Liquid Jet Impingement Normal to a Disk in Zero Gravity

The problem of determination of friction factor (or average fluid velocity) for a steady gravitational flow of fluid in an inclined open triangular groove is addressed. It is assumed that the angle of slope of the groove is known and the gravity is the only driving force of flow. The solution comprises of two parts. First, for a given groove’s basal angle, liquid-solid contact angle, and the Bond number, the shape of free surface is determined starting from the Young-Laplace equation. The optimization technique as developed formerly is applied. Then, having determined the shape of the free surface and slope of the groove, the distribution of fluid flow is determined. The boundary value problem is solved using boundary collocation method in least square sense. Given the distribution of fluid velocity the friction factor as a function of the other parameters of the model is analyzed.

A three-dimensional transient round liquid jet within a low-speed coaxial outer gas flow is numerically simulated and analysed via vortex dynamics ($\unicode[STIX]{x1D706}_{2}$ analysis). Two types of surface deformations are distinguished, which are separated by a large indentation on the jet stem. First, there are those inside the recirculation zone behind the leading cap – directly affecting the cap dynamics and well explained by the local vortices. Second, deformations upstream of the cap are mainly driven by the Kelvin–Helmholtz (KH) instability, unaffected by the vortices in the behind-the-cap region (BCR), and are important in the eventual atomization process. Different atomization mechanisms are identified and are delineated on a gas Weber number ($We_{g}$) versus liquid Reynolds number ($Re_{l}$) map based on the relative gas–liquid velocity. In a frame moving with the liquid velocity, this result is consistent with prior temporal studies. A simpler and clearer portrait of similarity of the atomization domains is shown by using the relative gas–liquid axial velocity, i.e. $We_{r}$ and $Re_{r}$, and avoiding the widely used velocity ratio as a third key parameter. A detailed comparison of vorticity along the axis in an Eulerian frame versus a frame fixed to a surface wave reveals that the vortex development and surface deformations are periodic in the upstream region, but this periodicity is lost closer to the BCR. In the practical range of the density ratio and for early times in the process, axial vorticity is mainly generated by baroclinicity while streamwise vortex stretching becomes more important at later times and only at lower relative velocities when pressure gradients are reduced. The inertia, vortex, pressure, viscous and surface tension forces are analysed to delineate the dominant causes of the three-dimensional instability of the axisymmetric KH structure due to surface acceleration in the axial, radial and azimuthal directions. The inertia force related to the axial gradient of kinetic energy is the main cause of the axial acceleration of the waves, while the azimuthal acceleration is mainly caused by the pressure and viscous forces. The viscous forces are negligible in the radial direction and away from the nozzle exit in the axial direction. It is interesting to note that azimuthal viscous forces are important even at high $Re_{l}$, indicating that inertia is not totally dominant in this instability occurring early in the atomization cascade.

A finite-element algorithm for computing free-surface flows driven by arbitrary body forces is presented. The algorithm is primarily designed for the microfluidic parameter range where (i) the Reynolds number is small and (ii) force-driven pressure and flow fields compete with the surface tension for the shape of a stationary free surface. The free surface shape is represented by the boundaries of finite elements that move according to the stress applied by the adjacent fluid. Additionally, the surface tends to minimize its free energy and by that adapts its curvature to balance the normal stress at the surface. The numerical approach consists of the iteration of two alternating steps: The solution of a fluidic problem in a prescribed domain with slip boundary conditions at the free surface and a consecutive update of the domain driven by the previously determined pressure and velocity fields. For a Stokes problem the first step is linear, whereas the second step involves the nonlinear free-surface boundary condition. This algorithm is justified both by physical and mathematical arguments. It is tested in two dimensions for two cases that can be solved analytically. The magnitude of the errors is discussed in dependence on the approximation order of the finite elements and on a step-width parameter of the algorithm. Moreover, the algorithm is shown to be robust in the sense that convergence is reached also from initial forms that strongly deviate from the final shape. The presented algorithm does not require a remeshing of the used grid at the boundary. This advantage is achieved by a built-in mechanism that causes a smooth change from the behavior of a free surface to that of a rubber blanket if the boundary mesh becomes irregular. As a side effect, the element sides building up the free surface in two dimensions all approach equal lengths. The presented variational derivation of the boundary condition corroborates the numerical finding that a second-order approximation of the velocity also necessitates a second-order approximation for the free surface discretization.

