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DIFFUSER DEVELOPMENT FOR A DIFFUSER AUGMENTEDWIND TURBINE USING COMPUTATIONAL FLUID DYNAMICS

D.G. Phillips, P.J. Richards, R.G.J. Flay

Department of Mechanical EngineeringThe University of Auckland

New Zealand

PHOENICS VERSION 3.2 and earlier.

Abstract

Vortec Energy Limited and The University of Auckland have carried out significantwork on developing the diffuser design for a Diffuser Augmented Wind Turbine

(DAWT). The use of PHOENICS has been an essential element in this development,as it has allowed rapid investigation of a range of concepts. In addition wind-tunneltests using both gauze screens and turbine blades have been performed to validate theuse of Computational Fluid Dynamics (CFD) in the design of the diffuser.

This work is a continuation of the CFD modelling of the Vortec 7 prototype DAWT.The CFD model developed for the Vortec 7 was successfully used to predict the flow

behaviour within the diffuser. Examination of proposed modifications to predict their potential performance improvement was carried out using PHOENICS andmodifications exhibiting the most potential were then made to the prototype Vortec 7.With several key features of the diffuser design identified in the Vortec 7 analysis,development of an advanced diffuser began. A controlled contraction of the flow intothe diffuser was incorporated, together with modifications to the inlet and secondaryslots. Various inlet and exit to throat area ratios were examined over a range of turbine disc loading coefficients. The best design identified from this work was usedin a series of wind-tunnel tests to evaluate the CFD modelling. Results from thewind-tunnel have shown up deficiencies in the CFD model and have led torefinements of the model and the development of further design concepts.

It is anticipated that as the diffuser design progresses computational modelling willcontinue to play an important role alongside wind tunnel and prototype testing.

1. Introduction

A diffuser augmented wind turbine (DAWT) has a duct that surrounds the windturbine blades and increases in cross-sectional area in the streamwise direction. Theresulting sub-atmospheric pressure within the diffuser draws more air through the

blade plane, and hence more power can be generated compared to a bare turbine of the same rotor blade diameter.

Several researchers have examined the benefits and economics of placing a diffuser around a wind turbine. A theoretical study was undertaken by Lilley and Rainbird at

Cranfield in 1956. Their work identified the benefits of a DAWT, however it was notuntil the experimental work by Foreman [1,2] that an economical design was

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developed. This design was the basis of the Vortec 7, a technology demonstrationunit built by Vortec Energy Limited. An extensive programme of experimental work

performed at the Grumman Aerospace Corporation in the 1970s and early 1980sidentified the use of external air jets to prevent separation within the diffuser [1]. Thehigh speed jet flow re-energised the boundary layer within the diffuser enabling a

short length-to-diameter diffuser with a large outlet-to-inlet area ratio to be developed[2].

The economical benefits derived from this design were not seen until 1997. It was theuse of High Tensile Reinforced Fibrous Ferrocement (HT Ferro) that enabled VortecEnergy to secure the rights to the design and the production of the first full-scaleDAWT [3]. The Vortec 7 has a rotor blade diameter of 7.3m and is situated near theFranklin west coast, 120 km south of Auckland, New Zealand. The as-built Vortec 7is shown in Figure 1 (a), with the main geometric features shown in Figure 1(b)..

Figure 1 . The Vortec 7. (a) As built: Flow is from left to right. The curved boundary layer slot is seen near the outlet on the right. (b) main geometricfeatures.

During the initial testing of the Vortec 7 it became apparent that the power generateddid not match that predicted by Foreman and his co-workers. Since altering theconfiguration of full-scale wind turbine is both difficult and expensive, Vortec EnergyLimited enlisted the help of staff and post-graduate students at the University of

Auckland to carry out research into the cost-effective alterations that could be made tothe design. This research primarily involved the use of PHOENICS to test out the

possible gains from a variety of alterations. Following on from this modelling,PHOENICS has also been used to investigate a number of new diffuser designs.

2. Modelling Techniques

The initial study investigated the effects of the diffuser shape and boundary layer control slots of the as-built Vortec 7 and subsequent modifications. The model isaxisymmetric with specified inlet conditions and reference length making the modeldimensionless. A body-fitted grid is used to reproduce the complex geometry of theDAWT with the turbine modelled as a flow resistance. The turbine has a specifiedthrust coefficient that produces a pressure drop across the blade plane proportional to

Modifications

Original Vortec 7

Nosecone Nacelle

Bullnose

Instep

Structural Inlet Ring

Inlet Truss

Turbine BladeInlet Slot

Secondary Slot

user u e ange

CL

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the local dynamic pressure. This is analogous to the use of a gauze screen in windtunnel testing [4]. Features such as tip vortices and flow swirl have been omitted inorder to reduce the computation time. In this regard it should be noted that the focusof these studies is on the diffuser design and not on the design of the rotor itself.Hence it is considered that a simple turbine model is sufficient in this situation. These

assumptions allowed the use of the CFD modelling for rapid development of diffuser designs. The inlet truss of the Vortec 7 was modelled by a shear stress acting in boththe axial and radial directions over the cells in the inlet boundary layer control slot.The k- turbulence model has been used with uniform inlet boundary conditionsspecified for k and . The values of k and have been calculated for the hub heightand terrain in which a turbine would be situated by using the recommendations of Richards and Hoxey [5]. Modelling of later diffuser designs has generally usedsimilar techniques with only the geometry altered.

