A mass-preserving interface-correction level set/ghost fluid method formodeling of three-dimensional boiling flows

B. M. Ningegowdaa,b,∗, Zhouyang Gea, Giandomenico Lupoa, Luca Brandta,c, Christophe Duwiga

aLinné Flow Centre and SeRC (Swedish e-Science Research Centre), KTH Mechanics, Stockholm, SE-100 44, SwedenbDepartment of Engineering, University of Perugia, Perugia-06125, Italy

cDepartment of Energy and Process Engineering, Norwegian University of Science and Technology (NTNU),Trondheim-7491, Norway

Abstract

We present an efficient numerical method for modeling of three-dimensional (3D) boiling flows. Theliquid-vapor interfacial dynamics is captured using an interface-correction level set method, modified toaccount for the interfacial mass transfer due to phase change. State-of-the-art computational techniques,such as the fast pressure-correction and the ghost-fluid methods, are implemented to accurately solvethe coupled thermofluid problem involving large density contrasts and jump conditions. The solver isthoroughly validated against four benchmark cases with increasing complexity. We also simulate a 3Dsaturated film boiling of water and its vapor at near critical conditions over a horizontal plane heatersurface, and a single spherical water vapor bubble at subcooled liquid temperatures of 5K, 12K and30K to demonstrate the potentiality of our method.

Keywords: Numerical simulation, three-dimensional, two-phase, boiling, Level set, Ghost fluid,interface jump,

∗Corresponding author.Email address: nbm@mech.kth.se (B. M. Ningegowda)

Preprint submitted to International Journal of Heat and Mass Transfer May 11, 2020

Nomenclature

D bubble diameter, m~g acceleration due to gravity, m/s2

hlv latent heat of vaporization, J/kgL reference length scale, mT temperature, KU reference velocity, m/su non-dimensional velocity vectorg dimensionless gravityk dimensionless thermal conductivityp dimensionless pressure~n the outward-pointing unit normal vectorm dimensionless interface mass fluxcp dimensionless specific heat at constant pressured dimensionless diametert dimensionless timex position vectorFr Froude number, U

2

gL

Gr Grashof number, gβρl(ρl−ρv)L3

µ2l

Ja Jacob number, cp,l(∆T )

hlv

Pe Péclet number, Re PrPr Prandtl number, µlcp,l

kl

Re Reynolds number, ρlULµl

We Weber number, ρlU2Lσ

Greek symbols

ρ non-dimensional densityκ interface mean curvatureφ level set shortest distance functionσ surface tension coefficient, N/mµ dimensionless dynamic viscosityβ coefficient of thermal expansion, 1/Tsat, 1/K

∆Tsup superheated temperature, (Tw − Tsat), K∆Tsub subcooled temperature, (Tsat − Tl), Kε interface finite thickness, 3/2∆hγ dimensional thermophysical propertyρ dimensional density, kg/m3

µ dimensional dynamic viscosity, Pa · s/m2

k dimensional thermal conductivity, W/m ·Kcp dimensional specific heat at constant pressure, J/kg ·Kθ non-dimensional temperature, T−Tref

Tw−Tref

Subscripts

sat saturatedsup superheatedsub subcooledpc phase changemc mass correctionc correctionl liquid phasev vapor phasew wall surfaceΓ liquid-vapor interface

Superscripts

n+ 1 new time stepn old time step

2

1. Introduction

Liquid-vapor phase-change phenomena are widely encountered in industrial applications due to theirhigh heat-transfer rates and the large energy-storage capability [1]. Examples include spray cooling andcombustion, high temperature heat exchangers [2], refrigeration or cryogenic fluid thermosiphon [3], fluidstorage under micro-gravity in space [4], thermal and nuclear power plants [5], cooling of electronics [6].The successful development of these multi-scale and multi-physics technologies requires not only well-managed system engineering, but also a detailed knowledge of the underlying phase-change processes.Experimental studies of such processes are usually costly, time-consuming and demanding in terms ofoperating conditions and instrument tolerance. It is therefore of paramount importance to developaccurate and efficient numerical algorithms able to capture the fluid dynamics and resolve the interfacialheat and mass transfer down to the microscopic scale.

Computational methods used for multiphase flow simulations typically fall under two broad categories,i.e. reduced-order models and direct numerical simulations [7]. The former can be further divided intoEulerian-Lagrangian and Eulerian-Eulerian frameworks. In the Eulerian-Lagrangian framework, thedispersed phase (e.g. bubbles) is tracked by point forcings based on constitutive equations of mass,momentum and energy [8]. The dispersed dynamics, such as the bubble-bubble interactions, are typicallymodeled under the assumptions of homogeneity and spherical shape. This approach allows the explicitcalculation of the bubble statistics; although topological changes such as breakup and coalescence aretypically neglected. In the Eulerian-Eulerian framework, the dispersed phase is described as a field, andboth the liquid and gas phases are solved simultaneously using separate conservation equations [9]. Thisapproach is suitable for modeling of large-scale industrial flow, such as bubble columns. However, theclosure laws for the conservation equations of mass, momentum, energy, and/or the bubble surface arearequire sophisticated modeling and are typically not valid on a general basis.

In contrast to the reduced-order descriptions, direct interface-resolved numerical simulations can cap-ture the bubble motion and deformation down to the grid resolution. Depending on the grid generatingtechniques [10], these methods can be broadly grouped into (i) body fitting methods, (ii) front trackingmethods, and (iii) interface capturing methods. Body fitting methods, also referred to as the Lagrangianmoving body-fitted mesh approaches [11, 12], are numerically complex and computationally expensiveas they require frequent re-meshing to handle large topological changes. Front tracking methods (FTM)[13, 14, 15, 16] are a semi-Lagrangian and semi-Eulerian approach where moving Lagrangian particlesare tracked inside an Eulerian fixed domain. Compared to body fitting methods, the FTM are com-putationally less expensive, but they still require a cumbersome treatment for interface merging andfragmentation.

Finally, interface capturing methods, such as the level set (LS) [17, 18], the volume-of-fluid (VOF)[19, 20, 21], the coupled level set and volume-of fluid (CLSVOF) [22, 23, 24, 25, 26], the Volume Offluid and level SET (VOSET) [27], and Lattice-Boltzmann Method (LBM) [28] are completely Eulerianand calculate the dynamics of the interface based on the advection of an indicator color function on afixed grid. Compared to (i) fitting or (ii) tracking, these capturing methods are probably the easiest toimplement, can automatically handle large topological changes, and are highly efficient when computedin parallel. Therefore, these capturing methods have gained increasing popularity in multiphase flowsimulations, especially if breakup and coalescence are frequent.

Along with the above interface capturing methods, various surface tension models have been proposedto impose the stress jump across the liquid-vapor interface. These include the continuous surface tensionforce (CSF) [29], continuous surface tension stress (CSS) [30], parabolic reconstruction of surface tension(PROST) [31], and ghost fluid method (GFM) [32, 33]. The difference between these methods lies in thenumerical treatment of the singular forcing; see Ref. [34] for a recent comparison on some of the methodsand further discussions.

