Transient Capillary Channel Flow Stability

Microgravity Sci. Technol. (2014) 26:385–396DOI 10.1007/s12217-014-9403-z

ORIGINAL ARTICLE

Transient Capillary Channel Flow StabilityExperiments on the International Space Station

Aleksander Grah · Peter J. Canfield ·Przemyslaw M. Bronowicki · Michael E. Dreyer ·Yongkang Chen · Mark M. Weislogel

Received: 18 December 2013 / Accepted: 21 October 2014 / Published online: 8 November 2014© Springer Science+Business Media Dordrecht 2014

Abstract Capillary channel techniques with free liquid sur-faces provide very reliable means for liquid management inspace. However, capillary channel flow is subject to lim-itation due to liquid surface instabilities when a criticalflow rate is reached. Steady flow rate limitation is a conse-quence of the choking effect and well understood. Criticalsteady flow rate computation with a one-dimensional modelis related to a numerical singularity which occurs at criti-cal flow. For transient flow the singularity does not occur.Therefore, a new transient stability model is defined. It isbased on the steady model, a simplified transient momen-tum balance, the consideration of the capillary pressureof typical observed surface shapes, and on a simplifieddynamic inside the channel. The balance and dynamicare defined by liquid and geometrical properties only andtherefore significantly easier to compute than a transientdifferential equation system. In 2011, experiments wereperformed in cooperation with NASA on the InternationalSpace Station (ISS) to confirm the model for steady flowand validate the new transient model. A new phenomenonis discussed, the flexibility effect, which provides signifi-cant additional transient stability for channels of sufficientlength. An undesired feedback effect, provoked by the reuseof the liquid in a circular loop of the experimental setup,

A. GrahEuropean Commission, Joint Research Centre,1755 ZG Petten, The Netherlands

P. J. Canfield · P. M. Bronowicki · M. E. Dreyer (�)University of Bremen, 28359 Bremen, Germanye-mail: [email protected]

Y. Chen · M. M. WeislogelPortland State University, Portland OR 97207, USA

and which influenced the measurements, is compensated bya semi-empirical model for a feedback ratio.

Keywords Capillary flow · Critical flow rate · Transientstability

Introduction

Capillary Channel Flow has been of great interest since thebeginning of space exploration because of its widespreadapplication in liquid systems in spacecrafts such as inlife support systems and propellant tanks. In propellanttanks, liquid fuel may be displaced from the tank outletdue to adverse disturbances. In such a scenario, the fuelsystem is at a high risk of feeding gas to the thrusterswhich could lead to off-nominal thrust, engine damageand ultimately to mission failure. This is why tanks areequipped with propellant management devices that ensurethe transportation of gas-free liquid fuel to the tank’s outlet(Rollins et al. 1985; Kostner and Sani 1990; Jaekle 1991;Gilmore 1994; Srinivasan 2003). Vane-type propellant man-agement devices consist of vanes which are typicallymounted along the inner wall of the tank’s hull as shown inFig. 1.

One of the most important methods used in propel-lant management devices is the capillary effect, whichis a combination of forces of adhesion between the liq-uid and the solid surfaces and surface tension, which iscaused by cohesion within the liquid. The effect is usedin capillary channels formed by the vanes of propellantmanagement devices. Capillary channels are designed toalways be in contact with the bulk liquid which maymove around inside the tank when the spacecraft performsmanoeuvres.

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386 Microgravity Sci. Technol. (2014) 26:385–396

Fig. 1 Vanes of a propellantmanagement device formcapillary channels along thespherical tank’s hull. Capillarychannels of various shapes canbe established. Liquid iscaptured within the channel andcan be transported toward thepump at the outlet. The freesurfaces must withstandpressure differences to preventgas ingestion into the liquidacross the interface

Capillary channelswith free surfaces

Vane(s)

Outlet to

managementdevice

Propellant

Tank hull

thruster

Flow

Liquid pool

A simple capillary channel consists of two parallel plateswith one or two free surfaces between them (Fig. 2). Thepressure difference between the bulk liquid and the liquid inthe channel provides an effective passive way to ‘capture’and transport the liquid to the outlet. In our model the liquidis transported through the vane by a pump which is locatedfurther downstream. In real propellant tanks, the flow isdriven by a pressure in the propellant tank which drives thepropellant through the piping and towards the thrusters. Theflow causes the pressure to decrease along the flow pathand the free liquid surfaces bend inwards to balance the dif-ference between the pressure in the liquid and ambient gaspressure.

A typical problem of this method of liquid managementis the stability of the free surfaces. For example, increas-ing the flow rate would decrease the liquid pressure causingthe free surfaces to bend further inwards. If the curvature

Fig. 2 The capillary channel is consisting of two parallel glass plates.The channel length is l, the distance between the plates is a and thewidth of the plates is b. The liquid is transported through the channeltoward the outlet (arrow). The free liquid surface is on the top

of the free surfaces fails to balance the pressure differ-ence between the liquid in the channel and the surroundinggas then the surfaces would collapse ingesting gas intothe propellant system in the process. This phenomenon iscalled choking. The collapse of the free surface is related tothe speed of a longitudinal capillary wave (Lighthill 1978;Rosendahl et al. 2004; Shapiro 1953; 1977; Wilson et al.1986). The computation of the critical steady flow rate facil-itates a numerical singularity which occurs at the collapseof the free liquid surfaces.