3. CFD comparison with Vortec 7 site data

The initial CFD model was of the as-built Vortec 7 and the results obtained werecompared with flow visualisation using smoke and spinnaker cloth tufts attached tothe walls of the diffuser. The CFD model correctly predicted separation downstreamof the nacelle and the diffuser outlet flange. Another important observation was thereversal of flow around the structural inlet ring of the Vortec 7 which is shown inFigure 1(b). This was also seen during smoke testing of the Vortec 7. It was foundthat the CFD model also predicted the high velocity speed up through the inlet slotand the radial variation of flow speed across the blade plane which was observed in

preliminary site measurements. Vortec Energy Limited modified the Vortec 7 byadding an elliptical nosecone, a bull-nose fitted to the inlet slot to prevent the flowreversal observed initially, and modified the secondary inlet slot and diffuser to directflow tangentially to the diffuser wall.

These modifications were incorporated into the CFD model together with an instep inthe centrebody to match the change in geometry from the circular spinner to the

pentagon shaped nacelle. Both quantitative and qualitative results were compared between the CFD model and the Vortec 7 site data. Flow visualisation showedregions of separation downstream of the spinner along the nacelle. Intermittentseparation was also observed on the trailing third of the primary diffuser with strongsecondary slot flow adhering to the secondary diffuser wall. The separated regionsagreed well with those predicted by the CFD model. The effect of the disc loadingwas studied and showed that at high disc loading the flow was forced out toward thediffuser and reduced the separated region on the diffuser. This effect was observedwith the Vortec 7 by varying the rotational speed of the turbine. Velocity anddirectional anemometry sensors located on a boom at the diffuser exit clearly showedthe separated region next to the nacelle and the high diffusion angles outside thatregion. These observations confirmed the streamline plots obtained with the CFDmodel, as illustrated in Figure 2.

The parameters of interest, which characterise the performance of the DAWT, are the

velocity speed up at the turbine plane and the power coefficient. Comparison betweenthe CFD model and site data of these parameters shows good agreement.

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Figure 2. Streamlines for the as-built Vortec 7.

The inlet speed up found from the CFD and Vortec 7 studies disagreed with the predictions of Grumman [1,6]. To obtain the maximum speed-up for power production the flow velocity through the diffuser must decrease and leave the diffuser at a slightly sub-atmospheric pressure. With the large separation along the nacelle,the change in effective area through the diffuser is reduced and hence lower speed-upsresulted. To improve the diffusion of flow, streamlining of the nacelle was examined.A smooth centrebody was incorporated into the CFD model having the same radius asthe spinner. The effect previously found due to varying the disc loading was alsoshown with this geometric configuration. For a local disc loading coefficient between0.8 and 1.0, the flow remained attached to both the diffuser wall and nacelle. A

doubling of the power coefficient at the optimal operating point was achieved byensuring the flow did not separate. An increase in power was found for all the discloadings modelled.

The Vortec 7 was modified to match the smooth nacelle tested in the CFD model bythe addition of foam sections. The modified Vortec 7 had reduced separation alongthe nacelle. A radial variation in flow direction across the exit plane was observedwith axial flow near the nacelle and a gradual turning in direction to be parallel to thediffuser at the wall. These agreed with the streamlines from the CFD model, and boththe model and Vortec 7 exhibited similar degrees of separation for various discloadings. The magnitudes of the power increase and inlet speed-up measured from

the Vortec 7 agreed with that predicted by the CFD. The velocity field predicted for the Vortec 7 with smooth nacelle is shown in Figure 3. The inlet velocity is set at

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unity and the vector magnitudes represent the velocity speed up caused by thediffuser. A 67% increase in velocity can be seen at the inlet boundary layer controlslot. A speed-up, over the blade plane, of 10% at the nacelle and up to 20% at the

blade tip has been found for the modified Vortec 7. This is a significant increasecompared to a bare turbine where for an ideal Betz machine, the velocity at the blade

plane is reduced to 66% of the free stream velocity. The effect of the secondary slotintroducing high speed flow is evident, although there is still some flow separation. Amore attached flow was predicted at higher disc loading.

Figure 3. Velocity vectors for the modified Vortec 7 at a disc thrust coefficientof 0.8.

4. Advanced diffuser designs

The CFD model of the Vortec 7 has been the starting point for the development of new diffuser designs. The effect of the inlet slot, diffuser outlet flange, nacelle shapeand size have been examined using the existing area ratio and diffuser exit angle. Anew design has been developed which introduces controlled contraction of the flowinto the blade plane. The primary diffuser has been extended upstream and the inner inlet ring removed. The area ratio and diffuser angle have been altered for this newgeometry and the diffuser exit flange removed. Increases in power coefficient by afactor of approximately four compared with the original design have been predicted.An example of the new geometry is shown in Figure 4. The new geometry withcontrolled inlet contraction displayed similar trends to those predicted by a one-

dimensional analysis [7]. Much larger inlet speed-ups were predicted with the

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optimal power coefficient around 1.9 occurring at a local disc loading between 0.3and 0.4.

