Symbolic regression-based genetic approximations of the Colebrook equation for flow friction

wa e Communica ion Symbolic Reg ession-Based Gene ic App oxima ions o he Coleb ook Equa ion o Flow F ic ion Pa el P aks 1,2,*ID and Dejan B ki´c 1,*ID 1Eu opean Commission, Join Resea ch Cen e (JRC), Di ec o a e C: Ene gy, T anspo and Clima e, Uni C3: Ene gy Secu i y, Dis ibu ion and Ma ke s, Via En ico Fe mi 2749, 21027 Isp a (VA), I aly 2IT4Inno a ions Na ional Supe compu ing Cen e , VŠB—Technical Uni e si y o Os a a, 17, lis opadu 2172/15, 708 00 Os a a, Czech Republic * Co espondence: [email p o ec ed] o [email p o ec ed] (P.P.); [email p o ec ed] (D.B.) Recei ed: 6 Augus 2018; Accep ed: 30 Augus 2018; Published: 2 Sep embe 2018   Abs ac : Widely used in hyd aulics, he Coleb ook equa ion o low ic ion ela es implici ly o he inpu pa ame e s; he Reynolds numbe , Re and he ela i e oughness o an inne pipe su ace, ε /Dwi h an unknown ou pu pa ame e ; he low ic ion ac o , λ ; λ = ( λ ,Re, ε /D). In his pape , a ew explici app oxima ions o he Coleb ook equa ion; λ≈ (Re, ε /D), a e gene a ed using he abili y o a i icial in elligence o make inne pa e ns o connec inpu and ou pu pa ame e s in an explici way no knowing hei na u e o he physical law ha connec s hem, bu only knowing aw numbe s, {Re, ε /D} → { λ }. The ac ha he used gene ic p og amming ool does no know he s uc u e o he Coleb ook equa ion, which is based on compu a ionally expensi e loga i hmic law, is used o ob ain a be e s uc u e o he app oxima ions, which is less demanding o calcula ion bu also enough accu a e. All gene a ed app oxima ions ha e low compu a ional cos because hey con ain a limi ed numbe o loga i hmic o ms used o no maliza ion o inpu pa ame e s o o accele a ion, bu hey a e also su icien ly accu a e. The ela i e e o ega ding he ic ion ac o λ , in in he bes case is up o 0.13% wi h only wo loga i hmic o ms used. As he second loga i hm can be accu a ely app oxima ed by he Padéapp oxima ion, p ac ically he same e o is ob ained also using only one loga i hm. Keywo ds: Coleb ook equa ion; low ic ion; u bulen low; gene ic p og amming; symbolic eg ession; explici app oxima ions 1. In oduc ion The Coleb ook equa ion o low ic ion is one o he mos used o mulas in hyd aulics, which is a b anch o ci il enginee ing, ha deals wi h he con eyance o liquids h ough pipes. I is also widely used in mechanical, pe oleum and chemical enginee ing, e c., whe e e low h ough pipes occu . I is an empi ical ela ion de eloped by Coleb ook [ 1 ] based on his expe imen wi h Whi e [ 2 ]. The expe imen deal wi h low o ai /liquid h ough a i icially oughened pipes; Equa ion (1): 1 √λ=−2·log102.51 Re ·1 √λ+ε 3.71·D(1) In Equa ion (1), λ is he Da cy low ic ion ac o , Re is he Reynolds numbe , and ε /Dis he ela i e oughness o inne pipe su ace (all h ee quan i ies a e dimensionless). In he Coleb ook equa ion, he low ic ion ac o λ is implici ly gi en, λ = ( λ ,Re, ε /D) whe e i can be exp essed in an explici way only app oxima ely [ 3 – 8 ], λ≈ (Re, ε /D) o o he wise he o iginal equa ion can be sol ed i e a i ely [ 9 , 10 ]. Today, i is impo an no only o ha e accu a e Wa e 2018,10, 1175; doi:10.3390/w10091175 www.mdpi.com/jou nal/wa e Wa e 2018,10, 1175 2 o 14 bu also compu a ionally e icien app oxima ions [ 11 – 13 ]. He e, we used he abili y o a i icial in elligence o connec inpu da a; in ou case he Reynolds numbe , Re and he ela i e oughness o inne pipe su ace, ε /Dwi h he ou pu pa ame e ; in ou case he low ic ion ac o , λ no knowing he s uc u e o he Coleb ook equa ion [ 14 – 17 ]. We used he abili y o a i icial in elligence o connec inpu wi h ou pu da a o o m pa e ns no knowing he na u e o he da a o he physical law ha connec s hem (a simila app oach is alid o o he b anches o hyd aulics [ 18 , 19 ]). In ha way, we ied o a oid he compu a ionally expensi e loga i hmic law on which he Coleb ook equa ion is based. As a inal p oduc we de eloped ew low-cos bu e y accu a e explici app oxima ions o he Coleb ook equa ion. 2. Me hods Used, P epa a ion o Da a and So wa e Tool, Resul s, S uc u e o App oxima ions, Accu acy and Compa a i e Analysis The main idea is o use he abili y o a