the derivation of final evolution equations for q and sj. Substituting the expansions of the streamwise velocity u (2-68), spanwise velocity w (2-69), cross-stream velocity v (2-70), temperature (2-71) into the streamwise momentum equation (2-48) , multiplying streamwise momentum equation by the weight functions Wj, which are equal to the trial functions (2-64)-(2-67), and integrating in the cross-stream direction gives four residuals that are set to zero.
25 Chapter 2 Theory Model
h Wyh( / )3 Re ( uuuvuwudy ) 0 j t x y z h Wyh( / ) 1 uCth 2 [2 uuw ( v | )] (2-78) 0 j yy x xx zz xz x y h 3 We ( xxx h xzz h ) dy ,
h where the Wj = fj (j = 0, 1, 2, 3). The term 0 Wj (/) y hyy u dy can be written in the following form through two integrations by parts. h h hy y 1 y 1 h y Wj yy udy W j y u W j u 2 W j udy , (2-79) 0h h h h h 0 h 0 0 where the ∂yu|h is obtained by the boundary condition (2-57), where the surface temperature θ is coupled through its gradient.
Finally, get four evolution equations for q and sj that we rewrite in the Shkadov scaling:
t q
30 90q 1050s1 3690 s 2 9066s3 6 q qz h 6 q r1 z h h 2 2 2 2 2 2 3131h 31 h 31 h 31 h 5 h 5 h 2124q r h 2648 q r h 6q s h 2124 q s h 2648 q s h 2z 3 z 1z 2 z 3 z 2015h2 3875 h 2 5 h 2 2015 h 2 3875 h 2 6q q 6s q 3006 s q 9074s q 6q q 6 r q z 1z 2 z 3 z z 1 z 5h 5 h 2015 h3875 h 5 h 5 h 2124r q 2648 r q 6 q r 2502 q r 5042 q r 6 q s 2z 3 z z 1 z 2 z 3 z 1 2015h 3875 h 5 h 2015 h 3875 h 5 h 2124q s 2648q s 6q2 h 12 q s h 4248 q s h z 2 z 3 x 1 x 2 x 2015h 3875 h 5 h2 5 h 2 2015 h 2 5296 qshqq12 12 sq 1026 sq 11722 sq 12 qs 3x x 1 x 2 x 3 x x 1 3875h2 5 h 5 h 403 h 3875 h 5 h
4626qx s2 1538 q x s 3 213 183q h 825 q h x z z x x 2015h 775 h 248 62 h 496 h 2289q h h 2103 q h 849 q h1569 q h 849 q() h 2 x z xz z z x x z 496h2 496 h 496 h248 h 496 h2 2 (2-80) 1569q()x h 3 q zz h 2847 q xx h 821 1069 2 xzq zz q xx q 248h 2 h 496 h 248 248 30 30 h h 3We h( h h ), 31x 31 xzz x x x
26 Chapter 2 Theory Model
t s1
1 3q 126s1 126 s 2 126s3 3 q q z h 54 q r1 z h h 2 2 2 2 2 2 10 10h 5 h 5 h 5 h 35 h 55 h 2511q r h 3 q r h 54q s h 2511 q s h 3 q s h 2z 3 z 1z 2 z 3 z 5005h2 35 h 2 55 h 2 5005 h 2 35 h 2 2q q 49s q 7146 s q 2 s q 3q q 54 r q z 1z 2 z 3 z z 1 z 35h 55 h 5005 h 35 h35 h 55 h 2511r q 3 r q3 q r 1107 q r 5 q r 54 q s 2z 3 z z 1 z 2 z 3 z 1 5005h 35 h11 h 2002 h 14 h 55 h 2511q s 3q s 3q2 h 108 q s h 5022 q s h z 2 z 3 x 1 x 2 x 5005h 35 h 35 h2 55 h 2 5005 h 2 6q s h 1 q q 103s q 9657 s q 1 s q 39 q s 3 x x 1x 2 x 3 x x 1 35 h2 35 h 55h 5005 h 35 h 55 h 10557q s 19 q s 3 81 q() h2 57 q h x2 x 3 z z z x 2 10010h 70 h 8 80 h 80 h 3q h 21q h h 63 q h 93 q() h 2 9 q h zz x z z x x x z 40h 16 h2 80 h 40 h 2 40 h (2-81) 69 xq xh27q xz h 9 21 q xx h 9 xzq xxq 40 h80 h 40 80 h 40 1 1 h h 3We h( h h ), 10x 10 xzz xxx
t s2
13 13q 39s1 11817 s 2 11817s3 2 q r1z h 9 q r 2 z h h 2 2 2 2 2 2 420 140h 5 h 140 h 140 h 11 h 11 h 19q r h 2q s h 9 q s h 19q s h 8 s q 3 z 1z 2 z 3 z 1 z 25h2 11 h 2 11 h 2 25 h 2 33 h 10s q 57s q 2r q 9 r q 19r q 4 q r 2 z 3 z 1z 2 z 3z z 1 11h 25h 11 h 11 h 25h 55 h 27q r 99 q r 2q s 9 q s 19q s 4 q s h z2 z 3 z1 z 2 z 3 1 x 385h 350 h 11 h 11 h 25 h 11 h2 18qsh 38 qsh 2 sq 19 sq 76 sq 6 qs 2x 3 x 1 x 2 x 3 x x 1 11h2 25 h 2 33 h 11 h 25 h 55 h 288q s 73 q s 13 3627q() h2 3029 q h x2 x 3 z z z z 2 385h 70 h 64 8960h 8960 h 299q h 559q h h 4927 q h 3211 q() h 2 zz x z z x x 17920h 1792 h2 17920 h 4480 h 2
27 Chapter 2 Theory Model
533q h 2613q h 5993 q h 559 x z x x xz q 17920h 4480 h 17920 h 2240 xz (2-82) 2847q xx h 559 13 13 3 xxq h x h We h( xzz h xxx h ), 8960h 2240 420 420
t s3
3 9q 27s1 8181 s 2 158709s3 171 q r2 z h h 2 2 2 2 2 868 868h 31 h 868 h 868 h 2015 h 4947q r h 171q s h 4947q s h 342 s q 4411 s q 3 z 2z 3z 2 z 3 z 3875h2 2015 h 2 3875 h 2 2015 h 3875 h 171r q 4947 r q 342 q r993 q r 171 q s 2z 3 z z2 z 3 z 2 2015h 3875h 14105h 54250 h 2015 h 4947q s 342q s h 9894 q s h 171 s q 9358 s q z 3 2x 3 x 2 x 3 x 3875h 2015 h2 3875 h 2 2015 h 3875 h 171q s 13653 q s 435 19665 q( h)2 x2 x 3 z x 2 2821h 10850 h 1984 55552 h 18735 q h15 q h 20241q h h 37581 q h z z zz x z z x 55552h 3584 h 55552 h2 111104 h 19953q( h)2 1041q h 19023 q h 35499 q h x x z x x xz 27776h2 111104 h 27776 h 111104 h
2973 17517q xx h 2973 xzq xx q 13888 55552h 13888 (2-83) 3 3 h h 3We h( h h ). 868x 868 xzz xxx Performing the same procedure for the spanwise momentum equation and energy equation gives other evolution equations for q , θ, rj, tj as follows.
