Wind Turbine

DIFFUSER DEVELOPMENT FOR A DIFFUSER AUGMENTED WIND TURBINE USING COMPUTATIONAL FLUID DYNAMICS D.G. Phillips, P.J. Richards, R.G.J. Flay Department of Mechanical Engineering The University of Auckland New Zealand PHOENICS VERSION 3.2 and earlier. Abstract Vortec Energy Limited and The University of Auckland have carried out significant work 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-tunnel tests using both gauze screens and turbine blades have been performed to validate the use 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 and modifications 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 into the diffuser was incorporated, together with modifications to the inlet and secondary slots. 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 used in a series of wind-tunnel tests to evaluate the CFD modelling. Results from the wind-tunnel have shown up deficiencies in the CFD model and have led to refinements of the model and the development of further design concepts. It is anticipated that as the diffuser design progresses computational modelling will continue 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 wind turbine blades and increases in cross-sectional area in the streamwise direction. The resulting 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 not until the experimental work by Foreman [1,2] that an economical design was

developed. This design was the basis of the Vortec 7, a technology demonstration unit built by Vortec Energy Limited. An extensive programme of experimental work performed at the Grumman Aerospace Corporation in the 1970’s and early 1980’s identified the use of external air jets to prevent separation within the diffuser [1]. The high 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 the use of High Tensile Reinforced Fibrous Ferrocement (HT Ferro) that enabled Vortec Energy to secure the rights to the design and the production of the first full-scale DAWT [3]. The Vortec 7 has a rotor blade diameter of 7.3m and is situated near the Franklin west coast, 120 km south of Auckland, New Zealand. The as-built Vortec 7 is shown in Figure 1 (a), with the main geometric features shown in Figure 1(b).. Diffuser Outlet Flange

Sec ondary Slot Struc tural Inlet Ring Bullnose Inlet Truss Turbine Blade

Inlet Slot

Original Vortec 7 Modific ations

Instep

Nosecone

Nacelle

C L

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 geometric features. During the initial testing of the Vortec 7 it became apparent that the power generated did not match that predicted by Foreman and his co-workers. Since altering the configuration of full-scale wind turbine is both difficult and expensive, Vortec Energy Limited 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 to the 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 is axisymmetric with specified inlet conditions and reference length making the model dimensionless. A body-fitted grid is used to reproduce the complex geometry of the DAWT with the turbine modelled as a flow resistance. The turbine has a specified thrust coefficient that produces a pressure drop across the blade plane proportional to

the local dynamic pressure. This is analogous to the use of a gauze screen in wind tunnel testing [4]. Features such as tip vortices and flow swirl have been omitted in order to reduce the computation time. In this regard it should be noted that the focus of 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 both the axial and radial directions over the cells in the inlet boundary layer control slot. The k-ε turbulence model has been used with uniform inlet boundary conditions specified for k and ε. The values of k and ε have been calculated for the hub height and terrain in which a turbine would be situated by using the recommendations of Richards and Hoxey [5]. Modelling of later diffuser designs has generally used similar 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 were compared with flow visualisation using smoke and spinnaker cloth tufts attached to the walls of the diffuser. The CFD model correctly predicted separation downstream of the nacelle and the diffuser outlet flange. Another important observation was the reversal of flow around the structural inlet ring of the Vortec 7 which is shown in Figure 1(b). This was also seen during smoke testing of the Vortec 7. It was found that the CFD model also predicted the high velocity speed up through the inlet slot and the radial variation of flow speed across the blade plane which was observed in preliminary site measurements. Vortec Energy Limited modified the Vortec 7 by adding an elliptical nosecone, a bull-nose fitted to the inlet slot to prevent the flow reversal observed initially, and modified the secondary inlet slot and diffuser to direct flow tangentially to the diffuser wall. These modifications were incorporated into the CFD model together with an instep in the 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 showed regions of separation downstream of the spinner along the nacelle. Intermittent separation was also observed on the trailing third of the primary diffuser with strong secondary slot flow adhering to the secondary diffuser wall. The separated regions agreed well with those predicted by the CFD model. The effect of the disc loading was studied and showed that at high disc loading the flow was forced out toward the diffuser and reduced the separated region on the diffuser. This effect was observed with the Vortec 7 by varying the rotational speed of the turbine. Velocity and directional anemometry sensors located on a boom at the diffuser exit clearly showed the separated region next to the nacelle and the high diffusion angles outside that region. These observations confirmed the streamline plots obtained with the CFD model, 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 between the CFD model and site data of these parameters shows good agreement.

