Reference

AlllDt.

NBS

Publications

\ 82-2478

Application of the Method of

Characteristics to Predict

Attenuation in Unsteady Partially

Filled Pipe Flow

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Building Technology

Washington, DC 20234

and

Drainage Research Group

Department of Building Technology

Brunei University

Uxbridge, U.K.

December 1 981

Issued March 1 982

—QC

100

.U5G

02-2478

5. DEPARTMENT OF COMMERCE

TIONAL BUREAU OF STANDARDS

1931

JfATlONAL BUIUIaO

01* fTAlTDABDS

UBBAST

NBSIR 82-2478

APPLICATION OF THE METHOD OF

CHARACTERISTICS TO PREDICT

ATTENUATION IN UNSTEADY PARTIALLY

FILLED PIPE FLOW

MAY 1 7 1982

Hot u cc

CiCfOO

, (o

Dr. J. A. Swaffield

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Building Technology

Washington, DC 20234

and

Drainage Research Group

Department of Building Technology

Brunei University

Uxbridge, U.K.

December 1981

Issued March 1982

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
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PREFACE

This report Is one of a group documentatlng National Bureau of Standards (NBS)

research 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. Swaf field. Senior Lecturer, Drainage Research

Group, Department of Building Technology, Brunei University, Uxbridge, U.K.,

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

Institute of Technology.

Ill
ABSTRACT

The mechanism of flow attenuation In partially filled unsteady pipe flow Is

presented and shown to have relevance to the design of gravity drainage

systems

.

The equations defining unsteady flow In partially filled pipe are derived and

shown to be capable of solution by means of the method of characteristics.

This technique as a method of predicting flow depth, velocity and wave speed

along a long drainage pipe at a range of pipe gradients, diameters and roughness coefficients was tested by means of numerical examples for a series of

simulations run on a digital computer. Additionally, limited experimental

verification of the analysis technique Is presented for the supercritical flow

response to a short duration Inflow surge.

Generally the technique developed was found to be applicable to the design of

drainage systems and further work Is proposed to both extend the experimental

verification and for the greater complexity of the multl-branched pipe system.

Key words: building drainage; computer model; surge attenuation; unsteady flow.

Iv
CONTENTS

Page

Notation vlx

1. INTRODUCTION 1

2. FLOW ATTENUATION IN LONG PARTIALLY FILLED PIPE FLOW 3

3. DEVELOPMENT OF THE NUMERICAL SOLUTION 6

3.1 Sunmary of the Method of Characteristics Solution of the

Equations Defining Unsteady Flow in Partially Filled Pipe

Flow 6

3*2 Partially Filled Pipe Flow Regimes 7

3.3

Application of the Method of Characteristics Solution to

Drainage Flow 11

3.3.1 Initial Steady Conditions Along the Pipe Length 12

3.3.2 Internal or Nodal Points 14

3.3.3 Entry Boundary Conditions, Supercritical Flow 14

3.3.4 Entry Boundary Conditions, Subcrltlcal Flow 14

3.3.5 Exit Boundary Conditions, Supercritical Flow 15

3.3.6 Exit Boundary Conditions, Subcrltlcal Flow 15

4. CALCULATION TECHNIQUES AND PRESENTATION OF RESULTS 16

4.1 Determination of Normal and Critical Depths 16

4.2 Numerical Integration for Surface Profiles 16

4.3 Choice of Time Step 17

4.4 Presentation of Results 17

5. DISCUSSION OF UNSTEADY FLOW SIMULATION RESULTS 19

5.1 Influence of Specified Time Interval Grid Size on Predicted

Flow Parameters 19

5.2 Influence of Surge Duration on Partially Filled Pipe Flow

Response 20

5.3 Influence of Pipe Slope, Diameter and Roughness on Predicted

Variations in Flow Depth and Velocity 21

V
CONTENTS (Cont.)

Page

5.4 Influence of Initial Steady Flow Value on Predicted Flow

Attenuation 22

5.5 Estimate of Computer Run Time 23

6. EXPERIMENTAL VERIFICATION 24

7. CONCLUSIONS AND FURTHER WORK 26

8. REFERENCES 27

Figures 28

Appendix I Program TRANSCC 52

Appendix II Typical Output TRANSCC, Supercritical Flow Case 71

Vi
LIST OF FIGURES

Page

1 Propagation of positive and negative surges in partially filled

pipe flow 28

2 Surge attenuation as observed in drainage pipe flow 29

3 Application of momentum equation to unsteady flow in a general

open channel 30

4 Modification to specified time interval grid for sub and

supercritical flow regimes 31

5 Application of specified time interval grid to subcritlcal

partially filled pipe flow with known entry and exit boundary

equations 32

6 Relationship between flow specific energy and flow depth,

Illustrating the boundary between sub and supercritical flow

regimes 33

7 Summary of pipe cross section and depth dependent parameter

calculations 34

8 Basis of gradually varied flow depth prediction 35

9 Schematic representation of numerical integration to determine

water surface profile 36

10 Influence of the choice of pipe section length on subcritlcal

flow parameter predictions 37

11 Influence of the choice of pipe section length on supercritical

flow parameter predictions 38

12 Depth of flow predictions for a surge introduced into subcritlcal

flow 39

13 Depth of flow prediction for a surge Introduced into

supercritical flow 40

14 Comparison of surge response to a 2 s peak duration inflow at

pipe gradients of 1/100 and 1/300 41

15 Comparison of surge response to a 2 s and 6 s peak flow duration

inflow at a pipe gradient of 1/300 42

vli
LIST OF FIGURES (Cont.)

Page

16 Comparison of surge response to a 6 s peak Inflow duration at

pipe gradients of 1/100 and 1/300 43

17 Flow attenuation along a 30 m pipe, 0.1 m diameter at gradients

1/40, 1/100 and 1/300 44

18 Flow depth predictions along a 30 m pipe at 1/40, 1/100 and

1/300 gradients In response to a 1.27/s peak Inflow surge 45

19 Continuation of figure 18 46

20 Peak flow rate predicted along a 30 m pipe for pipe gradients

1/40, 1/100, 1/300 and pipe diameters of 0.15, 0.1, and

0.075 m 47

21 Peak flow rate and depth predictions along a 15 m pipe, diameter

0.1 m illustrating the dependence on roughness coefficient 48

22 Effect of the initial steady flow on response to a 4.2 1/s peak

surge 49

23 Summary of experimental test rig and associated inflow and depth

recording Instrumentation 50

24 Comparison of observed and predicted water depth vs. time

profiles 51

vili
NOTATION

A

C+, C”

c

E

Fr

g

h

m

N,n

P

Q

Sq

S

T

t

V

Xl-4

X

a

At

Ax

e

p

Pipe flow cross section area,

Notation referring to the positive and negative characteristics

Wave speed, m/s

Specific energy h + v2/2g, m

Froude N* - V//ih

Acceleration due to gravity, m/sec^

Flow depth, m

Hydraulic mean depth, m

N“ of pipe length sections employed

Wetted channel perimeter, m

Flow rate, 1/s

Pipe slope

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

Surface width of flow within partially filled channel, m

Time, sec

Local mean velocity, m/s

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

time step

Distance, positive in Initial flow direction, m

Pipe slope, Sq * sina

Time step, sec

Pipe section length, m

At/Ax, sec/m; also subtended angle in figure 7

Fluid density, gm sec^/m^

lx
NOTATIONS (Cent.)

Wall to fluid shear stress » gm/m

Suffixes

A, B, C

c

n

P

R, S

U, D

Calculated points In an x-t grid at time t

Critical flow conditions

Normal flow conditions

Calculated points In an x-t grid at time t + At

Interpolated points In an x-t grid at time t

Upstream and downsteam conditions relative to the solid boundary

X
1 . INTRODUCTION

Observations of flow in long partially filled drainage pipes has shown that

the shape of an input surge wave is materially altered during its passage along

the system. In the absence of any downstream inflow the surge is observed to

attenuate, that is the depth of water in the pipe decreases and the time taken

for the surge to pass any station is seen to increase, indicating a reduced

flow rate at any downstream time and distance coordinates.

This effect is of course well known and has been observed in large scale open

channel flow and in flood routing studies. In the uniform channel case it is

dependent on frictional effects, while the similar observations in flood routing are also dependent on the variable storage characteristics of the natural

valleys followed by the flood.

In the design of drainage systems this flow attenuation can be of significant

Importance, particularly in the context of reduced water consumption. In simplifying some of the criteria commonly applied to drainage design it is possible to highlight two conflicting criteria that are affected by flow

attenuation:

(1) The prevention of pressure variations within a building’s drainage

pipework in order to avoid the diminution of appliance trap seals is

a major design criterion. If the flow in a portion of the pipe system

becomes full bore then positive air pressures are generated that can force odors past trap seals. Thus the maintenance of partially

filled pipe flow is a major consideration and has led historically to

conservative pipe sizing.

(2) The removal of waste solids discharged into the drainage system is to

a large extent dependent on the maintenance of flow depth behind the

solid during its transportation. It will be seen that this criterion

can be considered as being in possible opposition to the need to retain

a clear air passage through the pipe network.

An accurate prediction technique for flow attenuation that was capable of

dealing with both random inflows along the length of a drain pipe and a variable loading in terms of frequency of use of the appliances served by the

system would be capable of effecting design decisions under both the criteria

outlined above.

By definition therefore the technique would have to be a numerical solution

based on relatively small time increments and capable of modelling the hydraulic properties of the system as well as being applicable in a general way to

quite complex pipe networks.

The flow in the drainage system at any time may be generally described by the

unsteady flow differential equations. Based on the application of the principles of momentum and flow continuity to an element of flow a pair of hyperbolic partial differential equations may be shown to describe the flow in

terms of two Independent variables, distance x and time t, and two dependent

1
variables, flow depth and velocity. These equations, that have been fully

developed in an earlier report [1], may be solved numerically by performing

two basic steps, first a combination of the equations to yield a pair of

finite difference algebraic equations applicable along predetermined characteristics drawn in the x-t plane, and the solution of these equations with

representative system boundary conditions.

This solution technique, known generally as the method of characteristics,

meets the solution criteria set out above and indeed has been used extensively in the analysis of pressure surge in full bore complex pipe networks.

This report therefore presents the results of a series of computer simulations

for partially filled flow attenuation in drainage sized pipes subject to typical

prescribed inflow surges. In addition the mechanism of wave attenuation is

described together with a summary of the development of the necessary unsteady

flow equations. Details of the computer program TRANSCC are included together

with the data preparation instructions. Limited experimental verification of

the analysis is also presented based on transient flow depth measurements for

hydraulic characteristics of surge flows which have been conducted as part of

the test program for solid transport mechanisms in the Plumbing Research

Laboratory of NBS CBT.

2
2. FLOW ATTENUATION IN LONG PARTIALLY FILLED PIPE FLOW

Gravity driven flow in partially filled pipes or open channels may be

characterized by the observation that the flow depth may change in response

to both the applied loading and the frictional forces acting on the fluid

flow. In order to explain the mechanism of this depth change, or attenuation,

observed in long channels or free surface pipe flow it is necessary to consider

wave propagation in these flow regimes.

It may be shown [2] that in non rectangular channels the wave propagation

velocity is related to flow depth via the channel geometry.

where T is the flow surface width and A is the flow cross section area, both

functions of flow depth and channel geometry.

This implies that deeper waves travel faster than shallower waves and would

thus overtake leading shallower water or conversely outdistance following

shallower flow. Obviously this model is perceived in the flow by changes

in the observed depth profile shape.

Figure 1 illustrates the propagation of both positive and negative waves in

frictionless channels. If the wave front is considered to be made up of a large number of very small waves placed one on top of the other then it may

be seen that, in the theoretically frictionless channel, the wave profile

steepens for the positive wave and attenuates for the negative wave. The

deeper waves in each case travel faster than the shallower waves, thus in the

positive surge the crest overtakes the leading edge and the profile steepens.

Alternatively for the negative surge the crest outdistances the trailing edge

resulting in a spreading out of the wave.

The presence of frictional effects materially alters this situation. As the

forces generated depend on the square of velocity the introduction of frictional effects results in modifications to both the positive and negative

surge shape changes described above. Figure 1 illustrates the reduction in

negative surge attenuation as the deeper wave, travelling at the higher

velocity, is retarded relative to the shallower trailing waves and to its

frictionless performance.

Similarly the steepening of the positive wave front is reduced and, depending

on the channel friction, possibly attenuated as shown.

The effect of this mechanism on observed partially filled pipe flow is a progressively reducing depth downstream for a surge introduced into a long

pipe, figure 2, associated with a reduction in the peak flow rate recorded

at any downstream section.

This flow attenuation and reduced flow depth has direct implications on drainage system design, particularly in the context of water conservation.

c

3
Consider an Isolated finite discharge introduced into a long drainage pipe.

As this proceeds along the pipe, its depth decreases and the length of the

water wave increases, i.e., the flowrate at any section is attenuated.