t q
90q 1050 r1 3690 r 2 9066r3 6q q x h 6 q s 1 x h 2 2 2 2 2 2 31h 31 h 31 h 31 h 5 h 5 h 2124q s h 2648 q s h 6q r h 2124 q r h 2648 q r h 2x 3 x 1x 2 x 3 x 2015h2 3875 h 2 5 h 2 2015 h 2 3875 h 2 6q q 6 r q 3006 r q 9074r q 6 q q 6 s q x 1 x 2 x 3 x x 1 x 5h 5 h 2015 h 3875h 5 h 5 h 2124s q 2648 s q 6 q s 2502 q s 5042 q s 2x 3 x x 1 x 2 x 3 2015h 3875 h 5 h 2015 h 3875 h 6q r 2124 q r 2648 q r 6q2 h 12 q r h 4248 q r h x1 x 2 x 3 z 1 z 2 z 5h 2015 h 3875 h 5 h2 5 h 2 2015 h 2
28 Chapter 2 Theory Model
5296q r h 12q q 12 r q 1026 r q 11722 r q 3z z 1z 2 z 3 z 3875h2 5 h 5 h 403 h 3875 h
12q r 4626 q r 1538q r 213 183 zq x h z1 z 2 z 3 z 5h 2015 h 775 h 248 62 h 825q h 2289 q h h 2103 q h 849 q h 1569 q h x z x z xz x x z z (2-84) 496h 496 h2 496 h 496h 248 h 849q() h2 1569q( h)2 3 q h 2847 q h 821 x z xx zz q q 496h2 248 h 2 2 h 496h 248 xz x x
1069 30 30 3 zz q h z h We h( xxz h zz z h ), 248 3 1 31
t r1
3q 126 r1 126 r 2 126r3 3q qx h 54 q r1 x h 2 2 2 2 2 2 10h 5 h 5 h 5 h 35 h 55 h 2511q r h 3 q r h 54q s h 2511 q s h 3 q s h 2x 3 x 1x 2 x 3 x 5005h2 35 h 2 55 h 2 5005 h 2 35 h 2 2q q 49 r q 7146 r q 2 r q 3q q 54 s q x 1 x 2 x 3 x x 1 x 35h 55 h 5005 h 35 h 35h 55 h 2511s q 3 s q3 q s 1107 q s 5 q s 54 q r 2x 3 x x 1 x 2 x 3 x 1 5005h 35 h11 h 2002 h 14 h 55 h 2511q r 3 q r 3q2 h 108 q r h 5022 q r h x2 x 3 z 1 z 2 z 5005h 35 h 35 h2 55 h 2 5005 h 2 6q r h 1q q 103 r q9657 r q 1r q 39 q r 3 z z 1z 2 z 3 z z 1 35h2 35 h 55h 5005 h 35 h 55 h 10557q r 19q r 3 81 q( h)2 57 q h z 2 z3 x x x z 2 10010h 70 h 8 80 h 80 h 3q h 21q h h 63 q h 93q( h)2 9 q h xx x z x z z z x 40h 16 h2 80 h 40 h 2 40 h (2-85) 69 z q z h 27q xz h 9 21q zz h 9 xzq zz q 40 h 80h 40 80 h 40 1 1 h h 3We h( h h ), 10z 10 xxz zzz
t r2
13q 39 r1 11817 r 2 11817r3 2q r1x h 9 q r 2 x h 2 2 2 2 2 2 140h 5 h 140 h 140 h 11 h 11 h 19q r h 2q s h 9 q s h 19 q s h 8 r q 3x 1x 2 x 3 x 1 x 25h2 11 h 2 11 h 2 25 h 2 33 h
29 Chapter 2 Theory Model
10r q 57 r q 2s q 9 s q 19 s q 4 q s 2x 3 x 1x 2 x 3 x x 1 11h 25 h 11 h 11 h 25 h 55 h 27q s 99 q s 2q r 9 q r 19 q r 4 q r h x2 x 3 x1 x 2 x 3 1 z 385h 350 h 11 h 11 h 25 h 11 h2 18q r h 38q r h 2 r q 19 r q 76 r q 6 q r 2z 3z 1 z 2 z 3 z z 1 11h2 25 h 2 33 h 11 h 25 h55 h 288q r 73q r 13 3627 q() h2 3029 q h z 2 z3 x x x (2-86) z 2 385h 70 h 64 8960 h 8960 h 299q h 559q h h 4927 q h 3211 q() h 2 xx x z x z z 17920h 1792 h2 17920 h 4480 h 2 533 q h 2613 q h 5993q h 559 z x z z xz q 17920h 4480 h 17920h 2240 xz
2847q zz h 559 13 13 3 zz q h z h We h( xxz h z z z h ), 8960h 2240 420 420
t r3
9q 27 r1 8181 r 2 158709r3 171 q r2 x h 2 2 2 2 2 868h 31 h 868 h 868 h 2015 h 4947q r h 171q s h 4947 q s h 342 r q 4411 r q 3x 2x 3 x 2 x 3 x 3875h2 2015 h 2 3875 h 2 2015 h 3875 h 171s q 4947 s q 342 q s993 q s 171 q r 2x 3 x x 2 x 3 x 2 2015h 3875 h 14105 h54250 h 2015 h 4947q r 342q r h 9894q r h 171 r q 9358 r q x 3 2z 3z 2 z 3 z 3875h 2015 h2 3875 h 2 2015 h 3875 h 171q r 13653q r 435 19665 q( h)2 z 2 z3 x z 2 2821h 10850 h 1984 55552 h 18735 q h15 q h 20241q h h 37581 q h x x xx x z x z 555 5 2 h3584 h 55552 h2 111104 h 19953q( h)2 1041q h 19023 q h 35499 q h z z x z z xz 27776h2 111104 h 27776 h 111104 h
2973 17517q zz h 2973 xzq zz q 13888 55552h 13888 (2-87) 3 3 h h 3We h( h h ), 868 z868 xxz zzz
33B h 91 Prw 91B h 91 t w 1272 (FB ) 3855t1 80355 t 2 26085t3 27615 t 4 348 Pr r1 z F 13h 13 h2 91 h 2 13 h 2 13 h 2 13
30 Chapter 2 Theory Model
174 2024643 1275179 174Fq h 348 Fr h q F r F r F z 1 z 91 z 189282 z 7280 3 z 91h 13 h 2024643Fr h 1275179Fr h 1805 t q 446685 t q 2z 3z 3 z 4 z 18928h 7280 h 858 h 26026 h h
58Bw hF z q 9645 14789 261 qzt187 q z t 2 F z r2 F z r 3 91Bw h 1 2704 1040 91 h91 h 35603q t 53853q t 75 q 1665 r 700995 r z 3 z4 z 1 z 2 z 8008h 9464 h 182 h 52 h 5824 h 9134767r 174 348 2024643 1275179 3 z q F s F s F s F 29120h 91 x 131 x 18928 2 x 7280 3 x 174Fq h 348Fs h 2024643Fs h 1275179 Fs h 1805 t q x 1 x 2x 3 x 3 x 91h 13 h 18928h 7280 h 858 h
446685t4x q 58 B w hF x q 9645 14789 261 q x t 1 F x s2 F x s 3 26026h 91 Bw h 1 2704 1040 91 h 87q t 35603 q t 53853 q t 75 q 1665 s x2 x 3 x 4 x 1 x 91h 8008 h 9464 h 182 h 52 h 2 2 700995s2x 9134767 s 3 x 636B() z h 636 B() x h 5824h 29120 h 13 h 13 h 2 (2-88) 58 58 1272F() h 2 1272 F() h 2F h 2 F h Fh Fh z x z z x x91 zz 91 xx 13h 13 h
1272zh z 1272 xh x F zz h F xx h xx zz , 13h 13 h
Pr t t1 465 (FB ) 1395t 25925 t 7635t 8435 t 1 2 3 4 13h 13 h2 91 h 2 13 h 2 13 h 2
1685Bw ht 367 2871 6615979 Pr q z F r1 z F r 2 z F 2184 (1 Bw h ) 1456 832 302848 4709843 367Fq h 2871 Fr h 6615979 Fr h r F z 1 z 2 z 1164803 z 1456h 832 h 302848 h 4709843Fr h 11 5t q 51 t q 16537 t q 3z F q 1z 2 z 3 z 116480h 16z 112 h 112 h 48048 h
797623t4 z q 1685Bw hF z q 1 29989 73603 F z r1 F z r 2 F z r 3 208208h 2184 (1 Bw h ) 64 43264 16640 81q t 43 q t 31747 q t 91075q t 957 q z1 z 2 z 3 z4 z 182h 52 h 16016h 208208 h 728 h 101r 328563 r 1676281r 367 2871 1z 2 z 3 z q F s F 104h 11648 h 34944 h 1456 x 832 1 x
31 Chapter 2 Theory Model
6615979 4709843 367Fq h 2871 Fs h s F s F x 1 x 3028482x 116480 3 x 1456h 832 h 6615979Fs h 4709843 Fs h 11 5t q 51 t q 2x 3 x F q 1x 2 x 302848h 116480 h 16x 112 h 112 h
16537t3x q 797623 t 4 x q 1685 B w hF x q 1 29989 F x s1 F x s 2 48048h 208208 h 2184 (1 Bw h ) 64 43264 73603 81q t 43 q t 31747 q t 91075 q t F s x1 x 2 x 3 x 4 (2-89) 16640x 3 182h 52 h 16016 h 208208 h
957 q x 101s1x 328563 s 2 x 1676281 s 3 x 728h 104 h 11648 h 34944 h 2 2 465B() z h 465B() x h 1685 1685 zzFh xx Fh 26 h 26h 2184 2184 2 2 465F()z h 465F()x h 465 z h z 465 x h x , 13h 13 h 13 h 13 h
Pr t t2 675 (FB ) 2025t 40535 t 13465t 14945 t 1 2 3 4 13h 13 h2 91 h 2 13 h 2 13 h 2
995Bw ht 1717 25283 12214269 Pr q z F r1 z F r 2 z F 2184 (1 Bw h ) 1456 2496 302848 8776853 1717Fq h 25283 Fr h 12214269 Fr h r F z 1 z 2 z 1164803 z 1456h 2496 h 302848 h 8776853Fr h 1 21t q 7 t q 2539 t q 3z F q 1z 2 z 3 z 116480h 16z 80 h 240 h 11440 h
22623563t4 z q 995Bw hF z q 245 52525 F z r1 F z r 2 3123120h 2184 (1 Bw h ) 192 43264 72773 984q t 10943 q t219797q t 904601 q t F r z 1 z2 z 3 z 4 16640z 3 455h 5460h 80080 h 446160 h 995q 1171 r 491097 r 22642583r 1717 z 1 z 2 z 3 z q F 728h 104 h 11648 h 174720 h 1456 x 25283 12214269 8776853 1717 Fq h s F s F s F x 24961x 302848 2 x 116480 3 x 1456 h 25283Fs h 12214269 Fs h 8776853 Fs h 1 21 t q 1x 2 x 3 x F q 1 x 2496h 302848 h 116480 h 16x 80 h
7t2x q 2539 t 3 x q 22623563 t 4 x q 995 B w hF x q 245 F x s1 240h 11440 h 3123120 h 2184 (1 Bw h ) 192 52525 72773 984q t 10943 q t 219797 q t F s F s x1 x 2 x 3 43264x2 16640 x 3 455h 5460 h 80080 h
32 Chapter 2 Theory Model
904601q t 995 q 1171s 491097 s x4 x 1x 2 x 446160h 728 h 104 h 11648 h 2 2 22642583s3 x 675F()()z h 675 B z h 675 z h z (2-90) 174720h 13 h 26 h 13 h 2 2 995 675F()()x h 675 B x h 675 x h x 995 zz Fh xx Fh , 2184 13h 26 h 13 h 2184
Pr t t3
486(FB ) 1458t1 29700 t 2 9180t3 9450 t 4 459 Pr q z F 13h 13 h2 91 h 2 13 h 2 13 h 2 728
153Bw ht 5589 5157441 532005 459 Fq z h r1 z F r 2 z F r 3 z F 728(1 Bw h ) 832 302848 23296 728 h 5589Fr h 5157441 Fr h532005Fr h 3 t q 69 t q 1z 2 z3 z 1 z 2 z 832h 302848 h23296 h 70 h 140 h
17753t3z q 198933t4 z q 153 B w hF z q 27 97137 F z r1 F z r 2 40040h 74360 h 728(1 Bw h ) 64 43264 4629 7353q t 423 q t 1293q t 164601 q t F r z1 z 2 z 3 z 4 3328z 3 7280h 7280 h 10010 h 130130 h 459q 105 r 231615 r 574107r 459 z 1 z 2 z 3 z q F 728h 13 h 11648 h 58240 h 728 x 5589 5157441 532005 459Fq h 5589 Fs h s F s F s F x 1 x 8321x 302848 2 x 23296 3 x 728h 832 h 5157441Fs h 532005 Fs h 3t q 69 t q 17753 t q 2x 3 x 1x 2 x 3 x 302848h 23296 h70 h 140 h 40040 h
198933t4x q 153 B w hF x q 27 97137 4629 F x s1 F x s 2 F x s 3 74360h 728(1 Bw h ) 64 43264 3328 7353q t 423 q t 1293 q t 164601 q t 459 q x1 x 2 x 3 x 4 x 7280h 7280 h 10010 h 130130 h 728 h 2 (2-91) 105s1x 231615 s 2 x 574107 s 3 x 486 F() z h 13h 11648 h 58240 h 13 h 243B() h2 486 h 153 486F()() h2 243 B h 2 z z z Fh x x 13h 13 h 728zz 13 h 13 h
486xh x 153 xx Fh , 13h 728
Pr t t4 1386(FB ) 4158t 12100 t 26180t 26950 t 1 2 3 4 13h 13 h2 13 h 2 13 h 2 13 h 2
33 Chapter 2 Theory Model
187Bw ht 187 62377 4115313 Pr q z F r1 z F r 2 z F 312 (1 Bw h ) 104 2496 43264 2432617 187Fq h 62377 Fr h 4115313 Fr h r F z 1 z 2 z 166403 z 104h 2496 h 43264 h 2432617Fr h 1t q 259 t q 59819 t q 187 B hF q 3z 2z 3 z 4 z w z 16640h 12 h 104 h 4056 h312 (1 Bw h ) 7 173159 254639 561q t 509 q t F r F r F r z1 z 2 192z1 43264 z 2 16640 z 3 208h 624 h 113q t 3845q t 187 q 1309 r 189123 r z 3 z4 z 1 z 2 z 26h 1014 h 104 h 52 h 1664 h 1082345r 187 62377 4115313 3 z q F s F s F 4992h 104 x 24961x 43264 2 x 2432617 187Fq h 62377Fs h 4115313 Fs h s F x 1x 2 x 166403 x 104h 2496 h 43264 h 2432617Fs h 1t q 259 t q 59819 t q 187 B hF q 3 x 2x 3 x 4 x w x 16640h 12 h 104 h 4056 h 312 (1 Bw h ) 7 173159 254639 561q t 509 q t F s F s F s x1 x 2 192 x 1 43264x2 16640 x 3 208h 624 h 113q t 3845 q t 187 q 1309s 189123 s x3 x 4 x 1x 2 x 26h 1014 h 104 h 52 h 1664 h (2-92) 2 2 1082345s3 x 1386F()()z h 693 B z h 1386 z h z 4992h 13 h 13 h 13 h 2 2 187 1386F()()x h 693 B x h 1386 x h x 187 zzFh xx Fh , 312 13h 13 h 13 h 312
Here we need to emphasize the integral version of the continuity equation (2-50), which connects the film thickness with the streamwise and spanwise components of flow rates and completes the above system equations.
