NBS BUILDING SCIENCE SERIES 139

Application of Method of

Characteristics to Model

the Transport of Discrete

Solids in Partially-Filled

ipe Flow

. DEPARTMENT OF COMMERCE • NATIONAL BUREAU OF STANDARDS
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NBS BUILDING SCIENCE SERIES 139

Application of Method of Characteristics

to Model the Transport of Discrete

Solids in Partially-Filled Pipe Flow

J. A. Swaffield

Center for Building Technology

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

and

Drainage Research Group

Department of Building Technology

Brunei University

Uxbridge

U.K.

Sponsored by: Department of Housing and Urban Development

451 7th Street, SW

Washington, DC 20410

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued February 1982
Library of Congress Catalog Card Number: 81-600198

National Bureau of Standards Building Science Series 139

Nat. Bur. Stand. (U.S.), Bldg. Sci. Ser. 139,116 pages (Feb. 1982)

CODEN: BSSNBV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1982
PREFACE

This report is one of a group documenting 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/Office

of Policy Development and Research, Division of Energy Building Technology and

Standards, under HUD Interagency Agreement H-48-78.

The report was prepared by Dr. J. A. Swaf field, Senior Lecturer, Drainage

Research Group, Department of Building Technology, Brunei University, Uxbridge,

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

Institute of Technology.

Experimental results included in this report are drawn from the published work

of the Drainage Research Group at Brunei University.

iii
ABSTRACT

The flow depth and velocity changes across a moving solid in partially—filled

pipe flow are predicted by means of the application of the method of

characteristics to solve the unsteady flow equations.

Simplified force models are presented which, when used in conjunction with

empirical relationships linking leakage flow past the solid to upstream specific

energy, are sufficient to provide the required moving solid boundary conditions

that allow solid velocity prediction.

A wide range of simulated transport conditions are presented that confirm the

applicability of this technique as a basis for the future evaluation of more complex body force models.

The predicted solid velocity during drain transport is shown to be compatible

with laboratory observations of the influence of solid dimensions and position

in inflow profile on transport characteristics.

Key words: computer based model; drainage; solid transport; unsteady flow

iv
CONTENTS

Page

Notation ix

1. INTRODUCTION 1

2. DEVELOPMENT OF THE NUMERICAL SOLUTION 3

2.1 Summary of the Method of Characteristics Solution of the

Equations Defining Unsteady Flow in Partially Filled Pipe

Flow 3

2.2 Partially Filled Pipe Flow Regimes 5

2.3 Application of the Method of Characteristics Solution to

Drainage Flow 9

2.3.1 Initial Steady Conditions Along the Pipe Length 10

2.3.2 Internal or Nodal Points 11

2.3.3 Entry Boundary Conditions, Supercritical Flow 11 2.3.4 Entry Boundary Conditions, Subcritical Flow 12

2.3.5 Exit Boundary Conditions, Supercritical Flow 12

2.3.6 Exit Boundary Conditions, Subcritical Flow 13

2.4 Application of Method of Characteristics to Waste Solid

Boundary, Stationary Condition 13

2.5 Application of Method of Characteristics to Waste Solid

Boundary, Moving Condition 14

2.5.1 Model of Forces Acting on the Solid 14

2.5.2 Equations Governing Flow Past the Moving Solid 16

2.6 Simulation Cases Covered by the Proposed Model 17

3. CALCULATION TECHNIQUES AND PRESENTATION OF RESULTS 19

3.1 Determination of Normal and Critical Depths 19

3.2 Numerical Integration for Surface Profiles 20

3.3 Choice of Time Step 20

3.4 Presentation of Results 21

3.5 Choice of Calculation Constants 21

V
CONTENTS (Continued)

Page

4. DISCUSSION OF UNSTEADY FLOW SIMULATION RESULTS 23

4.1 Influence of Body Force Model on Predicted Solid Velocity 24

4.2 Influence of Solid Dimensions and Position in the Inflow

Profile on Predicted Solid Velocities 25

4.3 Comparison of Solid Motion from Both Initial Deposition and

Insertion at Water Velocity Models 25

4.4 Comparison of Predicted Solid Velocities to Empirical

Relationships 26

4.5 Limitations to the Simulation 27

5. CONCLUSIONS AND FURTHER WORK 29

6. REFERENCES 31

Figures 33

APPENDIX 1 Program TRANSCD 55

APPENDIX 2 Derivation of Solid Boundary Equations Used in Subroutine

SOLID 85

APPENDIX 3 Typical Output Program TRANSCD 89

vi
LIST OF FIGURES

Page

Figure

1 Application of momentum equation to unsteady flow in a general

open channel 33

2 Modification to specified time interval grid for subcritical

and supercritical flow regimes 34

3 Solution at pipe entry, exit and internal nodes for subcritical

and supercritical flow regimes 35

4 Measurement of the flow specific energy change across a solid

tethered in the waste pipe under steady flow conditions 36

5 Solution of (7*" characteristic with body equivalent B"*" characteristic 37

6 Equation of motion applied to a rectangular section solid 38

7 Calculation of buoyancy forces 39

8 Summary of alternative force models 40

9 Schematic representation of solid motion in the xt plane 41

10 .Modifications to xt grid adjacent to the solid position for

subcritical flow 42

11 Modifications to xt grid adjacent to the solid position for

supercritical flow 43

12 Propagation of steep fronted wave upstream from the solid

position 44

13 Comparison of maternity pad transport in UPVC and cast iron

100 mm diameter pipe 45

14 Influence of force model on the predicted solid velocity 46

15 Influence of solid position in the inflow profile on predicted

solid velocity 47

16 Influence of solid size on predicted velocity 48

vii
LIST OF FIGURES (Continued)

