AlllOO T=m73fi

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NBS

PUBLICATIONS

NBSIR 81-2308

An Initial Study of the Application

of the Numerical Method of

Characteristics to Unsteady Flow

Analysis in Partially Filled Gravity

Drainage Sized Pipes

Dr. J. A. Swaffield

National Engineering Laboratory

Center for Building Technology

U.S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

and

Drainage Research Group

Department of Building Technology

Brunei University

Uxbridge

U.K.

July 1981

DEPARTMENT OF COMMERCE

lONAL BUREAU OF STANDARDS

.U56

Ll-2308

1981

c. 2
'•A

i ^
NBSIR 81-2308

AN INITIAL STUDY OF THE APPLICATION

OF THE NUMERICAL METHOD OF

CHARACTERISTICS TO UNSTEADY FLOW

ANALYSIS IN PARTIALLY FILLED GRAVITY

DRAINAGE SIZED PIPES

H axiomAL or

LlBttAlTT JUL 3 0 1981

/y V'

Dr. J. A. Swaffield

National Engineering Laboratory

Center for Building Technology

U.S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

and

Drainage Research Group

Department of Building Technology

Brunei University

Uxbridge

U.K.

July 1981

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, D/rector
'i
Summary

#

The application of the numerical method of characteristics to the solution

of the differential equations defining unsteady flow in partially filled

drainage system sized pipes is outlined.

The derivation of the flow equations is presented, together with the necessary boundary equation formulation to represent variable inflow, system

discharge and leakage flow past a stationary deposited solid.

A computer program, written in Fortran, is included, together with typical

output, that establishes the applicability of this computational method to

unsteady flow analysis in gravity flow drainage systems.

Proposals for the extension of the described techniques to the prediction

of solid transport and flow attenuation in long pipes are also presented.

iii
Preface

This report is one of a group documenting National Bureau of Standards

(NBS) resear-ei; and analysis efforts in developing water conservation test

methods, analysis, economics, and strategies for implementation and

acceptance. This work is sponsored by the Department of Housing and

Urban Development /Of f ice of Policy Development and Research, Division of

Energy Building Technology and Standards, under HUD Interagency Agreement

H-48-'78.

Report prepared by Dr. J. A. Swaffield, Senior Lecturer, Drainage Research

Group, Department of Building Technology, Brunei University, Uxbridge, UK.

during a study leave period as a guest research worker at NBS/Stevens

institute of Technology.

Experimental results included in this report are drawn from the published

of the Drainage Research Group at Brunei University.

IV
Table of Contents

SUMMARY . . . r i i i PREFACE iv

List of Figures vi. NOTATION vii

1. INTRODUCTION 1

2. UNSTEADY FLOW IN OPEN CHANNELS j

2.1 Defining equation 3 2.2 Solution by method of characteristics 5 2.3 Application of solution to drainage flow 9 2. A Application to waste solid boundary condition 11

3. PROGRAM OUTPUT AND DEVELOPMENT 13

A. CONCLUSIONS 15

5. REFERENCES Id

Appendix 1 Print out of program TRANSCA plus flow chart and sample

input data \1

Appendix 2 Derivation of solid boundary equation for subroutine

SOLID A19

Appendix 3 Print out of typical TRANSCA output AJ

5

V
List of Figures

Figure 1

Figure 2

Figure 3

Figure A

Figure 5

Figure b

Figure 7

Figure 8

Fage

Application of momentum equation to unsteady flow in a general open channel

Specified time interval technique, Illustrating necessity

for base condition interpolation

Modification to specified time interval grid for sub and

super critical flow regimes

Application of specified time interval grid to simple channel

with known entry and exit boundary equations

Measurement of the flow specific energy change across a solid

stationary in a waste pipe under steady flow conditions

Flow rate vs time profiles at pipe entry, discharge and two

intermediate points along the 5m simulated drain pipe

Deptli profiles vs time along the simulated drainage pipe

following an inflow variation

Depth profiles along the 5m simulated drain pipe at lA time

intervals illustrating surface wave motion

vi
Notation

A Pipe flow cross section area

Notation referring to the positive and negative characteristics

c Wave speed

Fr Froude N6 = V/i^gh

g Acceleration due to gravity

h Flow depth

K Coefficient in solid leakage characteristics, Q KCSE-SE^)^

m Hydraulic mean depth

N,n Nc of pipe length sections employed

P Wetted channel perimeter

Q Flow rate

SE Specific energy = h + V^/2g

SEq Minimum specific energy required to initiate flow past solid

Sq Pipe slope

S Slope of energy grade line, defined by Manning's Equation

T Surface width of flow within partially filled channel

t Time

V Local mean velocity

Xl-4 Functions of h, V, c and S calculated at each base point at each

time step

X Distance, + ve in initial flow direction

a Pipe slope, Sq = sina

At Time step

Ax Pipe section length

0 At/Ax

vii
X Multiplier in combination equation yielding total differential

unsteady flow relationship = + /g~

/ AT

P Fluid density

Wall to fluid shear stress

Suf fixes

A, B, C

P

R, S

U, D

Calculated points in an x-t grid at time t

Calculated points in an x~t grid at time t + At

Interpolated points in an x~t grid at time t

Upstream and downstream conditions relative to the solid bounda

vii i
1 INTRODUCTION

Unsteady flow Is defined as fluid flow wherein the flow parameters vary with

time and as such will occur in all fluid carrying conduits if the system

boundary conditions are rapidly changed, either by design or as a result of some unforeseen event. The rate of change of the system boundary conditions

determine the severity of the flow parameter changes that result. In many

cases the resulting pressures generated in the flow system determine the

design conditions.

The most well documented area of transient analysis refers to the propagation

of pressure surges through full bore flow carrying pipe systems. For example

the pressure variations following inadvertent power failure to a pipeline

pumping station often provides the system design condition and leads to the

need to introduce safety systems such as air chambers, increased pump inertia

or by pass systems. Similarly the surges generated by load rejection by hydroelectric power station turbines lead to surge propagation in the water supply

tunnels, normally dealt with by the inclusion of surge shafts cut into the

rock surrounding the hydroplant supply tunnels.

The vast majority of research into unsteady flow has therefore been directed

towards the large civil engineering applications such as power stations, cross country oil, water or sewage pipelines. Currently these areas are well documented and several texts are available describing appropriate analysis and

design techniques [1,2].

The prediction techniques currently employed may be traced back to the late

19th century and for many years were based on the Bergeron graphical methods,

that incidentally also find application in electrical surge problems.

The widespread availability of digital computers from the 1960 's introduced or made practical, more versatile methods. Although translations of the graphical techniques were used initially, the numerical method of characteristics

employed to solve the differential equations defining the unsteady flow has now come to be widely accepted as the more versatile technique [3].

As mentioned the applications to be found in the literature refer almost

exclusively to large scale civil engineering applications. However, the problem of unsteady flow translates across the boundaries of scale, being dependent

on the relationship between the rate of change of boundary conditions and the

propagation rate of transient pressures within the system. Such applications

may be found in a wide range of small scale systems, including, for example,

diesel fuel injection systems. The analysis technique based on the method of

characteristics solution was successfully applied to a transient analysis of

the Concorde fuel system, both in the ground refuelling mode and the in flight

fuel transfer and emergency ejection modes [4]. Similarly the methodology has

been applied to airport hydrant refuelling networks.

1
Although this vast bank of experience and literature exists it is, as mentioned, primarily directed to full bore flow large civil engineering

applications.

Unsteady flowjnay, by definition, also occur in partially filled pipe or open

channel flow. Again the available literature applies to the large excavated

straight sided channels to be found in such applications as power station turbine discharge channels. In the open channel flow situation the full bore flow

transient pressure changes are replaced by channel depth variations, pressure

waves moving at the appropriate sonic velocity being replaced by surface waves. The simplification possible in air or vapor free fluid full bore flow that the wave propagation speed is constant, depending only on the liquid and pipe

materials and dimensions, no longer holds as the wave propagation speed depends

on flow depth and channel shape. Thus an extra variable is effectively added

to the local flow velocity and pressure, or depth, namely the local wave speed.