If the pipe is designed on the basis of a given number of such discharges

with no allowance for either flow attenuation or the probability of simultaneous discharge then it will be seen that the pipe is either oversized or the

maximum loading, allowing for a probability usage pattern for the appliances

contributing to the loading, will be underestimated.

In the U.K. particularly this discussion has been represented by the numerical

limit to the number of domestic dwellings that may be connected to a standard

100 mm drain. Current design recommendation could be substantially Increased

if flow attenuation and inflow probability were combined and Included in the

design calculations.

It has been shown [3] that the transport of solids through drainage systems is

substantially dependent on flow depth so that the exclusion of depth attenuation calculations also indicates an enhanced probability of solid deposition

in long pipes. These effects are naturally aggravated if water conservation

proposals are accepted for existing installed systems, and reinforce the view

that water conservation should also include a study of pipe sizing, leading

either to possible reductions in standard pipe diameters or to increased

appliance loading for existing pipework.

At present there is little experimental data on flow attenuation in long

pipelines. Burberry [4] presented limited laboratory test results together

with frequency of appliance operation tables that support the arguments above.

However in this case laboratory tests are limited in their usefulness due to

the complexity of the flow combinations possible in an installed system. For

example the attenuation of an appliance discharge, such as a water closet

(w.c.), can be measured in the laboratory in pipes at a range of gradients and

lengths, however in practice this attenuated flow could be joined at some downstream location by another discharge emanating from another appliance. Thus

if the benefits of flow attenuation are to be included in drainage design and

conversely the maintenance problems inherent in disregarding this effect in a water conserving system are to be avoided, then it is necessary to develop a predictive technique that would include random flow additions in both time and

location along the drainage system.

Due to the complexity of drainage systems in such building complexes as hospitals, this analysis would have to involve a computerized solution that

would be based on the full unsteady flow equations. Such a solution would have

to be capable of accepting multiple inputs to a branched system and would be

expected to yield predictions of flow depths and velocities, including backflow

at pipe junctions.

The current report presents a feasibility study based on the use of the method

of characteristics to solve the unsteady flow equations and predict flow

4
attenuation for a pipe without branches. It Is based on a proceeding study of

the use of this technique to determine the depth build up at a deposited

stationary solid In response to a discharge moving along a drainage pipe [1].

5
3. DEVELOPMENT OF THE NUMERICAL SOLUTION

3.1 SUMMARY OF METHOD OF CHARACTERISTICS SOLUTION OF THE EQUATIONS DEFINING

UNSTEADY FLOW IN PARTIALLY FILLED PIPE FLOW

Referring to figure 3, it has been shown [1] that

continuity may be expressed as the equations of motion and

g ill + g(S - s)+vi5L +

^ = 0

3x o 3x 3t

(1)

VT + T + A = 0

3x 3t 3x

(2)

These equations may be combined to form a total derivative expression

^+g^+g(S-s) - 0

dt c dt o

subject to

(3)

^ = V + c dt

The wave speed being defined by

(4)

c = /~iA

/ T (5)

where V = local average flow velocity

c = local wave speed

h = local flow depth

Sq = pipe slope

S = slope of the local energy grade line

coefficient and m the hydraulic mean

T = water surface width

A *= water depth cross sectional area

2^2

m Ti \

^ Manning

m4 / 3

depth

Referring to figure 4, if the variables V and h are known at R and S then four

equations may be written in terms of the unknowns

first order approximation,

at point P, by means of a

Vp - Vg + 8_ (hp - h^) + g(Sp - S^)At «= 0 (6)

Xp - Xr « (Vr + CR)At (7)

6
0 ( 8 ) Cc

xp - xs - (Vs - Cs)At (9)

It Is stressed that these equations are paired and that equations 6 and 8

only apply as long as equation 7 and 9 are satisfied. This Introduces the

characteristics lines and C“, shown In figure 5.

It is also necessary that the time step At Is sufficiently small for R and S

(points in the x-t plane) to fall within + Ax of point P as shown In figure 4.

From figure 5 It will be seen, provided boundary equations governing the

conditions at the extremities of the system are known, that an orderly solution

may proceed yielding flow depth and velocity at each pipe section at each time

Increment

.

The basis for the system boundary conditions are set out below.

3.2 PARTIALLY FILLED PIPE FLOW REGIMES

Two flow regimes may be Identified for open channels or partially filled pipes:

(1) Subcrltlcal flow, Froude N® « —— ^ 1 /ih

Here the local wave speed exceeds the flow average velocity, thus waves may be propagated both upstream and downstream In the flow, l.e., c > V.

(2) Supercritical flow, Froude N® > 1

Here the local wave speed Is less than the average flow velocity at that

section and hence waves cannot be propagated upstream, l.e., V > c.

The flow regime applicable to any partially filled pipe flow may be determined

by a comparison of the flow normal and critical depths as described, following

the equations 10-15.

Under steady uniform flow conditions the force balance equation for an element

of flow Is normally expressed by the Chezy equation

m = hydraulic mean depth A/P, m

Sq “ sin 0 -duct slope

V = mean velocity m/s

C Chezy constant.

P * pipe wetted perimeter.

The value of loss coefficient C was found by Manning to be dependent on hydraulic mean depth and duct surface roughness n. The Manning formula Is the

V » C m So ( 10 )

7
simplest of the open channel equations:

V . 1 ^2/3 sJ/2

Q i A m2/3 sl/2

— o

( 11 )

where Q is the flow rate m^/s

A is the flow cross sectional area, m^

The value of the Manning coefficient, n, varies with pipe or channel material.

Chow [2] suggests values in the range 0.009 to 0.020 for materials commonly

found in building drainage systems.

Equation 10 effectively determines the flow depth under steady, uniform

conditions, only one value of h yielding the values of A and m necessary to

satisfy the equation. As this depth is by definition constant downstream,

dh/dx =0, it must also be the terminal depth corresponding to the flow

terminal velocity at that channel slope.

This depth, h^^, is commonly referred to as the normal depth.

The specific energy of the flow may be defined as

E = h +

^

(12)

2g

where h = local flow depth, m

V = local average flow velocity, m/

s

It may be shown that for a rectangular channel, width w, that there are two

possible depths for any particular value of E above a minimum.

E = 2g 2gA^

with A = hw for rectangular case, then

E = h +

2g(hw)2

h^ - Eh2 + q2/2gw2 = 0

For E constant, this equation has three roots, two real, one imaginary.

Figure 6 illustrates the alternate depths possible in such a situation and

their significance in terms of the flow definition.

8
From equation 12 and figure 6 It may also be seen that the flow specific energy

has a minimum value below which the given flow conditions cannot exist. In a general » non rectangular channel this value may be determined;

. 0 - 1 - dA

Zh g^3 dh

(13)

From figure 3

dA » T dh (14)

where T Is the surface width at any depth, h.

From equations (13) and (14) the minimum value of E will occur at a depth

value, h(,, that satisfies the expression

1 - Q2T/gA3 = 0 (15)

where T and A are both f(h)

This value of h Is referred to as the flow critical depth h^..

If the normal flow depth hj^ exceeds h^, then the terminal flow would be termed

subcrltlcal, or tranquil flow. If h^ Is less than h^, then the flow Is termed

rapid or supercritical.

It should be stressed that h^ Is Independent of pipe slope and pipe surface

roughness; while the normal depth Is dependent on both. Thus the same volume

flow rate In any particular pipe may be rapid or tranquil depending on pipe

slope, and similarly the same flow rate In a series of constant diameter pipes

will be tranquil or rapid depending on roughness.

Pipes or channels In which rapid flow Is normal are termed steep, pipes or channels in which tranquil flow Is normal are termed of mild slope.

Figure 4 illustrates the importance of these two flow regimes on the solution

of equations 6 to 9.

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.

Both subcrltlcal and supercritical flow are encountered in the drainage

applications considered and the equations derived below cover both conditions.

9
Referring to figure 4 for subcritical flow:

and

- Vr

V

c -

~ ^R

\

''C *A

(Vr + Cr) ^

(Vr + CR) ^

(Vr + Cr) II

as xp “ and xp - xr (Vr + CR)At

S olution yields

,c + e (-v^c^ + cc v^)

Vr =

CR =

hR =

1 + e (Vc - v^ + cc - c^)

0^(1 - Vr e) c^Vr e

1 + Cc0 - c^e

(^C “ ^a)(®(^R * Cr))

where 0 = At/Ax

Similarly

„ _

- ecv^cg - c^Vg)

1 - 0(V,

CS =

ho =

_

c^ + V30(Cc - Cg)

1+0 (c^ - Cg)

hQ + 0(Vs - Cs)(hQ - hg)

(16)

(17)

(18)

(19)

( 20 )

(21)

For the supercritical flow regime the equations determining R in figure 4

remain unchanged. The interpolation equations for S' in figure 4 may be

determined by an identical technique to that shown above,

Vc - Vs* - (Vc - Va) 9(Vs’ - CsO

cc - Cg’ “ (cc - c/^) 0(Vg' - cgt)

and he - hs» * (h^ - h^) 0(Vs« - cgO

Solution yields

Ved - ^A®)

- VaCc®

Vs'

1 + 0(Vp " V. + c. - Cp)

( 22 )

10
cs' (23)

+ Vgl 0(c^ - Cj,)

From equations 7 and 9 It will be seen that

^ - 1 dx V + c (24)

hence if (V + c) becomes large then At becomes small for a constant Ax. This

Implies that the progress of the numerical solution could become prohibitively

slow for supercritical flow conditions. Fox [5] suggests a check within any

program that will terminate the solution if At falls below a specified value,

however this comment by Fox applies to the existing applications of the method

of characteristics, which have been limited to large civil engineering open

channel or river flooding flow problems. It is likely that the reduction in

At will not be a significant problem with the relative values of V and c encountered in drainage sized pipe applications due to their dependence on pipe geometry and flow rate. This point will be returned to later in the

discussion of the supercritical flow simulations.

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

steps (i-iv), for either sub or supercritical flow conditions

(i) All conditions known at time t for nodal points A, B, C etc,

figure 4

(ii) Values of V, h and c at interpolation points R, S, or S’ calculated

from equations 16 - 23.

(ill) 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 6 and 8.

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

equation 5. The value of flow surface width and cross sectional

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

relationships, figure 7.

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

3.3 APPLICATION OF THE METHOD OF CHARACTERISTICS SOLUTION TO DRAINAGE FLOW

For convenience the application of the solution developed above to drainage

pipe flow may be considered under two headings, namely boundary conditions and

characteristic equation solution at intermediate pipe sections. Both of these

headings must be further subdivided depending on whether the flow is termed

sub or supercritical.

The equations 6 to 9 may be restated as

11
Vp

1 for C+ (25)

- X2 - XI hp

“ XR * (Vr + CR)At

Vp - X4 + X3 hp

Xp “ Xs

where XI *

X3 -

X2 -

XA »

- (Vs - Cs)At

g/cR

g/cs

Vr + g - g(Sj^ - S^)Lt

CR

Vs - 8 ~ g(Sg - SQ>At

CS

for C" (26)

It will be noted that these equations apply in either subcritical or supercritical flow, the interpolated values of the conditions at S or S' being

sufficient to define the flow regime, figure A.

3.3.1 Initial Steady Conditions Along the Pipe Length

As will be seen from figure 5 the initial conditions along the whole pipe length

at time t = 0 must be known in order for the solution to proceed. It is therefore necessary to calculate the steady state flow velocity and depth throughout

the pipe length initially, together with the initial wave speed. This process

may be carried out by the following steps.

(i) Determine the steady flow normal and critical depths as set out

previously. This determines whether the flow is subcritical or supercritical.

(ii) For supercritical flow the normal flow depth may be assumed to

apply throughout the pipe length. As the velocity exceeds the

wave speed there is no effect propagated upstream from the pipe

discharge point. This implies that the flow leaves the pipe at

normal depth and for a known flow rate and pipe dimension allows

the local velocity and wave speed to be calculated along the whole

pipe length.

(iii) For subcritical flow the initial water surface profile is more complicated as the effect of the pipe discharge is propagated

upstream. In subcritical flow it has been found that the depth

of flow is at its critical value at the discharge point with the

water depth then rising upstream until the normal steady flow depth

is achieved. Calculation of this depth profile is presented in

[6] and summarized below in terms of the gradually varied flow

profile prediction technique.

12
Gradually varied flow is steady non-uniform flow of a special type. The flow

parameters are assumed to change slowly, If at all. In the flow direction.

The basic assumption In the treatment of this type of flow Is that the local

head loss at any section Is given by the Manning expression (11), for the Identical local flow depth and rate under assumed steady, uniform flow conditions.