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-ups resulted. 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 as the spinner. The effect previously found due to varying the disc loading was also shown with this geometric configuration. For a local disc loading coefficient between 0.8 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 by ensuring the flow did not separate. An increase in power was found for all the disc loadings modelled. The Vortec 7 was modified to match the smooth nacelle tested in the CFD model by the addition of foam sections. The modified Vortec 7 had reduced separation along the nacelle. A radial variation in flow direction across the exit plane was observed with axial flow near the nacelle and a gradual turning in direction to be parallel to the diffuser at the wall. These agreed with the streamlines from the CFD model, and both the model and Vortec 7 exhibited similar degrees of separation for various disc loadings. 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

unity and the vector magnitudes represent the velocity speed up caused by the diffuser. A 67% increase in velocity can be seen at the inlet boundary layer control slot. 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 increase compared 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 slot introducing high speed flow is evident, although there is still some flow separation. A more attached flow was predicted at higher disc loading.

Figure 3. Velocity vectors for the modified Vortec 7 at a disc thrust coefficient of 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 shape and size have been examined using the existing area ratio and diffuser exit angle. A new design has been developed which introduces controlled contraction of the flow into 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 new geometry and the diffuser exit flange removed. Increases in power coefficient by a factor 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 with controlled inlet contraction displayed similar trends to those predicted by a onedimensional analysis [7]. Much larger inlet speed-ups were predicted with the

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

Figure 4.

The CAD solid model and the Wind-tunnel model of the 2nd diffuser design.

To verify the predictions for the development geometry a series of wind tunnel investigations have been performed. Initially the testing was performed in the 3 by 2 metre 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, the velocity speed-up, and the base or exit pressure were measure with pitot-static probes and referenced to an upstream pitot-static probe in the free stream flow. These measurements 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 found that the blockage effect created by the closed wind tunnel test section accelerated the flow around the outside of the diffuser. The base pressure coefficient was artificially lowered as a result of this. When the blockage was accounted for in the CFD model with 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 tunnel flow visualisation that several regions of the flow exhibited separation. These were not predicted in the CFD model, probably due to the use of the k-ε turbulence model together 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) has improved the agreement with wind tunnel results. As a result of the wind-tunnel testing of the 2 nd generation diffuser, deficiencies in its design 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

efficient and can be manufactured at a reasonable cost. Figure 7 shows an artist’s impression of one such concept.

Wind Tunnel

CFD Predictions

Available Power Speed Up Base Pressure

2.5

Available Power Speed Up Base Pressure

2

Coefficients

1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.2

1.4

-0.5 -1 -1.5

Disc Loading

(a) Wind Tunnel

CFD Predictions

Available Power Speed Up Base Pressure

3

Available Power Speed Up Base Pressure

2.5

Coefficients

2 1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

-0.5 -1 -1.5

Disc Loading

(b)

1

Figure 5. Comparison of wind-tunnel results and CFD predictions: (a) with no blockage correction and (b) incorporating a solid boundary at the wind-tunnel hydraulic diameter.

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

Figure 7. Artist’s 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 used to interpret full-scale field results from the Vortec 7 prototype DAWT, to assess the value of proposed modifications and to show the limitations of the Vortec 7 design. In conjunction with wind tunnel testing computational modelling has allowed the rapid 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 to constantly cross-reference the CFD and wind tunnel data. In some cases the wind tunnel results have shown up limitations in the CFD models, which has led to refinements of the models, while in other situations the CFD model has proved valuable in assessing effects such as blockage on the wind tunnel data. It is anticipated that as the diffuser design progresses computational modelling will continue 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

2.

Foreman, K.M., Maciulaitis, A. and Gilbert, B.L., “Performance Predictions and 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 Wind Energy Association, First Annual Conference, Wellington, June 1997

4.

Gilbert, B.L, Oman, R.A. and Foreman, K.M. “Fluid Dynamics of DiffuserAugmented 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- ε turbulence model”, J. Wind Engng and Ind. Aerodyn., 46 & 47, 145-153, 1993

6.

Foreman, K.M., Maciulaitis, A. And Gilbert, B.L.,“Performance Predictions and 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.