As will be seen in the analysis presented this also introduces other difficulties in the numerical solution of the governing equations since the uniform x-t

grid system generally employed has to be modified to allow for variable characteristic slope as the local wave speed varies.

As with the scale translation from hydroelectric plant to aircraft fuel system,

so too does the basic flow analysis translate from large excavated open channels to partially filled drain piping systems. The boundary conditions change

to represent, for example, varying inflow dependent on the appliances connected

to the system. The channel cross sectional shape does not constrain the application of the fundamental analytical approach.

At present no numerical analysis technique has been documented to predict the

depth and flow rate along gravity driven unsteady flow in partially filled

drainage sized systems; however the analysis techniques incorporating the

method of characteristics together with boundary conditions expressed in terras of time dependent depth or flow rates may be employed.

The current report is designed to act as an introduction to the use of these

methods to predict depth-time profiles along a simulated drain. In particular

the analysis will attempt to predict the depth history upstream of a deposited

solid in the drain pipe as a prelude to a consideration of the forces acting on the solid and its subsequent motion along the drain pipe.

In addition the analysis techniques are shown to be applicable to determine

flow attenuation in long drain pipes, this topic being considered increasingly

important due to the probable reduction in overall system flow rates due to

water conservation proposals.

Similarly the techniques can be applied to pipe sizing calculations in order to

avoid the occurrence of local full bore flow and associated venting problems.

2
The differential equations defining unsteady flow are presented in this report,

together with the application of the method of characteristics to yield a total

differential equation which may be solved by numerical methods. The method of

specific time intervals is presented together with the necessary formulation of

boundary equations to represent variable Inflow, pipe discharge and leakage

flow past the solid. Finally a computer program designed to analyze the transient depth response to a stationary solid in a pipe supplied with a variable

inflow is presented together with typical output for flow and system variables

representative of drainage design values.

Extension of the methods described to flow attenuation and solid motion

prediction are also proposed.

2. UNSTEADY FLOW IN OPEN CHANNELS

2.1 DEFINING EQUATIONS

The equations defining one dimensional unsteady flow are presented for open

channels. Frictional losses are assumed to be adequately represented by the

Manning equation utilizing the local velocity and flow depth. Channel slopes

are assumed small enough for cosine slope (cosa) to be approximately 1.

Figure 1 illustrates an elemental fluid strip. Application of the unsteady

momentum equation yields:

(i) Net hydrostatic force: pg

^

Ax A

dx

(ii) Shear force on wetted:

perimeter P

-To P Ax

(iii) Gravity force in flow:

direction

pg A Ax sina

(iv) Net efflux momentum: 3_(pv2a) Ax

3x

(v) Increase of momentum within: 3 ( pAV ) Ax

3t

Thus the momentum equation becomes

-pgAxA -ili. - T PAx + pgAAx sina

( 1 ) = i_(pv2A) Ax + i_(pAV)Ax

3x 3t

3
Expanding and dividing by pMx

1q -

p m gsina 2V +

3 X

3A

"5 3ir

A 3 t 3 t

where m = A/P

I The continuity equation applied to the control volume of Figure 1 yields

-3(pAV) AX ^ (3)

Tx Jl

Expanding and dividing by pAx

^3A^3A^a3^=0

3 x 3 1 3ir

Multiplying (4) by V/A and subtracting from equation (2) yields

1!: + lit - gsina ^ ^ , 0

3x pm 3x 3t

(5)

Let = sin a and gS = Tq/pih

where S is the slope of the energy grade line as defined by the Manning

equation

S

( 0 )

Note that S = only under steady uniform flow conditions.

Equation 5 becomes

8

d_h

3x + g(S “ S„) + ^ + = 0

° 3x 3t

( 7 )

4
From Figure 1 3A = 9h. T.

where T is, surface width.

Hence lA = T ill. , 3x 3x

and = T

3 t 3 t

so that equation 4 becomes

VT — + T + A =0

3x 3t 3x

Thus equations 7 and 8 represent the unsteady flow conditions in terms of

local depth and average velocity. These equations may be solved by means of

the method of characteristics.

2.2 SOLUTION BY METHOD OF CHARACTERISTICS

Equation 7 and 8 may be combined as

Lj + XL

2 ~ 0

where Lj is equation 7 and L

2 equation 8

iX (V XA) + + XT ilL (V +_%) + iil

3tl

-• 2

+ g(S - Sq) = 0

(9)

In order to solve equations 7 and 8 it is necessary to transform the equations

into a total derivative expression.

For the terras in bracket [1] to become the total derivative dV/dt, it is

necessary for

V + XA = dt

and for the terras in bracket [2] to become the total derivative dh/dt then

5
— = V + g/XT

dt ^

hence V + XA_ = V + g/XT

therefore ^ ~ ~ ^

/ AT

and

^ = V ± /M

dt / T

The term

speed

dimensions of velocity and is identified as the local wave

c = /gA

T

Equation 9 may now be expressed as: ^ ^ ^ + g(S - S ) = 0

dt c dt o

subject to — = V ± c dt

( 10 )

( 11 )

( 12 )

Referring to Figure 2, if the variables V and h are known at R and S then four

equations may be written in terras of the unknowns at point P, namely

V_ - Vr g I i dh + r g(s-s^)dt = 0

•'ho ^ ''to

xp - Xg = ( V+c )dt

R

(13)

(

1 ^)

V D Vf

Xd -

+ g C

-L dh + r g(S-SQ)dt = 0

Jhs ^

i::

(V-c)dt

(13)

(lb)

6
It is stressed that these equations are paired, i.e. equation 13 only holds if

equation 14 is satisfied and 15 only applied if 16 is satisfied. Generally

these are referred to as C'*' and C“ characteristics as shown above.

In most cases "a first order integration is satisfactory, however attention must

be paid to the choice of time step to ensure a stable solution, as mentioned by

Fox [1].

In terms of Figure 2 the time step must be sufficiently small to ensure that

points R and S fall within ± Ax on either side of point P.

Generally a suitable rule is to set

At = Ax/2c (17)

where c is the wave speed appropriate to the initial flow in the channel prior

to the initiation of unsteady flow. As will be seen the solution requires an initial steady flow in the channel, although this may be infinitesimal.

Applying a first order approximation to equations 13 to 16, in terms of

Figure 2 yields

Vp - + i. (hp - hj^) + g(Sp - S^)ht = 0 (18) cr

xp - xr = (Vj^ + cr) At (19)

Vp - Vg - i_ (hp - hg) + g(Sj^ - = 0 (20)

Cs

Xp - Xg = (Vg - eg) At (21)

It will be noted that conditions at R and S at time (t-At) are determined by

interpolation between points A and C and C and B respectively. This interpolation introduces a dispersive element to the calculation as effects arriving at

A, C and B at time t-At are assumed, via the interpolation, to effect con- ditions at R and S at time t-At. This is comparable to increasing the

local wave speed.

Interpolation effects should be minimized by arranging for R and S to fall as close as possible to A and B by adjusting the time step At.

Two flow regimes may be identified for open channels defined in terras of the

Froude Number Fr = V//gh

7
1) Subcritical flow, Froude N“ < 1

Here the local wave speed is greater than the flow average velocity.

Thus waves may be propagated upstream.

2) Supercritical flow, Froude N* > 1

Here the local wave speed is less than the average flow velocity at

that section, hence waves may not be propagated upstream. Flow may

be transformed from supercritical to subcritical via a hydraulic

jump.

Figure 3 illustrates the importance of these two flow regimes on the solution

of equations 13 to 16,

If c > V then the conditions at P are determined by the intersection of the

C"*" and C“ drawn from P into the AC and BC sections.