When heat and mass transfer occurs across the liquid-vapor interface, the energy conservation isusually modeled via Rankine - Hugoniot jump conditions [35]. In the literature, various phase changeapproaches have been proposed for numerical simulations of boiling flows: the sharp interface basedmodel [36, 37, 38, 23], the ghost fluid method (GFM) based model [35, 39, 33], and diffused model [40].More specifically, to simulate saturated film boiling flow over a horizontal flat heater surface, variousnumerical methods such as FTM [36, 1, 41], LS [37, 39, 35, 33], VOF [38, 42, 43, 44, 45], CLSVOF[46, 23, 47, 24, 48, 25], VOSET [27, 49, 50, 51, 52] and LBM [53, 54], have been reported in theliterature. We note that the vast majority of these numerical studies considers mostly 2D saturatedfilm boiling flows; only limited studies of 3D film boiling flow are carried out in [1, 45]. In these

3

works, a one-dimensional Stefan problem [38] and the saturated film boiling of a single bubble over ahorizontal flat heater [37, 36, 38, 42] have been widely used as the benckmark test cases. Son and Dhir[37] simulated LS based 2D saturated film boiling of water and its vapor at near critical conditions ofsaturated pressure, psat = 21.9 MPa and saturated temperature, Tsat = 646.15 K. Gibou et al. [35]simulated 2D film boiling flows using a LS method with GFM for the interface jump conditions. Tanguyet al. [39, 33] simulated both droplet evaporation and bubble expansion processes similar to [35] using theLS method with smeared boundary conditions at the interface using a Dirac delta formulation. Recently,Scapin et al. [55] proposed the VOF method for phase change problems. Lupo et al. [56] proposed a newImmersed Boundary Method (IBM) for the interface resolved simulation of spherical droplet evaporationin gas flows. Lee et al. [57] investigated the smooth distribution of the sharp phase change velocity jumpcondition within a finite thickness of the interface. The improved implicit mass flux projection and thepost advection velocity correction step ensure the divergence-free condition of the velocity field across theinterface region. In the present study, a GFM based phase change model considered similar to Tanguy etal. [39, 33] to simulate the three-dimensional saturated film boiling of water and its vapor at near criticalconditions of saturated pressure, psat = 21.9 MPa and saturated temperature, Tsat = 646.15 K.

Furthermore, we shall here perform a numerical simulation of 3D subcooled boiling of a single sphericalbubble in a quiescent liquid to evalaute our GFM-based 3D phase change model at high densities andviscosities ratios. The subcooled boiling process occurs when the saturated or superheated vapor bubbleis surrounded by a quiescent subcooled bulk liquid. Initially, the quiescent liquid is at temperature, Tlbelow the saturation temperature, Tsat where Tl < Tsat. This situation is widely encountered in lightwater reactors (LWRs), boiling water reactors (BWRs) and pressurized water reactors (PWRs) [58].The safety and optimal design of water reactors depends critically on the presence of the vapor bubblewhich is significantly affected by the system pressure drop, rate of heat and mass transfer, and the flowstability [59]. Previous experimental studies of subcooled pool and flow boiling are presented in a numberof references [60, 61, 62, 63, 64, 65, 66, 67, 68, 69]. Kim and Park [65] experimentally investigated thesubcooled boiling of a single bubble at low pressure and provided a correlation for the interface heattransfer coefficient as function of the local Reynolds number, Re, Prandtl number, Pr and Jacob number,Ja of the subcooled liquid phase. Lucas and Prasser [66] experimentally studied the steam-water flowin a vertical pipe using novel wire-mesh sensors for high pressure and temperature conditions. Theyobserved that the bubble break-up is significantly influenced by the condensation process due to thechange in the surface area. In these experimental studies, it is however difficult to measure the complexflow physics and heat transfer mechanism of the bubble condensation.

Numerical simulations of subcooled boiling flow can be found in the literature, using either the VOF-CSF approach based on the open source code OpenFOAM [70, 71, 58], a modified user defined function(UDF) in the ANSYS-FLUENT CFD software [72, 73, 74] and a geometric reconstruction of piecewiselinear interface calculation (PLIC) based VOF method [75]. In particular, Jeon et al. [72] estimatedthe rate of bubble condensation in terms of the heat transfer coefficient at the interface obtained fromthe bubble velocity, liquid subcooled temperature and transient condensing spherical-bubble diameter.Bahreini et al. [70] numerically investigated the 2D single bubble dynamics with forced flow and observedthat the initial bubble size, liquid water subcooled temperature and inflow velocity of the bulk liquidwater all play an important role for bubble deformation. Due to the velocity gradients, the condensingvapor bubble moves towards the region of higher velocity and then back to lower velocity region accordingthe rate of subcooling. Samkhaniani and Ansari [71] accounted for the effect of (i) constant saturationtemperature and (ii) local saturation temperature of the vapor bubble as a function of thermodynamicpressure which is non-uniform in quiescent subcooled water due to surface tension and acceleration dueto gravity. These authors observed that for constant saturated temperature of the bubble, the bubblediameter reduces linearly due to the uniform temperature difference between the saturated vapor bubbleand the subcooled liquid phase. The various bubble regimes are related to the role of the additional forcesacting on the bubble such as capillary and buoyancy. Due to the condensation, the bubble reduces in sizegradually and eventually collapses [76]. Numerical modeling of multiple bubble condensation is usefulto understand and improve the continuum models of large scale subcooled boiling. Liu et al. [77, 74]developed a geometric piecewise linear interface calculation (PLIC) based on the volume of fluid (VOF)method to simulate multiple bubble condensation. These authors observed that the rate of condensationof a single bubble can be strongly influenced by the velocity of the fluid flow and the temperaturedifference between the saturated bubble and the subcooled bulk liquid. In the case of multiple bubblecondensation, the interaction between adjacent bubbles leads to an increase of the condensation rate ofthe lower bubble due to random perturbation induced by the upper surrounding bubbles. The interaction

4

with the surrounding bubbles can be neglected if the distance between adjacent bubbles is large enough.In the present numerical study, the mass-preserving, interface-correction LS/GFM method of Ge et

al. [78] is further modified to simulate 3D two-phase flows with phase change, such as saturated pool-filmboiling and subcooled pool-boiling flow. A projection method with a fast pressure Poisson equationsolver, similar to Dodd and Ferrante [79], is employed for the pressure-velocity coupling, based on fastFourier Transforms (FFT) and Gauss elimination. To model the interface normal velocity and thetemperature and mass fluxes at the interface due to phase change, the efficient Ghost Fluid method(GFM) as in Tanguy et al. [39, 33] is implemented. The present Mass-preserving Interface CorrectionLevel Set-Ghost Fluid Method (MICLS-GFM) based model is evaluated against various benchmark testcases in the literature. In particular, we present simulations of 3D pool boiling such as saturated filmboiling of water at near critical condition over a horizontal plane heater surface and of subcooled boilingof a spherical bubble in a quiescent liquid at atmospheric conditions.

2. Mathematical formulation

In the numerical modeling of boiling flows, the conservation of mass, momentum, and energy equationsare tightly coupled to each other. These governing equations require an additional modification toaccount for interface jump conditions as extensively discussed in the literature [37, 36, 38, 35, 39]. Thecontinuity equation is modified at the gas-liquid interface to account for the interface mass transfer, themomentum equations are modified to account for the surface tension force, viscosity jump and momentumexchange due to the phase change; finally, the energy equation is modified to account for the latent heatof vaporization [23]. In the present study, we investigate three-dimensional, unsteady, incompressible,Newtonian boiling flows.

2.1. Governing equationsIn the single phase incompressible bulk fluid region, the governing equations of continuity, momentum

and energy conservation are written in non-dimensional form [78] as

∇ · u = 0, (1)∂u

∂t+ u · ∇u =

1

ρi

(µiRe∇2u−∇p

)+

g

Fr, (2)

∂θ

∂t+ u · ∇θ =

kiρicp,iPe

∇2θ, (3)

where u = u(x, t) is the non-dimensional velocity vector, p = p(x, t) is the pressure field, g is a unitvector aligned with the acceleration due to gravity. The non-dimensional temperature is defined asθ =

(T−Tref

Tw−Tref

), where T is the dimensional temperature in K and the subscripts w and ref indicate

the wall surface and the reference fluid phase. The non-dimensional ratios of thermophysical propertiesinclude density ρi, dynamic viscosity µi, thermal conductivity ki and specific heat at constant pressurecp,i. In general, the non-dimensional ratios of any thermophysical property, γi, is defined as the ratio ofthe quantity in the fluid, γc,f in each cell to the corresponding reference fluid phase γref .

