Development of a CFD model and methodology for the

IN DEGREE PROJECT MECHANICAL ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2021

Development of a CFD model and methodology for the internal flow simulation in a hydrogen-powered UAV

ALESSANDRO PORCARELLI

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

SD281X DEGREE PROJECT IN AERONAUTICS, SEPTEMBER 20, 2021 1

Development of a CFD model and methodology forthe internal flow simulation in a hydrogen-powered

UAVUtveckling av CFD-modell och metodik för intern

flödesimulering i vätgasdriven UAVAlessandro Porcarelli

KTH, Royal Institute of Technology, Stockholm, Sweden

September 20, 2021

Abstract—In the context of an aviation industry

whose top priority is to face the sustainability chal-

lenge, the growing civil UAV branch is not an ex-

ception. Hydrogen-powered UAVs equipped with PEM

(Polymer Electrolyte Membrane) fuel cells are more

and more frequently identified as the most convinc-

ing and promising technology, particularly for long-

endurance mission requirements. However, the on-

board carriage of a hydrogen fuel cell leads to unex-

plored internal flow characteristics, including the in-

troduction of water vapour. The purpose of this master

thesis is to develop a valid CFD model and methodology

for the internal flow simulation of hydrogen-powered

UAVs. Given the strict environmental operational re-

quirements of PEM fuel cells, the intended application

of the model is to e�ectively assess the evolution of

the internal bay flow temperature and humidity fields.

An explicit-time fourth-order Runge-Kutta projection

method is tested successfully on a sample 2D case setup.

The case geometry and flow conditions are inspired

by the Green Raven UAV project conceived by the

Department of Aeronautical and Vehicle Engineering

at KTH.

Sammanfattning—I samband med en flygindustri

vars högsta prioritet är att bemöta hållbarhetsutma-

ningen är den växande civila UAV-sektorn inget un-

dantag. Vätgasdrivna UAV:er utrustade med PEM

(Polymer Electrolyte Membrane) bränsleceller beteck-

nas allt oftare som den mest övertygande och lovan-

de teknologin, särskilt för att de ska kunna utföra

långvariga uppdrag. Den ombordgående transporten

av en vätebränslecell leder emellertid till outforskade

inre flödesfenomen, inklusive alstrad vattenånga. Syftet

med detta examensarbete är att utveckla en lämplig

CFD-modell och metodik för intern flödesimulering

av vätgasdrivna UAV. Med tanke på de strikta mil-

jökraven för PEM-bränsleceller är modellens avsedda

tillämpning att e�ektivt utvärdera utvecklingen av de

inre flödestemperaturerna och luftfuktighetsfälten. En

tidsexplicit Runge-Kutta-projektionsmetod av fjärde

ordningen testas framgångsrikt på ett 2D-exempel.

Fallets geometri och flödesförhållanden är inspirerade

av Green Raven UAV-projektet som utförts på Farkost

och Flyg avdelningen på KTH.

I. IntroductionA. Hydrogen-powered UAVs

An unmanned aerial vehicle (UAV) can be defined asany aircraft without an onboard human pilot. It could beeither remotely piloted or controlled with an autopilot. [1]

During the last decade, civil applications of UAVs areexpanding their horizons, particularly in the surveillancefield. A growth of the UAV industry is following from thisdevelopment. The most promising solution to deal withthe long endurance requirement of surveillance missionsis combining batteries and hydrogen fuel cells in a hybridpropulsion system. On one hand, the high specific energyprovided by hydrogen fuel cells allows to reach a largeflight endurance, which common lithium polymer batteriescannot assure by themselves [2]. On the other hand, thebatteries cope with the limited specific power of fuel cells,providing additional power during critical flight phasessuch as take-o� and maneuvers [1].

There are many di�erent available fuel cell types. To-gether with several other hydrogen UAV design groups,both Lapeña et al. [2] and Herwerth et al. [1] identifyPolymer Electrolyte Membrane (PEM) fuel cells as thebest choice for long-endurance UAV mainly due to theirlow operating temperature (around 60 oC). Other advan-tages include a long life cycle, the ability to withstandlarge pressure di�erentials and low noise and vibrations.PEM use hydrogen as fuel and oxygen from the air forthe reaction. According to Lapeña et al. [2], consideringthe MTOW < 25 kg UAV category, PEM fuel cells at thecurrent maturity stage already meet the weight target ando�er competitive advantages.

B. Hydrogen for sustainable aviationThe hydrogen alternative for a more sustainable avia-

tion has been widely discussed in the last decade. Pereiraet al. [3] evaluate the environmental impact of alternativeaviation fuels including liquid hydrogen (LH2) using a


well-to-wake approach. Their study also considers theinfluence of the energy source from which LH2 is pro-duced. They show that the emissions of pollutants canbe reduced up to 60% considering LH2 produced throughelectrolysis with electricity from renewable sources. Hence,they conclude that renewable hydrogen is the less pollutingalternative.

As it will be further explained in Section I-C1, PEMfuel cells do not deliver any pollutant, given that the onlyproduct of their chemical reaction is water. Therefore, itis necessary to focus on the way hydrogen is producedand/or supplied on-site in order to assess the environmen-tal impact of PEM-powered UAVs. Considering hydrogenproduction, as also pointed by Pereira et al. [3], theprice and carbon footprint will follow the development ofrenewable energies, which is currently seeing a significantgrowth.

Taking into account hydrogen supply systems, thereare two main possible alternatives for PEM-powered un-manned aerial vehicles. The majority of UAVs rely oncompressed gas hydrogen delivered to the UAV user’slocation. The second option is producing hydrogen onboard by means of chemical hydride systems. Lapeñaet al. [2] suggest an exploration of the latter. However,the outcome related to both practical and environmentalsustainability of the system turned to be disputable. Theirwork exposed the immature stage of the technology, un-derlining particular system maintenance challenges.

C. The output of the fuel cell reaction: water1) Fuel cell reaction: In PEM fuel cells, hydrogen from

the fuel gas stream is consumed at the anode. Whilethe electrons produced go through an external circuit,the hydrogen ions (protons) move towards the cathodethrough the electrolyte. At the cathode, protons combinewith air oxygen to produce water. [2]

Water is the only output of the reaction and is expelledfrom the cathode into the oxidant gas stream. The lattertransports water in vapour state out of the fuel cell.

2) Importance of water disposal: Given the fuel cellchemical reaction, for each mole of reacting H2 one cor-respondent mole of H2O is produced. The molar massesof H2 and H2O are respectively around 2 g/mol and18 g/mol. Hence, the total mass of water vapour producedand exhausted in the UAV bay can be roughly estimatedto account for nine-ten times the hydrogen fuel "burnt" bythe fuel cell.

Clearly, a flying vehicle with a landing weight abovethe take-o� weight sounds itself very counter-intuitive,and would lead to unexplored flight mission managements.Moreover, the water vapour exhausted inside the aircraftinternal bay might condense at the walls, potentiallycausing the centre of gravity to shift and thus a�ectingthe overall vehicle stability. These two issues underline theneed to dispose this exhausted water outside the bay.

Considering the possible environmental consequences ofwater disposal, Khandelwal et al. [4] analyse the global

warming potential (GWP) of contrails and reassure aboutthe sustainability of the process.

3) Fuel cell humidification: Before being exhausted, thewater produced at the fuel cell cathode is used to contin-uously wet the fuel cell radiator to enable its evaporativecooling. This process is known as "humidification". Wernerat al. [5] remark the important role of a proper humidifica-tion for the fuel cell performance. Their study explores theinfluence of several aviation-relevant parameters a�ectingthe humidification process, such as operating pressure andtemperature. Moreover, the influence of dynamic loadsis further accounted by Gong et al. [6], and their studyultimately identifies in the limited operational robustnessthe major barrier to maturing fuel cells for commercialUAVs.

D. Ventilation of the UAV bayAs introduced in section I-C1, at the fuel cell cathode

protons combine with oxygen from air. Clearly, the air inthe proximity of the fuel cell intake should be su�cientlycool such that the cathode is always supplied with oxy-genated air. As the total volume of air inside the UAV’sbay is limited, it is necessary to develop an intake-exhaustsystem such that proper ventilation is allowed.

Furthermore, most fuel cells are designed to operateup to a limit ambient temperature. At the same time,the fuel cell exhaust air-stream temperature is generallylarger than the designed limited ambient temperature. Anexample manual where these temperatures are specifiedcan be found for the Horizon H-1000 [7]. Therefore, aproper ventilation is necessary also to avoid overheatingof the bay temperature, which could lead to overheatingof the fuel cell itself and thus automatic shut o�, as alsopointed by Gong et al. [6]. Barroso et al. [8] remarkthe importance of the internal cooling and perform avery interesting wind tunnel heat transfer analysis. Theyexplore the influence of geometrical and airflow parameterswith the goal of designing an optimal passive coolingsystem for a hydrogen-powered UAV.