Figure 4. The CAD solid model and the Wind-tunnel model of the 2nddiffuser design.

To verify the predictions for the development geometry a series of wind tunnelinvestigations have been performed. Initially the testing was performed in the 3 by 2metre closed section wind tunnel at Vipac Engineers and Scientists Limited,Melbourne, Australia. Various wire screen meshes were used to produce a range of

pressure drops across the blade plane. The total pressure drop across the mesh, thevelocity speed-up, and the base or exit pressure were measure with pitot-static probesand referenced to an upstream pitot-static probe in the free stream flow. Thesemeasurements provided the information required to characterise the DAWT

performance and allow comparison with the CFD predictions. Figures 5(a) and 5(b)show the wind tunnel results for power coefficient, velocity speed-up and base

pressure coefficient. These are compared with two CFD predictions. It was foundthat the blockage effect created by the closed wind tunnel test section accelerated theflow around the outside of the diffuser. The base pressure coefficient was artificiallylowered as a result of this. When the blockage was accounted for in the CFD modelwith the addition of a solid boundary at the hydraulic diameter of the wind tunnel,good agreement was obtained for the base pressure. It was noted during wind tunnelflow visualisation that several regions of the flow exhibited separation. These werenot predicted in the CFD model, probably due to the use of the k- turbulence modeltogether with wall functions on the diffuser surfaces. Consequently the power measured from the wind tunnel testing was lower than that predicted by the CFD.Refinement of the CFD model to include boundary layer calculation (Figure 6) hasimproved the agreement with wind tunnel results.

As a result of the wind-tunnel testing of the 2 nd generation diffuser, deficiencies in itsdesign have been highlighted. Hence further CFD and wind-tunnel studies are being

conducted in an effort to find a diffuser design which is both aerodynamically

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efficient and can be manufactured at a reasonable cost. Figure 7 shows an artistsimpression of one such concept.

(a)

(b)

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Disc Loading

C o e

f f i c i e n

t s

Available Power Available Power Speed Up Speed UpBase Pressure Base Pressure

Wind Tunnel CFD Predictions

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Disc Loading

C o e

f f i c i e n

t s

Available Power Available Power Speed Up Speed UpBase Pressure Base Pressure

Wind Tunnel CFD Predictions

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Figure 5. Comparison of wind-tunnel results and CFD predictions: (a) with noblockage correction and (b) incorporating a solid boundary at thewind-tunnel hydraulic diameter.

Figure 6. Streamline plot for the refined 2 nd diffuser design CFD model.

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Figure 7. Artists impression of a future DAWT design.5. Conclusions

Throughout the development of diffusers for a Diffuser Augmented Wind Turbine(DAWT), computational modelling has proved to be a valuable tool. It has been usedto interpret full-scale field results from the Vortec 7 prototype DAWT, to assess thevalue of proposed modifications and to show the limitations of the Vortec 7 design.In conjunction with wind tunnel testing computational modelling has allowed therapid development of new diffuser designs and the optimisation of these. However the models have not always been totally reliable and so it has been necessary toconstantly cross-reference the CFD and wind tunnel data. In some cases the windtunnel results have shown up limitations in the CFD models, which has led torefinements of the models, while in other situations the CFD model has provedvaluable in assessing effects such as blockage on the wind tunnel data.

It is anticipated that as the diffuser design progresses computational modelling willcontinue to play an important role alongside wind tunnel and prototype testing.

References

1. Foreman, K.M., Gilbert, B.L. Experiments with a Diffuser-Augmented Model Wind Turbine , J. Energy Resour Technol Trans. ASME, Vol 105, 3, 46-53,1983

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2. Foreman, K.M., Maciulaitis, A. and Gilbert, B.L ., Performance Predictionsand Recent Data for Advanced DAWT Models , ASME Solar Energy Division,Grumman Aerospace Corp., Bethpage, New York, April 1983

3. Nash, T.A. Design & Construction of the Vortec Seven , NZ WindEnergy Association, First Annual Conference, Wellington, June 1997

4. Gilbert, B.L, Oman, R.A. and Foreman, K.M. Fluid Dynamics of Diffuser- Augmented Wind Turbines , J. Energy, Vol 2, 6, 368-374, 1978

5. Richards, P.J., Hoxey, R.P. Appropriate boundary conditions for computational wind engineering models using the k- turbulence model , J.Wind Engng and Ind. Aerodyn., 46 & 47 , 145-153, 1993

6. Foreman, K.M., Maciulaitis, A. And Gilbert, B.L .,Performance Predictionsand Recent Data for Advanced DAWT Models , ASME Solar Energy Division,Grumman Aerospace Corp., Bethpage, New York, April 1983.