The non-dimensional parameters of the reference fluid phase are the Reynolds number, Re is definedas the ratio of the inertia force to viscous force as Re = ρlUL

µl, the Froude number, Fr is defined as the

ratio of the flow inertia to the external gravitational field as Fr = U2

gL, the Péclet number, Pe is defined

as the ratio of the rate of advection of a physical quantity by the flow field to the rate of diffusion of thesame quantity driven by an appropriate gradient, computed for the heat transfer as the product of theReynolds number and the Prandtl number Pe = RePr with the Prandtl number, Pr is defined as theratio of momentum diffusivity to thermal diffusivity, Pr =

µl ˜cp,lkl

. These dimensionless parameters are

based on the reference length L, velocity U , time t, temperature Tref , gravitational constant g, and theuniform thermophysical properties at the saturation condition of the reference liquid phase: density ρl,dynamic viscosity µl, thermal conductivity kl and specific heat at constant pressure cp,l. 1 Subscriptsv and l denote the vapor phase and the liquid phase, respectively. In the energy conservation equation,the effect of the viscous dissipation is neglected.

1The dimensional quantity of any variable, f is defined with a symbol tilde f throughout the paper.5

2.2. Dimensional jump conditionsAs the level-set (LS) method is a sharp interface capturing method, the governing equations are

carefully discretized using various jump conditions. These jump conditions are classified into threecategories: material property jump conditions, zero-order jump conditions and first order jump conditions[80]. To define the jump condition from the liquid phase, l, to the vapor phase, v, the jump operator,[·]Γ of any quantity (e.g. f) across the interface Γ is defined [33, 80] as

[f ]Γ = fv − fl. (4)

The material jump conditions of the thermophysical properties, [γ]Γ, such as density, dynamic vis-cosity, thermal conductivity and specific heat at constant pressure are generally written as

[γ]Γ = γv − γl. (5)

The enthalpy jump condition, [h]Γ = hv − hl is the latent heat of vaporization, hlv.The zero-order jump conditions include the jumps in the normal velocity, u, and in the pressure field,

p, across the interface Γ. The appropriate jump condition for the interface normal velocity u is expressed[35, 33] in the case of phase change as:

[u]Γ = ˙m

[1

ρ

]Γ

~n, (6)

where, ˙m is the rate of interface mass flux in kg/s ·m2 and ~n is the interface unit normal vector, directedfrom the liquid phase towards the vapor side of the interface. To impose the appropriate jump conditionfor the interfacial discontinuous pressure field, p one needs to consider the effect of the volumetric surfacetension force, viscosity difference and momentum exchange due to interface mass transfer, ˙m, see also[35, 33, 80],

[p]Γ = σκ+ 2[µ~nt · ∇u ·~n]Γ − ˙m2

[1

ρ

]Γ

, (7)

where κ is the interface mean curvature, ~nt is the transpose of ~n, and σ is the surface tension coefficientin N/m, which is here assumed to be constant, though it usually depends on both temperature andconcentration gradients [36]. Hence, the effect of the continuous tangential stress [37] along the liquid-vapor interface is neglected, ∇σ = 0 (i.e. no Marangoni convection effect).

The first-order jump condition is referred to as Rankine-Hugoniot jump [35] and is used to accountfor the balance between the conservation of mass and energy across the interface. According to the firstlaw of thermodynamics, the jump condition of the local heat flux normal to interface, [q′′]Γ · ~n can bewritten as

[q′′]Γ ·~n = [k∇T ]Γ ·~n, (8)

where the interface heat flux, [q′′]Γ = (q′′v − q′′l ) is computed using Fourier’s law as [k∇T ]Γ =(−kv∇Tv + kl∇Tl

)[38, 43]. The interface temperature gradient of the vapor phase, ∇Tv and of the liq-

uid phase, ∇Tl are numerically computed using second order accurate ghost linear extrapolation similarto literature [81, 35, 33].

Introducing the latent heat of vaporization, hlv, the interface heat flux [38, 33] can also be writtenas,

[q′′]Γ ·~n = − ˙m

(hlv + (cp,l − cp,v)

(Tsat − TΓ

))~n, (9)

where Tsat and TΓ are the saturation temperature and interface temperature in K, respectively. Variousapproximations have been considered to compute the interface temperature, TΓ; among those, a Dirichletboundary condition with constant saturation temperature of TΓ = Tsat is widely imposed [36, 37, 38, 43,46, 23].

Based on the uniform saturation condition at the interface, the mass flux normal to the interface issimplified to [43]

˙m = −

(−kv∇Tv + kl∇Tl

)hlv

·~n. (10)

6

2.3. Interface heat and mass transfer modelIn the phase change model, the mass jump condition, ˙m across the interface is written as

˙m = [ρ (u− uint)]Γ ·~n = 0, (11)

and the jump condition of the energy across the interface, Γ is[ρh (u− uint)

]Γ·~n = −[q

′′]Γ ·~n, (12)

where [q′′]Γ ·~n = (q′′v − q′′l ) ·~n is the net conduction heat flux across the liquid-vapor interface. Based onthe balance between mass and energy jump conditions, the interfacial mass flux, ˙m can be rewritten as

˙m =−[q

′′]Γ ·~nhlv

. (13)

Due to the interface phase change, the interface normal velocity field, [u]Γ = (u− uint) ·~n is computedacross the interface as

[u]Γ = ˙m ·~n[

1

ρ

]Γ

, (14)

where the reciprocal of the density jump condition,[

1ρ

]Γis calculated as

(1ρv− 1

ρl

).

Re-writing the mass flux, ˙m in (Eq. (14)) using uint in the moving reference frame, viz.

˙m = ρl(ul − uint) ·~n = ρv(uv − uint) ·~n, (15)

the velocity front, uint, defining the transient evolution of the interface in the moving reference frame,becomes

uint = ul −˙m

ρl~n = uv −

˙m

ρv~n. (16)

2.4. Modified non-dimensional jump conditionsThe non-dimensional form of the interface mass flux, m in Eq. (10), is expressed by Luo et al. [82] as

m =JaPe

(ki∇θv −∇θl) · ~n, (17)

where θ is the non-dimensional temperature, and the Jacob or Jakob number, Ja, is defined as theratio of sensible heat to latent heat absorbed (or released) during the phase change process with respectto the reference phase, Ja =

cp,l∆Tref

hlv. The interfacial temperature gradient, ∇θ, is calculated with a

second-order accurate ghost linear extrapolation similar to [35, 33].The dimensionless form of the interface normal velocity, [u]Γ, is written as

[u]Γ = m ·~n1− ρiρi

=1− ρiρi

JaPe

(ki∇θv −∇θl) · ~n, (18)

whereas the dimensionless form of the conservation of mass across interface region is [82]

∇ · u =1− ρiρi

JaPe

(ki∇θv −∇θl) · ~n. (19)

In the interfacial momentum equations, the pressure jump condition across the interface in Eq. (7)becomes in dimensionless form

[p]Γ =κ

We+

2[µ]ΓRe

nt · ∇u · n−(JaPe

(ki∇θv −∇θl))2

1− ρiρi

, (20)

where κ is the dimensionless mean curvature of the interface [78]. The Weber number, We is the ratioof inertial forces to surface tension forces and is defined as We = ρlU

2Lσ .

7

2.5. The classical level set methodIn the current numerical framework, we advect the liquid-vapor interface using a modified version

of the mass-preserving interface-correction level set method (MICLS) that was originally proposed in[78]. In the classical LS method [17, 83], the moving interface is represented as the zero iso-contour ofa shortest-distance function φ(x, t), i.e. Γ = x | φ(x, t) = 0. In the liquid phase, the LS functionis defined as the positive distance to the interface, φ(x, t) = +d, whereas in the discrete vapor-phaseregion, the LS function is negative, φ(x, t) = −d. The advection equation for φ in the case of phasechange problems is written as [37, 35, 33]

∂φ

∂t+ uΓ · ∇φ = 0, (21)

where uΓ is the dimensionless velocity of the moving front and is defined using Eq. (16) as

uΓ = ul −m

ρl~n = uv −

m

ρv~n. (22)

Note that the underlying velocity field is discontinuous across the interface, see Eq. (14). Due tothe non-uniform advection velocity, φ will therefore lose its regularity over time. To ensure numericalstability and accuracy, it is customary to regularly reshape φ into the signed distance function by solvingthe following Hamilton-Jacobi equation [84]

∂φ

∂τ+ S(φ0) (|∇φ| − 1) = 0, (23)

where τ is the pseudo-time, and S(φ0) is the mollified sign function (see e.g. [78] for its definition).Eq. (23) is typically iterated once per ten physical time steps. The numerical schemes for solving Eqs.(21, 23) are the same as in [78]. Specifically, we use a third-order Runge-Kutta (RK-3) scheme forthe temporal integration, a fifth-order Higher Order Upstream Central (HOUC-5) scheme [85] for theconvective terms in Eq. (21), and a fifth-order weighted essentially non-oscillatory (WENO-5) schemefor the spatial discretizations in Eq. (23).