E. Intake-exhaust systems in UAVsAt the current state of the art in hydrogen fuel cell

powered UAVs, there is very very limited informationregarding intake-exhaust system designs. Many of the pro-posed designs do not consider this issue at all, while otherimplement basic systems supported by rough calculationsor wind tunnel assessments.

A representative example of the way the problem hasbeen approached so far can be found in two studies byKim et al. [9] and Kim et al. [10], in which the emphasisis placed on the necessity to cool the stack to ensure itsperformance. In Figure 1, intakes have been introduced atthe fuselage nose of the glider-shaped UAV. Hence, theair flows through the bay and is exhausted at the rearof the fuselage, taking advantage of the pusher propellerwhich attracts the flow. Further, Kim et al. [10] clarifythat the intakes have been dimensioned such that they


Figure 1: Intakes of the UAV designed by Kim et al. [9].

(a) Inlets. (b) Outlets.

Figure 2: Manufactured inlets 2a and outlets 2b of theUAV built by Herwerth et al. [11].

introduce a mass flow rate equal to five times the massflow rate pulled by the fuel cell fan. Once the UAV hasbeen equipped with these roughly dimensioned intakes, thewind tunnel tests provided evidence that su�cient air wassupplied to cool down the stack. No optimization of thesystem is documented.

The most interesting example of an intake-exhaust sys-tem on hydrogen-powered UAVs is provided by Herwerthet al. [1] and is further developed by Chiang et al. [11].Figures 2a and 2b respectively show the equipped intakeand the exhaust ducts. The authors state that the inlethas been dimensioned to exceed the fuel cell fan flow rateby 25 % [1]. Such mass flow rate turned out to be su�cientto cool down the stack and guarantee the quality of thechemical reaction. This is significantly lower than the 500% suggested by Kim et al., thanks to a much more refinedbay flow management.

Considering non-hydrogen-powered UAVs, more de-tailed and complex passive cooling systems are doc-umented. For instance, Panagiotou et al. [12] dimen-sion cooling ducts to e�ciently supply the payload withproperly directed cooling air. The ducts are connectedto aerodynamically-e�cient NACA-shaped intakes whosepreliminary sizing methodology is validated through aninternal flow CFD simulation.

F. Goals and objectivesThe purpose of the present work is to develop a CFD

model and methodology to simulate the internal flow inthe "Green Raven" UAV. The "Green Raven" project is

Figure 3: Green Raven design with fuel cell and hydrogentank positioned onboard [14].

a multidisciplinary research project aiming to develop ahybrid-electric fixed-wing UAV powered by a hydrogenfuel cell [13].

The Green Raven falls within the MTOW < 25 kgUAV category and its design consists of a blended-wing-body (BWB) configuration (Figure 3), with a wingspanof 4m [14]. It is planned to be equipped with the Hori-zon H-1000 PEM fuel cell supplied by hydrogen storedonboard in a compressed tank. According to the usermanual [7], the fuel cell system is self-humidified throughan internal controller. Given that the UAV is intended tofly at operating altitudes around sea level, the operatingtemperature and pressure are not expected to vary in awide range. Therefore, for the purposes of this study, thehumidification system can be safely considered a "blackbox".

For its overall mission, the Green Raven is expectedto carry onboard a total of 80g of H2. According tothe estimation in Section I-C2, this means that duringthe flight around 800g of water vapour to dispose willbe produced. None of the studies available in literature(Section I-E) mention the way the fuel cell products areexhausted. However, water vapour disposal is more crucialin a BWB configuration like the Green Raven’s, since apotential near-wall condensation could shift the centre ofgravity and a�ect the BWB light stability.

The Horizon H ≠ 1000 is designed to operate up toan ambient temperature of 30 oC, and shut o� occurswhenever its inner operating temperature exceeds 65 oC[7]. Due to its narrow internal spaces, the BWB config-uration requires a more careful management of the heatreleased by the fuel cell. Moreover, the 1000 W fuel cell willintroduce in the UAV bay a greater internal dissipationheat flux in comparison to the 150 W of Herwerth et al.[1], thus probably requiring a larger flow rate of coolingair.

With these considerations in mind, the developed CFDsolver for the internal flow should consider both the heattransfer and the humidity. Given a sample 2D geometrywith a sample intake-exhaust system, di�erent ways of ac-counting these two combined flow features will be explored


and assessed.

II. TheoryA. Governing equations

Employing the continuum hypothesis, the fluid-dynamicbehaviour of air is described by the Navier-Stokes equa-tions, i.e. the mass scalar, the momentum vector and theenergy scalar conservation equations. At this stage, these5 partial di�erential equations (PDE) contain 7 unknowns.As the Green Raven cruise speed is U = 20 m/s [14], it issafe to assume the flow to be incompressible. Hence, theNavier-Stokes equations can be simplified, and the energyequation can be decoupled from the mass and momentumequations, which will form a new closed system. Fur-thermore, as the turbulent entities are a defining featurefor internal flows, it is necessary to employ a turbulencemodelling. To achieve a good quality of the mean turbulentflow without reaching a too high computational cost, aReynolds averaged Navier-Stokes (RANS) approach hasbeen chosen. RANS are based on a statistical modellingtechnique, that consists in a decomposition of the flowvariables (e.g. the velocity U) into a mean U and afluctuating contribution u, where the U corresponds to anensemble average:

U = U + u. (1)This so-called "Reynolds decomposition" is applied to theNavier-Stokes equations resulting in the RANS equations:

Ûi

ˆxi= 0; (2a)

Ûi

ˆt+ Uj

Ûi

ˆxj= ≠1

fl

ˆP

ˆxi+ ˆ

ˆxj(‹ Ûi

ˆxj≠ uÕ

iuÕj). (2b)

Whereby all Ui correspond to Ui, as the mean symbol hasbeen skipped for simplicity. In accordance with the UAVinternal flow CFD methodology employed by Panagiotouet al. [12], the gravity term has been neglected in themomentum equation, since the flow is expected to bedominated by the advection and the di�usion term. Fur-ther considerations on this assumption will be discussedin Section V.

B. TurbulenceThe last term in Equation 2b consists of the divergence

of the so-called "Reynolds stress tensor", and the lattershould be modelled in order to close the system. Mostof RANS turbulent closures are based on the Boussinesqhypothesis, where the traceless part of the Reynolds stresstensor is assumed to be proportional to the Strain RateTensor Sij with the coe�cient ‹T (Eddy Viscosity):

≠ uÕiu

Õj = 2‹T Sij ≠ 2

3k”ij . (3)

By dimensional analysis, it holds that ‹T is proportionalto the product of a turbulence velocity and length scale.These two parameters can either be prescribed (algebraicmodels), partially prescribed (one-equation models) or atransport equation can be solved for both of them or forrelated quantities (two-equations models).

C. Heat transferGiven the incompressible flow assumption (Section

II-A), the temperature T can be be solved as a passivescalar. Neglecting the radiation term, it follows a standardadvection-di�usion equation:

ˆT

ˆt+ Ui

ˆT

ˆxi= DT

ˆ2T

ˆx2

i

. (4)

One could object intuitively that a substantial change intemperature changes the flow thermodynamic pressure. Inan incompressible flow in an open domain box, however,this thermodynamic pressure is decoupled from the fluiddynamic pressure included in Equation 2b, so the solutionof the governing equations is not a�ected.

Furthermore, the constant-density assumption could ap-pear restrictive if a wide temperature range is experiencedin the flow domain, thus causing a non-negligible den-sity change. However, as otherwise the model complexitywould increase substantially, the density in the momentumEquation 2b is considered constant. The mean flow fea-tures should not change significantly, but the influence ofthis assumption on the solution should be further assessed(Section V).

The air thermal di�usivity DT is given by the sum of aphysical contribution – and a modelled physical turbulentcontribution –T , where – can be found as

– = k

flcP, (5)

whereby k is the thermal conductivity and cP is the spe-cific heat capacity. Given a constant thermodynamic pres-sure, k and cP only vary with T itself. As the maximumtemperature range should be limited (�Tmax ¥ 40 oC),– should not vary as substantially to a�ect the scalartransport significantly. Therefore, in order to simplify themodel, – is assumed constant. Considering –T , a constantvalue educated guess should be performed by means ofReynolds analogy, i.e. defining a turbulent Prandtl number

PrT = ‹T

–T. (6)

D. HumidityTo account for humidity in the computations, it is

necessary first to choose a proper humidity definition. Thespecific humidity could be pointed as the most indicatedparameter, and it is defined as the weight of water vapormv contained in a unit weight of moist air mv + md

q = mv

mv + md= r

1 + r[15], (7)

where md is the mass of dry air. As Equation 7 shows, thedefinition can be re-arranged as a function of the mixingratio r, which is defined as mv/md [16] and can be foundgiven the equilibrium water vapour pressure p0 in mbar as

r = 0.622 p0

patm ≠ p0

[15], (8)

for which p0 can be found for a given constant temperaturefrom Antoine’s equation [17].


