[american institute of aeronautics and astronautics 46th aiaa aerospace sciences meeting and exhibit...

American Institute of Aeronautics and Astronautics

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Convection Effects On Thermal Stratification Inside Enclosures Due To Wall Heat Flux

W.M.B. Duval1 and R. Balasubramaniam 2 Fluid Physics and Transport Branch 1, NASA Glenn Research Center, Cleveland,Ohio,44135

National Center For Space Exploration 2, NASA Glenn Research Center, Cleveland,Ohio,44135

We consider the effect of convection on thermal stratification inside a rectangular and hemispherical enclosure as a model problem for cryogenic storage tanks. We address the dynamical characteristics of the flow and temperature fields for Ra in the range of 1 to 107. We show that the flow field is dominated by conduction effects for Ra<105, a transition to boundary layer flow takes place for 105<Ra<106, and convective instability develops for Ra>106. However, the convective oscillations are shown to decay for Ra on the order of 107. This shows that quasi-steady state models can be used for Ra up to 107.

Nomenclature Ar = aspect ratio g = gravitational acceleration Gr = Grashof number H = cavity height k = thermal conductivity L = cavity width Pr = Prandtl number q” = imposed heat flux R = radius of hemisphere Ra = Rayleigh number S = stratification parameter α = thermal diffusivity β = thermal expansion coefficient ρ = density θ = polar angle

I. Introduction

The design of cryogenic storage tanks for the storage of cryogenic liquids or as a component of the propulsion system, for the new Crew and Cargo Launch Vehicles Ares I and V which houses the Crew Exploration Vehicle (CEV) and Lunar Lander, requires control of the rise of pressure inside the tank due to unavoidable absorption of heat from the environment. Experimental measurements have shown that for a typical cryogenic tank in which a vapor region overlays the liquid region, the temperature in the vapor region is greater than that in the liquid region. Thus, heat flows from the vapor to the liquid region which sets-up thermal stratification in the vapor and liquid regions. In order to control the rise of pressure various technologies are being used and tested, some examples include: cryocooler, thermodynamic vent system (TVS), spray-bar technology, and liquid mixing jets. These technologies are used to control the temperature in the liquid and vapor region thereby can be used advantageously to control the pressure rise without venting the system. We introduce a model problem to analyze

1 Sr. Research Scientist, Fluid Physics and Transport Branch, NASA Glenn Research Center, M.S. 105-1 2 Research Associate Professor, Case Western Reserve University / National Center For Space Exploration, NASA Glenn Research Center, M.S. 110-3

46th AIAA Aerospace Sciences Meeting and Exhibit7 - 10 January 2008, Reno, Nevada

AIAA 2008-821

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.


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the implications of creating a uniform local temperature sink or source at the bottom of the cryogenic tank through the use of one of the technologies, the TVS system with a heat exchanger, in order to control pressure. The phenomenon of thermal stratification in cryogenic storage tanks is fundamental toward understating how to control pressure. There have been numerous studies (Tellep and Harper1, Vliet et al.2, and Vliet3) that considers boundary layer analysis to analyze the implications of thermal stratification due to side1 and bottom heating3. One of the fundamental issue is whether or not the characteristic bulk temperature in the liquid remains time-invariant1. Experimental works of Evans et al.4 to complement the boundary layer analysis shows that thermal stratification can be reduced through bottom heating. The experimental study of transient effects of natural convection inside enclosed fluids by Evans and Stefany5 has shown that the heat transfer coefficient approaches a quasi-steady state. Numerical and experimental studies by Barakat and Clark6 have also confirmed the findings of a quasi-steady state through small amplitude oscillations. More recent numerical study by Lin and Hasan7 (L&H) which solve the steady state conservation equations for an axi-symmetric cylindrical enclosure has shown the effect of system parameters on the characteristics of thermal stratification. Panzarella and Kassemi8 (P&K) have introduced a more general model which couples the effect of thermal stratification in the liquid region to a lumped model of the vapor region to assess the effect of pressure rise on various heating conditions. Modeling of the experimental measurements of cryogenic tank pressurization of the Saturn S-IVB propellant tank have been carried-out by Grayson et al.9 using Flow-3D to predict self-pressurization and thermal stratification. We present a time accurate solution of the thermal stratification problem for a rectangular enclosure to assess the validity of steady-state models over a range of parameters. For the range of parameters for which the quasi-steady state assumption applies we contrast the rectangular model to a hemispherical model in order to investigate effects of geometry on convective characteristics.

The model problem addresses the effect of convection caused by the adsorption of heat in the liquid region on thermal stratification imposed by the operation of a TVS system or cryocooler. Within this context, we consider an enclosure partly filled with liquid hydrogen at saturation condition with fixed interface temperature and a uniform bottom temperature imposed via the TVS system or cryocooler. For typical use of the TVS system, we assume that the mass of the liquid removed is negligible in comparison to the bulk liquid region. We analyze the effect of convection, for a given temperature difference between the interface and bottom of the enclosure, on the temperature distribution in the liquid region due to imposed wall heat flux and contrast the difference between a rectangular and a hemispherical enclosure.

The absorption of heat in the liquid region destabilizes the equilibrium condition due to the stable thermal stratification and gives rise to flow. The dynamical motion of the flow field obeys the evolutionary equations of mass, momentum and energy. For an incompressible Boussinesq fluid we numerically solve the coupled set of governing equations consisting of the Navier-Stokes coupled with the scalar temperature field using finite-difference methods. We employ a direct explicit solver for the rectangular domain and an implicit solver for the hemispherical domain. We consider the range of Rayleigh numbers (Ra) of order 1 to 107. For the rectangular domain, we show that the temperature field in the liquid is dominated by the conduction mode up to Ra in the neighborhood of 105. The planform of convection is two counter-rotating elongated cells symmetric with respect to the central axis. This finding gives credibility to the axi-symmetric assumption traditionally used in cylindrical models for similar range of Ra numbers. At the low Ra threshold, convective motion is weak and does not homogenize the thermal stratification in the liquid region. As the Rayleigh number increases in the neighborhood of 106 the parallel shear flow inside the elongated cells becomes unstable against Kelvin-Helmholtz instability and bifurcates to two secondary cells inside the larger cells near the bottom and interface region. These co-rotating secondary cells increase in strength and interact as time increases and form homoclinic orbital structures which provide a mechanism for local mixing. Thus a localized mixed region is formed underneath the interface which overlays a thermally stratified region. We show that the thermal stratification can be partly homogenized via a bottom uniform temperature source. For the hemispherical domain, the free surface of the liquid hydrogen is at a saturation condition with fixed temperature. A portion of the curved surface of the tank wall at the bottom has a uniform temperature that is lower than the saturation temperature. A heat flux is imposed over the remainder of the tank wall; the surface areas of the two portions of the tank wall are equal. We compare and contrast the effect of the enclosure geometry on the dynamics of the flow and temperature fields and highlight the similarities and the differences.

The paper is organized as follows, we first postulate the physics of the rectangular and hemispherical model of

the cryogenic storage tank. We deduce the parametric space common to both geometries and discuss the particular


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numerical method used. We show results contrasting the global dynamics of the flow and thermal fields for the rectangular and hemispherical domain for the stable and unstable stratification. We use dynamical similarity to deduce the parametric set to compare the two models using scalar measures. Lastly, we show the relationship of the global behavior of the flow and thermal fields to its local dynamics by examining the dynamical behavior of scalar quantities at fixed points in the flow field.