If c < V then conditions in the downstream section BC cannot effect point P.

The slope of the C" characteristic, PS, becomes positive and both R and S lie

in the AC section as shown.

For the subcritical flow encountered in the drainage applications considered

and the equations derived below refer to this flow regime. Similar derivations

may be undertaken for supercritical flow.

Referring to Figure 2 for subcritical flow:

At

Ax

At

Ax

and

as xp = xq and xp - xp^ = (Vp + Cp)At

8
Solution yields

y ^ Vc Q (22)

- R 1 + e (V^ - + Cc - c^)

^ ^ CqC^ ~ Vr 9 )

-*• c^Vr Q (2J)

R 1 + " c^0

Hr = ~ (h^ ~ ^R^^ (^‘^)

Similarly

V = Vc

~ ~

( 2 :.

)

S 1 “ 0(V(^ - Vg - Cq + Cg)

c = ~

<^B) (2b)

S 1 * 0 - cg)

^S

^ *

®^^'s

The determination of conditions at P at time t + At requires the following

steps:

(i) All conditions known at time t for nodal points ABC etc.

(ii) Values of V, h and c at interpolation points k and S calculated fro;n

equations 21 - 27.

(iii) Using these values of V, h, and c the conditions at P, i.e. velocity

V and depth h, at time t + At are calculated by means of equations

18 and 20.

(iv) The value of wave speed c at P at time t + At is calculated from

equation 10. The value of flow surface width and cross sectional

area are calculated from flow depth, h, and the channel shape

relationships.

(v) The sequence is repeated at each time step.

9
2.3 APPLICATION OF SOLUTION TO DRAINAGE FLOW

Equations 18 to 21 may be expressed as:

Vp = X2 - XI hp

( 28 ) Xp ~ Xp ~ (^j( ^

Vp = X4 + X3 hp

C (29;

Xp - xs = (Vj^ - cs)At

where XI = g/cp

X3 = g/cg

X2 = Vp + g hp - g(Sp - S^)At

^R

XA = Vg - g hg - g(Sg - S^)At

Figure 4 illustrates a typical drainage pipe length to be analyzed in terms

of flow depth and velocity at each section.

The application may be dealt with in two distinct sections:

(i) Internal or nodal points.

The values of h, V and c at all points Ax apart between x = Ax and

x = (L - Ax) may be calculated by the sequence set out above, namely

by simultaneous solution of equations 28, 29 and use of equation I'J to

yield wave speed.

(ii) Boundary conditions.

In order to predict h, V and c at the system boundary it is necessary

to solve either 28 or 29 with an appropriate boundary condition.

At pipe entry a suitable boundary condition would be the inflow profile

Q

= f(t)

to be solved with the C characteristic:

Q(t) = VjA^ = f(t)

V^ = X4 + X3 h|

Q(t) = Aj (X4 + X3 hj)

10
where = fCh^)

Q(t) = f(hi)(X4 + X3 hi)

In the form

0 = Q(t) - f(hi) (X4 - X3 hi) ( 30 )

this relationship may be solved by bisection method as values of Q are known

at each time step. The channel cross section shape relationship is also

required so that no direct solution of equation 30 is available for circular

section channels.

At pipe exit in the subcritical flow regime the flow depth approaches the

critical depth value, given by zero value of the expression:

Solution may again be achieved by use of the bisection method together with the use of the area to depth relationship for the channel.

2.4 APPLICATION OT WASTE SOLID BOUNDARY CONDITION

Considering a stationary solid deposited at some point along the waste pipe,

the water depth and velocity upstream of the solid may be predicted if a suitable boundary equation may be written linking flow past the solid to upstream

conditions.

Figure 5 illustrates the relationship between flow past a stationary solid

and the specific energy upstream. These results were compiled during a Brunei

University Drainage Research Group study of solid transport in drainage

systems

.

where A and T are functions of depth, h.

This condition may be solved with the C*” characteristic

^N+1 - X2 - XI

where N = N° of pipe length sections, each of length Ax.

The boundary condition becomes

[(X2 - XI h^+i) %+l/8 - 1 0 (31)

11
From the experimental work summarized by Figure 5, the flow past the solid may

be expressed by the following relationship:

(32)

where SE = h + V^/2g, flow specific energy and SE^ is the flow specific energy

required for flow initiation past the solid (when SEq 0)

Equation 32 may then be solved with the C*" characteristic

Vn+1 = X2 - XI hN+i

where Q = V^+i

so that

An+1 (X2 - XI hf^+j) - K [h^+i +~ <X2 - XI hN+i>^ - SEq)^

This expression results in a quartlc in terms of water depth upstream of the

solid, hj^+i , see Appendix 2,

This quartic must be solved by an iterative technique as the flow area, A^j+i

,

is a function of hjq+j.

The Newton-Raphson method may be employed to carry out the necessary iteration

solution.

Once the value of hjij+i has been determined the value of Vfj+i and Cf^+i may be

determined from equation 28 and 10.

As mentioned the SEq term is the flow specific energy required to initiate

flow past the stationary solid. If the value of flow specific energy at time

t is less that SE^ then the value of flow velocity at the solid at time t + At

is set equal to zero. The flow depth then comes directly from equation 28 as;

This implies that the flow depth upstream of the solid must rise to SE^ prior

to the initiation of flow past the solid.

This solution is set out in detail in Appendix 2.

In this model no account is taken of flow downstream of the solid.

hN+i =* X2/X1 (33)

12
From Figure 5 the flow past the solid may alternatively be expressed as

where SE^ SEq are the specific energy values iitinie d i a t e ly upstream and downstream of’the solid.

This boundary equation may be used to link the flow conditions upstream and

downstream of the solid, for example the upstream flow is governed by the

equation 28 and that downstream is governed by the C~ equation 29.

This formulation of the boundary condition would allow a series of nodal

points downstream of the solid to be dealt with, the pipe being then terminated by the boundary condition already described, i.e. critical depth at

discharge.

The techniques described above have been included in a Fortran program

TRANSCA run on the NBS Center for Building Technology Perkin Elmer 732

computer.

This program is included in Appendix 1 together with a flow chart and

description and sample input data while representative program output are included in Appendix 3.

program output.

The program calculation technique requires a known steady uniform flow to be

set up along the channel prior to the calculation of unsteady conditions.

In the model presented this is assumed to be established with no solid in the

pipe. The calculation then assumes the presence of the solid from the first

time step onwards. This explains the depth increasing wave that is shown in

Figure 8 moving upstream from the solid boundary. This is merely a functior.

of the initial conditions chosen.

An alternative base condition would be either steady flow with the solid in

place, with the associated back water profile upstream of the stationary

solid, or stationary fluid trapped behind the solid. In both cases the fluid

depth would increase at the solid location, forming the expected backwater curve profile.

The former set of flow conditions could have been achieved in the current

simulation by holding the inflow rate constant, however this would have

required a longer computer run. The establishment of steady conditions is

demonstrated in Figure 6 as the inflow profile is assumed to become constant.

3. PROGRAM OUPUT AND DEVELOPMENT

Figures 6, 7 and 8 illustrate the predicted depth and flowrate at sections

along the simulated drain pipe considered, while Appendix 3 presents typical

13
The results demonstrate that the techniques described may be employed to

yield values of depth and flowrate under unsteady flow conditions in open

channel flow. In particular the results indicate that the concept and use of

specific energy to leakage flow characteristic, rather akin to a valve

pressure flow relationship, is capable of accurately describing the boundary

conditions at a stationary solid. The values of depth at the solid with

time obviously require experimental verification, however the depths predicted

are generally in line with observations of water build up behind stationary

solids made during the Brunei University drainage investigations.

Two main development paths may be proposed based on the analysis techniques

described:

(i) Application to the moving solid case.

