Numerical solutions of viscous flow through a pipe orifice al low Reynolds numbers

8/9/2019 Numerical solutions of viscous flow through a pipe orifice al low Reynolds numbers

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133

NUMERICAL SOLUTIONS OF VISCOUS FLOW T H R O U G H A

PIPE ORIFICE

AT

LOW REYNOLDS NUMBERS

By R. D. Mills*

Numerical

solutions

of the Navier-Stokes equations have been obtained in the low range

of

Reynolds numbers for steady, axially symmetric, viscous, incompressible fluid flow

through

an

orifice in a circular pipe

with

a fixed orificelpipe diameter

ratio.

Streamline

patterns and vorticity contours are presented as functions of Rcynolds

number.

The

theoretically determined discharge coefficients are in good agreement with experimental

results of Johansen 2).

INTRODUCTION

THE PLATE ORIFICE

has long been used as a device for

measuring flow-rates in pipes. Various specifications of

orifice shape and positions of the pressure tappings have

been put forward as ‘standard’ (e.g. reference r)t) .

The most convenient for theoretical investigation is the

orifice with ‘corner tappings’ since this allows different

diameter orifices to be compared on the basis of Reynolds

number in any given pipe. At high Reynolds numbers with

fully developed turbulent flow the coefficient of discharge

of the pipe orifice is well-established experimentally to be

nearly constant. However, at low Reynolds numbers under

conditions of viscous flow appreciable variation in the

value of this coefficient has been observed (e.g. reference

2)). This is due to the marked dependence of the flow on

Reynolds number at low values of this parameter.

It

has therefore been the purpose of the present paper

to investigate theoretically the nature of the flow through

an orifice of simple geometrical shape at low Reynolds

numbers. Streamline patterns and contours of constant

vorticity have been obtained as functions of Reynolds

number. The calculated discharge coefficients are in good

agreement with experimental results of Johansen 2) for

an orifice having corner tappings, even with the differences

in geometry which had to be permitted to render the

problem tractable from the computational point of view.

(The orifice used by Johansen was bevelled by 45” on its

downstream face, whereas the present calculations refer

to a square-edged orifice.)

It

has thus been necessary to solve the Navier-Stokes

equations numerically in the low to intermediate range of

Reynolds numbers for axially symmetric, incompressible,

The

MS.

of this paper was received at the Institution orz 16th

October 1967undaccepted

for

publication on2 Yth hrovember

1967.

2

*

Engineering Luborutory, Cambridge University.

f

R e f m w e s are

given

in the Appendix.

J O U R N A L M E C H A N I C A L E N G I N E E R I N G

S C I E N C E

viscous flow. The foundations of this type of work were

laid nearly 40 years ago by Thom (3). Considerable

attention is currently being given to this type of problem,

including time-dependent flows. Comparatively little

work, however, has been done for axially symmetric flow

even in the time-steady case, despite the importance of

such flows in engineering applications. Th e first numerical

solution for axially symmetric flow was that for creeping

flow through a sudden expansion in a pipe, given by

Thom 4) in 1932. Later work on flow with axial sym-

metry has been reported by Jensen

( 5 )

and Lester

6).

The method of solution employed is the ‘two-field’

method of Thom: replacement of the fourth-order non-

linear partial differential equation for the stream function

by two second-order simultaneous equations for 4 and

the vorticity

7;

these equations are then replaced by their

simplest finite difference equivalents. In the case of two-

dimensional flow, this coupled pair of finite difference

equations is now well known to exhibit superior conver-

gence properties compared with finite difference forms of

the original fourth-order equation when iterative methods

are employed for their solution; moreover, the boundary

conditions are more easily treated in rhis formulation of

the problem. There is little doubt that similar conclusions

hold for flow with axial symmetry.

Notation

D

CL

curl

d

grad

H

h

Radius and diameter of pipe (Fig. l}

Coefficient of discharge

of

orifice (equa-

Vector ‘curl’.

Diameter or orifice

Fig. 1).

Vector gradient.

p+9pqz , total head.

Finite difference mesh width.

tions (19), (20)).

Vol

10No 2

I968

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2/8

134

L ,

u

P

P o

Q

q

=

u, u, w )

Re

Reo

rJ

8J

t

r =

( 8 , b 51

P

V

P

0 2

X

Superscript

k

Subscripts

i i

R. D. MILLS

Characteristic length and velocity.