h u v w dy 0, (2-93) 0 x y z where
h v dy v| v | v | . (2-94) 0 y h0 h Using the kinematic boundary condition (2-56) and the definition of streamwise
y y and spanwise components of flow rates q 0 u dy and q 0 w dy ., the integral version of continuity equation can be written in the following form.
34 Chapter 2 Theory Model
h h ()()u v wdy u wdy huhwh 0x y z 0 x z t x z h h h udyuh wdywh (2-95) t0 x x 0 z z
th x q z q .
So the integral version of continuity equation becomes
th x q z q 0 . (2-96)
The above system equations (2-80)-(2-92) together with the continuity equation (2-96) will be referred to as the full second-order model the regularized model for the HF case.
2.4.3 Regularized model
The full-seconder order model is indeed simpler than the full size governing equations constituted by N-S equations, energy equation, and corresponding boundary conditions, but it has 13 unknowns and is cumbersome to perform the calculation due to its complexity It is hence necessary to obtain reduced models which also retain the dynamic characteristics of the full-size model. A certain simplification should be done. Two different ways are adopted for momentum equations and energy equations respectively. For the momentum equations, two measures of simplification are made in this paper without losing its second-order approximation. Firstly, the equations of sj, rj are truncated to first order approximation based on their nature. It is reminiscent of that sj, rj are respectively the first order correction to the Nusselt flat film parabolic velocity profile corresponding to the [16] residuals produced out of f0(y ). As justified in Ruyer-Quil et al. , they appear through inertial terms involving their space and time derivatives or through products with
2 derivatives of h and q , q , θ, which are terms of O() . Meanwhile, they are absorbed into the stream-wise flow rate and span-wise flow rate through the boundary conditions on surface in the viscous and conductive terms, which are already of O() 2 . So the amplitudes sj, rj can be approximated as functions of h, q , q , θ and their derivatives 2 truncated at O() . The resulting system of sj, rj is as follows,
12 19 2 s1 hq t () hqqqqhqh z z x 210 1925 (2-97) 74 17 1 2 2 hqx q hq zq h x O() , 5775 5775 4 0
35 Chapter 2 Theory Model
22 2 s2 () hq z q q q z h q x h hq x q 5775 17325 (2-98) 4 299 2 2 q z q h x O( ), 17325 53760 5 s h2 O( 2 ), (2-99) 3 3584 x
12 19 2 r1 hq t () hqqqqhqh x x z 210 1925 (2-100) 74 17 1 2 2 hqzq hq x q h z O() , 5775 5775 40
22 2 r2 () hq x q q q x h q z h hq zq 5775 17325 (2-101) 4 299 2 2 q xq h z O( ), 17325 53760 5 r h2 O( 2 ), (2-102) 3 3584 z
Now the expressions of sj, rj are much simpler, and substituting them into the streamwise flow rate equation (2-80) and spanwise flow rate equation(2-84) gives simpler system equations without unknowns sj, rj. However, the resulting equations are of no use since they have the finite-time blow up behaviour due to the high-order nonlinear terms originating from the inertial term[16]. To reduce the singularity due to the strong nonlinearity, a algebraic Padé-like regularization approach is introduced by Scheid et al.[19] Following the same method and introducing Pade-like regularization factor, the original high-order nonlinear term is reduced. Eventually, flow rate equations for q , q are obtained as follows.
2 9q 17 q 8 q z q 9 q z q 9 q q z h tq ()()2 x h x q 2 7h 7 h 7 h 7 h 7 h 5 5q 5 5 h h h 3 We h() h h 6 2h2 6x 6 xzz xxx 5 q 9 q h q 2 x x ( x hq xx ) 42 ( x h ) 6 xx h (2-103) 4 224 h 2 h h 9 13q h 43 q h 73 q h q h q z x x z xz z z 2xx 16h 16 h 16 h h 3q() h2 13q h h 23 q h z x z zz q 4h2 4 h 2 16 h zz
36 Chapter 2 Theory Model
7 5 x xzq / (1 q x h ), 2 70 56 h 9q2 17 q 8q q 9 q q 9 q q h q () h p x x x t 7h2 z 7 h z 7 h 7 h 7 h 2 5q 5 5 5 h h 3 We h() h h 2h2 6z 6 xxz zzz 4 z
q2 9z q z h q 9 13 xq z h 42 ( zh ) 6 zz h zz q (2-104) h2 h h 2 16 h 43q h 73 q h q h3 q() h 2 z x zx x x x 16h 16 h h4 h2
13qz h x h 23q xx h 7 2 xxq zx q , 4h 16 h 2 As far as the energy equations (2-88)-(2-92) are concerned, it is cumbersome to perform the regularization approach because of coupling between the different physical effects (momentum and energy advection, free-surface deformations and Marangoni [18] effect) . For the sake of simplicity, the corrections tj of the temperature distribution are neglected. This assumption is reasonable and still consistent with the gradient expansion at second order since the interfacial temperature is only coupled to the flow rate through its gradient (already of O() )[16]. Substituting the linear temperature distribution into energy equation (2-53) multiplying by the weight functions W0 (set equal to g0) and integrating across the film layer gives the final evolution equation for θ as follows. FB 1 3 h 5 Pr[() h F q F Fx F q h2t t 8 x h 8 x q 3h 5 q x q()] F Fz F q z h8z h 8 z h 2 (2-105) 1 h h 1 [2F h ( F B ) x x x h F F h x x2h h 2 xx xx xx 2 1 h h 1 2F h ( F B )z z z h F F h ] 0. z z2h h 2 zz zz zz Equations (2-103)-(2-105) and (2-96) constitutes the three-dimensional regularized model for the HF case, hereinafter referred to as RM.
2.5 Summary of This Chapter
The detailed derivation process of the regularized model has been shown in this chapter. With a physical model of the falling liquid film with several assumptions, the corresponding governing equations and boundary conditions were determined. For
37 Chapter 2 Theory Model analysis convenience, the Shkadov scaling was used for dimensionless processing. Considering the long-wave characteristics of the falling film, a gradient expansion strategy was performed by introducing a film parameter ε to estimate the amplitude order of each term in the governing equations and corresponding boundary conditions. Simplifications of the governing equations and boundary conditions were made by neglecting terms higher than Ο() 2 . Following the idea as in Prandtl’s boundary layer theory, the pressure P was eliminated, and the film layer equations were obtained. Subsequently, using the weighted residual method to perform the mathematical approximations of the film layer equations led to the full second-order model. However, the full second-order model is still too complicated. Therefore, further simplifications were carried out where the regularization procedure was adopted to avoid the singularity. Finally, the three-dimensional regularized model for HF case with liquid film thickness h, streamwise flow rate q , spanwise flow rate q , and liquid film surface temperature θ as variables was obtained.
38 Chapter 3 Model Validation
Chapter 3 Model Validation
Before conducting the three-dimensional numerical simulation, it is necessary to verify the accuracy of the RM. Firstly, perturbation expansion of RM is used to test whether RM is equivalent to BE at small Reynolds number to illustrate whether RM is correct at small Reynolds number. Secondly, the linear stability analysis of RM in temporal mode is performed and compared with the precise Orr-Sommerfeld equation to study the accuracy of RM at moderate Reynolds numbers. Meanwhile, the influence of Reynolds number, Marangoni number and the disturbance wavenumber on the instability of film flow is investigated by linear stability analysis. Finally, the comparison between the numerical simulation results and the experimental results is made to check the accuracy of RM in three-dimensional situations.