The pressure difference between the liquid and the ambi-ent gas is affected by steady and transient flow condi-tions. In steady flow, the pressure difference is caused bymolecular momentum transport and the change of the cross-sectional area of the flow. In transient flow, the pressuredifference is additionally influenced by the acceleration ofthe liquid. The influence of all these effects on transient flowstability are investigated in this work.

Steady flow stability is well understood and a shortoverview will be given. The experiments confirm themodel for steady flow which remains unchanged. Forunsteady flow, a new transient stability model is devel-oped and discussed here in detail. The new model providessignificant simplification. The studies are supported bythree-dimensional computations. The models are validatedwith capillary channel flow (CCF) experiments performedon the International Space Station in cooperation withNASA.

The Microgravity Science Glovebox (MSG) on the ISSprovides a unique experimental interface for long-termresearch under conditions of microgravity. Supervised byscientists at the Marshall Space Flight Center (MSFC) inHuntsville, Alabama, NASA astronauts installed the CCFhardware into the MSG. During the 50 day period of oper-ations the experiment is operated remotely from a ground

Microgravity Sci. Technol. (2014) 26:385–396 387

station. The communication, telemetry for real-time control,and data download is provided by NASA (Canfield et al.2013).

The CCF Experiment on the International Space Station

The technologies and instrumentation used in the experi-mental work have been tested extensively in drop tower testsand ballistic rocket flights (Dreyer et al. 1994; Rosendahland Dreyer 2007; Salim et al. 2010). The setup for the CCFexperiment on the ISS is shown in Fig. 3 (Canfield et al.2013). The key part of the apparatus is the test section withthe capillary channel. The channel consists of two paral-lel glass plates and its design resembles the channel in thetank in Fig. 1. The channel length can be varied within therange 0 < l′ ≤ 48 mm. The distance between the plates isa = 5 mm and the width of the plates is b = 25 mm as

shown in Fig. 2. Furthermore, the channel may be openedon both sides to form an parallel-plate channel or it maybe closed on one side to form a groove shaped channel. Inthis paper however, we consider a groove channel only asshown in Fig. 3. Generally, both geometries result in similarbehavior of the flow.

The channel length can be varied on both sides by meansof two sliders (S1, S2). In Fig. 3, one of the sliders (S1)is closed, which results in the groove channel configurationwith only one free surface (FS). The main liquid loop isdriven by a pump (P) which is connected to the channel’soutlet. From the pump, liquid and, potentially, gas is forcedfurther through a flow meter (F) and into the phase separa-tion chamber (PSC). When the free liquid surface collapsesand two phase flow occurs, the gaseous phase is held backby a porous screen (S). The gas bubble below the screen isheld in place by vanes (V) and may be removed by the gasplunger (K3). In the flow preparation chamber (FPC), the

Fig. 3 Experimental setup ofthe CCF experiment on the ISS(not drawn to scale). Thecapillary channel, which consistsof two parallel glass plates, islocated in the test section (TS).The channel length can bevaried on both sides by means oftwo sliders (S1, S2). Thenotation is: free surface (FS),pump (P), flow meter (F), phaseseparation chamber (PSC),porous screen (S), vanes (V),flow preparation chamber(FPC), perforated sheet (PS),FPC valve (C9), nozzle (N),liquid plunger (K2), gas plunger(K3), compensation tube (CT),valves for operation (C4 andC11), valves for auxiliary fluidcontrol (C1 and C3)

S

V

C3

D

TS

C4

F

P

S1S2

FS

PS

C11

CT

FPC

PSC

C1

N

Liquid

Gas

PSC9

MM

K2 K3

FS


liquid is forced into rotation by deflectors (D) and passesthrough a perforated sheet (PS) which acts as a flow recti-fier. This maintains an equal velocity distribution over thecross-section of the inlet. The PSC valve (C9, depicted ina half open position) is closed during shuttle launch andopened prior to the experiments. The nozzle (N) allows theformation of laminar velocity profiles within the liquid flowbefore it enters the test section. The liquid volume in theloop is controlled by the liquid plunger (K2). The compen-sation tube (CT) maintains a constant pressure in the systemthat is defined by a meniscus of the free surface (FS) witha radius of 30 mm. It also compensates displaced liquid,for instance when gas is ingested into the test channel. Theeffective distance between the CT and the TS is 388 mm.Valves C4 and C11 are open during operation. Other valves(C1 and C3) are used for auxiliary fluid control and they arenormally closed.

The test liquid used in the experiments is 3M NovecT M

Engineered Fluid HFE-7500. Its properties are densityρ = 1639 kg m−3, kinematic viscosity ν = 0.9 10−6 m2

s−1, and surface tension σ = 17.6 mN m−1 at 16 oC asestimated before flight. Due to illumination the temperatureduring experiment runs successively rose by 10–12 oC anda variation of the properties of up to approx. 5% is assumed.The temperature is monitored at various locations within theliquid and gas loops using PT1000 resistance temperaturedetectors.

The experimental unit was transported to the ISS onboardthe shuttle Discovery in April 2010 and installed into theMicrogravity Science Glovebox in December of the sameyear. During the experiments, the system parameters (flowrate, flow acceleration, channel lengths etc.) were controlledfrom a ground station to study the stability of the free liquidsurface.