II. Formulation A basic model which typifies the storage of a cryogenic fluid such as hydrogen inside tanks consisting of a

rectangular and hemispherical geometries is shown in Fig. 1. We consider a fluid under saturation thermodynamic condition in which a vapor region overlays a liquid region separated by a flat free surface in which the shear stress is zero. The characteristic dimension of the rectangular model is denoted by its width (L) and height (H); whereas the hemispherical model is characterized by its radius (R) and polar angle (θ). The interface temperature, Ts, corresponds to the saturation liquid temperature whereas the bottom temperature, Tb, of the enclosure can be maintained uniform by operating the TVS system with a heat exchanger (Ts > Tb). Heat leakage into the tank from the environment is denoted by an imposed heat flux q’’ over a portion of the boundary. A typical condition in which heat flows, Qvl , from the vapor to liquid region is shown. Note that the interfacial heat flux at the interface can be bi-directional locally due to local superheat which can induce local evaporation in the presence of condensation at the interface. The typical scenario shown in the Figure implies that thermal stratification is imposed in the vapor and liquid region. The vapor region is assumed to be at constant temperature equal to the saturation interface temperature (Tv=Ts), and we neglect phase change at the interface which has been found to be a good approximation to model convection in the liquid region as pointed out by L&H7. This basic model is designed to analyze the effect of convection on thermal stratification and the implication of curved geometries. We thus consider the liquid region with its free interface as the basic model.

The basis of the rectangular model is to provide a generic benchmark simple model which leads insight into

dynamical convective instability which can be used to guide more sophisticated models that incorporate all the external complexity associated with real tanks. It bridges the gap between ideal and real situations while preserving the basic physics. The simplicity of the rectangular model is ideal for the resolution of small scales associated with high Rayleigh number flows on the order of 1013 as found in cryogenic storage tanks operating under ground-based conditions; thus the model can be used for direct numerical simulation. The model is also ideal for probing into the dynamical characteristics of the flow field at fixed points. In addition, the concept of dynamical similarity, can be used to lead insight into the flow and thermal field characteristics as found in both the hemispherical and cylindrical geometries.

In order to address the issue of the effects of convection on thermal stratification we consider a simplified model

in which the vapor region is at fixed temperature corresponding to the saturation temperature of the interface (Ts). We thus consider the simplified problem of an enclosure with a prescribed heat flux (q”) along part of its boundary


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and fixed bottom temperature (Tb) in which a free surface exists between the vapor and liquid region. Equilibrium in this system implies that the liquid and its vapor be maintained at its saturation condition and (Ts=Tb) which could occur only if there is no thermal exchange with the environment (heat leakage, q”=0) implying that the tank is perfectly insulated. Unfortunately, in real situations this idealization can only be approached, this implies that for q”>0 non-equilibrium ensues. The absorption of heat along the walls gives rise to natural convection flows in the liquid region inside the enclosure, due to decrease of density near the wall, for which the dynamical motion of the fluid can be described via the conservation of mass, momentum, and energy. For an incompressible Boussinesq fluid, the motion in the liquid region may be described by its continuity, momentum, and energy equations as follows:

0*V∇⋅ = (1)

* *

2 * *DV p V T gDt

ν βρ∇

= − + ∇ − ∆ (2)

2*

*DT TDt

α= ∇ (3)

Since the liquid is incompressible density is taken to be independent of pressure and a function of temperature only, its linearization implies

( )[1 ]oT dTρ ρ β= −

in which β the coefficient of thermal expansivity is given as,

1

pTρβ

ρ∂⎛ ⎞= − ⎜ ⎟∂⎝ ⎠

The incompressibility of a fluid implies Dρ/Dt=0, thus the flow field is divergence free as implied by Eq. 1. The Boussinesq approximation as used in the momentum Eq. (2) implies that variation in density is significant only in the body force term. The steady component of the body force due to uniform density is taken into account in the pressure term which represents the motion pressure. The motion of the fluid is coupled to the increase of energy input into the system through the body force term in the momentum Eq. (2). The energy input (q”> 0) to the fluid into the system is constrained to obey the evolutionary Equation describing the temperature field in Eq. (3). The thermophysical properties of the fluid, its density ρ, kinematic viscosity ν, and thermal diffusivity α are taken to be at saturation conditions corresponding to liquid hydrogen at Tsat=20.0 K.

A. Description of Initial and Boundary Conditions The initial and boundary conditions will be stated for the rectangular geometry; the hemispherical geometry can

be treated similarly. Initially the temperature field inside the liquid region is approximated to be linearly stably stratified with Ts>Tb; this represents the basic problem in cryogenic storage tanks in which thermal stratification dominates. Thus, initially,

* * ** *0, ( , , ) ( / )( )b s bt T x y o T y H T T= = + −

The boundary conditions, for t>0, consist of no-slip along the boundary of the cavity Γ,

0, 0t V on Γ> = For the free surface condition at the interface, there is vanishing of the tangential shear stress and normal

velocity across the interface,


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* *, , / 0, . 0y H x u y V n= ∀ ∂ ∂ = =

in which n is the normal to the interface. The vanishing of the normal velocity at the interface implies that mass flux due to condensation and evaporation at the interface is negligibly small in comparison to local convective velocity in the liquid; a condition which has been shown to hold ( L&H7 and P&K8). The bottom boundary y=0 and free surface y=H has constant prescribed temperature,

** *0, , ( , , ) by x T x o t T= ∀ =

** *, , ( , , ) sy H x T x o t T= ∀ = The left and right walls have prescribed heat flux : * *0, , " , / /H x o Hx y q q T x q k== ∀ = ∂ ∂ =

* *, , " , / /H x L Hx L y q q T x q k== ∀ = ∂ ∂ = −

in which Hq is the prescribed heat flux leakage into the system. Similar boundary conditions may be stated for the hemispherical region.

III. Scaling Analysis and Computational Method: Rectangular Geometry The nonlinear set of evolutionary Equations (1-3) may be simplified and reduced by transforming to vorticity-

streamfunction set of Equations using the following definition,

* * * *

* * ** * * *

v, , uuy x x yψ ψ ξ∂ ∂ ∂ ∂

= = − = −∂ ∂ ∂ ∂

v (4)

The transformed set of equations is non-dimensionalized using characteristic length Lc= H, time Tc=H2/α, and velocity scale Uc= α/H based on height of the enclosure (H) and the diffusion of heat (α) through the wall from the prescribed heat flux qH. The characteristic length Lc depends on the geometry of the enclosure, for hemispherical and spherical enclosures the characteristic length is based on the radius R. The characteristic velocity can also be based on the buoyancy scale 2( ) /o oU Tng Hβ α= ∆ for mixing flows in which buoyancy instability occurs. This implies that depending on the prevailing conditions various groups of dimensionless parameters may arise in the problem. In the above the gravitational acceleration g is denoted by ong in which n is a ratio by which the

standard gravitational acceleration og can be reduced via a low gravitational environment such as, low Earth Orbit, the International Space station (ISS), the Moon and Mars surface. For the basic condition inside cryogenic storage tanks in which thermal stratification prevails the characteristics length and velocity scales based on H and α are appropriate. Scaling the x=x*/L, and y=y*/H directions independently allows the aspect ratio parameter Ar=H/L to arise as a scaling parameter, the superscript asterisk denotes dimensional values. The remaining set of variables may

be scaled as * * * * */ , / , / , / , /c c c c c c cu u U U t t T U L U Lψ ψ ξ ξ= = = = =v v and

*( ) /( )b s bT T T T T= − − . The resulting set of dimensionless equations become