Page

Figure

17 Initially stationary solid, predicted water and solid

velocities illustrating relative flow over the body 49

18 Initially stationary solid, depth upstream of the solid during

motion 50

19 Solid injected into flow, predicted water and solid velocities

illustrating relative flow over the body 51

20. Solid injected into flow, predicted depth differential across

solid during motion along simulated drain 52

21 Predicted solid velocity plotted vs /L/G, pipe slopes 1/40 and

1/100 53

22 Predicted solid velocity plotted vs /L/G, pipe slopes 1/40 to

1/300 , 54

viii
Notation

A Pipe flow cross sectional area,

B"^ Solid characteristic in xt plane

C^yC Notation referring to the positive and negative characteristics

c Wave speed, m/sec

Froude N° = V//gh

F Forces on solid, gm or kg

G Pipe slope or Pipe Gradient

g Acceleration due to gravity, m/sec^

h Flow depth, m

a Solid length, ra

L Distances in flow direction

m Hydraulic mean depth, m

N,n N° of pipe length sections employed

P Wetted channel perimeter, m

Q Flow rate, i/s

SE Specific energy = h + V^/Zg, m

SEq Minimum specific energy required to initiate flow past solid, ra

Sq Pipe slope

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

T Surface width of flow within partially filled channel, m

t Time, sec (also t = thickness dimension solid in forces acting

on body, m).

V Local mean velocity, m/sec

Vg Solid velocity, m/sec

ix
NOTATIONS (Continued)

w Solid width, m

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

step

X Distance, positive in initial flow direction, ra

a ~ sina - Angle of solid relative to pipe

At Time step, sec

Ax Pipe section length, ra

0 At/Ax, sec/m

p Fluid density gmsec^/m^

Tq Wall to fluid shear stress

Suffixes

A,B,C Calculated points in an x-t grid at time t

abs Absolute velocity

P Calculated points in an x-t grid at time t + At

R,S Interpolated points in an x-t grid at time t

re Relative velocity

U,D Upstream and downstream conditions relative to the solid boundary

X
1. INTRODUCTION

The prediction of solid transport in partially filled drainage pipe flow

requires a knowledge of the forces acting on the solid and their relative

importance. The treatment of the force and momentum equations for the body and

surrounding flow is analytically complex; however, a numerical model capable of

simulating the unsteady flow equations could be used to evaluate models of the

body forces for comparison to laboratory observations.

1
Previous reports [1,2] have developed the application of the method of

characteristics solution to the unsteady partially filled pipe flow equations.

The introduction of a moving solid into the simulation requires the knowledge

of leakage flow past the body as a function of upstream conditions, with a suitable model for the forces acting on the body. The passage of the simulated

solid through the pipe may then be treated as a moving boundary condition by a similar technique to that applicable to the tracing of a shock wave or steep

fronted wave motion.

The necessary computational techniques to allow inclusion of the solid boundary

conditions are presented in this report with computer simulations involving

three force models. The predicted solid motion characteristics are also com- pared to laboratory observed solid motion in drains which are set at a wide

range of gradients.

The development of the method of characteristics solution is also summarized,

and there is a statement of the necessary boundary conditions at the drain

inlet and exit in both subcritical and supercritical flow regimes.

It is stressed that the objective of the study was the development of the

necessary computational techniques to demonstrate the feasibility of the method

coupling the hydraulic and solid motion. The development illustrates the applications of the basic modeling approach and the simultaneous solution of the

governing equations of hydraulic interaction with that of the motion of the

solid. A basis has been established for future comparison with development of

other body force models representative of the test conditions for a wide range

of body configurations for which observed laboratory solid transport results

are obtained.

2
2. DEVELOPMENT OF THE NUMERICAL SOLUTION

2.1 SUMMARY OF THE METHOD OF CHARACTERISTICS SOLUTION OF THE EQUATIONS DEFINING

UNSTEADY FLOW IN PARTIALLY FILLED PIPE FLOW

Referring to figure 1, it has been shown [1] that the equations of motion and

continuity may be expressed as

g ill + g(S - S ) + V II + 11. = 0 (1)

8x o ax at

3
VT-ii+ T ill + A

^ = 0.

9x 8 1 3x

(2)

These equations may be combined to form a total derivative expression,

dV + g dh + g(g _ s ) = 0, (3)

dt c dt o

subject to

i2i = V + c , (4)

dt

the wave speed being defined by

c = gA

T (5)

where V = local average flow velocity

c = local wave speed

h = local flow depth

Sq = pipe slope

2„2

S = slope of the local energy grade line = ^ ^— , n the Manning

m^/3

coefficient and m the hydraulic mean depth

T = water surface width

A = water depth cross sectional area.

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

equations may be written in terms of the unknowns at point P, by means of a first order approximation,

Vp - + g_ (hp - h^) + g(S^ - S^)ht = 0 (6)

^R

xp - XR = (Vr + CR)At (7)

Vp - Vg - 8_ (hp - hg) + g(S^ - S )At = 0 (8)

Xp - Xg = (Vg - Cg)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 C"^ and C~, shown in figure 3.

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 2.

4
From figure 3 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 is set out below.

2.2 PARTIALLY FILLED PIPE FLOW REGIMES

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

(1) Subcritical flow, Froude N° = -^r ^ ^

Here the local wave speed exceeds the flow average velocity, thus waves

may be propagated both upstream and downstream in the flow, i.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, i.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.

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 (where A is the pipeflow cross section

area)

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

simplest of the open channel equations:

V C / m Sq (10)

V

Q (11)

where Q is the flow rate m-^/s

A is the flow cross sectional area, m'

5
The value of the Manning coefficient, n, varies with pipe or channel material;

values In the range 0,009 to 0,020 apply to for materials commonly found In

building drainage systems.

Equation 10 effectively determines the flow depth under steady, uniform

conditions with 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

From equation 12 It may 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:

E = h+-^

2gA 2

^=0 = 1 ^ , (13)

dh g^3 dh

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, he 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 h^^ 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.

6
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 2 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 affect point P.

The slope of the C~ characteristic, PS, becomes positive, and both R and S lie in the AC section as shown.

