Journal of Sound and Vibration (1974) 37(4), 507-519

WAVES IN BRANCHED HYDRAULIC PIPES

SHRI KANT, M. L. MUNJAL AND D. L. PRASANNA RAO

Department of ~lechanical Engineerhtg, Indian b~stitute of Science, Bangalore-560012, India

(Received 30 )llay 1973, and ht revised form 24 June 1974)

The paper deals with the classical problem of axi-symmetric transmission of low amplitude waves through a circular pipe containing a viscous liquid. Exact governing equations are identified and solved, the radial as well as the axial component of the velocity being considered. Attention is drawn to certain fallacies underlying the conventional approach. The parameters required in the formulation of the transfer matrix for a pipe have been evaluated. In order to evaluate the response at the terminal point of a branched system for a sinusoidal input at one of the ends, a general algorithm has been developed.

1. INTRODUCTION

While working on automatic pumping brakes for atitomobiles, a problem was encountered where a sinusoidal input is given at the master cylinder and wave energy is transmitted through a branched system of hydraulic pipes to the different wheel cylinders in order to operate brake shoes for decelerating the vehicle. In a system of this kind, the diameter of the con- stituent pipes is of such a magnitude that the transmission of energy is more or less through a plane wave. The effects of viscosity, however, cannot be neglected and the particle velocity is, therefore, a function of radius. A generalized analysis of such systems, in which the radial as well as the axial component of the velocity is considered, is presented here. The propagation constant and the characteristic impedance required for the transfer matrix of a pipe are evaluated explicitly and a general algorithm is developed for the analysis of branched hydraulic systems.

2. WAVES IN A VISCOUS FLUID

In some existing literature [1, 2, 3] the governing equations of the classical problem of transmission of low amplitude waves through a small diameter pipe containing viscous fluid do not seem to be justified. Viscosity is considered while the radial component of the velocity is neglected. Also, the continuity equation used is not compatible with the physics of the system. Here, in this section, the exact governing equations are identified and solved with the radial component of the particle velocity being taken into account.

2.1. Tim GOVERNING EQUATIONS The general Navier-Stokes momentum equations for the axial and radial components of

the velocity may be written as [4] (a list of notation is given in Appendix II)

[av~ av, vo av, av,~

a [a,,, av, a [1 a,,, avo a,o 1,

+ l , g 2-~-;-5~-~/+T-~-+ ~ + , (i) 50"/

508 SIIRI K A N T E T ,4L.

P ~ ~ - + " ; Tff + v ` Jz -

a_(Jv~ Jv=~ ~ J [Jr. v~ lay, I =pX.+t, jz~a z+ ar/+rEffkTr r + r ~ } -

----JP § 2 s l J/.). ~_=r/ "~ 2].~ - - a2/)t" 2 J /J/')r _lal)' J~. [)// Jr r k J r r i o r I Jr= 3"l ia r ( " ~ r + r ~ ' + ' ~ ' z + - ' ; " "1

(2)

In the absence of any body forces,

X~=O, (3)

z~t ~ 0 .

For an axi-symmetrie wave propagation in a pipe with rigid walls,

V e = 0 ,

(4)

(5)

a = 0. (6)

~0

For a low amplitude wave with zero mean flow equations (I) and (2) are reduced to

Jr= jp + p a2v, l pav, a2v= It Or= +41taZv~.~ (7) P~ J--T=- Js 3 b-;-~ § 3 r ~ + ~ ' - ~ + 7 T az~'

v) [a2v, lJ2v,~ 4 [aZv, l J v , _ ~ (8) po-ff=- o- 7 -

where Po is the density of the undisturbed fluid. The general continuity equation in cylindrical co-ordinates is given by [4]

1 a(pvo) i- J(pv:) + pv, = O. (9) r aO az r

For axi-symmetric wave propagation in a stationary fluid, and with second order terms neglected, equation (9) can be reduced to

ap [v, av, av,~ a--;+ po/7 + ~+ ~) =o. Oo)

The energy equation for the case considered would be

P-- = P -P~ = co ~ + 6, (11) P" P - Po

where 5 consists of the terms of the order of p(JvJar)2.. For waves in a stationary fluid ,5

would be insignificant and the energy equation reduces to

p - p o _ co~, (12) P - Po

WAVES IN BRANCttED IIYDRAULIC PIPES 509

o r

Op c20p T; = oT?.

