Journal of Scientific Computing, Vol. 19, Nos. 1–3, December 2003 (© 2003)
Computations of Explosive Boiling in Microgravity Asghar Esmaeeli 1 , 2 and Grétar Tryggvason 1 Received May 31, 2002; accepted October 31, 2002 Dynamics of the explosive growth of a vapor bubble in microgravity is investigated by direct numerical simulation. A front tracking/finite difference technique is used to solve for the velocity and the temperature field in both phases and to account for inertia, viscosity, and surface deformation. The method is validated by comparison of the numerical results with the available analytical formulations such as the evaporation of a one-dimensional liquid/vapor interface, frequency of oscillations of capillary waves, and other numerical results. Evolution of a three-dimensional vapor nucleus during explosive boiling is followed and a fine scale structure similar to experimental results is observed. Two-dimensional simulations yield a similar qualitative instability growth. KEY WORDS: Unstable boiling; front tracking; microgravity; liquid/vapor phase change.
1. INTRODUCTION Boiling flows are central to many natural and industrial processes. Heat transfer through boiling is the preferred mode in most power plants and in many engineering applications such as metal processing, chemical, petroleum, and nuclear industries the thermal-hydraulics of boiling flows is the basis for the accurate design and safe operation of these systems. Similarly, boiling is fundamental in microgravity applications such as design of energy generation systems of spacecrafts. In regular (i.e., heterogeneous) boiling, evaporation usually starts at a solid wall where the wall temperature is higher than the saturation temperature of the liquid. As a result, vapor bubbles are formed which grow and subsequently detach from the surface. While the bubble generation depends on the degree of the superheat; the difference between wall temperature and the saturation temperature, the vapor bubble formation and motion in turn affect the heat transfer rate. If the bulk of the liquid is subcooled, the motion of the bubbles is not affected by the heat flow and is controlled only by the liquid inertia. However, if the liquid is superheated, the bubble dynamics depends on the combined effect of heat transfer and hydrodynamics. Although a superheated liquid is thermodynamically at a 1
Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, Massachusetts 01609. 2 To whom correspondence should be addressed. E-mail:
[email protected]
163 0885-7474/03/1200-0163/0 © 2003 Plenum Publishing Corporation
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metastable state, the growth process may still be hydrodynamically stable if the degree of superheat is low enough. In that case, the bubble starts to grow as a result of evaporation of liquid at the phase boundary and the phase boundary remains smooth during the evaporation. Here, the bubble growth goes through three stages. In the first stage, surface tension is dominant and the bubble grows from a critical radius. This stage is followed by an inertia-controlled stage during which the bubble grows at a constant rate. In the final stage, growth is mainly controlled by heat transfer in which the radius grows as `t. The growth of a vapor bubble in superheated liquid has been extensively studied in the past. The inertia-dominated growth was studied by Lord Rayleigh (1917) who solved for the momentum equation of the ambient fluid in the absence of viscosity and surface tension. Plesset and Zwick (1954) and Forster and Zuber (1954) solved for the heat transfer-dominated stage using Rayleigh’s equation in conjunction with the energy equation to incorporate the variation of bubble pressure with its temperature. Their solution, however, was only valid for high Jacob numbers, i.e., when the thermal boundary layer around the bubble is thin compared with the bubble radius. This restriction was eliminated by Scriven (1959) who developed a solution which is valid for entire range of Jacob numbers. Mikic et al. (1970) derived a general relationship for spherical bubble growth in a uniformly superheated liquid which would tend to the limit set by Rayleigh and Plesset and Zwick. Experimental studies by Dergarabedian (1960), Kosky (1968), and Florschuetz et al. (1969), among others, have supported the asymptotic solutions developed by Plesset and Zwick and Scriven. A liquid cannot be superheated indefinitely and there is a temperature limit beyond which it starts to evaporate. This temperature is called the ‘‘superheat limit’’ Tsl and is about ten percent less than the critical temperature for most substances. At a given pressure, liquid is mechanically stable for temperatures less than the saturation temperature and is mechanically metastable for temperatures between the saturation temperature and the superheat limit. Rapid evaporation occurs when a liquid boils near its superheat limit. Boiling at supreheat limit can be violent and explosive. Explosive boiling is encountered, for example, if a cold and volatile liquid is brought into contact with a relatively non-volatile and hot liquid, or if the system pressure is suddenly decreased. This results in a large vaporization rate and destructive shock waves. An extensive summary of the occurrence of accidental vapor explosions in a variety of industries is found in Reid (1983). Vapor explosions are also a concern in microgravity where energy generation for spacecraft depends on the storage of low boiling point cryogenic fluids. Both the superheat limit and saturation temperature depend on the system pressure and increase with increase in the system pressure. The rate of increase in the saturation temperature, however, is faster compared with that of the superheat limit. This results in a decrease in the degree of superheat Tsl − Tsat with increase in the system pressure. Thus, explosive boiling can be avoided at high enough system pressures. In a series of experiments Sturtevant and collaborators [Shepherd and Sturtevant (1982), Frost and Sturtevant (1986), and Frost (1988)] studied evaporation of single droplets where heterogeneous boiling was suppressed by immersing the droplets in a hot, non-volatile liquid. They showed that at high enough pressures,