This case would require the introduction of flow relative velocity

into the boundary conditions describing the solid described above. It is proposed that, once this has been done, the stationary solid specific energy differential across the blockage to flow past relationship

is assumed to apply to the moving case.

A number of unknowns are required to obtain a solution, namely the

force balance necessary to be overcome to initiate motion and the

solid-flow momentum equation that would determine solid velocity and

distance traversed in each time step.

One simplifying assumption that could be made would be to assume that

the flow ahead of the solid, i.e. downstream, assumes the normal depth,

and velocity characteristic to the channel dimensions and the flow

past the solid.

This case is currently being considered.

(ii) Application of the techniques described to the prediction of flow

attenuation in long channels.

At first sight it would appear that the existing program would be

capable of dealing with this case, however the errors due to interpolation made necessary by the adoption of a fixed time-distance

grid, Figure 2, could introduce errors. The interpolation techniques

effectively disperse the moving wave fronts and would tend to over- estimate the attenuation produced.

A free grid in which both time and distance become calculated

variables is a solution to this problem. This is not normally done as

it results in a rather "untidy” output, however this method will be

investigated as part of the current investigation.

14
As mentioned the method of characteristics requires that there be an initial

steady flow condition along the channel, or that the flow is at rest at

known depth. This limiting condition is necessary to provide the base values

of depth, veJocity and wave speed at each calculating position. The influ- ence of magnitude of this initial assumed flow on the attenuation of a con- stant superimposed inflow will also be considered as part of (ii) above,

together with an assessment of the effect of the interpolation ratio employed

in choosing the appropriate calculation time step.

4. CONCLUSIONS

The program presented indicates that the techniques involved in the solution

of unsteady flow problems via the method of characteristics may be applied

to partially filled drainage pipeflow.

The methods presented will be employed to investigate further their application to solid waste transport analysis and to the study of flow attenuation

in long channels, typically representative of underground building drainage

systems.

5. REFERENCES

1. J. A. Fox, Hydraulic analysis of unsteady flow in pipe networks.

Macmillan Press London, 1977.

2. Wylie, E. B. and Streeter V. L, Fluid Transients, McGraw Hill, New York

1978.

3. Douglas, J. F., Gasiorek, J. M. , and Swaf field, J. A., Fluid Mechanics,

Pittman, London, 1979.

4. Swaffield, J. A., Application of the method of characteristics to a pressure transient analysis of the BAC/SNIAS Concorde fuel system.

Proceedings Institute of Mechanical Engineer London, 1972.

15
16

Hgure

1.

Application

of

nioiuentum

equation

to

unsteady

flow

in

a

general

open

channel.
CD

lO

CC<

0/tJ0eaVuC3o30

u.

17

Figure

2.

Specified

time

interval

technique,

illustrating

necessity

for

base

condition

interpolation.
eaVL 0ii C XI 30 aCO c 0 c 0 * ^ cr

«/» c V c y 0 T 0 r x» 4J w 0

J) O' (J) vi) 30

M-)0c0J5

J) 30

HO0• ^

XI

'CJ

0-3

«/)

V ^ Z

9f “i30L

Utl

18

Kigurt:

3.

Modification

to

specified

time

interval

grid

for

sub

and

supercritical

flow

regimes.
t

z

—^

N

C

0

Of J)

3

O

2u

V33

o

Ul00

19

Figure

4.

Application

ot

specified

time

interval

and

exit

boundary

equations-
Depth

of

flow

measurement

20

Figure

5*

Measurement

of

the

flow

specific

energy

change

across

a

solid
I'igure

6.

I'lowrate

vs

time

profiles

at

pipe

entry,

solid

boundary

and

inteniiediate

points

along

the

simulated

drain

pipe.
Figure

7.

Depth

vs

time

profiles

along

the

simulated

drainage

pipe

foiiowing
Figure 8. Depth profiles along the 5m simulated drain at 14 time intervals

illustrating surface wave motion.
figure C5 cont. Uepch protiies along the om simulated drain at 14 tine intervals

illustrating surface wave motion.
Appendix 1

Program TRANSCA

A-1
Program TRANSCA

This appendix presents a complete print out of this program, written in

Fortran, togefher with a flow chart and sample input data. The program was

run on the NBS Center for Building Technology Perkin Elmer 732 computer.

It was found necessary to employ double precision calculating techniques in

the program due to the inclusion of calculations involving the root of small

differences between large numbers. Otherwise no special facilities are required.

The program :*ccepts data in SI units with the exception of the inflow

profile, this is read in as litres/second and corrected to m^/s within the

program.
TRRn S C A.

^ irv*tva"V Cot\cA\^vof\^

”Tim.«. = 0.0

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^o^-*. S(Z<

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.^'VouJ ^ciS't.

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^ tS “ ^ ^ J4 c"t \orv S

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s"t.raijKt "lv^*

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IS * DESIGNfui TL APP.T THE flElMO^ OF :rtA‘^4CTERl STI CS S3LJUON 1C Ihf fcOUftTlONS OEFInI.nG

jnsteaoy flcw in PA^riACLY ricLEt cmannels.

jJu-lE PkECISILN JP ( 30 ) N< jO ) ,TI N( jO ) «VP ( is » ( 1S» , C? ( L5 ) JQU9LE PkECISlCh V(l->)tM(lM

JOUB.t PkECISICN YR(i.5),rlF(lS)»CMl5i,VSti5),rlS(i3),CS(l3»,Sj(l-'i

JQUELE Precision sr< is ) .xk i is > ,)(S( is* ,xn( isj tC (is i }OUB»E PKtCISICF TME, 0 ,1- *' , S 0 , PL » S£ J » XK , ( MAX t TF aC

JOUBlE Pt^cJSlfN H-<,HC»A,T,Pt P ,CN.SLJ»0

jQU3i.E PRECISION JX» )T ,OTO,LTKAT

CJMm3n/Cm 1/L7 C»CT « D< tOfSOtPLy SfO»/Ky<ri

,n.“.iL)N/Cn2 /N, Tmax common /C n3/NPTS»CM,r 1 m

i,jM'’'jN/Cr’. A/V^E1lC»'*Ry^K,CR»XF^SF yVSyUtCSyXSySSyXn

C0MM0N/Cm 5/E^C yEiN

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4

t

4 :

i. 3

1

4

'

t in

€

4

4

"10 ’

I

1 4 5 D

SET Initial ccncitions

r I ME = 0 .

0

READ PIPc DESCKIPTIO* JATA

r:AO( ^tlJCir. *kE ySjyPL

J-OU. R.m-manMNO COEFF. S('-PIPE SL'JPt. E^-PlPf LtNCr.-t

f •R**aT( mF 1C .<• * S^lIL [tSCE iPTI jh data

rE 4 M, 101 )SE GtXr

SEO-SPECIfIC cNEKOr CuUlPtL FTF PAST SuLlO*

<R-F_Cm TC specific Zn£PoY CLEFE. AT 3P YALJcS Ab^j^E Sro.

FQPMAT(2F IC.'t)

r£aO CALCOLATILN JATA

<EAu(Mylo3)NyT MAXy Tr AC

'*“NUnt£P CF PIFE SECTIONS CCNSICt'-ED, T ma <—

J

jr, A T 1 On jr CAuC.

TfAC-TiME STEP factor, 1-10.

F JP"aT(

I

jy 2F IC.A

J

rFAO INFLCW Data PR]F ILF. INbLf'N PROFILE USED IS =ASc^ 0.> A linear I NTE F POLA TI JN atTWftS TnESE DATA PjlNTS. NOT: JkTa

READ IN IN L/S BUT JSEO Ih PfCGlAM In mppj/S.