When a fluid liquid interface joins a solid boundary a common line is formed, this line will be referre to as the contact line. The main objective of this paper is to obtain explicit solutions of the governing equations of fluid mechanics for viscous flow of a liquid, in the vicinity of a stationary contact line. In each case we have indicated the procedure for developing locally consistent asymptotic expansions for the flow field and the shape of the free surface, with explicit solutions mostly for the first two terms only. Apart from arbitrary constants of the type f'(0; Aj+1) y which are to be determined by the distant driving mechanisms, the flow field and the shape of free surface are completely determined by the contact angle 0(1 and the local arbitrary parameter pQ. However, mainly due to difficulties in labelling the eigen-values and partly because of the size of the paper, in this part we present results for wetting liquids only.

In an oil hydraulic vane pump for an automobile, it is very important that the vane doesn't separate from the camring inner race during the operation of the vane pump. The vane generally has not only the oil hydraulic force acting on the bottom face to contact to camring inner race but there is also an inertial force and viscous force. Because the oil hydraulic force is much larger than the other forces, the contact state between the vane tip and the camring inner race is sufficient. However, the contact state between the vane tip and the camring inner race is only affected by the inertial and viscous forces during the delivery of the vane pump, because the oil hydraulic force acting on the vane is in equilibrium. If the inertial force is larger than the viscous force, which happens when the vane is separated from the camring inner race, the delivery of the vane pump can become unstable or the volume efficiency can become decrease rapidly. Therefore, in this paper, the state of the contact between the vane and the camring is considered. The results show that the rotating speed of the shaft, the operating temperature of the oil, the clearance between the vane and the rotor, and the mass of the vane exert a great influence on the state of the contact between the vane and the camring.

Free surface deformation due to a source and a sink of equal strength in Stokes flow

Equipment involx, ing the use of perforated (sieve) trays has met with great populari ty in extract ion operations, since perforated trays are s imple in design, show l i t t l e sensi t ivi ty to the presence of solid par t icu la te mat ter in the process fluids, and are compara t ive ly inexpensive. The effectiveness of extract ion in those equipments, as in other extract ion gear, is dependent in the first ins tance on the amount of surface ava i l ab le for phasecontac t between the liquids. It is, therefore, very important to know the mechanism governing the formation of that contact surface in the process of re la t ive fluid flow. This faci l i ta tes the study of the processes tak!ng p lace in bubb le -cap tray extract ion towers, and of ways for intensifying the processes. The amount of phase contact surface ava i lab le depends largely on the degree of dispersion of the liquids, which is in turn de te rmined by the relat ionship of the forces on a drop of l iquid during its continuous advance through the column. The dimensions of a l iquid drop are de termined by the equi l ibr ium prevai l ing between inert ia and surface forces. The iner t ia forces bring about a tomiza t ion of the drops as they col l ide with one another and with the trays, while the surface forces, charac te r ized by the value of surface tension, bring about coa lescence of the drops. The size of the droplets is therefore dependent on the extent to which inert ia forces a re capable of overcoming the surface tension of the l iquid phase. The qual i ty of the phase contact is charac te r ized by the relat ionship of forces determining the frequency at which l iquid is renewed in the tray orifices, s ince renewal of the contact surface occurs in the small layer of l iquid i m m e d i a t e l y ad jacen t to the orifices. The insufficient e f f ic iency of conventional perfora tedtray ext rac t ion towers owes to the fact that surface tension forces have a considerable edge over inert ia forces, s ince the value of the la t ter is de termined solely by the difference in the specif ic weights of the l iquid constituents. At the same t ime, theuse of s imi lar equipment in appl ica t ion to gas (vapor)l iquid systems, e.g. , in fract ional dis t i l la t ion, yields good results. The high ef f ic iency of f rac t iona l -d i s t i l l a t ion bubb le -cap tray equipment is due to the fact that iner t ia forces in the l a t t e r are proport ional to the difference in the specif ic weights of the vaporl iquid system, which is far in excess of the difference in the specif ic weights of two dropletphase liquids. A comparison of the operat ion of bubb le t r ay equipment in l i q u i d l i q u i d systems and gas (vapor) l iquid systems led m this concept : could not the energy of flow in an extract ion column be increased by introducing a system of gas into a bubblet ray extract ion tower through use of the forces act ing in fract ionat ing towers?