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the surface of the bubble remained smooth and the evaporation process was stable. As they decreased the ambient pressure, the boiling process first passed through a transitional state and eventually became unstable. The unstable boiling resulted in the development of wrinkles on the bubble surface (which grew with time) and dramatic increase in heat transfer and the evaporation rate. They attributed this high evaporation rate to the substantial increase in bubble surface area. Other features included bulging of vapor into the liquid and eventual detachment of protrusions from the bubble. Moreover, as a result of roughness developed on the bubble surface, the bubble appeared opaque in the photograph. This is the reason that some researchers refer to explosive bubbles as ‘‘black bubbles.’’ Sturtevant and collaborators proposed that the Landau instability, originally described in connection with the instability of laminar flames, applies to rapid evaporation in the superheat limit. Ervin et al. (1992) conducted microgravity experiments in a NASA drop tower. They studied the phenomenon by superheating a bulk of liquid and made similar observations as those of Sturtevant and collaborators. They also showed that the temperature distribution in the heated liquid characterizes the bubble dynamics and boiling regimes. There have been several attempts in the past three decades to compute liquid/vapor phase change numerically. However, these computations have, until recently relied on a number of simplifications. Examples of such computations can be found in Theofanous et al. (1969), Dalle Donne and Ferranti (1975), Lee and Nydahl (1989), and Patil and Prusa (1991) for the growth of a vapor bubble in superheated liquid. More advanced computations started with Welch (1995), who simulated a fully deformable, two-dimensional bubble using moving triangular grids. He was, however, only able to follow the bubble for a relatively short time due to the distortion of the grids. The limitation to modest deformation of the phase boundary was overcome by Juric and Tryggvason (1998) who developed a two-dimensional front-tracking method to study film boiling. Similarly, Son and Dhir (1998) and Welch and Wilson (2000) were able to resolve this issue using a level set and Volume-Of-Fluid method in their film boiling simulations. Numerical simulations using other techniques include the work of Legendre et al. (1998) who used a boundary-fitted coordinate to study evolution of an axisymmetric spherical bubble in a channel and the papers by Yoon et al. (1999), (2001) who studied bubble growth, departure, and rise in nucleate pool boiling using a mesh-free numerical method (MPS-MAFL). All the previous computations, however, were two-dimensional. Three-dimensional simulations of fully-deformable vapor/liquid phase change include film boiling computations of Esmaeeli and Tryggvason (2001) using a front tracking method and Shin and Juric (2002) using a front tracking/ level contour technique. Rapid evaporation was studied numerically by Juric (1997), using a front tracking/finite difference technique, who investigated the effect of the Jacob number on the instability and growth of a two-dimensional bubble. He showed that as the Jacob number is increased the growth is more rapid and more unstable. Here, we are concerned with fluid flow and heat transfer during unstable growth of a vapor bubble. Our aim is to study some of the aspects of the problem which have not been studied by previous investigators.
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2. FORMULATION AND NUMERICAL METHOD Consider a domain consisting of a liquid and its vapor undergoing a phase change. The material properties of the phases are different but constant within each phase. The governing equations are conservation of mass, momentum, and energy for each phase and jump conditions across the interface. Rather than writing the governing equations separately for each of the fluids along with the jump conditions at the interface, a ‘‘one-field’’ formulation is used which is valid for the entire flow field and takes the jump in properties across the interface into account. Away from the phase boundary, the one-field equations lead to the usual governing differential equations in each phase and integrating these equations over an infinitesimal volume (area in two dimensions) moving with the interface results in the proper jump conditions. The reader is referred to, for example, Peskin and Printz (1993) and Chang et al. (1996) for a mathematical justification of this approach in the absence of phase change. We assume that both the liquid and the vapor are incompressible and the only change of volume (area) is due to the phase change at the phase boundary. With these assumptions, the momentum equation in conservative form, valid for the entire flow field, is: “r u +N · ru u=−Np+N · m(Nu+Nu T)+s F of nf d(x − xf ) dAf . “t F
(1)
Here, u is the velocity, p is the pressure, and r and m are the discontinuous density and viscosity fields, respectively. d is a three- or two-dimensional delta-function constructed by repeated multiplication of one-dimensional delta-functions, and dAf is a volume (area) element of the interface. s is the surface tension, o is twice the mean curvature in three-dimensions and curvature in two-dimensions and n is a unit vector normal to the front and toward the vapor. Formally, the integral is over the entire front, thereby adding the delta functions together to create a force that is concentrated at the interface, but smooth along the front. x is the point at which the equation is evaluated and xf is the position of the front. The variables with subscript f are evaluated at xf . The thermal energy equation, in conservative form, valid for the entire domain, is “r cT +N · rcuT=N · k NT+K F q˙f d(x − xf ) dAf . “t F
(2)
Here, T is the temperature, c is the heat capacity, k is the heat conductivity, K=1 − (cv − cl ) Tsat /hfg , and q˙f =kl
:
:
“T “T − kv . “n l “n v
(3)
K is a constant which modifies the latent heat hfg as a result of unequal specific heats. The temperature at the phase boundary must be specified, and we assume a temperature equilibrium such that the temperature is continuous across the phase boundary. The temperature of the phase boundary can be found using GibbsThompson equation by assuming thermodynamic equilibrium. The Gibbs-Thompson
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equation contains terms that lead to corrections of the saturation temperature due to curvature, uneven heat capacities, and local kinetics. However, as shown by Juric and Tryggvason (1998), the corrections are usually small since length-scales resulting from the flow are considerably larger than those resulting from the thermodynamic conditions. We therefore take the temperature at the phase boundary to be equal to the saturation temperature at the system pressure Tf =Tsat (psys ).
