Numerical models in aquacultural engineering:two examples

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Numerical models in aquacultural engineering:two examples

Raman-Nair, W.; Colbourne, D. B.; Gagnon, M.; Bergeron, P.

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∗ ∗ _! ∗∗ ! ∗∗ ∗ ! " ∗∗ _# _$%" _& _{' (} _' ) " ! " ! # $ " % &' () *+ &)' , , & - . - . / "* 01 1)'21&'0 3 0" 4 5 , & -6 + . , 2 1 + & 7 -. * + 7 7

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Still Water Level

Seabed

Mussel Longline System (n=113)

0 A 2 A 108 A 113 A 6 Y Buoy Mussel sock Concrete block 113 = n m 5 . 15 m 142 m 5 . 15 Not to Scale 0 at Origin A x z 1 A 6 A 6 B 111 A 3 A 112 A 3 B B108 m 7 m 15 114 A 108 Y 111 B intervals m 1.392 at to NodesA6 A108 108 18 12 6, , ,..., nodes at blocks Concrete 18 AA A A Node m 14 14m 8 % + k A k b3 B k G B k e B k P B k Q k B Buoy k y3 Y k G Y k P Y k Q k Y Sock Mussel 1 − k A Ak+1 B Segment L Segment Y Segment R

Segment Main Line

R k t L k t B k t Y k t B k f Y k e Y k f 8 / Sk - . & , $ ! -$ 1 %9:;. 4 4 7 & # 7 " -+ %9:<. , 5 , & & ) 8 % O N −→N ,−→N ,−→N A An Sk k , . . . , n 8 / Sk Ak Bk - . Yk & Ak 7 k k AE ℓ A E ℓ = c c ζ × √ k × ζ . 7 > S , S , Sn− Sn + B Bn− A An− 7 > S S_n− B % / ? θi i , , -$ 1 1 %9:?. 5 , , − →_b i −→Ni i , , B − → N ,−→N ,−→N θ , θ , θ , −→bi − → Ni    − →_N − →_N − → N    NCB    − →_b − → b − → b    -%. , N_CB % / ? , -$ 1 1 %9:?. N_CB  c c s s c − s c c s c s s c s s s s c c c s s − c s −s s c c c   -/. si θi ci θi i , , & Bk Yk , , − →_b ik −→yik i , , Sk • ' Ak qAik i , , • 5 Bk qBik θBik i , , qB i ,k i , , • " + Yk qYik θY_ik i , , qY i ,k i , , % / ? -$ 1

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1 %9:?. • ' Ak uAik − →_vA −→_N i i , , −→vA Ak • 5 Bk uB ik −→ωB − →_b ik i , , uB i ,k −→vB − →_b ik i , , − →_ωB −→_vB Bk • " + Yk uY ik −→ωY −→yik i , , uY i ,k −→vY − →_y_ik _i _{, ,} −→_ωY − →_vY Yk. qS ik uS ik Sk qS_ik qA_ik uS_ik uA_ik i , , qS _i,k qB_ik uS _i,k uB_ik i , . . . , qS _i,k qY_ik uS _i,k uY_ik i , . . . , -?. $ & Ak @ # # - N N Ak F∗A r −→vAr −mAk−→aA -A. − →_aA −→_vA r mA k Ak − → F /A Ak # F_rA −→F /A −→vA_r r , . . . , -;. & Ak VkA Ak − →_FGB/A _ρ fVkA− mAk g − →_N _-B. g Ak 8 / L, R, B Y 8 , A_k− Ak ℓL k −−−−−→ AkAk− −→tLk ZL k Ak kLkZLk−→tLk kkL 7 ZL k , ZL k Z k− ℓLk Z k− ℓLk k , . . . , n -<. A_k− Ak Ak cLkZLk−→tLk cLk = ZL k A_k− Ak ZL_k i uA_i,k− − uAik tAik ZLk -:. k , . . . , n tA ik i −→tLk. * − →_FT /A − →_FSD/A _A k − →_FT /A _kL kZLk−→tLk kRkZRk−→t Rk kB kZBk−→tBk kkYZYk−→tYk -9. − →_FSD/A _cL kZLk−→t Lk cRkZRk−→tRk cB kZBk−→t Bk cYkZYk−→tYk -%C. 4 B Y Ak # − →_FD/A AkAk− AkAk − →