Humidity variation can a�ect the density in the mo-mentum Equation 2b, and varies itself with temperature,as p0 is dependent on T . Similarly to the assumptionperformed for the temperature influence on the density inSection II-C, in order to limit the model complexity, theinfluence of q on fl is not accounted in the model. Hence,q is similarly solved as a passive scalar, and its transportequation reads:

ˆq

ˆt+ Ui

ˆq

ˆxi= Dq

ˆ2q

ˆx2

i

. (9)

Moreover, in order to keep the two scalars separated andsubstantially simplify the model, the influence of T on q isneglected. As a consequence, the model will impreciselyestimate the same q in two points with the same mv

and di�erent T . However, with the goal of understandingthe humidity transport and distribution, and ultimatelyevaluate water vapour disposal, this assumption can beconsidered acceptable, as will be confirmed in SectionIV-D.

Under particular conditions, for instance in near-wall,low-speed or low-temperature flow, the water vapour cancondense into liquid water. Nonetheless, for the class ofinternal flows investigated, the moist air leaving the fuelcell will be invested by a moving flow of dry air with arelevant momentum. This flow feature, together with alimited particle resident time inside the bay, establishesnon-favourable conditions for condensation. Therefore, forthe considered class of UAVs internal flows, in order tolimit the model complexity, the water condensation is nottaken into account. Even if considering a larger fuel cellstack temperature and a lower ambient temperature, inthe wind tunnel model tested by Barroso et al. [8] there isno hint that condensation occurs, thus raising confidencein the reliability of the assumption.

In analogy to DT , the water vapour di�usivity in air Dq

consists of the sum of a physical contribution Dqp and amodelled turbulent contribution DqT . The former can befound for a given temperature as

Dqp = 0.2303

T

273

41.81

, (10)

according to Schirmer’s formula found in Lee et al. [18].Similarly to – in Section II-C, chosen a reference tem-perature, Dqp is taken constant. Considering the latter, aconstant value should be again educatedly guessed withReynolds analogy as for –T , defining a turbulent Schmidtnumber

ScT = ‹T

DqT

. (11)

E. Numerical schemes and solverIn order to numerically resolve the flow, the govern-

ing equations have to be discretized in space and time.The choice of the discretization numerical schemes willdetermine the consistency and the stability of the problemwhich are necessary conditions for its convergence.

The consistency property is related to how well the dis-cretized equations approximate the PDE. The di�erencebetween the exact PDE and its discrete representation iscalled truncation error. To allow the consistency of theproblem, this error needs to go to zero decreasing the gridspacing in space (�x, i.e. mesh spacing) and time (�t).On the other hand, stability is related to the sensitivity ofthe solution to disturbances, which can be errors from anysource. If a problem is unstable, the solution might blowup at some time. For convectively dominated transientproblems, the so-called CFL stability condition can bederived considering the advection term of Equation 2b

Co = UŒ�t

�xÆ 1, (12)

where Co is called "Courant" number. The presence of�x in the condition also points out the importance of themesh quality to determine the problem stability, as a singlepoorly shaped cell can blow up the simulation. Hence,particular attention should be focused on the fulfilmentof the standard mesh quality criteria prescribed by thesolver. [19]

For the Lax equivalence theorem, consistency and sta-bility are the necessary and su�cient conditions for theconvergence of the problem. The convergence propertylinks the numerical solution to the exact solution of thePDE. If a simulation is converged to very low residuals,the two solutions will be very close, i.e. the numerical simu-lation closely reproduces the real flow field. However, thereare some cases where the two solutions do not accuratelymatch even if numerical convergence is achieved. Thisgenerally occurs in one or more of the following conditions:

• the chosen order of accuracy is too low, i.e. thetruncation error introduced is too large;

• the grid spacings �x and �t are large, leading simi-larly to a large truncation error;

• the overall mesh quality is poor for complex flowgeometries discretized with unstructured grids.

In light of the above considerations, with the goal ofobtaining the best numerical solution, it might seem obvi-ous that one should employ the highest order of accuracypossible with very refined grid spacings. However, thehigher the order of accuracy, the higher is the possibilitythat instability and uncontrolled oscillations grow [19].Moreover, higher-order schemes and refined grid spacingsrequire larger computational time and are thus moreexpensive. Therefore, a reasonable compromise should befound.

F. OpenFOAM solversOpenFOAM is a an open-source C++ toolbox for the

development of customized numerical solvers, most promi-nently addressing CFD [20]. It provides a wide range ofbuilt-in solvers suitable for a broad variety of classes offlows. Its flexibility lies in the possibility to customizethese built-in solvers according to the particular featuresrequired by the flow case investigated. For instance, in


this work an incompressible PISO-based algorithm willbe adjusted to account for heat transfer and humidity,implementing the equations reported in Sections II-C andII-D.

The PISO algorithm (Pressure Implicit with Splittingof Operators, of Issa 1986 [21]) is a non-iterative transientcalculation procedure, where all time-dependent terms areretained in the momentum and continuity equations [22].PISO consists of an implicit second-order backward timeintegration method which couples pressure and velocityin each timestep [21]. The coupling involves one predictorstep, where an intermediate velocity is computed using thepressure of the previous timestep, and a number of correc-tor steps, where the intermediate and final velocity andpressure are obtained iteratively [23]. If solved for a longperiod of time with guessed initial conditions, a steadystate can sometimes be achieved. Even if computationallyexpensive, a transient solver is highly recommended for in-ternal flows accounting for heat transfer and other scalars.Further details on the PISO algorithm can be found inacademic textbooks, such as Versteeg and Malalasekra[22].

As mentioned in Section II-E, low-quality cells can causeinstability and potentially simulation divergence. Whilemost commercial solvers are equipped with numerical cor-rectors limiting mesh-originated instabilities, OpenFOAMis very sensitive to the mesh quality. Hence, particularattention should be focused on the mesh quality criteriaespecially the so-called "cell non-orthogonality".

G. PointwisePointwise is a commercial mesh generator software spe-

cialized in CFD applications. It covers all aspects of pre-processing, from import of geometric models, to meshesexport directly with the proper format for selected flowsolvers, including OpenFOAM. High-quality grids canbe created quickly, intuitively, and eventually throughscript programming. Its highly customizable grids andits quality-check tool are very helpful to allow a goodsimulation convergence, particularly given the complexgeometry and flow pattern investigated (Section III-A)and the strict quality limitations of the solver.

III. CFD caseA. Geometry

In order to properly test the developed CFD model, arealistic sample case should be built. The case geometryshould desirably fulfil the following requirements.

1) It should be computationally cheap to allow fastiterations of the model tests.

2) It should reproduce as close as possible the realfeatures of the Green Raven UAV, so that the re-sults will be representative of realistic internal flowfeatures.

3) The sample intake and exhaust should be placedand dimensioned appropriately, assuming that this

TT

1,7 m1,7 m

UUxx

zz

TankTank

Fuel cellFuel cellFans pullingthe flow

Fans pullingthe flow

IntakeIntakeExhaustExhaust

Stack outlet flowStack outlet flow

Figure 4: 2D mid-section of the Green Raven bay showinghow the fuel cell fans pull the air in the vertical direction.

configuration will be a starting point for a potentialfuture optimization.

4) In order to appreciate relevant temperature andhumidity changes, and assess the model functioning,the intake-exhaust system should introduce a coolantflow rate comparable with the stack flow rate. Theresulting strong temperature and humidity increaseinside the UAV bay will be deliberately exaggerated.

In order to substantially save computational cost, a 2Dconfiguration has been chosen. The mid-cross-section ofthe Green Raven has been reconstructed. It consists ofthe airfoil MH104 with chord c = 1.7m rotated accordingto the geometrical twist and the cruise angle of attack– = 1.357 o [14]. The fuel cell and the tank have beenplaced inside the internal bay according to a preliminarystability analysis performed in [14]. The resulting CADconfiguration is shown in Figure 4.

In the configuration by Herwerth et al. [1] (Section I-E),the intake-exhaust system takes advantage of the fuel cellfan to e�ciently convey the flow longitudinally towardsthe rearward outlet. In contrast, Figure 4 shows that theonly way for the fuel cell to fit inside the BWB is to liedown on the bay floor. Hence, the fans will pull the airthrough the fuel cell and out of the stack perpendicu-larly to the longitudinal direction. Therefore, the designedintake-exhaust system cannot exploit the external flowmomentum. On the other hand, a beneficial feature of theBWB configuration lies in its airfoil’s external pressuredistribution. In fact, the intake-exhaust system can takeadvantage of the higher pressure regions to introduce coolair inside the bay and of lower pressure regions to exhaustthe internal flow outside the bay. Therefore, as shownin Figure 4, the intake has been placed at the leadingedge close to the external flow stagnation point, and theexhaust is located in the upper surface in the proximityof the airfoil maximum thickness. The dimension of theintake and the exhaust has been kept as small as possiblein accordance with the fourth requirement listed above.The consequence of this choice will be discussed in SectionIII-B.