(4)
The major new aspect of computations of a fluid undergoing phase change is the local expansion at the phase boundary. The liquid and the vapor have been taken to be incompressible and to find the divergence of the total velocity field at the phase boundary, we write u=uv H+ul (1 − H). Here, H is a Heaviside function which is one in the vapor and zero in the liquid and it is assumed that the velocity in each phase has a smooth incompressible extension into the other phase. Taking the divergence of u and using N · uv =N · ul =0, yields: N · u=F (uv − ul ) · nf d(x − xf ) dAf .
(5)
F
To relate the difference between the velocity of the liquid and the velocity of the vapor to the evaporation rate and the velocity of the phase boundary, the normal velocity of the phase boundary is denoted by Vn , the normal velocity of the liquid next to the boundary by ul =ul · n and the velocity of the vapor by uv =uv · n. Since there is a change of phase at the interface, these velocities are all unequal. If the liquid is evaporating, Vn is smaller than ul and if the density of the vapor is much lower than that of the liquid, uv is much larger than ul . The rate of evaporation of liquid is equal to the difference in the velocity of the phase boundary and the liquid velocity, times the density of the liquid rl (ul − Vn ). Similarly, the rate of production of vapor is equal to the difference in the velocity of the phase boundary and the vapor velocity, times the density of the vapor, rv (uv − Vn ). Since mass is conserved, these two are equal, and the mass transfer rate at the phase boundary is m ˙ f =rl (ul − Vn )=rv (uv − Vn ).
(6)
The volume expansion per unit interface area is found by eliminating Vn uv − ul =m ˙f
1 r1 − r1 2 , v
(7)
l
where m ˙ f is found using the energy jump condition at the interface; m ˙ f =q˙f /hfg . Inserting the expression for the velocity difference across the phase boundary from Eq. (7) into Eq. (5) yields
1
2
1 1 1 − F q˙ d(x − xf ) dAf . N · u= hfg rv rl F f
(8)
The normal velocity of the phase boundary is easily found to be
1
1 q˙ 1 1 Vn = (ul +uv )+ f + 2 2hfg rl rv
2,
(9)
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and the new position of the front points can be found by integrating dxf =Vn nf . dt
(10)
The governing equations are solved using a front tracking/finite difference method similar to the one developed by Unverdi and Tryggvason (1992) for isothermal multifluid flows. The implementation of phase change and heat transfer to the fluid dynamics problem, however, requires extensive reformulation of the original method. Initially, the velocity and temperature fields are set and fluid property fields are found using the known position of the front. q˙f is found using two normal probes which originate at the phase boundary and extend a distance D into the vapor and the liquid. Numerical experiments showed that the results are insensitive to the length of normal probes as long as h [ D [ 2h, h being the grid spacing. The source term in the energy equation is computed and distributed to the grid points using Peskin’s distribution function (1977). Using the same interpolation function, the fluid velocity at the interface (first term in Eq. (9)) is found and added to the velocity due to phase change (second term in Eq. (9)). This results in the normal interface velocity Vn which is used to advect the front points to their new positions using simple Euler integration of Eq. (10). This advection of the points enables us to compute property fields such as r and c at the next time step. This is required due to the use of the conservative form of the momentum and the energy equation. It is also a necessary step for implicit computation of surface tension force which eliminates a capillary time step constraint. The energy equation is integrated using a second order predictor-corrector time integration and a second order space discretization, using centered differences on a fixed, staggered grids. Similarly, the Navier–Stokes equation is solved using a second order projection method. To do this, the surface tension is computed at the new position of the front and is distributed to the grid by Peskin’s distribution function. The momentum equation is integrated by dropping the pressure term to find a provisional velocity field u˜ and an elliptic equation for pressure is derived and solved by a multigrid method. The provisional velocity field is then corrected by the pressure to find the real velocity at the next time step. As the front deforms, surface markers are dynamically added and deleted. For a detailed description of the method for isothermal multifluids see Tryggvason et al. (2001). We have made three assumptions that simplify the simulation: (i) The flow is assumed to be incompressible except at the interface where we allow for volume expansion. While it is likely that acoustic waves may influence the instabilities, the agreement between our results and the experimental observations suggests that the incompressible model captures the essential aspects of the dynamics reasonably well. (ii) The interface temperature is assumed to be at the saturation temperature of the system pressure. Although this assumptions is a good approximation for most boiling systems [see, for example, Juric and Tryggvason (1998), and Son and Dhir (1998)], it may not be the case in explosive boiling. We note, for example, that Frost (1988) reported that in his experiments the interface temperature was midway between the saturation temperature and the superheat temperature. This needs to
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be investigated further. (iii) The surface tension is assumed to be constant. In reality, surface tension will be affected by temperature gradient, contaminants, and other effects such as electrostatic forces. The implementation of these effects in our code is a relatively simple task and while we have studied these effects in the absence of boiling, here we have elected to make the analysis simpler. The results depend on rl , rv , ml , mv , kl , kv , cl , cv , hfg , s, and DT=T. − Tsat where T. is the superheat temperature. This problem does not have a natural length scale and we define a capillary length scale as ls =DTscl /rl h 2fg . Similarly, a time scale is defined as ts =l 2s rl cl /kl and a velocity scale is found using the time- and the length-scale; us =ls /ts . When we present our results, time, length, velocity, and temperature are nondimensionalized by ts , ls , us , and hfg /cl , respectively. Nondimensionalization leads to rv /rl , mv /ml , kv /kl , cv /cl , Ja=c DT/hfg , Pr=mc/k, and Ca=s/rhfg ls as controlling nondimensional parameters. Here, Ja and Pr, and Ca are the Jacob number, the Prandtl number, and a phase change capillary number defined using the liquid properties.
