OSCILLATORY VISCOUS FLOW IN CURVED PIPES

OSCILLA':OHY VISCOUS FLOH IN CUTIVED PIPES

bv "

Arnold F. Bertelsen

Institute of r~the~atics, Department of Mechanics, University of Oslo, Norway

Abstract

The flov1 induced in a curved pipe of circular cross flection anc'l continuous curvature by a pressure gradient alon~ the pipe varying sinusoidally in tine is investicatefl theoreticA..lly. The oain

influence of the mtrvature on the oscillatory axial conponent of the flow is to cause a skew distribution of this conponent across the pipe. It is also found that, besides the secondary flows denonstrated by Lyne (1970), the chan~e of curvature will induce secondary flow effects qualitatively in accordance Hith the

observations of Bertelsen

&

Thorsen (1981).

Knov'lledge of time derh::ndent flowR through curved p1.pes is of considerable interest botll in engineering and physiology. In this paper we study the flm1 induced in an incompressible viscous fluid by an oscillatory pressure cradient alan~ a curved pipe in order to explain sane non-linear streaoin~ cffect8 observed by Bertelsen and Thorsen

(1931 ).

They found anon~ other thinrs stron~ secondary

strea~ing at the junction between a bend and a strai~ht nipe. There exist several theoretical nodels which depict tine dependent flows in curved pipes, for exanple Lyne

(1970, 1971),

Zalosh ~Nelson

(1973), Sinrr,h, Sinha~: Aggarwal (197£3) and f''1ulin /', Greated (19BO).

Unfortunately all these theories are build on too restrictive assunptions to explain the phenonenon nentioned above.

In order to retain the essential !"'hysical effects and at the sane tir1e sinplify the nathenatics as much as possible, He use the following assunptions: (1) The radius of curvature A(z) (sec firure 1) of the curved pipe is continuously differentiable with respect to

z.

⁽²⁾ The thickness of th~ oscillatory boundary layer

(Stol~es layer)

/v/(,_,

^(v, kinenatic viscosity of the fluid; u), circunferential frequency of oscillation) is snall conpared to the radius of the pipe a, i.e.

P)=~l2vlw<<1.

⁽³⁾ The aspr:ct ratio is everywhere snall, i.e.

ATZT

a <<1 • ^(lt) The anplitude of oscillation is sr:mll, L e.

basic axial oscillations.

where \1J

0 is the velocity anpliturlc of the

(5)

The neynolds nunher R associated with the secondary streanin~ is snall, i.e. R=A-2 6E 2<1.

Subject to the restrictions presented above it is possible to obtain an approxinate solution of the lJavier-Stolces equations

depicting non-linear streanins effects and other details of the flow field.

- 2 -

2. THE EQUATIOJJS OF I10TIOH

In order to describe the flow problem discuRsed in the first section, we introduce the coordinate systeo sketched in ^fi~ure 1 with

a ₌

JITOT ^0(6),

and therefore we write A(z)

6D(s),

s=o:?;/a. IJ:he essential chanf~e of A(z) is supposed to take place over a distance of a few pipe

radii a alonG the pipe around z=±z "'±21tA(O).

0 - In accordance with the experiMental results of Bertelsen f< Thorsen ⁽¹ 981 ) we expect an oscillatory flm.r field in such a pipe to approach the correspondin(j flow in a straight pipe when lzl>z0^+a/a and that in a torous when z -a/o:<z<-z +a/a.

0 0

With a view to further simplifications, the following dinen- sionless quantities seems convenient

r'

=

r/a,

z' =

z/a, ^1;

=

wt ( 1 ) u' ⁼ u/11_{0 }} v' - v/H _{0 ,}

w'

^{= vr/H}

0

pi = 1

pt.)a\J P

0

Inserting these quantities into the equations of notion ^~iven in Appendix A (equations

A5,

A6, A7 and AIJ) we ~et after dropping the prines,

(2)

ou

+ _"'[l1c.

.ou

+

_v ou __

_{- -}v² + o:KvJ~s

ou

-· 6KH 2 caseD s . ( ) ·1

01;

or

^r

oe

^{r u}

+ 6EcoAElD(s)~

or

( 3) _{-;:- +} OV _{v~ E- U-;:-}[ OV _{vr + -}V _{r ~e} OV _{u + -}UV _r + aKw-;:- + oh.w SlnOD( s . OV _vS ^r

2 • ) J

0 l;T [ 0 V'T V

o

^{W r ( ) •} 0 Vv ) ]

(4) B-t + ^E uor +

r

59+ tw oucoseD(s -OVSln8D(s)+aos .

= _

y,:£12. +

la2{[v2

+ ^J:1..

en( )L _

ol:sineD(s)

o

a 'as 2^1.) 2 ^v .cos . ' s 0 r r 0 e

(5) on + or ~ r + .l_ r

2.Y. oe

+ oED(s) [twose-vsine) + ^a 1'

'c

0w s

=

0

where ^{V 2} ₂ is eC'I_ual to V~ given by equation (A3) (see Appendix A) when r in that equation is regarded dinensionless, and

I:

=

1 +o_r_c_o_s-:::-e":;=;:D-r(-s~) 1

With recard to an invectigation of secondary flows, the vorticity and continuity equations are the ~ost suitable basic equations.

The non-dinensional vorticity equation can be written

- 4 -

( 6)

~

_v't c! X+!ew+! r z n) + E{[! cl r r

~e

^{v -} 6Ksin8D(s))

1

0

(~r

+ oKcoseD(s))

J[u~; + ~ ~~

+ I:vr(oucos9D(s)

+ !!Y. _r + KH(a~v + owsinGD(s))]

uS -

+ -t1 [ ( z .Q_ or +

l ) (

r u o v + v or r 158

o

^v

⁺

u v + K o v + r ^a 'w

o

s ^{J:. u n.w l '} 2 s n i 8 D ( ) ) s

l.

roe· or+ r ~(uou v au Be-

r

v^{2 +} a'Kw"§B-au

oi:H

2 ^cosOD(s))

J}

where the conponents of the vorticity are given as the following

a ov 6Hsin8D(s) ) l+orcoseD(s)

FS -

l+orcoseD(s)

+

! (-

ow + ^a

ou

owcoseD(s) )

e

^or 1+orcos8D(s) 'B-s =· l+orcosen(s) +

1 (ov

+

v _ l

au)

z or r r

oe

The right hand side of equation

(G)

can be inferred fron equations ( 2) , ( 3) and ( 4) •

In the following sections we seek solutions of equations (2),

(3), (4), (5)

(or equivalent equations derived herefron) which are asynptotic to the exact solutions in the linits a~o, B+O, 6+0, E+O and

R=o£

²

8-

²+0. ':'he nethod of natched asynptotic expasions Hill be used to ohtain the asymptotic solut:!.ons. The result is an inner and an outer solution. The gaup;e fnnctions of each tern of tb.e expansions are dcternined successively as the calculations proceert.