Depth profile predictions by numerical integration are based on this

assumption, expressed In terms of figure 8 by

dL ^ 2g

where (Zo _ ( in the

hence _ V dV

g dL

and as. Q

= VA

-I 1 ^ ,2/3

^ (27)

Am

+

dL

- (-2^)

Am

2/3- (28)

^

A + V

^ = 0

dL dL

and as ^ = T from equation 5 it follows that

dh

^ = _v^ = = - Qlill

dL A dL A dL A^ dL

and substituting in equation 28 yields

Q^T dh . 5 dh _ f n Q i ^

dL dL A

L = h

7 - ^ M

S — (nQ/Am^' ^)^

o o

where L is the distance between two known depths ho, h]^

.

Figure 9 illustrates this numerical integration, which may be conveniently

achieved by Simpson's rule.

The numerator and denominator of equation 30 will be recognized as the

equations determining the critical and normal flow depths in an open channel.

(29)

(30)

13
When the term (1 - Q^T/gA^) is zero the flow Is at critical depth, i.e., there

is no change in L for a change in h.

When the term - (nQ/Am ' ) is zero uniform flow depth is achieved, i.e.,

there is no change in h for a change in L.

The initial depth at each section Ax apart along the pipe may then be

calculated from the profile produced by the integration of equation 30. Flow

velocity is then calculated based on a constant flow rate through the pipe and

similarly wave speed is determined based on flow depth and channel geometry.

Once the initial flow depth, velocity and wave speed have been determined the

unsteady flow calculation procedure may begin.

3.3.2 Internal or Nodal Points

Simultaneous solution of equations 25 and 26 at all points Ax apart between

X = Ax and x = (L - Ax) will yield the required values of flow depth and

velocity at the end of each time step. Wave speed may then also be determined

from equation 5. This process applies to either sub or supercritical flow con- ditions as the particular regime is represented in the Interpolations required

to fix points R, S or S’, figure 4.

3.3.3 Entry Boundary Condition, Supercritical Flow

In this case the inflow profile alone determines the flow depth at pipe entry

as downstream conditions, that would have been represented by the C” characteristic in subcritical flow, cannot effect the flow conditions at the upstream

boundary, as by definition the flow velocity exceeds the wave speed.

Hence the boundary condition is obtained from the flow profile Q

= f(t)

solved with the normal depth expression

Q

= 1 A m2/3 S 1/2

n o

where A and m are both f(h). This equation may be rewritten in the form

1 (nQ)^

a2 m^/3

(31)

and this boundary expression may be solved at each time step with a known Q

by the bisection technique; this technique is described later.

3.3.4

Entry Boundary Condition, Subcritical Flow

For subcritical flow the downstream conditions do contribute to the entry flow

depth and velocity. In this case the inflow profile Q

= f(t) is solved with

the C“ characteristic

14
Q

- f(t) - ViAi

Vi - XA + X3 hi

Q(t) - Ai (XA + X3 hi) ( 32 )

where suffix 1 refers to the entry boundary location and A * f(h) dependent on the pipe cross sectional geometry.

In the form

this boundary equation may be solved by the bisection technique.

3.3.5 Exit Boundary Condition, Supercritical Flow

As the flow velocity exceeds the local wave speed the exit boundary condition

may be determined in the same manner as the upstream nodal points, namely by

the simultaneous solution of the CT^ and C“ characteristics. With reference ti figure A both the R and S' points lie upstream of the pipe exit.

The exit condition may be included in the nodal point calculations for the

supercritical flow case, no separate exit subroutine being necessary in the

program as equations 25 and 26 are sufficient.

3.3.6 Exit Boundary Condition, Subcritical Flow

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

critical depth value, given by zero value of equation 15:

where A and T are f(h)

This condition may be solved with the characteristic

V^l = X2 - XI h^i

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

The boundary condition becomes

Solution may again be achieved by use of the bisection method together with

the use of the area to depth relationship for the channel.

Q(t) - f(hi)(XA + X3 hi) * 0 (33)

'Tcrit “

I(X2 - XI hN+l) A^i]^ TtH-i - I - 0 (34)

15
4. CALCULATION TECHNIQUES AND PRESENTATION OF RESULTS

4.1 DETERMINATION OF NORMAL AND CRITICAL DEPTHS

The bisection method was used to solve the equation defining both critical flow

depth

X - 1 - Q2T/gA3

and normal flow depth

Y » Sq - (n Q/Am^''^)^.

that feature as boundary conditions.

It may be assumed that both X and Y have zero values for some value of depth h

in the range 0 < h < D for the pipe case.

This initial Interval is bisected and h = D/2 used to evaluate X, Y. If the

resulting values are positive then the root is less than the midpoint and the

upper limit is reset equal to the h value just used and the remaining interval

0 to D/2 bisected. The process repeats until a root is obtained. If the X or Y value had been negative then the root would be greater than the trial h

value, in this case the lower limit is reset to the trial h value and the

interval bisected.

Due to the need to include the area depth relationship this solution must be

undertaken by an iterative process. The time taken depends on the complexity

of the area-depth function.

4.2 NUMERICAL INTEGRATION FOR SURFACE PROFILES

The integration of the position vs depth profile

^2

L = f -j..

- dh

hj^ S^ - (nQAm^'

^1

is achieved by means of Simpson’s Rule. Let the integral X = / F(h) dh,

ho

than if the interval hj^ - hQ is divided into 2 equal increments, the value of X

is given by

X » ^dh [F(h^) + 4F(h^j + dh) + FCh^ + 2 dh)]

As the integration progresses the length traversed may be accumulated as

Ljj+i • Ln + X at the completion of each integration.

16
A. 3 CHOICE OF TIME STEP

Referring to figure A it will be seen that the time step chosen must be such

that the points R and S fall within + Ax of point P. In order for this to occur for all sections along the pipe it follows that

At « (35)

(V + c)jjj^x

This expression yields the smallest possible time step as the maximum values of

flow velocity and wave speed at any pipe section has been used.

In order to ensure that the computation proceeds as quickly as possible the

computer program presented calculates a new time step magnitude at each time

increment so that the time step increases when V and c decrease but decreases

to maintain a stable solution when V and c are Increasing in response to an Inflow surge.

A. A PRESENTATION OF RESULTS

The objective of the simulated pipe flow data developed from use of the Fortran

computer program TRANSCC, run on the NBS CBT Perkin Elmer 732 computer, was to

identify the potential of the method of characteristics solution for flow

attenuation prediction and also to highlight any limitations inherent to the

technique

.

The following test cases were investigated and are reported here:

(1) Investigation of effect of the number of nodes, i.e., pipe computational

section length Ax, for subcritical flow, pipe slope 1/300, diameter 0.1 m,

length 5m, and for supercritical flow, pipe slope 1/AO, diameter 0.1 m,

length 5 m.

(2) Investigation of effect of peak flow duration, either 2 or 6 seconds at

1.0 i/s at pipe slopes of 1/100 or 1/300, pipe diameter 0.1 m, pipe

length 5 m.

(3) Effect of pipe gradient on flow attenuation in a 30 m long, 0.1 m diameter

pipe, pipe slopes of 1/AO, 1/100, 1/300.

(A) Effect of pipe diameter on flow attenuation, pipe diameters 0.15 m, 0.1 m,

0.075 m at 1/AO, 1/100, 1/300 slopes, 30 m long pipe.

(5) All simulations above employed a constant Manning coefficient of 0.015.

The effect of varying Manning coefficient from 0.009 to 0.020 in a 0.1 m

diameter, 15 m long pipe at 1/100 slope was also investigated.

(6) The effect of the initial flow rate assumed in the pipe was studied,

varied from 0.2 A/s to 0.05 A/s in a 0.1 m diameter pipe, 15 m long, at

1/300 and 1/AO pipe gradient subjected to peak inflow rates of 1.2 and

A. 2 A/s.

17
The results of these simulations are presented in graphical

discussed.

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5. DISCUSSION OF UNSTEADY FLOW SIMULATION RESULTS

5.1 INFLUENCE OF SPECIFIED TIME INTERVAL GRID SIZE ON PREDICTED FLOW

PARAMETERS

In order to apply the solution technique described the pipe length must be split

Into a number of equal Increment at length Ax. The calculation time is

dependent on this subdivision length and on the local wave speed and flow

velocity via the expression

V + c

Hence for a given pipe length and duration of flow simulation the computing

time and consequently cost, is determined by the choice of the number of pipe

sections adequate to provide an acceptable prediction of the flow parameter

variation with time.

Figure 10 illustrates the effect of dividing a 5 m length of pipe, into 5, 10 or 30 equal computational subdivisions, Ax varying from 1 m to 0.167 m.

The variations in predicted flow rate at each node along the pipe length may

be explained entirely in terms of the interpolation required to yield the base

conditions for each computing time step, outlined in figure 4.

The interpolation in figure 4, necessary to determine conditions at R and S at

time t, from calculated values of depth, velocity and wave speed at A, B and C,

effectively assvimes that disturbances arriving at A, B or C at time t effect

conditions ahead of their true positions, namely at R and S.

Therefore the initial depth increase propagated in the pipe flow by the rise

in the Inflow curve, Q

= f(t), would be expected to register earlier at any

particular section if the number of pipe sections were small. With larger

pipe section lengths the apparent disturbance propagation speed is increased

due to this interpolation technique.

Similarly any decrease in depth at the inlet would also be recorded earlier at

a downstream section if the Ax chosen were large. As the depth and flow velocity predictions may be thought of as summations of positive and negative change

waves moving back and forth along the flow it is therefore to be expected that

the peak values predicted for flow depth and velocity will increase towards a true limit as the pipe computing section length decreases.

These effects are clearly seen in the simulated results presented in figure 10.

As the number of pipe sections increases from 5 to 30, so does the peak flow

rate predicted, although the change between 5 and 10 sections is several times

the change between 10 and 30 sections, indicating an approach to a true

prediction limit.

Figure 11 repeats this assessment for a supercritical flow in a pipe at a 1/40

gradient. Here there is little attenuation in the first 4 m length when

19
compared to the very pronounced attenuation in the 1/300 gradient pipe of

figure 10. This lack of attenuation is explainable in terms of the frictional

and mass forces acting. The effects of Ax choice however remain as described

for the 1/300 pipe case, earlier depth changes being detected with the larger

pipe sections together with reduced peak flow rate predictions.

Figures 12 and 13 illustrate the depth changes along both the 1/40 and 1/300

pipe lengths for the Inflow surge considered above. One point mentioned in the

derivation of the flow boundary conditions is clearly demonstrated by these

results, namely the Initial water surface profile and the associated exit

depth criteria.

For the 1/300 gradient pipe the flow is subcritical and hence the depth

decreases at exit to the appropriate critical value. Figure 12 illustrates

this profile for the steady initial flow of 0.1 f/s calculated by the gradually

varied flow analysis described. The flow in the 1/40 pipe, figure 13 is by

contrast supercritical and hence the flow depth along the whole pipe length

remains at its normal value at all pipe sections, this depth being by definition both less than the critical depth appropriate to this pipe slope,

diameter and flow rate and less than the corresponding normal depth in the

shallower gradient, 1/300, pipe.

The choice of incremental length and the associated time step is to some extent a matter of experience. In general the flow simulations presented in

this report were carried out with an incremental length of 0.5 m, this length

being felt to be a reasonable compromise between short computer running time

and an adequate representation of the flow depth and velocity time profiles.

5.2 INFLUENCE OF SURGE DURATION ON PARTIALLY FILLED PIPE FLOW RESPONSE

In a study of surge wave propagation Wyly [7] noted that the peak surge depth

was dependent on the duration of the surge flow. This effect is clearly demonstrated in figures 14, 15, and 16, by means of a comparison of the attenuation

in peak flow rates predicted at pipe slopes of 1/100 and 1/300 for two surge

durations

.

It will be seen that the effect of extending the duration of the inflow peak

rate of 1 Jt/s from 2 to 6 s is to increase substantially the peak flow rate

predicted at the downstream pipe sections. Examples have been taken at 2 and

4 m from the pipe entry.

This effect is to be expected and is in fact due to the flow establishment

response characteristic of the pipe system. Obviously if the peak inflow of

1 l/s was held constant for a sufficient period then the flow in the pipe

would become steady at this new flow rate, the water surface profile in the

subcritical regime assuming the new 1 £/s characteristic normal depth with

a progressive reduction to critical depth at the pipe exit.

20
5.3 INFLUENCE OF PIPE SLOPE, DIAMETER AND ROUGHNESS ON THE PREDICTED

VARIATIONS IN FLOW DEPTH AND VELOCITY

The attenuation In flow depth and flow rate predicted by the unsteady flow

analysis are caused by the balance of frictional and mass forces acting on the

fluid. Within the model these friction forces are represented by the appropriate steady flow loss calculated from Mannings equation 11, but utilizing

the flow velocity and depth at the particular pipe location and time step under

consideration. Thus the variations in attenuation would be expected to follow

the dependencies Inherent in Mannings expression, namely friction forces

dependent on pipe slope, roughness and diameter.