3.1 Derive the Benney equation from the RM
As mentioned in Chapter 1, the Benney equation (BE) is derived under the assumption of low Reynolds number and long-wave, and can be used as a benchmark to examine the accuracy of other models at small Reynolds number. So, if it provides that the BE can be deduced from the RM with the perturbation expansion procedure, it indicates that the RM is valid, at least at small Reynolds number, The BE for HF case is derived by Kalliadasis et al.[18] based on the long-wave theory with perturbation expansion procedure of full Navier–Stokes and energy equations and associated wall and free-surface boundary conditions. Here we write the BE for HF case in the Nusselt scaling for clarity. 2 h h2 h Re () h 6 h t x5 x x 32 3 (3-1) hh 1 2 h 2 xz Ct xz h M xz We xz xz h 0. 32 B Bw BB w h 3
One thing to mention here is that the BE gives clear distinction among the factors influencing the film evolution. The first term is the evolution of film thickness, the second term is the convective term owing to mean flow, the third term arises from inertia, the fourth term represents the hydrostatic head in the direction perpendicular to the wall, the fifth term is due to the Marangoni effect and the sixth term accounts for the
39 Chapter 3 Model Validation streamwise and spanwise curvature gradients associated with surface tension. The perturbation expansion was applied for the variables in RM as the following form,
(0) (1) 2 q q q O() (3-2)
(0) (1) 2 q q q O() (3-3)
(0) (1) O() 2 (3-4) where the different number in the superscript corresponds to different order solution. Substituting these variables into the RM, collecting terms of the same power in ε, and solving the resulting equations of the same order, one can get the formula for variables of different order successively. Solving the equations from terms of 0 , we get
h3 q(0) (3-5) 3
(0) q 0 (3-6)
1 (0) (3-7) B Bw BB w h
Substituting (3-5)-(3-7) into the equation from terms of 1 , and solving the equations, we get 1 2BMB h2 h 1 q(1) Cth3 h Reh 6 h w x 2 Weh 3 ( h h ), 3x 5 x 2 3 xzz xxx 2BBw hB w B (3-8) 1BMB h2 h 1 q(1) Cth3 h w z 2 Weh 3( h h ), (3-9) 3z 2 3 zzz zxx 2BBw h B w B
3 (1) BPrRehx h Bw h 1 B w (3 Bh 4) 12 B 3 . (3-10) 8Bw ( Bh 1) B
(0) (1) (0) (1) Then we substitute q q q and q q q into the equation (2-96), and neglect the terms of 2 , and get the evolution equation as follows,
40 Chapter 3 Model Validation
2 h h2 h Re () h 6 h t x5 x x 32 3 (3-11) hh 1 2 h 2 xz Ct xz h M xz We xz xz h 0, 32 B Bw BB w h 3 which is the same as the BE. It has proved that the BE can be obtained by applying the perturbation expansion of the RM, which indicates that the RM is valid and accurate at small Reynolds number. However, the validity of RM at moderate Reynolds number is still unknown. Thus detailed examination of the validity at larger Reynolds number will be performed by linear stability analysis.
3.2 Linear stability analysis
Flow stability has been one of the central issues of fluid mechanics for a long time. In nature, only a stable flow state can be maintained and easily observed. Studying the stability of the flow is to investigate the response characteristics of the flow to disturbances, that is, to study whether the flow is stable, under what conditions the flow turns from stable to unstable, and what direction the unstable flow will develop to. After more than a century of development, research on flow stability has formed a set of theories and methods. It is generally believed that the linear stability theory developed by the normal mode has matured, but the nonlinear stability theory is not yet perfect, and it is only successful for the weak nonlinear stability theory. In general, in the stability analysis, the disturbance imposed on the fluid can be arbitrarily small, the system of equations can be linearized, and the instability can be predicted by analyzing the behavior of the waves. If the prediction is unstable, the final flow pattern cannot be predicted by the linear equations. In this case, the complete nonlinear equations must be considered. Therefore, for the nonlinear flow instability phenomenon, the main research is through experiments and numerical simulation methods.
3.2.1 Definition of the flow stability
The physical definition of flow stability is determined based on the response characteristics of base state flow to disturbance. If for some reason the flow deviates slightly from the base state flow at the initial moment, this deviation is called as the disturbance to the base state flow. As time progresses, if this disturbance eventually
41 Chapter 3 Model Validation decays and the base state recovers, we call this base state flow asymptotically stable to such disturbance; if the amplitude of the disturbance remains the same, the base state flow is called neutrally stable to the corresponding disturbance; if the disturbance is eventually amplified, the base state flow is unstable to the disturbance; if the disturbance is damping at all times, the base state flow is said to be monotonically stable to the disturbance. For flows that are asymptotically stable rather than monotonically stable, the disturbances will grow for a certain period, but eventually they will decay. This growth is called transient growth. Unstable base state flow eventually evolves into multiple flow states. Compared with the base state, the symmetry of the new flow formed after disturbance is generally reduced. In addition to the formation of turbulence, there are several representative flow states after disturbance: steady flow will form a periodic flow; periodic flow will form a multiple frequency quasi periodic flow; two-dimensional flow will form a three-dimensional flow.
3.2.2 Method of linear stability analysis
Four steps are needed to study linear stability of the flow. First, the base state of the flow is needed, such as velocity field, pressure field, and temperature field, which are called base state flow. It is the object of the research of flow stability. Theoretically, the flow field should satisfy the governing equations and corresponding boundary conditions. The basic flow generally has symmetry in time or space, and the symmetry in time mainly manifests as steady flow and periodic flow. The symmetry in space mainly manifests as uniformity and spatial periodicity. Second, the equations need to be linearized in the vicinity of base state, which means the disturbance need to be infinitesimal to neglect the nonlinear terms. Then, the normal mode ansatz is applied to formulate the problem as an eigenvalue problem. Since the base state flow has temporal or spatial symmetry, these small perturbations can be expanded into exponential forms. For example, for the Nusselt flat film solution[2], since the disturbance of film thickness only changes in y direction, thus the disturbance can be expressed as hˆei(kx x k z z t ) , where the hˆ is the amplitude of the disturbance. Substituting the disturbance into the linearized equations, the eigenvalue problem about ω is given. And the dispersion relation among the wave number kx, kz and frequency ω is determined. Finally, the eigenvalue problem needs to be solved and analysis needs to be performed. For the temporal mode, the wave number kx, kz are assumed to be real number and ω is assumed
42 Chapter 3 Model Validation to be complex number. The imaginary part of ω, ωi, is called as the temporal growth rate. When the temporal growth rate ωi > 0, the disturbance increases with time, and the base state is unstable; when ωi = 0, the disturbance remains unchanged, and the base state is neutrally stable; when ωi < 0, the disturbance eventually disappears, and the base state is stable. For the spatial mode, the wave number kx, kz are assumed to be complex number and ω is assumed to be real number. When the spatial growth rate k(x,z), i >0, the base state is unstable; k(x,z), i=0 , the base state is neutrally stable; k(x,z), i <0, the base state is stable. In fact, the amplitude of the disturbance may change with time and space at the same time. The disturbance is not only different from the temporal mode but also different from the spatial mode. This case is called as spatial-temporal mode, where the frequency and the wave number are complex numbers. The disturbance of spatial-temporal mode has two kinds of different stability. One is that all growing disturbances only propagate to the downstream of the flow, which is called convective instability; the other is that the disturbances increase while they spread both downstream and upward, which is called absolute instability. The spatial model is only effective in the flow of convective instability. For the absolute instability of the flow, as the disturbance grows locally, the periodic disturbance source is submerged, making the spatial model ineffective.
3.2.3 Linear stability analysis of the falling film for HF case
The Benney equation and Orr–Sommerfeld equation are used here to check the accuracy of RM by performing the linear stability analysis. The Orr–Sommerfeld equation, obtained by linearizing the full governing equations and boundary conditions and decomposing the perturbations into normal modes, are accurate in predicting the linear stability of the falling film. Here the Orr–Sommerfeld equation is used as a benchmark to conduct the examination. Considering that under the actual working situation, the flow rate, heat flux density, and artificial disturbance are easily controlled and most significant parameters, and the corresponding dimensionless numbers are the Reynolds number Re, the Marangoni number Ma and the disturbance wavenumber k. Thus, the linear stability analysis is used to study the influence of Re, Ma and k on the instability of film flow in this part besides examining the accuracy of RM. The base state solution of the RM has been derived in 3.1, which are (3-5)-(3-7).
43 Chapter 3 Model Validation
And for the base state, the film is flat, thus the dimensionless film thickness h=1, which leads to the base state solution h=1 (3-12)
q 1/3 (3-13)
q 0 (3-14)
1 (0) (3-15) B Bw BB w
i(k x k z t ) ˆ i(kx x k z z t ) i(kx x k z z t ) x z By substituting h1 h e , q1/3 qˆ e , q0 qˆ e ,
ˆ i(kx x k z z t ) ˆ ˆ 1/ (BBB w (1 )) e into the RM, where h , qˆ , qˆ , are disturbance amplitudes, kx is streamwise wavenumber, kz is spanwise wavenumber, and ω is angular ˆ ˆ frequency, and linearizing for h 1 , qˆ 1 , qˆ 1 , 1 , we can get the dispersion relation. Taking ∂z = 0, and ∂x = 0, δ = 0 leads to the streamwise and spanwise dispersion relations, respectively. For the sake of simplicity, only temporal mode, which is taking k real number and ω complex number, is considered in this paper.
The imaginary part of ω, denoted as ωi, is the growth rate of instability of temporal mode. Here we only give the numerical solution rather than analytical solution due to the complexity of the results.