B. GridThe computational domain consists of the 2D internal

bay geometry shown in Figure 4 (Section III-A). The inlet


Fuel cellFuel cellInlet coneInlet cone

Inlet intakeInlet intake

Near-wall inflations

(a) Inlet mesh region.

Fuel cellFuel cell TankTank

Outlet coneOutlet cone

Outlet exhaustOutlet exhaustNear-wall inflations

(b) Outlet mesh region.

Figure 5: Mesh design in correspondence of the inlet (5a)and outlet (5b).

and the outlet are located respectively in correspondenceof the leading edge intake and the upper surface exhaust.Their presence allows to thermodynamically consider thedomain an open box, thus permitting to decouple ther-modynamic and fluid-dynamic pressure as mentioned inSection II-C. Moreover, the fuel cell bottom fans pull theflow out of the domain, while the top stack re-introduces inthe domain the same air flow rate. The rest of the internalsurfaces act as walls, including the tank and the airfoilinterior shell.

As mentioned in Section II-E, the spatial discretizationof the computational domain, i.e. the mesh designed, cana�ect stability, convergence and reliability of the simula-tion results. First of all, the mesh should be fine enoughto capture all the relevant flow features. As the intake andthe exhaust act as inlet and outlet and thus play a keyrole in the flow definition, their patch should consist ofa significant number of grid points. At the same time,as mentioned in Section III-A, the intake and exhaustdimension has been kept as small as possible. Therefore,the grid �x results to be very small, as shown in Figure5. Given the cost-e�ectiveness of a 2D configuration, fromthe spatial perspective this does not represent a majorexpense issue. However, as discussed in Section II-E, inorder to preserve the simulation stability, the �t shouldbe adjusted according to �x such that the CFL condition(Equation 12) is verified. In this case, the resulting nec-essary very small �t increases the computational expensesubstantially. Nevertheless, as it will be shown in SectionIV-A1, for the purposes of this study of assessing themodel validity and appreciating relevant temperature and

humidity changes, it is not necessary to run the tran-sient simulation for a long physical time and this trade-o� is thus accepted. In a future 3D optimization case,conversely, larger local span-wise intakes can be designed,thus allowing a reasonably large �x yielding a reasonablylarge �t.

One of the main mesh design requirements is that itsedges should be parallel to the gradients. Hence, in orderto capture the expected inlet jet-like flow and the outletconfluence flow, a cone-shaped domain fully consisting ofquadrilaterals has been designed in correspondence of theintake and the exhaust (Figure 5a and 5b).

Considering the near-wall region, since the most signif-icant gradients related to the boundary layer are in theperpendicular direction to the surface, layers of structuredrectangular high aspect ratio cells are inflated above thesurfaces. Even if the boundary condition is not "wall" (seeSection III-E), the inflation is e�ectively designed also incorrespondence of the fuel cell stack and fans. In fact, asit will be confirmed in Section IV, the order of magnitudeof the in-flowing and out-flowing perpendicular velocity ismuch smaller than the horizontal velocity of the "primary"flow.

The y+ is a non-dimensional wall unit calculated as

y+ = yu·

‹(13)

where y is the distance from the wall and u· is the ShearVelocity. In very close proximity to the wall, i.e. in theso-called viscous sub-layer where y+ < 5, the turbulentfluctuations are gradually suppressed by viscous e�ects.To allow a good near-wall resolution, the height of thefirst inflation cell should be small enough to fully encloseonly the viscous sub-layer region. For most external flows,a dimension estimation can be derived by means of the flatplate turbulent boundary layer formulas, given a referencelength L and velocity U . For the considered complexinternal flow, h1 = 0.5 mm is estimated in the samemanner. However, as it is very di�cult to realisticallychoose L and U , the estimation validity should be furtherverified by means of a y+ check, which will be laterperformed in Section IV-A3. Nevertheless, considering thecase geometrical features, the flow is not expected to besignificantly a�ected by the near-wall resolution.

The design of the mesh features is realized by meansof Pointwise T-rex algorithm. Given the inflation firstcell thickness y1 and a growth factor, Pointwise inflatesrectangular cells until a very smooth dimension transitionwith the parent mesh is achieved. The parent mesh consistsof triangles and quadrilaterals which are automaticallyshaped to accomplish high-quality criteria. Overall, themesh counts 50k cells. It can be roughly estimated thatthis cell count allows performing an e�cient problem spaceparallelization up to 4 logical processes in the computa-tion.

As mentioned in Section II-F, OpenFOAM is very sen-sitive to the mesh quality, and a single low-quality cellcan cause instability and simulation divergence. Pointwise


Figure 6: Trailing edge domain cut to avoid skew cells inthe cusp.

provides a very wide toolbox of functionalities giving thepossibility to prescribe strict quality criteria to be fulfilledby the algorithm, and a very e�cient quality check tool.The main checks generally to be performed for 2D meshesinvolve the skewness and the aspect ratio.

• The skewness is kept well below the suggested thresh-old value of 0.8 in the whole domain. Only a totalof 7 cells in correspondence of the concave cornersin the cavity between the two fans exceed the limit.Nevertheless, given the irrelevance of the flow in thatcavity, the solution stability is not a�ected. To achievethis high-quality, given the expected irrelevance of theflow in the trailing edge region, the rear concave cusphas been cut as shown in Figure 6.

• The aspect ratio is kept below 10 in the whole domain,including the boundary layer cells.

Furthermore, as introduced in Section II-F, OpenFOAMhas its peculiar rule of thumb to assess the mesh quality,i.e. the so-called "cell non-orthogonality" should be keptdesirably below 70. Its built-in tool "checkMesh" allows toquickly verify this condition. The checkMesh performedon the final design provided a maximum non-orthonalityof around 66, thus ultimately finding evidence of the meshhigh-quality.

In most of the CFD studies, a best-practice to provethe simulation consistency and convergence is to perform aso-called grid-convergence check. It consists of monitoringthe solution variation progressively decreasing the mesh�x. In the studied internal flow, however, there is nocomputed solution coe�cient to monitor the variationwith �x. Moreover, the sensitivity of OpenFOAM toinstabilities ensures that an improper mesh will cause thesimulation to diverge instead of providing artificial andnon-realistic converged results. Hence, for the purposesof the present study, the computationally-consuming grid-convergence check can be avoided.

C. Numerical methodsAs introduced in Section II-E, the temporal and spatial

discretization of the governing equations determines thesimulation convergence and the accuracy of its results.While higher-order schemes introduce a minor truncationerror and are thus more precise, lower-order schemes aremore dissipative and prevent instabilities to grow.

Considering the temporal discretization, two di�erenttransient integration methods have been used and com-pared. The first method consists of the default implicit-

time second-order backward PISO solver with two cor-rector steps (Section II-F). However, the large simulationtime caused by the very small �t (Section III-B) ledto the search for a more e�cient solver. The choice fellon the low-dissipative fourth-order explicit Runge-Kutta(RK) projection method published by Vuorinen et al. [23].Mathematically, RK is considered one of the standardmethods to solve transient problems of the form

dU

dt= f(t, U). (14)

The classical fourth-order RK-method (RK4) consists ofthe solution Un progression to Un+1 according to thefollowing update sequence:

Y___________]

___________[

k1 = f(tn, Un)�t,

k2 = f(tn + �t/2, Un + k1/2)�t,

k3 = f(tn + �t/2, Un + k2/2)�t,

k4 = f(tn + �t, Un + k3)�t,

Un+1 = Un + (k1 + 2k2 + 2k3 + k4)/6,

tn+1 = tn + �t.

(15)

In contrast to the operator splitting and corrector loop inPISO (Section II-F), the new solver employs the projectionmethod to couple pressure and velocity, as is often thecase when dealing with explicit integration time methods.The method consists of the projection of the velocity fieldonto its solenoidal counterpart using the pressure gradientin a single projection step [23]. The final OpenFOAMRK-projection code has been implemented following thealgorithm account, development and test provided byVuorinen et al. [23]. Their study also shows an overall algo-rithm e�ciency increase up to nearly 50% in comparisonto PISO, whilst not detecting any significant instabilitygrowth due to the higher order of accuracy.