Eliminating p from equations (10) and (13) yields

pocg + ~ + ~-'z/ at

Eliminatingp from equations (7) and (8) with the help of equation (14) gives

aZ v= o ( a2 v= 1 aZ v, 1 i avr Iav= 4 a2 v= l P~ a-V- = t' ~ _ a-;r+ 3 a-7~; + 3 7 ~ + r T~ + 3-~/ ) +

+ poCg ~,7~ + a--~ + - ~ ) '

(13)

(14)

(15)

aZv, a[32v~ lazy= 4[a2v, lav, o,\] P ~ = la -~t [-~z2 + 3 a'ff~r + -3 [-~r2 + r Or -~ ) J +

(lay_., v, a~v, a~v= I + P o ~ r ar r~ ~a-~-+a-7~z/�9 (16)

2.2. TIlE SOLUTION

For a sinusoidal wave, if the input is only axial, the steady state solution would be of the form

v= = VJr)e tt~r-k=), (17)

v, = V,(r)el(~"-k=L (18)

Substituting for v, and v, from equations (17) and (18) in equations (15) and (16), and cancelling e l(of-k') from both sides, gives

" t r d r r 2 + E2 V , - F 2 i k - ~ r = 0 ' (20)

where

4 ipo c 2k 2 ipo Co 2 k 2 C"

- "3 k2 + ~ /Leo (21)

ik poc~k D z = - - , (22)

3 peo

_ipeok 2 + eoz Po E 2 = (23)

i ~ / l e o + Po Co 2 '

510

F rom equations.(19) and (20),

where

SHRI KANT E T A L .

F2 _ ipoJ]3 + Po c2o

i~pco + Po co z " (24)

(d2 ,d )(d2 ld ) ~r~+;~+Cf ~-77:+7Tr+C~ ~5=0, (25)

l d , )(d2 ,d 1 ) -t I- C~ h + § Cz z V, --- O, (26)

r dr r 2 dr = r dr r 2

C~ + C~ = E 2 + C 2 - i k D 2 F 2, (27)

2 z E 2 C 2 Cl C2 = �9 (28)

Equat ions (25) and (26) are in s tandard Bessel form and their solutions would be

V, = A t So(C, r) + A2 So(C2 r) + As Yo (C, r ) + A, Yo(C2 r), (29)

V, = B, J,(C, r) -b B 2 Jl(C2 r) + a 3 Y , ( C , r ) + B, Y , ( C 2 r), (30)

but as r ---> O, Yo(Clr ) , Yo(C2r), Y I ( C I r ) and Y l ( C 2 r ) --->- - ~ . Hence A 3 = A, = B3 = B4 = 0. Solutions of equat ions (25) and (26), therefore, become

It'= = A 1 Jo(C* r) + A2 Jo(C2 r),

V, = Bt JI(C1 r) + B 2 J1(C2 r).

(30

(32)

By substituting solutions (31) and (32) in equat ion (20), one finds

BI = ml d l ,

B2 = m2 A2~

where

i kC l m l = C ~ - E " '

(33)

(3,1)

(35)

For most liquids

ikC2" m2 = C~ - E z "

Po Co z ~ I Ito9

(36)

(37)

and, therefore, f rom equations (21) to (24) (7, D, E a n d F c a n be approx imated as

C ' = ip o c~ (k 2 _ k2o) ' pm (38)

WAVES IN BRANCtlED IIYDRAULIC PIPF_,S

po cg k D2 ~ ~

Itto

511

(39)

E 2 = ko z, (4o)

F 2 ~ I .

Equations (27) and (28) then change to

(40

ipo to c , ~ + c ~ = - ~ ,

/1 (42)

-~"2 ~=r" -_ l p o t o ( k ~ _ ko2) '

P whence

C~, C~ = ipo/tto u p2

and, with the help of inequality (37), one obtains

4ipotol~ (M-ko2)} '/']

(43)

(44)

c ~ = - - - ipo to 7

/.t (45)

c I = kg - k s.