Appendix A. Sample Q1 File TALK=T;RUN( 1, 1);VDU=X11-TERM GROUP 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 PARAMETERS PI=4*ATAN(1);LREF=20 Axisymmetric Wedge THETAD=2;THETA=pi*THETAD/180;GRNDC=5/lref TURBINE PARAMETERS CT2=0.3;NACL=10/LREF;NACH=3/LREF;SPINNERL=1.5*NACH;TR=LREF/(2*LREF) PRIMARY DIFFUSER Inlet Area Ratio, Exit Area Ratio, Inlet Angle and Exit Angle IAR=1.80; EAR=1.6;IAD*PI/180;EAD=45;EA=EAD*PI/180 Upper Inlet Angle, Upper Exit Angle UIAD=10;UIA=PI/2+UIAD*PI/180;UEAD=40;UEA=UEAD*PI/180 Diffuser Axial Length, Inlet Slot Size, Nose Radius DFL=18.3/LREF;ISS=0.4/LREF;NR=1.3/LREF SECONDARY 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 Angle SEAR=4.0;SIAD=EAD;SIA=3*PI/2+SIAD*PI/180;SEAD=EAD;SEA=SEAD*PI/180 Upper Inlet Angle, Upper Exit Angle SUIAD=0;SUIA=PI/2+SUIAD*PI/180;SUEAD=0;SUEA=SUEAD*PI/180 Diffuser Chord, Nose Radius, Cell Band Width Above SDFL=6/lref;SNR=0.8/LREF;SASD=1/LREF REFERENCE POINTS AROUND PRIMARY DIFFUSER Lower Arc Radius YT=TR+ISS;ZN=0.;YN=(((TR+ISS)**2)*IAR)**0.5 Rb=YT*((EAR**0.5)-1)/(1-cos(EA)) Inlet Angle b2=2/lref;elpa2=2.0*b2;a=elpa2;b=b2-Rb;c=YN-YT-Rb coef4=(-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)

Lower Exit ZT=(Rb-b2)*sin(IA)+elpa2*cos(IA);YY3=YT+Rb*(1-cos(EA)) ZZ3=ZT+Rb*sin(EA) Upper Exit ZTE=-0.089/LREF*SIN(EA);YTE=0.089/LREF*COS(EA) ZZ2=ZZ3+ZTE;YY2=YY3+YTE Calculate Upper Arc Radius b1=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/B COEF5=ATAN(COEF);COEF4=pi/2-(IA+(COEF5**2)**0.5);RA=C/(2*cos(COEF4)) Nose Upper Tangent TIAD=45;TIA=TIAD*PI/180+IA COEF3=((ELPA1**4/B1**2*TAN(TIA)**2)/$ (1+ELPA1**2/B1**2*TAN(TIA)**2))**0.5 COEF4=(1-(COEF3/ELPA1)**2)**0.5*B1;COEF3=ELPA1-COEF3 ZNU=COEF3*COS(IA)+COEF4*SIN(IA);YNU=YN-COEF3*SIN(IA)+COEF4*COS(IA) Upper Inlet ZZ1=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 Tangent TIAD=45;TIA=TIAD*PI/180-IA COEF3=((ELPA2**4/B2**2*TAN(TIA)**2)/$ (1+ELPA2**2/B2**2*TAN(TIA)**2))**0.5 COEF4=(1-(COEF3/ELPA2)**2)**0.5*B2;COEF3=ELPA2-COEF3 ZZ4=COEF3*COS(IA)-COEF4*SIN(IA);YY4=YN-COEF3*SIN(IA)-COEF4*COS(IA) Lower Inlet YY5=YT+Rb*(1-cos(IA));ZZ5=ZT-Rb*sin(IA) Primary Diffuser Chord PCHORD=(DFL**2+(YY3-YN)**2)**0.5;COEF=(YY3-YN)/DFL;PALPHA=ATAN(COEF) PRIMARY DIFFUSER COEFFICIENTS FOR CUBIC Upper Surface COEF=UIA-PI/2;KU=1/(ZZ2-ZZ1);AU=2*(YY1-YY2)+(TAN(COEF)+TAN(UEA))/KU BU=3*(YY2-YY1)-(TAN(UEA)+2*TAN(COEF))/KU;CU=TAN(COEF)/KU;DU=YY1 Lower Surface Inlet COEF=2*PI+IA;IA-PI/2;KLI=1/(ZT-ZZ4);ALI=2*(YY4-YT)+TAN(COEF)/KLI BLI=3*(YT-YY4)-2*TAN(COEF)/KLI;CLI=TAN(COEF)/KLI;DLI=YY4 Lower Surface Exit KLE=1/(ZZ3-ZT);ALE=2*(YT-YY3)+TAN(EA)/KLE;BLE=3*(YY3-YT)-TAN(EA)/KLE CLE=0;DLE=YT REFERENCE POINTS AROUND SECONDARY DIFFUSER COEF=EA+PI/2;SZTE=0.1/LREF*COS(COEF);SYTE=0.1/LREF*SIN(COEF) Nose, Centre of Nose Circle SZN=ZZ3-SSZ;SYN=YY3+SSY;SZC=SZN;SYC=SYN+SNR Adjacent to end of Primary Diffuser SZZ5=ZZ3+YSLOTE*COS(COEF);SYY5=YY3+YSLOTE*SIN(COEF) Upper Inlet, Lower Inlet SZZ1=SZZ5+SZTE;SYY1=SYY5+SYTE SZZ4=SZC+SNR*COS(SIA);SYY4=SYC+SNR*SIN(SIA) Lower Exit, Upper Exit SYY3=(((TR+ISS)**2)*SEAR)**0.5;SZZ3=SZZ5+(SYY3-SYY5)/TAN(EA) SZZ2=SZZ3+SZTE;SYY2=SYY3+SYTE Secondary Diffuser Chord SCHORD=(SDFL**2+(SYY3-SYN)**2)**0.5 COEF=(SYY3-SYN)/SDFL;SALPHA=ATAN(COEF) SECONDARY DIFFUSER COEFFICIENTS FOR CUBIC Upper Surface COEF=SUIA-PI/2;SKU=1/(SZZ2-SZZ1) SAU=2*(SYY1-SYY2)+(TAN(COEF)+TAN(SUEA))/SKU SBU=3*(SYY2-SYY1)-(TAN(SUEA)+2*TAN(COEF))/SKU SCU=TAN(COEF)/SKU;SDU=SYY1