2 2

2 2x yψ ψ ζ∂ ∂+ = −

∂ ∂ (5)


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2 2

22 2

Prv Pr Ra TAr u Ar Art x y x y S xξ ξ ξ ξ ξ⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂+ + = + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

(6)

2 2

22 2vT T T T TAr u Ar

t x y x y⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂

+ + = +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ (7)

The parametric space Λ of the problem becomes, ( , , , Pr)Ra Ar SΛ = Λ (8)

in which the Rayleigh number, aspect ratio, stratification parameter, and Prandtl number are defined as

" 4 "

), , , Pr

(o

s b

q ng H H q HRa Ar Sk L k T T

β ννα α

= = = =−

(9)

The Rayleigh number based on prescribed heat flux at the boundary is referred to as a modified Rayleigh number

in the heat transfer literature. For natural convection flows driven by imposed heat flux, the flow field is often

characterized by the Grashof number " 4 2( / )oGr q ng H kβ ν= which represents the ratio of flow field intensity due to buoyancy force to its retardation due to viscous force. It is related to the Rayleigh number via Ra=GrPr; these two numbers can be used interchangeably for Pr=1 situations such as considered herein for liquid hydrogen. Another reason for using Ra as a parameter is that the problem can be conveniently extended to consider the case in which s bT T< for which buoyancy instability occurs; this situation is analogous to the Rayleigh-Benard10 problem for which buoyancy instability occurs for theoretical critical values of Rac=1100.65 and Rac=1707.72 corresponding to the cases in which one surface is rigid and the other free and two rigid boundary surfaces respectively. The aspect ratio parameter Ar has two degrees of freedom (H and L) which can be varied independently; however proper limiting solutions from the set of Eqs. 5-7 requires keeping one length scale fixed as the other is varied. The stratification parameter S represents the ratio of imposed heat flux q” which drives natural convection without a stability threshold to the degree of thermal stratification ( )s bT T T∆ = − controlled using a TVS system with a heat exchanger; it can also be interpreted as a dimensionless heat flux parameter. For the case in which heat flows ( )s bT T> into the liquid region 0S > , whereas when heat flows out ( )s bT T< of the interface 0S < which corresponds to the case for which buoyancy instability occurs. These two cases will be discussed further.

0S → implies either " 0q → , no heat leakage or T∆ →∞ in which thermal stratification becomes dominant.

On the other hand S →∞ implies either "q →∞ or 0T∆ → in which convection generated via heat leakage into the system is balanced by the thermal stratification thus local superheating of the liquid is minimized. The value S=1 has been coined as a critical transition value by L&H7 for Ra=1×106 for which heat leakage into the system is balanced by the thermal stratification thus local superheating of the liquid is minimized . For 1S > local superheating of the liquid near the interface can become significant such that the local heat flux through the interface can be bi-directional implying that both evaporation and condensation can occur simultaneously at the interface; this is illustrated for the hemispherical model. The solution of the above set of Eqs. 5-7 with its initial and boundary conditions yield the velocity and temperature field,

( , ; )V V r t= Λ (10) ( , ; )T T r t= Λ (11)


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which can be used to compare with other geometries such as the hemispherical geometry presented herein, and the spherical geometry of P&K8 which is similar to our hemispherical geometry as well as the cylindrical geometry of L&H7 . In order to compare results of other geometries to the rectangular model, the proper length scale Lc need to be selected depending on the spatial attributes r . This requires matching of the parametric space Λ, using the proper length scale, in order to achieve dynamical similarity. Examples of parametric matching for the hemispherical, spherical and cylindrical models will be discussed.

IV.Numerical Results: Rectangular Geometry

For the rectangular domain the coupled set of vorticity and streamfunction Equations (4-6) are solved using

finite difference methods. The difficulty encountered in the solution of this problem stems from the resolution of the boundary layer near the wall for large Rayleigh numbers. As the Rayleigh number increases the boundary layer

thickness scales as 1/ 4~ C Raδ − , for Ra O (109 ) yields (0.1 )cmδO . For a benchmark enclosure in which the length scale requires a linear grid density on the order of 500 grid points. Thus a square enclosure requires approximately 250,000 cells for adequate resolution; which demonstrates the challenge posed for computing large Rayleigh number flows in this system. In order to resolve the sharp interior gradients associated with the boundary layer, the Flux Corrected Transport (FCT) method11 is used. In addition to the ability of the FCT method to handle the nonlinearity in the scalar field Eq. 7, it also resolves sharp gradients and ensures positivity of the temperature scalar component throughout the domain of computation, thus eliminating false numerical oscillations. The set of Eqs. (5-7) are solved using a direct solver approach. The solution methodology consists of solving the stream function Poisson’s equation using matrix inversion which is followed by the solution of the energy and vorticity equations. The vorticity field is used subsequently for each time step. Since we are interested in time evolution of the flow field dynamics, a time accurate methodology is employed which uses third order Adams Bashforth for time discretization. A method prescribed by Roache12 is used to evaluate the vorticity at the boundaries. Grid refinement studies show that even 80×80 grid can represent the flow and temperature field with adequate resolution over the flow domain. The results reported for the rectangular domain uses 200×200 grid resolution in order to compare to the hemispherical domain on similar grid resolution.

A. Flow Field Dynamics We illustrate the global bifurcation of the flow field and the response of the temperature field for the basic

cryogenic storage tank problem in which heat can flow from the interface to the bottom of the tank. The application of the results of the rectangular model to hemispherical, and cylindrical geometries is illustrated using dynamical similarity. Typical thermal and flow field characteristics that occur in the Rayleigh number range of order 1 to 107 are shown in Fig. 2. As indicated by the initial conditions, all flow fields evolve from the basic stably stratified temperature field imposed initially. There exist three basic flow regimes: (a) conduction dominated regime for Ra<1×105, (b) transition to boundary layer flow 105 < Ra < 106 , and (c) convective instability for Ra > 106 .

The basic planform of the flow field for the conduction dominated regime, illustrated for Ra=2.5×102, is two

symmetric counter-rotating cells which is found to remain stable for Ra<105 . In the conduction dominated regime, convection is weak as indicated by the local maximum magnitude of the velocity field,

2 2max max ( ( , , ) ( , , ) )p pV V u x y t x y t= + v (12)

at fixed time t=tp , Vmax=7.5×10-5cm/sec. Thus the flow field has minimal effect on the distortion on the temperature field. The imposed heat flux along the wall boundary sets-up natural convection along the wall, due to local density decrease which forms a layer of homogeneous fluid underneath the interface. Since conduction is dominant, local superheating of the liquid occurs near the top wall region underneath the interface. The local superheated region can


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induce local evaporation in addition to condensation at the interface. However, these two effects conserve mass, thus the net mass flux across the interface can be zero.

The transition to boundary layer flow is illustrated for Ra=2.5×105. In this regime the local convective

intensity , as indicated by Vmax=0.027 cm/sec, increases which causes displacement of the central vortex core toward the interface region. Impending instability of the flow field is denoted by the distortions of the streamlines near the bottom region; however the planform of the flow field remains stable. The increases in intensity of the flow field underneath the interface causes a large homogeneous mixed layer with a corresponding smaller region of superheated liquid near the corner of the top wall. These trends are shown in the works of P&K8. Since convection becomes more dominant, the local stirring underneath the interface causes a decrease in the local superheat as


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indicated by a decrease in Tmax from the conduction regime. There is a corresponding decrease in size for the region in which local superheating occurs. Since the boundary layer thickness (δ) for natural convection scales as

1/ 4~ C Raδ − , the boundary layer thickness is quite large for this case (0.5 )cmδO which is indicated by the local maximum of the temperature field near the wall.