FDL/A −→_FDR/A _{& −}→_vf

k # Ak # Ak −→v(k −→vfk − −→vA AkAk− − →_v( /t k −→v(k −→tLk −→v( /nk −→v(k −−→v( /tk -%%. 4 AkAk− − →_FDL/A _ρ fATCDT −→v( /t_k −→v( /t_k ρ_fANCDN −→v( /n_k −→v( /n_k -%/. CDT CDN = AT πd ZLk AN d ZLk − →_FDR/A Ak − →

FD/A −→FDL/A −→FDR/A -%?.

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− →_t k . − → T_k & - −→N . T_kV, > -− → N −→N . T_kH, 8 T_kH, '_sT_kV, '_sW -%A. '_s = W Ak - . Ak − →_FC/A $ $ & , -4 - . Yk Fr∗NH/Y −→ωYr −→T∗Y −→v G r −mY k−→aG − →_T∗Y Yk −→ωYr −→v G r − →_aG mY k F∗NH/Y r −VrkYu S rk φYrk r , . . . , -%;. Yk − →_HA/Y _Y k -5 %9:/D 1 1 > %9;9. HA − A {aY} -%B. A , Yk {aY_} Yk Fr∗A/Y −→HA/Y −→vGr F∗A/Y r −WrkYu S rk ψYrk -%<. r , . . . , k , . . . , n Yk F∗Y r Fr∗NH/Y Fr∗A/Y -%:. r , . . . , k , . . . , n Yk FrGB/Y −→vGr ρfVkY − mY k g − →_N QY k Yk fY k , −→y k hY_k CY_,kqY_k CY_,kqY_k CY_,k qY_k f_kY -%9. CY ij,k i − j , / Yk RY k − →_N RY_k kE hYk − hYk -/C. kE 7 -. 4 RY k > hYk > FrT D/Y RY_k−→N −→vQr , PY k Ak * Yk F_rT SD/Y −kYkZYk− cYkZYk −→tYk −→vPr -/%. # Yk − →_U( k Uik( −→yik -//. U( ik 7 # Yk , Yk − →_y _k − →_y _k −→_U( k − →_y _k ₌ CDT CDN AT -π× × . AN -× . Yk − →_FD/Y i αY_ik−→yik -/?. FrD/Y −→FD/Y − →_vG r − →_HY _Y k # − →_HI/Y -5 %9:/D 1 1 > %9;9. HI ρ_fV_kY I A {aF} -/A. A , Yk VY k Yk I × , {aF_} _# Yk # FrI/Y −→HI/Y −→vGr

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$ ! ./ , Sk -8 /. F∗S rk Fr∗A Fr∗B Fr∗Y FS rk FrA FrB FrY -/;. r , . . . , k , . . . , n A, B Y S Sk & F∗S rk −MrkSu S rk bSrk -/B. r , . . . , k , . . . , n - sys. -$ 1 %9:;. F∗sys r Frsys r , . . . , n -/<. ur sys Mrsys Fsys r bsysr r , . . . , n -/:. 4 n × {x} {x} {q} sys {u}sys -/9. 7 x {f t, {x}} f qr sys -/:. " 1 5 ( $ A; ? 0 E()510" 4 8 ? 0 OA, AB, BC 7 k L WY WA FB ! ./ ( + 0 1 2 F FB− WA WY . φ c d c O A B C Buoy Sock Mussel 8 ? E -n . φ OA BC > F kL d − L L φ − φ -?C. T OA, BC T F φ ! $ , -a b θ weight submerged = W speed current = v x R z R