i icial in elligence o connec inpu da a se s; in ou case he Reynolds numbe , Re and he ela i e oughness o inne pipe su ace, ε /Dwi h he ou pu da a se ; in ou case he low ic ion ac o , λ ; {Re, ε /D} → { λ }, no knowing he physical law which connec s inpu o ou pu . Sign “ → ” p ac ically ep esen s he Coleb ook equa ion; Equa ion (1), bu he gene ic p og amming ool is no awa e o ha ac . To p epa e da a o eed he gene ic p og amming ool, we co e ed he whole p ac ical domain o applicabili y o he Coleb ook equa ion; which is o he Reynolds numbe , Re be ween 4000 and 10 8 (whole u bulen low co e ed) and o he ela i e oughness o inne pipe su ace, ε /Dup o 0.05 (pipe co e ed om p ac ically smoo h o he e y ough) [ 20 ] wi h a mesh which consis s o 90 housand in e sec ion poin s {Re, ε /D} o which we calcula ed e y accu a ely he low ic ion ac o , λ using he Coleb ook equa ion; Equa ion (1); Figu e 1a,b. Wa e 2018, 10, x FOR PEER REVIEW 2 o 14 inne pipe su ace, ε/D wi h he ou pu pa ame e ; in ou case he low ic ion ac o , λ no knowing he s uc u e o he Coleb ook equa ion [14–17]. We used he abili y o a i icial in elligence o connec inpu wi h ou pu da a o o m pa e ns no knowing he na u e o he da a o he physical law ha connec s hem (a simila app oach is alid o o he b anches o hyd aulics [18,19]). In ha way, we ied o a oid he compu a ionally expensi e loga i hmic law on which he Coleb ook equa ion is based. As a inal p oduc we de eloped ew low-cos bu e y accu a e explici app oxima ions o he Coleb ook equa ion. 2. Me hods Used, P epa a ion o Da a and So wa e Tool, Resul s, S uc u e o App oxima ions, Accu acy and Compa a i e Analysis The main idea is o use he abili y o a i icial in elligence o connec inpu da a se s; in ou case he Reynolds numbe , Re and he ela i e oughness o inne pipe su ace, ε/D wi h he ou pu da a se ; in ou case he low ic ion ac o , λ; {Re, ε/D}→{λ}, no knowing he physical law which connec s inpu o ou pu . Sign “→” p ac ically ep esen s he Coleb ook equa ion; Equa ion (1), bu he gene ic p og amming ool is no awa e o ha ac . To p epa e da a o eed he gene ic p og amming ool, we co e ed he whole p ac ical domain o applicabili y o he Coleb ook equa ion; which is o he Reynolds numbe , Re be ween 4000 and 108 (whole u bulen low co e ed) and o he ela i e oughness o inne pipe su ace, ε/D up o 0.05 (pipe co e ed om p ac ically smoo h o he e y ough) [20] wi h a mesh which consis s o 90 housand in e sec ion poin s {Re, ε/D} o which we calcula ed e y accu a ely he low ic ion ac o , λ using he Coleb ook equa ion; Equa ion (1); Figu e 1a,b. (a) Figu e 1. Con . Wa e 2018,10, 1175 3 o 14 Wa e 2018, 10, x FOR PEER REVIEW 3 o 14 (b) Figu e 1. (a) An example o Eu eqa [compu e so wa e] in e ace: Bes app oxima e solu ions o he Coleb ook equa ion wi h a ious complexi y, which we e au oma ically ound by Eu eqa. (b) An example o Eu eqa [compu e so wa e] in e ace: The esidual e o plo o a selec ed analy ical model o 200 pai s oge he wi h an accu acy s complexi y plo o solu ions. Ha ing 90 housand combina ions; in o de o es obus ness o he symbolic eg ession algo i hms we ed he gene ic p og amming ool wi h 200 iple s {Re, ε/D}i, {λ}i (see Supplemen a y Ma e ial a ached o his pape ) hoping ha i will connec {Re, ε/D}i→{λ}i accu a ely. The inpu sample was gene a ed acco ding o he uni o m densi y unc ion o each inpu a iable. The low- disc epancy Sobol sequences we e employed [21]. These so-called quasi andom sequences ha e use ul p ope ies. In con a y o andom numbe s, quasi andom numbe s co e he space mo e quickly and e enly. Thus, hey lea e e y ew holes. We used [compu e so wa e] Eu eqa by Nu onian, Inc., Bos on, MA, as a gene ic p og amming ool [22,23]. The symbolic eg ession app oach adop ed he ein [24–29] is based upon gene ic p og amming whe ein a popula ion o unc ions is allowed o b eed and mu a e wi h he gene ic p opaga ion in o subsequen gene a ions based on su i al-o - he- i es c i e ia [30]. The main goal o his s udy is o make accu a e and compu a ionally cheap explici app oxima ions o he Coleb ook equa ion, whe e compu a ionally cheap means o con ain he leas possible numbe o loga i hmic unc ions and non-in ege powe