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Numerical Calculation of Internal Blade Cooling UsingPorous Ribs

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Citation for published version (APA):Al-Aabidy, Q., Craft, T., & Iacovides, H. (2017). Numerical Calculation of Internal Blade Cooling Using PorousRibs. 1-15. Paper presented at 23rd ISABE Conference, Manchester, United Kingdom.

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ISABE 2017 - ID-21383 1

Numerical Calculation of InternalBlade Cooling Using Porous RibsQahtan Al-Aabidy�y, T.J. Craft and H. Iacovides

Turbulence Mechanics GroupSchool of MACEThe University of ManchesterM13 9PLManchester, UK

ABSTRACTThe study of �ow and heat transfer around and through porous ribs is promising in manyindustrial applications that require high cooling rates, such as gas turbine blade cooling, orrequire noise reduction. Turbulent �ow in porous media is usually investigated by usingmacroscopic transport equations, which are obtained by applying double (both volume andReynolds) averaging to the Navier-Stokes equations. In this study turbulence is representedby using the Launder-Sharma low-Reynolds number k � " turbulence model [1], whichis modi�ed via proposals by Nakayama and Kuwahara [2] and Pedras and de Lemos [3],for extra source terms in turbulent transport equations to account for the porous structure.Additional re�nements are introduced to account for e�ects close to the porous media/clear�uid interface. In order to investigate the validity of the extended model, two types of porouschannel �ows have been considered, for which there is available DNS and experimentaldata. The �rst case is a fully developed turbulent porous channel �ow, where the results arecompared with DNS work conducted by Breugem et al. [4] and experimental work conductedby Suga et al. [5]. The second case is a turbulent porous rib channel �ow to understand thebehaviour of �ow through and around the porous rib, which is validated against experimentalwork carried out by Suga et al. [6]. Cases are simulated covering a range of porous properties,such as permeability and porosity. Through the comparisons with the available data, it hasbeen found that the extended model shows generally satisfactory accuracy, except for somepredictive weaknesses in regions of either impingement or adverse pressure gradients.

Keywords: turbulence in porous media, interface porous-�uid region, turbulent �owaround a porous rib.

� Corresponding author: [email protected] Mechanical Engineering Department, College of Engineering, University of Kufa, Najaf, Iraq

ISABE 2017.

2 ISABE 2017

NOMENCLATUREb ConstantcF Forchheimer coe�cientc"1; c"2; ck Non-dimensional turbulence model constantc� Coe�cient in the eddy-viscosityDa Darcy number, Da = K=H2

Dp Pore diameterf �U ; f

�k ; f

�" Damping functions for source terms of porous media

H Channel heighth Rib heightGk Generation rate of k due to porous mediaG" Generation rate of " due to porous mediaK Permeability of porous mediak Turbulent kinetic energyN� ConstantP PressurePk Production termReb Bulk Reynolds numberRt Turbulent Reynolds numberUD Darcy or super�cial velocityU Velocity vector5V Total volume5V f Fluid volumey0 Normal distance from the nearest porous surface

Greek Symbol

" dissipation rate of the turbulent kinetic energy, k� kinematic viscosity�t kinematic turbulent viscosity�1 Porosity of homogeneous porous media�i j Kronecker delta unit symbol’ General quantity� Porosity of inhomogeneous porous media (= V f =V)� Fluid density

Acronyms/Abbreviations

DNS Direct Numerical SimulationLES Larg Eddy SimulationLSMPL Launder Sharma Modi�ed by Pedras and de Lemos [7]LSMNK Launder Sharma Modi�ed by Nakayama and Kuwahara [2]PPI Pore Per InchREV Representative Elementary VolumeRANS Reynolds-Averaged Navier-StokesSIMPLE Semi-Implicit Method for Pressure-Linkage EquationsTCL Two-Component-LimitUMIST Upstream Monotonic Interpolation for ScalarVAT Volume Averaging Theory

Qahtan Al-Aabidy et al. ID-21383 3

Special Characters

h i f Intrinsic averageh i Volume average

1.0 IntroductionTo improve combustion e�ciency, gas turbines are being designed to operate at increasinglyhigher temperatures. As a consequence, the air �owing over the blades can exceed the per-missible temperature level of materials. Protecting gas turbine components and increasingthermal e�ciency therefore require more e�cient internal and external blade cooling strate-gies to meet the demands of the modern gas turbine applications. One of the more e�ectivestrategies in internal blade cooling is the use of roughened internal serpentine passages withribs in di�erent con�gurations, to increase the mixing of the �ow that in turn enhances theblade cooling. As a result of the attractive thermal characteristics of metal foams in a vari-ety of applications in industry, the use of porous metal foams within these passages has beeninvestigated for turbine cooling applications [8, 9, 10, 11, 12].