RE A0( A , 102 )NETS

F'iPNAK I 3 ) Oj so I^lyNPTS

READ! A, lOA )Cl N ( I * , T H ( I

)

FOPmaT(2F10.M)

QIN(I)*0IN(I)/1000.0

CONTINUE

4

4

4

c

9

CALCULATION OF STEADY UNIFORM FLOW 0=PTH ttASEu ON iNiTlAL FLj"

Rate 0IN(1) plus calculation of initial HAVEsPEED ANu ^OSj

term slo

CALL 0£PTh(7I ME)

CALL SHAPE (hN,AyT,»-R)

CALL R AVSPD(HNyCN)

A-10
: _ ASS Ion T i ^-F Zt RC SlNJlTIJNS alCNC TEaT t'lPt.

Jj 51 I-l tN*!

VP(I)*QINJ1)/A

jp ( n =oiN

(

1 »*iooo.o

HP ( n *HN

tP II ) -CN

51. C On T i nUE

*r« I 1 ) =0 . vj jD 52 I» 2 ,Na 1 SRI I » =SLO

yRi i)>oiNi n/A

HRI I

) -HN

CR( I)*CN

<NII )*XNI I-n*FL/ELJ4TIN»

•)2 CONTlNUf

j .J 5 3 I = 1 , F i S I I ) = S L 'J /sm = oiN(n/A

HS I I ) =HN

0 S ( I J *CN

vIOnTInuF ALL AS3I CMN )

: CALCULATiL'^ Cf Tii" STEP ANL LENGTH itCTiON.

3<“°l/FlJAT (N)

.yT = / ( TF Af *CN ) ,;T-^=j T

: j'jTPjT test LlSU-IPTION PLGS initial lO-uITI^nS,

tki:. (3,2C2)L,Pr,i3,?L

nz f UI- T I ini , /i oX, 'T? j r PIPE ClNF ILJ-' ATI 0 »

I / LOX, ’01 -NFTf F =• Srlj.A,' r.',

1

• ^ANr INC CCtF' ICI -NT ,F IC.^,

1 /IOr,*PlPt SLLPl « '.rlO.A,' P!?r length * I F lO.A N .',/ / ) IF IHL .CT.NNT NF.ITE I 3. 2 00)NN,.- C

I FI HC .Lt .NN ) HRITi I 3,201 jNN,hL

Z33 rOP-A T( ioX , »FlRW S JP E < CP I T 1 Ca L , ',*N^^-AL OEPT- « *,rlO.

1 • aNC critical uE»TM = NFIO.A,' h.‘,//)

2)1 F-jP-A T I 1 OX , *FlCW SJICRITICAL, S'N'IPtaL i^EPTH - *,

i FI n. ANC critical 0£PTh * N.*,//)

n’t ITt I J,2C3)CINI1), 'i.CN

2 ji Fv.P'IaT I IC'X ,

• IN 17 I Al flow rate « SFlu.^,* W** 3 /S.',

1 • initial depth - •,FiO,‘f,' h.',

2 • initial wave SPEE) « '.FIC.N,* *</S.*,/)

N l«N»

1

WRITE I 3,2C A »UT ,Ex, Wl 2JA FORN.AT I lux , 'CALCUL ATI JN TINE STEP » ',Flu.A,* S.*,

I 'length INCFENENT - 'tFlO.A,' N.',' NGWREk OF NOufcS •

: 3 UTPJT TlST SINGLATI 3 N RtSLLTS.

«RITE l3,2C5)IXNin ,I'* 1 ,N 1 )

2)

5 format I/IE X, • position FkUN • Ilf 7.A, / ,lcx,' tNT->Y M.'«/

WPITEI3t2Cfenil-LflH?II)fI = ltM)

205 F0RMATI2X, 'TIME » 'Ft,.3t* S.'t* OE PT H* 7 X n- 1 IF 7. h I WRITc(3f2C7)(VKH I-i,Nl)

writ: l3»30 7M0PU),I>l,Ni)

3)

7 FUPMATI luXt* FLOW RATE L /S * * 1 1 F 7. A

)

WRITE 13, 208)1 CP II). I-lfNl)

2)7 FOPMaTIIoX.' VELlCITY h/S= * . 1 IF 7. <. J

13 ./)

)

A-11
»o

1)

’

I»OO

'JO

tJ

-*

''I

ooo

KAVt s;> 1. r>/S = ' : J3_ f j^*lA T ( 1 3 > , , i If 7,<«,/ ) *

C JPOAff TIhf AWC CrlEC< CALCL-L^TICN LE'«GTH.

o30 GUNTINUf

50: r 1*16 - T I .It -*CT ir < TI fir .GT.TtiAjr I GTTJ SOO

SET UP C aLCULATIo'iS fJP THIS TlfL STtP.

aTP AT = ",T/[ TC IFIOTSAT.L T.U.Cf.) GOrj ?oC

CALL INFLCMT Iht » J»

J 1*1

CALL INT:F(M

Jl-2

CALL LOSS<VF,VS.h<,-tS,N,SS«Si,Ff')

J 1 *3

CALL £NTRY(TIh£)

J 1 *A CALL NODAL ( K)

Jl*5

CALL SuLlLITIhE)

IFfiT.LT.LTC) CCIJ oJJ

JT=DTO

J 1* *> CALL ASS I Gf (N

)

J 1= 7 Wkl T-; I 3,C Cf ) T I Kt » < HP( 1 I » I * 1 »M ) iRIT£ n,iC7) ( VM i ) , I* i,Ni ) WK ITC t 3 , JC7 H tf ( 1 ) , I * L ,NU

wL I T£ ( 3,^C f ) ( LF ( I ) , I »1 ,N1 J uCTO 301

CUNTl NU6

c NO

subroutine LCSS<VR,VS,H»,hS,H,SF,SS,R.D

00U=i-6 PLECISICH /R ( ; 3 N VS (15 ) ,HR( lii ,-Sl IB) , SP ( 1 ) » j S( 13 ) THIS SUSRCL'TINE CAlCJLATcS The ECUIV4LE.NT STlAPY STaT!: LOSS EAScC CN THt lANNiNG CLlFFICIEnI.

double PKECISICN A.T.PERtPH

DO 1 1*2, K^l

CALL SHAPE (HM I ) ,A ,r,PtP)

SP(I)*((VklIJ«*2i*RH**2)/(A/PEFI**l,A33

CONTINUE

DO 2 I-1,H

call SHAP E ( H5 ( I » ,A , T.PERJ

SS{I)*( (vS (I J**2)*RNP*2)/(A/PEP)**1.3 j3

CONTINUE

RE TURN

cNO

A-12
jJSSOUTiNF Dt PTH< T I ) r-.IS SUi/vlCTINt u'jrS A ScCIIotx Cf- F‘^TkY Ti: CAlCUlATc HCktal

AN tJ IkITiCAL FLUW OfPTMS.

JGU3l£ PkiCISlUh J1N( iOJ ,T1NOO)

JOUBLc PKECIilCN HC.HN,TirF

C!J«MJN/C’^3/^^ TSt-M, r In

COf'iLiN /Ciii /ec »HN

CALL £NT«>mPt)

RETURN

cnD

iUBR ..UTIi'it INFLGM TI1C ,2AV J TMIS jUBkLL'TIKC LALCJLATES lNfLC*« CaIES AT PIPE cnTEY

/N Tr!' fNTEY FLU" ®^)FILt LaTA. NO T' THAT Tnt u CALC>Ji.ATij

IS AN AVEFACt VALJE F THIS TiNt STEP.