s [31–36]. We can see ha app oxima ions ound by Eu eqa [compu e so wa e] ha e he o m {R, K}→xsol, whe e he symbol R deno es he Reynolds numbe , K ep esen s ela i e oughness and 𝑥𝑠𝑜𝑙= 1 √𝜆. All accu a e models a e compu a ionally expensi e, as hey con ain many loga i hmic e ms wi h di e en a gumen s. Thus, we can see ha Eu eqa i sel equi es human knowledge, in o de o ob ain an accu a e bu s ill a compu a ionally cheap app oxima ion o he Coleb ook Figu e 1. ( a ) An example o Eu eqa [compu e so wa e] in e ace: Bes app oxima e solu ions o he Coleb ook equa ion wi h a ious complexi y, which we e au oma ically ound by Eu eqa. ( b ) An example o Eu eqa [compu e so wa e] in e ace: The esidual e o plo o a selec ed analy ical model o 200 pai s oge he wi h an accu acy s complexi y plo o solu ions. Ha ing 90 housand combina ions; in o de o es obus ness o he symbolic eg ession algo i hms we ed he gene ic p og amming ool wi h 200 iple s {Re, ε /D} i , { λ } i (see Supplemen a y Ma e ial a ached o his pape ) hoping ha i will connec {Re, ε /D} i→ { λ } i accu a ely. The inpu sample was gene a ed acco ding o he uni o m densi y unc ion o each inpu a iable. The low-disc epancy Sobol sequences we e employed [ 21 ]. These so-called quasi andom sequences ha e use ul p ope ies. In con a y o andom numbe s, quasi andom numbe s co e he space mo e quickly and e enly. Thus, hey lea e e y ew holes. We used [compu e so wa e] Eu eqa by Nu onian, Inc., Bos on, MA, as a gene ic p og amming ool [ 22 , 23 ]. The symbolic eg ession app oach adop ed he ein [ 24 – 29 ] is based upon gene ic p og amming whe ein a popula ion o unc ions is allowed o b eed and mu a e wi h he gene ic p opaga ion in o subsequen gene a ions based on su i al-o - he- i es c i e ia [ 30 ]. The main goal o his s udy is o make accu a e and compu a ionally cheap explici app oxima ions o he Coleb ook equa ion, whe e compu a ionally cheap means o con ain he leas possible numbe o loga i hmic unc ions and non-in ege powe s [31–36]. We can see ha app oxima ions ound by Eu eqa [compu e so wa e] ha e he o m {R,K} → xsol, whe e he symbol Rdeno es he Reynolds numbe , K ep esen s ela i e oughness and xsol =1 √λ . All accu a e models a e compu a ionally expensi e, as hey con ain many loga i hmic e ms wi h di e en a gumen s. Thus, we can see ha Eu eqa i sel equi es human knowledge, in o de o ob ain an accu a e bu s ill a compu a ionally cheap app oxima ion o he Coleb ook equa ion. Consequen ly, Wa e 2018,10, 1175 4 o 14 we will combine se e al app oaches in his pape : Eu eqa [compu e so wa e] [ 22 , 23 ], he ixed-poin i e a ion [ 9 ] and Padéapp oxima ion [ 10 ]. Acco ding o ou nume ical expe imen s, Eu eqa seems o be use ul especially o inding a compu a ionally cheap a ional app oxima ion o he Coleb ook solu ion, which se es as a good s a ing poin o he ixed-poin i e a ion me hod (accele a ion). Finally, he Padéapp oxima ion is used as a cheap bu e y accu a e app oxima ion o he loga i hm in he second and he success ul i e a ions o he ixed-poin me hod. 2.1. Inpu Pa ame e s in Thei Raw Fo m Using he inpu pa ame e s in hei aw o m {Re, ε /D} i→ { λ } i , Eu eqa, he used gene ic p og amming ool gi es a se o app oxima ions in polynomial o ms [ 12 ]. Knowing ha loga i hmic exp essions and non-in ege powe s a e expensi e o compu a ion, we hoped ha we ha e ully accomplished ou ask. Un o una ely, Eu eqa gi es a numbe o no e y accu a e solu ions and he e we show Equa ion (2) wi h he ela i e e o o λ0 e en up o 16.56% in espec o he accu a e λ , whe e he ela i e e o [ 5 , 26 ] is de ined as (| λaccu a e −λ |/ λaccu a e ) · 100%, whe e λaccu a e is calcula ed in an i e a i e p ocedu e using he o iginal implici ly gi en Coleb ook equa ion [ 6 , 9 ]; Equa ion (1), while λ is ob ained h ough he p esen ed app oxima ions; Equa ions (2)–(6). In Equa ion (2), “ ↔ ” means ela ed bu no su icien ly accu a e: 1 √λ0↔4.34·Re Re+129,000·Re·ε D+7,850,000 +781·Re 187·Re+133,000·Re·ε D+8,960,000 −20.5·ε D+4.85 (2) On he o he hand, we ound ha he accu acy can inc ease signi ican ly using one ixed-poin i e a i e cycle o accele a ion [9]; Equa ion (2a), a e which accu acy o λ1inc eases up o 0.98%. 