The porous foams contain highly tortuous �ow paths that can signi�cantly intensify themixing of �uid �ow and enhance the heat transfer as a result of extensive surface areas thatpermit the �uid to be in a contact with a large extended area. However, because of the com-plex �ow paths, treating the �ow through such media at the pore level requires huge computerpower and cost, and even when possible it is limited to simple cases. As a result, the Vol-ume Averaging Theory (VAT) is a common approach for the numerical modelling of �ows inporous media [13, 14].

In many traditional engineering applications the �ow in porous media is almost laminar,as a result of the small voids and high resistance that cause relatively low velocities [15].Turbulence may become appreciable at the pore level, however, if the �ow within or aroundthe porous structure is at very high speed, or if the pore scale is larger than the turbulencelength scale, i.e. when the pore Reynolds number Rep, de�ned as Rep = upDp=� where up ispore velocity scale (intrinsic velocity) and Dp is the pore diameter, is su�ciently high [16].This is what was visualized by Dybbs and Edwards [17] who conducted an experimentalinvestigation and found that the �ow in the porous media became turbulent when the poreReynolds number was greater than a few hundreds.

Numerical modelling of turbulent �ow in porous media is mainly based on a macro-scopic approach in which the double-averaging (volume and Reynolds averaging) is used.Macroscopic Navier-Stokes equations can be derived by two di�erent methodologies: eitherusing time-averaging of volume-averaged Navier-Stokes equations, or volume-averaging ofReynolds-averaged Navier-Stokes equations. Lee and Howell [18] and Getachew et al. [19]used the �rst methodology to obtain macroscopic k � " transport equations for treating theturbulence in a porous media. However, since a highly porous medium was considered byLee and Howell [18], no additional terms were included related to porosity in the turbulenceequations. The second methodology, which is based on using the Reynolds averaging �rst,has been more widely used in dealing with turbulent �ows, for example in the studies of Ma-suoka and Takatsu [20], Nakayama and Kuwahara [16], Finnigan [21], Pedras and de Lemos[3] and Nikora et al. [22].

Pedras and de Lemos [3] proved that the momentum equation is not a�ected by the orderof averaging, although the two approaches do lead to di�erent de�nitions of the turbulentkinetic energy. Nakayama and Kuwahara [16] believed that the Reynolds averaging shouldbe performed �rst in order to detect the turbulence level at the porous scale, since this levelof turbulence is unlikely be detected in the case of starting with volume averaging. The �nalforms of macroscopic turbulent transport equations obtained by Pedras and de Lemos [3] andNakayama and Kuwahara [16] shared similar terms to those found for the non porous media,but also contained extra production and dissipation rate terms due to the presence of porous

4 ISABE 2017

media.Kuwata and Suga [23] used a sophisticated turbulence model for treating three unknown

elementary second moment terms that appear in the Navier-Stokes equations after applyingdouble (both volume and Reynolds) averaging. These terms are namely the dispersive covari-ance and the volume averaged Reynolds stress, which is decomposed into the macro-scaleand micro-scale(sub-�lter scale) Reynolds stress. The TCL(two-component-limit) secondmoment closure and a one-equation eddy viscosity model were used for modelling macro-scale and micro-scale Reynolds stresses, respectively, while the dispersive covariant was mod-elled via a form based on the Smagorinsky scheme typically used to represent sub-grid-scalestresses in Large Eddy Simulation (LES). In addition to this model they proposed an eco-nomical multi-scale k � " model for industrial applications. M¤oßner and Radespiel [24] usedReynolds averaging of volume-averaged Navier-Stokes equations, justifying its use in theirstudy by showing good agreement with DNS data from Breugem et al. [4].