Static pressure of fluid.

Upstream pressure at section AC in

Poiseuille flow (Fig. 1 and equation

(21)).

Volumetric discharge rate.

Velocity vector.

w,a/v, Reynolds number based on pipe

radius and maximum pipe speed (Fig. 1).

G0d/v, Reynolds number based on orifice

diameter and mean speed I? through

orifice (as used by Johansen 2)).

Cylindrical polar co-ordinates.

Time.

Vorticity vector.

Viscosity coefficient of fluid.

Kinematic viscosity coefficient of fluid.

Density of fluid.

Stokes stream function.

Laplacian operator.

Scalar product.

Vector product.

Number of iterations.

Refer to location of mesh points in the

usual sense of matrix notation.

N.B. Field variables, operators and the mesh width are

made dimensionless according to the scheme of

equation

3).

Primed quantities are dimensional and

unprimed quantities dimensionless.

GOVERNING

EQUATIONS

In the usual notation the equation of steady, viscous, in-

compressible fluid flow is

1)

(q’.V’)q’

=

--grad’p’+vVf2q‘

P

where

q‘

is the velocity vector,

p’

is the static pressure,

p

the density, v =

p i p

the kinematic viscosity coefficient of

the fluid, and primes signify

dimensional

quantities. If the

variables and operators are made dimensionless with

respect to a representative length

L

and velocity U , then

equation 1) becomes in

dimensionless

form

1

(q.’?)q = -ggradp+-Vq

e . .

2)

where the Reynolds number of the motion is Re =

UL/v.

Take cylindrical polar co-ordinates (r , 0,

z

with velocity

components (u, v, w ) and vorticity components t,

,

5 .

Identify L with the pipe radius a and

U

with the maximum

velocity

w ,

in Poiseuille flow (Fig.

1).

The full non-

dimensional scheme is then

] 3)

r

=

r’/a z

=

z’ia

h =

h’/a u

=

u’iw, etc.

+ = f / w , a 2 .$

=

. ’a/w, etc. p = p‘/pwm2

whereupon the Reynolds number is defined naturally as

Re

=

w,a/v;

h’ is the mesh width. Henceforth all

un-

primed quantities are dimensionless.

For motion with symmetry about the z-axis the velocity

and vorticity vectors reduce to q

=

u, 0,

w )

and

r

=

(0,

7,

0)

respectively where

au

aw

”G 5

.

.

.

’

4)

If the Stokes stream function is introduced by

Fig. 1

T O U R N A L M E C H A N I C A L E N G I N E E R I N G S C I E N C E

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No 2 1968




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NUMERICAL

SOLUTIONS OF

VISCOUS FLOW

THROUGH

A PIPE ORIFICE

AT


135

then the equation of continuity,

L'u aw u

-+-+-=o . . . .

ar az

r

is satisfied automatically, and it can be shown (by taking

the 'curl' of both sides of equation

2)

thereby eliminating

the pressure) that the equations of motion reduce to the

pair of simultaneous equations

(7)

1 c a q la*&]

r

i.z Cr

r

8 r t k r2

Cz,

. .

(8)

Expression of equation 2) in the form

1

qxq

=

grad(p+:q2)+zcur11; . 9)

and resolution of components in the r and z directions

leads to the following two integrals relating to the total

head H = p + + q 2 :

Once the functions

I/

and

7

have been calculated equations

10) and 11) allow the pressure at any point in the flow to

be determined relative to some datum.

FINITE DIFFERENCE E QUATIONS

A

portion of the square mesh used for the finite difference

scheme is shown in Fig.

1.

Mesh points on the solid

boundaries will be referred to as boundary nodes and the

others as field nodes. By use of the simplest central-

difference formulae the stream function and vorticity

equations (7) and (8) can be written in the following

iterated forms:

*:., = ~ ~ ~ ~ ~ ~ l + ~ ~ - l * ~ + ~ ~ . ~ - l + ~ ~ ~ ~ . ,

h

2r,

:-

1 1

FF1'.J>-h t?lk

1

12)

where k is the iteration number. Apart from a difference

J O U R N A L M E C H A N I C A L E N G I N E E R I N G S C I E N C E

of notation these equations are identical with those given

by Lester

p. 5,

reference

(6) ) .

