Droplet Impingement on a flat surface

7/29/2019 Droplet Impingement on a flat surface

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Numerical Simulations of Microdroplet Impingement on a Flat Surface

Project Report: CFD-I

Submitted by:

Aritra Sur


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Introduction:

In the recent years the problem of dissipation of high heat fluxes in the electronics industry has

been investigated intensively. Liquid cooling technologies have proven to be the best solution interms of dissipation of such high heat fluxes. It is because of the high heat of vaporization of

certain liquids which can be used as a tool to remove ultra high heat fluxes. The liquid cooling

technologies include spray cooling, heat pipes, thermo-syphons, flow boiling, jet impingementcooling etc. All these technologies have their specific advantages and disadvantages. However,

the water spray cooling technology has been found to be the most attractive amongst all of them.

The reasons were described as: (1) Ability to uniformly remove large heat fluxes (2) use smallfluid volumes (3) take advantage of large heat of vaporization (4) use low droplet impact

velocity (5) provide control and spray regulation of system temperatures1.

The spray cooling technology involves spraying of micro-sized liquid droplets on the

heated surface through a nozzle. The droplets then spread on the surface and evaporate or form athin liquid film which enables them to remove heat from the surface even at temperatures lower

than the liquid saturation temperature. Horacek et al2

and Sodtke and Stephan3

found that the

increase in the heat flux removed from a heated surface is directly proportional to the three-phase

contact line length. Navedo4

found that the heat transfer coefficient, h, is affected by the dropletvelocity whereas the critical heat flux (CHF) is affected primarily by the number of droplets.Chen et al

5also found that the CHF is affected by the droplet flux and the impact velocity of the

droplets. This suggests that the mechanism of droplet spreading on a heated surface holds theanswers to developing very efficient heat transfer solutions.

Other than spray cooling the droplet impact problem is also relevant in the areas of spray

cleaning, ink jet technology, windshield films, aircraft icing, electronic-component welding etc.The extreme deformation of a droplet upon impact with a surface along with the surface tension

and wetting effects of the liquid makes it a very interesting topic for fluid dynamics research.

Moreover, the complexity of the physics behind the spreading phenomena has attracted many

researchers to work on this problem at the academic level. Other than experiments numerical

simulation techniques provide with very in depth understanding of the parameters that affect theprocess of droplet spreading such as the variation of pressure at the impact center, wall shear

evolution with time as the droplet spreads. They also help in establishing the correctunderstanding of the effect of different thermo-physical properties of the droplet liquid and that

of the surface material on the spreading process. Once the parameters for generating a correct

numerical simulation of the fluid dynamics of the droplet spreading is achieved, various

parametric studies can be conducted to predict the behavior of the different liquids impinging onsurface with varying surface properties.

Numerical simulation of a droplet impingement was first studied by Harlow and Shanon6.

They solved the full Navier-Stokes equations numerically in cylindrical co-ordinate system and

investigated the splash of a liquid drop on a flat surface, into a shallow pool and a deep pool.

They used the Marker-and-Cell technique to obtain the numerical solutions for the N-Sequations. However, their model did not incorporate the effect of surface tension and wettability

on the droplet impingement. Several other such simulations have been carried out till the 90swhere for the simplicity of the numerical simulation the viscosity effects and the surface tension

effects have not been considered. In the 90s different research groups approached the problem

using both and Eulerian and Lagrangian methodologies. The Eulerian approach includes thevolume of fluid (VOF) method, details of which are explained in following sections. The

Lagrangian approach developed by Fukai et al7 uses a moving mesh approach which allows an


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extremely precise tracking of the free surface deformation. They carried out their simulations

using the finite element method and were able to capture the droplet spreading, oscillations of thefree droplets and also the recoiling phenomena.

Recently, Briones et al8

et al numerically investigated the mircodroplet impingement

dynamics and evaporation on a flat surface. This project is an attempt to recreate the

impingement studies as reported in their article. However, only a part of the impigment dynamicshas been investigated and reported here due to time constrains. The section titled future work

will provide with details of furthering the present work and its applications in my PhD research.

Droplet spreading phenomena:

The wetting of a surface is described as the motion of the interface of two fluids (liquid

and gas) over a solid surface. A contact line, also known as wetting line, is formed which isdefined as the collection of all points around the periphery of the droplet where the solid and

both the fluid phases are in contact with each other. The phenomena of wetting is defined by a

spreading parameter, S, defined by

SG LS LGS ( ) . (1)

When S>0, the liquid spreads over the whole solid surface and this is referred to as totalwetting. When S0, 0=0) and (3) dissipation of the energy in the

close vicinity of the solid near the contact line. According to this view in case of partial wetting,

the viscous dissipation near the contact line described by the hydrodynamic theory and the


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dissipation near the contact line which is escribed by the molecular kinetic theory are the most

dominant processes that primarily are involved (Figure 2).