3. VALIDATION Although the front tracking technique as introduced by Unverdi and Tryggvason (1990) has been tested against various experimental, analytical, and numerical results [see, for example, Esmaeeli and Tryggvason (1998)], the implementation of the phase change adds a new dimension to the problem. In order to validate our method and code we have done a number of new tests. The purpose of these tests were to check the accuracy of the mathematical formulation and numerical implementations. The intention was to test the fluid flow solver, energy solver, and implementation of the singular terms such as surface tension and latent heat. Both two- and three-dimensional codes were tested. • The fluid flow solver and surface tension were tested by comparing the frequency of oscillation of a capillary wave with analytical results. We used a twodimensional 1 × 2 domain and a grid resolution of 64 × 128. The domain was periodic in the horizontal direction, confined by the wall from the bottom, and open at the top. Liquid was placed on top of the vapor. The initial temperature was set to a uniform temperature and the wall temperature was kept fixed at the same temperature. We picked rl =2.5, rv =0.25, s=0.3, ml =10 −4, mv =10 −5. The initial interface position was y=y0 +A0 cos(2px), where y0 =0.5 and A0 =0.01. This results in k=2p as the wavenumber of initial perturbation. For this problem, gravity was set to zero and interface started to oscillate as a result of capillary instability. The solution to this problem in inviscid limit can be found in Lamb (1993) 1 f= 2p
= rsk+r 3
l
.
v
For this set of parameters, the theoretical frequency was 0.8279 and the numerical one was 0.8286. This results in 0.085% relative error.
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• A more rigorous test for this purpose is to measure the linear growth of Rayleigh-Taylor instability and compare with the exact solution. Here, we tested our code against both viscous and inviscid theories. For both cases the cut off wave number is kc =((rl − rv ) g/s) 1/2. While a simple formulation exists for the growth rate of the inviscid instability, ks −r 5rr +r 6 − g(r +r ) 2
nA =(gk) 1/2
l
v
l
v
l
1/2
,
v
analytical expression for viscous growth rate is more involved and exists only for fluids with equal kinematic viscosity [Chandrasekhar (1961)]. For both cases we used the same setup as used for the capillary waves. For the inviscid test, the individual parameters were the same as those in capillary wave problem except g=8. For the viscous test, everything was unchanged except we increased the viscosities by two order of magnitudes. To test the inviscid limit, we needed to find the initial velocity corresponding to the interface perturbation. To do that, we computed the vorticity at the interface using the expression for perturbation velocity at the interface as given by Drazin and Reid (1981). We then distributed this vorticity to the grid using Peskin’s distribution function and solved for the initial velocity field using the vorticity-streamfunction equation. We then measured the amplitude of the interface A(t). If A(t)=A0 exp(nN t), nN being the numerical growth rate, then nN =1/t ln(A(t)/A0 ). nN originally increased (due to the contribution of decaying terms) and then leveled off to a constant value before changing again as a result of nonlinearity. For this set of parameters, nA =3.7512 and nN =3.7333 which results in 0.5% relative error. Similar results were found in the viscous limit. • We compared our code with a few one-dimensional benchmark problems. In these problems phase change takes place at a flat interface between a liquid and a vapor layer in a rectangular domain. Liquid may be at the top or the bottom and at t=0+ the wall is exposed to either a constant heat flux qw or a constant temperature Tw > Tsat . As a result, the liquid starts to evaporate. For these problems it is possible to derive a closed-form solution or to find a numerical solution by solving a simple partial or ordinary differential equation with a moving boundary. Here, we discuss one of these tests in detail. Consider a vapor layer which blankets a solid wall and is below a liquid layer in a rectangular domain. The domain is periodic in the horizontal directions and it is open at the top and confined by a solid wall at the bottom. Initially both vapor and liquid are at saturation temperature Tsat and the flow is quiescent. The temperature of the bottom wall is kept at Tw > Tsat . The system pressure psys is fixed by imposing a constant pressure at the top. Conservation of mass results in “v/“y=0 in both fluids. However, since the lower part is confined by the wall, no-through flow results in vv (t)=0. vl (t) is uniform in the liquid layer and is set by the evaporation rate. Energy analysis for the liquid layer results in T=Tsat and the energy equation is reduced to the following form for the vapor layer: “ 2T “T =av , “t “y 2
0 [ y [ d(t),
(11)
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subject to the following boundary and initial conditions T(d(t), t)=Tsat ,
(12)
T(0, t)=Tw ,
(13)
rv un hfg =−k
“T “y
:
.