BOUNDARY CONDITIONS

To

satisfy the condition of no slip at solid walls the

normal and tangential gradients of must vanish at these

boundaries. The tangential conditions are satisfied by

setting + = constant along these boundaries. Special

boundary formulae have to be developed, however, for

the normal conditions. In view of the axi-symmetric

nature of the flow it is no longer possible to use the same

boundary formulae for both vertical and horizontal walls

as can be done for two-dimensional flow referred to a

rectangular co-ordinate system. Each direction must be

treated separately, or more generally, as Thom and Lester

have done, by considering a boundary wall at inclination.

Essentially, they expand both + and

7

in Taylor series

about their boundary values at the wall and utilize the

governing equations to provide expressions for the higher

order derivatives. I t will suffice here to quote the results;

for a full derivation the reader

is

referred to Lester's

paper ( 6 ) .

6

2

* ~ ~ I , ~ - ~ ~ , J ) - ~ 2 * l ~ ] ~ ~ l , ~

r . 1 horizontal wall 15)

1171 3

h

hZ

2F---

r, 43

(upper sign

=

outer' wall, lower sign = inner' wall)

3

r,h

72.1 = *~,~~1-***,)-~17~.~*1 vertical walls

* 14)

Equations

15)

and

(16)

are accurate to order

h3;

equation

15)

is in fact exact for Poiseuille flow, as can easily be

checked. These boundary values are determined by

iteration as the solution proceeds.

The boundary conditions are such that a parabolic

velocity distribution is prescribed at

a

section A C a short

distance upstream of the orifice and also again 'far down-

stream' a t section

BE

(Fig.

1).

In reality the downstream

reversion to Poiseuille flow at low Reynolds numbers is

achieved in an asymptotic manner (as also is the change

upstream). For these numerical computations, however,

the author adopted the following approach: take the sec-

tion at a position which seems sufficiently far downstream

physically (actually, some guide in this matter can be

obtained from a study of the photographs in reference

2));

then repeat the calculations at a section farther downstream;

if

no change in the eddy structure at the orifice is observed

then the first position is regarded as 'sufficiently far down-

stream'. This sort of downstream boundary condition of

course becomes invalidated at high Reynolds numbers

when ultimately turbulence sets in.

The sharp intruding corners at F,

G

(Fig.

1)

were

treated in the manner suggested by Thom. There are

assumed to be two values of the vorticity at the boundary

nodes F, G. These are calculated from equations 15),

16)

utilizing the values of the vorticity at

K, N

and at

V d 10

NO

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4/8

136 R. D. MILLS

My S respectively. When the 'diamond' in the iteration

process is centred on

K,

M then the first pair of values of

the vorticity is used and when centred on N,

S

the second

pair of values is used. I n this way the effects of the corners

are forced to spread into the fluid

in

both the Y and

z

directions-as must occur physically when the fluid

'passes round the corners'.

ITERATION TECHNIQUE

The iterative routine used to solve the system

of

equations

(12)

and

13)

and boundary conditions (15) and (16) was in

the main the basic technique used by Thom (p. 115, ref-

erence

7)).

Many variants are possible but the present

author has found that a single iteration of the +field, setting

the boundary valuesby equations(15)and (16), followed by

asingleiterationof the 7-field was the most satisfactory 8).

Generally, the boundary values of vorticity were given one

half their 'theoretical' movements in each iteration. To

start the solution it is necessary to have a guessed or trial

solution at each node.

This

author started simply with

all field nodal values zero and the boundary nodes given

their Poiseuille flow values. Once the solution was ob-

tained for a given Reynolds number, then this was used

as the starting solution for the next higher value of Re.

This seems a satisfactory enough procedure, though other

workers in this field have found that greater stability

(leading to a higher attainable Reynolds number) can be

obtained by taking into account the aji't terms in the equa-

tions, and stepping on from a solution at time

t

to one at

time t+ by one of the standard procedures for diffusion

problems (e.g. references

( 9 )

and

10)).

Stability and convergence

Thom and Apelt

Ir)

have studied the effect of introduc-

ing a small disturbance into a two-dimensional vorticity

field. They have shown for a square cell that the distur-

bance will not grow in magnitude provided

where

L

is the characteristic length of the problem.