1 √λ1≈ −2·log10(y1) . . . 1 √λi+1≈ −2·log10(yi+1)          (2a) In Equa ion (2a), “ ≈ ” means easonably accu a e enough and a gumen s o loga i hms a e de ined by; Equa ion (2b): y1≈2.51 Re · 4.34·Re Re+129,000·Re·ε D+7,850,000 +781·Re 187·Re+133,000·Re·ε D+8,960,000 −20.5·ε D+4.85 ! | {z } 1 √λ0 +ε 3.71·D . . . yi+1≈2.51 Re ·1 √λi+ε 3.71·D                      (2b) The simple ixed-poin i e a i e p ocedu e [ 6 , 9 ]; Equa ion (2a) in case o he Coleb ook equa ion is as ; λ0→ 16.56%, λ1→ 0.98%, λ2→ 0.13%, e c. (Figu e 2). Thus, using only wo loga i hmic o ms, high accu acy o λ2→ 0.13% is eached. Resul s a e in he o m { λ } 0↔ {Re, ε /D} 0 , { λ } 1≈ {log 10 ( λ0 )} 1 , { λ } 2≈ {log 10 (log 10 ( λ0 ))} 2 , e c., whe e “ ↔ ” means ela ed bu no su icien ly accu a e, while “ ≈ ” is easonably accu a e enough. This app oach wi h accele a ion is widely used in de elopmen o app oxima ions o he Coleb ook equa ion [ 37 – 42 ]. The e o can be u he educed by using one mo e accele a ing s ep as shown, o using gene ic algo i hms [ 25 , 29 , 36 ], Excel i ing ool [ 27 ] o he Mon e Ca lo me hod [43,44]. Wa e 2018,10, 1175 5 o 14 Wa e 2018, 10, x FOR PEER REVIEW 5 o 14 Wa e 2018, 10, x FOR PEER REVIEW 6 o 14 Figu e 2. Dis ibu ion o he ela i e e o o λ o e he domain o applicabili y o he Coleb ook equa ion in oduced by polynomial Equa ion (2)—(up), Equa ion (2a) a e i s s ep o ixed-poin accele a ion—(middle), and he second s ep o accele a ion—(down); ela i e e o up o 16.56%, up o 0.98% and up o 0.13% espec i ely. 2.2. No malized Inpu Pa ame e s Un o una ely, in ou case using he inpu pa ame e s in hei aw o m he accu acy was no a a high le el wi hou accele a ion, so ha ing p e ious expe ience wi h he same p oblem whe e we used A i icial Neu al Ne wo k [15,16] o simula e esul s, we no malized pa ame e s a = log10(Re), b = −log10(ε/D), in o de o a oid disc epancy in he scale which a e in aw o m 1000 < Re < 108 and ε/D << 1 and a e no maliza ion 3.5 < a < 8 and 1.3 < b < 6.5 (Eu eqa, so wa e used a as gene ic p og amming ool also sugges ed o us a da a no maliza ion p ocess) [34–36]. The no maliza ion gi es ela i ely good esul s, and he gene ic p og amming ool gene a ed mo e accu a e esul s wi hou knowing ha he loga i hmic o m o he Coleb ook equa ion was o iginally used bu only knowing he p edic ed inpu and ou pu da ase s; Figu e 3: Figu e 3. Gene ic p og amming ool makes λ ≈ (a,b) wi hou knowing he physical law ha connec s a and b. Using ou p e ious expe ience [15] wi h he aining o he A i icial Neu al Ne wo k whe e e y good esul s we e achie ed h ough he no maliza ion o pa ame e s; a = log10(Re), b = −log10(ε/D), he gene ic p og amming ool gene a ed a dozen equa ions wi h di e en le els o accu acy and complexi y, bu o una ely none o hem con ain loga i hms o non-in ege powe e ms. He e, we p esen he ou mos success ul explici app oxima ions; Equa ions (3)–(6). Adding one addi ional loga i hmic o m o accele a ion using one addi ional ixed-poin i e a i e s ep [9,45]; Equa ion (2a), he accu acy o he app oxima ions inc eases signi ican ly (abou 10 imes); Equa ions (3a)–(6a). Figu e 2. Dis ibu ion o he ela i e e o o λ o e he domain o applicabili y o he Coleb ook equa ion in oduced by polynomial Equa ion (2)—( up ), Equa ion (2a) a e i s s ep o ixed-poin accele a ion—( middle ), and he second s ep o accele a ion—( down ); ela i e e o up o 16.56%, up o 0.98% and up o 0.13% espec i ely. Wa e 2018,10, 1175 6 o 14 2.2. No malized Inpu Pa ame e s Un o una ely, in ou case using he inpu pa ame e s in hei aw o m he accu acy was no a a high le el wi hou accele a ion, so ha ing p e ious expe ience wi h he same p oblem whe e we used A i icial Neu al Ne wo k [ 15 , 16 ] o simula e esul s, we no malized pa ame e s a=log 10 (Re), b=−log10(ε/D) , in o de o a oid disc epancy in he scale which a e in aw o m 1000 < Re < 10 8 and ε /D<< 1 and a e no maliza ion 3.5 < a< 8 and 1.3 < b< 6.5 (Eu eqa, so wa e used aas gene ic p og amming ool also sugges ed o us a da a no maliza ion p ocess) [ 34 – 36 ]. The no maliza ion gi es ela i ely good esul s, and he gene ic p og amming ool gene a ed mo e accu a e esul s wi hou knowing ha he loga i hmic o m o he Coleb ook