7. Phillips, D.G., Flay, R.G.J., Nash, T.A. Aerodynamic Analysis and Monitoring of the Vortec 7 Diffuser Augmented Wind Turbine, IPENZ Trans., Auckland,

New Zealand, November 1999.

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Appendix A. Sample Q1 File

TALK=T;RUN( 1, 1);VDU=X11-TERMGROUP 1. Run title and other preliminaries

TEXT(WT Model CT2=0.3 IAR=1.8 EAR=4.0 DA=45)INTEGER(NYNAC,NYTURB,NYPSLOT,NYPD,NYSSLOT,NYSD,NYASD,NYTOP,NYABOVE)INTEGER(NZUP,NZNAC,NZPD,NZSSLOT,NZSD,NZDOWN,NZT,NZIU,NZRDOWN,NZRUP)INTEGER(IYSTART,IYEND,IZSTART,IZEND,COUNTJ,COUNTK,COEFI)INTEGER(NYPDL,NZTI,SDIF)REAL(IAR,EAR,IAD,IA,EAD,EA,UIAD,UIA,TIAD,TIA,UEAD,UEA,DFL,INL,TR)REAL(ISS,NR,SEAR,SIAD,SIA,SEAD,SEA,SUIAD,SUIA,SUEAD,SUEA,SDFL,SNR)REAL(YTE,ZTE,SYTE,SZTE,SSZ,SSY,alphap,alphas)REAL(CT2,NACL,NACH,SPINNERL,TOPD,UPWND,DWNWND,GRNDC)REAL(LREF,PI,DEN,ZED,ZEDO,VELIN,VK,USTAR,EKVAL,EPVAL)REAL(TCELLS,THETAD,THETA,TOTPWR,COEF,COEF2,COEF3,COEFJ,COEFK)REAL(ZN,YN,ZNU,YNU,ZT,YT,ZZ1,YY1,ZZ2,YY2,ZZ3,YY3,ZZ4,YY4)REAL(KU,AU,BU,CU,DU,KLI,ALI,BLI,CLI,DLI,KLE,ALE,BLE,CLE,DLE)REAL(SZC,SYC,SZN,SYN,SZZ1,SYY1,SZZ2,SYY2,SZZ3,SYY3,SZZ4,SYY4)REAL(SYY5,SZZ5,SKU,SAU,SBU,SCU,SDU,SKL,SAL,SBL,SCL,SDL)

REAL(PCHORD,PALPHA,SCHORD,SALPHA,RDOWN,ZRUP,YTOP,YSLOTI,YSLOTE)REAL(VY1,VZ1,VY2,VZ2,VY3,VZ3,VY4,VZ4,VZ5,VZ6,VZ7,AV,BV,CV,DV,KV)REAL(VIA,VEAP,VEAS,VUEA,VUIA,VUEAS,YPSLTI,ZPSLTI,SASD)REAL(YPSLTE,ZPSLTE,YSSLTI,ZSSLTI,YSSLTC,ZSSLTC,YSSLTE,ZSSLTE)REAL(YSLTIA,ZSLTIA,YSLTEA,ZSLTEA,YSLTEB,ZSLTEB,YSLTPA,ZSLTPA)REAL(AA,BB,CC,DD,KK,HY1,HY2,HY3,HY4,HY5,HY6,HY7,HY8,YSLTPC,ZSLTPC)CHAR(RESP)REAL(RB,B2,ELPA2,B1,ELPA1,RA,ZAC,YAC,YY5,ZZ5,YPSLTA,ZPSLTA)REAL(A,B,C,COEF4,COEF5,COEF6)

REFERENCE PARAMETERSPI=4*ATAN(1);LREF=20

Axisymmetric WedgeTHETAD=2;THETA=pi*THETAD/180;GRNDC=5/lref

TURBINE PARAMETERSCT2=0.3;NACL=10/LREF;NACH=3/LREF;SPINNERL=1.5*NACH;TR=LREF/(2*LREF)

PRIMARY DIFFUSERInlet Area Ratio, Exit Area Ratio, Inlet Angle and Exit Angle

IAR=1.80; EAR=1.6;IAD*PI/180;EAD=45;EA=EAD*PI/180Upper Inlet Angle, Upper Exit Angle

UIAD=10;UIA=PI/2+UIAD*PI/180;UEAD=40;UEA=UEAD*PI/180Diffuser Axial Length, Inlet Slot Size, Nose Radius

DFL=18.3/LREF;ISS=0.4/LREF;NR=1.3/LREFSECONDARY SLOT SIZE

COEF=PI/2+EA;YSLOTE=0.3/LREF/SIN(COEF);ZSSLTE=YSLOTE*COS(COEF)YSSLTE=YSLOTE*SIN(COEF)

SECONDARY DIFFUSER

Exit Area Ratio, Inlet Angle, Exit AngleSEAR=4.0;SIAD=EAD;SIA=3*PI/2+SIAD*PI/180;SEAD=EAD;SEA=SEAD*PI/180

Upper Inlet Angle, Upper Exit AngleSUIAD=0;SUIA=PI/2+SUIAD*PI/180;SUEAD=0;SUEA=SUEAD*PI/180