In this study, the double-decomposition technique has been used to model the �ow througha porous media, with surrounding clear �uid. The Launder and Sharma low-Reynolds numberk � " turbulence model [1] has been used, with modi�cations proposed by Nakayama andKuwahara [2] and Pedras and de Lemos [3, 7] to mimic the �ow inside the porous media.To improve predictions close to the clear �uid/porous media interface, additional dampingterms are introduced to the porous source terms in this region, to relax the resistance of theporous media from the homogeneous porous region to the clear �uid zone across the distanceof the mean pore diameter. These damping functions are modi�ed forms of those proposedby Kuwata and Suga [23]. Results from the use of the current extended turbulence models arecompared with DNS data by Breugem et al. [4] and experimental data by Suga et. al.[5, 6] tovalidate the models.

2.0 Macroscopic Mathematical Formulation2.1 Macroscopic Navier-Stokes Equations

The turbulent �ow over porous walls or around porous blocks/ribs/ba�es is described by theconventional RANS equations, while the �ow through porous regions is described by theBrinkman-Forchheimer-extended Darcy model. This latter formulation accounts for viscouse�ects (Darcy term), form drag (Forchheimer term) and the Brinkman term (viscous di�u-sion),Vafai and Tien [25]. The continuity and mean momentum equations can be written as:

@UDi@xi

= 0 (1)

@UDi@t

+@@x j

UD jUDi

�

!= �

1�@�hPi f

@xi+

@@x j

"� @UDi@x j

+@UD j@xi

!� �huiu ji f

#�

f �U

"��K

UDi +cF�p

K

pUDkUDk

# (2)

where the symbol K is the permeability of the porous media, which is de�ned as a measureof the ability of the porous medium to permit �uid �ow through it, and cF is the Forchheimercoe�cient, �hPi f is the super�cial average pressure of the �uid, and � is the porosity of theporous media, which is de�ned as the pores’ volume fraction. K, cF and � are unique proper-ties of the porous media. UDi

�= �hUii f

�is the Darcy velocity which is super�cial velocity.

When the porosity and permeability are extremely high (and the porous matrix e�ectivelydisappears), the generalized model Equation 2 reverts to the traditions RANS equations. Therelationship between the super�cial h’i and intrinsic h’i f property is h’i = � h’i f .


The macroscopic Reynolds stress tensor, �huiu ji f , appearing in Equation 2 has been mod-elled in analogy with the Boussinesq concept for clear �uid (non-porous), as follows [7]:

�huiu ji f = ��t� @UDi@x j

+@UD j@xi

!+

23�hki f �i j (3)

where hki f is the intrinsic turbulent kinetic energy, and �t� is the macroscopic turbulent vis-cosity, modelled similarly to that in the clear �uid as:

�t� = f�C�hki f 2

h"i f(4)

where c� is a constant with the usual value of 0.09 and f� a near-wall damping term, heretaking the form proposed by Launder and Sharma [1] of exp

h�3:4= (1 + Ret=50)2

i

2.2 Macroscopic equations for k and "Nakayama and Kuwahara [16] and Pedras and de Lemos [3] conducted numerical experi-

ments for turbulent �ow through periodically arranged square rods and circular rods, respec-tively. The �nal forms they adopted for the macroscopic turbulent kinetic energy and itsdissipation rate equations, after applying the volume-averaging operator for microscopic k�"equations inside a Representative Elementary Volume REV, can be written as follows:

@�hki f

@t+@�UD jhki f

�

@x j=

@@x j

" � +

�t��k

!@�hki f

@x j

#+ P fk + f

�k Gk � �h"i

f (5)

@�h"i f

@t+@�UD jh"i f

�

@x j=

@@x j

" � +

�t��"

!@�h"i f

@x j

#+c"1 f1

h"i f

hki fP fk +c"2

f �" G" � f2

h"i f

hki fh"i f

!(6)

where P fk = ��huiu ji f@hUii f@x j

is de�ned as the production rate of hki f , f1 and f2 are viscousdamping terms for low-Reynolds-number k � " Launder and Sharma [1] model, and theyare, respectively, 1.0 and 1:0 � 0:3 exp(�Ret2). Gk and G" represent respectively the extrageneration rate of hki f and the extra production rate of h"i f due to the presence of the porousmedia. These extra source terms were derived in di�erent forms in the two studies, as shownin the Table 1, where ck and cD are constants equal to 0.28 and 0.09 respectively, and b =cF=p

K.