Lester

12)

nd Mills 8) have generalized this work to

include a rectangular cell, Lester considering also the

effect

of

introducing a disturbance into the -field. Lester

concludes that a stability criterion based on the latter field

is less stringent than on the vorticity field. Lester also gives

procedures for obtaining 'optimum' convergence. It has

not yet proved possible to derive a similar simple criterion

for axi-symmetric flow, but it has been found that condi-

tion (17) does in practice give some indication as to when

divergence is likely to occur. It may be worth observing

here that Fromm 13) has given a stability analysis of the

full time-dependent equations, showing that simultaneous

with a condition like (17) St'lh'2

Q

1 4 v must hold.

The following two empirical criteria were utilized for

terminating the iteration process

h'/L

<

%@]Re . . . (17)

where and irirnl are the largest values of

4

and 71

occurring in the undisturbed Poiseuille flow. The fields

were given

20

iterations and then tested every subsequent

10 iterations.

VERIFICATION O F

THE

METHOD

As a check of the above methods tests were made on

Poiseuille flow in a circular pipe. Mathematically this flow

represents an exact solution of the Navier-Stokes equations

for arbitrary Reynolds number, though physically the

transition to turbulence begins at a Reynolds number

based on mean speed and pipe diameter of about 2300.

Two different starting solutions were used: in the first the

field values were set equal to zero, and in the second

starting values appropriate to a linear velocity profile were

used. A 'square' section was used for each test, that is,

equal numbers of

i

and j-steps were taken. The results

quoted refer

to

the mid-plane.

A study of Table

1

will show that there is some advantage

to be gained by starting with a closer approximation to the

ultimate solution when using the smaller mesh. As a

further investigation, it might be worth while applying the

extrapolation and empirical error tests utilized by the

author in reference 14)n connection with the boundary

layer equations.

VI SCOUS FLOW THROUGH

A

The computational method outlined above was used to

determine solutions of steady, incompressible, viscous,

axi-symmetric flow through a square-edged orifice in a

circular pipe.

A

mesh

of

width

h'

=

a116 was used, this

being sufficiently small for resolving the details of the flow

in the corners, while the orifice wall was made one mesh

width in thickness (Fig. 1). The resulting flow patterns

are depicted in

Figs

2-6 for Reynolds numbers Re,

=

0-50

and fixed diameter ratio d / D = 0.5. Also presented in these

figures are the contours of constant vorticity. (For com-

parison purposes, the Reynolds number used by Johansen,

Reo,

is

occasionally used instead of the present Reynolds

number Re. For d / D= 0.5, Re, = 2Re.)

Fig. 2 shows that even for the 'creeping' (Re

= 0)

motion two corner eddies exist upstream and downstream

of the orifice. As the equations of motion for Re = 0 are

symmetrical in

z

(equations 7),

(8))

the solution should

show symmetry about the mid-plane of the orifice. This is

confirmed in the computational solution, with its near-

perfect symmetry (Fig.

2).

It is thus immaterial whether

the flow is from left to right or vice versa

at Re

=

0.

This

condition will never actually be realized in practice though

creeping motion is possible with Re arbitrarily near to zero.

As the Reynolds number is increased (Figs 3-6) the

downstream eddy lengthens and increases in size. Simul-

taneously with this process the upstream eddy shrinks in

size and at Re, = 50

it

is very small indeed. Note also the

characteristic stretching of the vorticity contours in the

direction of motion

as

the Reynolds number increases.

SQUARE- EDGED ORI FI CE

J O U R N A L M E C H A X I C A L E K G I N E E R I N G S C I E N CE Vo110 No 2

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NUMEKICAI, SOLUTIONS O F

VISCOUS

FLOW THROUGH A PIPE ORIFICE AT LOW REYNOLDS

NUMBERS

137

Table 1.

Numerical solution of Poiseuille f l o w in a circular p e

(I) Zero

field

values as

starting

solution.

11)

Field values correspondingto

a linear velocity

profile

as starting

solution.