Diffuser Chord, Nose Radius, Cell Band Width AboveSDFL=6/lref;SNR=0.8/LREF;SASD=1/LREF

REFERENCE POINTS AROUND PRIMARY DIFFUSERLower Arc Radius

YT=TR+ISS;ZN=0.;YN=(((TR+ISS)**2)*IAR)**0.5Rb=YT*((EAR**0.5)-1)/(1-cos(EA))

Inlet Angleb2=2/lref;elpa2=2.0*b2;a=elpa2;b=b2-Rb;c=YN-YT-Rbcoef4=(-2*a*c/b**2);coef5=(a/b)**2+1;coef6=(c/b)**2-1

coef3=(-coef4+((coef4)**2-4*coef5*coef6)**0.5)/(2*coef5)IA=ASIN(COEF3)

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Lower ExitZT=(Rb-b2)*sin(IA)+elpa2*cos(IA);YY3=YT+Rb*(1-cos(EA))ZZ3=ZT+Rb*sin(EA)

Upper ExitZTE=-0.089/LREF*SIN(EA);YTE=0.089/LREF*COS(EA)ZZ2=ZZ3+ZTE;YY2=YY3+YTE

Calculate Upper Arc Radiusb1=1.5/lref;elpa1=6*b1;A=YY2-(YN-elpa1*sin(IA)+b1*cos(IA))B=ZZ2-(elpa1*COS(IA)+b1*SIN(IA));C=(B**2+A**2)**0.5;COEF=A/BCOEF5=ATAN(COEF);COEF4=pi/2-(IA+(COEF5**2)**0.5);RA=C/(2*cos(COEF4))

Nose Upper TangentTIAD=45;TIA=TIAD*PI/180+IACOEF3=((ELPA1**4/B1**2*TAN(TIA)**2)/$(1+ELPA1**2/B1**2*TAN(TIA)**2))**0.5COEF4=(1-(COEF3/ELPA1)**2)**0.5*B1;COEF3=ELPA1-COEF3ZNU=COEF3*COS(IA)+COEF4*SIN(IA);YNU=YN-COEF3*SIN(IA)+COEF4*COS(IA)

Upper InletZZ1=ELPA1*COS(IA)+B1*SIN(IA);YY1=YN-ELPA1*SIN(IA)+B1*COS(IA)

Centre of Upper Arc

ZAC=ZZ1+RA*SIN(IA);YAC=YY1+RA*COS(IA)Nose Lower TangentTIAD=45;TIA=TIAD*PI/180-IACOEF3=((ELPA2**4/B2**2*TAN(TIA)**2)/$(1+ELPA2**2/B2**2*TAN(TIA)**2))**0.5COEF4=(1-(COEF3/ELPA2)**2)**0.5*B2;COEF3=ELPA2-COEF3ZZ4=COEF3*COS(IA)-COEF4*SIN(IA);YY4=YN-COEF3*SIN(IA)-COEF4*COS(IA)

Lower InletYY5=YT+Rb*(1-cos(IA));ZZ5=ZT-Rb*sin(IA)

Primary Diffuser ChordPCHORD=(DFL**2+(YY3-YN)**2)**0.5;COEF=(YY3-YN)/DFL;PALPHA=ATAN(COEF)

PRIMARY DIFFUSER COEFFICIENTS FOR CUBICUpper Surface

COEF=UIA-PI/2;KU=1/(ZZ2-ZZ1);AU=2*(YY1-YY2)+(TAN(COEF)+TAN(UEA))/KUBU=3*(YY2-YY1)-(TAN(UEA)+2*TAN(COEF))/KU;CU=TAN(COEF)/KU;DU=YY1

Lower Surface InletCOEF=2*PI+IA;IA-PI/2;KLI=1/(ZT-ZZ4);ALI=2*(YY4-YT)+TAN(COEF)/KLIBLI=3*(YT-YY4)-2*TAN(COEF)/KLI;CLI=TAN(COEF)/KLI;DLI=YY4

Lower Surface ExitKLE=1/(ZZ3-ZT);ALE=2*(YT-YY3)+TAN(EA)/KLE;BLE=3*(YY3-YT)-TAN(EA)/KLECLE=0;DLE=YT

REFERENCE POINTS AROUND SECONDARY DIFFUSERCOEF=EA+PI/2;SZTE=0.1/LREF*COS(COEF);SYTE=0.1/LREF*SIN(COEF)

Nose, Centre of Nose CircleSZN=ZZ3-SSZ;SYN=YY3+SSY;SZC=SZN;SYC=SYN+SNR

Adjacent to end of Primary DiffuserSZZ5=ZZ3+YSLOTE*COS(COEF);SYY5=YY3+YSLOTE*SIN(COEF)

Upper Inlet, Lower InletSZZ1=SZZ5+SZTE;SYY1=SYY5+SYTESZZ4=SZC+SNR*COS(SIA);SYY4=SYC+SNR*SIN(SIA)

Lower Exit, Upper ExitSYY3=(((TR+ISS)**2)*SEAR)**0.5;SZZ3=SZZ5+(SYY3-SYY5)/TAN(EA)SZZ2=SZZ3+SZTE;SYY2=SYY3+SYTE