Figure 17 Illustrates the attenuation in peak flow rate at four locations at

6 m intervals along a 30 m pipe at a range of gradients, 1/40, 1/100, 1/300.

For this simulation pipe roughness and diameter were held constant. Two

effects are clearly visible as gradient is reduced, namely the reduction in

peak flow rate and the progressively later occurrence of this peak flow,

indicating the dependence of flow velocity on pipe gradient.

Figures 18 and 19 illustrate the propagation of the surge wave at the three

pipe gradients. Initially the water surface profiles at the three gradients

are at normal depth with a reduction to critical depth at pipe exit in the subcritical flow cases, 1/100 and 1/300. As the wave travels along the pipes it

will be noted that the wave progresses most rapidly in the steeper pipe, again

this effect reinforces the forward movement of the peak flow rate noted in

figure 17 and is predictable from the Manning expression dependency.

Figure 20 presents the peak flow rate for the 0.1 m diameter pipe from

figure 17 in comparison with the predicted peak flows in 0.15 m and 0.075 m

diameter pipes. At each of the three gradients considered the attenuation

effect is least in the 0.15 m pipe and greatest in the 0.075 m pipe. Again

this effect would be expected from the frictional forces acting for constant

pipe slope and roughness.

Based on these observations it would be expected that attenuation would also be

dependent on roughness, as represented by the Manning coefficient. Generally,

in the simulations reported a constant value of Manning coefficient of 0.015

has been employed. This value is appropriate to cast iron pipe, Chow [2], and

as such would be representative of many drainage applications. Figure 21 illustrates the effect of reducing pipe roughness to a Manning coefficient of 0.009

representative of very smooth bore glass or plastic drainage piping, and

increasing to value of 0.02 representing aged piping. A pipe gradient of

1/100 was chosen to illustrate both the dependence of flow normal depth on pipe roughness and slope and the fact that subcritical or supercritical flow

may be established in a constant slope pipe dependent only on pipe roughness.

It will be noted from figure 21 that subcritical flow is established at Manning

coefficients of 0.015 and above, supercritical flow at Manning coefficients of

0.012 and below.

Figure 21 also indicates the depth of flow dependence on pipe roughness, the

smoother pipe featuring the shallower flow. Again this could be predicted for

21
a given flow rate in a pipe of constant gradient from the Manning equation for

steady uniform flow.

5.4 INFLUENCE OF INITIAL AND RESIDUAL STEADY FLOW ON THE PEAK FLOW AND DEPTH

PREDICTIONS

It has been shown that in order to utilize the method of characteristics

solution of the unsteady flow equations it is necessary to establish the flow

depth and velocity at all points along the pipe at time zero. In the simulations presented in this report the initial conditions have been chosen to

represent a steady low flow rate through the system. Initial conditions are

of course not limited to such a steady flow; however^ this is by far the simplest

base condition to employ.

It might appear therefore that the need to assume an initial flow in the system

is a major limitation to the application of this analytical technique, however

this may be shown not to be the case. The objective of the technique development reported was to provide a means of Including the beneficial effects of

flow attenuation on the design of long drainage pipes within, or collecting

flow from, complex building drainage networks. A cursory study of the usage

patterns of appliances within any complex building, such as a hospital, or domestic network serving a large number of living units, will indicate that

such a system would be consistently carrying water at some minimum flow rate

so that the response of the system to a surge input could be reasonably

modelled by means of the techniques described.

Thus the initial constant flow rate in the region bounded by 0.05 to 0.2 £/s

is reasonable as test conditions for the numerical solutions.

Similarly most appliance flow discharge profiles feature long, low flow,

"tails", this is especially the case for such appliances as bathtubs and basins,

thus the solution may be allowed to run for a given time period at a final

value of steady flow and still be representative of actual flow conditions.

These constraints apply rather more to the test simulations included in this

report than to the use of the techniques in a full system analysis as in the

latter case the usage pattern of the system appliances would provide a continuous unsteady flow input to the system.

Figure 22 illustrates the peak flow rate and depth response predicted along a 20 m long, 0.1 m diameter pipe set at gradients of 1/40 and 1/300, during and

following an input surge of 4.2 £/s. The initial and final steady flow values,

assumed in the simulation, were set at 0.05, 0.1 and 0.2 £/s for each pipe

slope. It may be seen that the maximum flow rate and depth achieved in the

system at, or close to, pipe entry remains unaffected by these steady flow initial and final conditions. There is a variation in peak flow rate and depth

achieved along the length of the pipe however, and this is entirely predictable

in terms of the final steady state value being imposed on the system. Obviously

the lower final flow rate of 0.05 l/s imposes a more rapid attenuation on the system when compared to the higher values of 0.1 and 0.2 £/s.

22
Theoretically there is no lower limit to the chosen value of final steady flow,

however It should be noted that the peak flow rate to be expected In a building

drain of these diameters would be In the range 8 to 10 i/s, so that 0.05 i/s

represents a very small percentage.

Although the necessity to provide an Initial and final flow rate along the

system is a limitation on the technique, it is unlikely that this will be In

anyway detrimental to the application of the developed technique to actual

system analysis.

5.5 ESTIMATE OF COMPUTER RUN TIME

The program TRANSCC is included in an appendix to this report. The run time of

the program is dependent on the number of pipe sections chosen to represent the

pipe being studied. During the duration of this study the program was run on

the NBS CBT Perkin Elmer 732 computer. It was found that a 30 m pipe made up of

60 pipe sections could be studied for a real time of 200 seconds for a computer

run time of 15 minutes if no interruptions due to other users programs were experienced. It is clear therefore that the shorter length and real time duration cases were run in 2 to 3 minutes of computer time. In view of the fact

that the machine used was by no means the fastest available it is considered

that the technique developed is practical in terms of usability costs as a design tool.

23
6. EXPERIMENTAL VERIFICATION

The simulated flow attenuation results presented and discussed In the

proceeding section have Indicated that the technique developed is capable of

providing flow rate, depth and velocity predictions as a surge wave moves along

a drainage pipe at a range of slopes, roughness coefficients and diameters.

Experimental verification of the accuracy of these predictions for both

subcrltlcal and supercritical flow regimes will be a further stage in this

work, however limited verification was possible utilizing an existing NBS CBT

test rig designed to study solid transportation under controlled solid geometry

conditions. Figure 23 summarizes the test rig and associated instrumentation.

The discharge profile from the supply tank into the test pipe was obtained from

the tank depth vs time curve as shown. This profile was then used as input to

the simulation program TRANSCC. The depth vs time recordings at four stations

along the test pipe were produced by means of an array of water sensing pins

set at opposite pipe wall locations providing alternatel)' 0.5 cm intervals

around the pipe circumference at four locations as shown in figure 23. A d.c.

current summing amplifier was employed to provide a depth sensitive signal, as illustrated in figure 23 and presented in figure 24 in comparison with predicted depth profiles at 4 adjacent points along the pipe. The program as used

employed 30 pipe sections to simulate the 5.5 m length, however predicted

results were only output at one tenth pipe length locations, although the program output could in the future be adapted to point predictions at a series of

predetermined locations as an alternative.

The comparison between observed and predicted depths is in general encouraging;

however^ a number of points require clarification. The pipe slope was measured

at 0.02, the Manning coefficient was assumed to be 0.012 based on available

data*, however, the results tend to indicate that the Manning coefficient was overestimated. The comparison indicates a progressive lag between observed

and predicted depth, most noticable at the 4.9 m station. A reduction in

Manning coefficient would also tend to reduce peak depth predictions, however

it is likely that the pins tend to yield overestimates of flow depth due to

meniscus effects.

Similarly errors in the inflow profile, represented in figure 23, may be seen

to contribute to the depth variations close to pipe entry, particularly in the

supercritical flow regime where depth is entirely dependent on the upstream

flow, represented in figure 24.

The depth comparison at the first pin station, at 0.64 m, indicates a rather

more rapid rise in depth than that predicted. It is likely that the local

flow close to pipe entry is subject to splashing and local waves not capable

of simulation in the current program. As expected these local effects subside

before the second pin station at 0.95 m is reached.

The Inflow profile employed to derive the predictions is shown in figure 23,

however in order to meet the simulation requirements the flow is assumed to

continue at 0.01 A/s beyond the completion of the supply tank discharge at

7 seconds, as shown in figure 24. To some extent this requirement contributes

24
to the depth lag between observed and predicted traces, particularly at the

final pin station. A final steady flow of 0.01 l/s In a 0.1 m diameter pipe

Is equivalent to a depth of 0.0027 m and accounts for a major portion of the

divergence between observed and predicted depths at this station beyond the

20 second time point. As discussed previously this effect Is a more serious

defect In the evaluation of the artificial single surge case In the laboratory

then It would be In practice In the study of a complex pipe network due to the

usage patterns of appliances discharging to the system.

In general the results of the limited experimental verification suggest that

the simulation and predictive techniques developed are capable of yielding

Information as to flow depth attenuation. A more detailed experimental program employing longer pipes, gradient adjustment, flow velocity and depth

measurements and possible variation of pipe diameters and materials will be

considered as an extension to the Initial study reported.

25
7. CONCLUSION AND FURTHER WORK

The simulation of flow attenuation presented In this study has shown that the

method of characterististic solution technique applied to the equations

defining unsteady flow in partially filled pipe flow is capable of providing

data on flow attenuation in long drainage pipes. A program of simulated flow

tests indicated the generally expected relationships between flow attenuation

and pipe diameter, gradient and roughness. Solutions to both subcritlcal and

supercritical flow regimes were developed and tested in flow simulations.

Limited experimental verification was also undertaken and, within the limits

imposed by pipe length and inflow profile measurement, the predicted depths

were found to agree well with the measured time dependent variations.

As a result of the study reported further development of the technique is

suggested to include multiple flow input to a branched pipe system. Similarly,

an extended experimental program is proposed to further verify the accuracy of

the flow attenuation predictions made practical by the reported techniques.

26
8 . REFERENCES

[1] Swaffleld, J. A., "An initial study of the application of the nximerical

method of characteristics to unsteady flow analysis in partially filled

gravity drainage sized pipes," NBS Report, August 1980.

[2] Chow, V. T., "Open channel hydraulics," McGraw Hill, 1970.

[3] Swaffield, J. A. and Marriott, B. S. T., "An investigation of the effect

of reduced volume w.c. flush on the transport of solids in above ground

drainage systems." CIB W62 meeting, Brussels, 1979.

[4] Burberry, P., "Techniques for the estimation of flows in foul drains in

the light of present water conservation proposals.” CIB W62 meeting,

1979.

[5] Fox, J. A., "Hydraulic analysis of unsteady flow in pipe networks."

Macmillan Press, London, 1977.

[6] Swaffield J. A., "Prediction of the hydraulic jump location following a change of slope in a partially filled drainage pipe," NBS Report, August,

1980.

[7] Wyly, R. S., "Investigation of the hydraulics of horizontal drains in

plumbing systems," NBS Monograph 86, 1964.

27
Figure 1* Propagation of positive and negative surges In

partially filled pipe flow

28

30

Figure

3.

Application

of

momentum

equation

to

unsteady

flow

In

a

general

open

channel
31

Figure

A,

Modification

to

specified

time

Interval

grid

for

sub

and

supercritical

flow

regimes
z

— N

C

9• m

JJj

J)

32

Figure

5.

Application

of

specified

time

Interval

grid

to

subcrltical

partially

filled

pipe

flow

with

known

entry

and

exit

boundary

equations
Figure 6. Relationship between flow specific energy and flow depth,

Illustrating the boundary between sub and supercritical flow regimes
'

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2 *tan * ( J K ^ ^-Vn)/ ("p/x -O)

I N

2L*tan' ( h-^/iy/CKC'V-K)) )

Figure 7. Summary of pipe cross section and depth

dependent parameters calculations

34
o^r C<«.JL Vn««J-

^\ow j

\o*s OL*t Section

^f«UvctioO ^ s= - ^

rr\o.tvcN‘nv J

\oss

Figure 8. Basis of gradually varied flow depth prediction

35
At

/ .,ur.-fc.c»^ oU^-tK

\_= (^S 1-^VilL.^, ajLK

i A<9c incr«a5«.s Cc^ctTVo.’t ion proceeds.

V. Ca^c tT\a‘^ien ^om

K = V\^ if\itvai\ con^'t ton . V« ^K « CK^ — V\^) f

r\ j

r\ in ran^-4 \0 -

Figure 9. Schenatlc representation of numerical integration

to determine water surface profile

30.