Lower Surface COEF=SIA-PI/2;SKL=1/(SZZ3-SZZ4) SAL=2*(SYY4-SYY3)+(TAN(COEF)+TAN(SEA))/SKL SBL=3*(SYY3-SYY4)-(TAN(SEA)+2*TAN(COEF))/SKL SCL=TAN(COEF)/SKL;SDL=SYY4 DOMAIN SIZE TOPD=5*SYY3;UPWND=4*SYY3;DWNWND=10*SYY3+SZZ3;RDOWN=SZZ3+0.8*SYY3 ZRUP=10/LREF;YTOP=1.5*SYY2 BOUNDARY LAYER AND TURBULENCE PARAMETERS DEN=1.0;ZED=EAR*(TR+ISS)+GRNDC;ZEDO=0.02/lref;COEF=ZED/ZEDO VELIN=1.0;VK=0.4;USTAR=VK*VELIN/(LOG(COEF)) EKVAL=3.33*USTAR**2;EPVAL=USTAR**3/(VK*ZED) CELLS NYNAC=5;NYTURB=25;NYPSLOT=28;NYPDL=33;NYPD=38;NYSSLOT=15;NYSD=4 NYASD=20;NYTOP=20;NYABOVE=40 NZNAC=10;NZPD=70;NZTI=20;NZT=35;NZIU=50;NZSSLOT=0;NZSD=60 NZUP=38;NZRUP=20;NZRDOWN=25;NZDOWN=45 TURBINE PARAMETERS TCELLS=NYTURB;THETAD=2;THETA=pi*THETAD/180;TOTPWR=360/thetad GROUP 2. Transience; time-step specification STEADY=T GROUP 3. X-direction grid specification GROUP 4. Y-direction grid specification YVLAST=TOPD GROUP 5. Z-direction grid specification ZWLAST=UPWND+DWNWND GROUP 6. Body-fitted coordinates or grid distortion BFC=T;VUP=T;WUP=T;NONORT=T;NX=1 NY=NYNAC+NYTURB+NYPSLOT+NYPD+NYSSLOT+NYSD+NYASD+NYABOVE NZ=NZUP+NZPD+NZSD+NZDOWN Set all coordinates to zero DOMAIN(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 for inclusion in this appendix. *********************************************************** Group 7. Variables: STOREd,SOLVEd,NAMEd SOLVE(P1 ,V1 ,W1 );STORE(VCRT,ENUT,WCRT);SOLUTN(P1 ,Y,Y,Y,N,N,N) Group 8. Terms & Devices DIFCUT=0.0 Group 9. Properties RHO1 = DEN;CP1 = 1.005E+03 ENUL = 1.500E-05/(8*lref);PRT (EP ) = 1.314E+00 TURMOD(KEMODL) Group 10.Inter-Phase Transfer Processes Group 11.Initialise Var/Porosity Fields INIADD=F PATCH(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