Anticipatinr~ the r,auge functions knovm · •

.re

can Hri te for the inner expansion (Stokes layer)

(8) [ ] u (~,e,s)fixed

=

ao [ Bu1110 +a 2 Bu2110+ ... ]

(a,B,o,e:,R)~o

(9)

_{[v]( ~,e,s )} fixeo

=

ao[v1010 + a²v2010+ •.. ]

(a,B,o,e:,R)~o

( 1 0)

[w](~ 3 ^B,s)fixed =

woooo + o[w0010+Bw0110+ ... ]

(a,g,o,e:,R)~o

where ~ is the relevant boundary layer coordinate

(11) ~

= e·-

¹ (1-r).

On the sane basis the outer expansion can be written

( 12 ) [u](r,B,s)fixect

=

^a6 [U1010+ a²U2010+ BU1110+ ... ]

(a,B,6,e:,R)~o

( 1

3)

[v](r,A,s)fixed

=

^{a6 [V1010+ a2}V2010+ BV1110+ ... ]

(a,6,6,e:,n)~o

( 1

4 ) [ "~'

¹

J (

r ,

e ,

s )

r

i x e ^d

=

^H

o o o

o + 6 [ VJ o o 1

o

+ [;)

vJ o

¹¹

o

+ · · ·

J

( a, f), 6 , e: ,

n)

^~o

+ a ² 0 [ \J 1 0 1 0 + B \J 111 0

+ · · · ]

((a,B,6,e:,n)~o neans a~o,B~o6~o,e:~o anrl R=A-²oe: ²~o). ~hese

exoansions will be insertect into the basic equations

(2),(3),(4),

(5)

and

(7)

in order to obtain the ~overnin~ equation for each term.

- 6 -

3.

SOLUTION OF' THE LINEARIZED F.QUATiotJS 3.1 The outer inviscid solution.

As mentioned in the sections above, we seek solutions of equations (2)~(3),(4) and (~) subject to

(15) a+O, B+O, 6+0, e+O, R+O

Since 6+0 a boundary layer approximation will be atter:1pted. This inplies that the time dependent flow field outside the Stokes layer can be considered as inviscid and depicted by a velocity potential

¢ satisfying ( 1 6)

( 1 7) r.£1] ⁺ .lsin'L"

Los ex s+±oo

where 11 2 is given by equation ( AL~) in Appendix A. ':'he velocity potential shall also natch a Stokes layer solution which will be used to describe the flow close to the wall. The asynptotic expan- sian of the velocity potential turns out to have the forn

Substitution of this expansion into equation (16) leads to, subject to ( 1 5)

( 1 9)

=

0

The solution of equation (19) subject to condition (17) and the demand of natchin~ with the Stokes layer solution, is

(20) ¢ 0000

=

s sin'L"

Inserting (20) into equation (16) and (17) and equalizing terns of O(ao) we find

( 21 ) _rcos

e

dsS~LD'L" dD • =

o

.

(22)

[04>1010] ⁼⁼ 0

or

r=1

o¢1 01 o

[

~ _OS

l

_{· S·+±co} ^-+ 0

This problen has the solution,

Equalizing terns of O(a 36) in equation (16) we get ( 2ll) 112"' _2~2010 + IT-1 ( 3 r

+

r 3)d3D . ~Slh ~ == 0

which, subject to

r 0<1>2010

l or J

r=1 ^{= 0 (natchin~ u} condition) (25)

have the solution

(26) ₄₁₂₀₁₀ = 192 ^{1 ( ')2 -~} r + 9 r 3 - r s) cos ed3D . ^d8Ts1n~

The terns given ahove reveal favourable nunerical coefficients with respect to ohtain an accurate solution even if a is not very

snall. In order to find ^$1110 the Stokes layer solution to the appropriate order must be known. The velocity field in the outer inviscid region is denoted by (U, V, IJ) and

3.

2 The Stokes lc.qer solution.

In this section we seek the inner asymptotic solutions (0),(9) and (10). EstiMatin~ the pressure cradient in the Stokes layer using the linearized Euler equation outside this layer, equations

(2),(3),(ll) and (5) e;ive subject to the connitions (15) and

TJ fixed,

-

⁸

-

(28)

^{ovi010 oVi010} 1 02 vi010

=

_{[ o-. ] r=1}

^{+ -}

_on2

o-. ^{' )} ,_

where 1=1,2, and

oVliOkO

o\J

1 02 wiOkO (29)

o-.

= [ 10k0] ^{o 1: r=l}

+ 2

on²

v.rhere i=O, 1 and 1c=O, 1 . The velocity components V i0 1 0 and VJiOkO are derived from the velocity potential, i.e.

(30) vio1o = -1 o4>io1 o r oe vJ oooo = sin,;

( 31) \voo1 o = -rcos eD ( s) sim;

\11010

oQ>1010

= os

The radial velocity conponents ui110 (i=1 ,2) are deternined by the continuity equation (5) which gives

(32) 011i 11 o ov i01 o owi-1 , 01

o

on +

oe

+

as

^{= 0}

The boundary conditions are

The rnatcing conditions are derived from

(34)

(u,v,w) _T)+c:o + (U,V,W) _r+ 1

Hhere a Taylor expansion of (U,V,H) at r=l must be perforned in orrler to obtain the required conditions. The solutions of equations

(28), (29)

and

(32),

subject to the boundary and natchin~ conditions

(33)

and

(34),

respectively, are

(35) u 1110 ^{= -} 1[1 ~ ~(cos-.-sin,;) + nsin-.