(72 is sufficiently small, and therefore, for pipes of small diameter,

(46)

Solution (31) becomes

Jo(C'2 r) = 1,

Jl(C2 r) = 0.

I/, = Ai Jo(Cl r) + A2.

Upon using the boundary condition F, = 0 at r = a, solution (49) becomes

V, = Al[Jo(Cl r) - Jo(C, a)].

(47)

(4S)

(49)

(50)

Solution (32), with the help of relations (48), (33) and (35), becomes

ikAl V,= JI(C~ r). (51)

C1

When o, and o, are substituted in the continuity equation (14), one obtains

pocgk p = - ~ AI Jo(Cl a ) e tt'~t-~. (52)

to

Interestingly, the pressure, p, is independent of radius whereas v= and v, are not. Qualitative profiles ofp, v, and v, are shown in Figure 1.

Incidentally, profiles of pressure and axial particle velocity tally with the ones given in the cited existing literature [2, 3] which, however, is too deficient in mathematical rigour and

512 SHRI KANT ET AL.

I !

!

in) (b) (c)

Figure 1. Profiles of(a) axial velocity, (b) radial velocity and (c) pressure, at some cross-section of the pipe.

logic. In these references the radial component of the particle velocity is omitted from the analysis and for the continuity equation

~ + po cg ~2 = 0 (53) dz

is used, which clearly shows thatp and v2 are similar functions of radius. Despite this pressure is presumed to be independent of radius, owing probably to experimental observations. The reason for such an anomaly is that v, and/~ are very much coupled and it is not possible to account for viscous effects without considering o,. Continuity equation (53) is valid only for inviscid fluids.

The volume velocity in the axial direction is

[2~ ] V= ~ vz2nrdr= ~ze't~'-k:) A, -~l J'(C* a) - a2 j~ a) "

0

The characteristic impedance is

(54)

P Y = - . (55)

P

Substituting from equations (52) and (54) forp and v in equation (55) gives

Po c~ kJo( C l a) r' = ( 5 6 )

Co/z a 2 Jo(C, a) -- ~ J r ( e l a)

If c is the velocity of wave propagation in the viscous liquid then

Po c Po c

cross-sectional area of the pipe ~za 2" Y= (57)

WAVES IN BRANCItED tlYDRAULIC PIPES 513

Comparing equations (56) and (57) gives

C ~ cg kJo(C1 a) a 2

~ [ a 2 J o ( C l a ) - ~ J x ( C ~ a ) ] " (58)

The complex wave number k = o~/c is

or

or

o'[o'':,a, k =

cgkJo(Cl a) a 2

kZ = k Z [ 1 2J,(C, a) ] Cx aJo(Cl a) ] '

(59)

(60)

2JI (CI a) ]1/2

k = + k o 1 CaaJo(Cla)J " (61)

Substitution o f k from equation (61) in equation (56) results in

[ 2Jl(C,a) ] -'/2 Y = + P~ C...--.-~~ 1 - na 2 Cl aJo(Cl a)

(62)

It can be shown that Jl(Cla)/Jo(Cla) = - i for values of [Cla[ > 10 and equations (61) and (62) will then simplify to

[ 1 k = + k o - , a a

(63)

y = + p o c o 1 + - - �9 (64) - na 2 a 2pco a

]'he two values of k and Y correspond to forward and reflected waves, respectively, and the ~eneral solution for the pressure, p, and the volume velocity, v, will be

p = (A e -lk-" + Be Ik-') e t'~', (65)

k" ,66, �9 vhere A and B are the amplitudes of the forward and reflected pressure waves, respectively, .tt z = 0 .

A generalized analysis of one dimensional systems like branched hydraulic pipes may conveniently be carried out in terms of transfer matrices [5]. With respect to pressure and

514 SIIR! KANT E T A L .

volume velocity as state variables the transfer matrix for a uniform pipe would be

cos(kl) i Ysin(kl)"

ySin (kl) cos(k/) , (67~

where i is the length of the pipe, k is the complex propagation constant given by relation (63), and Y is the characteristic impedance given by the relation (64).