The liquid loop is driven by a pump as shown in Fig. 3,which emulates a thruster’s demand for a supply of liquid assuggested in Fig. 1. Increasing the flow rate decreases thepressure in the liquid phase in the test channel. This causesthe free surface of the liquid in the capillary channel to bendinwards as shown in Fig. 4.

Figure 4 shows video snapshots of an experiment in agroove channel with a free liquid surface on the left-handside (only the left slider is open, its outline is visible in theupper-left corner). The sliders and the free surface appearblack (the surfaces due to refraction of the light source). Thedirection of the liquid flow is from bottom to top. The flowin Fig. 4a is subcritical and therefore stable. This meansthat the curvature of the free surface is sufficient to bal-ance the pressure difference between the two phases. Thedeformation of the surface increases along the flow path. Anindentation appears close to the outlet. In Fig. 4b the stabil-ity limit is reached and the surface collapses. Gas is ingestedinto the flow and carried away by the liquid stream as shown

Fig. 4 Experimental video images of a groove channel Image a showsa subcritical flow with a stable liquid surface. In image b, the stabilitylimit is reached and the surface caves into the flow. Gas ingestion isvisible in image (c). The surface has collapsed to the point where agas bubble detaches and is subsequently carried downstream with theliquid flow

in Fig. 4c. The ingested gas is held back by a screen inthe phase separation chamber and accumulates there. At thesame time the ingested gas displaces liquid and increasesthe level in the compensation tube.

Another effect in the presented setup may influence thestability of the free surface during the unsteady phase. Thisis the feedback effect of the fluid loop. Due to the circulationof the liquid as shown in Fig. 3, liquid is not only drawn outof the channel’s outlet but is also forced into the flow prepa-ration chamber at the same time. This may increase the localpressure at the channel inlet and consequently the stabilityof the surface during the transition. The feedback effect issignificantly influenced by the amount of gas in the phaseseparation chamber in Fig. 3. If there is no gas present, thecomplete feedback effect takes place. Otherwise, a certaindamping of the feedback occurs due to the compressibilityof gas.

State of the Art for One-Dimensional Modeling

The new model (which is presented in Section Formulationof a New Model for Transient Flow Stability) is based onthe existing model explained in detail in prior publication(Rosendahl et al. 2002; Rosendahl et al. 2004; Rosendahland Dreyer 2007; Rosendahl et al. 2010; Grah et al. 2008;Grah and Dreyer 2010); it remains unchanged for steadyflow. For transient application significant simplification isobtained with the new model. For the present model ageneral overview over the final equations is given here.

In this one-dimensional model the liquid is assumed to beincompressible, isothermal, Newtonian, and perfectly wet-ting with a zero degree contact angle. All distances in themodel are scaled by a/2 (in former publications x wasscaled with the channel length l′). The velocity v is scaled


by the characteristic velocity vc = √2σ/ρa. All cross-

sectional areas are scaled by ab, the volumetric flow rateQ by abvc, and the surface curvature h by 2/a. Time t isscaled by a/(2vc) and the change of flow rate over time δ by2bv2

c . The dimensionless channel length for frictional ductflow l̃ = Ohl′/(2dh) (Rosendahl et al. 2004) is defined withthe Ohnesorge Number Oh = √

ρν2/(2σa); the hydraulicdiameter is dh = 4a/(�−1 + 2) for the groove geometry.The aspect ratio is defined by � = b/a. The scaling factoris � = a/(2l′). The Reynolds Number is Re = 2vvca/ν <

103 and the flow is assumed to be laminar (Rosendahl et al.2004); hereby v is the dimensionless velocity and vvc is thephysical velocity.

The flow along the channel axis x is assumed to beone-dimensional and characterized by the mean velocity v,the pressure (surface curvature) h, the surface contour k

(Fig. 5), and the channel’s cross-sectional area A. Thedimensionless governing equations (Grah et al. 2008) arethe unsteady momentum (1) and the unsteady continuity (2)

− ∂t v = −∂xh + v∂xv + 0.5Kf l̃�v, (1)

−∂tA = v∂xA + A∂xv. (2)

For the steady flow model the left-hand sides of the equa-tion system are equal to Zero (∂tv = ∂tA = 0). In that caseduring computation a singularity occurs at the critical flowrate. Since the singularity is comparably easy to detect theeffect can be facilitated for critical flow determination (Grahand Dreyer 2010). The largest flow rate at which the sin-gularity does not occur is considered to be the critical flowrate. For the transient flow the singularity does not occurat all. The pressure loss factor (Rosendahl et al. 2004) isKf = Kpf + KSf . The viscous resistance factor is Kpf =76.4 for the groove channel (White 1986). It is assumed thatthe flow is not fully developed at x = 0 (Rosendahl et al.2004), and that the additional pressure drop KSf caused bythe transition to a fully developed laminar flow profile canbe computed (Sparrow et al. 1964).