-Tl

+

~-·

(sin( 't·-T)) -cos ( -.-n))] case dD

2 ds

( -) _{3b u 2110}

= -

34 ₁₉₂

[1

₂ccos~-sin~) + nsin~

(37) (]8)

(3 9)

(40)

w

₀₀₁₀

=

-[sin~-e-nsin(~-n)]cosBD(s)

(Ll2) ¹¹¹ -n - d 4 D

192[sin~-e sin(~-n)Jcosed04

(43)

w

₀₁₁₀

=

ncos8D(s)sin~

The velocity c0nponents ~iven above, 1Jfj_ll in the next section be used to calculate induced secondary flows.

l.i • SECOlJDARY FLOlJ.S 4.1 The boundary layer flows.

The ~overning equations of the secondary flows in the Stokes layer are derived by substituting the inner expansions (8),(9) and

(10) into the vorticity equation

(6)

and continuity equation

(5)

and the equalizinr; relevant terns of the sane order of nar;nitude.

Terns of order 0(6E) ~ive

( 114)

with the boundary condition

- 10 -

and [v₀₀₁₁ ]n+m shall also natch an outer solution. TI1e solution of equation (1~4) sub.]ect to boundary condition (LI5) is

5

-12,, -

^{+ 1 -2n ·} ₂

l .

2

~e cos12n ~e cos n_sln ~

r-- ;;·-

+ [ ^-~n.

.')

~t2n. ^{f '} 1 -2n .,

J

_{2 } ·} ·D()

e s1nn- ~c slnv2n + ~e sinLn cos-~ s1n8J s

It is obvious fran the denand of natching with an outer solution

The velocity conponent

v 0011

given above are indentical with the tangential velocity derived fran equation

(3.26)

in Lyne

(1971)

except for the ~adulation hy

D(s).

The radJ_P.l conponcnt u₀₁₁₁ is now deternined by the followinf\

equation of order

0(50

derived fron

(5)

(47)

+ ~----^ovoo11 = 0

oe with the boundary condition

o:='he solution is

{ !=i 1 e

-n

. 1 ~n

( 49)

u 0 .111 ::::

ff -

^~n - ~( S 1n11+COS'q) -

15

^e

,. r:f2

-n rJ/?e-/2n I?

+ [-

fG"

⁺

l6 -

^~

-(

s:l_nn~cosn)

+ ·

₁

g- (

sin/2-n-cos~ 2n) e -2n ·, · ')

" ) l ·

2

- lG~slnc.n-coscn _.sln -r

r n

~)/2

^{r:>-·n )} 5/2e-l2n(sin/-2 .... ,+cos/2"")

- , _ -::-7"1.) + --1^~ + .::--2 -(sinn+co~'>n lf , - , ''·

~ ¹tl 0 - )

~2n

+ e _{1 ( ) (} ~; ::L n2 n +cos 2 n )

1

cos 2 -r } cos

e

D ( s )

Equalizing tervs of order O(a6E) in the vorticity equation (6), vJe find

.(50) 03"'1011

()T)3

with the boundary condition

(w ₁₀₁₁ ) shall natch an outer solution.

T)->c> The solution of

and

H1 011 subject to these conditions is

(52) e ^-11 sJ.nT) -^• -yeasT) ^e-T)

+ 11 ~

-n

2 ^(-sin~ +cos 11) + n--e-n 2-(sinn+cosTl) + -{fe ^{7 /211} COSv ^~ c11

1 -211 C)

1 .

₂ , [ 3 -r1 . e -11 , . )

- 4e

COSeT) Sln.,; -,-

2e

^{Sll1T)- '11-}₂ ~SlnT)-COST)

7

/211 . r;s + 1 -2 n . ')

l ') }

"dD - 1Te Slnv~T)

4

^e Sl.nL.11_COSc"!; COSr:'dS

Now, as the leadin~ terns of the inner expansion of the secondary flow are lmoHn, the nethod 8.f-!plied rec)1Jires that sone terr.:s of the

correspondin~ outer expansion are calculated.

4.2 J:he floH outside the boundary layer.

It is easily seen fron the secondary flow velocities in the Stokes layer p:iven above~ thn.t sub,iect to n~co, the steady str·ean- ine; velocities only, Hill be non zero to leaclinr; order. rl'herefore only steady outer solutions will be considered. These solutions are obtained using the continuity equation

(5)

and vorticity equn.tion

(6). According to the denand of natchin~, the leading tern of the outer ex pans ion nust be of order 0 ( 6 E) • The vor•tici ty ec1Ua tion ( f,) r,ives subject to this condition and (r,A,s) fj_xed, a~o, 0~0, o~o,

·- 12 -

where, with reference to equations (7),(12) and (13) (54) _Q 0011 =

-;..iT

v ·. 0011

+ ^VOOll

or r

The nc;.tching condition l"equires (55)

r

(U0011) r+l+ O

l _l

^(V ₀₀₁₁ ⁾ _r+1 ^{+ rv} \0011 T)+co ⁾

fran which it follows that

(56) QOOll = HOOll (r,s)sinA

1 oU0011

r o8

'l'his result insePted into equat:':.on (53) gives the follmrinr, solution, rec;ular in r< 1,

(57) = B,(s)PsinEl

I

Eouation (54) and the continuity equation

(58)

derived from (5) [;lVe wne ^{• f-,} following solution, subject to

(55)

(59) U0011 ⁼ t(-l+r²)cos8D(s)

( 60) 11 0011 ⁼

~(1-3r

_(I ²

)sin9D(s)

v~1ich should be compared to

x

₀₀ ^~iven by equation

(4.4)

in Lyne (1971 ). His and our result differs only hy the nodulation by D(s).

According to the demand of natching between the inner and the outer' solutions, the outer expansion r1.ust have a tePn of order

O(a5E) to r:1a tch This tern is determined by the and

!

9- conponents of the vorticity equaiton to order ^{O(a6E) \~ich} give, respectively,

(v ~ 1 )" ^') .2_\11

( G

1 ) L

?

^{Al 011}

- _?