equa ion was o iginally used bu only knowing he p edic ed inpu and ou pu da ase s; Figu e 3: Wa e 2018, 10, x FOR PEER REVIEW 6 o 14 Figu e 2. Dis ibu ion o he ela i e e o o λ o e he domain o applicabili y o he Coleb ook equa ion in oduced by polynomial Equa ion (2)—(up), Equa ion (2a) a e i s s ep o ixed-poin accele a ion—(middle), and he second s ep o accele a ion—(down); ela i e e o up o 16.56%, up o 0.98% and up o 0.13% espec i ely. 2.2. No malized Inpu Pa ame e s Un o una ely, in ou case using he inpu pa ame e s in hei aw o m he accu acy was no a a high le el wi hou accele a ion, so ha ing p e ious expe ience wi h he same p oblem whe e we used A i icial Neu al Ne wo k [15,16] o simula e esul s, we no malized pa ame e s a = log10(Re), b = −log10(ε/D), in o de o a oid disc epancy in he scale which a e in aw o m 1000 < Re < 108 and ε/D << 1 and a e no maliza ion 3.5 < a < 8 and 1.3 < b < 6.5 (Eu eqa, so wa e used a as gene ic p og amming ool also sugges ed o us a da a no maliza ion p ocess) [34–36]. The no maliza ion gi es ela i ely good esul s, and he gene ic p og amming ool gene a ed mo e accu a e esul s wi hou knowing ha he loga i hmic o m o he Coleb ook equa ion was o iginally used bu only knowing he p edic ed inpu and ou pu da ase s; Figu e 3: Figu e 3. Gene ic p og amming ool makes λ ≈ (a,b) wi hou knowing he physical law ha connec s a and b. Using ou p e ious expe ience [15] wi h he aining o he A i icial Neu al Ne wo k whe e e y good esul s we e achie ed h ough he no maliza ion o pa ame e s; a = log10(Re), b = −log10(ε/D), he gene ic p og amming ool gene a ed a dozen equa ions wi h di e en le els o accu acy and complexi y, bu o una ely none o hem con ain loga i hms o non-in ege powe e ms. He e, we p esen he ou mos success ul explici app oxima ions; Equa ions (3)–(6). Adding one addi ional loga i hmic o m o accele a ion using one addi ional ixed-poin i e a i e s ep [9,45]; Equa ion (2a), he accu acy o he app oxima ions inc eases signi ican ly (abou 10 imes); Equa ions (3a)–(6a). Figu e 3. Gene ic p og amming ool makes λ≈ (a,b) wi hou knowing he physical law ha connec s a and b. Using ou p e ious expe ience [ 15 ] wi h he aining o he A i icial Neu al Ne wo k whe e e y good esul s we e achie ed h ough he no maliza ion o pa ame e s; a=log 10 (Re), b= − log 10 ( ε /D), he gene ic p og amming ool gene a ed a dozen equa ions wi h di e en le els o accu acy and complexi y, bu o una ely none o hem con ain loga i hms o non-in ege powe e ms. He e, we p esen he ou mos success ul explici app oxima ions; Equa ions (3)–(6). Adding one addi ional loga i hmic o m o accele a ion using one addi ional ixed-poin i e a i e s ep [ 9 , 45 ]; Equa ion (2a), he accu acy o he app oxima ions inc eases signi ican ly (abou 10 imes); Equa ions (3a)–(6a). Resul s a e in he o m { λ } 0↔ {a=log 10 (Re), b= − log 10 ( ε /D)} 0 , { λ } 1≈ {log 10 ( λ0 )} 1 , { λ } 2≈ {log 10 (log 10 ( λ0 ))} 2 , e c., whe e “ ↔ ” means ela ed bu no su icien ly accu a e, while “ ≈ ” is easonably accu a e enough. 1 √λ0↔3.13·b−1.56·b2 a(3) 1 √λ1≈ −2·log102.51 Re ·3.13·b−1.56·b2 a+ε 3.71·D | {z } accele a ed Eq.(3) (3a) 1 √λ0↔b+0.904·a+1.08·sin(0.937·a−b)−1.85 (4) 1 √λ1≈ −2·log102.51 Re ·(b+0.904·a+1.08·sin(0.937·a−b)−1.85)+ε 3.71·D | {z } accele a ed Eq.(4) (4a) α=1 √λ0≈a+0.61·b+0.28·a·b+0.51·sin(0.935·a−b)−0.894 −0.103·a2−0.158·b2(5) Wa e 2018,10, 1175 7 o 14 1 √λ1≈ −2·log102.51·α Re +ε 3.71·D | {z } accele a ed Eq.(5) (5a) β=1 √λ0≈1.15·a+0.569·b+0.292·a·b+0.478·sin(0.939·a−b) +0.122·sin2(0.939·a−b)−1.284 −0.12·a2−0.162·b2(6) 1 √λ1≈ −2·log102.51·β Re +ε 3.71·D | {z } accele a ed Eq.(6) (6a) Dis ibu ion o he ela i e e o o e he domain o applicabili y o he Coleb ook equa ion in oduced by hese ou app oxima ions is p esen ed in Figu es 4–7. Wa e 2018, 10, x FOR PEER REVIEW 7 o 14 Resul s a e in he o m {λ}0↔{a = log10(Re), b = −log10(ε/D)}0, {λ}1≈{log10(λ0)}1, {λ}2≈{log10(log10(λ0))}2, e c., whe e “↔” means ela ed bu no su icien ly accu a e, while “≈” is easonably accu a e enough. 1 √𝜆0↔3.13∙𝑏−1.56∙𝑏2 𝑎 (3) 1 √𝜆1≈−2∙𝑙𝑜𝑔10(2.51 𝑅𝑒 ∙(3.13∙𝑏−1.56∙𝑏2 𝑎)+ 𝜀 3.71∙𝐷) ⏟ 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑒𝑑 𝐸𝑞.