Secondary Diffuser ChordSCHORD=(SDFL**2+(SYY3-SYN)**2)**0.5

COEF=(SYY3-SYN)/SDFL;SALPHA=ATAN(COEF)SECONDARY DIFFUSER COEFFICIENTS FOR CUBICUpper Surface

COEF=SUIA-PI/2;SKU=1/(SZZ2-SZZ1)SAU=2*(SYY1-SYY2)+(TAN(COEF)+TAN(SUEA))/SKUSBU=3*(SYY2-SYY1)-(TAN(SUEA)+2*TAN(COEF))/SKUSCU=TAN(COEF)/SKU;SDU=SYY1

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Lower SurfaceCOEF=SIA-PI/2;SKL=1/(SZZ3-SZZ4)SAL=2*(SYY4-SYY3)+(TAN(COEF)+TAN(SEA))/SKLSBL=3*(SYY3-SYY4)-(TAN(SEA)+2*TAN(COEF))/SKLSCL=TAN(COEF)/SKL;SDL=SYY4

DOMAIN SIZE

TOPD=5*SYY3;UPWND=4*SYY3;DWNWND=10*SYY3+SZZ3;RDOWN=SZZ3+0.8*SYY3ZRUP=10/LREF;YTOP=1.5*SYY2BOUNDARY LAYER AND TURBULENCE PARAMETERS

DEN=1.0;ZED=EAR*(TR+ISS)+GRNDC;ZEDO=0.02/lref;COEF=ZED/ZEDOVELIN=1.0;VK=0.4;USTAR=VK*VELIN/(LOG(COEF))EKVAL=3.33*USTAR**2;EPVAL=USTAR**3/(VK*ZED)

CELLSNYNAC=5;NYTURB=25;NYPSLOT=28;NYPDL=33;NYPD=38;NYSSLOT=15;NYSD=4NYASD=20;NYTOP=20;NYABOVE=40NZNAC=10;NZPD=70;NZTI=20;NZT=35;NZIU=50;NZSSLOT=0;NZSD=60NZUP=38;NZRUP=20;NZRDOWN=25;NZDOWN=45

TURBINE PARAMETERSTCELLS=NYTURB;THETAD=2;THETA=pi*THETAD/180;TOTPWR=360/thetad

GROUP 2. Transience; time-step specificationSTEADY=TGROUP 3. X-direction grid specificationGROUP 4. Y-direction grid specification

YVLAST=TOPDGROUP 5. Z-direction grid specification

ZWLAST=UPWND+DWNWNDGROUP 6. Body-fitted coordinates or grid distortion

BFC=T;VUP=T;WUP=T;NONORT=T;NX=1NY=NYNAC+NYTURB+NYPSLOT+NYPD+NYSSLOT+NYSD+NYASD+NYABOVENZ=NZUP+NZPD+NZSD+NZDOWN

Set all coordinates to zeroDOMAIN(1,NX+1,1,NY+1,1,NZ+1);SETLIN(XC,0);SETLIN(YC,0);SETLIN(ZC,0)GSET(D,NX,NY,NZ,LREF,TOPD,UPWND+DWNWND)

--- Points --gset(P,D1,0.0,0.0,-UPWND);gset(P,D2,0.0,TOPD,-UPWND)gset(P,D3,0.0,TOPD,DWNWND);gset(P,D4,0.0,0.0,DWNWND)

--- Lines ---gset(L,Dom1,D1,D2,NY,1.0);gset(L,Dom2,D2,D3,NZ,1.0)gset(L,Dom3,D3,D4,NY,1.0);gset(L,Dom4,D4,D1,NZ,1.0)

--- Frames ---gset(F,F1,D1,-,D2,-,D3,-,D4,-) ;gset(M,F1,+j+k,1,1,1,trans)

***********************************************************The remainder of the grid generation coding is too lengthy forinclusion in this appendix.***********************************************************

Group 7. Variables: STOREd,SOLVEd,NAMEdSOLVE(P1 ,V1 ,W1 );STORE(VCRT,ENUT,WCRT);SOLUTN(P1 ,Y,Y,Y,N,N,N)

Group 8. Terms & DevicesDIFCUT=0.0

Group 9. PropertiesRHO1 = DEN;CP1 = 1.005E+03ENUL = 1.500E-05/(8*lref);PRT (EP ) = 1.314E+00TURMOD(KEMODL)

Group 10.Inter-Phase Transfer ProcessesGroup 11.Initialise Var/Porosity Fields

INIADD=FPATCH(START,INIVAL,1,NX,1,NY,1,nz,1,1)INIT(START,P1,0.0,1E-3)INIT(START,W1,0.0,VELIN);INIT(START,V1,0.0,0.0)INIT(START,KE,0.0,EKVAL);INIT(START,EP,0.0,EPVAL)

RESTRT(ALL);FSWEEP=201

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Group 12. Convection and diffusion adjustmentsGroup 13. Boundary & Special SourcesNacelle and spinner