The influence of Re on the growth rates ωi predicted by RM, BE and Orr-Sommerfeld (hereinafter referred to as the O-S) equation are shown in Figure 3.1. In Figure 3.1 (a), the streamwise growth rates increase as Reynolds number increases. Because inertial effect acts as a destabilizing factor for the streamwise flow. The influence of the inertial effect on the long-wave hydrodynamic instability increases with increasing Reynolds number. However, the result is totally different in spanwise direction as shown in Figure 3.1 (b). Spanwise growth rates decrease as Reynolds number increases. Because film flows in streamwise direction, and thus there is no flow in spanwise direction, which indicates that there is no long-wave hydrodynamic instability in spanwise at the initial stage ( q 0 ), and only long-wave Marangoni instability exists. And larger Reynolds number means thicker film, which weakens the Marangoni instability in spanwise direction. The increase of the Reynolds number also weakens the Marangoni instability in streamwise direction, but in comparison with the increase of hydrodynamic instability caused by an increase of the Reynolds number in
44 Chapter 3 Model Validation streamwise direction, the inhibiting effect on the Marangoni instability plays less role. Therefore, the increase of the Reynolds number mainly exerts a destabilizing effect in the streamwise direction.
(a) Streamwise growth rates versus the Reynolds number
(b) Spanwise growth rates versus the Reynolds number
Figure 3.1 Growth rates versus the Reynolds number. Ma =20, Γ = 3375, = π / 2, Pr = 7, Bi =
0.12, Biw= 0.6, kx=0.25 and kz=0.25
Furthermore, the streamwise growth rates predicted by RM, BE and O-S are almost the same at small Reynolds number, but BE diverges apparently at moderate
45 Chapter 3 Model Validation
Reynolds number. Because the BE is derived at small Reynolds number, where the magnitude order of inertial term is Ο(1). When the Reynolds number increase, the influence of inertial effect increases, and thus the magnitude order of inertial term is no longer Ο(1), which leads to the failure of BE at moderate and large Reynolds number. RM does not have this limitation and is applicable from small to moderate Reynolds number, which can also be seen from Figure 3.1 (a). Streamwise growth rates predicted by RM are in good agreement with the results predicted by O-S. Another advantage of RM is that it is much simpler than full Navier-Stokes and energy equations, and easy for mathematical and numerical calculation. In Figure 3.1 (b), the spanwise growth rates predicted by the O-S, BE and RM are almost the same from small to moderate Reynolds number. Because the inertial term does not exist in the spanwise direction at the initial stage ( q 0 ). The main reason for the failure of the BE that the inertia term is no longer a small amount under larger Reynolds number does not exist as well. The growth rates versus Ma are shown in Figure 3.2. The effects of increasing Ma on the growth rates in both streamwise and spanwise directions are same, all resulting in an increase in the growth rate of instability, indicating that the Marangoni effect destabilizes the film layer. As Ma increases, the surface tension gradient becomes larger, thus leading to larger thermocapillary stress, so that the instability in streamwise and spanwise directions is enhanced.
(a) Streamwise growth rate versus the Marangoni number
46 Chapter 3 Model Validation
(b) Spanwise growth rate versus the Marangoni number
Figure 3.2. Growth rate versus the Marangoni number. Re=1.25, Γ = 3375, = π / 2, Pr = 7, Bi =
0.12, Biw= 0.6, kx=0.25, kz=0.25
The growth rate curves as functions of wavenumber k at small Reynolds number are shown in Figure 3.3. The growth rates obtained by RM are fairly close to the growth rates obtained by O-S and BE. There exists a wavenumber interval where the growth rates ωi>0 in both the streamwise and the spanwise directions, which means the disturbance is unstable in this interval. The wavenumber corresponding to the maximum growth rate is called as the most unstable wavenumber (km), and the corresponding disturbance is called as the most unstable disturbance. The wavenumber is called cut-off wavenumber (kc) when the growth rate is equal to 0. km,x≈0.3891 is the most unstable streamwise wavenumber and km,z≈0.2997 is the most unstable spanwise wavenumber. kc,x≈0.5520 is streamwise cut-off wavenumber and kc,z≈0.4240 is the spanwise cut-off wavenumber. From the calculation we can find that km,x≈kc,x/ 2 , km,z≈kc,z/ 2 , which [40] coincides well with the formula km=kc/ 2 , indicating the validity of the RM. As can be seen from the changes of the growth rates of the flows in Figure 3.1, Figure 3.2 and Figure 3.3, the error of the RM increases slightly with the increase of Re, Ma and k, but still in good agreement. Therefore, the RM is used to study the three-dimensional evolution of the liquid film under HF condition in next chapter. Investigations on the change of heat transfer during the film evolution and the influence of perturbations on the heat transfer are also performed.
47 Chapter 3 Model Validation
(a) Streamwise growth rate versus the wavenumber
(b) Spanwise growth rate versus the wavenumber
Figure 3.3. Growth rate versus the wavenumber. Re=1.25, Γ = 3375, = π / 2, Pr = 7, Bi = 0.12,
Biw= 0.6, Ma=20
3.3 Three-dimensional verification
The linear stability analysis is based on the assumption that the disturbance is infinitesimal, and the interaction between the unstable waves is neglected. The influence of their development on the base state is also ignored, which limits the application of
48 Chapter 3 Model Validation linear stability analysis in the further development of disturbances. In that case, the nonlinear effects become non-negligible. At this time, it is important to consider three-dimensional numerical simulation that can demonstrate the nonlinear characteristics of the liquid film. For this purpose, the accuracy of the RM in the three-dimensional case needs to be tested. But many of the experimental results are unheated or non-uniformly heated. There is a lack of uniform heat flux heating experimental results. For this reason, the experimental result of the isothermal falling film is compared with our simulation results to examine RM. Just set the value of Ma=0 to calculate the isothermal falling film using RM. The details of the way to perform the three-dimensional simulation will be introduced in next chapter. The RM is validated by reproducing the three-dimensional horseshoe wave similar to the experimental one. The physical property parameters Re = 40.8, Γ = 3375, β =π / 2, [48] kx = kz = 0.3845 are the same with the ones in the experiment , and other parameters hN = 0.912, nx = nz = 3, M = N = 256. Figure 3.4 shows the comparison between the experimental experiments and our simulation results, and we can see very similar wave shapes are obtained.
Figure 3.4 Comparison of film wave patterns between the experimental results and our simulation results. Re = 40.8, Γ = 3375, β = π/2, hN = 0.912, Lx × Lz =2nxπ/kx × 2nzπ/kz, nx = nz = 3,
kx = kz = 0.3845, M × N = 256 × 256.
By comparison with the experimental results, it is proved that RM can simulate the three-dimensional falling liquid film correctly, which lays the foundation for us to study the three-dimensional heated falling liquid film with RM.
49 Chapter 3 Model Validation
3.4 Summary of this chapter
In this chapter, it has proved that RM is accurate at small Reynolds numbers by using perturbation expansion of RM, and RM is also accurate from small to moderate Reynolds numbers via linear stability analysis. Compared with the three-dimensional experimental results, the simulation result has proved that the RM can reflect the nonlinear characteristics of the liquid film correctly. In general, the RM not only agrees with the exact linear behavior obtained from Orr–Sommerfeld but also describes properly the nonlinear dynamics of the falling films. The linear stability analysis has shown that the streamwise instability growth rates increase with the increase of Reynolds number and Marangoni number; the spanwise instability growth rates increase with the increase of Marangoni number and decrease with the increase of Reynolds number. There is an unstable range of disturbance wave number in both streamwise and spanwise directions.
50 Chapter 4 Three-dimensional Simulation
Chapter 4 Three-dimensional Simulation
As mentioned above, flow rate, heat flux density, and external disturbance are the most significant factors, which not only have an important influence on linear instability but also have a significant effect on the evolution of three-dimensional film patterns. To this end, three-dimensional numerical simulations of the evolution of liquid film under HF condition and the investigations of the influence of the Reynolds number, the Marangoni number and the disturbance wavenumber on the film evolution will be carried out, and the preliminary results will be obtained. Various structures appearing during the evolution of the liquid film will affect the heat transfer. Scheid et al.[40] conducted three-dimensional numerical simulation with random disturbance as the initial disturbance in ST condition. It was shown that when the Reynolds number is low, the film developed into rivulet structures, upon which the three-dimensional solitary waves were formed. The heat transfers between the liquid film and the gas also changed as the structures of the liquid film changed. In order to enhance heat transfer by applying appropriate external disturbances, detailed investigations on the effect of external disturbance on heat transfer will be performed. Besides, the influence of Reynolds number, the Marangoni number on film patterns and rupture time will be studied to find a way to make the film stay continuous without rupture as long as possible. The structure of this chapter is organized as follows. First, the way to carry out three-dimensional numerical simulations with the RM is introduced to investigate the nonlinear evolutions of the liquid film and define a parameter to describe the change of the heat transfer between the liquid film and the gas. Second, the influences of different initial disturbances on the evolutions are discussed: the random disturbance is used as the initial disturbance to study the evolution process of the film under HF condition, where the random disturbance represents the natural noise which is inevitable in practical circumstance and includes a wide spectrum of disturbance wavenumber and plays an important role during the evolution of the film layer; the most unstable streamwise and spanwise disturbances in dominance are introduced in the initial random disturbance respectively; then both are introduced in the initial random disturbance at the same time. Further, the influence of Reynolds number, the Marangoni number on
51 Chapter 4 Three-dimensional Simulation nonlinear evolution of the falling liquid film is investigated. Finally, the conclusion is summarized.
4.1 Introduction to three-dimensional simulation
In streamwise and spanwise directions, the periodic boundary conditions are applied to perform the simulation. The spatial derivatives are calculated with the Fourier spectral method[49]. MN grid points are used here to discretize the computational domain LLx z . xi = iLx/M, zj =jLz/N (where i = 0, 1 · · · M, j = 0, 1 · · · N) are the corresponding coordinates. The wavenumber aliasing is treated by retaining the first 2/3 of the Fourier modes in each direction. Grid independence tests are performed by increasing M and N until there is no qualitative difference in the energies of deformation [50] Ex, Ez . The time dependence is numerically solved through a fourth-order Runge-Kutta scheme. When the thickness of the film h 103 , the film is adjudged to have ruptured and the computation is terminated. The following parameters are used as the initial parameters for the simulation.
hxz( , ,0) hAN x cos(2 nxL x / x ) A z cos(2 nzL z / z ) Arxz noise ( , ), (4-1)
1 1 q , q 0, , (4-2) 3BBBw (1 )
where hN is the dimensionless average film thickness at the initial stage, Ax, Az and
Anoise represent the amplitudes of streamwise disturbance, spanwise disturbance and random disturbance, respectively; nx and nz represent the number of the waves in streamwise and spanwise directions, respectively; and r is a random function, which takes random values in the interval [−1, 1][19].