As the Rayleigh number increases, there exists a critical value at which the single mode symmetric cells

bifurcate to two mode symmetric cells. The transition to convective instability is illustrated for Ra=2.5×106 . The bifurcation event which gives birth to the instability is shown by the streamlines. Local parallel shear flow inside the cell structure becomes unstable against Kevin-Helmholtz instability and causes the bifurcation to two cells inside the larger cell. These two co-rotating cells form a homoclinic orbital structure indicated by the pinching of the streamlines in which there is a sharp discontinuity. The birth of homoclinic orbital structures inside the flow field signals a transition to oscillatory flow. Supporting evidence that the flow field does indeed transition to oscillatory flow will be shown from the time history of the flow field at fixed points for Ra in the neighborhood of 106. Late time evolution of the flow field show that the two co-rotating cells remain stable and propagate toward the top and bottom of the cavity. The increase in intensity of the flow field causes the formation of a homogeneous layer underneath the interface and a decrease in the local superheating as indicated by Tmax. Note that the boundary layer thickness decrease (0.2 )cmδO as Ra increases. This illustrates the challenge posed for computation of high Rayleigh number flows, as Ra increases δ becomes thinner and thinner; thus the resolution of the boundary layer near the wall becomes exceedingly difficult to resolve. The flow and thermal field characteristics were also studied for Ar=1; for Rayleigh numbers which yield similarity matching with the hemispherical enclosure, similar trends were found to occur in comparison to Ar=1/2. These three distinct flow regimes serve as a planform for which flows in hemispherical and cylindrical geometries can be compared.

B. Mixing due to Buoyancy Flow

The basic thermal stratification inside cryogenic storage tanks, as illustrated in Fig. 2, causes unwanted

pressure rise. One method to temporarily decrease the pressure is to homogenize or mix the stable stratification. Various technologies exist which can be used advantageously, these include cryocoolers, spray-bars, and liquid jets. An alternative method to homogenize the temperature field using buoyancy instability is demonstrated. This can be accomplished in principle by using the TVS system with a heat exchanger to provide either a local uniform temperature source in the bottom region or local temperature sink near the interface. Although a local temperature sink near the interface is much more practical to induce mixing, we will illustrate buoyancy induced instability via local uniform temperature source at the bottom due to its analog to the Rayleigh-Benard instability. The results can be use interchangeably for a local uniform temperature sink at the interface. In order to induce mixing, consider the situation in which s bT T< ; experimental studies for this case has been addressed by Vliet3 as a practical means to reduce thermal stratification inside cryogenic storage tanks, while Evans et al.4 considered nonuniform wall heating. For the case s bT T< the density field in the liquid becomes unstable and at a critical Rayleigh number threshold (as a reference Rac=1100.65 for the Rayleigh-Benard case) buoyancy instability ensues. The critical Rayleigh number is much larger than the analog Rayleigh-Benard problem and is on the order of 105 . We demonstrate this in Fig. 3 for Ra=1×107, since heat flows from the bottom region to the interface, S becomes negative. For the case illustrated, S= -1 which means that the ratio of heat input to heat removed from the interface is balanced, thus local superheating is minimized. This fact is supported by examining the value of Tmax which remains in the neighborhood of 1 indicating no local superheating; thus the local superheating can be inhibited under these conditions. The planform for buoyancy instability is illustrated for t=40 sec in Fig.3. Since Tb>Ts, unstable density stratification gives rise to two counter-rotating cells at the bottom of the cavity, whereas the imposed heat flux at the wall drives two counter-rotating cells near the interface. The interaction of these independent counter-rotating cells, in which cross-wise interaction may occur, causes local mixing of the liquid region which is indicated by the destruction of the thermal stratification in which the temperature field exhibits a descending plume from the interface region and an ascending plume in the bottom region. This unstable thermal field configuration becomes nearly homogeneous over a short time scale at t=100 sec due to the intensity of the convective cells which drive mixing indicated by the large magnitude of Vmax. An analog to this problem is illustrated by P&K8 for mixing induced via a subcooled liquid jet in which they show that a thermally homogeneous liquid reduces the pressure rise in the tank. Thus an alternative effective means to mix is illustrated using buoyancy induced flows. We will now examine the implication of similar scenarios inside a hemispherical tank.


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V. Numerical Results: Hemispherical Geometry

A. Description of tank geometry:

We analyze the buoyant motion of the liquid in a spherical tank that has a 50% fill ratio. Figure 1b shows a schematic of the spherical tank. Thus the tank is half-filled and the free surface of the liquid is a horizontal plane passing through the center of the tank. We choose a spherical coordinate system with origin at the center of the tank. The liquid free surface corresponds to the polar angle θ =π /2. The temperature at the free surface is the liquid-vapor saturation temperature that corresponds to thermodynamic equilibrium at the prevailing pressure in the tank. We assume that a portion of the tank wall described by θ ≤π /3, for all azimuthal angleφ (0 ≤ φ ≤ 2π ) is maintained at a fixed temperature, which is lower than the saturation temperature, by a cryocooler external to the tank. Heat enters the tank through the wall in the region π /3≤θ ≤π /2 for all φ . The heat flux in this region is assumed to be constant. The rest of the tank (π /2≤θ ≤π for allφ) is adiabatic. The boundary conditions on the tank wall are thus symmetric about the axis θ =0. It is readily seen that the areas of the portion of the tank with the constant controlled temperature and the portion subjected to the heat flux are both equal to 2

0Rπ , and are thus equal to the area of the free surface.

A cryogenic liquid tank is surrounded by multi-layers of insulation (MLI). In spite of this, a heat leak into the tank is unavoidable, as the surroundings are at a much higher temperature. It is believed that heat leaks primarily occur in areas near the struts used to support the tank. A primary benefit for design of a tank with a spherical


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geometry is that the volume of the stored liquid is maximized for a fixed tank surface area. Thus, relative to tanks of other geometries, the same amount of heat leakage can be absorbed by more liquid in a spherical tank. If active means are not used for removal of the heat that leaks into the tank, the liquid temperature rises steadily, causing increasing rates of vaporization of the liquid at the free surface. This results in a continual increase in the pressure within the tank, which is usually not desired. Even when an external cryocooler is used to remove heat from the liquid, it is necessary to understand the temperature distribution in the liquid and on its free surface, so that the rate of vaporization of the liquid can be ascertained. The boundary conditions we have chosen on the tank wall mimics the influence of uncontrolled heat leaks into the tank as well as the heat removal by the use of a cryocooler. We have chosen the region where the heat leaks into the tank to be adjacent to the free surface, which is a worst-case scenario. B. Description of numerical method:

The steady, three-dimensional axisymmetric Navier-Stokes equations in primitive variables with constant physical properties that govern the liquid motion are solved numerically in this study. The vapor above the liquid is assumed to be passive in these computations. The Boussinesq approximation is used for the buoyancy terms. An equally spaced computational mesh is used in r and θ directions. The numerical velocities and temperature are located at the computational cell vertices, while numerical pressures are located at cell centers. The momentum and energy equations are discretized at the cell vertices by employing standard three point second order central differences for the convective and diffusive terms. The pressure gradient is discritized to second order by using the pressures at the four cell centers closest to the cell vertex in question. Likewise, the continuity equation is discretized to second order at the cell centers (where the pressures are located) by using the velocity components at the four vertices of the cell (see Balasubramaniam and Lavery13, 1989 for more details). Boundary conditions are applied at appropriate points of the discrete mesh. These conditions correspond to the no-slip velocity boundary conditions and the temperature conditions for various portions of the tank wall described above, symmetry conditions on the axis θ = 0, and zero normal velocity, shear stress conditions on the free surface, and the temperature corresponding to a saturated liquid-vapor at the free surface. The free surface of the liquid is assumed to be undeformed by the motion in the liquid.