In the LS method, the unit normal vector, ~n, and the interface mean curvature, κ, can be readilycomputed from the level set function

~n =∇φ|∇φ|

, κ = −∇ ·~n. (24)

The thermophysical properties, γ(ρ, µ, k, cp), can be written as

γ(φ) = γlH(φ) + γv (1−H(φ)) , (25)

where H(φ) is a smoothed Heaviside function obtained from the continuous LS function, φ as

H(φ) =

1 if φ > ε,12 + φ

2ε + 12π sin (πφε ) if |φ| ≤ ε,

0 if φ < −ε.(26)

For a uniform grid spacing, ∆h, the interface is assumed to a have finite thickness ε = 32∆h. The

Heaviside function, H(φ) effectively smears the discontinuity across the liquid-vapor interface, avoidingnumerical instability [86].

Solving both LS advection and re-distancing does not guarantee mass conservation [87, 76, 88] due tothe numerical errors arising in the discretization schemes. To overcome this problem, various global masscorrection schemes have been proposed [76, 88, 78]. Here, the interface correction LS (ICLS) method of[78] is modified to simulate 3D phase change problems.

2.6. MICLS-based global mass correction methodThe main advantages of the level set method are its simplicity and accurate curvature calculation,

which reduces erroneous velocities and spurious currents [29, 19] commonly occurring in interface cap-turing methods [78, 89]. The disadvantage, however, is the inherent mass loss when applied to incom-pressible, multiphase flows. As discussed earlier, many efforts have been devoted to overcoming the issue

8

of mass loss by e.g. coupling with the VOF method, so called CLSVOF [23] and VOSET [27] methods.Note, however, that the geometric reconstruction and advection procedure of the VOF method becomesnumerically difficult and computationally expensive for 3D problems [20, 21].

In this study, the MICLS is applied across the finite thickness of the interface region to obtain globalmass conservation for boiling flows. To improve the LS method to conserve mass, the interface velocitycorrection approach of [78] is extended to phase change processes. The modifications proposed here takeinto account the velocity jump due to phase change, while maintaining the simplicity of the standardlevel set method. The detailed algorithm is described below. This velocity correction is not required atevery time step and has a limited computational cost. The interface velocity correction is based on thesolution of a PDE as for the LS advection equation.

As in [78], we solve an additional advection equation to compensate the mass loss, viz.

∂φ

∂t+ uc · ∇φ = 0, (27)

where uc is the interface correction velocity field, satisfying∫Γ

~n · ucdΓ =δV

δt. (28)

Here, δV/δt denotes the dispersed volume (e.g. bubbles) that is numerically lost over some time.Since the bubble volume can change physically due to boiling or condensation, the actual rate of massloss at time n is (

δV

δt

)(n)

=

(V (n) − V (n−Nmc)

)− δV (n)

pc

Nmc∆t, (29)

where δV (n)pc is the volume change due to phase change at time level n, and Nmc is the number of time

steps between each correction, solution of Eq. (27). Here, we choose Nmc = 20, that is, we perform themass correction every 20 physical time steps.

In Eq. (29), V (n) and V (n−Nmc) can be obtained easily by integrating the Heaviside function, Eq.(26), over the whole domain, e.g.

V (n) =

∫Ω

(1−H(φ(n)))dΩ. (30)

δV(n)pc depends on the integral of the interface velocity, from the last time the mass correction was

performed to the present time, i.e.

δV (n)pc =

∫ t(n)

t(n−Nmc)

∫Γ

~n · uΓdΓdt′. (31)

Numerically, the surface integral in Eq. (31) can be approximated by the sum of the interface velocitymultiplied by a Dirac delta function,∫

Γ

~n · uΓdΓ '∑

(~n · uΓ)δ(φ)∆x∆y∆z (32)

where δ(φ) is the derivative of Eq. (26)

δε(φ) =

0 if φ > ε12ε

(1 + cos(πφε )

)if |φ| ≤ ε

0 if φ < −ε.

Finally, the algebraic form of the correction velocity is expressed in the same way as in [78],

uc(φ) =δV

δt

κ(φ)

AΓ∇H(φ), (33)

where Af =∫

Γκ(φ)δ(φ)|∇φ| dΓ. The interface correction velocity uc becomes large for highly curved

interfaces, because the local information of the interface is used to obtain the global mass conservation.The mass correction is used to correct the LS function near the interface, before the LS re-initializationis performed.

9

3. Numerical model validations

The ICLS method was extensively validated by Ge et al. [78] for simulating 3D two-phase flowswithout phase change. Here, we present validations for flows with phase change. In the numericalverifications of the present model, we consider (i) the LS advection, re-distancing and modified LS masscorrection algorithm of [78], (ii) a linear ghost extrapolation of the non-divergence free velocity field asin [33], (iii) a second order accurate extrapolation of the temperature as in [81, 90] and (iv) a secondorder extrapolation of the phase-change mass flux [33] across the finite thickness of the interface region.

3.1. Evaluation of the LS advection algorithmHere, a simple analytical benchmark test case is used to evaluate the LS advection, re-initialization

and MICLS based mass correction algorithms for simulating two-phase flows with phase change. Weconsider a spherical bubble undergoing uniform expansion as discussed in [33, 91]. In detail, an initiallyspherical gas bubble linearly expands by an externally-imposed constant interface mass flux, with uniforminterface normal velocity, [u]Γ = 0.1. This test case does not require the solution of the Navier-Stokes andenergy equations, only of the LS-related algorithms. Also, the effects of the thermophysical propertiesof the working fluids and the acceleration due to gravity are neglected. Initially, a zero velocity fieldis imposed in the whole computational domain. The evolution of the dimensionless spherical bubblediameter, d(t) function of the non-dimensional time, t is given by d(t) = d(0) + [u]Γt, where, d(0) = 1is the initial dimensionless bubble diameter. In the simulations, we consider the spherical bubble atthe centre of a cubic domain of length, l = 4d(0). The cubic domain is uniformly discretized using anincreasing grid resolution of 163, 323, 643, 1283 and 2563, with corresponding grid size, ∆h = d(0)/Nd= 1/4, 1/8, 1/16, 1/32 and 1/64. A constant time step of ∆t = 10−3 is used.

The non-dimensional spherical bubble diameter at dimensionless time, t = 5 obtained from threedifferent algorithms is compared with the exact analytical solution in Fig. 1. In particular, we con-sider a classical LS method implementation (denoted LSM), an interface-correction level set using theanalytically known exact volume (MICLS-Exact) and the same approach with corrections obtained inte-grating from the interface normal velocity (MICLS-Modified). A simple visualisation like that in panela) does not enables us to differentiate the various mass corrections even for the coarse grid resolution of∆h = 1/8. Hence, we report in panel b) of the same figure the ratio between the bubble diameter att = 5 from the simulations and the theoretical value, dr = d(t)num/d(t)exact, for the different resolutionsconsidered. The data show that the correct and modified MICLS well approximate the exact solutionalready at low resolution (4 and 8 grid points per bubble diameter), whereas a classic LS would require32 points to achieve a similar accuracy. For a simple test case like this one, the present LS advectionwith mass correction algorithms becomes independent of the grid resolution unlike classical LS withoutmass correction.

The absolute error or L1 error norm is the difference between the computed value and true value, isdefined here as E = |d(t)num− d(t)exact|, is reported in Fig. 1 (c) for the different grid resolutions underinvestigation in logarithmic scale. The error at final time at t = 5 from the different implementationsindicates that the order of convergence for the classical LS method without mass correction varies betweenfirst and second. The LS method with MICLS based mass correction displays an order of convergencebetween first and second for the coarser grids and close to second order accuracy for finer grids. Hence,the LS method with MICLS based mass correction is able to conserve mass already at coarse gridresolution, which implies a lower computational cost than for the classical LS method.