Table 1:Expressions for extra production terms for turbulent kinetic energy and its dissipation rate

Gk G"Pedras and de Lemos [7] ck �hki

fp

K

pUDkUDk ck �h"i

fp

K

pUDkUDk

Nakayama and Kuwahara [2] b (UDkUDk)3=2 bq

cD2�K (UDkUDk)

2

6 ISABE 2017

2.3 Modi�cation for Porous-Fluid Interface Regions

The equations introduced in the previous section are valid for the porous media, whilst in theclear �uid region the additional terms in the momentum and turbulence terms clearly vanish.However, the additional terms are based on spatially uniform porous media properties, whilstthere will be a thin layer at its interface with the clear �uid where the e�ective porosity willincrease as the clear �uid region is approached. This is accounted for in the present modelby relaxing the porosity and additional source terms in the macroscopic Reynolds averagedNavier-Stokes equations across a thin transitional layer at the edge of the porous region. Theporosity across this layer is taken as:

� = �1 + (1 � �1) exp��N�y0=Dp

�(7)

where �1 is the porosity of the homogeneous porous media and y0 is the normal distancefrom the nearest porous surface. The coe�cient N�, which is taken as 4, is chosen so thatthe porosity varies within the distance of the mean pore diameter Dp. The drag terms, in themomentum Equation 2, and other additional terms in the turbulence transport Equations 5 &6, are respectively multiplied by the following functions:

f �U = 1 � exp��1:03

�y0=max

�0:004;

pK��1=2�

(8a)

f �k = 1 � exp��1:21

�y0=max

�0:004;

pK��5=2�

(8b)

f �" = 1 � exp�� (1:0 + Rt=58:51)

�y0=max

�0:004;

pK��2�

(8c)

where Rt(= (�hki f )2=(��h"i f )) is the turbulent Reynolds number. These functions are modi-

�ed forms of those suggested by Kuwata and Suga [6], re-optimized for use within the presentk � " framework, and also designed to vary between 0 and 1 within the distance of the meanpore diameter to relax the e�ect of porous resistance from the homogeneous porous region tothe homogeneous clear �uid region.

3.0 Results and DiscussionsThe calculations presented in this study have been carried out by using an in-house �nitevolume code STREAM developed in Manchester by Lien and Leschziner[26]. It employsa non-orthogonal and body-�tted grid system in which all transported properties are storedin a fully collocated manner. The SIMPLE pressure-correction algorithm of Patankar andSpalding [27] is used to evaluate the pressure �eld in addition to Rhie and Chow interpolation[28] to avoid pressure oscillations. Advective volume-face �uxes are approximated using thesecond order-accurate UMIST scheme [29]. Results are presented from the use of the currentextended turbulence models, referred to as LSMNK in case of that based on the Nakayamaand Kuwahara turbulence model [2] and LSMPL in case of that based on the Pedras and deLemos model [3, 7].

3.1 Turbulent Porous Channel Flows

The case considered is that of a fully developed plane channel �ow over a porous media,which was examined using DNS by Breugem et al. [4] and experimentally by Suga et al. [5].Such �ows are encountered in a wide range of engineering and environmental problems suchas metal foam heat exchangers, catalytic converters, �ows in oil wells, �ows over forests andporous river beds.


Figure 1: Schematic of turbulent porouschannel �ow.

Figure 1 shows the channel geometry, with asolid top wall and a permeable lower layer. Thetotal channel height is H, and the bottom imper-meable wall covered by a porous layer of halfthe channel height. Simulations have been per-formed covering a range of porosity, permeabil-ity and Reynolds numbers, matching the availableDNS and experimental data. Due to space limi-tation reasons, only a selection will be presentedhere, with the results below concentrating on threecases: the DNS study at Reb = 5500 with poros-ity � = 95%, cF = 0:295, Da= 4:75 � 10�5 andDp=H = 0:011, the experimental studies are at Reb = 5259 and Reb = 10200 with porosity� = 81%, cF = 0:1, Da= 9:93 � 10�6 and Dp=H = 0:048, where Da is the dimensionlesspermeability. For simplicity, in these cases, the porosity and Reynolds number are used todistinguish them, and they will be referred to below as E95 , E81L and E81H , respectively.The bulk Reynolds number is de�ned as Reb = UbH=(2�) based on the bulk velocity Ub ofthe �ow in the clear channel region and its height H=2. Periodic boundary conditions are im-posed between the upstream and downstream boundaries with no-slip boundary conditions atthe impermeable walls. The imposed streamwise pressure gradient is adjusted in an iterativefashion to give the desired bulk �ow rates.