The energies of deformation Ex, Ez in each direction are defined as
NM /2 1 1 2 1/2 Ex() t ( |((,)|), a m z j t (4-3) MN j1 m 1
MN /2 1 1 2 1/2 Ez() t ( |((,)|), b n x l t (4-4) MN l1 n 1 where the spatial Fourier coefficients are defined as M 1 i2ml am( z , t ) h ( x l , z , t )exp( ), (4-5) l0 M
52 Chapter 4 Three-dimensional Simulation
N1 i2nj bn(,) x t h (, x z j ,)exp( t ), (4-6) j0 N where i is the imaginary unit, and Ex, Ez are the energies of deformation in streamwise and spanwise directions, respectively. The values of energies of deformation represent the degree of deformation of film. For example, when EEx z , it indicates that the deformation of film in the spanwise direction is very large and the deformation of film in streamwise direction is very small, which represents the emergence of the rivulet structures. To study the change of heat transfer during the evolution of the film, we define a heat transfer coefficient describing the heat transfer between the liquid film and the gas based on the temperature scale TN , denoted as H. Thus the corresponding overall heat transfer from liquid film to ambient gas is
HAw T N , (4-7) where Aw is the heated area of the film. The instantaneous amount of heat transferred from the liquid film to the ambient gas based on the film surface temperature is
ATTs( s a ), (4-8) where As is the surface area of the film, which changes due to the deformation of the film and is calculated by surface integral, and Ts is the average surface temperature of the film. Combining the equation (4-7) and (4-8), and eliminating leads to ATTA H s s a s . (4-9) ATAw N w
Since , Aw , Ta and TN are all constants, H is only affected by Ts and As . For the Nusselt flat film solution,
ATTTTw s a s a HNN , (4-10) ATTw N N
which is a constant, where N is the film surface temperature in Nusselt state and
N1/ (BBB w (1 )) . Thus the ratio of H to H N can be used to describe the change of heat transfer of wavy film compared to the flat film
53 Chapter 4 Three-dimensional Simulation
H As EABBBHT ( w (1 )), (4-11) HAN w N
where AAA s/ w is the ratio of instantaneous surface area of wavy film to its surface area in the Nusselt state. is the dimensionless instantaneous average surface temperature. EHT can be easily used to describe the change of heat transfer during the evolution of the film.
4.2 Random disturbance
The length of computational domain is taken as 8 times the wavelength of the most unstable streamwise disturbance wave and the width is taken as 6 times the wavelength of the most unstable spanwise disturbance wave, i.e., Lx = 2nxπ/kx (nx = 8), Lz = 2nzπ/kz
(nz =6) to include a sufficiently wide wavenumber interval to allow different interaction of different modes. Figure 4.1 shows the evolution of the liquid film at different times, from which four phases can be seen. In the initial stage, the random disturbance is amplified with the hydrodynamic instability. The ripples become parallel to the spanwise direction after a while, as shown in Figure 4.1 (a). The ripples can also appear without heating, because it is caused by the hydrodynamic instability. At this stage, the hydrodynamic instability is dominant, and the development of the flow is very similar to that in isothermal case. In the second stage, these ripples gradually split and merge due to the competition between hydrodynamic instability and Marangoni instability, as shown in Figure 4.1 (b). After a period of evolution, these ripples gradually become parallel to the flow direction under the influence of Marangoni effect, as shown in Figure 4.1 (c, d). In the third stage, the Marangoni effect gradually accumulates fluid and rivulet structures are formed, and the amplitude of the rivulet structures continues to grow, as shown in Figure 4.1 (e). In the final stage, when the amplitude grows sufficiently, the rivulet structures are again unstable to the disturbance in the streamwise direction, as shown in Figure 4.1 (f), and three-dimensional solitary waves are formed on the crests of the rivulet structures, with large amplitude and phase velocity. Capillary ripples precede the hump of these solitary waves. From the above stage, it can be seen that the falling film under HF condition has similar evolution process as under ST condition[40].
54 Chapter 4 Three-dimensional Simulation
(a) t = 1070{0.9280, (b) t = 1750{0.9040, (c) t =2475{0.8581, 1.0716} 1.1091} 1.1514}
(d) t = 3690{0.7121, (e) t =4625{0.1894, (f) t = 5165{0.0023, 1.2724} 2.4084} 2.9056}
Figure 4.1 Evolution of the film under random disturbance computed with the RM. The extrema of the film thickness are given in the brackets. Re=1.25, Γ =3375, =π / 2, Pr = 7, Bi =
0.12, Biw= 0.6, Ma=20, Lx×Lz=2nxπ/kx × 2nzπ/kz, nx =8, nz = 6, kx= 0.3891, kz= 0.2997, Ax = 0, Az
= 0, Anoise = 0.001, M×N=128×128
(a) t = 1070{0.7666, (b) t = 1750{0.7628, (c) t =2475{0.7587, 0.7811} 0.7837} 0.7887}
55 Chapter 4 Three-dimensional Simulation
(d) t = 3690{0.7464, (e) t =4625{0.6471, (f) t = 5165{0.6038, 0.8049} 0.8683} 0.8937}
Figure 4.2 Evolution of the film surface temperature under random disturbance computed with the RM. The extrema of the film surface temperature are given in the brackets. Re=1.25, Γ =3375,
=π / 2, Pr = 7, Bi = 0.12, Biw= 0.6, Ma=20, Lx×Lz=2nxπ/kx × 2nzπ/kz, nx =8, nz = 6, kx= 0.3891, kz=
0.2997, Ax = 0, Az = 0, Anoise = 0.001, M×N=128×128
Figure 4.2 shows the surface temperature of the liquid film at different times to help us understand the Marangoni effect more clearly. The surface temperature is low where the film is thick, while the surface temperature is high where the film is thin. Figure 4.3 shows detail profiles of film thickness and surface temperature. The phase difference between the film thickness and surface temperature is π. As we know, increasing temperature decreases the surface tension of the film. Therefore, a flow from hot film to cool film is induced by the surface tension gradient, leading to the formation of the rivulet structures.
(a) Streamwise profiles along y=53
56 Chapter 4 Three-dimensional Simulation
(b) Spanwise profiles along x=80
Figure 4.3 Profiles of film thickness and surface temperature at t=5130. See also caption of Figure 4.1
(a) Deformation energies
57 Chapter 4 Three-dimensional Simulation
(b) Change of heat transfer.
Figure 4.4 Deformation energies and change of heat transfer as a function of time. Letters refer to the snapshots of Figure 4.1 and Figure 4.2. See also caption of Figure 4.1
It can be seen from figure 4.4 (a) that before t=3300, the instabilities in streamwise direction and spanwise direction gradually increase, and both Ex and Ez increase slowly.
After t=3600, Ez increases rapidly and Ex decreases, indicating the appearance of rivulet structures and the disappearance of the wave. At t=4415, Ex increases again, reflecting the appearance of three-dimensional solitary wave structure. In the meantime, the value of EHT rapidly increases with the appearance of rivulet structure, indicating that the heat transfer is rapidly enhanced in a short period. At the same time, the value of A* also increases rapidly, indicating that the enhancement of heat transfer is mainly due to the expansion of heat transfer area.
4.2 Random disturbance superposed with the most unstable spanwise disturbance
Random disturbance superposed with the most unstable spanwise disturbance in dominance (Az>Anoise) is used as the initial disturbance. The evolution of the film is shown in Figure 4.5. As time proceeds, the dominant spanwise disturbance grows and rivulet structures form earlier, and small amplitude solitary waves appear on the crests of the rivulet structures as shown in Figure 4.5 (b). Corresponding Ex is small as shown
58 Chapter 4 Three-dimensional Simulation at point b in figure 4.6 (a). Under the continuous effect of the Marangoni instability, the amplitude of the spanwise wave is larger and larger, and the small solitary waves riding on the rivulet crests disappear, as shown in Figure 4.5 (c). When the amplitude of the rivulet structures increases to a certain degree, the solitary waves reappear on the rivulet crests, as shown in Figure 4.5 (d, e). These solitary waves continue to evolve and finally form three-dimensional solitary waves, as shown in Figure 4.5 (f). This phenomenon can be explained as follows. On the one hand, the increase of local film thickness leads to an increase in the local Reynolds number, so that the flow becomes unstable due to the hydrodynamic instability. On the other hand, the increase of amplitudes of the rivulet structures leads to an increase in the spanwise curvature. The capillary pressure is larger in the crests than in the troughs, which creates a pressure gradient from crests to troughs and thus reduces the amplitudes[40].
(a) t = 10 {0.9952, (b) t = 765{0.9576, (c) t =1910{0.4138, 1.0058} 1.0471} 1.6718}
(d) t = 2215{0.2539, (e) t =2320{0.1979, (f) t = 3245{0.0027, 2.1283} 2.4748} 3.0542}
Figure 4.5 Evolution of the film under random and most unstable spanwise disturbance computed with the RM. The extrema of the film thickness are given in the brackets. Re=1.25, Γ
=3375, =π / 2, Pr = 7, Bi = 0.12, Biw= 0.6, Ma=20, Lx×Lz=2nxπ/kx × 2nzπ/kz, nx =8, nz = 6, kx=
0.3891, kz= 0.2997, Ax = 0, Az = 0.005, Anoise = 0.001, M×N=128×128
59 Chapter 4 Three-dimensional Simulation
The increase of film thickness has both destabilizing and stabilizing effects, and these two effects compete with each other during the evolution of the liquid film. When the stabilizing effect is strong, the waves riding on the rivulet structures disappear; these waves reappear when the destabilizing effect is strong. As the rivulet structures develop, these unstable waves develop into three-dimensional solitary waves.