Considering spatial discretization, particular attentionhas been focused on divergence schemes. OpenFOAM pro-vides a variety of options ranging from a highly-stable buthighly-dissipative first-order scheme (Gauss upwind) anda more accurate but more unstable second-order scheme(Gauss linear) [24]. A quick test for the analysed flow caseshowed that the Gauss linear divergence scheme for thevelocity field is too unstable and brings the simulation todivergence. In contrast, the Gauss upwind scheme quicklyconverged to steady conditions due to its high dissipationfeature. Hence, a reasonable compromise should be foundin order to achieve su�ciently accurate results whilstavoiding the simulation divergence. Following the Open-FOAM guide [24], a set of preliminary comparative sim-ulations between divergence schemes has been performed.The blend Gauss linearUpwind scheme has been provedto be the most reasonable compromise, providing stableresults whilst not suppressing the most relevant turbulentstructures through numerical dissipation. This scheme isthus employed for the velocity, temperature and humidityfield. For the rest of the fields, the tests showed that theGauss linear scheme can be retained without generating


stability issues. Considering the discretization schemesfor the remainder spatial operators, some tests showedthat they do not significantly a�ect the stability and theaccuracy of the solution for the analysed case. Hence,the default setup has been kept considering a standardOpenFOAM tutorial for the PISO solver ("pitzDaily", [25])and is reported in Table I.

Operator Scheme

Gradient Gauss linear

Laplacian Gauss linear corrected

Interpolation linear

Surface-normal gradient corrected

Table I: Spatial discretization schemes employed for eachoperator.

D. Turbulence modellingFor the internal flow investigation of the present study,

the two-equations k ≠ Ê SST has been selected as themost indicated RANS-based turbulence model. In fact, themodel is recognized as the most reasonably general andreliable for a wide class of flows [26]. Furthermore, thischoice is in accordance to the UAV internal flow simulationprovided by Panagiotou et al. [12].

The k ≠ Ê SST model consists of solving a transportequation for the turbulent kinetic energy k and the turbu-lent frequency Ê. The turbulent frequency is defined as

Ê © Á

Cµk, (16)

whereby Á is the so-called "turbulent dissipation rate" andCµ = 0.09 is a model constant. There follows the modelexpression for the Eddy Viscosity

‹T = k

Ê, (17)

which allows to close the system of Equations 2. A com-plete most-recent illustration of the model, including thetransport equations for k and Ê, can be found in Menteret al. [27].

E. Boundary conditionsIn order to correctly define and solve the simulation

problem, a realistic definition of the physical boundaryconditions (BC) should be performed. For internal flowcomputations, it would be good-practice to extend thedomain and introduce an artificial external co-flow in theproximity of the real boundaries in order to develop somenumerical dissipation and make the solution more stable.However, the co-flow introduction in a transient and low-�t simulation complicates the case setup significantly andsubstantially increases the computational expense. Hence,as the well-thought-out grid design and numerical setup ofthe present case allows the simulation to converge regard-less of the co-flow, its addition is skipped. An overview ofthe final geometrical patches associated to each boundarycondition name is shown in Figure 7.

airfoilfuelcellstack

inlet

outlet

fuelcellhotwalls

fuelcellwallsfuelcellfans

tank

Figure 7: Outline of the case boundary conditions names.

1) Boundary velocity: The boundary conditions for thevelocity field U for each patch are reported in Table II.

Patch BC type U uniform value [m/s]

inlet fixedValue (20, 0, 0)

outlet zeroGradient –

fuelcellstack fixedValue (6.42 · 10≠5

, 2.71 · 10≠3

, 0)

fuelcellfans fixedValue (1.22 · 10≠4

, 5.14 · 10≠3

, 0)

airfoil

fuelcellhotwalls

fuelcellwalls

tank

noSlip (0, 0, 0)

Table II: Boundary conditions for U .

While a noSlip condition has been prescribed at thewalls, the fuel cell stack and fan velocities have beenestimated considering the regime flow rate processed bythe fuel cell of Q̇ = 10 l/min [7]. Their resulting valueis orders of magnitude smaller than the primary flow. Asdiscussed in Section II-D, this flow feature establishes non-favourable conditions for condensation, thus allowing toreasonably neglect its consideration in the model.

Moreover, Table II shows that the inlet is assumed toconsist of a uniform horizontal velocity of 20 m/s, i.e.equivalent to the external flow cruise speed. On one hand,this "ideal" intake guess can be considered reasonable,as the inlet has been placed in very close proximity tothe airfoil stagnation point. On the other hand, the realinlet velocity profile is not expected to be uniform and aslarge, because the very small intake dimension introducesconsistent viscosity and blockage e�ects. Hence, the in-troduced flow rate in the simulation may be significantlyover-estimated. Nevertheless, for the model developmentpurpose of the present study, the inlet velocity BC iskept uniform to 20 m/s in order to simplify the casesetup. Further research on the inlet boundary condition isrecommended if a future 3D intake-exhaust optimizationstudy will be performed.

2) Boundary pressure: Considering the pressure field(p), a zeroGradient boundary condition is set up for allthe patches except for the outlet. In fact, at the outleta fixedValue condition should be prescribed in order forthe problem to be well-posed [19]. In most of the external-flow outlets, a uniform 0 value is conventionally applied.In contrast, in this internal-flow case study, the exhaustis placed to take advantage of the suction enhanced bythe low-pressure region on the upper surface of the airfoil,as discussed in Section III-A. Hence, a realistic negative


Figure 8: Contour plot at the mid-cross-section planecomputed by the external flow simulation.

relative pressure value to prescribe should be estimated.The estimation is performed by means of a preliminarysteady-state 3D external flow simulation over the model incruise conditions. A contour plot of the resulting pressurefield at the mid-cross-section is shown in Figure 8. Therelative pressure found in the proximity of the exhaustposition and prescribed as a uniform boundary conditionfor the internal flow is

pout = ≠144.3 Pa. (18)

Similarly to the inlet-intake aforementioned (SectionIII-E1), this outlet condition is "ideal" and is consideredacceptable even if its uniformity and precision should befurther assessed for a real flow case.

3) Boundary turbulence quantities: As described in Sec-tion III-D, the k ≠ Ê turbulence model consists of solvinga transport equation for k and for Ê. Hence, a boundaryvalue for these two parameters should be prescribed aswell. The value k is defined as

k © 32(IUŒ)2, (19a)

I = urms

UŒ, (19b)

where urms is the flow characteristic turbulent velocityscale. A rough guess for the inflow I can be estimated andthus the boundary k can be derived. According to Wallin[28], in non-turbomachinery internal flows

I ≥= 0.3 %, (20)

and this boundary guess is not expected to a�ect thesolution mean flow field significantly, given its small mag-nitude [28]. Considering the fuel cell inflow conditions, incontrast, it is reasonable to predict a larger turbulenceintensity created by its internal flow path, i.e.

I ≥= 3 %. (21)

Nevertheless, the much smaller inflow speed (see Table II)will yield a very small inflow turbulent kinetic energy.

The inlet boundary Ê can be further obtained fromEquation 17 given a reasonable ‹T . A good estimate for ‹T

can be again provided from the aforementioned externalflow simulation (Section III-E2). In the intake proximitythe computed parameter reads

‹T≥= 0.2 m2/s. (22)

On the other hand, to estimate the fuel cell inflow con-ditions, it is more convenient to introduce a turbulentlength scale L which can be assumed around 10% of thegeometrical length of the patch [28]. Given that

L = k1/2

Á, (23)

the following expression for Ê can be derived from Equa-tion 16:

Ê = k1/2

CµL . (24)

The final turbulent boundary inflow conditions are sum-marised in Table III.

Patch BC type k [m2/s2] Ê [1/s]

inlet fixedValue 5.4 · 10≠3

1.8 · 104

fuelcellstack fixedValue 9.9 · 10≠9

4.2 · 10≠2

fuelcellfans fixedValue 3.6 · 10≠8

2.3 · 10≠1

Table III: Inflow boundary conditions for the turbulentparameters. All the reported values are uniform across thepatch.

Furthermore, the implementation of the k ≠Ê SST modelin OpenFOAM requires also a file prescribing the bound-ary ‹T . For the inflow patches, however, the condition is oftype "calculated", i.e. the model computes ‹T itself givenk and Ê (Equation 17).

The boundary wall patches include all those where thenoSlip condition has been prescribed in Table II. Given theuncertainty on the first cell thickness estimation providedin Section III-B, in order to safely resolve the near-wallflow region, a wall function should be set up at theboundary wall patches for k, Ê and ‹T . The default wallfunctions of the most popular PISO tutorials (pitzDaily)have been retained and are reported in Table IV.

Quantity BC type BC value

k [m2/s2] kqRWallFunction 5.4 · 10

≠3

Ê [1/s] omegaWallFunction 3.2 · 104

‹T [m2/s] nutkWallFunction 0

Table IV: Wall boundary conditions for the turbulentparameters. All the reported values are uniform across thepatch.