36
8p

37

Figure

10,

Influence

of

the

choice

of

pipe

section

length

on

subcrltlcal

flow

parameter

predictions
Figure 11. Influence of the choice of pipe section length on supercritical flow parameter predictions

38
Flow

DcrTH

Figure 12. Depth of flow predictions for a surge

introduced into aubcritical flow

39
Figure 13. Depth of flow prediction for a surge introduced

into supercritical flow

40
/\

P<p«mSI^ ? w

»»

41

Figure

14.

Comparison

of

surge

response

toa2s

peak

duration

inflow

at

pipe

gradients

of

1/100

and

1/300
s0«

'Si

•» 6

42

Figure

15.

Comparison

of

surge

response

toa2s

and ns

peas.

ixo»

Inflow

ata

pipe

gradient

of

1/300
%p

o

6

r

43

Figure

16.

Comparison

of

surge

response

to

a6

s

peak

Inflow

duration

at

pipe

gradients

of

1/100

and

1/300
' 44

Figure

Flow

attenuation

along

a

30m

pipe,

0.1m

diameter

at

gradients

1/40,

1/100

and

1/300
45

Figure

18.

Flow

depth

predictions

along

a

30m

pipe

at

1/400,

1/100

and

1/300

gradients

In

response

to

a

1.27

1/s

peak

Inflow

range
46
Figure 20. Peak flow rate predicted along a 30 m pipe for pipe gradients

1/400, 1/100, 1/300 and pipe diameters of 0.15, 0.1, and

0.075 m

47
Figure 21. Peek flow rate and depth predictions along a 13 pipe, diaaeter

0*1 illustrating the dependence on roughness coefficient

46
49

Figure

22.

Effect

of

the

initial

steady

flow

on

response

to

a

1.2

1/s

peak

range
Figure 23, fuuery of e*perl»ent«l test rig end associated Inflow and

depth recording lostruaentatlon

50
Figure 24. Coaparlson of observed and predicted water depth

vs. tlse profiles

51
APPENDIX I

Program TRANSCC

52
Program TRANSCC

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

Fortran, together with a flow chart and sample input data. The program was run on the NBS Center for Building Technology Perkin Elmer 732 computer.

The program accepts 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.

The program is written in terms of a series of subroutines that deal with

specific aspects of the numerical solution presented in the report. In order

to aid the understanding of the program each subroutine is discussed individually, with the calculation methods outlined in each case. The titles used

refer to the program print out included in this appendix.

Note that the notation employed in this appendix in describing the various

subroutines is compatible with that used in the main report and not the program

notation, although in most cases the variation is small.

The program as written applies only to simple straight pipes with choice of

constant gradient, diameter and roughness, however these subroutines can obviously be utilized in the modelling of more complex pipe networks in

future programs.

Subroutine TIMING

This subroutine determines the maximum values of wave speed and velocity at

any section along the pipe at each time step. This ensures that the next time

increment

At -

(V + c)

is always sufficiently small to ensure a stable solution. This effectively

removes the need for a time step factor, but this is retained in the program

and normally set to unity.

Subroutine LOSS

This subroutine calculates the equivalent steady flow friction loss over each

pipe section by means of the Manning equation, defined as

S = V |vl n2/m^/3

Note the use of V * absolute value of V to ensure that frictional forces always

oppose the local flow direction.

53
Subroutine DEPTH

This subroutine is only called at the initial time zero to calculate the normal

and critical depths based on the initial assumed flow. The values of h^^ and h^^ are then used to guide the solution whenever a subcritical or supercritical

flow calculation choice arises.

Subroutine INFLOW

This subroutine calculates the inflow to the pipe at any time during the

simulation, based on linear Interpolation between successive pairs of Q, T

coordinates

.

Subroutine SHAPE

This subroutine calculates flow surface width, T, area A and wetted perimeter

depth.

It is also used, at the initial time zero to calculate the terms necessary to

provide the subcritical water surface profile.

It should be noted that this subroutine could be rewritten for any other pipe cross sectional shape and that the program in general is not restricted to

circular cross section pipes. SHAPE is called from many of the other

subroutines

.

Subroutine WAVSPD

This subroutine namely simply local wavespeed based on

and utilizes SHAPE

Subroutine PROFIL

This subroutine is only fully employed if the initial flow is shown to be

subcritical as a result of the normal and critical depths calculated via

DEPTH.

The water surface profile is calculated from an assumed critical depth at

pipe exit by means of the techniques described in the report. The calculation

utilizes SHAPE. If the flow is supercritical then PROFIL sets all depths

equal to the normal flow depth.

c gA

T

54
Subroutine INTER

This subroutine carries out the necessary Interpolations to fix conditions at

points R and S In subcrltlcal flow, and R and S* In supercritical flow,

figure 4 main text refers.

Subroutine ENTRY

This subroutine deals with the flow depth, velocity and wave speed at the

upstream boundary. The Inflow at any time Is calculated from INFLOW and this

flow rate Is then employed. In the subcrltlcal case. In conjunction with the

appropriate C~ characteristic to yield the required parameter values, figure 5 and the section 3.3.4 of the main text.

In the supercrltltcal flow case. Identified In terms of the Initial flow normal

to critical flow depth comparison, the inflow curve is solved in conjunction

with the normal depth relationship as the downstream conditions, represented

by the C” characteristic previously employed, can no longer effect conditions

at pipe entry.

Subroutine NODAL

This subroutine solves the C*" and C~ characteristics at all internal pipe

sections between x = Ax and x = L - Ax, figure 4 main text refers, for both

sub and supercritical flow.

In addition for supercritical flow it also calculates the flow exit condition

as these are based solely on upstream conditions.

Subroutine EXIT

This subroutine is only utilized for the subcritical flow case. The exit

boundary conditions are determined by solution of the appropriate C"*" characteristic, figure 4 main text, with the critical depth expression

ilii - 1 = 0

g

and Q

= VA « A(X2 - Xlh)

As this is effectively a loss coefficient concentrated at pipe exit the

boundary equation must be solved in the form

Q lol T -1-0

g or (X2 - Xlh) |(X2 - Xlh) | JL - 1 g A

55
where | | indicate the absolute value of the term enclosed.

Solution is performed by means of the bisection technique described in the

main text.

Subroutine ASSIGN

This subroutine reassigns the values calculated at the end of a time step,

points P In figure 4 main text, to allow the calculation procedures to move on one time Increment. This Is necessary to avoid program storage space problems.

FLOW CHART PROGRAM TRANSCC

Set up Initial conditions.

Time =0.0

Read pipe data, diameter D, Manning coeff RM, slope S0, length PL.

Read calculation data, N - N“ pipe sections,

TMAX - calculation duration

TFAC - time step factor

Read inflow profile, NPTS - N® of coordinate pairs

Flow - QIN, time TIN.

Adjust flow units from \/s to m^/s.

Calculation of initial steady conditions at time zero.

CALL DEPTH - calculation of normal and critical depths at initial flow.

he < hj^

Subcritical flow

CALL PROFIL - calculate water

surface profile by gradually

varied flow analysis. Interpolate

to yield depth at N sections.

hn < hj.

Supercritical flow

CALL PROFIL - set all pipe flow

depths equal to normal value.

CALL ASSIGN - set up base arrays H, V, C

CALL TIMINC - identify maximum V, C

Calculate time and pipe length increments.

Ax = PL/N

At = Ax/ (TFAC X (V + C)max)

Output - Initial conditions and pipe description

- Table headings

56
- Depth, Velocity, Flow, Wavespeed at each Ax Increment if N £ 10,

at 1/10 pipe length points length points if N > 10 and a multiple

of 10.

A - Update time and commence unsteady flow simulation

CALL TIMIC - identify maximum V, C. Calculate next time increment At.

Time Time + At

Check Time vs TMAX

Time < TMAX Time TMAX

Go to B Go to E

B - unsteady flow calculations.

CALL INFLOW - calculate pipe inflow value

CALL INTER - interpolate to obtain base conditions HR, HS etc. Contains

choice based on flow regime.

CALL LOSS - calculation of equivalent steady loss terms SR, SS.

CALL ENTRY - solves upstream boundary dependant on flow regime

CALL NODAL - solves CT^, C“, characteristics based on output of INTER and

LOSS. For supercritical flow also calculates exist boundary

conditions

.

Supercritical flow. Go to C

Subcritical flow, CALL EXIT - solves exit boundary equation with CT*" characteristic

C - CALL ASSIGN - S€t up base arrays H, V, C.

Output - H, V, Q, C at appropriate points.

Go to A

E - END

57
SAMPLE INPUT DATA, PROGRAM TRANSCC

Line 1

.

Pipe diameter, Manning coeff., slope, length

Format 4F10.A

V7VV 0.1000 VVVV 0.0150 V7VV 0.0100 VV7V 5.000

Line 2.

N® calculation sections. Run time, Time step factor (Note; N*

sections either < 10 or multiple of 10 to match output format,

factor normally set = 1.)

Format 13, 2F10.4

V 10 V7 200.0000 VV7V 1.0000

Line 3.

N® pairs of coordinates on inflow-time curve. Format 13

77 5

Line 4 to 8.

Inflow QIN at time TIN

Format 2F10.4

7777 0.0500 7777 0.0000

7777 2.5000 7777 1.0000

7777 2.5000 777 10.0000

7777 0.0100 777 12.5000

7777 0.0100 77 200.0000

Units - SI units used in data fields except for inflow QIN in

to m^/s in program output in SI, except flow rate in i/s.

calculation

time step

i/s, converted

58
CCCCCL

CCCLCC

N N 'V ^

^ N <N ^ ^

M ^

^ ^

^ *M

^ 'X

:So*09990 m o o o o o o o o o o |8 ®§8

o o e 9 e o o o e 0 0 0 0 9

0 0 9 o o o o

o o

o o

9 O O

0 9 0 0

9 0 0 0 0 9 0

9 0 9 9

O 9

0 9 0

0 0 0 0 0 3 0 0 0 9 0 0 9 0 0 0 0 0 9 0 0 9

9 0 9 9 9 0 9 9 0 0 0 0 0

0 9 0 0 0

9 0 0 9 0 0 0

e 9

_ o o

9 O O

O O O o

o o o o 0 9 0 0

9 0 0

9 O

WOW (O u u

19

19 9 9

«9 <9 U

(9 U <9

«9 «9 <9 19 <9 (9 U 9

w w 9 9

9 9 9 <9 9 9

9 9 9 9 9

o c « sex •sssssssso

S

* « « c s O « s « s OS S S 9 0 S 9 9 9 S S • *«*«9'S9ST 9 9 9 S 9 9

9 9 9 9 9 9

9

99

9 9 9 9

9 9 9 9 990999999 «9u999999

o o © o o 000000393 0000009000 0 0 0 9 0 0 9 9 0 0 9 0 0 9

3 3 0 9 3 3 3 e o o o o o o oooooooooo eoooeoooo

0 3 0

O O

^ ^

V -V >.

9>

t/*

(/*

V* »/*(/< t/' ^ (/* li^ (/* l/f

l/» l/»

O O O O O o o o o e o o

O O O O o o o

o o o o o

e o o o o e o o o e

o o o o 0 0 0 0 0 3 3 o o e o o o

e

9

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o o

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3 3

3 3

0 3 0

O 3

^ -r * * r

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9 9

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o o o ocoocoeoo 03030333'33

3 0 3 3 3 0 3 3 0 0 9 0 O O

O O O O O 3 0

O O 3 0 0 3 3 333333 3 333 000303330O O C O 3

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r X X X X X X X X X o o o

X X z X X X X X z z 3 o 3 3 3 3 3 3 3 •A •9% ^

X X z X o e o e 3 O 3 O 3 O «r IT IT *r m

X X z o e c o O O O r% z X X o o 3 3 O O 3 1/^ m <9^ X X X o 3 3 O 3 3 3

X E X 3 3 e 3 3 O 3 li> fn z z X X e o o o 3 o O O O O IT 0^ X z X z z z X X X X o o 3 3 o 3 O O 3 r\ IT' m

r X X X X X z X X X 3 3 O O o

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X X

9

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9 9 9 9 9 9 9 9 9 9 9

9 9 9 9

9 9 U99999W9 99 99999999

9 9 9 9 9 9 999 999999 9 9 9 9 9 9

9 9 9 9 9 9

9 9 9 9 9 9

9 «r 9 9 9 9 9999999 9 99 99999999

3393?»'»'W^^

3 3 9 9 3 3

3 3 3 3 3 3

3 3 3 ? ^ y 3 3“ S' 3“ 3 3

3 3 3 3 3 3

3 3 3 3

O O

o o o ooeoeccoo 0030000000

3 3 3 3 3 3 3

0 3 0 3'“' 3 3

O O 0 0 3 O O

3 3 3 0 3 3 3 3000033333 000000030O O O

O O

59
iBATCH

C TRANSC ARE A SERIES 3F PRCGRA»-S DESIGNED TO APPLT Tnt NETHGD OF

c characteristics solution tc the eouations defining

C UNSTEADY FLCN IN PARTIALLY FILLED CHANNELS.

