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