^'- 08"1011 = 0

( 17 ~ 1 )w ^{' )} 0 y

(G2) + ^{'- :: 0}

'-

?

:t.l 011

r7

Be"lOll

vrher8

(63) ^v 1

au,

011 oV0011

"cl 011

=

r

oG

cs

( 6

I+) 1J.I1 011

o\.Jl 011

+ oUoo11

- - _{or os}

The soluti6n of these equations, regular in r<1, and satisfyin~

( G 5 ) ( \rl 1 0 1 1 ) r-+ 1 ~ ^{( H 1 0 11 )}

T)-+oo

is

( 6 6

1\ 1 - ., 3 ) dD

H1 011

=

¹⁾⁽ c_r+r cos8 ds

\lith reference to (66), the continuity equation (5) requires terns u_{1 011} and

v

_{1 011} of order ^O(a:25E:) \J"hich nust satisfy the

followin~ equations and conditions

U7)

where (70)

oU1011 TJ1011 ¹ oVlOll oUlOll

+ - - - + - + =0

or r r as os

{·\V 1 011 or

v

1 011

+ _r ^{1 oU1011} _{r 08}

and v₁₀₁₁ are ~iven ^by equation (R3) in Appendix B. ~he solutions of equation (67), (6R) and (70), rcsular in r<l anci satisfyinv

(69), are

r ¹ 5r^{2 L c12 D}

.. J 1 0 l 1

=

%(10

-

^- 5r f)r;os8r}s2

( 71 )

-

(72) v1o11

=

%(-10 1 ^± 3r² + r ^{4) . Sln.~ An2D}

CAS

The calculations sketched above can be continued systenatically. This:

has been done to sone extent ancJ tl1e r>e3u.l ts are r:j_ ven j_n anpendix FL

- 14 -

~). HUfTERICAL R.E.SUL'l'S AND DI,SCUSSIO!l

In this section He give sone nuner'ical exanples of the flow field described by the analytical results obtained above. First we choose a specific function for the radius of curvature of the bend A(z). In the following exanples ^~e use

(73) ^{A ( z) =}

a(z+z 0 ) tanh--- -a

o:(z~zo)

tanh---a

which with different values pf A0^{~ a and z 0} give the different bends sho;,m in fir;ure 2. The a;:tn.l oscillatory conponent of the flovr is depicted by the followin~ unifornly valid expression

+

_{a o ,,} ^2-q _{1 01 0 -}

c

'0 ""'

(

fl , ... ' ^p s , ^{'t" )}

'

( o=i!:_)

A .

0

where

c

₀^,H is the conJ•1on pc-trt of the inner and outer expansion ann

(75)

c

₀ vr = o(-coseD(s) -1f.coseD2(s))sin"L"

}

The function r,iven ^by equation ^(71!) is plotted in ficure 3¹ 4 ~ 5, 6 and 7. gradual change from a 1miforn velocity diGtribu- tion outside the Stokes layer in the strair;ht pipe to the slceH in the bend is easily seen in these fi~ures. The skewness is caused by

a

hit;lwr prersure gradient alone: the j_nsic'!e of the bend t.h.an alonr;

the outside. 'T'he effect of j_ncreaslnr: a~vahws rmy be understooc1 by comparinr: f!r;ures

3} 4

and

5.

Of course, the chanr;e fron a uni- fern to a skew profile takes place over a shorter distance (z) for large a-values than for snall. r::'he sl:eHness is established (.:md snoothed out) by a transverse flmr hePe exam.plifiec1 ^j_n fisure R by

v6t)

r;iven as

(76)

v<

^{t) =}

0

Hhere the conmon part is (77)

(78)

The steady secondary streaninr is depicted by

TJ ( s) _{. 1}

=

~ (P,1(s) +U 'PTT + 2U +-"-lJ c e: .d .. 0 1 11 0 0 1 1 ^{T _;::- '0} 0 1 1 1 a 1 0 1 1 · ^u 0 0 2 ·1

(7^{( ! '}

v<s) =

O ( (s) + 2 7(8) ^C1 (s) ~ (s)

v

^21!

7 J 1 e: v 0 011 a \ 1 0 11 + l·> v 0 111 + ^u v 0 0 21 + () 011 +a ' l 0 1 i

where the cor:1non parts are

c,

^'u

(B2) C = "" [ 1 + "(1 . 3 )

J .

"D( ) a²6e: . "d²D 1 , v u £ -

4

^D

2

^-r

4

^{11 . s1nu s -} --n;-sln~'CfSZ

(83)

^{[ l , l}

5 ,

~

l

dD

C 1 ,w ⁼ a

o

e: r. -u L ( y o -,-

.:rrn )

u -co s 8 -"-as

The nain features of the secondary steady streaning in the axial syrmetry plane of the pipe bP.nc'l.s are shown in the streanlinc diagrans in figure

9.

These diagra~s indicate that the nosition of the core of tl•e recircula tine rep;ions are heavily dependent on o:,

More detailed information about the steady secondary flow field ~ay

be drawn from the velocity distributions plotted in figures 10, 11 ,l 12, --- and 21[. Here the eadial diRtribution of the radial, anr;ular and axial velocity conponents ar2 plotted at different axial posi-

- 16 -

tions, \:Je notice that in nodel 2,3 ,ll and 5 the radial and aw~ular

velocity obtain maxina around r=O not at z=O, but at ^Z"'Z ~a/a.

0

This neRns that the Heynolrls stresses associated Hith the cllan~e of curvature and those associated with the curvature itself (Lyne type) induce outer· secondary flous in the sarte direction,

Finally we renark that the velocity profil~s shown in the fir;ures

J,Ll,5,6

and

7

are in qualitative ap.;reenent vrith the obser•- vations of Bertelsen and Thorsen (1981). The sane is true for the secondary flow field, but the recirculating vortex syste~ observed close to the vmll at the junction between bend and straight pipe did not appear in the strean line diagrans in figure

9.