3. ANALYSIS OF A BRANCIIED SYSTEM OF PIPES

Analysis is very simple when there is only a single pipe and all the properties of the fluid and the dimensions of the pipe are known. For a simple branched system of three branchc~ and one junction, Streeter and Wylie [6] have given a computer program which makes use o( the characteristics method. The method is good enough for a system with only a few branches. When the number of branches increases, this method is not very efficient. Streeter and Wyl~ [6] have also suggested an impedance method for a simple system of three pipes with one junction, when all the end-impedances except that of the excitation end are known.

Thus, the present literature lacks a general algorithm which could be used to find the response at any point for a given input at some branch of a general hydraulic system. In thi~ section, a general algorithm in the form of sub-Pr0grams for a digital computer is presented for the purpose. The algorithm can also be adapted for an inverse problem where the input is to be evaluated for a given desired output at some end. The algorithm has been used to determine the required input at the master cylinder for the desired output at the wheel cylindcr of the automotive pumping brake system.

3.1. SUBSCRIPTING OF THE BRANCHES Subscripting of the various branches is illustrated in Figure 2, where one typical s~stem

has been subscripted in two possible ways. It may be noted that in Figure 2(a) subscripts 1

3 ~ .> 9 (a)

Figure 2. Subscripting the branches of a system.

/ 8

(b)

and 2 as well as 4 and 5 are interchangeable whereas 6 and 7 are not. Also the branch sub- scripted as 7 in Figure 2(a) cannot be subscripted as 1.

3.2. EQUIVALENT IMPEDANCE OF A TERMINAL BRANCH

For a pipe with characteristic impedance Y, length 1 and propagation constant k, the transfer matrix already has been given as expression (67). The analog circuit for the pipe and its end impedance, Z, is given in Figure 3. By the transfer matrix approach, for state

WAVES IN BRANCIIED tlYDRAULIC PIPES

: _r_, ~ j _ _ _ . _ [ ~ ~,,, " t- t

"i~ (a) (b)

Figure 3. (a) A terminal branch; (b) analogous circuit. D, Lumped element, i.e., end-impedance; distributed element, i.e., the pipe.

variables pressure and volume velocity,

~- - - [ -~ s'n<''> c~ ~, .,.o

[.cos(k/) Zco. s(kl)+iYsin(kl)][O]

=[~sin(k/) Z--~ySin(k/,+cos(k/)JLv, j " The equivalent impedance, ~, is given by

P2 Zcos(kl) + i Ysin(kl) v z i

Z ~- sin (kl) + cos (kl)

515

(68)

(69)

3.3. Tim ALGORITtlMS The two algorithms, which are readily convertible to FORTRAN IV statements, are given

| < J =(I-2),(I-3), ..., I ~<---n +- Z ( i ) = Zeta ( [ - J ) * Z e t a ( J )

Zeta ( I - - J ) + Z e t a ( J ) Kode ( I = l ) = 0

Kode ( J ) = O

.I Z e t o ( 1 ) = Z ( I I , c o s l K L ( l l ) + i y ( I ) * s i n l K L . ( I I ) J

( i / Y ( I I ) * Z ( I ) * s i n (KL(I) I+r Kode ( I ) = I

Figure 4. Flow diagram for the sub-program SUBROUTINE EQUIMP (N, Y, K L, Z, K , ZETA).

516 SIIRI KANT ET AL.

_ ~ r . IVel(N) = I

F --.I Presur (N) FI >l -- L I J Zeta (N) �9 Ve (N) J

' I""-Zeta (Jl*Vel(L) L ve [M)-zeta (M)4-Zeta(d) Vel (d)= Vel (L) -Vel (M) J ~1 Presur (M) = Presur (LI i Call PEEVEE Presur (d)=Presur(L) I~ I (L,Presur,Vel) I"

) = 0 ~ Call PEEVEE ( M, Presur, Vel)

Call PEEVEE (d, Presur, Vel)

1

. . . . .