Figure 2. A liquid drop partially wetting the surface: hydrodynamic parameters are shown on the left and themolecular-kinetic parameters are shown on the right11

.

The velocity of the contact line according to the assumption that the dissipation of theunbalanced tension forced is governed by the viscous dissipation [reference] is given by

-1

3 3LGCL d 0

s

LV = - ln

9 L

(4)

Whereas, the microscopic kinetic theory uses the displacement length of the molecules from one

adsorption site to another with a jump frequency Ko to model the contact line velocity[11]

.According to this theory the contact line velocity is given by

2 0 dCL 0

B

cos -cos

V =2K sinh 2k T

(5)

These two approaches have been proved to predict the droplet spreading rate differently

for low d and 0 approaching 0. That is, for a scenario of total wetting the dynamic radius of the

drop as a function of time has found to be totally different 12 However, the existence and validity

of both the theories have been intensively investigated and agreed upon. Several researchers

have, thus, proposed various models in combination of both the theories so as to accuratelysimulate the hydrodynamics of the droplet spreading problem[11].

Problem statement:

In this work, I have simulation of droplet C from Briones et als work has been

undertaken. However, there are some simplification to the problem that has been incorporated aswell. For their work, they simulated the impingement and evaporation of micro droplets on a flat

surface. They used the dynamic mesh for capturing the movement of the gas liquid interface andthey refined it immensely. They also used a model based on both the hydrodynamic theory and

the molecular kinetic theory to implement the dynamic contact angle for both the impingement

and the evaporation case. In this work, I have used a very fine mesh to obtain similar results forthe droplet C as defined in Table 1 of their work. A static contact angle method has been used

and the contact angle specified is the equilibrium contact angle. The study compares two


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different schemes for evaluating the same problem and the effect of different mesh sizes on the

solution of the problem. After determining the most efficient mesh size simulation of spreadingof a droplet with a very low equilibrium contact angle that is very good wetting case and the

results are reported. The important parameters relevant to the problem being investigated is

provided in Table 1.Table 1. Simulation Parameters

D0 (m) V0 (m/s) We Ca x 10-

Re Oh x 10-

0

55.4 2.45 4.6 34 135.5 22 50.32

55.4 2.45 4.6 34 13.5 22 20.00

Methodology:

Numerical model:

The explicit VOF model in FLUENT is used to track the time-dependent volume

fractions of liquid (l) and gas (G) throughout the computational domain. In this approach, the

gas-liquid interface movement is described by the distribution ofG, the volume fraction of thegas phase in the computational cells. For instance, the cell is full of liquid ifG = 0, the cell is

full of gas ifG = 1, or an interface exists if 0 < G < 1. In general, the isocontour ofG = 0.5can be used to identify the interfacial location for computation and visualization purposes. Thefields for all variables and properties are shared by the phases and represent volume-averaged

values. The liquid phase is composed only of water, whereas the gaseous phase is composed air.

The axisymmetric governing equations of continuity, momentum, energy, and species are solvedusing the segregated pressure-based solver.

A single set of mass and momentum equations are solved throughout the computational

domain to obtain the velocity field shared by both the gas and liquid phases.

u 0t

(6)

T

uuu p u u g F

t

. (7)

where and are the volume-fraction-averaged properties, given as

L L G G . (8)

L L G G (9)

The volume fraction of the gas phase is computed by solving the scalar convection equation

GGu 0

t

. (10)

The liquid volume fraction can be then obtained from

L G1 . (11)

The surface tension force in equation (7) is given by


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G G

LG

G L

2F

(12)

This force is computed from the continuum surface force (CSF) model proposed by Brackbill13

.

In Fluent, a formulation of the CSF model is used, where the surface curvature is computed fromlocal gradients in the surface normal at the interface. If n is the surface normal, defined as the

gradient of the G, which is the volume fraction of the gaseous phase i.e.,

Gn , (13)

then the curvature, , is defined in terms of the divergence of the unit normal,

n (14)The solid surface wettability effects are introduced by using the surface tension and walladhesion model in FLUENT. The equilibrium contact angle that the liquid is supposed to make

with the wall is used to adjust the surface normal in cells near the wall. This is specified as a

boundary condition and it results in the adjustment of the curvature of the surface near the wall.

The surface normal at the live cell next to the wall is

w 0 w 0n n cos t sin (15)

where, wn and wt are the unit vectors normal and tangential to the wall, respectively. Thecombination of this contact angle with the normally calculated surface normal one cell away

from the wall determine the local curvature of the surface, and this curvature is used to adjust the

body force term in the surface tension calculation.