CC

DIHENSION CP(61)«ClN(30),TIM3C),VP(oI) yhP (61 1 f CP (61)

DINENSIOn V(61)«H(61)

01 PENSION VF(6l)tHR(6l)«CE(6I}«VS(61)tHS(6I)*CS(6l)«SS(bl)

DIMENSION SR( 61) «XR(6I ) «XS (61 ) «>N(6l) «C(c>l)

:* COMHON/CMl/DTO,CT.OX,0*SO,PLtS£C*XK,RM ; C0mmqn/CM2/n,TMAX

C0MM0n/Cm3/NPTS*CIn,TIn

COMMON / CM^/ V, H,C * Y^J^HRtCRtXR SF» vs * HS , C S , X S » S S * XN

C0MM0N/CM5 /HC*FN

COMMON/CM6/OP ,VP*HP,CP

C0MM0N/CM7/CL

c

CC

CC

CC SET INITIAL CCNCITIONS

TIME»C.O

C READ PIPE CESCRIPTICS DATA

REA0(^,100)D,RM.S0»»L

C D-DIA. RM-HANMNG COEFF. SO-PIPE SLOPE. PL-PIPE LtNOTn

ICO FQChaT(<.F1C.A ) C READ CALCLLA7ICN DATA

RE AD( A,103 )N,TNAXtTFAC

IF(N.LE.IC) N2«l

IF(N.GT.IO) N2-N/10

C N2 INCREMENT BEThEEN QUTPLT N CD E S , ALL OHS MORE THAN 10

C CALCULATICN NODES TO BE USED.

C N-NUMBER DF PIPE SECTIONS CCNSIDEREDt TNAX-uU RATION uF CAlC.

C TFAC-TIME STEP FACTOR, 1-lC.

103 FORMAT! I 3,2FlC.<i)

C READ INFLCH DATA PROFILE. INFLCH PROFILE USED IS BASED On

C A LINEAR INTERPOLATION BETWEEN THESE DATA POINTS. NOTE DATA

C read in IN L/S BUT USED IN PROGRAM IN M**3/S.

REAO( ^ 102 INPTS

102 F0RMAT(I3)

00 50 I-i,NPTS

R£AD(<», 10^)CIN( I ),TIN( I ) IC^ F0P.MAT(2F10.^)

QIN( I )-QIM I) /ICCO .0

50 CONTINUE

Cc

c

cc

C CALCULATICN OF INITIAL CONCITIGNS ALONG THE PIPE

C BASED ON GRADUALLY VARIED FLUK ECUATIONS. THE

C terminal CCNCITIOHS In The PIPE ARE BASED ON

C CRITICAL DEPTH AT «» I ? E EXIT.

CALL OEPTH(TIHE)

CALL PROF iL(CINd) ,HN,HC,XN)

CALL ASSICN(N)

C CALCULATION CF TIME STEP AND LENGTH iECTIJN.

CALL TIM1NC(N,VP,CP,VN,CN)

60
JX-PL/FLOATfh)

OT-DX/CTFAC*(Ch^VN» I OTO-OT

C

cc

cc

C OUTPUT TcST DESCRIPTION PLUS INITIAL CONDITIONS.

NRITcf 3t2C2)D«RN«S3«PL

202 FOPHATIlrtl»/lCX,«TEST PIPE CLNF I CURAT I ON s- '

,

I /lOX, *01 APETER - •*F10.^,« p.»,

I lOX , •panning COEFF ICIEnT;-*,F1C.4i.

I /lOX.'PIPE SLOPE • •tFlC.^t* PIPE LENGTH » •,

1 F10.4 » • rt . •// ) IF«HC.CT.HN) NRITE(3,2001NN,NC

IF(HC.LE.NN) NRITE n,20l)NN,hC

200 FORPATC 10X,*FLCN SUPERCRITICAL, •,*NOR«AL DEPTh »

I

• M.»,* and critical DEPTN - •,F10.A,« N.*,//»

2C1 FORHAT(10X,*FLCN SUBCRITICALf •,*NORr»AL DEPTH - •,

I FIO.^** N. AND critical DEPTH « •*F10.^,* N.*,//J

QIN(i}*QlNm»1000.C

NRITE f 3,2C3}C1N(1) ,HN,CN

QIHd )-GlN( 1)/1GC0.C

203 FORMATCIOX, 'INITIAL FLOW RATE - •fFlO.^,* L/S.*,

1

• initial depth » *,F10.4,» h.',

2 • initial nave speed - •,F10.'|,* N/S.*,/l

Nl»N»l

WRITE 13,20^ IDT, DXfNl,TF AC

2C<i FORNATC lOX, 'CALCULATION TINE STEP • 'tFiG.^,* 5.*,

1 ‘LeNGTH INCFEHENT » N.*,* nUNBER OF NODES ^ *,I3,

2 • TFAC -•,F5.2,/I

WRITE ( 3,7)HP»N41), VP(N^l),OP(N^l),CPtN-,l)

7 FGRMAT(/iCX, •H-*, c1^.6,5a, ‘V- •,E l4.6,/,10X,»0«»,cl^.o,5X,»C»*,

I EIA.6,//}

C OUTPUT TEST SIPULATI3N RESULTS.

WRITE C 3,2C5M XM 1 1 , I»1,N1 ,N2)

205 FQSNATI/ifcX,* POSITION FRCP •,/,l“X,* ENTRY n. • ,c X , li.F 7.3 , / ) WR ITE l3,2Cfc)TIP.E,lHP( I I ,I-1,N1 ,N2J

20o F0PHAT(2X,*TIPE - *,F6.2,* S.*,* DEPT H* , 7X ,

'fl *

• , 1 IF 7. A

)

WRITE (3,2C7HVP(1 I ,I« I,N1,N2)

WRITE f 3,3C7) (CP(l) ,I-1,N1,N2)

3C7 FQPMATCldX,* FLOW RATE L/ 5 « • , 1 IF 7. I WRIT£(3,20e)(CP(I),I*l,Nl,N2)

2C7 F0RNAT(13>,* VELOCITY N/S-* , 1 If 7. A

I

2C3 FORNATI IbX,* WAVE SP3. P/S «

* * 1 IF 7. W ,/

I

C

C

C

C UPDATE TIP.E AND CHECK CALCU’LATICN LENGTH.

6C0 CONTINUE

5ci continue

CALL TIPINC(N,VP,C'»,YN,CN)

OT-OX/ ITF AC*( CN^VN I

OTO-OT

T IHE-TIPE CT

IF ITINE.GT.TPAXI GOTO 500

C

C

rc

cC SET UP CALCULATIONS FOR THIS TIPE STcP.

61
ooo

or>oopoot*>

rj

oooo

ooonoo

OTRAT-DT/CTC

IF I0TRAT.LT.0.C6) OOTO . X

CALL INFLOWCTIHE ,Q

)

Jl-l

CALL 1NTER<F)

Jl -2

CALL LOSSIVRvVSthRvHStNfSRtSSvRf)

Jl«3

CALL ENTRYm^tE)

Jl«A

CALL NCOAL(h)

Jl -5

IF(HN.CT.HC)CALL EXITCTIrtE)

IF (DT.LT.tTC) GCTO 600

OT-OTO

Jl-6

CALL ASSICH(N)

J 1*7

HRITE( 3 , 20 fc)TIHE*MP( I »<RITE 13 * 207 ) (VP m . 1*1 ,Nl,h 2 ) i)RITE( 3 « 307 MQP<l)«I«I,Nl*N 2 ) WRITE

(

3 , 20 e)«CP ( I ) I»l.Nl,h 2 ) GOTO 501

SCO CONTIWUE

END

SUBROUTINE TIMNC(N*VP,CP.VN,CM

THIS SUekCUTlNE IDENTIFIES THE HIGHEST WAVE SPEEu

AND THE HIGHEST AVERAGE FLOW VELOCITY IN

THE SIMULATEL FLCW IN ORDER Tl ENSURE THAT TnE Ti nE STEP

CHOOSEN IS THE SNALLEST, HENCE ENSURING STABILITY.

DINENSION VPItol), CP(61)

CN«0.

C

DO 1 I»l,N^l

IF (CP n l.GE.CN) CN-CP(I)

CONTINUE

VN»0.

0

DO 2 1*1. N-H

IF ( VP ( I ) .GE.VN) VN«VP(I)

continue

RETURN

END

subroutine L(jSS(VR,VS*HR,HS*N*SF,SS.kn)

OINEnSICN VR(tl) */S(oI ) *HR (6l)*HS( 6U*SR(6l)«SS(tl)

CQNN0N/CH5/HC *HN

THIS SUBROUTINE CALCULATES THE ECuIVALENT STEADY STATE

LOSS BASEL ON THE WANNINC COEFFICIENT.

00 I l*2.N^l

CALL SHAPE ITIHE.HR ( I) .A,T,PER )

62
onooooo

>U»

OOO

OOOOOO

oo

oooooo

SR( !)•( ( VF (n*A 6 S( VR( 1 ) )»Rh»A 2 ))/( A/,>Eo 1,^33

1 CONTINUE

NZ«N

NY-l

iFCHN.LE.HC) NY«2

IFCHN.LE.HC) N2-N^l

00 2 I»NY»H2

CALL SHAPE CTiHEtHSC I) ,A«T«PEk)

SS C I) -n VS C I )^ABSl VS( I ) )*PP^*2 ))/C4/PER)**l.333

2 CONTINUE

return

END

SUBROUTINE DEPTHCTInE)

This subrcutine uses a section cf entry to calculate norhal

AND CRITICAL FLCN CEPTHS.

OINENSION CIH(30),TIN(30)

C0NM0N/CM3/HPTS*CIN,TIN

C0MM0N/CN5 /HC *HH

CALL ENTRYITIhE)

RETURN

END

subroutine iHFLCHCTI.iE^CAV)

THIS SUBRCU7IHE CALCULATES InFLCH RATES AT PIPE tNT^t BAS=D

ON THE ENTRY FLOh PROFILE DATA. NOTE THAT THE C CALCJLATEJ

IS AN AVERAGE VALUE FOR THIS TIHE STEP.

OINENSION CI30) «T( 30) fCXOC)

COMNON/CNl/CTC,DT*OX*D,SOtPL ,SEC,XK.Rrt

CGMH0N/Cn3/HFTS.C,T

Tl-TIME

TO«TImE-uT

J«1

TX»Tl

00 3 I«1*HPTS-1

IFCTX.CE.TC I) .AND.TX.LT.TII-H) ) GOTO 4

CONTINUE

UXC J) «2( 1 )«(C ( I«1)>Q( I ) )A (TX-TCI n/(T ( I*i )-T( I) ) QAV«OX

(

1 ) RETURN

END

SUBROUTINE SHAPE (TINE A, T, PER

)

OINEnSICn C1N(3G)«TI 4(30)

C3NM0N/Cnl/CTC.DT.CX,C.S0 .PL •SEC.XK.kN

C0MN0n/C«3/HPTS,CIN. TIN

C0H«UH/Cfl7/LL

63
oooooo

oooooooo

*—

oooo

THIS SUBRLtTl^E CALCJLAI S FLbh AREA,SURFACE KIDTH AhD

WETTED PEPII-ETER BASEL Oi- ? DEPTH

THIS SLBROLTIhE ALSQ CALCULATES »<ATER SURFACE PROFILE

AMO SETS UP BASE CONDITIOriS ALL^JC THE PIPE AT TIME 2ERC.

R-0/2.0

PI»3« 1A2

IF(H.LT.K)THETA»2. 3AATAN4SQRT d-A(O-H)

IF(H.EC.R)TH£TA-PI

IFIH.CT.R ) THETA-PI *2. C*AT AM ( H-R I / { S J RT (HA ( 0-h )» I

)

A-( (DAA2 )/e.C» A CTw-TA-S IN CTHETA)

i

PER-0ATHETA/2.C

T«2.0Af(nA CC-H))A*0,3)

IF (TIME.CT.C.O) C3TC 1 Q-0IN( 1)

C-9.81

C0N-RMAA2/SC

HCR IT-l.O-f CAA2 |at/CGAaa«3)

HN0RM-1.0-<C*A2)aCCN/ ((AAA3,333)/(PE.<AAi.i33)i

DL-HCRIT/ IHNLRPASO

I

CONTINUE

RETURN

END

SUBROUTINE NAVSPC(M,C)

THIS SU6RCUTINE CALCULATES NAVE SPEED BASED ON DEPTH AND CROSS

SECTION SHAPE.

CALL SHAPE (TIME, Ht area, T, PER

)

C-SQR T (9.EIAAREA/T

)

RETURN

END

SUBROUTINE FRCFILI C,HN,HC,XN)

This SU6RCUTINE CALCULATES THE INITIAL HATER SURFACE

PROFILE BASFL ON CRITICAL DEPTH AT PIPE EXIT.