(a) Deformation energies.
(b) Change of heat transfer. Figure 4.6 Deformation energies and change of heat transfer as a function of time. Letters refer to the snapshots of Figure 4.5. See also caption of Figure 4.5
As shown in Figure 4.6 (a), deformation energies become large in advance
60 Chapter 4 Three-dimensional Simulation compared with the case with random disturbance, which indicates that rivulet structures appear earlier due to the introduction of the most unstable spanwise disturbance in dominance. The heat transfer is also enhanced earlier due to the appearance of the rivulet structures as shown in Figure 4.6 (b), which is beneficial for engineering application. However, the film ruptures earlier due to the earlier appearance of rivulet structures, which is unfavorable for engineering application.
4.3 Random disturbance superposed with the most unstable streamwise disturbance
Random disturbance superposed with the most unstable streamwise disturbance in dominance (Ax>Anoise) is used as the initial disturbance. The evolution of the film is shown in Figure 4.7. The most unstable streamwise disturbance is amplified by the hydrodynamic instability, resulting in two-dimensional traveling waves with constant velocity, as shown in Figure 4.7 (a). These two-dimensional waves are unstable to the spanwise disturbance. Then, the unstable spanwise disturbance contained in the random disturbance is amplified with the Marangoni instability and starts to develop. Thus the three-dimensional waves appear and are staggered on the adjacent wavefront. This staggered structure is called subharmonic instability, that is, the wavenumber of the structure is half of the traveling waves, as shown in Figure 4.7 (b). The wave crests move faster than the troughs, and the wave crests on the latter wave catch and merge with the wave troughs on the previous wave, which results in the formation of a teardrop-shaped structure, as shown in Figure 4.7 (c), so that the original wavefronts cannot maintain. At certain moments, these coherent drops reform a parallel wavefront, as shown in Figure 4.7 (d). In order to explain this, let’s make a detailed analysis of the deformation energies. From point c to point e in Figure 4.8 (a), we can see that Ex and
Ez are almost of the same size and fluctuate. Because this is a stage where the Marangoni instability and the hydrodynamic instability compete with each other. At point f in Figure 4.8 (a), Ez is greater than Ex, which means that Marangoni instability dominates at this stage. The corresponding film structure is shown in Figure 4.7 (f). Rivulet structures appear and develop, as in Figure 4.7 (g). Finally, a three-dimensional solitary wave structure is formed, Figure 4.7 (h, i). This process is similar to the process in Figure 4.1.
61 Chapter 4 Three-dimensional Simulation
(a) t =500{0.9349, (b) t =1470{0.9170, (c) t =1645{0.9095, 1.0679} 1.0877} 1.0950}
(d) t =1885{0.9167, (e) t =2580{0.8984, (f) t =3710{0.7705, 1.0860} 1.1074} 1.2467}
(g) t =4460{0.3027, (h) t =4735{0.1845, (i) t =5385{0.0044, 1.9599} 2.5585} 2.9545}
Figure 4.7 Evolution of the film under random and most unstable streamwise disturbance computed with the RM. The extrema of the film thickness are given in the brackets. Re=1.25, Γ
=3375, =π / 2, Pr = 7, Bi = 0.12, Biw= 0.6, Ma=20, Lx×Lz=2nxπ/kx × 2nzπ/kz, nx =8, nz = 6, kx=
0.3891, kz= 0.2997, Ax = 0.005, Az =0, Anoise = 0.001, M×N=128×128
62 Chapter 4 Three-dimensional Simulation
(a) Deformation energies
(b) Change of heat transfer
Figure 4.8 Deformation energies and change of heat transfer as a function of time. Letters refer to the snapshots of Figure 4.7. See also caption of Figure 4.7
Figure 4.8 (b) shows that when the most unstable streamwise disturbance in dominance is applied simultaneously to the random disturbance as the initial condition, the heat transfer of the liquid film is enhanced. At the same time, the duration of the liquid film from the beginning of evolution to the final rupture is also comparable with
63 Chapter 4 Three-dimensional Simulation that of the random disturbance situation. They are beneficial for engineering application.
4.4 Random disturbance superposed with the most unstable streamwise and spanwise disturbances
Random disturbance superposed with the most unstable streamwise and spanwise disturbances in dominance is used as the initial disturbance. The evolution of the film is shown in Figure 4.9. The checkerboard structures appear due to the amplification of the dominant disturbance, as shown in Figure 4.9 (a). Since the crests velocity is faster than the troughs, the checkerboard structures become wavy structures, Figure 4.9 (b). As the crests continue to move, the latter crests occupy the positions vacated by the front crests and reform checkerboard structures, Figure 4.9, such that the checkerboard and wavy structures alternate as shown in Figure 4.9 (d, e). The Marangoni effect in this process brings the hotter fluid at the trough to the cooler crests. As the thickness of the liquid film at the crest increases, Figure 4.9 (f), the waves riding on the rivulet structures disappear when the stabilizing effect is strong, as shown in Figure 4.9 (g); when the destabilizing effect is strong, the waves riding on the rivulet structures reappear, as shown in Figure 4.9 (h). Finally, three-dimensional rivulet structures with solitary waves are formed, as shown in Figure 4.9 (i).
(a) t =115{0.9817, (b) t =480{0.9313, (c) t =630{0.9304, 1.0199} 1.0777} 1.0827}
64 Chapter 4 Three-dimensional Simulation
(d) t =700{0.9119, (e) t =1035{0.8926, (f) t =1230{0.8578, 1.1009} 1.1233} 1.1564}
(g) t =1750{0.5592, (h) t =2315{0.2109, (i) t =3285{0.0027, 1.4730} 2.3305} 2.9839}
Figure 4.9 Evolution the film under random, most unstable streamwise and spanwise disturbance computed with the RM. The extrema of the film thickness are given in the brackets.
Re=1.25, Γ =3375, =π / 2, Pr = 7, Bi = 0.12, Biw= 0.6, Ma=20, Lx×Lz=2nxπ/kx × 2nzπ/kz, nx = nz =
8, kx= 0.3891, kz= 0.2997, Ax = 0.005, Az = 0.005, Anoise = 0.001, M×N=128×128
65 Chapter 4 Three-dimensional Simulation
(a) Deformation energies
(b) Change of heat transfer Figure 4.10 Deformation energies and change of heat transfer as a function of time. Letters refer to the snapshots of Figure 4.9. See also caption of Figure 4.9
From Figure 4.10 (a) and (b), it can be seen that the trend of the spanwise deformation energy and the heat transfer of the liquid film are similar to that in the case of random disturbance superposed with the most unstable spanwise disturbance. The rivulet structures appear in advance and the heat transfer is enhanced in advance, but the film ruptures earlier.
4.5 Effect of disturbance on heat transfer
In actual engineering applications, we are more concerned about the ability of heat transfer and the period that liquid film can remain continuous. The effects of above four kinds of disturbance on liquid film are shown in Figure 4.11. As can be seen from Figure 4.11, the introduction of the most unstable spanwise disturbance enhances the heat transfer, regardless of whether the most unstable streamwise disturbance is introduced or not, but at the same time, the liquid film ruptures prematurely due to the earlier appearance of the rivulet structures, which is unfavorable for practical engineering applications. Under the circumstance that only the most unstable streamwise disturbance is introduced, the heat transfer is enhanced and
66 Chapter 4 Three-dimensional Simulation the time that film maintains continuous is comparable to the case of random disturbance, which is beneficial for engineering applications.
Figure 4.11 The change of heat transfer under different disturbance. 1: random disturbance; 2: random disturbance superposed with the most unstable spanwise disturbance; 3: random disturbance superposed with the most unstable streamwise disturbance; 4 random disturbance superposed with the most unstable streamwise and spanwise disturbances
4.6 Effect of Reynolds number and Marangoni number on the wave patterns
As mentioned above, flow rate and heat flux are two other important factors for the film flow. Figure 4.12 (a)-(d) depict the wave patterns for different Reynolds number with Ma=20, while Figure 4.12 (e) shows a snapshot of the evolution at the same moment in time with the same Reynolds as Figure 4.12 (d) but with Ma=0, which means it is an isothermal falling film situation. To show the detailed structures of the film structure, the film profiles at selected positions are plotted in Figure 4.13. The spanwise positions are determined by crossing the middle of the rivulet, and the streamwise positions are determined by crossing the solitary wave riding on the rivulet. When the Reynolds number increases from Re=1.25 to Re=4, where the wave patterns are shown in Figure 4.12 (a)-(c), the width of the crests increases as the Reynolds number increases. Figure 4.13 (b) shows that the valleys between the rivulets become narrower as Reynolds number increases. Some local troughs caused by spanwise instability appear at the wide crests. From Figure 4.13 (a), we can see some
67 Chapter 4 Three-dimensional Simulation uniform waves riding on the rivulets, but most waves are not as uniform as in Re=1.25. Increasing Reynolds number even leads to some disorder waves on the crests, which can be seen in Figure 4.12 (b) and (c). Actually, if we increase the Reynolds number further Re=6, three-dimensional disorder waves appear (Figure 4.12 (d)), similar as the wave patterns in isothermal condition (Figure 4.12 (d)), indicating that the hydrodynamic instability is dominant. We can still see a rivulet-like line in the middle of Figure 4.12 (d), which is due to the existence of the Marangoni instability
(a) t = 5165, Re=1.25, (b) t = 6195, Re=3, (c) t = 9325, Re=4, Ma=20{0.0023, 2.9056} Ma=20{0.0049, 3.4575} Ma=20{0.0606, 3.6935}
(d) t = 12000, Re=6, (e) t = 12000, Re=6, Ma=20{0.6657, 2.5091} Ma=0{0.7256, 1.7728}
Figure 4.12 Wave patterns near rupture for various Reynolds number. Γ =3375, =π / 2, Pr = 7,
Bi = 0.12, Biw= 0.6, Lx×Lz=2nxπ/kx × 2nzπ/kz, nx =8, nz = 6, kx= 0.3891, kz= 0.2997, Ax = 0, Az = 0,
Anoise = 0.001, M×N=128×128
68 Chapter 4 Three-dimensional Simulation
(a) Streamwise profiles of rivulets
(b) Spanwise profiles of rivulets
Figure 4.13 Profiles of film at selected positions for various Reynolds number. (a) Spanwise positions are y=72, y=67, y=63 and y=54 from top to bottom. (b) Streamwise positions are x=60, x=75, x=78 and x=80 from top to bottom
During the simulation, it is found that unlike the evolution of the film in Figure 4.12 (a)-(c), the film in Figure 4.12 (d) doesn’t reach a rupture time, at least before t=20000. So we analysis the deformation energy of this film and the result is shown in Figure 4.14.