The cross-sectional area for the flow in the groove is

A ={

1 − R20

4�arcsin 1

R0+ R0−�+k

4�; k ≥ � − 1,

12�

(k + 1 − π4 ) + 1

2 ; k < � − 1,(3)

The pressure is related to the curvature of the free surfaceby the scaled Young-Laplace-Gauss equation (Landau andLifschitz 1959) with the principal radii R1 and R2

h = R−11 +R−1

2 = R−11 +Γ 2 d2

xxk[1+Γ 2(dxk)2]−3/2. (4)

The radius for the cross-sectional area R0 (i = 0) and thefirst principal radius of the surface curvature R1 (i = 1) are(Rosendahl et al. 2004)

Ri =⎧⎨

⎩

1+(�−k)2[1+�2(dxk)2]i

2(�−k)[1+�2(dxk)2]i/2 ; k ≥ � − [1 + �2(dxk)2

]−i/2,

1; k < � − [1 + �2(dxk)2

]−i/2.

(5)

The radius of the cross-sectional area for the flow R0 isparallel to the y-z plane. The radius R1 is perpendicular tothe free surface. At the point of minimum cross-section (x =x∗) both radii are identical R0 = R1 (compare Fig. 5b). R2

is located in the x-z plane.The boundary conditions are given by the flow velocity

at the channel outlet v1 = Q (due to scaling of Q with thechannel dimensions a and b), the pressure (surface curva-ture) at the inlet, and the position of the liquid surface at thechannel inlet and outlet (where the surface is pinned at theedges of the closed ducts) (Rosendahl et al. 2004)

v(x = 1) = v1,

h(x = 0) = h0,

k(x = 0) = k(x = 1) = �. (6)

The scaled pressure at the channel inlet h0 depends on thesystem pressure K0, the pressure loss in the nozzle (defined

Out

let

Inle

t

Free liquid surface

yx

z

l

11

f l f0

R

l

1

1R

z

yk

Λ

R2

Fig. 5 Typical shape of the free liquid surface. Image a shows thecross-section of a channel in the symmetry plane (y = 0). The surfacein the vicinity of the inlet is typically almost linear along the x axiswith one principal radius. Downstream, it forms an indentation closeinto the outlet with two principal radii (dimensions due to scaling with

half of the channel’s width). Image b displays the cross-section of thechannel at the point of minimum surface contour height k (at the bot-tom point of the indentation, plane x = x∗). The scaled radii at thestability limit are R0 = R1 = 1. Due to symmetry reasons the secondradius is assumed to be R2 = 1 as well


by K1 and K2), and on the properties of the liquid col-umn between the capillary channel and the compensationtube (Grah et al. 2008). The inertia of the liquid columnis defined by its length l0 and the viscous pressure loss ofthe liquid column is defined by the factor K3. The pressure(surface curvature) at the channel inlet may be expressed byGrah et al. (2008)

h0 = l0dt v0 + K0 + K1

4Ohv0 + K2

2v2

0

+K3Oh [v0 − v1], (7)

with the velocity difference v0 − v1 between channel inlet(x = 0) and outlet (x = 1). This difference occurs duringunsteady flow and causes liquid to move from the channelinto the compensation tube (Fig. 3) and vice versa. Dur-ing the flow rate increase, the liquid column, which has ascaled effective length of l0 between the compensation tubeand the channel inlet, accelerates. The effective lengths ofthe liquid column l0 is defined by the nozzle lengths lN(between the FPC and the channel inlet) and by the weightedlength of the compensation tube line lCT L (between the FPCand the CT): l0 = lN + lCT L/ACT L = 155.2 (388 mm);the shape of the nozzle is neglected. The scaled systempressure

K0 = 2/RCT (8)

is maintained by the meniscus in the compensationtube (CT, Fig. 3). The (scaled) radius of the compensationtube RCT = 12 (30 mm) yields the system pressure K0 =0.167. The coefficients for the pressure loss of the noz-zle, i.e. along the flow path from the perforated sheet untilx = 0, were computed and published long before the flightto be K2 = 1.4, and K3 = Oh−1, respectively (Grah et al.2008); K1 = 680 is obtained from three-dimensional com-putation presented in Section Three-Dimensional Model ofthe Channel.

The equation system of the one-dimensional model forsteady flow is evaluated by a Newton method. Centraldifferences of second order are applied for the spacialdiscretization.

Three-Dimensional Model of the Channel

For additional comparison and analysis of the feedbackeffect three-dimensional computations are performed usingOpenFOAM (2011). The solver interFoam (volume of fluidmethod) is used to compute the unsteady two-phase flowin the capillary channel, the nozzle, and the compensationtube. The model (Fig. 6) is based on a hexahedral gridcreated from a STL geometry with the SnappyHexMeshtool (OpenFOAM 2011). Grid studies with more than onemillion cells have been performed to determine a suitable

p

outlet

CT

FS

FS

FPCinlet

[Pa]

Fig. 6 Three-dimensional model of the capillary channel with oneinstable free liquid surface (FS), the compensation tube (CT) withanother free surface, and the flow preparation chamber (FPC). A super-critical flow regime is shown; after surface collapse and gas ingestionthe bubble is carried upwards with the current. The pressure fielddepicts the pressure decrease in flow direction and along the linetowards the compensation tube (a consequence of liquid displacementafter gas ingestion)

density of the numerical mesh (approx. half a million cells).The uncertainty of the critical flow rate computed with thethree-dimensional model is estimated to be within ± 5 %.At the inlet (bottom of the image) uniform flow is assumed;at the inlet and at the outlet (top of the image) the flow rateof the pump is applied to model the circulating loop of theliquid as shown in Fig. 3. The PSC is assumed to be filledwith liquid only. The effect of the gas bubble is thereforenot modeled. Rather, the model represents one special casewhich is discussed in detail in Section Formulation of a NewModel for Transient Flow Stability. The part of the circulat-ing loop between the outlet and the inlet, consisting of pipesand vessels with rigid walls (including the PSC), and whichare filled with incompressible liquid only, are not subject tomodeling. The model assumes that the flow rate increase ofthe re-entering circulated liquid at the inlet is identical to theflow rate increase at the outlet (with opposite sign).