(14)
d(t)
Here, d(t) is the position of the interface and un =dd(t)/dt is interface velocity. The above problem can be solved analytically using similarity solution by defining T(y, t)=F(t), where t=y/`t [see, Alexiade and Solomon (1993)]. Thus, the interface position and the temperature in the vapor is d(t)=2l `av t
(15)
−T 2 y 1 Terf(l) 2, erf 1 2 `a t
T(y, t)=Tw +
w
sat
(16)
v
where erf(y) is the error function and l is a solution to the transcendental equation c (T − Tsat ) . l exp(l 2) erf(l)= v w hfg `p
(17)
The liquid velocity ul can be found from the mass jump condition (Eq. (6))
1
ul = 1 −
2
rv un , rl
(18)
where un =l `av /t. Momentum equation for vapor results in pv =const and for the liquid a linear pressure profile pl (y, t)=psys − rl
dvl (y − yl ), dt
yl being the combined height of the liquid and the vapor layers. Thus, psys sets the pressure in the liquid and pressure in the vapor is found using the momentum jump condition at the interface: m ˙ (u¯v − u¯l )=(Tv − Tl ) · n¯+son¯.
(19)
T
Here, T=N · m(Nu¯+Nu¯ ) − Ip is the stress tensor, the first term is the deviatoric part and the second term is the isotropic part, I being the idemfactor. Since the interface remains flat, o=0, and since the fluid velocity is uniform the deviatoric part of the stress tensor is the same for both fluids. Therefore, the momentum jump condition in the normal direction yields pv − pl =−m ˙ 2(1/rv − 1/rl ),
(20)
where, m ˙ is the mass flow rate. Since m ˙ =rl (ul − un )=−rv un , therefore
1
pv − pl =−rv 1 −
2
rv u 2n . rl
(21)
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Thus, regardless of the direction of phase change (m ˙ > 0 or m ˙ < 0), for this onedimensional problem, the pressure is higher in the denser fluid. Finally, the analytical heat flux at the wall qw =−kv “T/“y|w is qw =−kv
−T 2 1 1 Terf(l) . `pa w
sat
v
To simulate this flow, we specify the initial conditions using the analytical solution at time t0 , where t0 is the time at which the computation is started. The interface is initially placed at d(t0 )=2l `av t0 and the initial temperature field is therefore −T 2 y 2 1 Terf(l) erf 1 . 2 `a t
T(y, t)=Tw +
sat
w
v 0
The computations are performed in a 1 × 1 domain with grid resolution of 34 2, 64 , and 128 2. The other parameters are t0 =0.1032, d(t0 )=0.2, Tw =1, Tsat =0, hfg =10, rv =1, and rl =2. Other fluid properties are the same for both fluids; ml =mv =2, kl =kv =2, cl =cv =1. Although the problem is one-dimensional, the computations are carried out in two and three dimensions and one-dimensional results are extracted by averaging the results over the grids. These parameters yield Ja=cv (Tw − Tsat )/hfg =0.1 for which l=0.22 from Eq. (17). In Fig. 1(a) the instantaneous interface locations from the computations at grid resolution of 32 2 is compared with the exact interface location. The exact and computed solutions at 32 2 grid are very close and the results for the finer grids (not shown here) were essentially the same as that of the 32 2 grid, reflecting the fact that the solution was well-converged at 32 2 grid. Figure 1(b) shows a similar comparison between the numerical and the exact liquid velocity. The liquid velocity is initially high because of the steep temperature gradient at the interface which results in a high evaporation rate. However, it slows down as the vapor layer expands and the temperature gradient decreases. In Fig. 1(c) we compare the numerical and the exact temperature field at three different times, 20.78, 201.6, and 310.94 for the 32 2 grid resolution. The difference between the numerical and the exact solution is very small at the early time and at the later times these results are basically indistinguishable. Figure 1(d) compares the numerical and the exact Nusselt number, defined based on the wall heat flux as Nu=−ls “T/“y|w /(Tw − Tsat ), for the 32 2 grid resolution. Comparison of numerical and the exact pressure difference, pl − pv , gave reasonable results as well. Three-dimensional simulations of this problem led to the same results. One of the important issues in any numerical simulation is the convergence of the results under grid refinement. We checked the actual convergence rate of our code by computing the L2 =`; i, j (Tij − Texact ) 2/N 2 norm for the temperature field at an early and a late time. We plotted the L2 norm versus the grid spacing in a log − log scale. The method showed linear convergence. A log − log plot of the error 2
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12
10
2
0.03
32
exact
δ(t)
ul
322
8
exact
0.02
6 0.01 4
2 0
100
200
0 0
300
100
200
Time
Time
(a)
(b)
0.1
300
0.12
2
0.1
32
exact
0.06 0.08
T
Nu
t=310.94 t=20.78
0.06
0.02 t=201.6 0.04
–0.02 0
4
8
12
0.02 0
100
200
y
Time
(c)
(d)
300
Fig. 1. Comparison of numerical and exact results for the one-dimensional evaporation problem. The frames proceed from the left to the right and the top to the bottom. (a) Interface position, (b) Liquid velocity, (c) Temperature field, (d) Nusselt number.