3.2. Non-divergence free Interface velocityIn our approach, the modified momentum equations for the simulation of phase change are modeled

similarly to the non-divergence free conditions with the interface velocity extension approach consideredby Tanguy et al. [33]. In this new test, the previously evaluated MICLS advection algorithm is mergedwith the projection method to obtain the pressure and velocity coupling. We still consider the caseof a uniformly expanding spherical bubble in the absence of gravity, a case which does not require thesolution of the energy equation. The growth of the spherical bubble is due to the imposed uniform masstransfer rate and the non-divergence free condition of the simple velocity extension is considered withinan interface region of finite thickness, ±6∆h.

Tanguy et al. [33] performed a 2D simulation of this problem with a dimensional reference bubblediameter of D = 0.002m placed at the center of a square domain of length l = 4D. Following theseprevious studies [39, 33], we assume the physical properties of liquid water and air at 1 atm with density of

10

1 1.5 2 2.5 3 3.5

Y-axis

1

1.5

2

2.5

3

3.5

Z-a

xis

Initial at t=0

Analytical

LS

Exact-MICLS

Modified MICLS

(a) Bubble radius at grid resolution of 1/8

0.998

1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

0 10 20 30 40 50 60 70

Bu

bb

le d

iam

ete

r ra

tio

, d

r

Grids per unit diameter, Nd

AnalyticalLSM

MICLS-ExactMICLS-Modified

(b) Bubble diameter ratio, dr

10-5

10-4

10-3

10-2

10-1

10-2

10-1

L1 e

rro

r n

orm

s

Uniform grid size, dx

dx2

LSMMICLS-Exact

MICLS-Modifieddx

(c) Absolute error

Figure 1: Grid resolution study for the test case of uniform expansion of a spherical bubble.

11

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

0 5 10 15 20 25 30 35 40 45

Dim

en

sio

nle

ss b

ub

ble

vo

lum

e

Non-dimensional time

d(0)/16d(0)/32d(0)/64

d(0)/128

(a) Bubble volume

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 5 10 15 20 25 30 35 40 45

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en

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ss b

ub

ble

dis

pe

rse

d v

elo

city

Non-dimensional time

d(0)/16d(0)/32d(0)/64

d(0)/128

(b) Dispersed velocity

Figure 2: Uniform expansion of a spherical bubble at constant mass rate. Expanding bubble volume and dispersed bubblevelocity for the different grid resolutions examined.

liquid, ρl = 1000 kg/m3, density of gas, ρv = 1 kg/m3, dynamic viscosity of liquid, µl = 1.0×10−3 kg/msand dynamic viscosity of gas, µv = 1.78 × 10−5 kg/ms, with a constant surface tension coefficient ofσ = 0.07N/m and uniform mass transfer rate of 0.1 kgm−2s−1.

Here, we present a 3D validation on a cubic computational domain discretized using an uniform meshwith 643, 1283, 2563 and 5123 grid points, corresponding to a grid size ∆h in units of the initial bubblediameter, d(0)/Nd = 1/16, 1/32, 1/64 and 1/128. Periodic boundary conditions are imposed in the xand y directions and homogeneous Neumann boundary conditions on the top and bottom boundaries.The constant time step is ∆t = 1.0 × 10−6 owing to the high density and dynamic viscosities ratios,ρ = 1.0 × 10−3 and µ = 1.78 × 10−2. The non-dimensional parameters defining the test case arebased on a reference length equal to the initial bubble diameter, L = D is 0.002m, reference velocity,

U =

√Dg = 0.14m/s and reference time scale, t = D/U = 0.01427s. Using the liquid as the reference

phase, the Reynolds number is Re = 280 and the Weber number is We = 0.56. Here, the qualitativecomparison is carried out in terms of averaged bubble dispersed velocity, udisp, defined as

udisp =√u(x)2 + u(y)2 + u(z)2, (34)

and reported in terms of ud = udisp/U . The time evolution of the non-dimensional expanding bubblevolume, V (t), and of the dimensionless bubble dispersed velocity, ud are displayed in Fig. 2 (a) and(b) for the different grid sizes under consideration. The results show that for the fine grid size, ∆hsmaller than 1/32, both the bubble volume and the average dispersed velocity obtained from the presentMICLS method becomes independent of the grid resolution. Deviations are observed for the coarse gridresolution of 1/16. To accurately capture the interface dynamics and flow physics, a minimum grid sizeper unit bubble diameter of ∆h = 1/32 is therefore required.

The order of convergence, defined by the absolute or L1 error norm at the various grid resolutions,is shown in Fig. 3. Recalling that the order of convergence of the LS with mass correction is betweenfirst and second order, see Fig. 1 (c), coupling the LS advection with mass correction (MICLS) withthe projection method reduces it to first order accurate. However, the simple interface ghost velocityextension of the present 3D problem is computationally cheaper than the 3D extension in Tanguy et al.[33], due to the iterative solution at each time step of a Poisson equation for the ghost velocity extension.This requires approximately twice the computational cost than without the ghost velocity extension [37].

3.3. Extrapolated interface temperature for two-phase flowsAt the liquid-vapor interface, both heat flux and temperature are continuous. The continuous tem-

perature across the liquid-vapor interface is written as TΓ = TΓ,l = TΓ,v. The corresponding continuouslocal heat flux is q′′Γ = q′′Γ,l = q′′Γ,v. The heat flux, q′′ = −k∇Tn is computed using Fourier’s law of heatconduction. The temperature gradient, ∇Tn is therefore discontinuous across the interface, the con-tinuous temperature gradient is computed independently from either side of the liquid-vapor interface[45]. Hence, the interfacial temperature gradient calculated using a one-sided temperature difference [33]

12

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

10-2

10-1

L1 e

rro

r n

orm

s

Uniform grid size, dx

dx2

LSMMICLS-Exact

MICLS-Modifieddx

Figure 3: Absolute error versus grid resolution for the test case of a bubble expanding at constant mass transfer rate.

is a better approximation of the two-phase flow heat transfer. For simplicity, we first consider the 1Dequilibrium across the interface, when the local heat flux q′′Γ,v = q′′Γ,l,

−kvTΓ − Tv

∆nv= −kl

Tl − TΓ

∆nl, (35)

From the above relation, the 1D interfacial temperature for the two-phase flow is written as

TΓ =kvTv∆nl + klTl∆nv

kv∆nl + kl∆nv, (36)

which becomes in dimensionless form, θ,

θΓ =kiθv∆nl + θl∆nvki∆nl + ∆nv

. (37)

It is difficult and inaccurate to extend the above 1D temperature equation (37) into numerical studyof 2D or 3D problems. Hence, to solve the heat flux normal to the interface for 3D problems, PDEbased ghost temperature extrapolations such as constant, linear and quadratic approaches are proposedin the literature [90, 35, 33]. The main advantage of these ghost extrapolation approaches is that bothmomentum and energy equations are solved independently in the liquid phase and vapor phase [33]. Inthis numerical study, a second-order accurate linear ghost-temperature extrapolation is adopted, similarlyto Aslam [90].

To validate the linear extrapolation of the temperature across the interface, a simple plane interfacewith 1D steady state is solved similarly to the 2D numerical study considered by Son and Dhir [76].Here, a 3D unsteady, laminar, two-phase flow with heat transfer without phase change is used to obtainthe steady state solution. From this steady state solution, the linear temperature variations inside thesuperheated vapor phase and subcooled liquid phase are obtained and compared with the temperatureprofile of the analytical solution.