The three-dimensional model is validated with mea-surements for the stability of steady free surface flow inSection 3. The new transient model does not depend onthe results of the three-dimensional computation. However,three-dimensional computation turned out to be helpful tounderstand and, consequently, eliminate the undesired feed-back effect of the experimental setup as presented in thefollowing section.


Formulation of a New Model for Transient FlowStability

Compared to steady flow, additional effects take place whenthe flow is transient. The most significant effect for flow sta-bility in transient flow is caused by an acceleration δ of theliquid at the channel’s outlet which is defined as the tran-sition from one steady flow rate Qt=0 to a higher flow rateQ = Qt=0 + �Q (with �Q > 0) within a time interval �t

δ = �Q/�t. (9)

Due to the aggregate of the steady effect (convective accel-eration) and the transient effect (local acceleration) it maybe expected that the maximum stable flow rate Q decreasessignificantly with increasing acceleration δ. In other words,the lower the flow rate Q at which the transition ends, thehigher the maximum stable acceleration is. With this inmind, we may assume a maximum stable acceleration δ0

is attainable for Q = 0, which requires a negative initialflow rate Qt=0 = −�Q. As explained above, the pressureat the channel inlet is given by Eq. (7) and is time depen-dent. At t = �t the steady flow effects vanish, becauseQ = 0. However, because the derivative of the ramp func-tion is undefinable at t = �t , we consider the pressurefunction at t → �t and consequently Q → 0, wheresteady flow effects are non-zero but negligibly small. Con-sequently, the model assumes negligible flow (v0 ≈ 0,v1 ≈ 0) infinitesimally close to the end of the transitionphase for the maximum possible stable acceleration. Thus,neglecting the terms for the velocities in Eq. (7) yields thepressure at the channel inlet

h0 = l0dt v0 + K0, (10)

for the maximum possible stable acceleration of the liquidcolumn (situated in front of the inlet) of length l0; K0 is thecorresponding system pressure as defined in Eq. (8). Inte-gration of Eq. (1) between the inlet (x = 0, h = h0) and thepoint of minimum cross-section at the bottom of the free-surface indentation (x = x∗, h = h∗, Fig. 5) yields forv = 0 (and with Eq. (10))

l0dt v0 +∫ x∗

0dt vdx = h∗ − K0. (11)

Since l0 >> x∗ (and since the liquid in the vicinity ofthe channel inlet plays another role, as will be shown later)the integral in Eq. (11) is neglected. An additional benefitof this simplification is that Eq. (11) does not depend on thelength of the channel, which is variable. Furthermore, theacceleration may be simplified as dt v0 = δ0 which yieldsfor the momentum change

l0δ0 = h∗ − K0. (12)

The acceleration δ0 of the liquid column in front of thechannel inlet is actually the system response to the tran-sient pressure drop at the outlet induced by the pump. Theacceleration of the liquid column may be considered to cor-respond to the flow rate acceleration at the outlet (with acertain delay) if the free surfaces remain stable during thetransition.

The momentum change is balanced by the free-surfaceindentation at the channel outlet. A typical shape of the freesurface as a result of the flow effects is shown in Fig. 5. Thesurface is located at the top; the flow direction of the liq-uid is from the inlet at the left to the outlet at the right. Thesurface in the vicinity of the inlet is typically almost lin-ear along the x axis with one principal radius. Downstream,towards the outlet, it curves inwards and forms an indenta-tion close into the channel’s outlet with two principal radiiof curvature. All dimensions are scaled with a/2 (half ofthe channel’s width). The longitudinal length of the curvedpart of the surface at the outlet is lf 0. For a sufficientlylong channel with length l > lf 0 the surface in the regionx < lf is almost linear along x with one principal radius ofcurvature.

Since a free liquid surface tends to minimize its surfaceenergy it may be assumed that for small flow velocitiesvt=�t the indentation at the outlet is of almost hemispher-ical shape. This can also be confirmed by observation asshown in Fig. 4. Thus, the dimension of the indentation inflow direction is assumed to be equal to the channel’s width(which is 2 due to scaling). The transition from the linearpart of the surface to the indentation is assumed to take placewithin a distance of one half of the channel’s width (whichis 1). Consequently, the longitudinal length of the curvedpart of the surface is estimated to be

lf 0 = 3. (13)

The stability limit is assumed to be at the maximum curva-ture of the indentation (minimum radii) as shown in Fig. 5a.Due to the discussed geometry of the channel the principalradii for a hemispherical indentation in this case are equal tohalf of the scaled channel’s width R1 = R2 = 1. Accordingto Eq. (4) this surface geometry yields a capillary pressureat x = x∗

h∗ = 2 (14)

for the stability limit. The balance in Eq. (12) yields thestability limit

l0δ0 < 2 − K0 (15)

for the hemispherical shape of the indentation at the chan-nel’s outlet. As discussed for Eq. (8), the scaled systempressure K0 is defined by the radius of the meniscus in thecompensation tube RCT (Fig. 3). It is assumed that the tran-sient pressure drop in the channel is a result of the inertia of


the accelerated liquid column l0 which is located betweenthe channel’s inlet and the compensation tube (Fig. 3) fromwhich the required liquid is supplied.