IYSTART=1;IYEND=NY1IZSTART=NZ1+nz14+1;IZEND=NZ1+nz14+nz2+nz3+nz4+nz5+nz15CONPOR(0.0,CELL,1,1,IYSTART,IYEND,IZSTART,IZEND)

patch(nacnor,swall,1,1,IYEND+1,IYEND+1,IZSTART,IZEND,1,1)coval(nacnor,w1,grnd2,0.0)coval(nacnor,KE,grnd2,grnd2);coval(nacnor,EP,grnd2,grnd2)patch(naclow,hwall,1,1,IYSTART,IYEND,IZSTART-1,IZSTART-1,1,1)coval(naclow,v1,grnd2,0.0)coval(naclow,KE,grnd2,grnd2);coval(naclow,EP,grnd2,grnd2)patch(nachig,lwall,1,1,IYSTART,IYEND,IZEND+1,IZEND+1,1,1)coval(nachig,v1,grnd2,0.0)coval(nachig,KE,grnd2,grnd2);coval(nachig,EP,grnd2,grnd2)

Primary DiffuserIYSTART=NYNAC+NYTURB+NYPSLOT+1;IYEND=IYSTART+NYPD-1IZSTART=NZUP+1;IZEND=NZUP+NZPDCONPOR(0.0,CELL,1,1,IYSTART,IYEND,IZSTART,IZEND)

patch(pdifsou,nwall,1,1,IYSTART-1,IYSTART-1,IZSTART,IZEND,1,1)coval(pdifsou,w1,1.0,0.0)coval(pdifsou,KE,grnd2,grnd2);coval(pdifsou,EP,grnd2,grnd2)patch(pdifnor,swall,1,1,IYEND+1,IYEND+1,IZSTART,IZEND,1,1)coval(pdifnor,w1,1.0,0.0)coval(pdifnor,KE,grnd2,grnd2);coval(pdifnor,EP,grnd2,grnd2)patch(pdiflow,hwall,1,1,IYSTART,IYEND,IZSTART-1,IZSTART-1,1,1)coval(pdiflow,v1,1.0,0.0)coval(pdiflow,KE,grnd2,grnd2);coval(pdiflow,EP,grnd2,grnd2)patch(pdifhig,lwall,1,1,IYSTART,IYEND,IZEND+1,IZEND+1,1,1)coval(pdifhig,v1,1.0,0.0)coval(pdifhig,KE,grnd2,grnd2);coval(pdifhig,EP,grnd2,grnd2)

Pressure Loss Across TurbineIYSTART=1;IYEND=NYNAC+TCELLS;IZSTART=NZUP+NZT;IZEND=IZSTARTPATCH(TURBINE,LOW,1,1,IYSTART,IYEND,IZSTART,IZEND,1,1)COVAL(TURBINE,W1,GRND1,0.0)

Secondary DiffuserIYSTART=NYNAC+NYTURB+NYPSLOT+NYPD+NYSSLOT+1;IYEND=IYSTART+NYSD-1IZSTART=NZUP+NZPD-4;IZEND=NZUP+NZPD+NZSDCONPOR(0.0,CELL,1,1,IYSTART,IYEND,IZSTART,IZEND)patch(sdifsou,nwall,1,1,IYSTART-1,IYSTART-1,IZSTART,IZEND,1,1)coval(sdifsou,w1,1.0,0.0)coval(sdifsou,KE,grnd2,grnd2);coval(sdifsou,EP,grnd2,grnd2)patch(sdifnor,swall,1,1,IYEND+1,IYEND+1,IZSTART,IZEND,1,1)coval(sdifnor,w1,1.0,0.0)coval(sdifnor,KE,grnd2,grnd2);coval(sdifnor,EP,grnd2,grnd2)patch(sdiflow,hwall,1,1,IYSTART,IYEND,IZSTART-1,IZSTART-1,1,1)coval(sdiflow,v1,1.0,0.0)coval(sdiflow,KE,grnd2,grnd2);coval(sdiflow,EP,grnd2,grnd2)patch(sdifhig,lwall,1,1,IYSTART,IYEND,IZEND+1,IZEND+1,1,1)coval(sdifhig,v1,1.0,0.0)coval(sdifhig,KE,grnd2,grnd2);coval(sdifhig,EP,grnd2,grnd2)

InletPATCH(INLET,LOW,1,NX,1,NY,1,1,1,1)COVAL(INLET,P1,FIXFLU,VELIN*RHO1);COVAL(INLET,W1,ONLYMS,VELIN)COVAL(INLET,KE,ONLYMS,EKVAL);COVAL(INLET,EP,ONLYMS,EPVAL)COVAL(INLET,V1,ONLYMS,0.0)

OutletPATCH(OUTLET,HIGH,1,NX,1,NY,NZ,NZ,1,1)COVAL(OUTLET,P1,FIXP,-1.0);COVAL(OUTLET,W1,ONLYMS,SAME)COVAL(OUTLET,V1,ONLYMS,SAME)COVAL(OUTLET,EP,ONLYMS,SAME);COVAL(OUTLET,KE,ONLYMS,SAME)