In our test, a plane interface with non-dimensional film thickness of height, δ = 0.25 is placed insidea unit cube computational domain. The domain is discretized with a uniform grid size of 1283, withperiodic boundary conditions imposed in both X and Y directions. The bottom boundary consistsof a no-slip wall with superheated dimensionless temperature θw = 1 imposed as Dirichlet condition,whereas the top boundary is a no-slip wall with the subcooled non-dimensional temperature θw = −1.Initially, both phases are at saturation temperature, θ = 0, with zero velocity in the whole domain.The unsteady temperature equation is solved independently in the liquid and vapor phase similarlyto the ghost approach of [33] and the numerical simulation carried out until the steady state solutionis reached. The temperature variations inside both superheated vapor and subcooled liquid water atits near critical conditions of psat = 219 bar and Tsat = 646.15K are considered. The comparison ofthe linear temperature obtained from the numerical and analytical solution is shown in Fig. 4. Thegood agreement in the figure indicates the PDE-based second-order accurate linear ghost temperatureextrapolation can be further used to simulate 3D two-phase flows with heat and mass transfer.

13

Figure 4: 1D Steady solution of the two-phase flow with heat transfer without phase change.

3.4. Ghost fluid method (GFM) based phase change modelNext, we examine the ghost fluid method (GFM) for interface heat and mass transfer. The present

GFM model is validated against the benchmark case of the one-dimensional Stefan problem consideredin [38, 42, 35, 23]. In this test, the interface normal velocity due to the phase change is varying non-uniformly both temporally and spatially and it is calculated based on the temperature gradient acrossthe finite thickness of the interface.

We compare the transient evolution of the plane interface thickness obtained from the present nu-merical study with the analytical solution of the 1D Stefan problem. Here, we consider constant ther-mophysical properties of liquid water and its vapor at near critical conditions of psat = 219 bar andTsat = 646.15K as in [46, 24]. The non-dimensional variables are based on the capillary length com-puted with constant thermo-physical properties [1]. A film of initial thickness δ = 0.15 is placed insidethe unit-cube computational domain shown in Fig. 5(a). The 3D computational domain is uniformly dis-cretized using a grid size of 1283 and the simulations run with a uniform time step ∆t = 10−5. Periodicboundary conditions are imposed in the wall-parallel plane. The bottom boundary is a no-slip wall atsuperheated temperature ∆Tsup = 25K and corresponding dimensionless temperature θw = 1, whereasthe top boundary is a zero gradient/open boundary. Initially, liquid is at saturated condition, θ = 0, anda linear temperature profile is initialized in the thin layer of vapor at the superheated bottom wall. Theinitial velocity field is set to zero in the whole domain. Continuous heat transfer takes place betweenthe superheated bottom wall and the adjacent quiescent vapor phase so that a thermal boundary layerdevelops inside the vapor region which drives the plane interface upwards due to heat and mass transfernormal to the interface, see Fig. 5(b).

The transient evolution of the plane interface thickness and the corresponding interface normal ve-locity obtained from the present 3D MICLS-GFM based phase change are reported in Fig. 6(a) and(b). Both the interface film thickness and normal velocity display close agreement with the analyticalsolution. Concluding, the present GFM based interface heat and mass transfer model can be used tocapture the flow physics and heat transfer rate for 3D boiling flows.

4. Simulations of three-dimensional boiling flows

The GFM based phase change model considered in the present numerical study is further validatedusing 3D boiling flows such as saturated film boiling over a horizontal plane surface [1] and subcooledboiling of a spherical vapor bubble inside a quiescent liquid column [70].

4.1. Saturated 3D film boilingEven though the surface heat transfer rate obtained from nucleate boiling is higher and most

favourable for industrial applications, studies of film boiling are carried out in the literature mostlyto understand the accidental situations encountered in industrial reactors and boilers [1]. Here, we simu-late a single-mode (one bubble dynamics) saturated film boiling of water over a horizontal plane surface.A thin layer of vapor film completely covers a superheated flat surface and the mass transfer takes place

14

(a) Initial plane interface at t=0 (b) Plane interface at t=50

Figure 5: Visualization of the interface at different instants from the numerical solution of the 1D Stefan problem withnon-uniform phase change.

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 10 20 30 40 50 60

Dim

ensio

nle

ss inte

rface thic

kness

Non-dimensional time

AnalyticalMICLS-GFM

(a) Plane interface thickness

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0 5 10 15 20 25 30 35 40 45 50

Dim

ensio

nle

ss inte

rface v

elo

city

Non-dimensional time

AnalyticalMICLS-GFM

(b) Interface normal velocity

Figure 6: The transient evolution of the interface thickness and velocity field for the 1D Stefan problem.

15

(a) (b)

Figure 7: Initial condition for the simulation of 3D film boiling. (a) Thin vapor film thickness and (b) Linear temperatureprofile.

at the liquid-vapor interface, see initial configuration in figure 7. This thin vapor layer acts as a blan-keted insulating layer between the heated surface and the quiescent saturated liquid. As above, periodicboundary conditions are imposed in the wall-parallel directions whereas the bottom boundary is a no-slip wall with uniform superheated temperature Tw = Tsat + ∆Tsup and corresponding non-dimensionaltemperature of θsup = 1. At the top boundary, we impose outflow conditions with zero temperaturegradient. Due to the evaporation of the liquid, the liquid-vapor interface becomes unstable and a bubbleis generated. The spherical vapor bubble grows in size, detaches from the interface and rises upwardsdue to buoyancy.

In our numerical simulations, constant thermophysical properties are assumed for water and its vaporat saturation conditions psat = 219 bar and Tsat = 646.15K. The superheated temperature of the bottomplate, ∆Tsup = 10K is used similar to numerical studies of 2D film boiling in [37, 38, 43, 46, 23, 24, 25]and 3D film boiling in [1, 41, 45]. For the results presented here, the characteristic reference quantities arebased on the thermophysical properties of the working fluids [38, 1, 43]. Based on dimensional analysis

[1, 38], the capillary length scale is λo =√

σg(ρl−ρv) , the time scale to =

√λo

g and the velocity scale

Uo =

√λog. Rayleigh-Taylor instability (RTI) arises due to the presence of a high-density liquid phase

over the low-density vapor phase, whose critical length-scale is expressed in terms of the capillary lengthas λc = 2πλo. In 2D and 3D film boiling flows, the most dangerous unstable wavelength is related to thecritical wavelength, λd2 =

√3λc and λd3 =

√2λd2, respectively. The size of the computational domain

for film boiling problems is therefore typically based on this most dangerous Taylor wavelength [38, 43].As in [1], the rectangular computational domain we use has size 1 × 1 × 2 in terms of the 3D mostdangerous Taylor wavelength, λd3. Moreover, initially, the unstable thin vapor layer has a perturbedthickness, δ(X,Y )Z

δ(X,Y )Z = δo + aw

[cos

(2πNX

L

)+ cos

(2πNY

W

)], (38)

where, δo = 0.125L is the initial unperturbed vapor film thickness, aw = −0.05 is the perturbation waveamplitude and N = 1 is the perturbation wave number. A linear temperature profile is initialised insidethe superheated vapor film. The bulk liquid phase is at saturation temperature, θl = θsat = 0. Theperturbed vapor film thickness, δ(X,Y )Z and initial linear temperature profile, θ are displayed in Fig. 7.

For boiling flows, the grid resolution and time step are dependent on the operating conditions of theworking fluids [1, 23]. In this study, the rectangular domain is discreatised using a uniform grid size with32 × 32 × 64, 64 × 64 × 128, 96 × 96 × 192 and 128 × 128 × 256 cells. The time step, ∆t is calculatedwith a constant CFL number, c = 0.001 and uniform wall superheat, ∆Tsup = 10K. The instantaneousliquid-vapor interface morphology at non-dimensional time, t = 34.77 is shown in Fig. 8 for the variousgrid resolution considered. The interface represented by the Heaviside function value H(φ) = 0.5, shows

16

Figure 8: Interface morphology obtained at non-dimensional time, t = 34.77 from the grid sensitivity study of 3D saturatedfilm boiling.

smaller deviations at the two highest grid resolutions considered 96× 96× 192 and 128× 128× 256. Themesh domain with 128× 128× 256 grid cells is therefore adopted for the following simulations.

In computations, the selected time step must satisfy the CFL condition as well as the capillaryconstraint [46]. We therefore examine the convergence for various CFL numbers, c = 0.01, 0.004, 0.002and 0.001 for the uniform grid of size 128 × 128 × 256. Figure 9 displays the bubble interface obtainedwith the different values of the CFL number. The differences are negligible for CFL numbers c = 0.002and c = 0.001. For the next simulations, we therefore choose a CFL number equal to 0.001.