Conversely, glucose can reach the epithelium via di¬usion in laminar ow (Figure 14.1) [3]. If the di¬usion rate of glucose in the lumen is lower than its transepithelial absorption rate, the overall absorption rate of the glucose should be proportional to either its di¬usion rate in the lumen or its membrane transport rate. ´e slower of these two factors is the rate-limiting factor for the overall absorption process. Considering the mentioned short-chain fatty acid gradient and heterogeneous distribution of microbiota across the intestinal lumen [6,7], the di¬usion rate in the lumen should be slower than the membrane transport rate. ´erefore, the di¬usion rate of glucose in the lumen should correlate with the overall absorption rate in laminar ow (Equation 14.2):( ) ( ),Overall absorption rate Diffusion rate in lumen= b (14.2)where b is a constant. ´e ow behavior of digesta in the lumen can be estimated using the Reynoldsnumber, which expresses the ratio of inertial forces to viscous forces in a uid [8]. ´e inertial force is the tendency of the uid to stay in motion or at rest unless acted upon by an outside force [9]. ´e viscous force is an internal property of a uid that o¬ers resistance to ow [9]. ´e Reynolds number is used to determine whether a ow will be dominated by inertial or viscous forces, i.e., whether the ow is turbulent or laminar. A Reynolds number below 2300 indicates that viscous force predominates over inertial force to keep the ow laminar (Figure 14.2a), which results in poor micromixing along the transverse axis. Conversely, a Reynolds number exceeding 2300 indicates that inertial force dominates and that the ow has become turbulent [8] (Figure 14.2b), which completely mixes digesta at the molecular level [4].

Thinning and breakup of liquid filaments are central to dripping of leaky faucets, inkjet drop formation, and raindrop fragmentation. As the filament radius decreases, curvature and capillary pressure, both inversely proportional to radius, increase and fluid is expelled with increasing velocity from the neck. As the neck radius vanishes, the governing equations become singular and the filament breaks. In slightly viscous liquids, thinning initially occurs in an inertial regime where inertial and capillary forces balance. By contrast, in highly viscous liquids, initial thinning occurs in a viscous regime where viscous and capillary forces balance. As the filament thins, viscous forces in the former case and inertial forces in the latter become important, and theory shows that the filament approaches breakup in the final inertial-viscous regime where all three forces balance. However, previous simulations and experiments reveal that transition from an initial to the final regime either occurs at a value of filament radius well below that predicted by theory or is not observed. Here, we perform new simulations and experiments, and show that a thinning filament unexpectedly passes through a number of intermediate transient regimes, thereby delaying onset of the inertial-viscous regime. The new findings have practical implications regarding formation of undesirable satellite droplets and also raise the question as to whether similar dynamical transitions arise in other free-surface flows such as coalescence that also exhibit singularities.

The liquid spreading on structured packings plays an essential role in affecting gas–liquid mass transfer in separation columns, yet the synergistic mechanism of surface wettability and textured geometries remains insufficiently understood. This study integrates experimental and computational methods to systematically investigate the liquid spreading characteristics on textured surfaces. The synergistic combination of hydrophilic modification and surface textures markedly enhances liquid spreading performance. Compared with the hydrophilic plane surface, the spherical cap texture increases the interface area and wetted area by 25.2% and 49.6%, respectively, while the pyramid-shaped texture leads to improvements of 24.5% and 48.9%, respectively. Based on Weber number analysis, it is identified that the competition between inertial force and surface tension governs the evolution of liquid spreading regimes. In addition, the results suggest that variations in liquid viscosity and density may further influence spreading behavior by modifying the balance among inertial, viscous, and surface tension forces. The geometric parameters of spherical cap textures are systematically examined, and it is revealed that a spherical cap with a non-uniform staggered configuration (Mode III) enables the efficient liquid spreading. A new non-uniform spherical cap texture is designed to enhance liquid spreading, which enhances spreading performance compared with the original plate, increasing the interface area by 27.3% and the wetted area by 47.4%. Although the liquid film thickness increases slightly, the wetted area ratio is significantly improved, indicating enhanced effective surface coverage. Both simulations and experiments confirm that the new textured structure further enhances liquid spreading performance on the textured surface. This research unveils a strategy to improve liquid spreading through tailored surface textures, opening up new possibilities for the design of efficient packings.