Numerical schemes:

An unsteady pressure based, first order implicit in time unsteady model has been used forthe simulations. The multiphase flow is incorporated using the VOF model. Both the Green-

Gauss cell based and node based models have been used to calculate the derivatives and the

gradients of the governing equations in each cell and the results along with the numerical

efficiency has been compared. The pressure velocity coupling has ben achieved using the PISO

algorithm. The VOF equations have been discretized using the Geo-Reconstruct scheme whereasthe momentum equation has been discretized using both the second order upwind scheme and the

QUICK scheme and compared in terms of accuracy and efficiency.

Geometry and mesh:The computation domain was chosen to be a square with the sides being ten times the

size of the droplet. The whole domain was meshed with quadrilateral elements (Figure 3). Grid

independence study was conducted to optimize the solution. However, a dynamic mesh was notadopted in this study due to the lack of time and computation resources as is done by Briones[8].

The whole domain was refined multiple number of times to study the effect on the droplet

spreading and the oscillations. The parameters are listed in Table 2.

Table 2. Grid independence study parameters

Case No. D0 (m) V0 (m/s) Domain Size Grid Spacing Result

1 55.4 2.45 0.55 mm x 0.55 mm 0.01 Diverged

2 55.4 2.45 0.55 mm x 0.55 mm 0.005 Inaccurate

3 55.4 2.45 0.55 mm x 0.55 mm 0.005 gradient Acceptable Accuracy

4 55.4 2.45 0.55 mm x 0.55 mm 0.001 Accurate

5 55.4 2.45 0.55 mm x 0.55 mm 0.0005 Accurate


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Time stepping:

For the time advancement of the unsteady problem a first order accurate implicit schemehas been used. Variable time stepping has been adopted in order to achieve better computational

stability. Since the grid sizes are in microns time steps in the order of 10 -210-4s has been

used. This ensures that with the time advancement the Courant number, which is the parameter

for the stability is kept below 1. This ensures the accuracy and convergence of the computation.Both the iterative and non iterative time advancement (NITA) schemes were tested. It was found

that the solution process was less computationally expensive using NITA but however the

accuracy of the solution was not as was desired. Hence, the iterative time advancement has beenemployed.

Boundary conditions:

Figure 3. Computation domain and implied boundary condition

The boundary conditions implied on the problem are as shown in figure 3. There are three main

boundary conditions:1) Wall Boundary Condition:

L 0, u 0, v 0z

(16)

2) Axisymmetric Boundary Condition:L v0, u 0, 0

r r

(17)

3) Pressure Outlet Boundary Condition:P 0 (gauge). (18)

AxisymmetricB.C

PressureOutletB.C

Pressure Outlet B.C

Wall B.C

z

r


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Problem Setup:After specifying the models, boundary conditions and the solver controls the problem is

initiated. A region is marked where the liquid droplet is patched along with the required velocity.

The iterations are then started with the variable time stepping scheme after setting the desired

residual criteria for convergence. The solutions are saved after every 10 time steps for 2500 time

steps.

Results and Discussion:

Effect of grid size and model options on the efficiency of the solution:

The different grids used for the determination of the optimal meshing criteria have been

stated in table 2. Both uniform and gradient meshes have been evaluated. For the 0.01 mm gridspacing the solution diverged. Then the whole grid was constructed using 0.005 mm as grid

spacing. For the iterative time advancement scheme, with the criteria for convergence as 10 -6,

FLUENT nearly one hour to solve for 2500 time steps. Further a gradient mesh, refined in the

vicinity of the droplet spreading zone with a successive ratio was generated. The refinement led

to creation of 0.002mm square grids near the droplet spreading zone. Other grid sizes that havebeen used in the grid independence study are mentioned in the table 2. It was found that for

0.001mm uniform grid spacing the solutions were in agreement with that presented by Briones.However, the small droplet that was observed both in experiments and in their numerical

simulations at the center of the droplet was not observed. This is because the grid size was not

small enough to capture the mirco-bubble. Briones[8] had a grid size of 0.415 m and only then

they could resolve the gas bubble. But because of the computational time required to simulatethis problem with such small grid size, it was not done. They were able to address this issue by

employing a dynamic mesh and refining the region just around the droplet interface.

Table 3. Parametric study of effect of the numerical schemes on the efficiency of the numerical model.