Even if both ‹T = 0 and k = 0 at the walls, inthe very near-wall region k increases suddenly. Hence, itis suggested to initialize the wall boundary value withthe same k as the inlet [29]. In addition, the followingexpression dependent on the grid first cell thickness h1 isrecommended for the wall boundary Ê initialization:

Êwall = 10 6‹

—1h1

, —1 = 0.075 [30]. (25)


Further, during the simulation, k and Ê are solved acrossthe iterations by means of the wall function and their wallvalue is updated.

At last, k and Ê BCs are set to type "zeroGradient" atthe outlet similarly to U , while the outlet ‹T is set to type"calculated" and its value is thus computed from k and Êacross the simulation.

4) Boundary temperature: The intake supplies the in-ternal bay with external air. Hence, as the Green Ravenflies in cruise conditions close to the sea level, a standardT = 20 oC has been prescribed at the inlet. Furthermore,according to the fuel cell manual [7], the operating stacktemperature is set to T = 60 oC.

The fuel cell has an operating e�ciency of 40% [7].This means that, known the fuel cell rated power of1000 W , approximately 600 W are dispersed into heat.This thermal power is introduced inside the internal baythrough the fuel cell hot walls and should therefore beaccounted for in the model. In order to provide a roughestimate for a thermal flow boundary condition, this powershould be divided by the surface of the hot walls. InOpenFOAM, this thermal flow condition is prescribedthrough the "fixedGradient" BC type.

In the remainder of the patches, a "zeroGradient" BCis set up. A summary of the boundary conditions for T isreported in Table V.

Patch BC type BC value

inlet fixedValue 293.15 Kfuelcellstack fixedValue 333.15 K

fuelcellhotwalls fixedGradient 3800 W/m2

airfoil

fuelcellfans

fuelcellwalls

outlet

tank

zeroGradient -

Table V: Boundary conditions for T . All the reportedvalues are uniform across the patch.

5) Boundary humidity: The only source of moist airin the domain is the fuel cell stack. Given the stacktemperature T = 60 oC, the equilibrium water vapourpressure p0 can be found with Antoine’s equation (seeSection II-D). Hence, considering the atmospheric pressurep = 1 atm, the mixing ratio r can be found by means ofEquation 8. Finally, the specific humidity q is found fromEquation 7 and can be prescribed as a "fixedValue" BC atthe fuel cell stack:

q = 0.13155 kg/kg. (26)

In these calculations, particular attention should be fo-cused on the units of measurement. For the remainder ofthe patches, a "zeroGradient" type condition is again setup.

F. Initial conditions and flow propertiesThe remainder of the flow properties and initial con-

ditions for the internal flow initialization are reported in

Property Symbol Value

Viscosity ‹ [m2/s] 1 · 10≠5

Velocity U [m/s] (1,0,0)

Pressure p [P a] 0

Temperature T [K] 293.15

Humidity q [kg/kg] 0

Turbulent kinetic energy k [m2/s2] 5.4 · 10

≠3

Turbulent frequency Ê [1/s] 1.8 · 104

Turbulent viscosity ‹T [m2/s] calculated

Time step �t [s] 2.5 · 10≠6

Air thermal di�usivity DT [m2/s] 2.82 · 10≠4

Water vapour di�usivity Dq [m2/s] 2.86 · 10≠4

Table VI: Flow properties and initial conditions.

Table VI.The turbulent parameter initialization values have beenkept equal to the ones estimated at the inlet (see TableII) as suggested by most of OpenFOAM tutorials [29]. Thetime step �t has been calculated from Equation 12 giventhe grid inlet �x and the inlet speed U = 20 m/s.

Considering Equation 6, to estimate the turbulent con-tribution –T to the thermal di�usivity DT , it is necessaryto make a reasonable guess for PrT . In the flow casesinvestigated by Kays [31], the computed PrT is alwaysaround 1, or usually slightly less. Hence, for the presentstudy

PrT = 1 (27)

is assumed to be an acceptable guess. Even if the realvalue is slightly overestimated, the order of magnitudeof the resulting –T will not be a�ected. Malhotra et al.[32] further assess that this guess is acceptable for flowswith Prandtl numbers below 1, whilst not being safe forhigher Pr values. Thus, given Pr = 0.72 < 1 calculatedfor the investigated flow case, their study raises confidencein the hypothesis reliability. Last, according to Equation6, a constant value for ‹T should be selected. A domain-averaged value has been thus computed from a preliminarysimulation of the present case without accounting for thescalars T and q:

‹T = 2.6 · 10≠4 m2/s. (28)

As PrT = 1, the above ‹T value corresponds to –T .To obtain DT , this contribution should be summed to– = 2.17·10≠5 calculated at T = 20 oC with Equation 5. Itcan be immediately observed that the turbulent contribu-tion is predominant and defines the order of magnitude ofDT . Hence, even if the variation of – had been accountedaccording to the temperature distribution as objected inSection II-C, the total thermal di�usivity and thus thescalar transport would not have been a�ected significantly.Physically, a predominant –T over – means that the scalartransport follows the turbulence.

Considering the water vapour di�usion in air Dq, ac-cording to Equation 11, to estimate its turbulent contri-bution DqT it is necessary to guess a reasonable turbulentSchmidt number ScT . The Lewis number is defined torelate the transport of any scalar to the transport of


temperature as Le = Pr/Sc. Using Reynolds analogy, aturbulent Lewis number can be defined as

LeT = PrT

ScT. (29)

Since it is physically reasonable to assume that the trans-port of water vapour in air will follow the turbulencebehaving similarly to the temperature di�usion, LeT = 1is assumed, hence leading to

ScT = 1. (30)

Furthermore, Goldman et al. [33] achieve an estimationclose to ScT = 1 considering the transport of similargases in air, thus raising confidence in the assumptionvalidity. As ScT = 1, the magnitude of DqT will be equalto the one of ‹T reported in Equation 28. To obtain Dq,this contribution should be summed to Dqp = 2.57 · 10≠5

calculated from Equation 10 given T = 20 oC. Once again,the turbulent contribution is predominant and defines theorder of magnitude of Dq, thus confirming that the watervapour di�usion will follow the turbulence. Moreover, assimilarly discussed for –, accounting for a variable Dqp

with T would not have significantly a�ected the total Dq

and thus the scalar transport behaviour.

IV. Results

A. Quality

1) Convergence and computational cost: In light of thevery careful simulation setup discussed in Section III,both the RK4 and the PISO simulation converged to lowresiduals without displaying any instability issue. In fact,the maximum Courant number safely oscillates in theproximity of 1, thus fulfilling the condition for stabilityaccording to Equation 12. On the other hand, due to theaforementioned very small �t, both the solvers requireda computational time of the order of magnitude of daysto display appreciable transient flow features. The spaceparallelization gave a significant contribution to speed upthe iterations performed at each time step, but the limitedspatial extent of the problem makes ine�ective any furtherincrease of the number of processes. Furthermore, even ifRK4 improves the performance at each iteration in com-parison to PISO, it is not itself su�cient to significantlyscale down the simulation time expense. On one hand,as discussed in Section III-A and III-B, the particularneeds dictated by the model development process andthe limited amount of simulations necessary have madethis high computational cost acceptable. In fact, a limitedphysical time extension of t = 10 s has been proved tobe su�cient to achieve relevant results to draw significantconclusions on the validity of the models. On the otherhand, in a future 3D optimization project the problem canbe addressed with a more e�ective space parallelizationand a mesh allowing a larger �t.

Figure 9: Contour plot of the velocity field at t = 10 scomputed with RK4 (upper) and PISO (lower).

2) Steady state: Apart from a few flow regions wheresome intrinsically unsteady vortex structures keep devel-oping in time, the PISO simulation achieves an overallsteady state after around ten seconds of physical timefor all its solution fields. As shown in the lower part ofFigure 9, the primary flow path is stabilized surroundingthe fuel cell. In contrast, the flow solved with RK4 showsa solution dominated by turbulent swirling motions whichdo not display any steadiness trend (upper Figure 9). Thisdiscrepancy can be explained by the higher dissipationperformed by the lower-order scheme (second order intime) implemented in PISO. A more dissipative solver ismore likely to achieve steady state, whilst introducing alarger truncation error and thus trading some reliabilityof the solution. As a matter of fact, the more unsteadysolution provided by the higher-order RK4 solver looksmore realistic considering the geometrical complexity ofthe investigated internal flow.

3) y+ check: As discussed in Section III-B, a y+ checkis necessary to assess the quality of the estimation of h1.The solutions show that the y+ is kept below 5 in themajority the domain, i.e. the first cell fully encloses theviscous sub-layer. However, a small region exceeds thethreshold in the proximity of the intake, where the localvelocity magnitude is still large. Nevertheless, the primaryflow features are mainly determined by the free-shearvortices rather than by the wall-bounded flow. Moreover,the wall functions introduced in Section III-E3 ensure anacceptable resolution of the higher y+ zones. Therefore,the estimated h1 = 0.5 mm can be considered acceptable.