Next, we investigate saturated film boiling of water and its vapor at saturation condition psat =219 bar and Tsat = 646.15K with a wall superheat, ∆Tsup = 10K, corresponding to a vapor phaseJacob number, Jav = 12.735. The transient evolution of the vapour film thickness during the formationof the first bubble is shown in Figure 10. The thin perturbed vapor layer gradually grows into a largespherical bubble, which detaches and rises upwards due to buoyancy. The formation of a stable vaporcolumn is observed at ∆Tsup = 10K, with the periodic bubble release cycle occurring at the tip/end ofthe vapor column. The formation of similar stable vapor columns has been observed in the experimentalstudy [92] and in several 2D numerical studies [43, 46, 35, 24, 93]. The non-dimensional temperaturedistribution at the centre xz-plane is shown in Fig. 11 for the same test case with time step as in Figure10. The temperature distribution inside the vapor phase is superheated whereas the surrounding liquidphase is at the saturation condition. The temperature field displays variations corresponding to theperiodic bubble release cycle.

For the heat transfer analysis, we compute the variations of local Nusselt number, Nux,y, a spaceaveraged Nusselt number, NuA and the time averaged Nusselt number, Nut over the superheated

17

Figure 9: Interface morphology obtained at non-dimensional time, t = 34.77 from the time step convergence study of 3Dsaturated film boiling.

surface[37, 24] as

Nux,y =λo

∆Tsup

∂T

∂z

∣∣∣∣∣w

=λo

λd3

∂θ

∂Z, (39)

NuA =1

A

∫ A

0

Nux,ydA, (40)

Nut =1

t

∫ t

0

NuAdt. (41)

In laminar film-boiling studies, the semi-empirical correlations of Berenson [94] for NuB and Klimenko[95] for NuK are usually considered [45],

NuB = 0.425

(GrvPrvJav

) 14

, (42)

NuK = 0.19 (GrvPrv)13 . (43)

For the conditions of our film boiling simulations, the averaged Nusselt numbers calculated from thecorrelations of Berenson [94], NuB and Klimenko [95], NuK are 3.696 and 7.935, respectively. Thetransient variations of the space averaged Nusselt number NuA and the time averaged Nusselt numberNut obtained from the present 3D film boiling are therefore compared with the semi-empirical correlationof Berenson [94] in Fig. 12. This result indicates that the space averaged Nusselt number fluctuates overthe horizontal plane superheated surface due to variations in the vapor film thickness, with the peakof the space averaged Nusselt number is corresponding to lower thickness of the vapor film. The timeaveraged Nusselt number, Nut obatined from the MICLS-GFM method is Nut = 3.9 which slightlydeviates from the semi-empirical correlation of Berenson [96] by 5.4%.

4.2. Subcooled boiling of a spherical vapor bubbleThe mass conservative LS method is further used to simulate 3D subcooled pool boiling flow for high

density and viscosity ratios. In subcooled boiling, a saturated spherical vapor bubble is placed inside a18

(a) 17.58 (b) 23.44 (c) 29.3 (d) 31.25

(e) 33.2 (f) 35.16 (g) 35.94 (h) 36.33

(i) 37.12 (j) 38.29 (k) 39.06 (l) 46.87

Figure 10: Transient evolution of the interface morphologies for the 3D film boiling of water and its vapor at psat = 219 barand Tsat = 646.15K using an uniform wall superheat of ∆Tsup = 10K, with corresponding Jacob number, Jav = 12.735.

19

(a) 17.58 (b) 23.44 (c) 29.3 (d) 31.25

(e) 33.2 (f) 35.16 (g) 35.94 (h) 36.33

(i) 37.12 (j) 38.29 (k) 39.06 (l) 46.87

Figure 11: Transient distribution of the non-dimensional temperature during 3D film boiling of water and its vapor atpsat = 219 bar and Tsat = 646.15K using a uniform wall superheat of ∆Tsup = 10K, corresponding to a Jacob number,Jav = 12.735.

20

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

0 50 100 150 200 250 300 350 400 450

Ave

rag

ed

Nu

sse

lt n

um

be

r, N

u

Non-dimensional time

NuA at 10K

Nut at 10K

NuB at 10K

Figure 12: Space and time averaged Nusselt numbers obtained from 3D film boiling of water and its vapor at wall superheatof 10K. NuB is the prediction from the semi-empirical correlation of Berenson [96], whereas the space and time averagedNuA and Nut are defined in the text.

pool of quiescent subcooled liquid water. The water and its vapor are studied at two different operatingconditions of saturated pressure and temperature of psat = 1 bar, Tsat = 373K and psat = 30 bar,Tsat = 506.7K for three liquid subcooled temperatures, ∆Tsub of 5K, 12K and 30K. The evolution ofthe condensing bubble volume and its relative or dispersed velocity are examined below.

Following the experimental study of Kim and Park [65] and numerical study of Bahreini et al. [70],we consider a spherical vapor bubble of diameter, D = 4.7mm and is equivalent to the initial Sautermean diameter (SMD). The non-dimensional unit diameter of the initial bubble is considered using theinitial reference diameter, D. The bubble size reduces due to the subcooled surrounding liquid.

We examine different grid resolutions, ∆h = d(0)/Nd = 1/16, 1/24, 1/32, 1/48 and 1/64, whileneglecting the acceleration due to gravity. The time step, ∆t is calculated at uniform CFL number,c = 0.001. A static spherical bubble is initially placed inside the non-dimensional cubic computationaldomain of size, 2d(0) × 2d(0) × 2d(0). The time history of the non-dimensional bubble radius, r(t)obtained at various grid resolutions is shown in Fig. 13 (a). From these results it is observed that thecondensation of the static subcooled sphreical bubble is significantly affected by the grid resolution;in particular, the condensation rate is overpredicted by a coarse grid with insufficient grid resolution.The ratio of surface area to condensing bubble volume, ab/vb is higher for a small bubble. Hence, thesmall bubble lead to very fast rate of condensation when the bubble diameter becomes close to twoor three grid sizes, ∆h [71]. Therefore, the bubble radius reduces faster when using the coarser grid.The steep gradients of the interface heat and mass flux become inaccurate, which needs to be addressedwith adequate grid refinement. Figure 13 (b) shows the variation of the bubble radius at different gridresolutions at fixed dimensionless times, t = 228.415 and 456.83, it indicate that the results tend toconverge at the highest resolutions considered. Thus, to obtain accurate flow and heat transfer data at areasonable computational cost, the resolution of 48 cells per bubble diameter is used for the simulationsof subcooled boiling flow.

The effect of the buoyancy-driven rising motion and condensation of bubbles inside a quiescent sub-cooled liquid is important for the cooling of condensers, boilers and nuclear reactors [97]. We thereforeexamine the potentiality of our method for the case of subcooled pool boiling of buoyancy-driven upwardrising bubbles in water at saturation conditions psat = 1 atm and Tsat = 373K with subcooled tem-perature, ∆Tsub = 30K. The time history of the condensing bubble radius, r(t) and the correspondingnon-dimensional dispersed bubble velocity, ud are shown in Fig. 14 (a) and (b) for a case without andwith gravity. As expected, the data show that the condensation of the bubble is faster in the presence ofgravity, as the increased magnitude of dispersed velocity field. Owing to the buoyancy-driven motion, thebubble undergoes larger deformations, which also further enhance the condensation, as clearly observedin Fig. 14 (a).

Next, we consider the effect of temperature and simulate a system of saturated water and its vaporphase at psat = 1 atm and Tsat = 373K, for the subcooled temperatures, ∆Tsub of 5K, 12K and 30K, asshown in Fig. 15. A high pressure case, with saturation conditions psat = 30 atm, and Tsat = 508.15Kat various subcooled temperatures, ∆Tsub of 5K, 12K and 30K, is shown in Fig. 16 (a) and (b).The condensation rate of the spherical bubble increases with increasing subcooling of the bulk liquid.