Both subcritical and supercritical flow are encountered in the drainage

applications considered and the equations derived below cover both conditions.

Referring to figure 2 for subcritical flow:

Vr * OR)

and

-\ _ ^C ^R _ ^c - ^A ^C " ^A

^c " ^R _ ^C ~ ^R _ ^c - ^A ^C " ^A

he - H _ (Vr + cr) - - ^A

At

Ax

as xp = -KQ and xp - xr = (Vr + CR)At.

The solution yields

R

1 + e (V(, - + cc - c^)

. c,(l - Vr 9) + C,V, 0

^^^^ ^ I + - CpQ

hR = he - (he - hA)(0(VR + cr)) (18)

where 0 =At/Ax.

Similarly,

7
i-e(v^-v, -cc+c,) ^^^^

cs = 11^^

^""^ ^

(20) 1+0 (cc - Cg)

hg = he + 0(Vs - cs)(hc - hg) (21)

For the supercritical flow regime, the equations determining R in figure 2 remain unchanged. The interpolation equations for S' in figure 2 may be

determined by an identical technique to that shown above,

Vj. - Vs« = (Vc - V^) 0(Vs' - cs')

cc - CS' = (cc - ca) 0(Vs« - cs')

and he ~ hs' = (h^ - h^) 0(Vs' - cs')«

The solution yields

V^(l - c,0) - V,cp0

Vet = ^ ^ ^ (22) l+0(Vc-V^ + c^- Cc)

S

1 + c^0 - cc0

From equations 7 and 9 it will be seen that

dt = 1 (24)

dx V + c

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 [3] 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 Lheir dependence on pipe geometry and flow rate.

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

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

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

(figure 2).

8
(li) Values of V, h and c at Interpolation points R, S, or S' calculated

from equations 16 - 23.

(iil) 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

.

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

2.3 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

subcritical or supercritical.

The equations 6 to 9 may be restated as

Vp = X2 - XI h

C+ (25)

Xp - XR = (Vr + CR)At

Vp = X4 + X3 hp

C- (26)

Xp - Xg = (VS - cs)At

where XI = g/cR

X3 = g/cs

X2 = Vj^ + g g(S^ - S^)ht

CR

X4 = Vg - g ]^ - g(S5 - S^)At.

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 2,

9
2.3.1 Initial Steady Conditions Along the Pipe Length

As will be seen from figure 3, 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 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

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

profile prediction technique.

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,

-;^{ ^+ (Zo - SoD + h 1 =

^ (27)

dL 2g ^^2/3

where (Zq - SqL) is the elevation at distance L along the channel, measured

in the downstream direction; Sq is sin 0, channel bed slope.

hence _ V dV + s dh = [ nQ J

2

g dL ° dL Am2/3

and as, Q = VA

10
dV A + V dA = 0

dL dL

and as = T from equation 5, it follows that

dh

dV = V dA = - VT dh = _ QT dh

dL A dL A dL A^ dL

'

and substituting in equation 28 yields

Q^Tdh . c _ dh ^ J n Q i gA^ dL dL A (29)

A

dL = { 1 - Q^T/g a3

I S - (nQ/Am2/3)2

hi

1 - Q^T/gA^ L = /

^

h S - (nQ/Am2/3)2

o o

dh (30)

where L is the distance between two known depths hg, hi

.

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.

2.3.2 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 2.

2,3.3 Entry Boundary Conditions, 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.

11
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

2

1 - —(nQ) = 0 (31

a2 s o

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

the bisection technique, this technique is described later.

2,3.4 Entry Boundary Conditions, 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

Q = f(t) = ViAi

Vi = X4 + X3 hi

Q(t) = Ai (X4 + 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

Q(t) - f(hi)(X4 + X3 hi) = 0 (33)

this boundary equation may be solved by the bisection technique.

2.3.5 Exit Boundary Conditions, 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 C^ and C~ characteristics. With reference to

figure 2, 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.

12
2.3.6 Exit Boundary Conditions, 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:

T . = 1 o ^crit ^» 8 crit

where A and T are f(h).

This condition may be solved with the C*" characteristic

V^+i = X2 - XI hfj+i

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

The boundary condition becomes

[(X2 - XI hN+i) An+i1^ Tn+L -1 = 0. (34)

8^N+1

The solution may again be achieved by use of the bisection method with the use

of the area to depth relationship for the channel.

2.4 APPLICATION OF METHOD OF CHARACTERISTICS TO WASTE SOLID BOUNDARY,

STATIONARY CONDITION

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

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

conditions.

Figure 4 illustrates the relationship between flow past a stationary solid and

the specific energy upstream. These results were compiled during a Brunei University Drainage Research Group study of solid transport in drainage systems.

The flow past the solid (experimentally determined) may be expressed as

Q = K(h + SE„)2 (35)

2g o

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

required for flow initiation past the solid.

Equation 35 may then be solved with the CT*" characteristic, of figure 5,

Vx = X2 - XI hi

13
where Q = Aj

so that

At(X2 - XI hj) = K hj + — (X2 - XI hj)^ - 2g

This expression results in a quartic in terms of depth upstream of the solid,

hx (see appendix 2). This quartic must be solved by an iterative technique as the flow area, Aj^+i , is a function of hj. The Newton Raphson method may be

employed to carry out the necessary iterative solution. Once the value of hj

has been determined, the value of and cj may be determined.

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

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

less than that of SEq, then the value of flow velocity at the solid at time

t + At is set equal to zero. The flow depth then comes directly from equation

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

to the initation of flow past the solid. This solution is set out in detail in

appendix 2, and it has been shown to be capable of simulating depth increase

upstream of a stationary solid [11.

It should be noted that the analysis above applies to both subcritical and

supercritical flow regimes, as only the C"^ characteristic is involved.