7 . . o 0.25 050 0-75 1 oo

1 h Re

k

-values

8 10

20 0

-0.030790

-0.110 492

-0.202 992 -0.250 000

8

10 100

0 -0.030528

-0.109 747 -0.202

381

-0.250

000

16 10

50

0

-0.029560 -0.108 316

-0.202 482 -0.250 000

16 10

100 0 -0-030441

-0.109 739

-0.202 388 -0.250 000

8

20

20 0

-0'030400 -0-109 448

-0.202 254 -0.250

000

8 20

100

0

-0.030540

-0.109 776 -0.202 402

-0.250 000

16 20

50 0

-0.028097 -0.104 324

-0.199 431 -0.250 000

-0.202 141 -0.250 000

6 20 100

0

-0.030277 -0.109 342

8

8

16

16

8

8

16

16

R

8

16

16

8

8

16

16

8

8

16

16

8

8

16

16

10 20

10

50

10 50

10 100

20 20

20

50

20 50

20 100

0

-0.030 606

0

-0.030528

0 -0'031 226

0

-0.030 342

0 -0.030616

0

-0.030540

0 -0.031 036

0

-0.030 359

-0.109 935

-0.109 747

-0.1 11 609

-0.109 477

-0.109 967

-0.109 776

-0.111 160

-0.109 524

-0.202 469 -0.250 000

-0.250 000

0.202 381

.203 520 -0.250 000

-0.202 197 -0.250 000

-0.202 512 -0.250 000

-0.202 403 -0.250

000

-0'203 236 -0.250

000

-0.202 235 -0.250 000

Exact 0

-0.030273 -0.109 375 -0.202 148 -0.250 000

?-values

I0

20

0

0506 089 1.007 39 1

1.487 303 1.939 197

10 100

0

0.499 344

0.997 059

1.492 019 1.984

150

10 50

0

0,576 993 1.157 902

1.643015 1.894 199

I0 100

0

0.501 253 0.999 953 1-493330 1.985 105

20 20

0

0.501 131 1.003 136 1,503 257 1.982 480

1.982 723

0 100

0

0.500 547 0.998 756

2.084 726

0

50

0

20 100

0

0.500 113 1 000 845 1.501 477 1.997 151

1.492 660

0.499 132 1.042 35 1 1.635 715

10 20

0

10

50

0

10 50 0

10

100 0

20 20 0

20 50

0

20 50

0

20 100 0

0-496 611

0.499 345

0-499 841

0.498 926

0.500

467

0500 547

0502 419

0.500 154

0.990 094 1.482 477 1.984

138

1.492 021 1.984

148

.997 062

0.983 419 1.446 749 1.936 452

0.997 291 1.496 033 1.996 959

0.997 152 1.487 464 1.979 045

1.982 723

.998 755 1.492 660

1.950

342

,992 121 1.460 389

1.994 365

.999 246 1.496 630

Exact 0 0-500000 1 ooo 000 1

500

000 2.000 000

0

q=

1 6

0

Fig. 2.

Streamlines and vorticity contours: Re, = 0

Fig.

3.

Streamlines and vorticity contours: Re,, = 5

V d 10 h o 2 1968

O U R N A L M E C H A N I C A L E N G I N E E R I N G S C I E N C E

4




6/8

138

R

D.

MILLS

Fig.

4.

Streamlines and vorticity contours: Re, = 10

Fig.

5. Streanzlines and vo rti city contours:

Re,

=

20

c

Fig.

6. Streamlines and vorticity comours: Re,

= 50

l O U R N A L M E C H A N I C A L E N G I N E E R I N G S C I E N C E

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1968




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NUMERICAL SOLUTIONS OF

VISCOUS

FLOW THROUGH A PIPE

ORIFICE A T


139

Pressure calculations

The usual relationship for the discharge of an incompres-

sible fluid through a pipe orifice is

where

Q’ is

the volumetric rate of flow,

C

the coefficient

of discharge and

p‘ , -p ‘ ,

is the pressure drop across the

orifice. In general the coefficient of discharge is a function

of Reynolds number, diameter ratio d / D , and orifice

geometry. The pressure drop p , -p , was calculated from

equations (10) and (11) for each Reynolds number and the

coefficient of discharge determined from

since

r =

~Ll*z 3nd p’,-p’,

=

(p1-p,)pwrn2, The

positions of the pressure points used for this calculation

and the path of integration (broken line) are shown in

Fig. 1. Integration along lines containing an even number

of mesh widths was performed with Simpson’s rule while

Gregory’s formula was used for the other lines. The pres-

sure drop was recalculated in each case using a neigh-

bouring path; this not only provided an immediate check

but also a more general one on the results as

a

whole for

the pressure changes between any

two

points

in

the field

are

independent of the path linking them. The results of

these calculations are shown quantitatively in Table

2,

and

graphically in Fig. 7 alongside the experimental results

of

Johansen 2).