It turns out to

2 (s) 2

be necessary to include the terns a E6Eu1111 ^{and a-G8EU}1111 ⁱⁿ the asynptotic expansions to dipict such details of the secondary flow field. Figure 25 shows a strean line diaFrarn based on a veloc- ity field with these terns included. ~he ~ap between the start and the end points of a strean line wll.ich shonld close, is to be j_nter- preted as a neasure of the Rccuracy of the asynptotic expansions used. In spite of this unaccurac~r, the vortex systen close to the wall in figure 25 is a significant feature of the flow field calcu- la ted above" Unforuna tely a quantitative cor1parison Hi th the

experiuents nentioned above is iupossible since the experiMents were perfornecl in pipes with abrupt change fran bend to straight pipe. On the other hand, the physical nechanisn inducing the

secondary floH around the juncture behreen the strai(-r;ht and curvccl pipe sections, is probably retained in the theory above (see

equation 50).

List of references

Bertelsen, Arnold F

&

Thorsen, Leif K.,

1981.

Preprint Series.

Institute of I1athematics, Dept. of I'lechanics, University of Oslo, Horvvay, No 1} April

1981 .

Lyne, H.II.,

1970.

Ph.D.Thesis, UniV(·rsity of London.

Lyne,

vJ.J-I.,

1971. J.Fluidr1ech.

45,

13-]1.

Hullin T.

e,

Greated, C.A.,

1980.

J.Flnid f1ech.

9B, 383-396.

Lingh, rLP., Sinha, P.C. E: Aggarwal, TL, 1978. J.Ji'luid !1ech. 87, 97-120.

Zalosh,

n..G,

p, Nelson, H.G.,

1973.

J.Fluid Hech.

59, 693-705.

·- 18 - Figure captions

Figure 1. The coordinate system refereed to in the text (nodel 1 : A0=6, a=0.2, z0 =10a)

Fisure 2. Pipe models used in the numerical exanples (nodel 1 : see figure 1 ) .

a) model 2: A0 =6, a=0.5 and z 0=9.5a b) no del ^~. _~· A0 =6, a=CL 75 and zo=9.5a c) no del

4:

A0 =4, a=0.5 and zo=6.5a d) no del 5: A0=6, cx=0.75 and zo=5a

Figure

3.

Nornalized tine-dependent axial velocity component in model 1 for B=0.08 and £=0.5 in positions z=O, z=8.6a, z=11 .4a and z=15a. Decreasin~ skewness with increRsin~ z.

Fi~ure 4. Normalized tine-dependent axial velocity component in nodel 2 for B=0.08 and £=0.5 in positions z=O, z=8.075a, z=10.925a ans z=14.25a. Decreasinc skewness with increasinG z.

Figure

5.

Nornalized tine-dependent axial velocity component in model

J

for B=0.08 and z=0.5 in positions z=O, z=8.075a,

z=10.925a and z=14.25a. Decreasing skewness with increasing z.

Figure 6. IJornalized tir1e-dependent axial velocity conponent in model

4

for B=O.OO and £=0.5 in positions z=O, z~5.07a,

z=7.93a and z=9.75a. Decreasing skewness with increasing z.

Figure 7. Normalized tinP-dependent axial velocity conponent in model

5

for B=0.08 and £=0.5 in positions z=O, z=3.6a, z={). 4a and z=7. 5a. Decreasinc sl:cvmess ~vi th increasing z.

Figure 8. Normalized tine-dependent an~ular velocity conponent in nodel 3 for A=O.OS and £=0.5 in positions z=9.5a (upper curve and z=10.925a (lower' curve).

Fir,ure 9. Stream line diagran of the time averaged secondary flov-1 in the axial synnetry plane of: a) model 1, b) nodel 2,

c) model 3. (8=0.08; E=n.5).

Figure 10. Radial component of the (.5=0.08, E=0.5) in model 1 at z=11 .4a(D) and z=15a(E).

time averaged secondary flow z=O(A), z=G.6a(B)," z=10a(C),

Figure 11. Radial conponent of the tioe averaged secondary flow (8=0.08, E=0.5) in r10del 2 at z=O(A), z=B.075a(D), z=9.5a(C), z=10.925a(D) and z=14.25a(E).

Figure 12. Radial component of the tine averaged secondary flow (.5=0.08, E=0.5) j_n nodel

3

at z=O(A), z=8.075a(B), z=9.5a(C), z=10.925a(D) and z=14.25a(E).

Figure 13. Radial component of the (8=0.08, E=0.5) in nodel 4 at z=7.93a(D) and z=9.75a(E).

Figure 111. Radial conponent of the (8=0.08, E=O.S) in nodel 5 at z=6.4a(D) and z=7.5a(E).

Figure 1 50 Angular conponent of the model 1 (parc:meter values, see Figure 1 6

⁰

Angular conponent of the r.1odel

2

(paraneter values, see Figure 1 7 . Angular conponent of the model

3

(parar:1eter values, see Figure

¹

g. Angular conponent of the r!odel

¹⁺

(paraneter values, see

tine averaged secondary flow z=O(A), z=5.07a(B), z=6.5a(C),

tine averaged secondary flow z=O(A), z=3.6a(B),

~=5a(C),

tine averaged secondary flow figure 1 0)

⁰

time averaged secondary flow figure 11 )

⁰

tiDe averaged secondary flow

.p• ~

1e;ure

^{1 2) •}

tine averaged secondary flovr fj_gure 1

3) •

in

in

in

in

Fi~ure

19. Angular conponent of the tine averaged secondary flow in

model 5 (paraneter values, see

fi~ure

14).

- 20 -

Figure 20. Axial conponent of the tine averaged secondary flow in nodel 1 (B=0.08

3

E=Oo5) at z=8.6a(A), z=10a(I3) and

z=11.4a(C).

Figure 21.

Axi~l

conponent of the tine averaged secondary flow in model 2

(S=O.OS~

E=0.5) at z=8.075a(A),

z=9.5~(B)

and

z=10.925a(C).

Figure 22. Axial conponent of the tine averaged secondary flow in r.10del 3 (13=0.08, E=0.5) at z=8.075a(A), z=9.5a(n) and

z=10.925a(C).

Fi~ure

23. Axial conponent of the tine averaged secondary flow in model 4 (B=0.08

9

E=0.5) at z=5.07a(A), z=6.5a(R) and

z=7.93a(C).

Figure 24. Axial component of the tine averaged secondary flow in model 5

(f~=0.08,

E=0.5) at z=3.6a(A)

3

z=5a.(D) and

z=6.4a(C).