The dimensionless parameters that appear in the system of equations are the Grashof number 30

2

g TRGr βν∆

= ,

the Prandtl number Pr= να

, and the stratification parameter 0qRSk T

=∆

which is equivalent to a dimensionless

heat flux, where R0 is the radius of the tank, ∆T =Tsat −Tc is the extent of supercooling of the controlled temperature portion of the tank wall, β,ν,α and kare the volumetric expansion coefficient, kinematic viscosity, thermal diffusivity and thermal conductivity of the liquid, and g is the magnitude of the gravitational acceleration. The Rayleigh number Ra is the product of the Grashof and Prandtl numbers.

The nonlinear system of equations that results from the above discretization is solved using Newton’s method. A band-matrix direct solver is used to solve the linear system in each iteration of Newton’s method. The iterations are

carried out until the relative l1 norm of current values to previous values |ψi, j

cur −ψi, jpre |

|ψi, jcur |i, j

∑ is less than a specified

value, which is typically 1×10−6. Quadratic convergence of the Newton iterations was observed in all the computations.

The numerical code was validated by computing the flow due to natural convection in the spherical annulus between concentric spheres, where the walls are isothermal, and the inner wall is hotter than the outer wall. Garg14 (1992) has reported steady state computations of the streamfunction and temperature fields when the ratio of the outer to inner wall radius is 2, the Prandtl number Pr = 0.02, 0.7 and 6, and Rayleigh number Ra up to 106. Our computations are in excellent agreement with those reported by Garg not only for the flow pattern and isotherms, but also quantitatively for the magnitude of the extrema of the streamfunction: ψmax = 0.104 (present calculations)


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vs. 0.105 (Garg, 1992) for Ra = 2.5 ×105, Pr = 0.7; ψmax = 0.045 (vs. 0.046) for Ra = 9 ×104 , Pr = 6;

ψmax = 0.024 (vs. 0.024) for Ra =106 , Pr = 6 and ψmax = 0.699 (vs. 0.649) for Gr = 9 ×104 , Pr = 0.02. We have used grid points varying from 81 by 41 (r and θ directions; 1≤ r ≤2, 0≤θ ≤π ) to 101 by 101 in these validations. C. Results:

We show the results for the temperature and streamfunction fields in the liquid in the hemispherical region in the tank. The vertical axis passing through the center of the spherical tank (θ =0) is an axis of symmetry. The computational region is therefore chosen to be 0≤ r ≤1 and 0≤θ ≤π /2. The computational results presented below have used 101 and 76 grid points in the r and θ directions, respectively, unless noted otherwise. Results are presented for three regimes: (i) a conduction dominated regime with Ra ≈103 (ii) a convective regime with Ra ≈106 −107, and lastly (iii) a mixing regime with Ra ≈107, that is obtained when a temperature stratification is not present in the tank. In all these calculations, the Prandtl number in the liquid is set to 1, which approximates the value for liquid hydrogen.

Figure 4 shows the isotherms and streamlines for a regime where conduction is dominant. What are plotted are equally spaced contours of the scaled temperature distribution

T−Tc∆T and the scaled streamfuction ψ /(gβ∆TR0

5)1/2. The streamfunction is obtained by post-processing the numerically computed velocity field via

ψ =−r2 sinθ udϑ0

θ

∫ , where u is the radial velocity. For Ra= 103 and S=4, the temperature field is seen to be

dominated by conduction. The liquid is superheated near the region of the wall where heat enters the tank. The liquid free surface, which is at the saturation temperature, is cooler than the liquid beneath it in this region. Consequently there is a heat flux from the liquid to the vapor above it in this region. Away from the wall region, the free surface is warmer than the liquid beneath it, whose temperature is influenced by the portion of the wall that is

supercooled. The net heat transfer from the free surface is computed to be Qfs = 2πr∂T∂θ

(r,π /2)0

1∫ dr= −3.977,

which is a negative quantity, implying that heat is transferred from the liquid to the vapor for the case considered. The streamlines show the presence of two counter rotating vortices. The main vortex near the free surface has a counter clockwise rotation, and is driven by the heat flux entering the tank wall. The streamfunction has a value ψmin = −0.00483 at the center of this vortex.The weak vortex near the bottom of the tank has a clockwise rotation, and ψmax = 0.000216 at the center of this vortex. It is caused by the horizontal component of the radial temperature gradient prevalent on the cooled portion of the tank wall.

Figure 4: Isotherms and streamlines for Ra= 103, Pr=1, S=4, with a constant heat flux

in the region π /3≤θ ≤π /2. The region 0≤θ ≤π /3 has a constant temperature below the saturation temperature. The free surface at θ =π is at the saturation temperature.


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Figure 5 shows the isotherms and streamlines when convective effects are predominant. Three cases are shown –

Ra = 106, S = 4; Ra = 107, S = 4 and Ra = 107, S=1. In all three cases, the isotherms and streamlines reveal a boundary layer structure where steep gradients are present near the tank wall. Outside the boundary layer, the temperature distribution is essentially stratified. When S = 4, the isotherms reveal a region of superheated liquid beneath the free surface close to the wall where the heat flux enters the tank. Outside this region, the temperature gradient at the free surface is quite weak. The net heat transfer at the free surface is from the liquid to the vapor, with Qfs =−4.51 and −5.28 for Ra = 106 and 107, respectively. When S = 1, the superheated liquid region near the free surface is absent. The free surface is warmer than the rest of the liquid in the tank, and the net heat transfer at the free surface is from the vapor to the liquid, with Qfs =2.79. From a tank pressure control point of view, it is only is this case that there is a heat transfer from the vapor to the liquid, implying that the vapor must condense, and therefore there must be a decrease in the tank pressure. In the rest of the cases where there is heat transfer from the liquid to the vapor, the liquid must evaporate, causing the tank pressure to increase. Our computations suggest that optimal control, where there is no net heat transfer from the free surface, and consequently there is no net mass transfer, is achieved for S between 1 and 4. Preliminary calculations, which we do not report here suggest that the exact value is S=2.1 for Ra=103.

The streamlines show the two counter rotating vortex structure discussed earlier. The streamlines in the upper vortex are no longer rounded as in the conduction dominated case, but are elongated along free surface as well as the tank wall. For Ra = 106, S = 4, ψmin = -0.00485 and ψmax = 0.00163. For Ra = 107, S = 4, the lower portion of the upper vortex shows the beginnings of the transition of the flow to a different spatial structure, that likely will be observed when the Rayleigh number is larger. For this case, ψmin = -0.00248 and ψmax = 0.000958. When S = 1 and Ra = 107, the upper vortex has a complicated structure where there are multiple co-rotating eddies near the stagnant region in the vortex core. These are likely due to Kelvin-Helmholtz instability of the elongated core of the upper vortex. ψmin = -0.000288 at the center of the lower eddy of the upper vortex and ψmax = 0.000874 at the center of the lower vortex.

(a) (b)


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(c) (d)

(e) (f)

Figure 5: (a) Isotherms and (b) streamlines for Ra= 106, Pr=1, S=4; (c) isotherms and (d) streamlines for Ra= 107, Pr=1, S=4; (e) isotherms and (f) streamlines for Ra= 107, Pr=1, S=1.