(3) (3a) 1 √𝜆0↔𝑏+0.904∙𝑎+1.08∙sin(0.937∙𝑎−𝑏)−1.85 (4) 1 √𝜆1≈−2∙𝑙𝑜𝑔10(2.51 𝑅𝑒 ∙(𝑏+0.904∙𝑎+1.08∙sin(0.937∙𝑎−𝑏)−1.85)+𝜀 3.71∙𝐷) ⏟ 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑒𝑑 𝐸𝑞.(4) (4a) 𝛼= 1 √𝜆0≈𝑎+0.61∙𝑏+0.28∙𝑎∙𝑏+0.51∙𝑠𝑖𝑛(0.935∙𝑎−𝑏)−0.894−0.103·𝑎2−0.158·𝑏2 (5) 1 √𝜆1≈−2∙𝑙𝑜𝑔10(2.51∙𝛼 𝑅𝑒 +𝜀 3.71∙𝐷) ⏟ 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑒𝑑 𝐸𝑞.(5) (5a) 𝛽= 1 √𝜆0≈1.15∙𝑎+0.569∙𝑏+0.292∙𝑎∙𝑏+0.478∙sin(0.939∙𝑎−𝑏)+0.122∙ sin2(0.939∙𝑎−𝑏)−1.284−0.12∙𝑎2−0.162∙𝑏2 (6) 1 √𝜆1≈−2∙𝑙𝑜𝑔10(2.51∙𝛽 𝑅𝑒 +𝜀 3.71∙𝐷) ⏟ 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑒𝑑 𝐸𝑞.(6) (6a) Dis ibu ion o he ela i e e o o e he domain o applicabili y o he Coleb ook equa ion in oduced by hese ou app oxima ions is p esen ed in Figu es 4–7. (a) (b) Figu e 4. Dis ibu ion o he ela i e e o o λ o e he domain o applicabili y o he Coleb ook equa ion in oduced by Equa ion (3)—(a), and accele a ed Equa ion (3a)—(b); ela i e e o up o 20% and up o 5.35%, espec i ely. (a) (b) Figu e 5. Dis ibu ion o he ela i e e o o λ o e he domain o applicabili y o he Coleb ook equa ion in oduced by Equa ion (4)—(a), and accele a ed Equa ion (4a)—(b); ela i e e o up o 60% and up o 6.29%, espec i ely. Figu e 4. Dis ibu ion o he ela i e e o o λ o e he domain o applicabili y o he Coleb ook equa ion in oduced by Equa ion (3)—( a ), and accele a ed Equa ion (3a)—( b ); ela i e e o up o 20% and up o 5.35%, espec i ely. Wa e 2018, 10, x FOR PEER REVIEW 7 o 14 Resul s a e in he o m {λ}0↔{a = log10(Re), b = −log10(ε/D)}0, {λ}1≈{log10(λ0)}1, {λ}2≈{log10(log10(λ0))}2, e c., whe e “↔” means ela ed bu no su icien ly accu a e, while “≈” is easonably accu a e enough. 1 √𝜆0↔3.13∙𝑏−1.56∙𝑏2 𝑎 (3) 1 √𝜆1≈−2∙𝑙𝑜𝑔10(2.51 𝑅𝑒 ∙(3.13∙𝑏−1.56∙𝑏2 𝑎)+ 𝜀 3.71∙𝐷) ⏟ 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑒𝑑 𝐸𝑞.(3) (3a) 1 √𝜆0↔𝑏+0.904∙𝑎+1.08∙sin(0.937∙𝑎−𝑏)−1.85 (4) 1 √𝜆1≈−2∙𝑙𝑜𝑔10(2.51 𝑅𝑒 ∙(𝑏+0.904∙𝑎+1.08∙sin(0.937∙𝑎−𝑏)−1.85)+𝜀 3.71∙𝐷) ⏟ 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑒𝑑 𝐸𝑞.(4) (4a) 𝛼= 1 √𝜆0≈𝑎+0.61∙𝑏+0.28∙𝑎∙𝑏+0.51∙𝑠𝑖𝑛(0.935∙𝑎−𝑏)−0.894−0.103·𝑎2−0.158·𝑏2 (5) 1 √𝜆1≈−2∙𝑙𝑜𝑔10(2.51∙𝛼 𝑅𝑒 +𝜀 3.71∙𝐷) ⏟ 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑒𝑑 𝐸𝑞.(5) (5a) 𝛽= 1 √𝜆0≈1.15∙𝑎+0.569∙𝑏+0.292∙𝑎∙𝑏+0.478∙sin(0.939∙𝑎−𝑏)+0.122∙ sin2(0.939∙𝑎−𝑏)−1.284−0.12∙𝑎2−0.162∙𝑏2 (6) 1 √𝜆1≈−2∙𝑙𝑜𝑔10(2.51∙𝛽 𝑅𝑒 +𝜀 3.71∙𝐷) ⏟ 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑒𝑑 𝐸𝑞.(6) (6a) Dis ibu ion o he ela i e e o o e he domain o applicabili y o he Coleb ook equa ion in oduced by hese ou app oxima ions is p esen ed in Figu es 4–7. (a) (b) Figu e 4. Dis ibu ion o he ela i e e o o λ o e he domain o applicabili y o he Coleb ook equa ion in oduced by Equa ion (3)—(a), and accele a ed Equa ion (3a)—(b); ela i e e o up o 20% and up o 5.35%, espec i ely. (a) (b) Figu e 5. Dis ibu ion o he ela i e e o o λ o e he domain o applicabili y o he Coleb ook equa ion in oduced by Equa ion (4)—(a), and accele a ed Equa ion (4a)—(b); ela i e e o up o 60% and up o 6.29%, espec i ely. Figu e 5. Dis ibu ion o he ela i e e o o λ o e he domain o applicabili y o he Coleb ook equa ion in oduced by Equa ion (4)—( a ), and accele a ed Equa ion (4a)—( b ); ela i e e o up o 60% and up o 6.29%, espec i ely. Wa e 2018,10, 1175 8 o 14 Wa e 2018, 10, x FOR PEER REVIEW 8 o 14 (a) (b) Figu e 6. Dis ibu ion o he ela i e e o o λ o e he domain o applicabili y o he Coleb ook equa ion in oduced by Equa ion (5)—(a), and accele a ed Equa ion (5a)—(b); ela i e e o up o 6% and up o 0.28%, espec i ely. (a) (b) Figu e 7. Dis ibu ion o he ela i e e o o λ o e he domain o applicabili y o he Coleb ook equa ion in oduced by Equa ion (6)—(a), and accele a ed Equa ion (6a)—(b); ela i e e o up o 2% and up o 0.17%, espec i ely. 2.3. Discussion—Compa a i e Analysis In gene al, he ela i e e o o app oxima ions o he Coleb ook equa ion is no uni o mly dispe sed o e he domain o applicabili y o he Coleb ook equa ion [7,29], whe e he mo e complex is no always mo e accu a e which can be no iced by compa ing Equa ion (3) and Equa ion (4) and ela ed Figu es 4 and 5, espec i ely. Acco ding o Eu eqa, he so wa e package which gene a ed he app oxima ions, Equa ion (5) is h ee imes mo e complex compa ed wi h Equa ion (3), while Equa ion (4) is 