Case Model/Scheme Parallel Iterations (approx) Time (hrs)1 Cell based/QUICK No - -

2 Cell based/QUICK No 38000 1

3 Cell based/QUICK No 35000 1.45

4a Cell based/QUICK No 56000 22

4b Node based/QUICK Yes 77000 29

4c Cell based/QUICK Yes 56000 18

4d Node based/2n Order Upwind Yes 80000 32

4e Cell based/2n

Order Upwind Yes 65000 22

5 Cell based/QUICK Yes 100000+ 48 +

Table 3 shows the effect of using different numerical schemes on the numerical solution. The use

of cell based or node based computation does not make any change in the accuracy of thenumerical solution as square mesh was used. However, it can be seen that the time required for

the two schemes are different. The node based solution scheme reconstructs exact values of a

linear function at a node from surrounding cell-centered values by solving a constrained

minimization problem, preserving a second-order spatial accuracy. Whereas, the cell basedscheme computes the gradients of the flow field at a given cell by taking the average of the


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values at the neighboring cell centers. This reduces the computation time. For a structured and

block-structured meshes composed of quadrilaterals or hexahedra, the number of cells isapproximately equal to the number of nodes, and the spatial resolution of both storage schemes is

similar for the same mesh.

Also for discretization of the momentum both the second order upwind scheme and the

QUICK scheme has been used. It was found that the QUICK scheme provides with the samelevel of accuracy as that of the 2nd order scheme and is faster at achieving convergence thus

reducing the overall computation time. Parallel processing was later used to increase the

computation speed. The effect of the different schemes studied in this work is summarized intable 3.

Simulation Results:

Figure 4. Numerical solution images of the droplet impingement process

Figure 4 shows the spreading of the droplet achieved with the numerical simulation attempted inthis work for different time steps. Upon comparison with Briones et als

[8] work it can be seen

that the simulated droplet very closely resembles the experimental observations. The small

droplet that was observed by them and also numerically obtained was not observed in this casebecause of the grid spacing constraints as explained earlier.

0 s 5 s 20 s 50 s

65 s 80 s 97 s 111 s

250 s


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Figure 5. Results of Briones et al [8].

Figure 6 shows the droplet spreading diameter as a function of time. The spreading diameter

evolution with time is shown for three different cases. These cases are explained in table 2 & 3.It is observed that for the 0.001 mm uniform grid the droplet spreading resembles the values that

reported by Briones [8] in their work. The effect of the grid size on the accuracy of the value

depends heavily on the grid that is used. This is because the droplet size in itself is very smalland hence the ratio of the grid dimension to the droplet radius becomes critical.

From figure 7 it can be seen that the oscillations of the free droplet surface is heavily

dependent on the grid size. The average droplet height for the 0.001 mm case is similar to that

observed by Briones in his work using the static contact angle model. But the oscillationspredicted by the rougher and the gradient mesh cases are too high as compared to the

experimental observations.

From the simulations of the case with 20 static contact angle it was observed that the

droplet gets pinned very early and the oscillations of the droplet also vanish after spanning only a

short period of time. This is in accordance with the theory that for small contact angles or betterwetting surfaces the hydrodynamic dissipation of the unbalanced surface forces is the primary

channels of energy dissipation and its successful simulation in FLUENT proves the same.


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Figure 6. Variation of dimensionless spreading radius with time.

Figure 7. Droplet dimensionless height variation with time.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10 12

R/D0

t x V0/D0

Case 2

Case 3

Case 4

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12

H/D0

t x V0/D0

Case 2

Case 3

Case 4


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Conlcusion:

Numerical simulation of a microdroplet impingement has been performed. The partial

wetting phenomenon of a surface by a liquid droplet has been investigated by the use of

FLUENT. The VOF model has been implemented to simulate the same and it was found that the

static contact angle method provides a close approximation to the microdroplet spreadingphenomena. Different numerical schemes, mesh sizes have been tested to determine the most

efficient and accurate model for simulation this physical process. The pinning of the liquid

droplet while spreading has been identified and also the oscillations of the free interface beencaptured. It has also been seen that the FLUENT static contact angle model provides a very good

agreement with the theory behind the spreading on a highly wetting surface.

Future Work:

This work will form the basis for future project which will include the incorporation of

both hydrodynamic theories and molecular kinetic theories which describe the process of droplet

spreading. Dynamic adaptation of the gradient of the VOF parameter would be done to refine themesh near the interface and thus enabling to use a coarser mesh elsewhere, hence reducing

computational expense. Also various evaporation models would be investigated to study theeffect of surface properties on the heat flux of boiling and evaporation.

References:

1Kim J IJHFF 2007

2 Horacek et al3 Sodtke and Stephan4

Navedo5

Chen et al6

Harlow and Shanon J App Phy 1967 38 3855}7

Fukai et al8

Briones et al9

Attinger10

de Gennes11

Seveno Langmuir 200912

Ruitjer 1999 droplet spreading Langmuir]13

Brackbill

Droplet Impingement on a flat surface

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