2.5 APPLICATION OF METHOD OF CHARACTERISTICS TO WASTE SOLID BOUNDARY, MOVING

CONDITION

Prior to application of the moving boundary conditions representing the solid

motion, it is necessary to determine the solid motion initiation time. This may

be accomplished by monitoring the net force acting on the solid.

2.5.1 Model of Forces Acting on the Solid

Figure 6 illustrates the forces assumed to act on the rectangular section

solid, namely, hydrostatic pressure forces, Fpj^ , Fp2, acting on the trailing and

leading edge projected areas, body weight force, frictional force due to wall

to body contact, Fg, and a bouyancy force, Fg, dependent on the solid position

in the flow.

The net force acting on the solid may be expressed as, figure 6,

25 as:

hi = X2/X1. (36)

Fpi - Fp2 + mg sine - Fg = FgoDY (37)

where Fg = f(mg cose - Fg)

14
and f is the wall to solid friction factor, its value being reduced from the

static friction coefficient to the sliding friction value if F30DY becomes

positive.

If FgoDY ^ ^> then solid motion is initiated and the solid velocity at the end

of a computational time step At is given by:

where AV = V|.+^^ ~ ^t TM ~ mass of body. (39)

(note values of V^. = 0 for the first time step)

Hence. AV - V^^^ = ^t^W" (*°>

The distance traveled by the solid in the time step may be approximated by

= X + 1(V ^ ^ + V )A (41)

B o 2 t + At t t

where Xq is the solid position at time t and Xg the final position at the end

of the time step.

Equations 37 to 41 apply to subsequent time steps with the mentioned

modifications to the value of wall to solid friction coefficient in equation

37, The net force on the body may become negative in subsequent time steps;

however, this represents solid deceleration and no modification to the equations

is necessary until the predicted value of V^- + (equation 40) becomes

either zero or negative (the condition for the body coming to rest).

Figure 7 illustrates the forces assumed to act on the solid. The calculation

of the buoyancy force, Fg, (equation 37) requires the solution of the body force

system. Values of Fj^, F2 and Fp may be approximated by the hydrostatic equation; however, the force F^ is not readily estimated due to the curvature of the

solid to pipe boundaries.

Laboratory observations have shown, that the model solids, maternity pads [4] , due to their flexibility, tend to take up the shape shown in figure 7, Taking

moments about the leading edge point A, as shown in figure 7, allows F^ to be

estimated.

The bouyancy force Fg may then be determined as

j^B ^ ^^n ~ ^d) cosa (42)

where a is the slope of the solid surface relative to the pipe wall.

15
Figure 8 illustrates three force models investigated in this study:

Model 1 - buoyancy forces neglected, the value of Fg = 0

Model 2 - downthrust force Fj) = 0, The justification for this model

comes from test observations. In the boundary equations for

the solid leading edge, i.e. downstream, the leakage flow is

assumed to take up its normal depth at once. Laboratory observations indicate a downstream transition length so that the

flow depth immediately ahead of the solid is less than that

predicted.

Model 3 - all forces shown in figure 7 included.

2.5.2 Equations Governing Flow Past the Moving Solid

It may be assumed that the relationship between specific energy and flow past

the solid may be employed in the moving solid case, provided that the absolute

fluid velocity employed in equation 35 is replaced by a relative water to solid

velocity. The value of SEq, the specific energy term to initiate flow past the

body, remains unchanged.

Equation 35 is therefore rewritten as

(V - V^)2 2 Q = K(h + —^

SE^) (43)

2g o

where Q is the leakage past the moving solid, because

. Q = V^gA = (V^bs - Vb)A = (X2 - Xlh - Vb)A (44)

where A is the flow area upstream of the solid and V^^g is the fluid velocity

expressed by the C"*" characteristic, figure 5.

Appendix 2 presents the full solution to this boundary condition in a general

form, applicable to both the initially stationary and moving solid.

The prediction of solid velocity allows the solid path to be drawn in the x-t

plane as shown in figure 9. The B"*" lines drawn in the plane are the equivalent

to the fluid characteristics; the gradient of B"*" is hence given by

dx/dt = Vg. (45)

Figures 10 and 11 illustrate the necessary techniques to allow the solution to

proceed with a moving solid. A slightly different solution is required,

depending on whether the flow is subcritical or supercritical.

Figure 10 presents the subcritical case. Assume that the solid was at point C

at time t and is predicted to move to P' by time t + At. In order to calculate

16
the conditions at t + At, It is necessary to set up the interpolated base values

at R and S; therefore, for conditions immediately upstream and downstream of the

solid, a new Ax grid A'P' to P'C must be set up.

Using the position P' predicted from solid velocity, the points A' and C are determined. The conditions at R and S are then calculated by interpolation

(equations 16-21).

Point R may be used to yield the characteristic RP ' that may be solved with

the solid leakage equation 43 to yield depth and flow velocity upstream of the

solid.

The conditions on the downstream face of the solid are calculated by a similar

technique applied to C'B'. The C~ characteristic is solved with the flow rate

at the solid by the technique utilized at pipe entry in subcritical flow

(equations 32-33).

Points W, X and Z, figure 10, may be dealt with by the nodal point equations,

25 and 26, as the necessary interpolations are not affected by the presence of

the solid at P '

.

Conditions at P and Y, however, cannot be readily calculated due to the B"*" characteristic. Conditions at these points, however are required as base con- ditions for the next time step. As the Ax and At values are small, it is rea- sonable to determine conditions at P and Y by interpolation between X and P'

and P' and Z respectively.

Figure 11 illustrates the solution for the supercritical case. It remains

necessary to approximate conditions at P and Y and the conditions downstream of

the solid are determined by the use of equation 31 (the normal depth calculation).

2.6 SIMULATION CASES COVERED BY THE PROPOSED MODEL

Two types of solid motion must be covered by a model of the type presented.