Group 12. Convection and diffusion adjustments Group 13. Boundary & Special Sources Nacelle and spinner IYSTART=1;IYEND=NY1 IZSTART=NZ1+nz14+1;IZEND=NZ1+nz14+nz2+nz3+nz4+nz5+nz15 CONPOR(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 Diffuser IYSTART=NYNAC+NYTURB+NYPSLOT+1;IYEND=IYSTART+NYPD-1 IZSTART=NZUP+1;IZEND=NZUP+NZPD CONPOR(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 Turbine IYSTART=1;IYEND=NYNAC+TCELLS;IZSTART=NZUP+NZT;IZEND=IZSTART PATCH(TURBINE,LOW,1,1,IYSTART,IYEND,IZSTART,IZEND,1,1) COVAL(TURBINE,W1,GRND1,0.0) Secondary Diffuser IYSTART=NYNAC+NYTURB+NYPSLOT+NYPD+NYSSLOT+1;IYEND=IYSTART+NYSD-1 IZSTART=NZUP+NZPD-4;IZEND=NZUP+NZPD+NZSD CONPOR(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) Inlet PATCH(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) Outlet PATCH(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)

Group 14. Downstream Pressure For PARAB Group 15. Terminate Sweeps LSWEEP = 1000;SELREF = T RESFAC = 1.000E-03 RESREF(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 Iterations Group 17. Relaxation RELAX(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. Limits varmax(ke)=20*ekval;varmax(ep)=40*epval varmax(p1)=den*velin*velin;varmin(p1)=-5*den*velin*velin varmax(V1)=5*velin;varmax(W1)=5*velin Group 19. EARTH Calls To GROUND Station RSG16=DEN;RSG17=CT2;RSG18=TOTPWR RSG19=CTcoef;RSG20=CT2cbh;RSG21=cbh Coordinates of Primary Diffuser ISG1=NYNAC+NYTURB+NYPSLOT+1;ISG2=NYNAC+NYTURB+NYPSLOT+NYPD ISG3=NZUP+1;ISG4=NZUP+NZPD Coordinates of Secondary Diffuer ISG5=NYNAC+NYTURB+NYPSLOT+NYPD+NYSSLOT+1 ISG6=NYNAC+NYTURB+NYPSLOT+NYPD+NYSSLOT+NYSD ISG7=NZUP+NZPD-4;ISG8=NZUP+NZPD+NZSD GENK = T Group 20. Preliminary Printout ECHO = T Group 21. Print-out of Variables OUTPUT(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-Out IXMON=1;IYMON=nynac+nyturb+1;IZMON=nzup+nzt UWATCH = T;USTEER = T roup 23.Field Print-Out & Plot Control YZPR=T;IXPRF=1;IXPRL=1 IYPRF=1;IYPRL=NYNAC+NYTURB+NYPSLOT+NYPD+NYSSLOT+NYSD+NYASD IZPRF=1;IZPRL=NZUP+NZPD+NZSD+NZRDOWN NXPRIN=1;NYPRIN=1;NZPRIN=1 ITABL = 3 No PATCHes used for this Group Group 24. Dumps For Restarts SAVE=T LABEL QUIT STOP

Appendix B. Extracts from the GROUND file. C FILE NAME GROUND.FTN--------------------------------130995 SUBROUTINE GROUND C***************************************************************** C--- GROUP 1. Run title and other preliminaries C 1 GO TO (1001,1002,1003),ISC C * -----------GROUP 1 SECTION 3 --------------------------C---- Use this group to create storage via GXMAKE which it is not C necessary to dump to PHI for restarts 1001 CONTINUE CALL MAKE(GRSP1) CALL MAKE(GRSP2) C***************************************************************** C C--- GROUP 13. Boundary conditions and special sources C Index for Coefficient - CO C Index for Value - VAL 13 CONTINUE GO TO (130,131,132,133,134,135,136,137,138,139,1310, 11311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321),ISC 130 CONTINUE C------------------- SECTION 1 ------------- coefficient = GRND C 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 CONTINUE C------------------- SECTION 2 ------------- coefficient = GRND1 C 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) C C Calculates Wind Power Through Turbine CALL GTIZYX(10,IZ,TAREA,50,1) CALL SETYX(GRSP2,TAREA,50,1) write(14,*)'TAREA=',GRSP2 CALL FN3(GRSP1,W1,0.0,0.0,1.0) write(14,*)'W1=',GRSP1 CALL FN25(GRSP1,RSG17*RSG16*RSG18) write(14,*)'W1=',GRSP1 CALL FN25(GRSP1,4/3.1415927) write(14,*)'W1=',GRSP1 CALL FN20(PWR,GRSP1,GRSP2,W1) C C WRITE(14,*)'TURA=',TAREA,'RSG19=',1 WRITE(14,*)'PWR=',PWR,'1',1 RETURN