For illustration, Figures 2a and 3a show the mean streamwise velocity across both porouslayer and �uid region, normalized by bulk velocity in the clear channel for the lower Reynoldsnumber cases E95 and E81L. Figures 2b & 3b and 2c & 3c show corresponding pro�lesof turbulent kinetic energy and Reynolds shear stress, normalized by friction velocity U� atthe (impermeable) top wall for both cases. Since �uid, and some turbulence structures, canpenetrate across the porous boundary, there is less of a wall-blocking e�ect there (particularlyon the wall-normal velocity component) than would be found at a solid wall. As a result,Suga et al. [5] noted that the �ow becomes turbulent at a lower Reynolds number than wouldbe expected in a clear plane channel.

As can also be seen in Figures 2 and 3, the measurements show higher turbulence levels inthe clear �uid near the porous interface than near the solid wall, with the peak levels increasingas the porosity increases, together with a highly asymmetric mean velocity across the clear�uid part of the channel. These features can be explained by noting that, since the �ow is fullydeveloped the momentum equation implies that the total shear stress must increase linearlyacross the clear �uid part of the channel (as it would for a purely clear �uid channel �ow), andthe results show the turbulent shear stress doing so across the core region. Clearly, the meanvelocity and turbulent shear stress must both fall to zero at the solid wall, however, they donot do so at the porous interface. Consequently, there is not a rapid growth of viscous shearstress as the interface is approached, and the velocity gradient here is lower than near the solidwall, leading to the asymmetric pro�le seen in the data.

Considering �rst the predictions of the higher porosity case 2, the original form of the Pe-dras and de Lemos [3] model (without the near-interface damping terms described in Section2.3 ) clearly returns too high levels of turbulence around the porous/clear �uid interface, withconsequent high levels extending out into both regions. The addition of the proposed near-interface damping terms brings the predicted levels down to closely match the DNS data.Although not shown, the additional terms have a similar e�ect when incorporated within theNakayama and Kuwahara [2] model, and the modi�ed form again shows good agreement withthe DNS data.

A similar level of agreement to that shown above was found from both modi�ed modelforms in most of the other cases examined. However, when the porosity (and permeability)was reduced in the case E81L, as in Figure 3, the LSMPL scheme does produce a reductionin turbulence levels near the interface, again broadly matching the data. The LSMNK model,

8 ISABE 2017

(a) Streamwise mean velocity pro�le

(b) Turbulent kinetic energy pro�le

(c) Reynolds shear stress pro�le

Figure 2: Comparison between the current predictions and the DNS data [4]for the high porosity plane channel �ow, case E95

on the other hand, predicts a signi�cant increase and overprediction of near-interface turbu-lence levels, resulting in a more asymmetric mean velocity pro�le, as seen in Figure 3a. Inattempting to understand this discrepancy in LSMNK model behaviour, it is worth noting thatthe results shown are at fairly low bulk Reynolds numbers; simulations at higher Reynolds





Figure 3: Comparison between the currentpredictions and the experimental data [5] forthe low porosity plane channel �ow, caseE81L




Figure 4: Comparison between the currentpredictions and the experimental data [5] forthe low porosity, and higher Reynolds num-ber, plane channel �ow, case E81H

numbers and similar porosity (and permeability) levels gave quite close agreement betweenboth models and available data. An example of this can be seen in Figure 4, where LSMNK

10 ISABE 2017

model does generally produce better predictions than in the lower bulk Reynolds number caseE81L. The cause of the di�erence in model behaviour at low Reynolds numbers would appearto lie in the forms adopted for the modelled source terms Gk and G". Based on arguments ofhow terms in the turbulent kinetic energy equation should scale, Nakayama and Kuwahara[2] formulated their additional source terms to depend primarily on the mean Darcy velocity,whereas the forms proposed by Pedras and de Lemos [3] are dependent on both the Darcyvelocity and the local turbulent kinetic energy and its dissipation rate levels. From the resultshere, it would appear that the latter responds better to the �ow changes seen as permeability(or porosity)is decreased at quite low Reynolds numbers, whereas the former shows too greata sensitivity to the permeability (or porosity).