1.5 imes mo e complex compa ed wi h Equa ion (3). Using gene ic algo i hms [25,29] o he Excel i ing ool [27], he e is a possibili y o inc ease he accu acy o he p esen ed app oxima ions by changing hei pa ame e s nea he cu en alue. I he s uc u e o he app oxima ions emains unchanged, he complexi y emains also unchanged while accu acy can inc ease [4,12,13], o in o he wo ds, he s uc u e o he equa ions emains unchanged while he alues o he coe icien change in o de o i be e alues ob ained h ough he Coleb ook equa ion. In ha way e o om Figu es 4–7 can dec ease bu also he dis ibu ion o he e o o e he domain o applicabili y changes [7,29]. To summa ize ou indings, we made Table 1, in which he maximum ela i e e o s o app oxima ions compa ed o he accu a e λ a e exp essed as a unc ion o complexi y. Table 1. The maximal ela i e e o s o app oxima ions as a unc ion o complexi y. Complexi y: Numbe o log Func ions Accu acy 1 2 3 High Equa ion (2a)—λ2→0.13% Equa ion (6a)—λ1→0.17% Equa ion (5a)—λ1→0.28% Figu e 6. Dis ibu ion o he ela i e e o o λ o e he domain o applicabili y o he Coleb ook equa ion in oduced by Equa ion (5)—( a ), and accele a ed Equa ion (5a)—( b ); ela i e e o up o 6% and up o 0.28%, espec i ely. Wa e 2018, 10, x FOR PEER REVIEW 8 o 14 (a) (b) Figu e 6. Dis ibu ion o he ela i e e o o λ o e he domain o applicabili y o he Coleb ook equa ion in oduced by Equa ion (5)—(a), and accele a ed Equa ion (5a)—(b); ela i e e o up o 6% and up o 0.28%, espec i ely. (a) (b) Figu e 7. Dis ibu ion o he ela i e e o o λ o e he domain o applicabili y o he Coleb ook equa ion in oduced by Equa ion (6)—(a), and accele a ed Equa ion (6a)—(b); ela i e e o up o 2% and up o 0.17%, espec i ely. 2.3. Discussion—Compa a i e Analysis In gene al, he ela i e e o o app oxima ions o he Coleb ook equa ion is no uni o mly dispe sed o e he domain o applicabili y o he Coleb ook equa ion [7,29], whe e he mo e complex is no always mo e accu a e which can be no iced by compa ing Equa ion (3) and Equa ion (4) and ela ed Figu es 4 and 5, espec i ely. Acco ding o Eu eqa, he so wa e package which gene a ed he app oxima ions, Equa ion (5) is h ee imes mo e complex compa ed wi h Equa ion (3), while Equa ion (4) is 1.5 imes mo e complex compa ed wi h Equa ion (3). Using gene ic algo i hms [25,29] o he Excel i ing ool [27], he e is a possibili y o inc ease he accu acy o he p esen ed app oxima ions by changing hei pa ame e s nea he cu en alue. I he s uc u e o he app oxima ions emains unchanged, he complexi y emains also unchanged while accu acy can inc ease [4,12,13], o in o he wo ds, he s uc u e o he equa ions emains unchanged while he alues o he coe icien change in o de o i be e alues ob ained h ough he Coleb ook equa ion. In ha way e o om Figu es 4–7 can dec ease bu also he dis ibu ion o he e o o e he domain o applicabili y changes [7,29]. To summa ize ou indings, we made Table 1, in which he maximum ela i e e o s o app oxima ions compa ed o he accu a e λ a e exp essed as a unc ion o complexi y. Table 1. The maximal ela i e e o s o app oxima ions as a unc ion o complexi y. Complexi y: Numbe o log Func ions Accu acy 1 2 3 High Equa ion (2a)—λ2→0.13% Equa ion (6a)—λ1→0.17% Equa ion (5a)—λ1→0.28% Figu e 7. Dis ibu ion o he ela i e e o o λ o e he domain o applicabili y o he Coleb ook equa ion in oduced by Equa ion (6)—( a ), and accele a ed Equa ion (6a)—( b ); ela i e e o up o 2% and up o 0.17%, espec i ely. 2.3. Discussion—Compa a i e Analysis In gene al, he ela i e e o o app oxima ions o he Coleb ook equa ion is no uni o mly dispe sed o e he domain o applicabili y o he Coleb ook equa ion [ 7 , 29 ], whe e he mo e complex is no always mo e accu a e which can be no iced by compa ing Equa ion (3) and Equa ion (4) and ela ed Figu es 4and 5, espec i ely. Acco ding o Eu eqa, he so wa e package which gene a ed he app oxima ions, Equa ion (5) is h ee imes mo e complex compa ed wi h Equa ion (3), while Equa ion (4) is 1.5 imes mo e complex compa ed wi h Equa ion (3). Using gene ic algo i hms [ 25 , 29 ] o he Excel i ing ool [ 27 ], he e is a possibili y o inc ease he accu acy o he p esen ed app oxima ions by changing hei pa ame e s nea he cu en alue. I