So

that the creeping flow results obtain in the limit

Table

2

0 0-220

2.5

0.216

0.207

0.184

0.138

I

5.0

heory

0.5

Exper imenta l

resu l ts

f r o m reference ;2)

L

0

4 6

dRec

Fig. 7 . CoefJicient of discharge as a function of Reynolds

number

Re

0,

Re(p,

-p2)

was evaluated rather than ( p ,

p2)

in

equations (10) and (11). This means Cd12/Re is logically

evaluated in equation (20).

(For

the justification of this in

exlensu see reference IS), p. 168.) The first item in Table 2

can be expected to hold in a small neighbourhood

of

Re

=

0 . It

is seen that at low Reynolds numbers the

coefficient

of

discharge is very nearly proportional t o the

square root of the Reynolds number, a result confirmed

by Johansen’s experiments for a wide range of d / D ratios.

In Fig. 8 are shown the axial pressure distributions as

functions of Reynolds number, calculated from equation

(11). Some care is needed in evaluating 87jar and

numerically on the axis, for

7

+ 0

as r

(see reference

6) for details). Also shown

for

comparison is the linear

pressure drop

of

Poiseuille flow

Note that the pressure distribution correctly becomes

parallel to this line downstream of the orifice. Once again,

Re(p-p , ) has been evaluated rather than ( p - p , ) so that

the Stokes creeping flow results are obtained as Re

.

Fig. 8 . Axial pressure distributions as functions

of

Reynolds number

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8/8

140

R. D. MILLS

The pressure minima which begin to appear just down-

stream of the orifice at about

Re

=

5

are evidently asso-

ciated with the gradual appearance, as

Re

increases, of

significant kinetic energy in the flow along the axis. The

pressure drop across the orifice, viewed as a device

incurring a ‘loss of head’, can be determined from Fig.

8.

CONCLUSIONS

Numerical solutions

of

the Navier-Stokes equations have

been obtained for axi-symmetric, viscous, incompressible

flow through a square-edged orifice in a circular pipe for

Reynolds numbers

Re

= 0-50 and fixed diameter ratio

d / D

=

0.5. It has been found that there are two eddies

symmetrically located upstream and downstream of the

orifice for Re

=

0 (creeping flow). As the Reynolds

number increases the downstream eddy lengthens while

the upstream eddy shrinks in size and become almost

imperceptible at Re

=

50.

The phenomenon of eddy-lengthening with increase

of

Reynolds number is well known for other physical situa-

tions, e.g. the flow behind spheres

( 5 )

or backward-facing

steps. It is difficult to give a simple physical explanation of

this phenomenon. However, in the present case it must

originally be connected with the shedding of vorticity

from the sharp corners of the orifice. Ultimately, a vortex

‘sheet’ arrives at the pipe wall which has insufficient con-

vective speed to enable

it

to proceed downstream against

the counterflow effect of the vorticity itself at the wall.

A

backward flow is induced at the walls and this leads in turn

to a closed recirculating region in which both diffusion and

convection of vorticity occur. As the Reynolds number

increases and convection of vorticity becomes more

important than diffusion this limiting vortex sheet is

carried farther and farther downstream and

so

the eddy

lengthens.

In all these cases, at a certain critical Reynolds number,

laminar flow breaks down into periodic eddy shedding

( K i r m h vortex street, in two dimensions) and ultimately

turbulence sets in. The critical Reynolds number in the

present case must depend on the diameter ratio d / D and

also on the detailed geometry of the orifice. Johansen’s

experimental results indicate that periodic flow does not

occur below the value

Re

= 150.