DIMENSION HPIbl),^® (hi) ,QP(6i ) ,CP(61J

DIMENSION X(30) ,DEP(30) ,XN(El

)

DIMENSION Xl( 30) ,DE®1( 30

C0MM0N/CM1/DT0,DT,DX,D,S0, PL ,SEC ,XK,RM

C0MM0N/CM2/N, TMAX

COMMON/CMfc/CP , V P,H o ,CP

C0MHQN/CM7/CL

IF(HN.LE.HC) goto 900

DH»(MN-HC )/30.0

IS = 1 Hl-HC

CALL SHAP£(T1ME,H.1, A,T,PER ) X( 1)»PL

DEPd )-HC

SL-0.0

C HATER SURFACE PROFILE CALCULATIONS.

DO 80 I-I,2C0,2

IS-IS^l

h2»HC*0HAFlCAT ( !!

)

H3«MC«CHAFL0AT(1)

64
CALL SHAPEITlPEfHl «A,T »P£F

)

OLl-DL

CALL SHAPE(TlnE«H2»A,T«PEP)

0L2-0L

CALL SHAPE 4Tl^EfH3«A,T«P£P

)

DL3-OL

OXP-OHP(DL1^DL2^^.CPOL3»/3.0

SL-SL-DXP

HI-H2

IF(SL.GE.PL) CCTC 01

X< IS! -PL-SL

IFCHl.CE.HNJ COTC 93

OEP(

80 continue

81 X(IS)«0.0

NIS-IS

IFCHl.GE.hM GCTC 93

DEP (I S l«nl

GOTO 6A

83 DEP(IS)»HH

GOTO 8^

0^ IF(X( ISI.CT.O.OI C3TJ 05

GOTO 86

95 X( IS»1I»0.0

DEP(IS^n«hN

NIS«IS^I

86 CONTINUE

C

C

C INTERPOLATION FEOUIREO TO CETEfPlNE GEPTM AT EACH NGJfc.

9CC CONTINUE

0X-PL/FL0AT(M

XN ( 1) *0.0

DO 87 I»2,N4l

<N( I J»XNC 1-1)4CX

a? CONTINUE

IFCHN.LE .HC) CCTC 901

N2»NIS*1

DO 9^ J«lfM5

N2-N2-1

Xl( 3)«X(N2 ) OE«»l( J )»OEF IN2 ) HRITE(3f93)XiCJ),0EPl(J)

93 format I lOX ,2F 1C. 5)

95 CONTINUE

00 95 J-l,MS

X( J1«X IIJ)

DEP IJ) «DEP1 ( J)

9 5 C ONT I nUE

HP(l)-CEP(l)

HP(N^l)«HC

DO 88 I«2,N

DO 89 K»L,NIS-1

IF { XN ( I) .CT .X (K J .ANO. XN( I ) .LE .X (K»l ) ) CQTj 90

99 CONTINUE

90 Hp(n-DEPCKj^ccEP(<»i)-CE:fCKn*(xsu)-xi»vn/(x{K^i)-xi'<n

98 CONTINUE

C SET UP 3ASE CCNCITIDNS AT TIME 2EP0.

9C1 CONTINUE

00 91 I-l,N4l

IFCHN.LE.hCI hPCII-HN

CALL SHAPE CTI ME

(

I I ,A ,T *PfcP ) XP C I I *0/A

65
rioor>ooor»o

u*

vn

^

t\i

*->

rioo

oooooooo

QPCI)«Q*10CC.C

CALL wAvsPcihPc ij«CP( I

n

<51 CONTINUE

RETURN

ENO

subroutine IhTERCNJ

D I HEN SI ON V(61)9H(^L) ,C(6l)*VR(tl),Hk(bl) ,CR<bl).VS(oi)

OIHENSION HS(61),CS<3l),SR(bl),SSibl)tXN(6l)

OIHENSIOn XS(el)«XR(bL)

C0MM0N/CHl/CT0»CT»3X,D,S0,PL,SECtXK,Rrt

COHHON/CH^/V,H,C.VR^^R^CR ,XR*SR ,VS *HS t CSt XS *SSt XN

C0HH0N/CH5/I-C »HN

This subroutine sets -up* ev interpolati ON t the base conditions

FOR the next TIHE STEP.

THETA-CT/CX

Nl»N^i

DO 1 1-2,

M

VR( I) - ( V( I I^THETA* (C( I )*V< I-l )-Vn )«C ( I-i ) ) ) 1 / { i.o^thet A*< V (I )- vu-ii^c( n-cn-in ) CRm«(C(I)P( 1.0-VR 1 1 IPThETA) €( i-1 ) VR(i ) TricTA)

1 /{ l.O^CI I )*THETA-C( i-i»*thetai

HR ( n -H( I )-<H (I )-HC I- 1) »» THE TAPIVRI I) CRl I »

)

xR( n-xNi I )-( VR( I ) CR c n )*CT

continue

IFIHN.LE.HC) goto a 30 2 I«l,h

VS( n» (V( I J-THETA* (V(I)PCU^1)-C(I)PV<I»1)))

I /(1.0-TH£TAP(V< I>-V(I ll-CUJ CCI^lin

CS(I l-<C ( I }^vs m»THETAP(CU )-CI l^im

1 /ii.o^THET/*ccm-C(i^in)

hsn-H( 1 )+THETA*( vsm-cs m )p(hi n-Hc !! n

xsm-xN(i)^cvs(ii-cs(m*CT

CONTINUE

GOTO 3

DO 5 1-2, N+1

VS(I)-(VII)*( l.O + THET A^C( I-li )-V( I-l) »TH£TA*C( 1)1/

I < l.O^THETA*! V C I )-y ( I-l )4CU-i)-C( I n ) cs( I

j »

(

Cl I ) -»vs m» (C( I i-cu-i) ) )/( 1.0 THETAPi Cl i-i *-c( I n ) HSI I l-HI I )-<h( I )-H I I-I n*TH£TA*l vs (I l-CSl I I 1 CONTINUE

CONTINUE

RETURN

END

subroutine FnTRYITIHE)

THIS SUBRCUTINE CALCULATES ThE LPSTREAH faOUNOARY CONO‘ITICnS

at EACH TIHE STEP BASED ON A KNCV-N InFL.'m PROFiLt.

OIHENSION OP(6ll,CIN(30),TIM3C),VP(t)i),hP(bl),CP(6li

66
OIMENSICN

3 I HEN SI ON V»46l)tHRfbn«Ci:(6l )«VSf61i *HS(6L)tCSl6i)»^S(6l ) 01 HEN SI ON S^(6l)«XR(6l)«XSf6l}f)'M6l)fC(bl)

C0HH0N/CHl/CT0«0Tt3Xf CfSOvPLtSFCtXKfKi

C0HM0N/Cn2/N«TNAX

COHMON/Crt3/NPTS*OIH»T IN

C0HH0N/Cn4/V«h*C«XR,HRtCR«XR«SF fVStHStCSvXStSSfXN

C0HM0N/CH5/PC»hN

C0HH0N/CH6/CP ,VP *HP, CP

C

C-9,81

C0N-RM^*2/SC

IFCTIHE.CT.C.OCALL lNFLCN«TirF*C)

IFITIhE.GT.O.O) goto bOC

IF(TI PE.E C.C.0)0-CIN( I

)

C CALCULATION CF CRITICAL DEPTh.

UP»0

ON>0.0

HC»«UP^ON) /2.0

7 CONTINUE

CALL SHAPE(TlHE,HC.AREA,TtPER»

MCRIT-1.0«(CAA2 )«T/(GAAR£a*«3

)

IF(HCRIT)3,A»5

3 ON»HC

GOTO 6

5 UP-HC

6 MCN»( UP^OM/i .C

IF(ABS< (HCN-hC)/NC » .LE.C.COl) CCTO d

HC«HCN

GOTO 7

8 HC-HCN

<» continue

C CALCULATICN CF NORMAL DEPTH.

UP = D

0N»0.0

HN=(UP^CN)/2.C

i CONTINUE

CALL SHAPEfTlNE,HN,AREA,T,PEK

I

HN0RH-1.0-ICAA2 )AC3N/{ < ARE A*» 3 . 3 33 » / ( P E k 1 . 3 3 3 )

i

IF(HNCRH) 10*11*12

1C ON>HN

GOTO 13

12 JP»HN

13 riNN»I UP+DM/2 .0

IF (AeSUHNN-HN)/HN ).L£ .O.CCl) GLTO l<i HN»HNN

GOTO 9

14 HN«HNN

11 CONTINUE

IF(TINE.EC.C.O) COTC 500

c

• c

cC CALCULATICN CF BOUNCARY DEPTn.

fcCO CONTINUE

IFCHN.LE.HC) CCTD 7Cj

UP»0

0N«0.

0

HB-f UP^ON) /2.C

15 CONTINUE

CALL SHAPE (T1NE*H3 .AREA, T*PER

)

X3-C/CS(i)

X4-VS (1)-C*ISSC1I-S0)*0T-X3*HS C 1)

67
HFLOH-0-( X<»-*X3*HB » A<£ A IF(HFLOH Jlfcfl7fl8

Ifc UP-HB

GOTO 19

18 ON-HB

IS HBe»(UP^ON )/2.C

IF CABS* (HBE-HB )/H3 I.LE •O.OCl) CCTO 20

H8-H6 B GOTO 15

20 HB-HBB

17 CONTINUE

HP(1)»HB

VPC 1 J-X<i^X3*hP (1)

CALL SHAPE(TlNE*M3.AREA,T*PEk)

QP (1 1 -ARE A* VP m*100 0.0

CP( l»-SORT(C*AREA/T)

500 CONTINUE

GOTO 800

7C0 CONTINUE

C CALCULATION CF NCRIAl DEPTH.

UP-D

0N»0.0

HB»(UP^DN )/2.C

7S CONTINUE

CALL SHAPE IT1PE,H3, AREA, T,P£k

)

HN0RM»1.0-(C^*2)*CQN/1 ( AR£AA*2.333)/(PER«*1.333>)

IF (HN0Rf1}eC,81,82

ac ON«HB

CCTO 83

82 UP-HB

33 HBB-(UP+CN)/2 .0

IF (ABS ( (HfaB-HE )/H3 ).lE .O.CCl) GCTC 8 ^ GQTC 79

3<i HB»HBB

91 CONTINUE

HP ( i)=He

CALL Shape (TINE, H3 , AREA,!, PER)

VP( U-O/AREA

opm «c*iccc.o

CP ( LI »SGRT(G«AR£A/T » aOO CONTINUE

RETURN

END

C

CCc

c

c SUBROUTINE NOCALINI

DIMENSION CP(tl),aiN(30),TlN(3C),VP(oI),HP(61l,CP(hl>

DIMENSION V(tl),H(51)

DIMENSION VR(el),HR(oll,CP<61l,VS(61l,HSl6U,CS<ti),SStbi)

DIMENSION SRitl),XR(bll,XSC61),>M61),C(6i)

common/cmi/ctc,dt,cx,o,so,pl,sec,xk,r

C 0MM0N/Cn3/NPTS,UIN,TIN

COMMON/CM A /V,H,C,VR, HR, C R, XR,SR , VS ,r*S,CS,XS,SS,XN

C0MMCN/Cm5/HC,HN

C0MM0N/CM6/CP ,VP,H“,CP

: THIS SUBROUTINE CALCULATES THE FLOW ^ELOCITr AND DEPTH

C AND WAVE SPEED AT EACH OF Tht NODES BEThEEN THE uPSTkEAn

C and downstream BOUSCARIES EY SCLUTI3.< CF THE TmC -AVE

68
oooooo

r»ooooo

C EQUATIONS.

NZ-N

1 FCHN.LE.HC) NZ-N^l

00 I I* 2 *HZ

Xl-C/CP (I)

X 3 -C/CSC I ) X2»X1*HRC I )¥»( I )-QP( SRC I )-SC)*CT

X^-VS Cl l-X3»HSC I )-S*C SSC I )-SO)*CT

HPCII-CX2-XA)/CXl^X3l

VPC 1 1 -X<»*X3 *hPC I ) CALL MAVSPDCHPC I I f CPC I )

)

CALL SHAPE CTIPE,hP( I) *Xl,X3,XA)

QPC I) -Xl^VPC I )P1000.3

I CONTINUE

RETURN

END

subroutine EXITCTIPE)

OINEnSION CF(bl),VO( 6 l)tHPCei) 9 CP(^I)

OlHtNSION VC61)«HC^1)

OINENSICN VFCbl),HR<ol)»CRCtl),VSCSl),HS{6l},CS(en,SS(bi)

DIMENSION SF(fcl)*XRCbl),XS(blJ*XNC6l),C(bl)

COMNON/Cttl/CTC,DTtDX,D,SO*PL*SEC,XK,RN

C3 HMCn/C«2/N» TNAX

C0NNCN/CM5/HC tHN

CCHMOh/CMA/ V,H,C* VRf 1R fCR,XF * SP »vs tHS ,CS* XS *S S»XN

COHMON/Cnt/OP,VP,H»*CP

THIS SUBROUTINE CALCULATES THE FLCW OcPTH aT PIPE OISChAKGE

BASED ON THE ASSUMPTION OF CRITICAL DEPTH AT SuCH A djUnDA^Y.