21

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

0 50 100 150 200 250 300 350 400 450 500

Dim

en

sio

nle

ss b

ub

ble

ra

diu

s,

r

Non-dimensional time

d(0)/24d(0)/32d(0)/48d(0)/64

(a) Time history of bubble radius

0.6

0.65

0.7

0.75

0.8

0.85

0.9

20 25 30 35 40 45 50 55 60 65

No

n-d

ime

nsio

na

l B

ub

ble

ra

diu

s,

r

Grid size per unit bubble diameter, Nd

At NDT=228.415At NDT=456.83

(b) Convergence at fixed time

Figure 13: Transient evolution of a spherical bubble radius in subcooled boiling at various grids resolution.

0

0.1

0.2

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0.4

0.5

0.6

0.7

0.8

0.9

1

0 200 400 600 800 1000 1200 1400 1600

Dim

en

sio

nle

ss b

ub

ble

ra

diu

s,

r

Non-dimensional time

With gravityWithout gravity

(a) Bubble radius

0

0.5

1

1.5

2

2.5

3

0 200 400 600 800 1000 1200 1400 1600

Dim

en

sio

nle

ss d

isp

ers

ed

ve

locity,

ud

Dimensionless time, t

With gravityWithout gravity

(b) Bubble dispersed velocity

Figure 14: Subcooled boiling of saturated water at psat = 1 atm and Tsat = 373K with subcooled temperature, ∆Tsub =30K without and with gravity

The bubble dispersed velocity magnitude decreases when increasing the subcooling of the water. Dueto change in thermophysical properties of water at different saturation conditions, the condensationrate decreases when increasing the saturation pressure of water and is known to significantly affect thepool-boiling flow.

Finally, we compare the dimensionless bubble collapsing time at different liquid subcooled temper-atures of water at different saturation pressure conditions in Fig. 17. The non-dimensional bubblecollapsing time for the saturation condition psat = 30 bar is higher than at psat = 1 bar as alreadynoted. It is also observed that at fixed operating conditions, the non-dimensional bubble collapsing timedecreases when increasing the liquid subcooling, ∆Tsub.

5. Conclusions

In this numerical study, we present a mass-conserving interface-correction level-set (MICLS) al-gorithm, based on the ghost-fluid method (GFM), to perform direct numerical simulations of three-dimensional boiling flows. For the velocity-pressure coupling, a projection method with fast pressurePoisson solver is employed. The interface jump conditions due to phase change are modeled by a second-order accurate ghost extrapolation approach. The divergence condition at the interface (non-zero forphase change) and corresponding normal velocity is computed using a ghost-velocity extension approach.The present GFM-based phase-change model is extensively evaluated based on both qualitative and quan-titative analyses. Validations of the advection algorithm, ghost velocity extension, ghost temperatureand ghost interface mass flux extrapolation, and different phase-change benchmark cases are presented.The overall results are summarized below.

22

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 200 400 600 800 1000 1200 1400 1600 1800

Dim

en

sio

nle

ss b

ub

ble

ra

diu

s,

r

Non-dimensional time, NDT

5K12K30K

(a) Bubble radius

0

0.5

1

1.5

2

2.5

3

0 200 400 600 800 1000 1200 1400 1600 1800

Dim

en

sio

nle

ss d

isp

ers

ed

ve

locity,

ud

Non-dimensional time, NDT

5K12K30K

(b) Bubble dispersed velocity

Figure 15: Subcooled boiling of water at psat = 1 atm and Tsat = 373K at different subcooling temperatures, ∆Tsub

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Dim

en

sio

nle

ss b

ub

ble

ra

diu

s,

r

Non-dimensional time,NDT

5K12K30K

(a) Bubble radius

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Dim

en

sio

nle

ss d

isp

ers

ed

ve

locity,

ud

Non-dimensionless time, NDT

5K12K30K

(b) Bubble dispersed velocity

Figure 16: Subcooled boiling of water at psat = 30 atm and Tsat = 508.15K at different subcooling temperatures, ∆Tsub

1000

1500

2000

2500

3000

3500

4000

4500

5 10 15 20 25 30

No

n-d

ime

nsio

na

l b

ub

ble

co

llap

sin

g t

ime

Liquid Subcooled temperature, in K

psat = 1 barpsat = 30 bar

Figure 17: The spherical bubble collapsing time at various liquid subcooled temperatures for 3D subcooled boiling of waterand its vapor at various pressure conditions, psat of 1bar and 30bar

.

23

1. To evaluate the modified LS advection, re-distancing and curvature-dependent mass correctionfor phase change problems we consider the expansion of a spherical bubble at uniform rate. Wecompare the evolution of the dimensionless bubble diameter and absolute error or L1 error normsobtained using a classical LS method, the MICLS with exact reference volume and the MICLSwith modified reference volume with the analytical solution, reporting close agreement with theanalytical solution. The modified MICLS accurately captures the bubble dynamics even undercoarse grid resolutions, thus offering lower computational costs than the classical LS method.

2. We validate the MICLS with the projection method for velocity-pressure coupling, with GFM sur-face tension and the ghost velocity extension across the interface for the case of uniform phasechange. On each cell, a momentum equation for the fluid velocity and ghost fluid velocity compo-nents are independently solved with a continuous jump condition. Comparing with the analyticalsolution, we conclude that a minimum dimensionless grid size of 32 cells per unit diameter isnecessary to correctly capture the interface morphology and flow physics.

3. To validate the computation of the non-uniform interface normal velocity due to heat and massfluxes across the interface, a one-dimensional Stefan problem is considered, for which an analyticalsolution exists. Comparisons are presented for the non-dimensional plane interface thickness andinterface non-uniform velocity.

4. To further validate the MICLS-GFM method for 3D boiling flows, we present results for 3D single-mode saturated film boiling of water and its vapor at near-critical conditions of saturated pressure,psat = 219 bar and saturated temperature, Tsat = 646.15K over a horizontal plane heated surfaceat wall superheat of ∆Tsup = 10K, with corresponding vapor Jacob number, Jav = 12.375. Thedimensionless interface unstable film thickness, non-dimensional temperature distribution, space-averaged Nusselt number and time-averaged wall Nusselt number show trends similar to 2D and 3Dfilm boiling studies in the literature. The time averaged Nusselt number obatined with the presentMICLS-GFM method is found to deviate by 5.4% with respect to the value from the semi-empiricalcorrelation of Berenson [96].

5. To evaluate the present MICLS-GFM model for cases with high density and viscosity ratios, weconsider water and its vapor at saturation conditions of psat = 1 bar and psat = 30 bar. Wetherefore simulate 3D condensation of a saturated spherical water vapor bubble placed inside apool of subcooled liquid water. The rate of bubble condensation is evaluated with and withoutgravity at different subcooling temperatures. The time history of the bubble dynamics is computedbased on the non-dimensional bubble shrinking diameter. We observe that

• the subcooled boiling shows lower rate of bubble condensation and lower magnitudes of bubbledispersed velocity occurs in the absence of gravity;

• the increase of liquid subcoolng increases the bubble condensation rate due to higher massflux, and leads to faster bubble collapsing with lower magnitudes of dispersed bubble velocity.

• The interface mass flux is depends on the thermophysical properties and lantent heat. Theratio of thermophysical properties and latent heat of vaporization of the water and its vaporat saturation condition of psat = 1bar is higher as compared to saturation condition of psat =30bar. Hence, the bubble condensation rate is higher for the water at saturation conditionof psat = 1bar. However, the magnitudes of the condensing bubble dimensionless dispersedvelocity is lower for water and its vapor at saturation condition of psat = 1bar.

• Finally, the non-dimensional bubble collapsing time for the saturation condition of psat =30 bar is higher due to the lower rate of bubble condensation. At fixed operating conditions,the non-dimensional bubble collapsing time decreases by increasing in the liquid subcooling,∆Tsub.

Acknowledgements

This research is supported by the Linné Flow Centre at KTH Mechanics, Stockholm, Sweden. Thecomputing time is provided by SNIC (Swedish National Infrastructure for Computing), in particularresources at PDC, KTH, Stockholm, Sweden.

24

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