Figure 6 shows the isotherms and streamlines in the mixing regime, where there is a lot of fluid motion in the liquid with very little temperature stratification. This regime of flow is obtained when the boundary condition of a temperature controlled portion of the tank wall (for θ ≤π /3) is replaced by an adiabatic condition. The

temperature difference ∆T is now redefined to be ∆T =qR0

k. Thus 0 1qRS

k T= =

∆. This expression for ∆T must

be used in the expression for the Rayleigh number. We have used 101 by 101 grid points in these computations. Away from the boundaries, the temperature distribution closely resembles the streamlines, indicating the predominance of convective effects. The flow is now caused solely by the horizontal temperature gradient induced by the presence of the heat flux at the tank wall. The bulk of the flow occurs as a single vortex with a counterclockwise rotation. The core of the vortex, however, is elongated and bent, and co-rotating eddies due to the Kelvin-Helmholtz instability are evident. ψmin = -0.00433 at the center of the upper eddy in the core of the vortex. The free surface is cooler that the liquid beneath it everywhere, and the net heat transfer from the liquid to the vapor is Qfs =−3.51.

Figure 6: Isotherms and streamlines for Ra= 107, Pr=1, S= -1, with a heat

flux in the region π /3≤θ ≤π /2. The region 0≤θ ≤π /3 is adiabatic. The free surface at θ =π is at the saturation temperature.


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VI. Comparison of Global Bifurcation of Rectangular and Hemispherical Model

For practical applications, cryogenic storage tanks are designed to have spherical as well as cylindrical

geometries. We consider a model tank with a hemispherical enclosure in order to contrast with our results for a rectangular enclosure. Similar to the results presented for the rectangular model, we contrast the global behavior of the thermal and flow fields for various regimes illustrating conduction, convection, and mixing flows.

Scaling analysis shows that the parametric space for the hemispherical geometry may be represented by

( , , Pr)H H H HRa SΛ = Λ Unlike the rectangular model, the aspect ratio (Ar) parameter does not arise in the parametric set because there is a single characteristic length scale, Lc=R, which is the radius of the hemisphere. The subscript (H) in the parametric set indicates that the parameters for a hemispherical enclosure is dependent on a different length scale defined as,

" 4 "

),

(o

H Hs b

q ng R q RRa Sk k T T

βνα

= =−

The parametric set may also be conveniently defined based on ( )s bT T T∆ = − instead of q’’ in the Rayleigh number definition. In this case, using the definition RaH=GrHPr, it can be shown that the earlier definition of the Grashof number for the hemispherical enclosure is recovered.

Basic results show that the conduction regime in a hemispherical region is similar to that of a rectangular region. However, due to curvature effects at the bottom of the enclosure where the constant temperature boundary condition is imposed, radial gradients drive a weak secondary cell. Thus the planform of a single mode cell as found in the rectangular geometry is modified. This bifurcation trend is due to the effect of the boundary condition, as P&K showed, for a comparable hemispherical system (spherical geometry half full) for which heat is injected near the top wall and an insulated bottom sector, only a single mode bifurcation occurs similar to the rectangular model.

For the convectively dominated region, we showed the bifurcation behavior of the flow and thermal field as Ra increases and the effect of the stratification parameter S in Fig. 5. Note that even though there are two cells similar to the rectangular model, there is a transition to boundary layer flow for Ra=1×106, the increase in the convective intensity drive two local convective cells which are counter-rotating; thus there is no interaction between the cells. The feature of the distortion of the cells due to the flow field that occurs for Ra=2.5×105, is shown to occur for dynamically equivalent RaH=1.0×106 . Again the bottom cell is generated by the boundary condition as inferred from the works of P&K which shows a single mode bifurcation with distortion of the cell. An increase in the Rayleigh number to Ra=1.0×107 shows that the effect of the hemispherical geometry is to suppress the convective instability which gives rise to two co-rotating modes; note that even though there are two modes present, these modes are stable since these are counter-rotating cells. The trend that convective instability is suppressed is also supported by the works of P&K . The trend that the local superheat decreases as convection becomes important agrees with the rectangular model. Note that the secondary mode at the bottom does not have much effect on the thermal stratification. Convective instability does occur for the hemispherical geometry for equivalent Rayleigh numbers and a decrease in S, as shown for Ra=1.0×107, S=1. Note that the basic feature of Kelvin-Helmholtz instability which causes the bifurcation to two co-rotating cells inside a larger cell occurs at the top boundary . The secondary cell at the bottom boundary does not have much effect on thermal stratification. These trends show that it is possible to duplicate the behavior of the flow and thermal fields inside a rectangular enclosure with proper matching of the governing parameters between the two models.

In parallel to the mixing scenario for the rectangular model, we present a generic scenario for the hemispherical region shown in Fig. 6. In this case, there is no parametric matching, since the boundary conditions are different than the one used in the rectangular model. There is an imposed heat flux at the top boundary and an insulated bottom. Since the top surface has a fixed imposed temperature, the heat input at the top boundary is not preferentially withdrawn at the bottom boundary. Thus the convective flow in this case homogenizes the thermal stratification by generating a flow field which generates a downward plume near the center-line similar to that found in the upper


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portion of the rectangular domain. There is one main cell generated with interior instability which generates two interior cells. This mixing scenario is similar to the works of L&H who presented mixing scenarios inside a cylindrical geometry. Note that even though, one would expect that these results would be similar to those presented by P&K for the liquid heating condition, the fact that the free surface has a prescribed temperature boundary condition that can change in time as part of the solution allows thermal stratification to become dominant in this case. However, P&K demonstrated that thermal stratification can be partially homogenized by introducing a subcooled mixing jet at the bottom boundary which is shown to be effective for mixing the thermal field which leads to a reduction in pressure.

VII. Scalar Measures: Comparison between models The parametric space for the rectangular geometry was given as,

( , , , Pr)Ra Ar SΛ = Λ whereas the hemispherical geometry, when based on a single length scale, Lc=R, removes the aspect ratio dependence, so that

( , , Pr)H H H HRa SΛ = Λ The parametric space can be further simplified by defining a modified Rayleigh number (RaH

*) so that

* 1H HRa Ra

S=

Thus,

* * *( , ,Pr)H H H HRa SΛ = Λ It can be shown that, for the matching of flow and thermal field characteristics between rectangular and hemispherical models, dynamical similarity requires for Ar=1/2,

* 1 , / 2H HRa Ra S SS

= =

whereas for Ar=1,

* 16 ,H HRa Ra S SS

= =

For the cylindrical geometry used by L&H the parametric matching is one-to-one since similar scales are used between the rectangular and cylindrical geometry, though the initial conditions differs in comparison to the thermally stable condition considered.

We compare the results of the rectangular and hemispherical geometry using two scalar measures, the local superheat indicated by Tmax, and the maximum local velocity Vmax. We consider the cases which represents the basic state inside cryogenic tanks, that is, the thermal stratification cases Ts>Tb . Qualitative comparison is also made with the works of P&K for the hemispherical geometry and that of L&H for a cylindrical geometry since the boundary conditions do no match exactly to our model. These comparisons provide a means to gauge our results, with the understanding that variation between their model and ours can stem from the boundary condition used.