The motion of the solid subsequent to injection into the flow with a downstream

velocity must be dealt with, as this represents the introduction of solids with

finite velocities in the drain from water closet discharge. Similarly, the

motion of a deposited solid in response to a flow must be considered, as this covers the partial system clearance that results in any long drainage pipe

exhibiting a series of depositions along its length, each of which is moved on slightly by each incoming flow.

Both cases may be dealt with by the proposed model. The injection case is

covered by assuming that, at some instant of time and position, the solid

appears in the flow, moving at the local flow velocity that was calculated at

that section by the characteristics solution run up to this time with no solid

present

.

17
The deposited solid case, at any downstream location, is simply dealt with hy

the technique described above and more fully in [11. In this case, the solid

is assumed to be present from the first computing time step.

Similarly, the model must be capable of dealing with both subcritical and

supercritical flow regimes. The techniques for these cases have been set out;

however, care must be taken in the supercritical case.

Figure 12 illustrates the problem encountered in supercritical flow. In the case of a stationary, or even moving solid, the depth increase upstream of the

solid may be sufficient to produce a subcritical zone in an otherwise supercritical flow condition. In order for the upstream supercritical flow to

become subcritical behind the solid, a depth change wave, or series of depth

change waves must be propagated upstream. If the depth change upstream of

the solid is rapid, as would happen with a rapidly increasing inflow profile,

the depth change waves propagating upstream at the wave speed /g A/T combine

to form a steep fronted wave that moves at a velocity greater than /g A/T, and

hence the solution breaks down.

If the inflow profile is not sufficiently rapid, then the transition from

supercritical to subcritical flow may be accomodated by introducing a flow

condition check into the calculations at the stage represented by figure 2. If

the velocity at a section at time t is less than the calculated wave speed at

that section in an initially supercritical flow condition, then the flow locally

is assumed to have become subcritical and the interpolated values, (figure 2)

switch from R, S' to R and S.

The speed of propagation of a steep-fronted wave may be calculated and included

in the model in a manner similar to that described for the solid motion. Further studies will include this facility. The test cases presented in this

report were not affected by this effect, due to the inflow profile shape and

the initial position of the solid in the pipe.

18
3. CALCULATION TECHNIQUES AND PRESENTATION OF RESULTS

3.1 DETERMINATION OF NORMAL AND CRITICAL DEPTHS

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

depth

X = 1 - Q2T/gA3

19
and normal flow depth

Y = - (n Q/Am2/3)2

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, 0 < h < D is bisected and h = D/ 2 used to evaluate X, Y.

If the resulting values are positive, the root is less than the midpoint; then,

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

interval 0 to D/2 bisected. The process is repeated, with the upper limit

replaced until the value of X and Y become zero, i.e., the solution desired

for the critical and normal flow depths. 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 would be intially reset to the trial h value, D/s (D/s < h < D) , and the

interval bisected to a new value of 3/4D, The process is repeated until a root

is obtained.

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.

3.2 NUMERICAL INTEGRATION FOR SURFACE PROFILES

The integration of the position vs depth profile

h2

L = / 1 - QT^^/gA^ dh

h^ S^ - (nQAm2/3)2

^1

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

^0

then, if the interval - hQ is divided into 2 equal increments, the value of

is given by

X = ^ dh [F(h^) + 4F(h^ + dh) + F(h„ +2 dh)].

2 o o o As the integration moves on, the length traversed may be accumulated as the

added interval X with the prior L, L = L + X, at the completion of each

integration.

3.3 CHOICE OF TIME STEP

Referring to figure 4, 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

20
This expression yields the smallest possible time step, as the maximum values

of flow velocity and wave speed at any pipe section have 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.

3.4 PRESENTATION OF RESULTS

The objectives of the numerical method for computation of the transport

mechanisms of solids impartially filled pipe flow were to: (a) identify the

potential application of coupling the method of characteristics solution for

the hydraulic phenomena with the solution of the equation of motion for the

solid based upon modeling the liquid/solid interface forces; and (b) highlight

any limitations inherent to the technique. The simulated pipe flow/solids

motion numerical data were developed from use of the Fortran computer program

TRANSCC, run on the NBS CBT Perkin Elmer 732 computer.

The following test cases were investigated and are reported here

(1) Solid types — two cases represented by figure 4, 270 mm x 20 mm x 70 mm

and 270 mm x 20 mm x 35 ram with saturated weights of 250 g and 125 g

(2) Pipe gradients — 1/40 to 1/300

(3) Inflow profile — constant profile employed with peak flow of 4.2 il/s,

overall duration 9 seconds

(4) Pipe roughness coefficient — constant at 0.015

(5) Solid friction factors — 0.10 sliding and 0.15 static

(6) Solid position in flush, 3 s and 4 s from flow initiation in the moving

solid insert case, at 2.7 m from entry

(7) Solid position — 4.2 m from entry for deposited solid

(8) Force models — three models as illustrated in Figure 8,

3.5 CHOICE OF CALCULATION CONSTANTS

In order to undertake the calculation procedure described. It is necessary to

have values of leakage constants and initial specific energy for the solid.

In the simulations presented, the values of these constants are drawn from the

21
results illustrated in figure 4, referring to transport tests at Brunei using

large deformable solids.

The surface to pipe wall friction factors are also required. No values are at

present available for the deformable solids tested; however, similar tests con- ducted in the Plumbing Research Laboratory at the National Bureau of Standards

utilizing impermeable cylindrical solids suggest values for friction factors

in the range 0.6-0.8 for the sliding and static cases. It is likely that the

deformable solids will have lower values, due to the presence of the material

saturating water that will tend to provide a degree of lubrication. All the

simulations presented were carried out with values of 0.1 sliding friction

factor and 0.15 static friction coefficient.

22
4. DISCUSSION OF UNSTEADY FLOW SIMULATION RESULTS

As outlined in the introduction, the objective of the study was the development

of a technique, based on the method of characteristics, that would allow the

transport of discrete solids in partially filled pipe flow to be modeled

mathematically.