L = lcode ( I , I ) M= [code (2,[) J= Icode (3,1)

(a)

Save . Presur (d) * cos (KL(J)) - iY(J)*Ve l (d).sin (KL(J))

Vel (J)=-( I /Y{J) )*Presur(d)*s in (KL(d)) + Vel (d).cos (KL(J) )

Presur (J) = Save

(bl

Figure 5. Flow diagrams for the sub-programs. (a) SUBROUTINE RESPON (N, K, NN, ZETA, KL, Y, PR~UR, VEL); (b) SUBROUTINE PEEVEE (J, PRESUR, VEL).

2 ~ -Z2 lc )

(a)

~.k/3

I

(bl

Y, k~

r~,k/3

(d)

(el

Figure 6. Calculation of the equivalent impedance. (a) Branched system; (b) analogous circuit; (c), (d) and (e) three stages during the computation of the equivalent impedance.

WAVES IN BRANCilED IIYDRAULIC PIPES 517

in the form of flow diagrams in Figures 4 and 5. The first algorithm finds the equivalent impedance of the systemat the excitation point and also stores intermediate values of various equivalent impedances. The second algorithm finds the response of the system for the given input. The two algorithms given as sub-programs have been named as EQUIMP and RESPON. Description of the parameters thereof is given in Appendix I.

With all the end impedances, characteristic impedances and lengths of the pipes known, the equivalent impedance of the system at the input end is calculated by means of the first algorithm. If the input is known in terms of pressure, the volume velocity at the input end can be calculated making use of the equivalent impedance; similarly, if the input velocity is known, the input pressure can be calculated. Knowing the pressure and velocity at the input end, one can calculate pressure and velocity at all the junctions by means of the second algorithm which makes use of the transfer matrix technique. For example, for the branched system of Figure 6(a), the equivalent impedance is calculated step by step as shown in Figure 6t. For getting the pressure and velocity at the ends, the process is reversed. In this case the pressure and velocity are calculated beyond the distributed element. Using continuity of volume velocity and pressure at pipe junction, the velocity is further distributed in the inverse proportion of ~x and ~2 and so on. The two operations are affected by means of the two subroutines EQUIMP and RESPON, respectively.

3.4. SOME COMMENTS ON TIIE ALGORITIIMS The algorithms are not valid if a junction has more than three branches. Nevertheless, it

does not reduce the generality of the algorithms as such a junction can be broken into two or more junctions, each with three branches, and with zero length kept for some of the branches. Figure 7 shows how a junction having four branches can be converted into two equivalent junctions, each of them having only three branches.

0 0 ~I A2

(o) (b)

Figure 7. Handling of a junction with more than three branches. (a) A junction with four branches; (b) the required modification of the system. At A, = 0.

The algorithms have been prepared primarily for the use of a digital computer. Execution time increases linearly with the number of branches. Total execution time of the two sub-" programs EQUIMP and RESPON on an IBM System 360/44 for a system with, say 31 branches, with double precision complex arithmetic, is of the order of one second.

It is recommended that double precision arithmetic be used when real and imaginary parts of the end-impedances are not of the same order.

When any of the ends is closed, its end-impedance may be represented by a "sufficiently large" number. It is immaterial whether such a number is real, imaginary or complex. Overflow, during computations, can be avoided by modifying the expression for calculating G in the sub-program EQUIMP as follows:

COS (KL(1)) + i(Y(I)/Z(I))SIN(KL(I)) ZETA(I) = (i/Y(I)) SIN(KL(I)) + COS (KL(I))/Z(I)" (70)

t The peculiar way in which a branch with an end impedance has been represented in Figure 6(b) has been adopted for convenience. It is not, of course, the conventional way.

518 SHRI KANT ET AL.

4. CONCLUDING REMARKS

In this paper, the exact governing equations, including both axial and radial components of the velocity, for waves in a pipe containing a viscous medium have been identified and the solution for this two dimensional case has been attempted. It has been shown that in some of the existing literature the momentum and continuity equations for the waves in a pipe with viscous medium have been over-simplified. The radial component of the particle velocity has been neglected while viscous effects have been considered. This leads to an anomaly inasmuch as the resulting continuity equation shows that pressure and axial velocity are similar functions of radius, which is not true in practice. Despite all this, interestingly enough, the values of wave propagation constants and characteristic impedances are about the same as those determined in this paper by a more rigorous analysis of the problem.