No solutions were attempted by the present methods

for Reynolds numbers

Re

in excess of

50

for the following

reasons. It is extremely difficult to disentangle the effects

of instability in the numerical procedure used to solve the

finite difference forms of the equations of motion and

boundary conditions and the hydrodynamic instability to

small disturbances of the equations of motion themselves;

and

to

ascertain to what extent these are associated with

the actual physical instability. Furthermore, it is now known

that ‘stable’ solutions obtained at high Reynolds numbers

by finite difference methods can stem from truncation

error in the finite difference representations of the con-

vective terms. The truncation error, which increases with

the mesh width, gives rise to an enhanced viscosity, i.e.

stabilizing, term. These high Reynolds number ‘solutions’

J O U R N A L M E C H A N I C A L E N G I N E E R I N G S C I E N C E

are thus open

to

question as strict solutions of the viscous

flow equations.

The discharge coefficients calculated by the present

methods show good agreement with the values obtained

experimentally by Johansen even though there was not a

complete similarity

in

regard to orifice geometry and

location of pressure tappings. Thus it seems that at low

Reynolds number these considerations are not soimportant

as the Reynolds number itself in determining the flow

characteristics.

It

might be worth while doing some further

theoretical work on different orifice geometries to confirm

this (including different

d / D

ratios), though the handling

of the boundary conditions would then present rather

greater difficulties.

ACKNOWLEDGEMENTS

The author gratefully acknowledges the receipt of the

necessary computer time for this investigation on the

Science Research Council Atlas computer, Harwell, and

on the Cambridge University Ti tan computer. H e would

like also to thank Dr L.

C.

Squire for reading the

manuscript critically. This work was made possible by the

award of

an

I.C.I. Fellowship.

APPENDIX

RE FE RE NCE S

I)B.S. 1042:

Part

1: 1964. Methods fo r the measurement of

fluid f l ow in pipes.

Part 1:

Orifice plates nozzles and

venturi tubes

1964

(British Standards Institution, London).

‘Flow through pipe orifices at low Reynolds

numbers’, Proc.

R.

SOC. 930

126

(Series A), 231.

‘The flow past circular cylinders at low speeds’,

I’roc. R. SOC. 933 141

(Series A),

651.

‘An

arithmetical solution of certain problems in

steady viscous flow’, Rep. Memo. aeronaut. Res. Comm.

1475, 1932.

‘Viscous flow round a sphere at

low

Reynolds

numbers’, Proc. R . SOC. 959 249 (Series A), 346.

‘The flow past a pitot tube a t low Reynolds

numbers’, Rep. Memo. aeronaut. Res. Comm.

3240, 1961.

Field computations n engineering

and physics 1961

(Van Nostrand).

‘Numerical solutions of the viscous

flow

equa-

tions for a class of closed flows’,J. R. aeronaut. SOC.

965

69,

714

(erratum

880).

‘Dynamics and heat

transfer in the von Kirman wake of a rectangular cylin-

der’, Phys. Fluids 1964 7, 1147.

‘A computational method for viscous flow

problems’,J.

Fluid Mech.

1965

21,

611.

‘Note on the convergence of

numerical solutions of the Navier-Stokes equations’,

Rep. Memo. aeronaut. Res. Coun. 3061, 1956.

‘Some convergence problems in numerical

solution of the Navier-Stokes equations’, Rep. Memo.

aeronaut. Res. Coun. 3239, 1961.

13) FROMM,.

E.

Article in

Methods of computational physics

1965 Vol. 3 (Academic Press).

14)

MILLS,

.D.

‘The steady laminar incompressible boundary

layer problem as an integral equation in Crocco variables:

investigations

of

the similarity flows’,

Aeronaut. Res.

Coun. 28116, 1966

(to be published in

the Rep. Memo.

series).

IS) ROSENHEAD,. Ed.) Laminar boundary layers 1963 (Oxford

University Press).

Vol I0

No 2 1968

2)

JOHANSEN,

. C .

3) THOM,.

4)

THOM,

.

(5) JENSEN,

.

G.

(6)

LESTER,

. G. S.

7) THOM,. and APELT,C. J.

(8 )

MILLS,

R. D.

(9)

HARLOW,. H. and FROMM,

.

E.

10)

PEARSON,

.

E.

11) THOM,

.

and APELT,C. J.

12)

LESTER,. . .


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Numerical solutions of viscous flow through a pipe orifice al low Reynolds numbers