It was recognized that insufficient data on the frictional characteristics of

model solids used in laboratory test programs at Brunei and NBS were available

to attempt to predict actual solid transport velocity profiles. However, the

23
models of the forces acting on the solid were based on current research at

Brunei and NBS, and the values of friction factors assumed were, as already

discussed, based on the best available Information.

It is more important in this effort to demonstrate the ability of the analysis

techniques to yield the solid velocity and the associated flow depth and velocities during the transport along the pipe. Subsequently, modifications based

upon improved force models, friction factors derived from tests and other

empirical adjustments, can be introduced to fit predicted data to experimental

results

.

From the examples computed for the conditions of 3.4, it is worth noting the

form of the relationships of solid velocity to pipe length and gradients.

Figure 13 represents typical empirical results for the single maternity pad

solid tested at Brunei. This solid has dimensions 270 mm x 20 mm x 70 mm, a saturated weight of 250 g, and a specific energy to leakage flow relationship

in a 100 mm pipe as shown in figure 4. This solid, with a half width version,

was used as a basis for the computer simulations presented.

It will be seen from figure 13 that the velo city of the solid over the major

length of the drain is governed by the /L/G term; i.e., the square root of

distance travelled divided by pipe gradient. This effect will be studied by

plotting the computed velocity results against this term. Wherever possible,

simulations are presented at pipe slopes of 1/40 and 1/100 in order to represent both supercritical and subcritical flow regimes. In addition, these gradients represent the common spread of slopes employed in drainage system design.

It should also be noted that the water depth and velocity results presented

are immediately upstream and downstream of the solid, and thus refer to a location that moves down the pipe at the solid velocity.

4.1 INFLUENCE OF BODY FORCE MODEL ON PREDICTED SOLID VELOCITY

Figure 14 illustrates the predicted solid velocity for the three force models

presented in figure 8. It was found from the values of force predicted by

these models that, for the range of model sizes simulated, the downthrust on

the leading surface of the solid materially increased the surface frictional

force. This is represented in figure 14 by the observation that at both 1/40

and 1/100 gradients, the model 3 results yielded the lowest solid velocity. As

expected, therefore, the exclusion of the downsthrust force, Fq, (figure 8)

yielded the highest solid velocity. The omission of Fp is reasonable on the

basis of laboratory observations. The water depth tends to require a transition length downstream of the solid to achieve the normal depth appropriate

to the flow past the solid. Hence, the water depth predicted by the analysis,

the effective normal depth, will be an overestimate of the depth of this

location.

Figure 14 is also based on the assumption that the solid appears in the flow

at a particular x,t coordinate traveling at local water speed. This is a

24
reasonable assumption that leads inevitably to a rapid deceleration of the

solid accompanied by a buildup of water depth immediately upstream of the

solid. The adjustment to a new water-driven velocity is rapid, and any non- realistic effects due to the insertion model are likely to decay rapidly.

All force models were, used in the collection of the data presented; each figure

carries a note as to the applicable model.

4.2 INFLUENCE OF SOLID DIMENSIONS AND POSITION IN THE INFLOW PROFILE ON

PREDICTED SOLID VELOCITIES

Laboratory experiments and data analysis presented in [4] have shown that the

volume available behind a solid in the flush significantly determines its

transport properties.

Referring to figure 15, it will be seen that introducing the solid at 2.7 m

from entry at 4 seconds into the flush, as opposed to 3 seconds, does support

these observations. Although the predicted differences are small, they are consistent.

Similarly, laboratory tests have shown that base area, defined in the case of

a rectangular model solid (figure 6), as thickness, t, times width, w, is a major factor in determining transport performance. Small base area solids

(e.g., tampons or sheets of toilet tissue) travel badly when compared to larger

solids. This is due to the increased flow past the solid that reduces the

depth buildup behind the solid, and from equation 37, leads to smaller body

forces. Figure 16 clearly demonstrates this effect for both the subcritical

flow at 1/100 and the supercritical flow at 1/40.

4.3 COMPARISON OF SOLID MOTION FROM BOTH INITIAL DEPOSITION AND INSERTION AT

WATER VELOCITY MODELS

Figures 17 and 18, and 19 and 20 compare (for slopes of 1/40 and 1/100) the flow,

solid velocity, and depth changes for a solid accelerated from rest to that for

a solid assumed to enter the flow at local water speed.

A number of general observations may be made from these results that find

confirmation in previous observations of laboratory tests:

(1) At both pipe gradients, the maximum depth upstream of the solid occurs in

the acceleration from rest case.

(2) Perhaps more surprisingly, the maximum depth occurs in both test cases in

the steeper pipe. This is a direct result of the application of

equation 36.

In the supercritical flow at slope 1/40, the velocity term is greater than

at 1/100, and consequently, the flow depth is less. Thus the "destruction"

of the flow momentum by the partial blockage formed by the solid results

25
in a greater depth change in the steeper pipe. Similarly, this enhanced

depth change leads to a higher force acting on the solid and results in

the earlier motion of the solid in the 1/40 pipe case. Figure 17 indicates solid motion at 1.4 s for 1/40 and 4.2 s at 1/100 pipe gradients.

(3) Both the depth upstream of the solid as it moves along the pipe and the

associated solid and water velocities display oscillations. This is

entirely attributable to the choice of time step, and the fact that the

simulation equations 40, 41 are only first approximations; no return

iterative technique has been introduced. Subsequent time steps may then

underestimate the solid velocity. These oscillations are reflected in

the water depth upstream of the solid. This link between upstream depth

and solid velocity is clearly demonstrated in figure 18 by the sharp dip

motion at t = 1.4 seconds. All simulations involved a grid length of

15/50 m, time step dependent on wavespeed.

4.4 COMPARISON OF PREDICTED SOLID VELOCITIES TO EMPIRICAL RELATIONSHIPS

Figure 13 indi cated that solid velocity is dependent on the experimentally

derived group /L/G,

Vg = ci - C2 /l7g ,

where C^ and C2 are experimentally measured.