Understanding how the shape and motion of an aquatic animal affects the performance of swimming requires knowledge of the fluid forces that generate thrust and drag. These forces are poorly understood for the large diversity of animals that swim at Reynolds numbers (Re) between 10(0) and 10(2). We experimentally tested quasi-steady and unsteady blade-element models of the hydrodynamics of undulatory swimming in the larvae of the ascidian Botrylloides sp. by comparing the forces predicted by these models with measured forces generated by tethered larvae and by comparing the swimming speeds predicted with measurements of the speed of freely swimming larvae. Although both models predicted mean forces that were statistically indistinguishable from measurements, the quasi-steady model predicted the timing of force production and mean swimming speed more accurately than the unsteady model. This suggests that unsteady force (i.e. the acceleration reaction) does not play a role in the dynamics of steady undulatory swimming at Re approximately 10(2). We explored the relative contribution of viscous and inertial force to the generation of thrust and drag at 10(0)<Re<10(2) by running a series of mathematical simulations with the quasi-steady model. These simulations predicted that thrust and drag are dominated by viscous force (i.e. skin friction) at Re approximately 10(0) and that inertial force (i.e. form force) generates a greater proportion of thrust and drag at higher Re than at lower Re. However, thrust was predicted to be generated primarily by inertial force, while drag was predicted to be generated more by viscous than inertial force at Re approximately 10(2). Unlike swimming at high (>10(2)) and low (<10(0)) Re, the fluid forces that generate thrust cannot be assumed to be the same as those that generate drag at intermediate Re.

Spray drying is a widely used unit operation for producing particulate products directly from a liquid feed. Important processes that occur inside the spray dryer are droplet formation, droplet drying and interactions between droplets and recycled fines. Various studies have looked at the first two processes, but the latter phenomenon has received less attention. Literature on droplet-particle interaction which aims at quantitatively describing agglomeration in spray drying is scarce and mainly qualitative. For product quality the formation of agglomerates is often desirable. This thesis models and investigates the collisions of individual particles with single droplets. The surface tack of drying droplets has been identified as an important variable for the formation of agglomerates. In this thesis a novel method for measuring tack from the liquid phase has been further improved. The improvements are a more accurate load measurement, an automated control of the tack probe and an improved layout of the sample holder and probe. The key feature of the device is its ability to measure tack of drying droplets, whereas other devices measure tack by wetting a powder. Using our method the tack of a commonly spray dried product, yeast extract, has been measured. From these experiments it was found that with decreasing average moisture content the surface tack increases to a maximum. Below a critical average moisture content the surface of the droplet is dry and the tack rapidly decreases upon further drying. Another important parameter in determining the degree of agglomeration is the degree of penetration. If the particle penetrates the droplet too deeply, the agglomerate structure becomes too dense. To predict the penetration depth, a non-dimensional model has been developed. It describes the penetration of a particle into a liquid droplet during a head-on collision. It is based on a force balance and incorporates surface tension force, viscous force and capillary pressure force. The important parameters determining the collision outcome are the contact angle, the size of the droplet relative to the particle, the Reynolds and Weber numbers. For each contact angle an equilibrium penetration position exists, at this point the surface tension force vector is perpendicular to the penetration direction. Five different penetrations regimes are identified. At low Reynolds numbers, viscous forces dominate and the particle asymptotically travels towards the equilibrium position. Reducing the viscous drag force by increasing the Reynolds number results in initially overshooting the equilibrium position, but the surface tension force pulls the particle back, to attain the equilibrium in an oscillating motion. At even higher Reynolds numbers the particle fully penetrates the droplet, and reaches the centre of the droplet for even higher values for the Reynolds number. The ejection regime is found at high Reynolds number and low Weber numbers and the liquid should be non-wetting. Using the regime maps one is able to identify in which region a spray dryer is operating. Although the full penetration regimes are useful for capturing fines, it should be avoided when agglomeration is desired. The ejection regime should be avoided as well. To validate the model, impact experiments were carried out by dropping glass spheres on the surface of different liquids. These validation experiments were the first attempt to experimentally validate the collision of a single particle with a liquid surface. Besides yeast extract, which has non-Newtonian rheological properties, silicone oils with constant viscosities of 100 mPa•s and 1 Pa•s have been tested. The penetration over time for different impact velocities was determined by analysing high speed camera recordings. The typical penetration times ranged from 0.2 s to 2 s. To obtain accurate location data was recorded at frame rates up to 38 000 frames per second. Glass spheres, with a size of 2 mm were used to allow the visual tracking. Modelling the impacts showed that the model consistently predicted faster penetration times than were observed experimentally. The relative difference increased with increasing viscosity. A parameter fitting exercise showed that better agreement could be obtained by using a higher viscosity and a higher contact angle in the model. With this knowledge the most likely factor influencing the model-experiment mismatch was identified as being the dynamics of wetting of the particle surface. It was also found that using the dynamic contact angle in the model would improve its results. The non-Newtonian characteristics of the yeast extract resulted in the particle rebound and the formation of an air cavity upon impact. The tack measurement technique and penetration model presented in this thesis will be useful tools for the design of spray dryers. Recommendations are made for further model improvement. The experimental validation is the first attempt to validate the presented model. Future improvements are recommended and suggestions are presented.