69 Chapter 4 Three-dimensional Simulation
Figure 4.14 Deformation energies as a function of time. See also caption of Figure 4.12 (d)
Note that for the case, Re=6 and Ma=20, the deformation energies Ex ≈Ez≈ const, indicating that the snapshot in Figure 4.12(d) corresponds to a fully developed 3D wave regime. The liquid film keeps continuous without rupture, which gives us an idea that we can increase the flow rate a little to avoid the film rupture and heat transfer deterioration.
(a) t = 5165, Re=1.25 (b) t=2280, Re=1.25 (c) t=1214, Re=1.25 Ma=20{0.0023, 2.9056} Ma=30{0.0017, 2.8408} Ma=40{0.0128, 2.1265}
Figure 4.15 Wave patterns near rupture for various Marangoni number. Γ =3375, =π / 2, Pr = 7,
Bi = 0.12, Biw= 0.6, Lx×Lz=2nxπ/kx × 2nzπ/kz, nx =8, nz = 6, kx= 0.3891, kz= 0.2997, Ax = 0, Az = 0,
Anoise = 0.001, M×N=128×128.
70 Chapter 4 Three-dimensional Simulation
(a) Streamwise profiles of rivulets
(b) Spanwise profiles of rivulets
Figure 4.16 Profiles of film at selected positions for various Marangoni number. (a) Spanwise positions are y=73, y=64 and y=54 from top to bottom. (b) Streamwise positions are x=80, x=60 and x=80 from top to bottom
Figure 4.15 shows the wave patterns at different Marangoni number and Figure 4.16 shows the detail profiles of the film at some positions. The maximum of film thickness decreases with increasing Marangoni number, as the extrema shown in Figure 4.15. Because the increase of Marangoni number leads to early rupture of the film and
71 Chapter 4 Three-dimensional Simulation the insufficient development of film, which give not enough time for the solitary wave riding the rivulets to develop. In Figure 4.16 (b), it can be seen that increasing Marangoni number leads to more rivulet structures and narrower valleys due to the stronger Marangoni instability.
4.7 Summary of this chapter
In this chapter, three-dimensional simulations of the falling film under HF condition have been performed, showing that under HF condition the liquid film eventually formed into three-dimensional rivulet structures with solitary waves under random disturbance. Then the influence of different disturbances on the film evolution and heat transfer has been studied. Different disturbance can lead to different processes of film evolution, mainly due to the competition between the hydrodynamic instability and Marangoni instability. Under the circumstance that only the most unstable streamwise disturbance is introduced besides random disturbance, the heat transfer is enhanced and the period that liquid film can maintain continuously is comparable to the case of random disturbance, which is beneficial for engineering applications. Finally, the effects of Reynolds number and Marangoni number on the wave patterns were investigated. The crests of the rivulets become wider and the valleys between the rivulets become narrower as Reynolds number increases. Some local troughs caused by spanwise instability appear at the wide crests. When the Reynolds number increases to a certain degree value, three-dimensional disorder waves appear, showing similar features as the wave patterns in an isothermal condition. The increasing Marangoni number leads to the early rupture of the film and the insufficient development of film, thus reducing the maximum of film thickness. More rivulet structures and narrower valleys appear due to the increasing Marangoni instability.
72 Chapter 5 Conclusions and Further Research
Chapter 5 Conclusions and Further Research
5.1 Conclusions
The falling liquid film is widely used in engineering, and its flow instability phenomenon and mechanism have profound theoretical significance as well. In order to study the liquid film flowing down a uniformly heated vertical plate with constant heat flux, a three-dimensional second-order model has been obtained. First, the governing equations and boundary conditions of the falling liquid film were determined with several assumptions. Then, considering the long-wave characteristics of the falling film, a gradient expansion strategy was performed by introducing a film parameter ε to estimate the amplitude order of each term in the governing equations and corresponding boundary conditions. Simplifications of the governing equations and boundary conditions were made by neglecting terms higher than Ο() 2 . Following the idea as in Prandtl’s boundary layer theory, the momentum equation in cross-stream direction was integrated along the film layer to obtain an expression of pressure P, and then P was substituted into the momentum equations in the other two directions to eliminate the pressure terms. The film layer equations were then obtained. Subsequently, using the weighted residual method to perform the mathematical approximations of the film layer equations led to the full second-order model. However, the full second-order model is still too complicated. Therefore, further simplifications were carried out where the regularization procedure was adopted to avoid the singularity. Finally, a three-dimensional regularized model (RM) for HF case with liquid film thickness h, streamwise flow rate q , spanwise flow rate q , and film surface temperature θ as variables was obtained. The validity of RM has been verified and detailed investigations of the falling film under HF condition have been performed via linear stability analysis and three-dimensional simulations. Some conclusions are summarized here. (1) Performing perturbation expansion of RM leading to Benney equation and good agreement of linear stability characteristics of RM with Orr–Sommerfeld prove that RM is accurate from small to moderate Reynolds numbers. Comparison with the three-dimensional experimental results shows that RM not only agrees with the exact linear behavior obtained from Orr–Sommerfeld but also describes properly the
73 Chapter 5 Conclusions and Further Research nonlinear dynamics of the falling films. (2) The linear stability analysis shows that the streamwise instability growth rates increase with the increase of Reynolds number and Marangoni number; the spanwise instability growth rates increase with the increase of Marangoni number and decrease with the increase of Reynolds number. There is an unstable range of disturbance wave number in both streamwise and spanwise directions. (3) Three-dimensional numerical simulations show that the liquid film flowing under HF condition eventually forms a three-dimensional rivulet structures with solitary waves under random disturbance, similar to the three-dimensional structure formed under ST condition. The various patterns appearing during the evolution of the film are caused by the hydrodynamic instability and the Marangoni instability. The heat exchange between the liquid film and the gas is also enhanced by the appearance of rivulet structure. The most important reason is that the rivulet structures cause the expansion of heat transfer area. (4) When the most unstable spanwise disturbance in dominance is introduced besides random disturbance, the evolution of the film shows that the increase of film thickness has both destabilizing and stabilizing effects, and these two effects compete with each other during the evolution of the liquid film. The rivulet structures with solitary waves arise in advance and film also ruptures earlier. When the most unstable streamwise disturbance in dominance is introduced besides random disturbance, a subharmonic instability shows up during the evolution and rivulet structures with solitary waves are formed, and the heat transfer is enhanced while the film does not rupture earlier. When both the most unstable streamwise disturbance and spanwise disturbance are introduced besides random disturbance, the checkerboard and wavy structures appear alternately, and finally, rivulet structures with solitary waves are formed in advance thus resulting enhancement of heat transfer and earlier rupture of the film. (5) The crests of the rivulets become wider and the valleys between the rivulets become narrower as Reynolds number increases. Some local troughs caused by spanwise instability appear at the wide crests. When the Reynolds number reaches a certain level, three-dimensional disorder waves appear, showing similar features as the wave patterns in an isothermal condition. (6) The increasing Marangoni number leads to the early rupture of the film and the
74 Chapter 5 Conclusions and Further Research insufficient development of film, thus reducing the maximum of film thickness. More rivulet structures and narrower valleys appear due to the increasing Marangoni instability. (7) The three-dimensional simulations give us some inspirations for the use of liquid film in engineering applications: the heat transfer can be enhanced by applying the most unstable streamwise disturbance in dominance and we can keep the film continuous without rupture by increasing the flow rate to a reasonable degree.
5.2 Further research
The evaporation effect is not considered in the current study of this paper, which should be considered when the liquid film temperature is high. In the future study, we hope to take into account the evaporation effect to further refine our model. We are also building an experimental facility that will be used to perform experimental research on falling liquid film under HF condition. In the future, the simulation results will be compared with the experimental results under HF condition.
75 References
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78 Acknowledgements
Acknowledgements
The three-year graduate study is drawing to a close, and on the occasion of the completion of this master's thesis, I would like to express my deep gratitude to all those who have given me support and help. I am deeply thankful to my supervisor, Associate Professor Duan Riqiang, for his enthusiasm, excellent guidance, and sustained encouragement during my years as a graduate student at Tsinghua University. I am grateful to my co-supervisor, Associate Professor Ma Weimin, for giving me the opportunity to study in the laboratory and for his academic guidance when I studied at KTH. Thanks to our coordinator, Professor Waclaw Gudowski, for his contribution to the dual master degree program and his help to us in KTH-Royal Institute of Technology. We had abundant and wonderful experiences at KTH. I am also very grateful to Professor Sun Yuliang and Professor Zhou Zhiwei for their help to us double degree students. I thank postdoctor, Liu Jiein, for fruitful discussions and suggestions about the simulation works. I would also like to thank all my friends, both in China and Sweden, for their help in various aspects and their encouragement. Finally, thanks to the people who take the time to comment on this thesis.
79 Resume
Resume
Personal Details Place / Date of birth: Henan Province of China/28-02-1993. Bachelor’s degree in Nuclear Engineering and Technology from Harbin Engineering University, Harbin, June 2015. Dual Master’s degree candidate in nuclear energy related disciplines both at Tsinghua University and KTH-Royal Institute of Technology, September 2015-Present.
List of Publications
[1] WANG Meng, LIU Jie-bin, JIANG Sheng-yao, et al. A study on a vertical falling liquid film on a uniformly heated plate with constant heat flux (in Chinese). Chinese Journal of Computational Mechanics (Chinese Core Journal, accepted)
80