in the position of the interface (at a few times) versus grid spacing showed linear convergence as well. • A more stringent test is to compare our results with other available numerical simulations. We tested our code by comparing our results with the simulation in Fig. 12 of Welch and Wilson (2000). Since for this particular simulation, they try to validate their code by comparing their results with a similar simulation by Juric and Tryggvason (1998), this test implicitly serves as a comparison with Juric and Tryggvason’s code as well. We used Welch and Wilson’s parameters rl /rg =5.18, ml /mg =3.46, kl /kg =2.37, cl /cg =0.864, Pr=1.92, Mo=1.0 × 10 −6, and Ca= 0.020. Here, Mo=ml 4g/s 3rl and Ca=Tsat cl s/rg h 2fg l0 are the Morton and the Capillary number, respectively, l0 being a length scale. With the exception of the Morton number, these nondimensional numbers correspond to para-Hydrogen at 8 atm. In this simulation, initially the flow is quiescent and the temperature everywhere is the saturation temperature. Heat is supplied to the system from the bottom wall by a uniform constant nondimensional heat flux of qw =20.0. We used a
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domain size of 100 × 200 and a grid resolution of 144 × 298. The initial interface perturbation was y=20 − 5.0 cos(2px/100). This simulation resulted in a mushroomlike bubble which did not pinch off and was sustained by a hot vapor jet supplied to the bubble. We observed similar temperature and velocity fields as Welch and Wilson. More specifically, we measured the dimensions of the mushroom (i.e., height, length of arms, length of stem, width) at nondimensional time t=42.2 (corresponding to the second frame of Fig. 12 of Welch and Wilson) and found a very good agreement with their results. • We also tested our code against available empirical correlations for film boiling such as Berenson (1961) and Klimenko and Shelepen (1982). Although these correlations are based on static models and do not account for the time variations of the film thickness and the flow field, they are still valuable in checking the results qualitatively. According to Berenson, the Nusselt number defined based on heat flux at the wall is Nu=0.425(GrPr/Ja) 1/4, where Gr is the Grashof number. We compared our Nusselt number with Berenson’s Nusselt number for two two-dimensional simulations of film boiling at Gr= 0.5 × 10 6, Pr=1, and Ja=0.01 and 0.005. These tests were carried out for a very long time during which the instantaneous Nusselt number went through many cycles of oscillations as a result of bubble generation and pinch off at the phase boundary. Berenson’s correlation predicts Nusselt numbers 5.99 and 7.12 for these parameters which were within the range of our minimum and maximum numerical Nusselt number. We also did a three-dimensional simulation for a shorter time and at different parameters and found similar agreement [Esmaeeli and Tryggvason (2001)]. We note that Welch and Wilson (2000) found similar agreements when they compared their two-dimensional results with Berenson’s correlation. 4. RESULTS Our aim is to study explosive boiling of a vapor bubble. We start our analysis by considering the growth of a vapor nucleus in the center of a cubical liquid pool. Initially, both the vapor and the liquid are at a superheated temperature T. . The domain is periodic in the horizontal direction and is confined by a no-slip/nothrough-flow wall at the bottom and an open boundary at the top (to allow volume expansion of the fluid). The domain size is 1 × 1 × 1 and it is resolved by a 64 3 grid. In order to trigger instability, the surface of the bubble is perturbed randomly ri =r0 +E(0.5 − ran(i)). Here, ri is the distance of point i from the centroid of the bubble, r0 is radius of the unperturbed sphere, E is perturbation amplitude and ran(i) is a random number between zero and one. For this simulation, r0 =0.1 and E=0.005. The front resolution is determined by the size of the bubble and the grid and is initially resolved by 2592 triangular elements. The nondimensional parameters are rv /rl =0.1, mv /ml =0.025, kv /kl =0.025, cv /cl =1, Ja=0.02, and Pr=0.5. These property
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ratios are similar to the property ratios of cryogenic fluids such as Oxygen and a volatile material such as Butane used in the experiment. Figure 2 shows the initial condition of the vapor bubble and its subsequent evolution at five equispaced times. Initially, the instability develops at the rear of the bubble where temperature gradient is higher due to its proximity to the wall. This is evident from the second frame where all of the bubble surface is corrugated except for a small spherical cap at the top of the bubble. Because of superheat depletion, the growth of the bubble is slowed down over time. This is seen by comparing the bubble growth rate during
Fig. 2. The unstable growth of a nearly spherical vapor nucleus. The first frame shows the nucleus at t g=0 and the subsequent frames show the front at equispaced times. Here, Ja=0.02, Pr=0.5, rv /rl =0.1, mv /ml =0.025, kv /kl =0.025, and cv /cl =1.