he s uc u e o he app oxima ions emains unchanged, he complexi y emains also unchanged while accu acy can inc ease [ 4 , 12 , 13 ], o in o he wo ds, he s uc u e o he equa ions emains unchanged while he alues o he coe icien change in o de o i be e alues ob ained h ough he Coleb ook equa ion. In ha way e o om Figu es 4–7can dec ease bu also he dis ibu ion o he e o o e he domain o applicabili y changes [7,29]. To summa ize ou indings, we made Table 1, in which he maximum ela i e e o s o app oxima ions compa ed o he accu a e λa e exp essed as a unc ion o complexi y. Wa e 2018,10, 1175 9 o 14 Table 1. The maximal ela i e e o s o app oxima ions as a unc ion o complexi y. Complexi y: Numbe o log Func ions Accu acy 1 2 3 High Equa ion (2a)—λ2→0.13% Equa ion (6a)—λ1→0.17% Equa ion (5a)—λ1→0.28% Mode a e Equa ion (2a)—λ1→0.98% Equa ion (6)—λ0→2% Low Equa ion (5)—λ0→6% Equa ion (4a)—λ1→6.29% Equa ion (3a)—λ1→5.35% I is clea ha polynomial app oxima ion was accele a ed h ough he ixed-poin i e a i e p ocedu e; Equa ion (2a) shows be e pe o mances compa ed wi h hose wi h no malized inpu pa ame e s; Equa ions (3a)–(6a). Fo example, Equa ion (2a) wi h wo loga i hmic unc ions (used o accele a ion) gi es a ela i e e o o no mo e han λ2→ 0.13% compa ed wi h he app oxima ely same e o o Equa ion (6a) wi h h ee loga i hmic o ms ( wo o no maliza ion and one o accele a ion); Equa ion (6a)— λ1→ 0.17%. Also, he exp ession o λ0 in case o Equa ion (2) is a polynomial while Equa ions (4)–(6) con ain sinus igonome ic unc ion [ 46 ]. A e his ca e ul analysis i can be concluded ha i is be e o use compu a ionally expensi e loga i hmic unc ions o accele a ion h ough Equa ion (2a) and no o no maliza ion. All app oxima ions can be classi ied as highly accu a e, mode a e and wi h a low le el o accu acy: • Highly accu a e: Compa ed wi h he simila app oxima ions o he Coleb ook equa ion, accele a ed Equa ion (5a); wi h he ela i e e o up o 0.28% and accele a ed Equa ions (2a) and (6a) wi h he ela i e e o up o 0.13% and 0.17%, espec i ely, a e accu a e as app oxima ions by Ba [ 47 ] (0.2%), Chen [ 38 ] (0.36–0.18%), Zig ang and Syl es e [ 41 ] (0.14–0.08%, simple : 1–0.775%), Fang e al. [ 48 ] (0.61–0.56%), Se ghides [ 38 ] (0.14–0.0026%, simple 0.35–0.27%), Buzzelli [ 49 ] (0.14–0.08%), Sonad and Gouda [ 50 ] (0.8–imp o ed by Va ankhah and Kouchakzadeh [ 51 ]: 0.15%) and Romeo e al. [ 52 ] (0.14–0.008%); whe e he highe epo ed accu acy is achie ed h ough gene ic op imiza ion [ 25 , 29 ]. These app oxima ions a e among he mos accu a e a ailable o da e [ 3 – 8 ], bu a he same ime in many cases much mo e complex compa ed o he app oxima ions p esen ed in ou pape [ 4 , 11 – 13 ]. Fo example; app oxima ions by Ba [ 47 ] and by Chen [ 38 ] con ain wo loga i hmic exp essions and wo non-in ege powe s; by Romeo e al. [ 52 ], h ee loga i hmic exp essions and wo non-in ege powe s, e c. which means ha hey in oduce a highe compu a ional bu den o achie e he same accu acy. In his case, ou Equa ion (2a), which a e wo s eps o accele a ion, con ains only wo loga i hmic o ms. • Mode a ely accu a e: Ou Equa ion (6) wi h he ela i e e o up o 2% does con ain only wo loga i hmic exp essions used o no maliza ion and no non-in ege powe , and i s accu acy can be compa ed wi h app oxima ions by Swamee and Jain [ 53 ] (2.18–1.75%), Manadili [ 54 ] (2–1.5%), B ki´c [ 42 , 55 – 57 ] (2–1.3%), Haland [ 58 ] (1.4–1.1%), e c., all wi h he same o highe complexi y as Equa ion (6). Equa ion (2a) a e he i s s ep o accele a ion wi h only one loga i hmic unc ion and wi h he ela i e e o o up o 2.6% is e en mo e e icien . • Low accu acy: Ou accele a ed Equa ion (3a) is e y simple wi h he ela i e e o up o 5.35% bu wi h only one peak o high e o (o he wise up o 3% as can be seen om Figu e 4); i is mo e accu a e compa ed wi h app oxima ions by Round [ 59 ] (10.9–5.5%), Eck [ 60 ] (8.2–5.7%) and A ci and Ka agoz [61] (4.8–3.1%), Wood [62] (23.7–16.6%), Moody [63] (21.5–18.1%), e c. 3. Possible Simpli ica ions As al eady no ed, he main goal is o p oduce no only accu a e, bu also compu a ionally low cos [ 11 – 13 , 29 ] explici app oxima ions o he Coleb ook equa ion. T igonome ic unc ions