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The effect of convective intensity characterized via the Rayleigh number, on local temperature superheating is shown in Fig.7a. The results show that for Ar=1/2, the local superheat decreases as the convective intensity increases. Even though thermal stratification is dominant for these cases, local convective flows which create a homogeneous layer underneath the interface are effective at lowering the local superheat. This trend occurs since an increase in the aspect ratio implies that the bottom boundary is twice the distance from the interface thus less heat is available to increase the local temperature of the mixed sub-layer underneath the interface. The works of L&H for the cylindrical enclosure show opposite trends, however they induce mixing for their cases which is different from the thermal stratification cases considered herein. The data for the hemispherical enclosure show that even though the trend of the local superheat is consistent with the rectangular model, there is a decrease in the magnitude of the local superheat. This effect is potentially due to aspect ratio and the stratification parameter (S) matching; it is quite possible that a direct matching of the stratification parameter does not lead to equivalent results between the rectangular and hemispherical model. This implies that there is an equivalent parameter, S=CSH, between the two models whose constant needs to be determined in order to obtain direct matching.

Qualitative comparison is made with the hemispherical model of P&K and the cylindrical model of L&H . The data for the liquid, uniform, and vapor heating conditions of P&K shown in Fig. 7a correspond to heating of the liquid region similar to our hemispherical model but with an insulated bottom boundary, in the uniform heating case the heat load is distributed uniformly on the boundary, whereas the vapor heating case correspond to heat release in the vapor region with a corresponding insulated boundary. The condition closest to our hemispherical model is the liquid heating condition, however the temperature of the free surface is not imposed rather an energy balance which takes into account the discontinuity in heat flux due to phase change at the interface is used to couple to the vapor region. We use dynamical similarity to match the reported Rayleigh number of P&K to the rectangular model for comparison (Ra=RaH/S). In spite of the differences between our hemispherical model and that of P&K the local superheat for the liquid and uniform heating conditions match our results fairly close. Whereas the vapor heating case is comparable to the results for Ar=1 of the rectangular model. For the vapor heating case, since all the heat is released in the vapor region and the boundaries are insulated, it is expected that the local superheating would be minimized for this case. Note that from the data of P&K Tmax is to be defined as Tmax=(Tl-max-Tref)/(Tl-min-Tref), in which Tl-max and Tl-min are the local maximum and minimum temperatures, whereas Tref is the local saturation temperature. Since P&K’s model does not have imposed temperature conditions, the dynamical similarity based on Ra is approximate. Even though there is no direct matching between P&K and our model, through the S value, the results of P&K can be gauged against the hemispherical model. The results show that P&K’s uniform heating condition compares favorably well to our hemispherical model in which S=SH=4. The trend of the data indicates that matching of the stratification parameter using SH=8 would translate the data of P&K to match that of the rectangular model. In addition since we consider an instantaneous configuration in time for the rectangular model, the approach to steady state will affect the magnitude of the local superheat.


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We also show the results of the cylindrical model of L&H in Fig. 7a. Dynamic similarity indicates that based on the definition used by L&H for their system there is direct matching between the Rayleigh number, stratification parameter, and aspect ratio, i.e., Ra=Rac, S=Sc, Ar=Arc in which subscript c indicates cylindrical, however the Prandtl number is slightly larger than ours, Prc=1.25. We emphasize that the conditions considered in their model forces local mixing and is not directly equivalent to the rectangular model for the cases shown in Fig. 7a in which thermal stratification prevails. However, the mixing case considered by L&H is used to emphasize local flow and thermal field characteristics which maximizes the local superheat. The data reported for Ar=1 shows that the trend of their local superheat is similar to the rectangular case for Ar=1; their data for Ra=1.0×105 lies between our data for Ar=1/2 and Ar=1. Since L&H considered mixing conditions which appears to maximize the local superheating, it is not surprising that their local superheat is slightly greater than that of the rectangular model. It is somewhat surprising that that the results are so close to the rectangular model for high Ra despite the differences between the two models. However, this shows that the rectangular model can be used as a generic model to infer flow and thermal field behavior in different geometries. We also show from L&H a case in which S=1, which means that the heat input into the system balance the heat removed by the TVS system. This case shows that the local superheat is minimized, this is expected since the TVS system becomes more effective for heat removal as S decreases in comparison to the case in which S=4. Note that the case for S=1 lies within the neighborhood of local superheat predicted by the hemispherical model as well as the model of P&K. This implies that our earlier conjecture that there is a constant C which provides optimal matching of the stratification parameter (S =C SH) between the rectangular and hemispherical enclosure is exemplified by the trend of the data as S increases from 4 to 10. Another scalar measure which provides comparison between the various models is the local maximum velocity Vmax. In Fig. 7b, we show typical magnitude of the flow field as a function of Rayleigh number. The results for the rectangular model, Ar=1/2, shows that the local maximum velocity of the flow field increases as Ra increases. An increase in aspect ratio to Ar=1, results in a decrease in Vmax . This stems from the fact that as Ar increases, for a fixed Ra, there is greater volume available for an equivalent amount of heat to be absorbed, in order for the system to conserve energy the velocity of the flow field has to decrease. The hemispherical model shows excellent agreement with the Ar=1/2 case which is the value used for dynamical similarity; this shows that Ar=1/2 provides an adequate value for dynamical similarity between the hemispherical and rectangular models. The results of P&K for the liquid heating condition show excellent agreement with both the hemispherical as well as the rectangular model. The uniform heating model agrees with the rectangular model for Ar=1, whereas the vapor heating case in which thermal stratification becomes dominant yields the smallest velocity magnitude. Since L&H did not provide local velocity data, a comparison could not be made. However, these comparisons show supportive evidence of the usefulness of a rectangular model to deduce trends in hemispherical and cylindrical enclosures using dynamical similarity.


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VIII. Local Dynamics of Flow Field at Fixed Points Having described the global dynamics of the flow and thermal fields, we now consider its local behavior at

fixed points in the flow field in order to characterize its transient behavior. We consider the generic case for the stable stratification in which Ar=1. The dynamical characteristics of the flow and temperature fields are shown in Fig. 8. The corresponding behavior of the velocity and temperature field for the conduction dominated regime is illustrated for Ra= 4.0×103 . The velocity field at a fixed point (0.2, 0.8) below the interface increases monotonically with time and approaches an asymptotic constant value for long times. This indicates that the flow field approaches a steady state behavior. On the other hand, the trend of the temperature field indicates a continual increase with time; the approach to steady state, if it exists takes much longer than the flow field. This lends insight into why the local magnitude of the velocity shows closer agreement to the hemispherical model as compared to the local superheat. The fact that the temperature field has not reached a steady state affects the asymptotic value of the superheat. Local superheating due to absorption of heat from the boundaries starts in the neighborhood of t=500 sec and increases thereafter. Local dynamics which illustrate transition to boundary layer flow are illustrated for Ra= 4.0×106 at a point near the center of the cavity (0.4, 0.5), in this case the flow field shows damped oscillatory behavior which decays and becomes steady beyond t= 500 sec . The flow field exhibits a long period oscillation of the order of 100 sec. This trend is generic to the birth of oscillations in this system. The corresponding temperature field show minute oscillations in the interval 0<t<200 sec which decays thereafter; the general trend of the temperature field is a continual monotonic increase over the time interval considered, there is no approach to steady state.