JJU 3 Lt PktClSlGF. J n J ) ,T( jf.) «L> I 30 ) precision JAY.ijTOfr. T^uX^SufPLfSF‘*^xKf/<“

J.IUolF Pf^ECISILN T 1 »Tj,Ta

> j -“.• n/CnI/LT 0,LT . J ^ . u .S0» Fl ,S F ( tY. K , < C 'Jfi “ jN/CN 3 /NFT 5 »wtT

TI=TIhE

I j = T i .N f —j T J* 1 TXaTl

Jj 3 I*i,rPTS-l

I F ( Tx .CE . T I I ) .ANC . T)<. LT. T ( I ! ) ) GOT. A

: JHTI NOE

-x(Ji=u(i)-»(L(l4i)-Jti))9<TX-T(IJ)/(I(I*l) -T( i ) i J A V » 0 X ( 1 ) < £ Tij» N

t NC

SUaRGUTlNE SHAPE (H*A,T»PtK

)

PRECISION 0 T3 , J T , l)X , C t SC « PL . S= 0, Xf , R n, ^ , P

I

GOUBlE precision H, a , T ,P£ »Thc TA CaMfljN/CHl/CT0»DT,0X,U»S0»FLiSEC«XR,ir1

THIS SUBROLTINE CALCJLATES FlO A®. A, SURFACE WlDlH An- WETTED PEFIHETFP flASiiy On FL-H CtPTH

R-D/2.0

PI-3. IA2

IF (rt.LT.K)THETA»2.0»'JATAN(rSGPT<H*O-H) )/ tR-Hi ) IFIH.cO.R >THETA-PI

IF M. CT.k )TH£ TA-PI 2.0*nAT AN( (H-n) / ( J S Ot T ( H» I ./-N » ) ) ) A-((OT«2)/t,C)*(lH=TA-OSIN(T.-i£TA))

PcR-0*THETA/2.C

T-?.CJA< (rt«tC-H) )o#0.j ) RE TUkN

ENO

'^A-1 ”3
SUBi?OuTlr(£ »/lVSPL(^^C)

03U3LE PR£ClSIfjN H , C . AR E A , T , P E ( SUBkLCTH'E cALCJUATcS ^-AVE SPEEj based jn DtPTn AnD CRJiJ

inCTiJN ShAFt*

CALL SHAhE (hfAFEAt T*Pck

)

C«DSJRT( v.61PaR£a/D

RETURM

END

SUP.RQUTInE IMEFt-R)

jJU=_E PfECISILN V(iS*,H<lE),C<lSJ,Vt(15)«rlR(15),Co(it)),»j{L5i

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JJUBlE PRECISILN THFTA t 0 N 1 w N / C j11/LTO»LT* )x»JiSOfPL«S( C»XK»*'. *1 Cjf1“10f'i/CNA/v,h»C»'^R»HP*CR»XK»SF ^xS»^5»CS»xS*SS»x^»

TmIS SUb^CLTlNE SETS JP , !•. Y 1 nT f r.P Jl A T I On , Mb oAS= -jN^iTin'iS

r jV rt-i‘ n£XT TIht STEP.

TH£TA=DT/C>

> I * H I OU 1 I*2»F1

( I ) * ( v« I ) 4Th£TA* < C( I ) *v ( I-l )-V (I )«c ( 1-1 ) ) ) I / ( i.v.

-

ftheta* 1 vm-/(i-i)*c(n-c(i-i)))

:a(ij = (ciI)p<i. 0-/ •»(npTF(iTA»»c<i-uevR(n#T-itTA)

I / ( 1 .a»C ( I J *Th£ TA-C ( I-i > »TF<F TA J r»R(IJ*H(l)-(h(l)—((I-l)»»Tl-LTAP(VRm^CR(I))

aR( I ) »XN 1 1 )-( VR ( 1) «'C< ( I n f n

CONTI NUE

J3 2 1*1,

K

VS(I) = <V(l)-Th£TAP(/(I»PC(Mi»-CU)P^(Mli)l

I / ( 1.0 -Tm£TA*( V (I J-/( I ! MC < I ) €< I *1) » ) cs(n = (cm^vs(ii'»THTA*(C(i)-c(Mi)))

L / (1.0^TP*£TAP(C (li-Cd !>))

rtS( I ) »H( I )fTl-6TAP( VS ( 1 ) -CS ( I ) )*<n( I )-n( Ml) ) XS(I)-XN4l)-»(VSm-CSll))«LT

CONTI NUE

return

END

^ SUSROUTInE ENTRY(TlNt)

: This subroutine calcjlates ti-.e upstream oounjapy conjitiuns ^

C at each time step based UN A KNChN MFLOn PRJflLt. " DDUBlc PkECISION JP ( i 0 ) . u 1 N 1 3 0 ) , TI N ( iO ) . V P { 1 j ) ,mM 1 S) , C P » IS ) double precision V(15),H11S) « double PRECISILN V R ( 1 S ) ,.“iR ( It ) , CRI IS ) . V S t 1 5 ) , MS ( I S ) S < I > ) . S j t 1 ? J

A- 14
c

J-.U5LL PKlCmON lit ) «XS( 16) t»N( 16),L( 16)

jaU = L£ PkfCItlO TH;

JJUbwE PKtCISlOh JT 0 , 0 T,u> »L»SC,P l .S£ 0,XK,in

jou’Li pptcisiON riA<

Jo'JdwF PktCISlDh .^C.^N

double Pi«.tCISlCN G.CjN,3,UP,GN»AKEA,T,PER,HC<iIT,tiCN

OOUBLt PRECISION i"IJP.n,HNN,Hfc ,X3,X<i,lFL0h ,HSi

CUriNON/Cfll/r TC,L:T, 0 X,U» SOtPL , Sf C,XK..-I.’i Cl"f*3N/Cni/N»lHAX

C0H*i0N/Cn3/kFTS»QIN,r IN CCMrtUN/Cfi^/Vf)i»C»')^*“)RfCKtXk$jF»Vj*HifCS*XS»jS»X'i

COflHON/Crt /hi ,HN

C0«f10N/Cfit/CF VP,iP»3P

G-9.fll

C0N«Rrt*P2/ SC

IFITIhE.GT.O.OCALL InFLUP(TIPF,C)

IFITI-E.CT.C.C) Giri 60)

I F ( T 1 nt . c C . C . 0 ) C* J

I

--( ( 1)

CALCJLATICN CF CKiriCAL uEPTh.

J“*D

jN»0 .

j

^C»IJP^Cn)/Z.O

7 CONT I NUk

GALL SrtAP F (HC »ARt A , Tf PER

»

HC®IT*1.0-(C**i)*T/(l»AFkA**j)

IFInCP. ITJ3,^»5

g

J jr«aHC

GOTO 6

5 jP»^iC

^CN»( JP^uN J /£ .C IF (AcSI (lCK-hC)/riC I.Li. .O.OCi) Gl'TO 6 HC»HCN

GJI- 7 “

-i

<* HC =HC N

C GNTI NUt CALCOLATiCN LF Mj’IAi. .DEPTF-. JP-D

JN*0.0

^N-^ UP-»0n ) /2.

0

C3NTl,-(Ue

CALL SHAPE IhN ,ARtA ,T . PE®

)

HN 0 R»'.“ L. 0-1 C*A 2 jPCl'l/ (IAKE)«P3.3j3)/(PE^**1.333))

IFlHNG®rmC , 11 , I 4E. 1 J G.ITC 13 ^ i2

i3

OP *Hr>4 HHN-I UP«UN )/2 . C IF IA3SI (HHH-Hh)/h'^) .uE.O.OCl) CLIO 16

HN«HNN

GOTO 9 - M

• ll HN-HNN

CONTINUE

IFITINE.EO.C.O ) GOTO 600

c ^

c ® 600

CALCULATICN OF BOUNDARY OtPTh,

CONTINUE

JP-O

c

16

ON-O.