Fir;ure 25. Strean line diagr8J'l of the tiDe averaged secondary floH in the axial synnetry plane of a bend with

a=l,

A0=6a,

z 0=9 .5a and

^{B=O .OJ~,}

E=O .1.

----

----

---

Fi~ure

1

a)

-+·-·

^-rr

t

-t--+

- --

--

--

- - - _--- ----

- - o . -

Figure 2

• • •

~ /

/

/

- 22 ...

~~ T

T

\. I

^d)

+-

1

.

\

I l

/

I

1.0

0.8 0.6

e

=11

.... rcoae

-·

¹

^•

N *Sll

•

^{. t}

' ^•

~

-

( " y

... • •

-

^{ed c.o}

-

^{co a;Q}

^-

^{~ b}

I

Figure 3

\ol(t)

0

1.0

... 24

^~

w<t>

D

0. 2

1.2 w<t>

₀

1.0

- .

.

I

Figure

7

Fi ·:t:Ure

13

f) ;:

o

^{8:.: ;;- ...} rcos&

t . t---+---+~·-·+-+-·-+--t--+--·+--+--+- i

-·t

I

f.&:> ...,. ""'-~ llSl '"'-.; - · l.O co

~ ~ ~ G~ ~ ~ ~ ~

I I I

\ ' _l

e.:;::J

- 26 -

F1gur.e 9

u{s}

1

Figuro 10

~.002

-0.002

u<s)

1

Figure 11

-rcose

II:; 7i'

0. ~04 ^I

u<s}

1

-8.BI2

_F1~ure

₁₂

Figure 1)

0. 004 t

^{TT ( S )}

I

^vl

t I B.

~02.

- 28 -

...,.. E • '""

---~---~

s:S Ed ~ ~

I I

---~·-·--·---D

-rcos&

~ C"'>..J ...

i!!l'I:J -- ~--·---~

---

0.115.

-0.915

0.205

\ \

I

-0.005 t

lf:::!ii

f • ---h4.i~+

Figure ~5

v<s)

1

Fltiure 16

L'l-li

"7-A...

--

rs1n9

~~+--+--+-+·

~ ... -~

_~~·

^. --~"'

:rsine - .

·---·---

y(s}

1

UISt ~

t

0.010

0.005

-0.005

-0. 010

Figure 17

-0.015

- )0 -

rsinG

9.810

A

-UI5t

-lui

- .

rsine

~

32 ...

e.

^~1s

0. 010

~+~+-~~~~~-4-4~~~+-~rl-~~~-+~--;1

tS::)

rs1ne

-0.005

-0.010

F1qure 19

-0.015

e~o

0.014

A

Ul61,

0. ~04 ^I wf

^{8 >}

-rCOSQ

3.0~2

I

22

- '34 -

8.002

B

^{a 0 i)~}

;r

.

^-

• •

^t--~

^.

^-t--4

«..K;'!ll

....

I'""'..J ^{•"'¢ t.O} co

.

_ced

.

_s.i ed cs:i

\'!!IE& ^e.;;;)

I

' _-rcoee

0.002

f9:::0

.J---4--l~-t-+t

-·-+---+-·+---+t----+--

^{-+---+-1 _....-+-.} ··--+--f-+-1 ~,.__.+-.-t.+--~

C"'.,J ...,.. f...C:) 00

d ~ ~ s5

-rcose

--B

_{v ..} ^fltl\.,) _Vti'-

1.

-· ^.,.,._

F1sure 25

\ _\

)

' i

- 36 -

APPENDIX A

'::'he coordinate svsten and the hasic equations

An or'tho,ronal cur-vilinear coordinate systen as that sl-::etchcd in figure 1 secns to be the Most convenient for discription of the flow problcn considered jn this paper" The r:radj_C:nt operator ^j_n this coordinate systeu is

(A 1 )

~nd

r

(/\:?)

givinr;, (A3)

where

o!

_{c cocO}

=

^t

Oz ll.(z)+rcose-"-z

o1

-t _{8 sine -t}

::

-·

A(z)+rcosfl¹z oz

~-t -·i cos e+1₀ sine

Ol z r ,

oz

=

A(z)+rcose

]\ ( z)

1\(z)+ccosO

ol

-t _r

= -t ]_e

oe

0-t 18 _-t

=

^~l

or

^t'

,-,2 n2 + 1 (

"o

sinO 0

v' = 'I 2 - - - - -A+rcoso c 0 G " -or ·- --r- ~8 ^u

(A+rcose) ^{3 (}Z 0 Z}

+ - - - -1\2

Arcos0

+ - - - - · - - ^dA

o

+ 1 0 r i:)r

1 ^:>.2

+ ^u

0

oz

Hon-dincnsionalizinc the Laplacian by neans of the radius a. of the pipe, ue find,

(A4)

^{n2 -}v - ¹¹, 'J ^{2 J.} ' ^s: u 1 , ^{s: D(s)} ,.. oi)l--::\-⁽ c 0 s ^nO o~

'- •urCOo,·V\Sj u r

sinfl

c

= - r - o8)

2o rcos8

- a n- r;::-'1

--1 +o rcos

en\

^s;

-1- _ _ _ _ a2 ;:_:_ _ _ _ _

(l+orcos8D(s))2

Using the rliffcrential operatiors (Al) and (AJ) and the rcla- tions (1\2)) the Navier 8tokes equations c~n he written,

(A5)

(A6)

(A7)

(AS)

o~

+ au +

y_

ou +

I'n·r

au

o t

uo r r a

e

A+ r

cos e o

z

- - v2

_r

^-·

= _

~

op ⁺ v{[v2

¹

^sine o ^{+ case o}

P

or

2 -

? - ··d A+rcos

G)

iJe ·A+rcose or A2cos

²0

ArcosG

~A

_o + A2

0

2 - c A+ rcos. e )

""4

+ ' lA+rcose )

3 ^~z

oz (A+rcose)-

2 ~z ^v

]

u

+ [- 2

L ^{+ sine + sinecose} 1

?