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the first interval (the first and the second frames) with that of the last interval (the fifth and the sixth frames). The fine scale structure at the bubble surface during the last stage of growth is similar to the structure seen in the experimental results of Frost, 1988. As the phase boundary evolves, we use a regriding algorithm to enforce an adequate grid resolution on the fronts. Thus, large elements or elements with large aspect ratios are divided into smaller ones and small elements are deleted. The interface in the last frame consists of 162006 triangular elements. Figure 3 shows the temperature field and the streamlines in a midplane through the computation domain as well as the bubble at an early time (the second frame of Fig. 1) and near the end of the simulation (the fifth frame of Fig. 1); t g=8.44 × 10 10 and 3.4 × 10 11, respectively. Although the liquid and the vapor are initially in thermal equilibrium, the liquid is at a metastable state. Thus, heat is transfered from the liquid to the vapor at the phase boundary. As a result, a thin thermal boundary layer starts to develop at the liquid/vapor interface. This boundary layer eventually vanishes on the vapor side where vapor attains a uniform temperature Tsat . However, it continues to grow on the liquid side until it eventually occupies the whole liquid. Initially, the flow at the front and the rear of the bubble is nearly symmetric (first frame). However, over time it becomes more asymmetric as a result of mass flowing out through the top boundary. The bubble, therefore, experiences a slight upward motion due to this effect. In order to evaluate the effect of initial perturbations on the growth rate, we carried out another simulation with a symmetric perturbation given by r=r0 +E cos(nh h) cos(nf f). Here, r0 =0.1, E=0.005, nh =4, and nf =4. This introduces four lobes on the surface of the bubble in both h and f directions. Figure 4 shows two frames from
Fig. 3. Temperature field, streamlines and the bubble at t g=8.44 × 10 10 (a) and 3.4 × 10 11 (b) at a midplane through the computation domain. These times correspond to the second and fifth frame of Fig. 1, respectively. The figure shows the depletion of the superheat and the growth of the thermal boundary layer.
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this simulation at an early time (t g=1.7 × 10 11) and at the end of the simulation (t g=4.2 × 10 11). Here, the interface remains stable for a longer period of time and the instability develops initially at the boundaries of the lobes (the first frame). As time progresses, secondary instabilities are also triggered and at the end little traces remain of the primary perturbation. In order to evaluate the growth rate, in Fig. 5(a) and (b) we plot the volume and surface area of the bubble versus time for these simulations. Since higher evaporation rates at unstable boiling is attributed to a larger surface areas (as a result of corrugations; see, Shepherd and Sturtevant, 1982, for example), we have included the result of a simulation with the same set of parameters (i.e., dashed-point lines) where instabilities are suppressed in the course of the simulation. This is done by calculating the radius of an equivalent sphere using the computed volume and reinitializing the front using this radius at sufficiently short intervals of time. The figure shows that the volumes of the bubbles in the simulations with random- and symmetric-perturbations are very close and they are slightly higher than that with the stable growth. Comparison of the surface areas for these simulations shows larger differences between the three cases with the random-perturbation case having the highest area increase and the stable-growth one having the lowest area growth. The figures show that a nearly six-fold increase in the area of the randomly perturbed bubble compared with that of the stable one leads to only about 15% increase in volume. This suggests that the area increase due to corrugations does not have as much effect on evaporation rate as one would initially expect. Figure 5(c) shows the Nusselt number Nu=−“T/“y|w ls /DT versus time. The Nusselt number is initially very low while the temperature is uniform but increases as the liquid superheat begins to deplete. The Nusselt number has a close correlation with the volume of the bubbles. It is initially the same for all three cases where the volumes are nearly the same and takes on different values as the growth rates start to differ. We also computed a Nusselt number based on the difference between
Fig. 4. Temperature field, streamlines, and the bubble at t=1.7 × 10 11 (a) and 4.2 × 10 11 (b) at a midplane through the computation domain for a simulation with symmetric initial perturbation. The nondimensional numbers are the same as the corresponding ones in Fig. 1.
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liquid and vapor temperature gradient at the phase boundary. This Nu number was initially high and then decreased due to the same reason. In order to compare the growth rate in explosive boiling with that in classical theory, we calculate the average radius of the bubble in the first simulation using its computed volume. Figure 6 compares the numerical-, inertial-, and diffusiongrowth. In the inertial-dominated period R(t)=At, where A=(2/3hfg rv DT/rl Tsat ) 1/2 and R(t) is the bubble radius. In the diffusion-growth period R(t)=B `t, where B= (12al /p) 1/2 (rl /rv ) Ja. Note that the numerical result shows R(t) − R(0) where R(0) is the bubble radius at t=0. The figure shows that the numerical growth falls between the two classical limits. We note that Shepherd and Sturtevant (1982) found a similar result in their experiments of explosive boiling of Butane at atmospheric pressure. A key question, here, is the degree of dependency of our results on the domain size. The relatively long required time for the three-dimensional computations prevents us, at the present, to address this question using three-dimensional computations. However, we initiated such studies in two dimensions. Although two- and three-dimensional results lead to quantitatively different values, in most cases they yield qualitatively similar results. We therefore, performed two two-dimensional simulations with the same