When the intensity of the flow field increases, Ra= 4.0×107 , the flow field at the same location of (0.4, 0.5)

as before shows a decrease in the period of oscillation of the flow field or an increase in frequency. The time signal of the flow field shows that it takes longer (t=900 sec) for the oscillations to decay. This case illustrates typical scenarios in the convective instability regime. The corresponding temperature field shows that, even though there is an increase in the frequency of oscillation of the flow field, the trend of the temperature field is virtually unaffected and behaves similar to the case for Ra= 4.0×106 . This indicates that the flow field does not destroy thermal stratification of the temperature field as we had shown from the global dynamics. These behavior serve as a template to illustrate typical scenarios inside cryogenic storage tanks for the parametric range considered 1<Ra<108 . These trends also lend insight into why steady state models of the cryogenic storage tanks are applicable over the parametric range considered. Since the oscillations decay, steady-state models will always converge to a solution. This model also lends insight into the data of the Saturn IV experiments as addressed by Grayson et al.9 . The experimental temperature field data at a local point in the flow field indicates a monotonic increase in temperature, whereas the prediction using Flow-3D show oscillations around a mean value. Even though Flow-3D predicts the correct magnitude of the temperature field, the oscillations around a mean value is aphysical. The trend of our data indicates that even though there is an increase in Rayleigh number there is no significant oscillation of the temperature field. Qualitative comparison with our model seems to agree with the trends of the Saturn IV experimental data. However, it remains to show that for Rayleigh numbers in the neighborhood typical of the Saturn IV experiments that the temperature field does indeed remain monotonic.


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. A. Mixing Flows The dynamical behavior of mixing flows (Ts<Tb) show that the flow and temperature fields exhibit aperiodic

oscillations for 7(10 )RaO in contrast to thermally stable case (Ts>Tb) considered above which shows periodic oscillations. The dynamical characteristics of the flow and temperature fields for the initially unstable density stratification (Ts<Tb) shown in Fig. 3 and its corresponding power spectrum are shown in Fig. 9. The power spectrum for the velocity field is evaluated from,

22

0

1( ) ( )bT

i f tV

bP f V t e

Tπ−= ∫ (13)

in which Tb represents the time interval of the data; the corresponding smooth estimate PV(t) is obtained from a convolution relationship ( ( ) ( ) ( ))V H VP f W f P f= ∗ using a Hanning spectral window (WH(f)). The behavior of the velocity field (v(x,y,t)) at a point (0.25, 0.75) show that the flow field is aperiodic for the interval 0<t≤100 sec and decays periodically for t>100sec and approaches a constant value indicating a quasi-steady state. This indicates that mixing is a low period event, the buoyancy induced instability is effective for homogenizing the temperature


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field as was shown in Fig. 3. In response to the flow field the temperature field also exhibits aperiodic oscillations over the interval 0<t≤100 sec, decays periodically with low amplitude oscillations for 100<t<400 sec, and exhibits a monotonic increase for t> 400 sec toward an asymptotic state. The corresponding power spectrum for the flow field shows that there is a low frequency peak 0.01f Hz= and a broadband cascade over a frequency interval 0.01 0.1f Hz< < . This indicates that there are short and long period events; the short period event indicates that fast mixing of the temperature field occurs and the long period event ( 0.01f Hz= ) signifies the decay or damping of the velocity field. The fact that the long period event represents the decay of the velocity field is supported by the power spectrum for the thermally stable case (Ts>Tb), Ra= 4.0×105, S=4, which shows indeed that the decay of oscillations occurs at a fundamental damping frequency of 0.01f Hz= . Note relative to the Saturn IV experiments, these results indicate the conditions for oscillations in the temperature field. The implications of the results is that regardless of the technology used (cryocooler, liquid jet, spray-bar, TVS with heat exchanger) to homogenize the thermal field in order to reduce pressure, the induce mixing is a low period event. For practical applications to cryogenic storage tanks, the fact that mixing is a low period event is advantageous since it implies that the pressure rise can be reduced fairly quickly.

.

IX. Conclusion A time accurate rectangular model, to analyze the effect of convection on thermal stratification inside cryogenic

storage tanks, is developed in order to assess its applicability against a hemispherical model which employs an axi-symmetric steady-state formulation. The time-accurate model which does not assume symmetry conditions show that the traditional axi-symmetric assumption used to analyze cryogenic storage tanks is valid. In addition the time accurate model also show that the basic steady-state assumption used in most cryogenic storage tank models is valid over the parametric range considered (1<Ra<108) since oscillatory motion of the flow field decays. Using dynamical similarity, we show that the rectangular model predicts the characteristics of the flow and temperature fields for hemispherical, and cylindrical geometries.


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For typical conditions inside cryogenic storage tanks, thermal stratification cannot be homogenized or mixing cannot be induced unless special measures are taken to mix the thermal field. The operation of a TVS system with a heat exchanger or the use of a cryocooler to provide a uniform temperature can be used advantageously to induce buoyancy instability. We show that buoyancy instability, which is a low period event, can be used advantageously to homogenize thermal stratification which implies the lowering of pressure rise inside cryogenic storage tanks.

Acknowledgments This work is performed in support of the Cryogenic Fluid Management (CFM) program at NASA Glenn

directed by S.M. Motil, M.P. Doherty, and M. E. Moran. Technical discussions with M. Kassemi., S. Barsi, and M. Hasan greatly added to the approach taken. On behalf of the CFM program, we thank J.P. Moder for suggesting the problem.

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3Vliet,G.C., “Stratification with Bottom Heating,” J. Spacecraft, Vol. 3, No.7, 1966, pp.1142-1144. 4Evans,L.B., Matulevicius, E.S., and Pan, P.C., “Reduction of Thermal Stratification by Nonuniform Wall Heating,” AIChE

Journal, Vol. 17, No.3, 1971, pp. 752-753. 5Evans,L.B., and Stefany, N.E., “An Experimental Study of Transient Heat Transfer To Liquids In Cylindrical Enclosures,”

Chem. Eng. Prog. Symposium Ser. Vol. 62, No. 64, pp 209- 215. 6Barakat, H.Z., Clark,J.A., “Analytical and Experimental Study of The Transient Laminar Natural Convection Flows in

Partially Filled Liquid Containers,” Proceedings of the Third International Heat Transfer Conference, Vol. II, Chicago, Illinois, Aug. 7-12, 1966, Papers 41-80.

7Lin,C.S., Hasan, M.M., “Numerical Investigation of The Thermal Stratification in Cryogenic Tanks Subjected to Wall Heat Flux,” AIAA-90-2375, NASA Tech. Memo. 103194, 1990.

8Panzarella, C.H., Kassemi, M., “On The Validity of Purely Thermodynamic Descriptions of Two-Phase Cryogenic Fluid Storage Tanks,” J. Fluid. Mech., Vol. 484, 2003, pp. 41-68.

9Grayson, G.D., Lopez, A., Chandler, F.O., Hastings, L.J., Tucker, S.P., “Cryogenic Tank Modeling for the Saturn AS-203 Experiment,” 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, 9-12 July 2006, Sacramento, California, AIAA 2006-5258.

10Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Oxford University Press, Oxford, UK, 1961, pp. 9-71. 11Boris, J.P.,, abd Book, D.L., “Flux-Corrected Transport I: SHASTA-A Fluid Transport Algorithm That Works,” J. Comp.

Phys., Vol. 11, Jan. 1973, pp. 139-174. 12Roache, P.J., Computational Fluid Dynamics, Hermosa, Alburquerque, NM, 1972, pp. 139-174. 13R. Balasubramaniam and J.E. Lavery, “Numerical Simulation of Thermocapillary Bubble Migration Under Microgravity

for Large Reynolds and Marangoni Numbers,” Numerical Heat Transfer, Part A, Vol. 16, 1989, pp. 175-187. 14V.K. Garg, “Natural Convection Between Concentric Spheres,” Int. J. Heat Mass Transfer, Vol. 35, No. 8, 1992, pp.1935-

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