During the acceleration phase, liquid actually may bedrawn from two sources, the compensation tube and thephase separation chamber (Fig. 3). However, it can beshown that the pressure drop at the screen in the phase sepa-ration chamber is sufficiently high to enable decoupling; theaccelerated liquid is assumed to be drawn from the compen-sation tube (during transition). Consequently, δ0 is definedby geometry and fluid properties only and may be consid-ered to be a system constant. The stability limit (15) yieldsthe transient stability constant

δ0 = (2 − K0)/ l0, (16)

which is the maximum stable acceleration with the finalflow rate Q = 0. This is clearly beyond the capabilitiesof the measurements with the presented setup; the transientstability constant is considered to be a theoretical value forthe model.

In addition, a new effect is observed in the experiments.Augmented stabilization occurs in channels longer than acertain value (l > lf 0, Fig. 5). This effect is assumed tobe based on the flexibility of the part of the free surfacethat is almost linear along x (situated in vicinity of the inletof the channel) as shown in Fig. 7a, which shows a steadystate before the transition (t < 0) with almost no flow. InFig. 7b the transition has begun (0 < t < �t) and the flowrate increases due to increased demand at the outlet. Thetypical hemispherical indentation occurs at the outlet. Thesurface at the inlet moves inwards (arrow). Liquid is relo-cated towards the outlet which satisfies part of the demandand decreases the capillary pressure at the indentation. Thisincreases the probability for the surface to remain stable.At the end of the acceleration(t = �t), shown in Fig. 7c,the surface is close to the stability limit but remains sta-ble. It can be observed that a surface of a shorter channel,

where no flexible linear region is present, collapses at loweracceleration.

The model postulates that the transfer of liquid from theinlet area towards the outlet results in a reduction of capil-lary pressure during transition. However, this flexibility alsohas its limit, which is defined similarly as for the hemispher-ical indentation. The flexible free surface at x ≤ lf = l−lf 0

(Fig. 5) is supported by one radius. This is one half of themean curvature in comparison to the surface area at theindentation with two radii where the stability limit δ0 isdefined. This results in a flexibility coefficient

f∞ = 0.5 δ0, (17)

the maximum additional stability due to surface flexibilityfor a sufficiently long channel (lf → ∞). Hence, a flexibil-ity function f (f = 0 for l > lf 0) for a particular channellength can be defined as

f = f∞lf

l. (18)

As discussed in Section The CCF Experiment on theInternational Space Station the so called feedback effectof the fluid loop in the presented setup may influence thestability of the free surface. The circulation of the liquidas shown in Fig. 3 may increase the local pressure at thechannel inlet. As discussed in Section Three-DimensionalModel of the Channel, the circulating liquid loop betweenthe channel outlet (where the pump is situated) and the pointof re-entry into the system at the PSC consists of ductswith rigid circumference filled with incompressible liquidonly. In the most simple case there is no gas present inthe PSC and it can be counted to the rigid loop. Conse-quently, a flow rate increase at the outlet of the channelresults in an identical flow rate increase of the re-enteringcirculated liquid into the FPC. It may be assumed that liq-uid is partly forced into the CT and partly into the inletof the channel. The local pressure in the channel increases.

Fig. 7 Image sequence from theexperiment (processed pictureswhich contain only contourdata) to demonstrate theflexibility of the part of the freesurface at the inlet duringtransition. Image a: steady statebefore the transition. Image b:the transition begins and theindentation at the outlet occurs.The flexible surface at the inletstarts to move inwards (arrow),liquid is displaced towards theoutlet. Image c: end of thetransition; the surface is fullystretched but remained stable


The increased demand in the channel is partly covered bythe forced liquid re-entry. Significant stabilization of theflow may be assumed. The feedback is an undesired effectwhich influences the measurement. To compensate it andobtain genuine measurement results a semi-empirical modelis developed.

In the model the feedback effect is considered to be theincrease of the transient stability constant δ0 (representingthe maximum possible stable acceleration) by an enhance-ment factor r . The effect is influenced by the amount of gasVg in the phase separation chamber. For Vg = 0 the com-plete feedback effect takes place as discussed above. ForVg > 0 a certain damping of the feedback occurs due to thecompressibility of the gas bubble in the PSC with volumeVg . This effect is not instantaneous and results in a gradualchange of the pressure. Consequently, for Vg > 0 the feed-back effect can be assumed to be reduced, which coincideswith observations.