0

HB»IUP»DN) /2.0

CONTINUE

f

CALL SHAPE ()-£, AREA ,T, PEP)

A- 15
X 3 X ', / C S ( i. ) 1 l-SO) 0T-/ j*MS< 1)

HFLOh-O-I X<f»X 3 *Hc » •A.<£a

IF(HFL0*^»16,17,16

16 JF-HB

wCJTQ it

Id On*M=

H rtdB»( UP>uN )/Z .0

IF ( A& S ( ( rtt P-hb )/«< ).LE .0. 001) con 20

3 GOTO 16

20 HB-HbS

17 continue

HP« 1) -HS

vPlD^x^t^xO^MPll)

CALL SHAPE (l-fc »AktA , r. PER ) OPm-AKLA*Vf(l)*100 0.0

CP ( 1 ) *I)SyP T (C*AREA/T ) 500 CUNTINUE

i? E TU->. N END

C

1

SUdKiiuTlNE N0LAL('«»

PrlCISICN J“(ij»»-IMjO)»IIN(3o)«vp(15i,r.p(i5),CP(*5)

OJUdLL PKtCISlCN «/ ( 15 ) fi4 15 ) JUU-iL£ P-nECISION V^(l5)fHt.(l5),CMl6)»VS(15»ihS(lS),^S<l5)»Sj(l5)

F,.iriSlGN S? ( l 5 ) ,xt ( 15 ) *XS< 15 i .XN( 15) ,c U 5 » JuUPLh **KtCISlCN OTOtOTtJXtLtSCtPLtSiUtXK^^iithCtHMN

PRECISION GtXl»X3fX^tX^

COM-iQN/CNl/CTC^DTtOX, j,SO*PLt SFCtXX.Xr

C0H*"jH/C,lj/NPTS»Ul4»TI:'«

COrt'*ON/C^i^/V»h»Ct'^^»H(<«Ck»XK* SRtiSyHSfCStXStSStXN

C0NM0N/Cn5/PC tHN

COMn-N/Cnt/Cp,VP,HP,CP

THIS SUBkCLTINE calculates Th£ FLOW VELOCITY ANO JCPTH

AnO rVAVE SPEED AT each QF THt NCLE3 oET.»fc EN THE LPST^EA•^

AND DO-vnSTFEAH EuUNOAklES BY SCLLTIOn OF THE ThU iVAVc

t OUaT IOnS.

v» * ^ • d 1 1 1*2,

N

xl-C/CP( I

)

X3*G/CS ( 1 X2*X1AhO( I )-»VR(1)-GA(SR« I )-SO)*CT

X4-VS ( I )->3*HS ( 1 )-GA( SS II »-S0 )*CT

HPIl )-(X2-XA)/(X1 x<3I

VP I I J *XA*X3*HP I I ) call wavspc (hp( I ), CPI m

CALL SHAPE INF I I ) ,Xl,X3»XA)

JPI I )-Xl»VPII )*1000.0

CONTI HUE

RETURN

End

A-16
subroutine SOLlCtTM£)

DOUBLE PRECISION UP ( 30 ) t Q I N ( 30 ) ^ TI M I 3 0 1 « VP ( IS ) , HP ( 1 E J , C P ( i S

»

DjUSwE PkECISICN

DOUBLE PRf CISICN ^ R U S » ,riP I IS » , CR U S ) , VS ( 1 5 ) » HS ( I S » »CS I l3 I , S S « 1

*) J OJUPLf PRECISION FS(2 )»SR< 15) *>MlSliXS(lS».XN( 1S)»C(15I

double precision 3T0,oT«JX,L'»SCtPL,ScO,XK,RN,TlHt

OOUPlE precision T»1A<tHC*NN,C.«SEV,H0,Al,T»PER .XI DOUBLE PRCriSICN t P. S £ , U , f , R, Y . 2 . Z A . i J . Z E , Z 1 , Z 0 . E DOUBLE PRECISION 3 F . J n, H I , * 2 , S E

2

C')NrtON/CHl/CTO»DT,DX.O.SO,PL.Sf C,XK,-<rt

C0HHLN/Cn2 /N, TNAX

C0«H0N/CN3/NPTS»uI.i,r IN COMMON/C KA / V.h, C. VR, H R. C R. XR, SP * VS. HS.CS.XS.SS.XN

C0MM0N/Crt5/HC »HN

CONHON/Crtt/CF.VP.rtP.CP

; this subroltine calculates the flow depth, leakage rate and

£ «AV£ SPELT AT THE OOiNSTkEAH bOUNOARf FOkHED oY aN li'tlTlMLLY

: stationary solid.

9. <3

1

I Z-0

StV-SE 0 J» 1 N«-l J S3 CONTINUE

J “ 1 30 33 CALL SHAPE IHC.Al.T, PER)

Xl-G/CR (N-* 1 ) X2-Xi<’HR(N'»l) VK(N*1)-G»{ S? (N-^1 )-SO)PCT

SE*h(n»U-»<V(N-»1)aa2»/(2.C*C)

IF ( Sl .LT. SEC ) OCTJ 6 >5 CONTINUE

U»X’A Al/XK

3*yi»Al/XK

W^tXl*P2)/(2.0PG»

Y-(X2A»2»/I2.0PG)-SE3

Z«( l.a-X2*Xl/G)

Z<I*H*A2

Z3-2.0*w»2

22«Z*P2-»2.C*H*Y

Zl*2.0*Y*Z-»E

Z0*Y**2-U

F»ZA*hO**A-»Z3*HO** 3»Z2*H0**2* ZIVNO^Z J ESI J)

IFIJ.EQ.2) GOTO 7000

J »2

H0»H0*0 .D001*HC

GOTO BOOO

703 J OF -(FSt2)-fS(l))/(H0/ lOOOC.OI

3H-FS ( I )/CF

Hl«HO-Oh

1 F ( A9S ( (Hl-HC ) /HO) .L£ -0.005 ) GOTO 70

IF (Hl.CT.C )H1-1.1*H0

H0«H1

IZ-IZ^l

IF(IZ.GT.fcO) TIME-TNAX

IF(IZ.CT.tO) GOTO 301

IF(I2.CT.I0) GOTO 805

COTC 60

70 CALL SHAPE (HI, A2,T ,P£R ) IF (ABS ( (A2-AI )/Ai) .LE .0.005) COTL BO

AI»(A1^A2)/2.0

HO-Hl

GOTO 65

A-17
^ 30 continue

HP(N4l)=hl

“ VP (N* I

J

' JP IN* i f N^ 1»«A2*1000.0

GOTO 5 - t> VP(N*L)*O.C

* rtPCN* D-X2/X1

JPIN*l)=O.C

- GOTO 5 * 5 CALL WAViFC(MP(N*l » ,CP(N*1 )

)

S£2-HP(N*l )*<VP(N*U**2)/(2.0*C)

- IF(SE2.GT.StC.ANG.S£.LT.SE0JGCTC 300

* GOTO 801

30) SEO-SE

. HO«H(N*l)

* GOTO

301 CONTlNUt

- S£0=»S£V

' 0T=«07G

GOT; bOh

- 305 CONTlNUt

' TI.-E=*T iNE-t T i)T=*0T/2.0

- >,OTn tjcc * 30b CONTINUE

3 E TURN

_ ENO

iURSOUTlNt /SSIGnCNI

C TriIS SUPKCUTINE SETS uP InE N£w cASE LCNuITIJnS AlONv, TNt

C PIP. In fHPAFATIlN FOP TP£ NtXT Tint STEP.

jOUEuE PKECISiCN 3“( 3 Jl ,UIN( jC ) »1 INI JO ) *VP ( 15 ) ,HP I I5i ,CPJ 15)

jGU 3L£ PftfCISICN VIli),HI15)

0uU3LE PPtCISICN VRIlbl»HPIlt>)«CKll5»,VSll5),MSIlbl.:S(l3)»S^tl5)

JOUPlE PP£CISICN S<?t I 5) ,xM lb ) *XSt 15) *XNi 15) ,C 1 15 ) C0f1“0N/Cn3/NPTS»UI N. T IN '^GHNON/CNA/VfHtCyVRfiKtCKfXPfSFvVStHStCSfXSfbSvXN

CQNNJN/Crit/QF,VP»NP,CP

00 1 I»1,N-*1

VI I )*VPI I ) Ml I )*HP( ; ) Zl 1 ) = CPl I ) I ^uNTlNUc

*^ £ TUnN

END

V3ENJ

A-lfi
Appendix 2

Derivation of solid boundary equations

used in Subroutine SOLID

A-19
cct.\on O^ ^cT\vcASou\'J>.