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Group 14. Downstream Pressure For PARABGroup 15. Terminate Sweeps

LSWEEP = 1000;SELREF = TRESFAC = 1.000E-03RESREF(P1)=1.E-3;RESREF(V1)=1.E-3;RESREF(W1)=1.E-3;RESREF(EP)=1.E-3;RESREF(KE)=1.E-3;

Group 16. Terminate IterationsGroup 17. RelaxationRELAX(P1 ,LINRLX, 0.3);RELAX(V1 ,FALSDT, 0.01)RELAX(W1 ,FALSDT, 0.01)RELAX(KE ,FALSDT, 0.1);RELAX(EP ,FALSDT, 0.1)KELIN = 0

Group 18. Limitsvarmax(ke)=20*ekval;varmax(ep)=40*epvalvarmax(p1)=den*velin*velin;varmin(p1)=-5*den*velin*velinvarmax(V1)=5*velin;varmax(W1)=5*velin

Group 19. EARTH Calls To GROUND StationRSG16=DEN;RSG17=CT2;RSG18=TOTPWRRSG19=CTcoef;RSG20=CT2cbh;RSG21=cbh

Coordinates of Primary DiffuserISG1=NYNAC+NYTURB+NYPSLOT+1;ISG2=NYNAC+NYTURB+NYPSLOT+NYPDISG3=NZUP+1;ISG4=NZUP+NZPD

Coordinates of Secondary DiffuerISG5=NYNAC+NYTURB+NYPSLOT+NYPD+NYSSLOT+1ISG6=NYNAC+NYTURB+NYPSLOT+NYPD+NYSSLOT+NYSDISG7=NZUP+NZPD-4;ISG8=NZUP+NZPD+NZSDGENK = T

Group 20. Preliminary PrintoutECHO = T

Group 21. Print-out of VariablesOUTPUT(P1,Y,Y,Y,Y,Y,Y);OUTPUT(V1,Y,Y,Y,Y,Y,Y)OUTPUT(W1,Y,Y,Y,Y,Y,Y);OUTPUT(ENUT,N,Y,Y,N,Y,Y)OUTPUT(WCRT,Y,Y,Y,Y,Y,Y);OUTPUT(VCRT,Y,Y,Y,Y,Y,Y)

Group 22. Monitor Print-OutIXMON=1;IYMON=nynac+nyturb+1;IZMON=nzup+nztUWATCH = T;USTEER = T

roup 23.Field Print-Out & Plot ControlYZPR=T;IXPRF=1;IXPRL=1IYPRF=1;IYPRL=NYNAC+NYTURB+NYPSLOT+NYPD+NYSSLOT+NYSD+NYASDIZPRF=1;IZPRL=NZUP+NZPD+NZSD+NZRDOWNNXPRIN=1;NYPRIN=1;NZPRIN=1ITABL = 3

No PATCHes used for this GroupGroup 24. Dumps For Restarts

SAVE=TLABEL QUITSTOP

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Appendix B. Extracts from the GROUND file.

C FILE NAME GROUND.FTN--------------------------------130995SUBROUTINE GROUND

C*****************************************************************C--- GROUP 1. Run title and other preliminaries

C1 GO TO (1001,1002,1003),ISC

C * -----------GROUP 1 SECTION 3 ---------------------------C---- Use this group to create storage via GXMAKE which it is notC necessary to dump to PHI for restarts

1001 CONTINUECALL MAKE(GRSP1)CALL MAKE(GRSP2)

C*****************************************************************CC--- GROUP 13. Boundary conditions and special sources

C Index for Coefficient - COC Index for Value - VAL

13 CONTINUEGO TO (130,131,132,133,134,135,136,137,138,139,1310,

11311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321),ISC130 CONTINUE

C------------------- SECTION 1 ------------- coefficient = GRNDC Calculates Loss Through Inlet Truss

CALL FN3(CO,W1,0.0,0.0,1.0)CALL FN35(CO,V1,0.0,2.0)CALL FN51(CO,0.5)CALL FN25(CO,RSG15*RSG16/2)RETURN

131 CONTINUEC------------------- SECTION 2 ------------- coefficient = GRND1C Calculates Pressure Drop Across Turbine

CALL FN3(CO,W1,0.0,0.0,1.0)CALL FN51(CO,0.5)CALL FN25(CO,RSG17*RSG16/2)

CC Calculates Wind Power Through Turbine

CALL GTIZYX(10,IZ,TAREA,50,1)CALL SETYX(GRSP2,TAREA,50,1)write(14,*)'TAREA=',GRSP2CALL FN3(GRSP1,W1,0.0,0.0,1.0)write(14,*)'W1=',GRSP1

CALL FN25(GRSP1,RSG17*RSG16*RSG18)write(14,*)'W1=',GRSP1CALL FN25(GRSP1,4/3.1415927)write(14,*)'W1=',GRSP1CALL FN20(PWR,GRSP1,GRSP2,W1)

CC WRITE(14,*)'TURA=',TAREA,'RSG19=',1

WRITE(14,*)'PWR=',PWR,'1',1RETURN