The algorithms suggested here in the form of two sub-programs are general in application and can be utilized for any branched system of pipes where response at various points needs to be calculated for a given sinusoidal input. The transfer function of the system, which is very useful for both direct and inverse problems, can be determined by giving a unit input. These algorithms can be used as a powerful tool in the analysis of a system of hydraulic pipes, a typical example of which is an automotive pumping brake system--a potential anti-skid device.

REFERENCES

1. R. J. ALFREDSON and P. O. A. L. DAVIES 1970 Journal of Soundand Vibration 13, 389--408. The radiation of sound from engine exhaust.

2. L. E. KINSLER and A. R. FREY 1950 Fundamentals of,4coustics. New York: John Wiley and Sons. See pp. 232-242.

3. R. W. B. STEPHENS and A. E. BATE 1966 ACOUStics and Vibrational Physics. London: Edward Arnold (Publishers) Ltd. See pp. 677-682.

4. S. W. YUAN 1969 Foundations of Fhdd Mechanics. New Delhi: Prentice Hall of India Limited. See pp. 115-123.

5. M.L. MUNJAL, A. V. SREENATH and M. V. NARASIMHAN 1973 Journal of Soundand Vibration 26, 173-191. Velocity ratio in the analysis of linear dynamical systems.

6. V. L. S'rRE~rErt and E. B. WYLtE 1967 Hydraulic Transients. New York: McGraw-Hill Book Company, Inc. See pp. 38-39, 109, 282-283, 301-302.

APPENDIX I: DESCRIPTION OF PARAMETERS USED IN SUBPROGRAMS "EQUIMP" AND "RESPON"

N total number of branches NN number of branches other than the terminal ones

K an integer array of dimension N. K~ = 1 if ith branch is a terminal one: K~ = 0 if ith branch is not a terminal one. Values of all the elements need to be supplied a working integer array of dimension N an integer array of dimension (3 x NN). It is used in "RESPON". All the values are supplied by the user. It contains information about the net of branches. First row of NN elements contains subscripts of all the non-terminal branches in the descending order. Elements ICODE(2,J) and ICODE(3,J) are the subscripts of two branches in which the branch ICODE(I,J) is sub-dlvided. Position of the elements of the second and third row, but of the same column is interchangeable a complex array of dimension N. Values are calculated and stored during the execution of "EQUIMP" and are used in "RESPON" a complex array of dimension N. It is meant for storing end-impedances Z. If K~ = 1, value of Z~ must be supplied a complex array of dimension N. It stores values of the product of propagation constant and length of pipe for a particular branch. It is supplied by the user

KODE ICODE

ZETA

Z

KL

WAVES IN BRANCHED t l Y D R A U L I C PIPES 519

Y a complex array of dimension N. It stores values of characteristic impedance for a particular branch. It is supplied by the user

PRESUR a complex array of dimension N. On return it contains pressure on the output side of each branch

VEL a complex array of dimension N. On return it contains velocities on the output side of each branch

Note: Non-zero value of either PRESUR(N) or VEL(N) must be supplied as an input.

APPENDIX II: NOTATION

a internal radius of the pipe e velocity of wave propagation in the viscous fluid

Co velocity of wave propagation in the ideal fluid i

k complex wave propagation constant ko k for non-viscous fluid, ~/eo

1 length of the pipe Po pressure of the undisturbed fluid p ' fluctuating pressure p total pressure, Po +P"

p~ pressure at the output end of ith branch r radial co-ordinate t time co-ordinate

rz axial velocity v, radial velocity v0 angular velocity v~ volume velocity at the output end of ith branch Vz Vz(r), axial velocity of fluid particles at radius r, at z = 0, t = 0 1I, V,(r), radial velocity of fluid particles at radius r, at z = 0, t = 0 X~ body force in axial direction X, body force in radial direction Y complex characteristic impedance of the pipe, defined with respect to volume velocity z axial co-ordinate Z end-impedance of pipe 0 angular co-ordinate ~u coefficient of viscosity

Po densityofundisturbed fluid p ' fluctuations in density p density, po + p ' a~ circular frequency ~" equivalent impedance