Mussel sock modelled as cylinder of length b and diameter a

O

x

z

8 A + + b a W O v 8 A θ = - # . CDN AN ab = -# . CDT. AT πab 4

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-πa / 7 # # ρ_f α α W ρ_fabCDNv -?%. θ θ − −α √ α -?/. > Rx, Rz O Rx W −α β θ θ -??. Rz W α β θ θ − θ θ -?A. β πCDT CDN -?;. 4 α -?%. θ -?/. Rx Rz -??. -?A. ! ! + 0 * ( +2 -. m mm . kg/m, MP a d m 0 Bk Yk m, . m. Bk kg/m Yk kg/m = CDN . , CDT . . 7 k . N/m F . N 8 c -8 ?. . m . m φ . rad T 7 , T . m -?C. φ . rad T F φ . N 8 m/s x − →_y . , , − . -?/. θ . θ, , − θ . , , − . ! 3 * + 5 6 5 , & 5 , 8 % m . m - . . kg/m GP a n Ak, k , . . . , n . 5 Bk Yk A A + - kg kg/m . A , A , A , . . . , A kg f . 0 Yk m . m - B B . " . kg . m = . . kg , 5 , , , . . . , . kg . m = . . kg - . kg kg. - . B B ) B B . kg . m = . . kg . m kg/m kg. = '_s . . m/s y 8 ; -A;C

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0 50 100 150 200 250 0 5 10 15 20 25 30 35 0 2 4 6 8 10 12 14 16 18 20 x axis (m) 3D profile at equilibrium : Main Line and Bottom of Mussel Masses

y axis (m)

z axis (m)

Current : 0. 21 m/s in y direction

Main Line

Bottom of Mussel Masses

8 ; 0 . -x, y z . T   . . .   × N -?B. T  − .. .   × N -?<. m y -Y Y . . m " Y . 1 1 -?/. m , . , . m . x 8 B 8 < A ") 01 )8 8& F '0 Lα α , . . . , ν Jk _k _{, . . . , n} - . Akα 8 -:. -. A ,α_{, A}n ,α _α _{, . . . , ν} Ak, _{, A}k,ν _k _{, . . . , n .} ₊ t. 4 + 0 50 100 150 200 250 0 2 4 6 8 10 0 2 4 6 8 10 12 14 16 18 20 x axis (m) 3D profile at 100 sec : Main Line and Bottom of Mussel Masses

y axis (m)

z axis (m)

Wave propagating at 67.5 deg. to x axis

Wave height = 3 m, wave period = 8.5 sec, wavelength = 99.6 m

Main Line

Bottom of Mussel Masses

8 B & %CC 10 20 30 40 50 60 70 80 90 100 0 500 1000 1500 2000 2500 3000

Magnitude of Tension at Left Anchor

Time (s) Tension (N) 10 20 30 40 50 60 70 80 90 100 0 500 1000 1500 2000 2500 3000

Magnitude of Tension at Right Anchor

Time (s)

Tension (N)

Tension due to wave propagating at 67.5 deg. to x axis

Tension due to wave propagating at 67.5 deg. to x axis Wave height = 3 m, wave period = 8.5 sec, wavelength = 99.6 m

Wave height = 3 m, wave period = 8.5 sec, wavelength = 99.6 m

8 < 4 Akα ckα i k , . . . n α , . . . ν i , , qr r , . . . , nν ckα_i t q _{n α−} _k− i k , . . . n α , . . . ν i , , -?:. ur q_r r , . . . nν . - . > 4 nν {x} /9. x {f t, {x}}

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α L Line al Longitudin k J Line Transverse _Segment_Skα α k A Node α k R Segment ) ,..., 1 (α = ν ) ,..., 1 (k= n α , 0 A α , 1 + n A 0 , k A 1 ,+ ν k A 8 : ' " 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 x−axis (m) y−axis (m) z−axis (m) 20 x 20 net profile Current : 2 m/s in x direction

Wave height = 2 m, wavelength = 3 m, water depth = 7 m, travelling in x direction

8 9 ' 8 -9. , × -. m/s x m m x 7 ; +)'+1* &)' ' $ ! = 5 , & 6 + 4 , > , , & ) @ G G !& 6 G - ) &". -6 + . EH -" E 6. G G & 0, -" 0&0.

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(

5 + 0 %9:/ ( " 8 & 8 ( +( :/ C%C ' + 0 1 E F + + $ %9:< F )7 - I . $ ( 1 %9:; - " 2 F & . $ ( 1 E 4 1 %9:? -" 2 F & . ?C 1 1 1 > 0 " %9;9 8 " -E E . ?; " 1 5 /CC< " 4 & ' " * ( ' 4 + 5 /CC? I /< " 1 0 %9% /%/