^oe

r(A+rcose) (A+rcose)2

v r

2A 2cose o Arcos 2 e dAl

¹

- L (

A+rcos)

3"

Sz + TA+rcos

e)4f

dz-

H J

OV

+

UOV + V OV

+

UV

+

A-H oV

+ VT2sin8

ot or r oe r A+rcose oz A+rcose

= 1

pr sj_ne o

r(A+rcose) 'Fe A

²

sin 2e + Arcose dA

Q_

+ A2

0

2 ] - (A+rco88T4 (A+rcose)j

dz

oz (A+r~os8)7 ~-v

+ [

2sinOA 2 o + Arsin8cose dA]

( A+rcos

)3 ~ (

A+rcos

o) 4

dz

w

+ [2 o sine + A2sin8cose J

l

?

^~-

(A+rcose)r (A+rcose)Zf

u,

~vt~. _u

+

u~'r'v _u

+ vr

~v_u

8

^{·~ +}

^uwcose _A+rcose ^vwsine _A+rcosO ⁺ A+rcose oz ^Aw

^ow

A

o~

1 {[v2 ⁺ COEO o sinA o

-p-"7(-::-A-:+-r...:.c..:....o_s_G~--)

o z -

v · 2

A+rcose or - r( A+rcos

e)

or

1

+ Arcos8

c'lA

o

A

2

_~

hr

-

~(~A-+_r_c_o_s_e~)~2

(A+rcoso)

³

dz oz + (A+rcoso)2 oz -

+ [ 2 A²

cos

e o

(A+rcos)3 oz

+ [

2A

²

c}in8 o

(A+rcos)

³

oz

+ Arcos2e dA A2cose dA]u (A+rcose)

⁴

dz - (A+rcose)

⁴

dz- Arsin8cos0 dA +

A

2sin8 dA]v}

(A+rcose)

⁴ d."Z

U+rcose)

⁴

dz ou +

u

+ l ov + ucose

~

r r

~

A+rcos8 vsine

A+rcose

+ A oH

=

^O

A+rcose oz The vorticity expressed in this coordinate systen is}

(A9)

Vx ( i u+i

₈

v+i

u) =

1 (

^1-

oN

r z r r oO A ov _ wsinB l

A+rcose oz ^A+rcose

+ 1 _e f-

^oVT

_or

⁺

_A+rcos8 ^A

_FZ

^ou - _A+rcose

^{HCOSO }}

+

^-t _l

z or {ov

_- +

v r -

^-1 ^l'

_otq co

^- (-t ^l

^r

^J> ;~+ -t VI-I--t 1 E.' "·

^.

1

^z

Q) '

..,o

- .)0 -

P.PPP.NDIX B

Eir;her order terms of the asynptotic solutions

In this section we shall solve the governin~ equations of some hiGher order terns of the asynptotic expansions discussed in sec- tions

(3)

and

(4)

of this paper. ~he boundary and natchin~ condi- tions and the solutions of these terms will be given without any further details about their derivation.

Applications of ecmation (69) requires knowledce about v 1 011 (see equation C:1)). The inner expansion r;ives to order ^O(a2oe:)

( B1 ) o3v1011 ⁰²v1011

a

ov1010 on^{3 2} _ono't ⁼

~[woooo as ]

The boundary conditions are

and v 1011 nust be bounded subject to n+oo (matching re~uirernent).

The solution of

v

₁₀₁₁ subject to these conditions can nost easily be inferred fror equations

(37),(3S),(44)

^nnd

(45)

which give

d²D (B3)

v,

⁰¹¹

⁼

^{Tfv0011 1 1'5(S) ds2}

where v 0011 are ~iven by equation (4~). The continuity equation

(5)

cive to order O(a²

aoe:)

=

0

which, subject to

have the solution

( 6 ) _ { 19 5 1 -11 ( ') . ,. ) 3 -n . 1 -2n B) u1111- ·- 32" + fFTl +

rre

'-Slnn+)cosn

+

rrne Slnn -~ 32e

+

1112 _U

_{- e}

12 n ( .

sln c:n-cosvcn

1

^{~ !;') )} - Tfe ^{1 -} 2n ( · 2 · Sln .n-cos -n .. sln _, 2 )

1 ·

²

1 [ q 13/2 r, -n ( . ) 3 -n . +

If 1f -·

^~

-

^"-e SJ nn+cosn +

2

^{r1e slnn}

13/2 ...

/2nc

/'!!:" ;-::; ) 1 -27)

J }

^ct2n

+ --g--e sinl¹2n+cosv2n- Ife - (sin2n+cos2n) cos2't coseds2

Higher order effects due to the finite aspect ratio

o

^{can be}

depicted by including terns of order

O(o

²E) which are roverned by .the following equation

(B7)

_-:.-=-r---

^{02vo021 2}

°

^2vo021

^c

0

c =

2-"-[2vrOQ{)OH00 1 0sin9D(s)

uT] 't

•n

on ) . '

'I'he solution of v 0021 must also satisfy the followin~ conditions

{

(v0021 )n=O = 0 (B8)

( v 0021 ) n+cc + constant (bounded)

The solution of v 0021 can rrtost easily be inferred fron equations

(39),(40),(44)

and

(4S)

fron which we have.

3sin2e ( , 2sin8 v0011D 3 '

:novr as two nevr terns of the inner expansion are kno~m, we proceect to calculate hisher order terno in the outer expansion. In fact, this has already been done in section (4.2) where thP result

(P3)

was anticipated in orner to deterDine

u

_{1 011} and 1r1 011 r;iven by equa- tions (71) and (72). The next step is therefore to seek terns of order O(o²£) in the outer solution. The c;overnin~ equations are

(

^\'2Q '2''0021 ⁼ 0 (B10)

1

^{oUo021 or +}

^u

^{0021 r}

vrhere ( Bll )

~v

0 '0021 0 0021 = or +

- 40 -

+-1 r ^oVoo21 oe +

cu

0011 cose-v 0011 sin9)D(s)=O

v

00~!1

__.,___

__ -

r

l 0 U00?1

r oe

The natchin~ condition is ( B1 2)

The solution of (D10) and (B11 ), re~ular in rc1 and satisfying ( B12), is

( D1 3) ^{

U0021 = [k<r-r^{3 )} +

-h><r-~

³

)cos2e]D2(s)

v ( 5 11 3\ . 2 D2( )

, 0021 = - "ffr + l""t)r 1sln

e

s

The Reynolds nunber R is usually not very snall in real flow problEms of this kind. Therefore v~e Hj_ll study terns of order 0 ( 5.-26 2e:3 ·) in the outer solution. The governing equations are

(E14)

(D15) where

(Bl G)

v0011 oUoo11

+ - - - -

0

=

^{oVo122 + vo122}

' 0122 or r

r o8

1 oUo122 r o8 The boundary concH tions are

(B17) (U0122)r=1

=

(V0122)r=l = 0

The solutions are (B18)

(Rl9)

These results correspond to _{x 01} ~iven by eauation

(4.G)

in Lyne (1971).