3.2 Turbulent Porous Rib Channel Flows

As a further test of the present models, �ow along a channel again half-�lled with a porouslayer, but now with a porous square rib, of half the clear channel height, mounted on top ofthis layer, as shown in Figure 5, has been considered. The porous layer and rib are made ofthe same material, and �uid can both penetrate through the rib, and by-pass through the layerunderneath it, as a result of the low pressure behind the rib. Such �ows can be encounteredin the chip cooling of electronic packages and vegetative canopies. The two cases consideredhave been studied experimentally by Suga et al.[6] and numerically by Kuwata et al.[30]. The�rst is at Reb = 9800 with PPI = 20, � = 82%, cF = 0:17, Da=6:2 � 10�6 and Dp=H = 0:03,whilst, the second is at Reb = 10600 with PPI = 6, � = 80%, cF = 0:095, Da= 2:6 �10�5 and Dp=H = 0:065, where PPI denotes Pores Per Inch and is used to characterize thepermeability of porous meal foams. Because the two cases have almost the same porosity,PPI (characterising, but inverse to, permeability) is used to distinguish them, denoting asCase #20 and Case #6 the former and latter cases, respectively.

Figure 5: Schematic of porous rib chan-nel �ow.

To ensure su�cient upstream and downstream�ow development lengths, the computational do-main extends from 12h upstream of the porous ribto 51h downstream of the rib. Fully developed�ow pro�les (from a separate calculation) are im-posed at the inlet, with zero streamwise gradientsapplied at the outlet, and no-slip conditions at theimpermeable walls. A block-structured grid ofaround 275(x) � 240(y) cells was used, with gridnodes concentrated towards the porous-�uid inter-faces and solid wall. Tests with more re�ned gridscon�rmed the distribution adopted to be su�cient.

Streamlines from LSMPL model predictions of �ow over and through the porous ribmounted on the porous layer are shown in the Figure 6. Figures 7a and 8a show predictedpro�les of the mean streamwise velocity, turbulent kinetic energy and Reynolds shear stressfor the �ow within and around the porous rib with low and high permeability, Case #20 andCase #6 respectively, compared with the experimental measurements carried out by Suga etal.[6]. From Figure 6 it can be clearly seen that, due to the resistance of the porous rib, someof the �ow is directed towards the upper wall. However, the permeability of the rib and lowerchannel layer permit a certain amount of �uid to pass through them (not surprisingly, to agreater extent as the permeability is increased), and some of that through the latter subse-quently bleeds back into the clear �uid layer behind the rib. As a result of the �uid travellingthrough and under the rib, little or no �ow separation and recirculation is in the clear �uidimmediately behind the rib in these cases.

Although there is now no signi�cant �ow separation immediately behind the rib, the �uidseeping back into the clear �uid region downstream of the rib does result in an adverse pres-sure gradient in the porous layer, which together with the �uid being entrained into the clear


region leads to a weakly recirculating area (starting at around x=h = 5 in the lower perme-ability Case #20). As the permeability increases (Case #6), the �uid bleeds back into theclear channel more slowly and the recirculation within the porous region is weaker and oc-curs further downstream. Such a feature was also deduced to be present by Suga et al. [6]from analysing their measured mass �ow rates in the clear �uid and the streamwise pressurevariation.

(a) Streamlines of Case #20

(b) Streamlines of Case #6

Figure 6: Streamlines of �ow over and through a porous rib-mounted porous layer layer withdi�erent permeabilities, as predicted by the LSMPL model. Note that non-uniform stream-function step sizes are displayed, in order to make the distribution within the porous mediavisible.

The vertical distributions of turbulent kinetic energy and turbulent shear stress normalizedby the bulk velocity are shown in Figures 7b & 7c and 8b & 8c, it can be seen that high levelsof turbulent kinetic energy generally occur close to the upstream edge of the porous rib, andthen in the shear layers downstream, corresponding to the regions where the mean strains willresult in signi�cant turbulence generation.