governing parameters used in the three-dimensional simulation (Fig. 1). The domain size in the first run was a 1 × 1 square resolved by a 128 × 128 grid. The domain size for the second run was increased to 5 × 5 while keeping the radius of the initial vapor bubble the same. The grid was increased to 640 × 640 in order to have equal grid resolution for both runs. Figure 7 compares the area of the bubble for these two cases. For a relatively long time both bubbles have the same growth rate but around t g=2 × 10 5 the growth rate of the bubble in the larger domain starts to surpass that in the smaller domain. Examination of the bubble surfaces before this time showed that they had nearly identical small-scale structures. The primary reason for the difference in growth rate at the late time is the depletion of the superheat in the smaller domain. Moreover, the growth rate of the bubble in the smaller domain is influenced by the outflow when its size is comparable with the domain size. We conclude that for a nearly four-fold increase in 11
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the bubble area the results do not depend on the box size. We expect this to be even better for three-dimensional computations since a three-dimensional bubble at this radius has an order of magnitude lower volume fraction. The nondimensional parameters that govern this problem are Ja, Pr, and Ca along with the ratio of the material properties. A parametric study of the problem may shed some light on the degree of influence of each of these parameters. We hope to address this issue in a future publication. Here, we undertake a twodimensional study to explore the effect of Prandtl number on the bubble growth and instability. Since Pr number is an indication of the fluid type, this study helps to determine the growth dynamics of a nucleus in different fluids during explosive boiling. For two-dimensional simulations the interface is perturbed according to r=r0 +E cos(nh), where r0 , E and n are defined as before. We take r0 =0.1, E=0.001, and n=8 in a 1 × 1 domain with a grid resolution of 130 × 130. With the exception of Pr, all the nondimensional numbers are the same as the corresponding three-dimensional ones. The Pr numbers are 0.01, 0.5, and 10. Figure 8(a) compares the interface shapes at time 3.8 × 10 11. The initial interface is shown as a nearly circular shape in the center of the domain. Initially eight lobes of equal size are introduced at the surface of the bubble. As the bubble grows, the lobes grow and secondary instability leads to tip splitting. Comparison of the areas of the bubbles (not shown here) showed nearly identical growth rate. Similarly, comparison of the perimeters (not shown here) showed no major differences between the Pr=0.5 and Pr=0.01 cases. However, it showed a five-fold increase in the perimeter of the Pr=10 case compared with the others at t g=4 × 10 11. Thus, it appears that although higher Prandtl number does not have a strong effect on the growth rate, it contributes to the interface instability. Figure 8(b) shows the vector plot of heat flux for Pr=10 at t g=3.9 × 10 11. It is clearly seen that heat transfer from the liquid to the vapor drives the flow.
Fig. 8. (a) Growth of a two-dimensional vapor nucleus at different Prandtl numbers. The other nondimensional numbers are the same as the corresponding ones in three-dimensional simulations. (b) Vector plot of heat flux for Pr=10.
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Juric (1997) numerically studied the effect of Jacob number on the instability and growth of a two-dimensional vapor nucleus and showed that as the Jacob number is increased, the interface grows more rapidly, more wrinkles are formed, and the wavelength of the surface instability decreases. At low Jacob numbers, the variation of interface length with time was nearly similar to `t for smooth circles, but the area fraction grew linearly with time. For higher Jacob numbers both the interface length and the area grew linearly with time. In three dimensions, we expect to see higher growth rate and larger interface convolution at higher Jacob numbers. 5. CONCLUSION A numerical method for direct numerical simulation of liquid/vapor phase change has been presented. The method is based on the front tracking/finite difference technique of Unverdi and Tryggvason (1992) but the implementation of heat transfer and phase change required extensive mathematical and numerical reformulation. Two- and three-dimensional explosive boiling were simulated to study the unstable growth of a vapor nucleus. The simulations showed some of the experimentally observed features of the phenomenon such as roughening of the bubble surface as a result of protrusions and subsequent increase in evaporation rate as a result of this effect. A study on varying the Prandtl number showed that the interface instability increases as Pr is increased but evaporation rate is relatively insensitive to this parameter. ACKNOWLEDGMENT This work was supported by NASA Microgravity program under grant numbers NAG3-2162 and NAG3-2583. REFERENCES 1. Alexiades, V., and Solomon, A. D. (1993). Mathematical Modeling of Melting and Freezing Processes, Hemisphere, Washington, D. C., pp. 92–94. 2. Chang, Y. C., Hou, T. Y., Merriman, B., and Osher, S. (1996). A level set formulation of Eulerian interface capturing methods for incompressible fluid flows. J. Comput. Phys. 22, 745–762. 3. Dalle Donne, M., and Ferranti, M. P. (1975). The growth of vapor bubble in superheated sodium. Int. J. Heat Mass Transfer 18, 477–493. 4. Dergarabedian, P. (1953). The rate of growth of vapor bubbles in superheated water. J. Appl. Mech. 20, 537–545. 5. Drazin, P. G., and Reid, W. H. (1981). Hydrodynamic Stability, Cambridge University Press, Cambridge. 6. Ervin, J. S., Merte, H., Jr., Keller, R. B., and Kirk, K. (1992). Transient pool boiling in microgravity. Int. J. Heat Mass Transfer 35, 659–674. 7. Esmaeeli, A., and Tryggvason, G. (2001). Direct Numerical Simulations of Boiling Flows, Proceedings of the Fourth International Conference on Multiphase Flow, ICMF-2001, New Orleans, Louisiana. 8. Esmaeeli, A., and Tryggvason, G. (1998). Direct numerical simulations of bubbly flows. Part 1. Low Reynolds number arrays. J. Fluid Mech. 377, 313–345.
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