OU.VN^ \A.CL"t \or\S u.s-®-‘ \n

^ V «-[\ a «c c* VA.a.“tConS A.

l') U. Q.g^g

^ ^ o \JU 0l% jNicl

^ Oor\"tintxi‘tc( o uj o \4 ja-r So

C Kq Y‘q cJ^Ar t S ir I c Q- C| \xet't\'oi^ C*'* R oCt -fc to

P a-t

C"** A M.«.t ion e>^ J-ot-m

V = XI - X 1 K

N + «

A-20
Va^uA Y\ s. Vc

r rOTtv

V. 2. = \( - <^ ( S - S \ At

R 0 R o

5 ct't- scTWcA Woc*.*^oL a r

:i ©.= VF^ = ( X2-x> K'^R - XK Se -

= Xv^ f K-^- _ se I ^ ^ (XX-XI.K')^ ,|k^(xzXK L

XI K

L- se.

3

^ K

I

(X2 +X\*K* -2X2.X1.K

I

t a'N'X^c.t >yv« t^r'rv\§

a = xz. ft / X K • w = XI

V

^^3 "

3

se„r

J

XI, ft /XK • V = x^*/ - Se . /zj ® ^ z =. \ - 2.X1.X2./

/;^3

t^-BK ^fz-K+V+YvK*]

= + C'l ^ \NX")' +2 2 K c V + NNK'-) = 2*K' f 'y’’+

vm’K'»+ Z'/VSK*" <-227K + ^ ^ * 2ZW { z^ + axw)!-*^ -c(z2y + 6)h + 'l^- a = o

A-21
^VvJa a S

OL c t

^

u. Q.y't i c. I r\ \s-«. ^o\\jac^ cJ^vr-a c*t\ ^ ^

CK'^ a.v\cA t.V^ Ar-e^or-^ ^

cx^rvcX VA_ zr ^

CK^

^c^.YV\So^^ rn^'tKoc^ ^

«ayNcA. JJ. eX v S u \? r o <-<. "t i ^OU.\'p>

4-90
Appendix 3

TRANSCA output

A-23
CALCUl/IItN

lli'

SUP

'

S.LtNOrM

IrtCHIdtNI

=

O.SdOU

(1.

MOM

Ic H

.Jl

NlllJfS

9* iT O O « ^ ^ ^ ^ o ^ O 1*^

^ ^ ^ C ^ ^

“>J — ^ «N O C O — *>rf -NJ o o 9 <r

^ 9^

%i ^ ^ 9* N< ^ *W N O O 9 «>

O C rs< ^ « r» % — >i ^

9 0 0 4-

9 9 9 ^

^ ^ iJ 4* l*»

-Si ^ <N -• 9 9 C <r 9 0

4' 9

9 ^ O 4-

•4^ o -r « ^

9 0 4 9999 9090 9990 9000 9 9 0 9 9 9

9 4> 9 9 9^00

9 O 4% 9 ^ psi ^

0 9 0 0 9 4* O O — O 9 4V 9 -Si 9999 C999

4 9 9 9

9

9 9 0 0

N ^

9 O 9 1% 9 —

9 4 ^

— — y ^ ^ 4 O 9 — -n

» 9 4 4 — if' C —

*>* •% S <“4 ^

Q O -

o o ^ o

9 1^ 4 O

o o o o o o

> O 9 *^ '’^ * 4 r 9 9 -xj 9 %

O - ^ 0 9-

^ O O

-9 4 O — ^04 ^ ^ (V —

T 9 O O

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9 C ^ © — ->w -

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O — — - 9 —

09 9900 OOwCJ

-»4 OJ9-» *9 — ^*r-9 *s-9r*^

0 0 9 9 > o o X 4 O — ^ O J-s -SJ —

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o o o o >900 **4 0“^ — ^04 O — PV -

9 9 0 0 0 4 0 —

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o — N»

0 0 0 9

O O X ->J 9 O O C 9 0 9 90.0?- o ^ - 0003 oooo oooo oooo oooo oooo oooo o-*^o oo-*o

>rOO OOOO >900 0900 0*00 0900 or-oo '•3 0— 0 3N.N,

—>4-*^ -»4'^^ 040"^ — 40'^' 0400 — 40*^ — 4^0 — ^ ^ — 00 4 — 00 4 — 004 — 004 — OO 4 — 00 4 —00 4 — 00 4 — “^0 4 O — -Si*'’ O — %•’ O — PNir» o— «N*' o— o — o — *%<•• o — '*— O — -•w'— 0000

O 4» O O

0 4 0 0 — 004

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0000

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0 9 0 0 ©400 -^>040 9 0 0 ©400 — 004

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0000 o o o o O^ — "• 0*^4 — 94 — 0 ©9 — 0 — ^ O 4 —004 O— Si* 0—4*

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0 4 3 0 © 4 O — O O 4

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O •.% SJ ©400 — 004 O — NJ -* ©9*0 — 004

3 — ^ — COOO 0009)

O 4» O O ©40* -004

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0000

0900 0 4*0

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0 3 0 0 9 •' r 9) 0 0 9 9 9 0 0

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0 4 0 0 ©409 — 004 O — *sj -I

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OCOO 0090

0 4 0 0 ©400 •^9 0^'- O — PS^ —

r* o 9 4» 9 O

•>i 4 Ai 4» O 4J N

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3 "'g 000 *

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0000 0000 0000 0000 0000 o o o « 0 0 « o

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Z 4^ >.4 4

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>• — C — ^ ft. — X i4 X 9 Z 9 — OX— — o

^ — O > ^9 — — 9 < — —

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444 N, •«», N» z 9 r

— « ft — * 4

X 9 — O X — ^ 9 O > 9 9 9 < O > — X

Z 4 4 4

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ft. 9 0 > — — 9 < O » X

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A-24

L»I“IH

M.

O.tlj/V

0.

..«'.(»

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U.O/'iS

II.

dl

*/

ll.uldu

o.dl

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3 A-31
NBS»n4A (REV. 2 »ec) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR 2. Performing Orgsui. Report No. 3. Publication Date REPORT NO.

NBSIR 81-2308 July 1981 4.

TITLE AND SUBTITLE

AN INITIAL STUDY OF THE APPLICATION OF THE NUMERICAL METHOD OF CHARACTERISTICS TO

UNSTEADY FLOW ANALYSIS IN PARTIALLY FILLED GRAVITY DRAINAGE SIZED PIPES

5.

AUTHOR(S)

John A. Swaffield

6.

PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No. H-48-78

S. Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP)

Department of Housing and Urban Development

451 7th Street, SW

Washington, D.C. 20410

10.

SUPPLEMENTARY NOTES

|^~| Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

The application of the numerical method of characteristics to the solution of the

differential equations defining unsteady flow in partially filled drainage system

sized pipes is outlined.

The derivation of the flow equations is presented, together with the necessary

boundary equation formulation to represent variable inflow, system discharge and

leakage flow past a stationary deposited solid.

A computer program, written in Fortran, is included, together with typical output,

that establishes the applicability of this computational method to unsteady flow

analysis in gravity flow drainage systems.

Proposals for the extension of the described techniques to the prediction of solid

transport and flow attenuation in long pipes are also presented.

12.

KEY WORDS (S/x to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

building drainage; numerical solutions for transient depth; unsteady partiallyfilled pipe flow; waste solid transport.

13. AVAILABILITY

^31 Uni imited

I I For Official Distribution. Do Not Release to NTIS

I I Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

^31 Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

64

15. Price

$8.00

(JSCOMM-OC 6043.P80