The governinG equations of terns of order O(aB-²62 £3) of the outer solution are

(B20)

( B21 )

vvhere (D22)

(D23)

( 2 1 )v 2 o':¥1122 1 0 oi11011

112 -

?

-''1122 -

? oe =

2 [

r

^B'S(U0011

or

V 0 0 1 1

°

^{\I 1 0 1 1}

+ r

oe )

( II 2 l )w + 2 2 -

?

¹1122

?

VOOll 0111011

o

^{oUOOll "ooll}

+ r oe ) + FS(U0011-~- + r

i!2

'0011 )]

r

v 1

./'.,, 22 ⁼⁼ -

r

\1;' _-1122 =

-

o\11122 oVo122

oe os

Mill 22

+ ^oUo122 - - - -

or os

The boundary and r:1a tchinc conditions are

(\Ill 22) r=l

- 42 -

The solution of (B20),(D21)j(B22),(B23) and ^(B24) regular in r<l is (J325)

With a view to the calculation of hirher order boundary layer effects we need to find terns of order O(a66) in the outer linear- ized solution ~overned by the following equation

(D26)

and natching condition

(D27)

The regular solution is

(B28)

3

dD

r.rcose---d (sin~-cos~)

lJ s

The tangential velocity conponent of the linearized equations to order O(aG6) can now ^be found fron the following eauation

(B29) ( 1 2

W- o

^{2 ~}

o )

^VlllO

=

tfsln·_,cfs Sl!l't" ^{3 . 0} dD ( . + ^COS~ )

and the boundary condition

(B30)

The solution is

( B31 ) v_{111 0}

=

IT 1 [ -., . ^-.)SJ.n~ +..., .)COS'tT ' 2 TJSln~ . +3 e ·s1n -fl . ( ^~-TJ )

-r1 -TJ ( )] dD

-3e ·cos(~-n,)-TJe sin ^-r-T) sineds

where terns of exponential growth have been excludccL S:'he radial velocity conponent u₁₂₁₀ of order O(aB²6) can now easily be derived fron the continuity eauation

(B32) ou121n ov111n

- ~-- + ¹¹111o + ~e- ov1010 ow0110

+ ^TJ oe +

os

^{== 0}

The solution subject to (B33)

is

1 {

J ')

^{2 .} 3 -11 ( ) U1210

=

[~COS~ + Jn Sln~ - ~e COS ~-11

The tern ^H0100 of oroer 0(5) is alr,o of interest and it can be inferred fron Lyne's result (Lyne 1971, equation

3.36)

that

(B35)

Equalizing terns of order

O(B6s),

we find

(D36)

11ith the following bounoary and r::atching conditions on the steady ( s )

part v0111

l ^{cv6n,\=o =}

⁰

(D37)

( v ( s) ) ₀₁₁₁ 11~"' ~ ~/.1

1

^{11 sine D ( s )}

Since we are nainly interested in steady se6ondary flows, only the steaoy part of the solution is given,

i.e.

(B38)

11e -n . n 2 n

+ --_2--s1n11 + ~e- }sinBD(s) The axial velocity conponent

w

₁₁₁₁ by the follouinG equation

(B39)

of oroer

O(aBos)

is governed

As Mentioned above we ar~ especially interested in the steady sec- ondary streaning and therefore we have solved the time independent part of

(D39),

only, which shall satisfy the following conditions

(B40)

( BLI1 )

( .(s)) 0

"'1111

n=O = -

( ( s) )

3

"dD w_{1 lll .} ^{n+oo + -} UT)COS,-, ds The tine independent solution is

( Bl~2)

+ T)e-TJ (3sinT)-cosn)} coseddD - s

The tir:le independent terns of the outer solution of order 0(

roE)

are governed by the followin~ equation (R43)

(B44) where

(B45)

n2n v2""0111

= ()

U0111 1 oV0111

+ - - - + - - - = 0

r r oe

VOlll

+ _r

The solution, recular in r<1, and natchinr the inner solution to appropriate order is

(B46) U Q lll =

h-(

¹

^{9 - 9}

^{r 2 ) COS 8 D ( s )}

(B47) _V0111

=

^{tr(-19 + 27r2)sin8D(s)}

Equalizin~ tine independent terns of order O(aB6E) in the outer solution, we find

( 2 1 ) ,,

v2 -·

rz A,, - ?

^{2 o1'1111}

ao =

⁰

(B49) ^{1 2}

ox,,,,

(V~ ~ ~)~llll + ~ ~--

=

⁰

where

(BSO) ^y 1 0 \rJ1111 cV0111

"•1111

= - _{r ae os}

( B51 ) '¥1111

oHnn ⁰ uo111

= - _{or as}

The matching concH tion is,

(B52) 1 S dD

0.11111 ) r= 1

= -

^{;-tc o s}

e

^d s

The solution of equations

(D47)-(B51)

is

(D53) H1111 ^{= -} ftC2r+3r )coseds ^{~ 3 dD}

It turns out to be of interest to calculate terns of order O(a2B6E). In the Stokes layer

we

^find

(R54)

The boundary condition is (B55)

0

v,

^{01 0}

- 2

"'oooo

os

while natching the steady part v1111 ^(s) of vl11l with V10ll (B56)

The solution of

v,,

^{( s) is}

(B57)

Vllll ^{( s )}

=

^lf{-1 If~ ¹

1

^!) 2^{11- e -r1 . SlTIT) +~COST] e -1')}

give