4) Inflow boundary conditions: As it will be later dis-cussed in Section IV-B and shown in Figure 10, the solu-tions suggest that fuel cell stack and fan patches are lappedby vortical structures. The inflow "fixedValue" boundaryconditions prescribed in their correspondence artificiallyprevents these vortices to interact with the surfaces, i.e.


they are precluded to exit from the domain through theirpatches. However, given the very low suction and blowingboundary speed prescribed (see Table II), this is somethingwhich is likely to happen in a real case considering the highenergy carried by the vortices. Nevertheless, accordingto the fuel cell manual [7], the real stack consists of agrid whose distributed holes act as internal flow exhaustsinto the internal UAV bay. Hence, the remainder of thesurface is in all respects a wall which actually prevents thevortices from entering into the fuel cell interior. Similarly,the fans are covered by a protective grate. Therefore,from the vortices perspective, the chosen boundary inflowconditions can be considered to model as close as possiblethe real setup. On the other hand, the presence of theaforementioned grids raises a question mark on the validityof the turbulent inflow conditions. In fact, as a comparisonof Tables III and IV highlights, the very low speed set atthe stack and the fans leads to a substantial discrepancybetween their inflow turbulent boundary values and theones prescribed at the walls. In a real case, in contrast,the lattice-shaped presence of a wall above their interfacesis expected to generate larger turbulent boundary valuesthan the ones reported in Table III.

B. Velocity fieldIn order to achieve a better understanding of the flow

velocity field, the colour bar scaling has been changedto logarithmic and the resulting contour plots are shownin Figure 10. It can be immediately observed that thesolution di�ers substantially between the two solvers.

On one hand, RK4 shows that the flow is directed up-wards downstream of the inlet, lapping the internal airfoil-shaped upper surface. Hence, the flow di�uses above thefuel cell stack displaying appreciable large-scale turbulentstructures on top of it. Finally, the primary flow is eitherpulled out by the low-pressure outlet or di�used further

Figure 10: Logarithmic contour plot of the velocity field att = 10 s computed with RK4 (upper) and PISO (lower).

downstream above the tank or in the area between thetank and the fuel cell. In this last case, it runs in theproximity of the fans and thus rejoins the primary floworiginated at the inlet. Overall, as mentioned in SectionIV-A2, the solution is unsteady since the vortical entitiesevolve across the physical time.

On the other hand, the PISO solution shows a flowwhich is directed downwards downstream of the inlet.After passing the fuel cell fans, it is pulled by the low-pressure outlet from the bottom to the top. Hence, mostof the flow laps the airfoil-shaped upper surface in theproximity of the stack moving towards the leading edge.An overall steady state is achieved where the primary flowsurrounds the fuel cell in a counterclockwise direction. Incontrast to RK4, there is less flow transfer towards thetrailing edge. Moreover, the solution field displays a "bub-ble" of stagnation above the stack and the right hot wall. InRK4, conversely, the flow laps their surfaces with turbulentstructures. The reason for this discrepancy probably liesin the higher dissipation performed by the PISO solver,which tends to suppress the turbulent structures. As itwill be proved in Section IV-C and IV-D, the presence ofthis stagnation bubble will not be favourable for the scalartransport.

From a preliminary assessment, it already seems thatthe RK4 higher-order solution is closer to what one couldintuitively expect. In fact, the flow travels across theshortest path from the inlet to the outlet. Moreover,it displays more relevant turbulent structures which areexpected to be a constitutive feature in an internal flowwith the given geometrical complexity.

C. Temperature fieldThe contour plot of the temperature field after ten

seconds is shown in Figure 11. At first glance, it is imme-

Figure 11: Contour plot of the temperature field at t = 10 scomputed with RK4 (upper) and PISO (lower).


diately clear that the suppression of the turbulent struc-tures observed in the PISO solution (Section IV-B) leadsto an overall more consistent heating. In fact, the flowcounterclockwise circulation around the fuel cell createsa closed pattern where the air increases its temperatureprogressively in time under the influence of the fuel cell hotwalls and stack. As a consequence of this ine�cient coolingflow pattern, once a steady thermal state is achieved, theoverall temperature observed in the region around the fuelcell is higher than the one computed by RK4.

Furthermore, PISO’s bubble of stagnation discussed inSection IV-B results in a very high-temperature zonein its correspondence. Hence, in the proximity of thefuel cell stack and the right hot wall, there is a criticaloverheating region which, if realistic, could lead to thefuel cell overheating and shut o�. On the other hand, theturbulent structures developed in the solution computedby RK4 around the fuel cell lead to a much more e�cientcooling flow, confirmed by the lower temperature regionshown in the upper Figure 11.

With the goal of correctly predicting the transport ofscalars, these results point out the substantial importanceof an accurate computation of the turbulent structuresin the model solution. Therefore, given the lower-orderand turbulence-suppression features of PISO, it can beconcluded that the RK4 model appears more reliable andsuitable for the purposes of the investigated case.

D. Humidity field

The contour plot of the humidity field after ten secondsis shown in Figure 12. As introduced in Section II-D, thecomputed value of q is not exact in the whole domainbecause its variation with T has not been accounted.Nonetheless, relevant conclusions on the validity of themodels can be drawn evaluating the overall water vapour

Figure 12: Contour plot of the humidity field at t = 10 scomputed with RK4 (upper) and PISO (lower).

distribution across the domain, which is an output of areliable scalar transport advection-di�usion simulation.

Similarly to the temperature field, it is immediatelyvisible that PISO’s lack of turbulent structures and re-circulation flow pattern lead to a consistent humidity stag-nation around the fuel cell up to the airfoil-shaped walls.In addition, in correspondence of the bubble of low speedabove the stack, a region of water vapour concentrationclose to saturation is observed. Under these conditions, thenon-condensation assumption discussed in Section II-D isless robust and most likely non-realistic.

Considering the solution of RK4, in contrast, the highvapour concentration region near the stack is much morerestricted, and the humidity drops moving away from itspatch. In fact, the high-momentum primary flow investsthe stack with its turbulent structures and entrains thewater vapour, which is transported either towards the out-let or further downstream above the tank or below the fuelcell. Hence, the computed scalar concentration around thefuel cell is much more limited indicating an overall moree�cient ventilation. These considerations suggest that theresident time of the water vapour particles inside thedomain will be short, thus raising confidence in the non-condensation assumption. The only stagnation observedconcerns the region above the tank, but a more optimizedfuture 3D intake-exhaust system should be designed toavoid their occurrence, thus being suitable to be safelysimulated with the RK4 model developed.

Furthermore, comparing the T and q fields computed byRK4 (upper Figures 11 and 12), a very close similarity intheir distribution is observed, particularly on the top of thestack. The reason lies in the fact that the di�usivity of thetwo scalars is very similar, as its predominant contributionis given by the turbulence (Section III-F).

V. Model improvements: the density variationand the gravity term

Having developed the model and methodology and anal-ysed the results, there are two remaining assumptionswhich can still represent a weakness in the model defi-nition.

1) As stated in Section II-C and II-D, the influenceof T and q on fl in the momentum Equation 2b isneglected;

2) as mentioned in Section II-A, the gravity term in thesame momentum equation has been ignored.

It is very di�cult to provide any suggestion to copewith the former hypothesis without completely turningthe model upside down and increasing consistently itscomplexity. Nonetheless, the RK4 solution shows thatabove the stack where a higher fixedValue scalar boundarycondition is prescribed, both T and q tend to be quicklyrestored to the primary flow values (upper Figures 11and 12). Therefore, it is reasonable to expect a limiteddensity variation under the influence of the stack scalarBCs, which would not a�ect the output mean flow andthe scalar transport significantly. Of course, this solution


feature will depend on the flow rate of cooling air whichis introduced in the domain and invests the stack surface.Nevertheless, considering the goal of applying the modelto develop an optimized 3D intake-exhaust configuration,such flow rate should be large enough in any e�cientdesign to test. Hence, from this perspective, the RK4model can be concluded to be acceptably reliable for theintended applications.

Considering the second assumption listed above, thegravity term neglected on the right-hand side of Equation2b reads

fl̄

flgi, (31)

where gi is the gravity acceleration vector and fl̄ is thedensity adjusted with the contribution of the scalars T andq. From the inspection of the results obtained in SectionIV, it is not possible to predict whether the temperatureand humidity gradients will lead to density variationsenough significant to originate buoyancy forces, for whoseconsideration the gravity term addition is necessary. Giventhis uncertainty and since it is in principle possible toaccount for the gravity term in the model with a reason-able e�ort, an implementation attempt is made, with theultimate goal of evaluating the influence of the buoyancyforces on the solution.