The three-dimensional computations outlined inSection Three-Dimensional Model of the Channel providean estimate for the limiting value of the feedback ratior = r0 = 12, which is only valid when no damping occurs,i.e. when Vg = 0. For a fully decoupled system no feedbackis assumed when the PSC is filled with gas (r → r∞ = 1,Vg → ∞). A semi-empirical model for the feedback ratio r

is based on CFD simulations and experiment results and isformulated as

r = r0 − r∞1 + Vg/Vc

+ r∞. (19)

The model for the feedback ratio is depicted by the solidline in Fig. 8. Vc = 60 ml is a calibration coefficient for thesetup. It is defined by fitting the model (between r0 and r∞)to calibration measurements, obtained by intensive gaugingof Vg and measuring the corresponding feedback (Fig. 8,open symbols, slightly more scattered). The exes depictthe transient stability measurements presented in Section

0

4

8

12

0 1 2 3 4 5

r

Vg Vc

MeasurementCalibration

Model

Fig. 8 The feedback ratio due to the damping effect of the gas Vg inthe PSC. The model (solid line) is based on Eq. (19). The calibrationdata are measurements performed to define the gradient of the model.The exes represent the actual feedback ratio of the performed transientstability measurements. At Vg/Vc = 0 calibration and measurementoverlap with r = r0 = 12

Experimental and Computational Results. A standard pro-cedure preceding every measurement is used to determineVg using calibration data and the measured feedback effect.The error of Vg for the transient stability measurementsshould therefore correspond to the scatter of the calibrationdata.

Finally, the transient stability limit can be formulated forthe maximum stable acceleration as a function of the flowrate. For r = 1 (no feedback) and f = 0 (‘short’ channel, noflexibility effect), and for Q = 0 (no steady flow effects) thestability limit is defined by the transient stability constantδ = δ0 (point A in Fig. 9). For increasing flow rates thestability limit is δ = δ0(1 − Q/Qcrit ) and it decreases dueto increasing steady flow effects (line a in Fig. 9) until thecritical steady flow rate Qcrit is reached for δ = 0.

The flexibility effect provides additional stability whenthe length of the surface is sufficient. Since the flexibilityeffect is a function of the geometry, the stability limit isshifted upwards by f for all flow rates. This results in afunction for the stability limit δ = δ0(1 − Q/Qcrit ) + f

(line b in Fig. 9).For r > 1 the feedback effect provides enhanced stabil-

ity. For Q = 0 and f = 0 the stability limit is δ = rδ0

(point C in Fig. 9) and for increasing flow rates the stabilitylimit is δ = rδ0(1 − Q/Qcrit ) (line c). Since the feedbackeffect is a property of transient flow only, the value of thelimit at the critical steady flow rate Qcrit remains the same.For f > 0 the stability limit is δ = rδ0(1 − Q/Qcrit ) + f

(line d in Fig. 9). The scaled transient stability model yields

δ − f

rδ0= 1 − Q

Qcrit

. (20)

Q / Qcrit1

C

A

} f

δ

0

b

d

c

a

Fig. 9 The transient stability model defines the maximum stableacceleration δ as a function of the flow rate Q (normalized with thecritical steady flow rate Qcrit ). The maximum possible stable acceler-ation for a simple system, i.e. without the flexibility effect (f = 0) andwithout the feedback effect (r = 1), is defined for Q = 0 by the tran-sient stability constant δ = δ0 (point A) and it decreases for increasingflow rates (line a). The flexibility effect f > 0 (line b), the feedbackeffect r > 1 (point C for Q = 0, line c for Q > 0), or a combinationof both effects (line d) can provide enhancement of the stability


0.5

1

1.5

0 2 4 6 8 10 12 14 16 18 20

Qcrit

l

EX3D1D

Fig. 10 The critical steady flow rate as a function of channel’s lengthfor the groove channel. For increasing channel length (l → ∞) thecritical flow rate seems to decrease linearly (Qcrit → 0). Experimentalvalues (EX) are shown in comparison with one-dimensional (1D) andthree-dimensional (3D) computations. The one-dimensional model isassumed to be valid for l ≥ 6; the error exceeds 6% below this limit

For r = 1 the system is fully decoupled from the feedbackeffect (line a for f = 0, line b for f > 0 in Fig. 9) andthe model applies e.g. for a capillary channel of a propellantmanagement device as shown in Fig. 1.

Experimental and Computational Results

Steady flow measurements are performed by a stepwiseapproach of the critical steady flow rate. The step is definedas an increase of the flow rate by 0.05 ml s−1 per second.That is about 1 % of the lowest observed critical flow rate.The critical steady flow rate is defined as the maximummeasured flow rate which results in a stable free surface.

During the experiments the channel length is varied andmeasurements in the range of 1 to 48 mm (0.4 ≤ l ≤ 19.2)are performed.

Figure 10 shows the critical steady flow rate versusthe scaled channel length l. The critical flow rate typ-ically increases significantly for l → 0 and decreasesasymptotically for l → ∞. The one-dimensional modelprovides satisfying results for l ≥ 6. For shorter chan-nels, the assumption of the existence of the radii R0

and R1 in Eq. (5) appears to be inadequate. The three-dimensional model covers the entire investigated range.Both results suggest that the stability of steady flow iswell understood. Furthermore, the steady measurementsserve to validate the three-dimensional computation appliedfor studies of the feedback effect as discussed in SectionFormulation of a New Model for Transient Flow Stability.