In the low permeability case, Case #20, Figure 7, the predicted turbulence pro�les aregenerally in satisfactory agreement with the measured data, although rather too high levels ofturbulent kinetic energy are predicted by both models ahead of, and around, the rib (slightlymore so by the LSMPL variant). Since it is well-known that the Launder-Sharma model(and most other linear EVMs) tend to overpredict turbulence levels in normally strained andimpingement �ow regions, the high levels of predicted turbulent kinetic energy just ahead of,and around, the rib are perhaps not unexpected. The higher levels predicted by the LSMPLvariant within, and immediately beneath, the rib are then a result of that model’s porous sourceterms depending on the (already high) turbulence levels, whereas the LSMNK schemes sourceterms depend purely on the mean Darcy velocity, as noted earlier. Further downstream ofthe rib, both models tend to slightly underpredict the peak turbulent kinetic energy levelsin the clear �uid region, although peak shear stress levels are slightly overpredicted. TheLSMPL model does also rather overpredict turbulence levels around the interface betweenporous and clear �uid regions between 5 < x=h < 8, and returns higher turbulence levelsthan the LSMNK variant within the porous media here. This region corresponds to the area

12 ISABE 2017

(a) Streamwise mean velocity pro�les

(b) Turbulent kinetic energy pro�les

(c) Reynolds shear stress pro�les

Figure 7: Comparison between the current predictions for the low permeability Case #20 andthe Experimental data [6] for turbulent porous rib channel �ows, Red lines represent LSMPLmodel, broken black lines represent LSMNK model and symbols represent Experimental data


(a) Streamwise mean velocity pro�les

(b) Turbulent kinetic energy pro�les

(c) Reynolds shear stress pro�les

Figure 8: Comparison between the current predictions for the high permeability Case #6 andthe Experimental data [6], Red lines represent LSMPL model for turbulent porous rib channel�ows, broken black lines represent LSMNK model and symbols represent Experimental data

14 ISABE 2017

of �ow recirculation seen in Figure 6, as a result of the fairly weak, though non-negligible,adverse pressure gradient in this region, as �uid bleeds from the porous media back into theclear �ow stream. The same mechanisms as above can then lead to a slight overprediction ofturbulence levels by linear EVMs, with the LSMPL scheme magnifying this e�ect because ofits additional source terms being directly dependent on k and ".

For the higher permeability case, Case #6, Figure 8, the agreement in the turbulent quan-tities is generally satisfactory. The much weaker impingement around the rib in this casemeans that turbulence levels are not so overpredicted here (although the LSMPL variant doesagain produce higher turbulent kinetic energy levels than the LSMNK version within the rib,for the same reasons as noted above). The pro�les downstream of the rib are also generallybetter captured than in the lower porosity case and the weaker recirculation in the porous re-gion, which now occurs further downstream at aroundx=h = 8, does not result in a noticeableincrease in turbulence levels here..

4.0 ConclusionsTwo di�erent modi�cations of the Launder-Sharma model, those proposed by Nakayamaand Kuwahara [2] and Pedras and de Lemos [7] have been tested, representing widely-usedschemes for turbulent �ow in porous media applications. These have been combined withre-optimised forms of near-interface damping terms for the modelled porous source terms,which have been shown to improve signi�cantly the predictions around the porous/clear �uidinterface region. The results of the turbulent porous channel �ows show that the predictionaccuracy of both modi�ed models is generally satisfactory, although at low Reynolds numberand low permeability the LSMNK results become poorer, with turbulence levels quite signi�-cantly overpredicted. In the porous rib channel �ows the prediction are generally satisfactory,especially in the higher permeability case. In the lower permeability case both models tend tooverpredict turbulence levels around the impingement region at the front of the porous rib, as aresult of the underlying weakness in the linear eddy-viscosity formulation in such �ows. Theform of the modelled porous source terms then leads to the LSMPL scheme returning higherturbulence levels within and beneath the rib than the LSMNK model. The LSMPL schemealso predicts higher near interface turbulence levels further downstream, where a weak recir-culation occurs, for similar reasons.

AcknowledgementsThe �nancial support from the Ministry of Higher Education and Scienti�c Research of Iraqand the University of Kufa is gratefully acknowledged. The authors would like to acknowl-edge the assistance given by IT Services and the use of the Computational Shared Facility atThe University of Manchester.

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IntroductionMacroscopic Mathematical FormulationMacroscopic Navier-Stokes Equations

Results and DiscussionsTurbulent Porous Channel FlowsTurbulent Porous Rib Channel Flows

Conclusions

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