The most simple implementation consists of employingthe Boussinesq hypothesis, according to which the densityis assumed to vary linearly with the temperature. A similarconsideration can be drawn for the density dependenceon the specific humidity. Hence, Equation 31 can be re-written as

[1 ≠ —T (T ≠ Tref ) ≠ —qq]gi. (32)

The higher is the temperature range, the more inaccuratethe hypothesis will be. Nevertheless, in the investigatedcase, the solution’s �Tmax is reasonably limited (Figure11, Section IV-C), so the assumption can be consideredacceptable with the goal of assessing the contribution ofthe buoyancy forces.

A PISO-based solver implementing the Boussinesqhypothesis for the temperature called "buoyantBoussi-nesqPisoFoam" is available in OpenFOAM [34], and adetailed description of the equations and the solutionprocedure can be found in OpenFOAMWiki [35]. Further,the model source code can be easily modified to accountfor the contribution of q according to Equation 32 and thenew solver has been named "doubleDi�usionPisoFoam".For the setup of a simulation test, the expansion coe�-cients —T (thermal expansion coe�cient) and —q have beenestimated from a reference temperature of Tref = 20 oCand are reported in Table VII.

—T 3.4 · 10≠3

—q 1.45 · 10≠1

Table VII: Expansion coe�cients.

The results of the model run on the investigated case couldbe in principle compared with the ones of PISO reported

Figure 13: Diverged cells in the velocity field of thedoubleDi�usionPisoFoam solution.

in Section IV. However, surprisingly the case resolutionwith the new solver encountered convergence problems.After an inspection of the diverged partial results, thespatial origin of the divergence has been found in the cor-respondence of the fuel cell lower-right corner, as shown inFigure 13. Given that the setup is exactly the same as theprevious case converged with PISO, it can be concludedthat the gravity term addition increases significantly thenumerical sti�ness of the problem. Hence, even furtherattention should be focused on the numerical setup andthe mesh quality. Considering the former, some attemptsto introduce more dissipative space discretization schemesrevealed to be insu�cient to reach the simulation conver-gence. Therefore, the focus has been shifted to the mesh.By means of the examination tool of Pointwise, a criticalarea ratio above 10 has been detected in correspondence ofthe diverged cells, as shown in Figure 14. Even if the arearatio is not generally pointed as one of the most crucialquality parameters, in the presence of the gravity term,it turned to be itself su�cient to cause the simulationdivergence. Therefore, it is recommended to constructan improved mesh, whose design should get rid of high-area ratio cells, and to test and assess its convergence.Once convergence is achieved, it will be possible to draw

Figure 14: Evaluation of the area ratio of the divergedcells.


conclusions on the influence of the gravity term on thesolution of the PISO-based solver. Nonetheless, in orderto allow a consistent comparison, it will be necessary torun the new mesh also with the original PISO solver, thusresulting in a very time-consuming task.

Given the limited reliability demonstrated by the PISOmodel in the scalar transport and the extensive timerequested to e�ectively get it running, it is rather worthtrying to implement the Boussinesq hypothesis and thegravity term in the more advanced RK4 model. In fact,the comparison of the solutions provided by RK4 andRK4+gravity is expected to be more realistic and to allowto draw more robust conclusions on the importance of thegravity term. Moreover, considering the completely di�er-ent resolution method, it is possible that the gravity termwill not increase the numerical sti�ness of the problemenough significantly to cause the simulation divergence.Thus, the prospective RK4+gravity solver may in prin-ciple work with the original mesh without requiring anycell quality improvement. Following the guidance of thebuoyantBoussinesqPisoFoam solver [35], the addition ofthe gravity term in the governing equations has been easilyimplemented successfully. However, the development ofthe model failed while dealing with the pressure correctionstep. In fact, the guidance of the implicit-time buoyant-BoussinesqPisoFoam code [34] could no longer be followed,as the explicit-time RK4 solver employs the projectionmethod to couple pressure and velocity rather than theoperator splitting and corrector loops. Several implemen-tation attempts have been tested on the most simple casestudies provided by OpenFOAM’s heat transfer tutorials[36], but none of them succeeded. As the development ofa double-di�usion RK-based solver represents a researchinterest beyond the investigated UAV case, in the futureit is worth exerting a further e�ort to implement and test aworking code, perhaps with the help of expert OpenFOAMprogrammers and researchers in numerical methods.

VI. ConclusionsIn conclusion, a complete and successful CFD model has

been developed, tested and discussed. The RK4 solver hasbeen proved to be the most reliable choice with the goal ofanalysing the internal flow of a hydrogen-powered UAV,where the di�usion of both temperature and water vapourshould be taken into account. However, so far it has notbeen possible to understand whether or not the gravityterm consideration e�ectively influences the solution. To-gether with the development of a working RK4+gravitysolver, this topic will be object of future studies. Never-theless, the examples in literature (e.g. Panagiotou et al.[12]) show that the internal flows are primarily dominatedby the advection and di�usion terms of the momentumequation. Therefore, the RK4 solver can be consideredready to be employed for the design of an optimal UAVintake-exhaust 3D system.

Furthermore, the CFD methodology employed for theinvestigated sample test case can be safely used as a basefor the setup of more complex prospective 3D optimization

cases. However, there are still some additional or uncertainaspects which are recommended to be considered in futureapplications.

• The inflow turbulent boundary conditions at the fuelcell stack and fans are recommended to be further in-vestigated. In fact, the parameters k and Ê prescribedintroduce very weak turbulence at the boundaries,which could be non-realistic considering the actualgrid-shaped geometries of the patches. Nevertheless,in an optimized intake-exhaust configuration, they arenot expected to significantly influence the solution.

• In order to e�ectively reduce the simulation time,the 3D inlet and outlet patches should be meshedwith a su�ciently large �x to allow a reasonablylarge �t, whilst fulfilling the CFL condition. This�x feature can be achieved only if the intakes andexhausts designed are themselves not as small as theones investigated in this study, which is often the casein the examples found in literature (e.g. Panagiotu etal. [12] and Chiang et al. [11]).

• To achieve overall higher-quality boundary conditionsat the intakes and the exhausts, it is recommendedto combine the internal and external flow in onesingle simulation. However, running a large externalflow domain with a small-�t transient simulationwould be highly time-consuming and ine�cient. Apossible way to cope with this issue would be to firstbuild a mesh for the external flow domain, wherethe internal flow inlet and outlet patches are alreadyprepared. At these patches, a wall BC should beinitially prescribed, so that a steady-state simulationcan be performed on the external domain within areasonable computational time. Thus, after combin-ing the previous external mesh with the internal one,the patches can be converted into free exterior-interiorinterfaces, and a new global transient simulation canbe set up, using the earlier results as initial conditionsfor the external domain.

• So far, the influence of the propeller has not beenconsidered. Nonetheless, the intake-exhaust systemcan potentially take advantage of the flow attractionprovided by the Green Raven’s pusher propeller, assimilarly shown in the configuration provided by Kimet al. [9]. The influence of the propeller on the externalflow can possibly be modelled as a momentum sourcedisk, as suggested by Panagiotu et al. [12].

AcknowledgementsA heartfelt thanks goes to the Green Raven project

coordinator, prof. Ra�aello Mariani, who involved me inthe team and in correlated research projects. His abilityto recognize students potential and to informally and re-spectfully relate with them has been very stimulating. I amlooking forward to continuing our research collaboration inthe future.

A very special thanks goes to my supervisor LucaBrandt for his unconditional support. Thank you also for


coordinating the help provided by all the members of hisresearch team. Last but not least, thank you for involvingme in restoring football matches.

I would also like to express my heartfelt gratitude to dr.Matteo Montecchia for patiently supporting me with thecase setup and the solver development, sharing with me allhis crucial extensive experience with OpenFOAM. Thanksalso to Nicolò Scapin for teaching me the fundamentals ofmoist air flows and for reviewing the drafts of the relatedsections. Thank you to all KTH researchers of the 6th floorin the Mechanics department for allowing me to breathea research enthusiast environment.

Another special thanks goes to my company supervisorFaranggis Bagheri for being confident in my capabilitiesand for her friendly management in these two wonderfulyears at neptech AB. Thank you also for sharing with methe experience with Pointwise and for allowing me to usethe software for this master thesis.

Last but not least, I would like to thank all my friendsat KTH for sharing with me this amazing experience andfor the inspiring conversations we had talking about ourfuture.

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Appendix AAdditional contour plots to achieve a better flow physics understanding

Figure 15: Shaded contour plot of the complete velocity field at t = 10 s computed with RK4 (upper) and PISO(lower).

Figure 16: Logarithmic contour plot of the complete velocity field at t = 10 s computed with RK4 (upper) and PISO(lower).


Figure 17: Logarithmic contour plot of the complete temperature field at t = 10 s computed with RK4 (upper) andPISO (lower).

Figure 18: Logarithmic contour plot of the complete humidity field at t = 10 s computed with RK4 (upper) and PISO(lower).

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