In this paper, deformation and breakup of Newtonian liquid droplets at elevated pressures have been studied. Detailed physics pertaining to four different breakup regimes, oscillatory, bag, multimode and shear breakup modes , has been investigated using an incompressible interface tracking methodology. The accuracy and efficiency of the code was enhanced by incorporating an adaptive mesh refinement (AMR) techni que. In general, the aerodynamic drag force exerted by the ambient fluid causes the droplet to deform. The deformation is resisted by viscous and surface tension forces. The breakup mechanism becomes progressively violent as the We number increases, and moves from oscillatory to shear breakup regime. Quantitatively, the droplet lifetime decreases as the inertial force is increased in comparison to the deformation resisting, surface tension force. A criterion to mark the beginning of breakup has been quantified in terms of surface and kinetic energy associated with the droplet. Critical We numbers for the three regimes have also been identified for 100 atm pressure conditions. A generalized regime diagram, for Oh < 0.1, was developed to predict the various breakup modes, taking into account the pressure effect on critical We number, using data from previous experimental investigations , and simulations conducted during the current study.

Optical resonances in electrically charged particles and their relation to the Drude model

A three-dimensional unsteady compressible viscous flow solver has been developed, and applied to a low speed centrifugal compressor configuration. Good agreement with experiments have been obtained for a compressor tested at NASA Lewis Research Center. Comparisons with measured surface pressures, and performance data are given. For this compressor, stall was found to occur in the diffuser region. It is demonstrated that this stall may be eliminated, and stable operation may be restored by the use of bleed valves located on the diffuser walls. INTRODUCTION In many army applications (tanks and rotorcraft) small gas turbine engines are often used. The mass flow rate in these engines is small (below 10 Ibs/sec). The compression is provided by one or two centrifugal compressor stages, sometimes in combination with an axial compressor stage. Centrifugal compressors are well suited for handling such low mass flow rates. They are rugged and compact. However, the geometric complexity of the blades and the twisting blade passages give rise to a host of aerodynamic problems such as unsteadiness, establishment of cross flows, flow separation at the trailing edge of the compressor blade, separation over diffuser guide vanes, and unsteady meandering motion of a 'tip vortex' that originates in the vicinity of the gap between the rotor and the casing. In many instances, this unsteadiness leads to rotating stall and surge. Considerable interest exists in the engine community in understanding and eventually controlling these phenomena, particularly compressor stall and surge. Computational fluid dynamics methods provide an efficient way of studying the complex flow phenomena within centrifugal compressors. Thanks to the availability of post-processing and scientific visualization software, the computed flow fields can be extensively analyzed. It is also convenient to test within a numerical model various control strategies such as bleeding, energizing of incoming flow using jets, excitation /wiggling of guide vanes, etc. Passive mechanisms such as slots, slats and flaps can also be easily studied within the scope of a numerical method. These simulations are thus a useful and economical first step to detailed flow control experiments. Finally, CFD simulations provide a wealth of numerical and flow physics information which are useful in developing reduced order models. For these reasons, a 3-D CFD code capable of analyzing centrifugal compressor configurations has been developed by the present investigators. Rather than solving the flow equations in a rotating coordinate system, the present calculations are carried out in an inertial system, with grid velocity terms used to simulate a rotor. Inflow and outflow boundary