C»<5.81

Xl-C/CR CN^ 1 ) X 2 -X1 *HR(N ^1 )^VPCN * 1 )-GP 4 SMN ^1 )-SO)*OT

UP -0

ON-O.

0

HB-(UP^ON)

/

2.0

5 CALL SHAPECTIM£,H3 ,A,T,PER)

H6 XIT-l. 0 -CABSCX 2 -Xl*H 9 ) CX2 -X 1 «HB )}»T/CG»A}

IFCHEXIT) 1 . 2*3

1 0 N-H3

GOTO A

3 - UP-H9

GOTO ^

A H 3 B-CUP+0 H }/ 2.0

IFCABSC CHEB-Hfc}/H 3 >.LT.C.CCCC 1 ) GTTO 2 HB-HBE

GOTO 5

2 CONTINUE

VPCN» 1 )»X 2 -X 1 PHE

HP CN^ 1 )-HE

CALL SHAPE CTI ME *H 3 ,A. T *PEF

)

CALL NAVSPC CHB fCPCN»i I

QP CN»l J-ICOC.OPAPV?CN» 1 ) HRIT£C 3 * 7 )HFCN^ 1 )*V«»(N* 1 ),CP(N 41 )*CP(N^ 1 ) 69
r>

ooo

oo

7 FQS1AT(/1CX, 'H-* fEl^- 6v5s, ®V-*,fi<..6*/,10X»*U-*t£1^.6,5X,*C-*,

I ei4,6*//)

RETURN

cNO

subroutine ASSIGN(N)

C:- THIS SUoRCUTIhE SETS UP THE NEW EASE CONDI TIONS -ALONG THE

C_ PIPE IN PtEFARATIQN FOR THE NEXT TIrtt STEP.

OI»®ENSIOn CP(61)»aiN(30) «TlN(3C)«VP(oI)tHP(6i)tCP(^l)

OIMENSICN V<bl),H(6ll

OIHENSION VF(fcl),HR(6l)*CR(6l)fVS(6i)*HS(ol),CS(fcl»fSS(bl}

DIMENSION SR(bl)tXR(bl)«XS(6l)«XN(&I)«C(bI)

C0NNDN/Cr{3/NPTS»0IN,T I N

CONflON/CNA/V,h,C»VR,-iR,CR*XR* SP VS , HS t C S XS . S S * XN

CONWGN/Cnt/CF,VP,HP,CP

00 1 V( I)-VP ( I ) H< I J»HPU ) Cl I I-CP C I > I CONTINUE

RETURN

END

SBEND

70
APPENDIX II

Typical Output Program TRANSCC

71
FLUN

^tFlhCFITlCAt.*

NlIRhAL

flFTH >

O.ulli

H.

Ahl;

CRITICAL

OLPIrl >

U.OIJF c o3Z

u»3«

XICrzc

xtKUz M f a3u c c c o ^ -r

•«• < • • ^ O — o o o • u o

-• o

->rf o

«• o• • o o z P Qo9Oe9PO

m «• o « p # ^ ^ 9 ^ P *sj • • • • POPP ^ o ^ o < — o o ^ ^ • • • • o a e o ^ 3 ^ — ^ P # ^ «i O O

n i« o ^ o ^ p P ^ N >i t • • • O P o o A O ^

mt mt

O ^

mm ^ o P ^ N ^ • •ft e o o o ^ op # o p # -v

n •« c

•• -•

© ^ p

p ^ • • • • POPPo ^ o ^

«« ««

p ^ • • • • POPP ^ ^ p ^

•f «l

p # O ^ <M o ^

-9

p #

•f ^ p ^ p ^ ^ ^ • • • • P P o P ^ p ^

mm o ^

mm

m 9 f O ’€ >4 • • • • P P P P P 1% ^ ^ P ^

«• «• o ^ p ^ ^

ff|

M o •»

mm ^ 9 ^ ^ ^ 9 ^ P ^ V ^ • • • ‘I 9 P O P m m 9

«•

—

O ^

«• « P O *N *N • • • • o C P P m «• o — — p ^ — o ^ o -r M ^ mm

O r^

mm mm 9 ^ ^ •m 9 P ^ -V N

m ««

O fi»

— -• o ^

•mm 9 ^ P #

PPPP OPPP

m mm 9

*m ^ 9 ^ ^ ^ 5 - p # • • • P P 7

,•% -•

O

M « p ^

«• •• o ^ o *r •*v N • • • • P P p o ^ ^ o •«» — p ^

•M ^ o o < M ^

m-*p^ M««o^ mm m 9 ^ m ^ 9 ^ mm ^ 9 f% P^' 4 >A • ••• •••• POOP PPPP ^ ^ P ^ ^ mm 9 — — P ^ 3

«ri ««

C ^ P O ’t U 9 ^ ^ Nmm e o p p o a P a a a a a P a o u 0 9 o o o C o p p p P o art a P o o o h* 9 o P n» P ••rt art p •rt P P mt -m n «• 9 # P -r P z •4 a^ P Z art .M P z p Z art P z

mm

p P h» «i4 9 1^ mm 9 1^ mt ^ P hrt «M

© •u art art

p 1^ art© • e •SJ e M -g 9 nj •u p nj nj o z nj nj e z nj P z nj nj z nj nj p 9 p n P p 9 n n P p p P p P 9 9 n p p p 9 n n p P P P p P ft o m e iw o « O fn p n» .in mm e *z ia« a n» o n» 9 iw o mm P «• mm P < iM o Z mm mt P z P z -rt Z «rt ft Z 9 P 9 P •y p mm ^ P 1^ P p rt P • o n* 9 M o > 9 < <N 9 * na n# P z N •u P z N W z >* 9 o P P 9 9 O 9 O p 9 o 9 o 9 9 p p O P P p 9 P 9 '9 P P p p P 9 O mm o o -•n Ml o 1^ p mm O fn ^ O 9 •rt art art O <n ft o O mm O < o mm O mm o art M P z art art 9 Z art M P z ft c Z

«• mm P ««

p h* mm i«

O h% «« •IO mm mm 9 art P .•^ art o art ft 9 • P r *v M 9 •r >• 9 r N nj 9 •u 9 a •u •u P z N r z 'g *g 9 z •J O c s o e P e P 9 p 9 P O p 9 9 0, 0. 9 P c p 9 9 9 9 9 C P - o mm P <n P P 9 •• ^ art P rt« © •u n w P •a^ M p P a« o 9 «u a« e 9 z art 3 z art P z art ft 3 o «a« mm O Ml o h« « o mm M P 1^ mm ^ 9 n> Mi Mi O 1^ art irt © h* mm M © Art • P % *>J 9 Nl % P Pi P nj o .r nj nj P z nj 9 z ng >J T z >g o o o o O O O O O 9 9 o o 9 P P 0. 0. O O 9 9 P O 9 9 O 9 P 9 9 9 o o 9 hi* P mt o 1^ .n a* O n O •u M^ c •irt n P 9 p •y P •r mm •• 3 •r 9 Z «u «• 9 z 3 Z p z -rt «M P z

tr> o o iM «* P « 9 h* art artO art art P «Mp art P • P «c ^^• <M o Ni <N» O -N 9 *%! nj o <r fSJ nj P z ng P z Ni P z n* % 9 9 p 9 p P 9 P P P P P P P P P P P P P 9 p P P 9 p P P p P p o r> mm p m P m O 1^ m MM P fw ^ art P hrt in P 3 p % 3 X •M p •« mm p «• «• P P •r 0m P 9 Z art P Z z A Z n* r> r p p P P tm P art ^ 9 «rt •rt P rta rt O P •u n © z • P •Nl P < -w -sj P 'V O •r >< P ^ nj nj 9 z nrt n. z % P * 9 P p P O P 9 P O o P 9 o 9 o o P P p P P p P P P © P P 9 9 p 9 c c hi-. h* c m «« o •* m rt» z M« % 3 3 «M X c mm M c < •4 Ml P mm «0 o •Z mm c iT Z p IS* p <n C Z 3 P s. Z c p P

mm c •«

O mm

Dp

— 9 • nj nj P •m n. • p n<

N * • p r •w P N 9 r P <r % P *n P O n 9 p r o c c O o c O P o o 9 c 9 C o 9 a 9 9 O c P P 9 p = - - - C - z

in mm

p •n P •>m

m z N >• 9- ri» ^ r Z P J- rt P z z O .- «• mm p < «* P Md % < sr h% M P P rt ' hrt a ' «i mm hi* MM O mm «* < rfN iT- P P in IN 9- P rn .>• « w n p P • p < n< 9 .>• P ^ r' 9 9- a « % Z P ->• z P % z p p o 9 c P a a a a a a P a P P a a a a •rt P art P - - 9 P — p p p — - c V* r %n z Jh X *1^ K •n wn c an r r - V V V V S* V ft c mJ z r P X z z Z z z P Z z P r z T ft P • P • UJ • UJ • Urf • P • • aa . • V a e o z to o z o z V C . z < 0. mm &. « ft. « z z ft. rta Z ft « ft w «

•M <9t w or «• « «a or tn Mi

X «n •• * • >» X w X 9 • o X u X a X a p z P « z mm O Z p a z Ml a X mt ^ P z M rt» z w X P X a

••

p

p > €ma > P o > a

a Z ^ art

p z ft P p Z ft art Z ft rta > z p < tmd •y P < w mt rtJ M w < rti p < «rt n «M art W * 9 > z 9 > irf» z a z ly X a > z a > z a z — z — z z art art • • • • • • • • ¥* •n z» P M< 9 M. z

«« e fZ r» o p • • • • • • • • o mm n z z • n i 1 • •

UJ iy Ui UJ UJ UJ *M- u; T r « X Z X z z

tm •» n» t» n» Na ••

72

7^1
tiivo

inv e fte 91 rt o m •M

O K»

m mm e ri

M

O r» rtrtO r» rtrt o #n mm o art o art 9 •> rt rtrt

O r» a«

C Prt r> art c rt 9 9 mm •< o 9 tm o 9 mm a o 9 a a O a mm e 9 •m 9 9 art art o 9 «rt art O 9 rtrt o 9 art art O 9 •rt « 9 •rt art o rt 9 M 9 m a o 9 mm 0 o •rt o Prt rt rt O rt rt O o rt • o rt rt o > r 9 9 N O 9 1^ o 9 o 9 N 'V e 9 N o 9 o 9 <N o 9 % o 9 •rt o 9 o 9 *rt 'rt 3 9 *rt ' • • • t « • • f • • • 0 • t 0 • • • • • • • • • • • • • • • • • • • • • f • • • • • • • • • • • • • 9 e o o 9 9 o o e e o o e o O 9 O 9 e o o e 0 0 0 0 o o o o o o 0 0 0 0 o O 9 O o O o o o O o O o - o frt c

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HBS-114A (REV. 2«>o) U.S. DEPT. OP COMM.

BIBLIOGRAPHIC DATA

SHEET (See in struct/on sj

1. PUBLICATION OR

REPORT NO.

NBSIR 82-2478

2. Porforminc Organ. Report No. 3. Publication Date

March 1982

4. TITLE AND SUBTITLE

Application of the Method of Characteristics to Predict Attenuation in

Unsteady Partially-Filled Pipe Flow

5. AUTHOR(S)

John Swaf field

fi. 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

I. 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, DC 20410

10.

SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11.

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

bi bliography or literature survey, mention it here)

The mechanism of flow attenuation in partially filled unsteady pipe flow is

presented and shown to have relevance to the design of gravity drainage

systems

The equations defining unsteady flow in partially filled pipe are derived and

shown to be capable of solution by means of the method of characteristics.

This technique as a method of predicting flow depth, velocity and wave speed

along a long drainage pipe at a range of pipe gradients, diameters, and roughness coefficients was tested by means of numerical examples for a series of

simulations run on a digital computer. Additionally, limited experimental

verification of the analysis technique is presented for the supercritical flow

response to a short duration inflow surge.

Generally, the technique developed was found to be applicable to the design of

drainage systems and further work is proposed to both extend the experimental

verification and for the greater complexity of the multi-branched pipe system.

12.

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

building drainage; computer model; surge attenuation; unsteady flow.

13.

AVAILABILITY

fy I Unlimited rn Fo'’ Official Distribution. Do Not Release to NTIS

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

20402.

rTT] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

88

15. Price

$ 10.50

USCOMM-DC 6043-P60


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