Figures 21 and 22 present simulated solid velocity results plotted against the

/L/G term for two force models and pipe slopes from 1/40 to 1/300.

The _results indicate that the solid velocity in each case is linearly dependent

on /L over the major portion of the pipe length. The dependence on G~l/2 is

not clearly defined by the results; however, it is clear that a pipe gradient

terra is present and would have an index greater than 0.5. Alternatively, and

more probably, the deviation is due to factors not yet included in the force

models, for example, the surface to water shear force arising from the water

flow over the solid has not been included, as the surface shear stress is

unknown. However, the results are encouraging in that the general form of

the predicted solid velocity curves approximates the observed laboratory

results. This also indicates that the force models employed were reasonable,

as well as the values of surface to wall friction coefficients assumed.

The forces acting on the solid have not been presented in graphical form, as they were generally found to remain roughly constant during any simulation,

although exhibiting the oscillations mentioned previously. Typical values for

the zero downthrust model, figure 14, at 1/40 were -0.2 N during the initial

deceleration, falling to -0.01 N during the subsequent motion. At 1/100 the

comparable values were in the range -0.2 N initially, falling to around -0.02 N.

26
4.5 LIMITATIONS TO THE SDIULATION

As fully discussed in [2] , the method of characteristics solution requires an initial flow in the pipe that continues beyond the termination of any inflow

profile. This effectively means that a simulated solid will achieve a terminal velocity in the pipe, or alternatively, will be deposited and moved on continuously as the residual flov/ acts to increase upstream depth. This effec

is not readily apparent from the results presented due to both the relatively

short pipe length, 15 m, and curtailed run duration of 30 seconds. For the

purpose of investigating the force model to be used, this limitation is not

major and, indeed, could be duplicated in any parallel experimental work.

27

5. CONCLUSIONS AND FURTHER WORK

The objective of this study was the evaluation of the coupling of the method

of characteristics with the equation of motion of a solid based upon an assumed

force model to provide a numerical analysis base for modelling the transport

of discrete solids in partially filled pipe flow.

The results presented show that the method is applicable, and that the motion of

the solid may be satisfactorily tracked through an x-t grid representing the

29
pipe by introducing a solid characteristic whose slope in the plane is governed

by the forces acting on the solid.

A wide range of simulations in both subcritical and supercritical flow

regimes yielded solid velocity results that were compatible with laboratory

observations

.

Further work is required to establish the true values of the solid sliding

friction coefficient, with further study of the force models to be used as the

moving solid boundary condition. Similarly, the specific energy vs. relative

flow rate over the solid requires further investigation to extend the range of

solid geometry available at present.

30
REFERENCES

Swaffield, J. A. "An Initial Study of the Application of the Method of

Characteristis to Unsteady Flow Analysis in Partially Filled Pipe Flow.

NBS Report August 1980.

Swaffield, J. A. "Application of the Method of Characteristis to Predi

Attenuation in Unsteady Partially Filled Pipe Flow." NBS Report Octobe

1980.

Fox, J. A. "Hydraulic Analysis of Unsteady Flow in Pipe Networks."

Macmillan Press, London 1977.

Swaffield, J. A. "Dependence of Model Waste Solid Transport

Characterstics in Drainage Systems on Solid Geometry, Mass and Pipe

System Parameters." NBS Report August 1980.

31

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A 'I \ oo Qradi^ftt

t ^ o s a= o.:i-f|s

O r> = 1.0 = 1^. 2

® ='=15 = o. z

o

- 30.0 s ' o.-x-?j$, 000 g,

^A

AAA ^

10 20 30

7.^

Figure 21. Predicted solid velocity plotted vs /L/G, pipe slopes

1/40 and 1/100

53
6 0 O ^ -c

II n n n ^ r> /Z

<n ^ i/i

II II K » U ^

-P V

V II i 4J .£ ^ ^ vo

t6

'o0 Jii

V

It) -Hd

'5 3

o

>

II

L A

^c D- +j

01 4)

e

J)c o

. 0

-p

II «

+>

c

to

o

0

II 0

0-

0> C 0- ^

_1 vn

" J. « ^ O >P "

/A

^ ^3 ^

Q

/ a

«3 Q

1 o ^ a ^

/

a 13 «3 / o / o

/ / o

54
APPENDIX I

Program TRANS CD

55
Program TRANSCD

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

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,

which 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 printout 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 subrutine calculates the equivalent steady flow friction loss over each

pipe section by means of the Manning equation, defined as

S = V |V1 n2/ra^/3.

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

always oppose the local flow direction.

56
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 0, 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

In this subroutine, the local wavespeed is simply 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

57
Subroutine INTER

This subroutine carries out the necessary interpolations to fix conditions at

points R and S in subcritical flow and R and S' in supercritical flow

(figure 4).

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 subcritical case, in conjunction with the

appropriate C~ characteristic to yield the required parameter values referred

to in figure 5.

In the supercrititcal 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 CT*" and C~ characteristics at all internal pipe

sections between x = Ax and x = L - Ax (figure 4) for both subcritical and

supercritical flow.

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

which is 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) with the critical depth expression

g a3

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 IqI T _ 1 = 0

g a3

or (X2 - Xlh) |(X2 - Xlh) | = 1 g ^

58
where I I indicate the absolute value of the term enclosed.

The 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 to allow the calculation procedures to move on one time

increment. This is necessary to avoid program storage space problems.

Subroutine INSERT

This subroutine ensures that the solid upstream water velocity, depth, and wave

speed are equated to those calculated at the insertion time and pipe section

preset in the input data. The solid velocity is set equal to the fluid velocity at this stage. INSERT is only called once; at subsequent time steps,

subroutine SOLID calculates the conditions at the solid.