Results for a representative transient flow stability mea-surement are shown in Fig. 11a, where the normalizedacceleration δ/δ0 is shown versus the flow rate Q. The solidline represents the new transient stability model in Eq. (20).As expected, the stability limit increases significantly whenthe flow rate decreases. This is a consequence of the aggre-gate of the two effects causing the instability, the steadyflow pressure (due to convective acceleration) and the tran-sient pressure (due to temporal acceleration). The steadyflow effect increases with the flow rate and consequently thesurface collapses at lower accelerations. The transient sta-bility model in Eq. (20) provides a satisfying prediction forthe stability behavior of the liquid in the capillary channel.The slope is described accurately as well as the additionalstability, which is defined by the flexibility function f . The

Fig. 11 Transient stability measurement in comparison to the tran-sient stability model. Image a shows the flow rate acceleration (nor-malized with the transient stability constant) vs. the scaled flow ratefor a channel length of l = 10 (l′ = 25 mm) and r = 4.57 (Vg =125 ml). Experimental data for critical (CR) and stable (ST) transi-tions are given. The solid line represents the transient stability modelin Eq. (20). The dashed line represents the stability enhancement thatis ascribed to the (normalized) flexibility function f/δ0 in Eq. (18).

Image b shows the applied scaling that provides a common descrip-tion of all stability limits within one diagram. Each point represents anaverage of two measurements, a stable one and the corresponding crit-ical neighbor measurement (at the same flow rate) in image (a). Thescaling disposes of the variance due to different channel’s lengths anddue to the feedback effect. The measurements are sorted by the feed-back ratio r which obviously no longer influences the experimentalresults


model’s value for the flow rate at δ = f corresponds tothe critical steady flow rate given in Fig. 10 (l = 10,Qcrit = 0.7). For low critical flow rates (Qcrit → 0, whichapplies for l → ∞ as demonstrated in Fig. 10) the flexibil-ity function approaches f/δ0 → 0.5 according to Eqs. (17)and (18).

Furthermore, the applied scaling provides a commondescription of all stability limits within one diagram asdemonstrated by the feedback ratio r in Fig. 11(b) (differentchannel lengths and over 1000 data points are displayed). InFig. 11(b), each point represents an average value of a sta-ble acceleration measurement and a corresponding criticalneighbor (at the same flow rate) as seen in Fig. 11(a). Theinfluence of the undesired feedback ratio is obviously ade-quately modeled and, consequently, eliminated. For r = 1,which corresponds to a fully decoupled system, the phaseseparation chamber in Fig. 3 is virtually filled with gas. Thisresults in a large gas bubble that tends to oscillate, whichcauses a certain distribution of the measurements, particu-larly for high flow rates close to the origin of the diagram.However, for decreasing flow rates the observations seemto converge to the model from both sides and consequentlyconfirm the analytical prediction r → 1 for Vg → ∞ inEq. (19).

Summary

During the evaluation of experimental and numerical datasome new facts presented themselves. To begin with, itwas confirmed that numerical predictions of critical steadyflow rates are possible with a high accuracy. For three-dimensional computations of the critical steady flow rate,an error of about 5 % was observed in the predictions. Thecomputation time is in the range of days or weeks. Thepredictions with the one-dimensional model result in a sim-ilar accuracy within the range of 8 ≤ l ≤ 19.2, wherethe error is about 2 %. The application limit of the one-dimensional model may be considered to be at l = 6; forshorter channels the error exceeds 6 %. Nonetheless, theone-dimensional model covers the range that is interest-ing for technical applications. Furthermore, when using theone-dimensional model, the computation times of the steadycritical flow rate are in the range of few seconds. This is106 times faster than 3D computations. Consequently, theone-dimensional model may be considered as the tool ofchoice.

Likewise, the new stability model provides simple andreliable predictions for transient surface stability. It is basedon steady flow computations, which define one axis of thecorresponding parameter space. It has been extended to atransient stability model, in which the inertia of the accel-erated liquid column and the geometry of the surface are

taken into account. Due to the aggregate of steady andtransient flow, the critical flow rate and the critical acceler-ation are inversely proportional to each other. Furthermore,an important parameter of the flow system was identified,i.e. the transient stability constant. It defines the maxi-mum possible stable acceleration when flow rate is zero(at the end of the transition) and steady flow effects areassumed to have no influence on the stability of the freesurface.

The transient stability measurement may be significantlyinfluenced by the feedback effect, which results from liq-uid circulation in the presented experimental setup. A modelfor decoupling is developed, because the observed feedbackeffect is generally not present in technical applications. Asa result, the ‘real’ transient stability constant for the flowsystem was identified.

Finally, the flexibility effect is explained. This appliesto channels whose lengths exceed a certain limit. Theflexibility effect results in additional stabilization and itis provided by the part of the free surface which hasalmost zero curvature in the direction of flow and seemsto behave passively. It was shown that this part of thesurface acts as a damping system which relaxes thestrain of the surface at the outlet during the accelerationphase.

Acknowledgments The authors would like to acknowledge the sup-port provided by the NASA employees at Marshall Space FlightCentre. We thank NASA astronauts Scott Kelly, Catherine Coleman,and Mike Fossum, who installed and removed the experiment hard-ware on board the ISS. We also acknowledge the technical staff atAstrium for manufacturing the experiment hardware and for techni-cal support during the experiments. And we thank Lars Kiewidt, R.Jenson, and W. Blackmore for the help during the CCF experiment.

This experiment is realized in cooperation between NASAand the German Aerospace Center (DLR). The project is sup-ported by the DLR through funds from the German Federal Min-istry of Economics and Technology (BMWi) under grant numbers50WM0535/0845/1145.

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Transient Capillary Channel Flow Stability

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