conditions appropriate for internal flows in general and centrifugal compressors in particular are used. This paper is organized as follows. The flow solution algorithm is first described. Next, results in the form of velocity fields, surface pressure data and performance map are presented for a NASA low-speed centrifugal compressor. The performance of this compressor for off-design low mass flow ratio conditions is subsequently studied. Methods for improving the compressor performance during such conditions are discussed. NUMERICAL FORMULATION The 3-D unsteady compressible Reynolds averaged Navier-Stokes equations are solved numerically using a time marching scheme. This involves solving the governing equations at each time step by marching in time from an initial flow condition with appropriate boundary conditions. The governing equations are cast in a curvilinear body-fitted coordinate system (£,T|,£) and solved using an Alternating Direction Implicit (ADI) scheme 1. This implicit scheme has been validated for a number of fixed wing and rotary wing configurations both in steady and unsteady flight conditions^'*. The partial derivatives in the governing equations are calculated using finite difference approximations. The Graduate Research Assistant, Student Member AIAA. * Regents' professor, Associate Fellow AIAA. Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. inviscid flux terms are calculated using a third order upwind scheme and the viscous stress terms are calculated using standard central difference representations. The details of the upwind formulation are given by Roe. The time derivative is approximated using a two point first order backward difference. The finite difference representation of the governing Navier-Stokes equations at time level 'n' is components at the inflow boundary were set to zero, assuming that there was no swirl at the inlet. It is possible within the framework of the present boundary conditions to prescribe swirl, non-uniform total pressure and total temperature at the inlet. At every time step, the speed of sound a and the normal component of velocity un were found at the inflow boundary from the relations: AT 2a 2a — u = n — u n = I 8 f R + S S + S , T (1) Interior where Aq* is the change in the flow property vector q during adjacent time levels and At is the time step. The operators 8£, 5r| and 8^ are the standard central difference operators. The viscous terms are evaluated explicitly and the inviscid flux vectors F, G and H are calculated implicitly at time level n+1. A three factor approximate factorization is used to factorize the implicit coefficient matrix operator into block tri-diagonal matrices. The factorized system of equations is solved using the Thomas algorithm. Turbulence effects are modeled using the SpalartAllmaras model. INITIAL AND BOUNDARY CONDITIONS The centrifugal compressor is shown in figure 1. The boundary grids are also shown for a single blade passage. The volume grid is made of three segments: inlet region, the rotor, and the diffuser. The following boundary conditions were applied: Periodicity: In the present study, the flow through only a single blade passage has been studied, assuming blade-toblade periodicity. At the periodic boundaries, the flow properties are computed by averaging the properties on either side of the periodic boundary. Solid walls: At the solid walls, the no slip boundary condition was used. The velocity on the walls of the inlet and the diffuser were set to zero, while the velocity for grid points on the compressor blades and the shaft were set equal to Q x r. The pressure, density and temperature values at the solid surfaces were extrapolated from the interior using relationships such as 9p/3n=dT/9n=3p/3n=0, where n represents the direction normal to the solid surface. Inflow Boundary: At this boundary, the stagnation temperature T0 and the total pressure p0 were assumed known. In the simulation of a compressor drawing the air from the atmosphere, these quantities may be set based on ambient conditions. In a multistage compressor analysis, these quantities will be known at the downstream boundary of the previous stage. The tangential (vand w-) (2)