Subroutine SOLID

This subroutine calculates the water velocity and depth upstream of the solid

by means of the specific energy vs. relative flow rate boundary equation. The

flow conditions downstream of the solid are determined in the same manner as ENTRY, dependent on flow regime.

Subroutine FORCE

This subroutine calculates the force acting on the solid. Initially it

determines the inception of motion by monitoring the force acting on the solid.

Various force models may be investigated by modifying this routine.

FLOW CHART PROGRAM TRANS CD

Set up initial conditions.

Time = 0

Read pipe data - Line 1 data table

Read calculation data - Line 2 data table

Read inflow profile - Lines 3-8 in example data

Read solid position - Line 9 in example data

Read solid dimensions - Line 10 in example data

Read specific energy to flow over solid coefficients - Line 11 in example data

Read force model indicators and friction coefficients - Line 12 in example data

Adjust inflow rate from lis to m-^/s solid dimensions from mm to m solid

saturated weight from g to kg.

59
Calculation of initial steady conditions at time zero.

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

he < hn

Subcritical flow

CALL PROFIL - calculate

water surface profile and

interpolate for section

depths

.

^n <

Supercritical flow

CALL PROFIL - set all depths

to normal value.

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

CALL TIMINC - identify maximum V, C to yield stable time step.

A" = PL/N

At = Aoo/(V + c)n^x

Output - Initial conditions and pipe description

- Solid dimensions and model indicators

- Depth, velocity, wavespeed at each increment if N < 10,

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

A - Update time and commence unsteady flow simulation.

CALL TIMINC - Identify new time step based on (V, C)i^x

Check Time vs. TMAX

Time < TMAX Time > TMAX

Go to B Go to G

B - Unsteady flow calculations.

CALL INFLOW - Calculate pipe inflow rate from data profile.

CALL INTER - Interpolate to obtain base conditions HR, HS or HS ' etc.

Contains choices based on flow regime and presence of the

solid beween any two computing points.

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

CALL ENTRY - Solves upstream boundary condition for either sub or supercritical flow.

CALL NODAL - Solves C+C~ characteristics based on output of INTER and LOSS.

Omits the two nodes bracketting a moving solid and the intial

location node of a stationary solid. In supercritical flow,

NODAL also provides exit conditions.

60
For subcritical flow only, CALL EXIT - exit depth based on critical depth at

pipe discharges.

Check solid model being simulated:

Injected solid, time < injection time, Go to C

Injected solid, time _> injection time, on the first occasion, Go to D

Initially stationary solid. Go to E

Injected solid, time > injection time, following first occasion, Go to E

D - CALL INSERT - assigns water velocity at solid entry station and time

to be solid velocity.

Go to C

E - CALL SOLID - calculate conditions on upstream and downstream solid faces.

Downstream velocity given by same technique as ENTRY dependnet on flow

regime.

Go to C

C - CALL ASSIGN - sets up base arrays for next time step.

Output of simulation results.

Velocity, depth, flow and wavespread at solid and at 2 points each 1/10

pipe length upstream and downstream of the moving solid.

Solid velocity and body forces.

Solid position.

Check presence of solid, no solid Go to B, solid present Go to F.

F - CALL FORCE - calculation of forces acting on solid over next time step.

Calculation of solid position and velocity at end of next time step.

Check solid still in pipe at end of next time step:

Solid position > pipe length, set solid indicators to no solid and continue

calculation to TMAX, Go to B.

G - Program terminates.

SAMPLE INPUT DATA, PROGRAM TRANS CD.

Line 1.

Pipe diameter. Manning coeff., slope, length.

61
Format 4F10.4.

WW -.1000 WW 0,0150 WW 0.0100 WW 15.0000

Line 2.

N° calculation steps, Run time, Time step factor. (Note, N° calculation

sections < 10 or multiple of 10 to match output format, time step factor,

normally set = 1 .

)

Format 13, 2F10.4

V 50 WV 30.0000 WW 1.000

Line 3.

N° pairs of coordinates on inflow-time curve.

Format 13

W 5

Line 4 to 8 (note actual number depends on Line 3).

Inflwo QIN or time TIN

Format 2F10.4

WW 0.2000 WW 0.0000

WW 4.2000 WW 1.0000

WW 1.2000 WW 4.5000

WW 0.2000 WW 9.5000

WW 0.2000 WV 50.0000

Line 9.

Solid initial position in pipe, must

location. Anplie'? to both initially

Format 13

V 10

be at a section not an intermediate

stationary and injected models.

62
Line 10.

Solid dimensions, length, width, thickness, saturated weight.

Format 4F10.4

VV 270.0000 VVV 70.0000 WW 20.0000 VV 250.0000.

Line 11.

Solid minimum specific energy to allow bypass flow. Specific energy vs. Q

coefficient.

Format 2F10.4

WW 0.0200 WW 0.6000

Line 12 .

Buoyancy force indicator, 0. force omitted, 1. force included. Downthrust

force indicator, 0. force omitted, 1. force included. Solid surface shear

stress, set 0. Sliding friction factor, static friction coefficient.

Format 5F10.4

WW 1.0000 WW 0.0000 WW 0.0000 WW 0.1000 WW 0.1500

Line 13.

Solid insertion time, zero for stationary solid. Solid insertion model

indicator, zero for stationary solid, > 0 for moving solid.

Format FlO.4 13

WW 4.0000 W 4

Units - SI units used in data field except for inflow QIN in £.s, converted to

m^/s in program. Outut in S.I., except flow rate in Is. Solid dimensions

and saturated weight read in mm and g, converted to m and kg in program.

63
1^

r>t r>t I

H

^ *V ^vj "V -N

N "v* "M og fV (V ^ <N N % IV

fg N IV

fV N N IV IV

IV >V N IV IV f» N "V (V "V «J IV IV N IV "V % IV N IV

IV 'V ^ IV IV IV IV

IV IV IV IV IV 'V

IV "V N IV r» N

IV IV % <v

oeooocc — o O O O 3 OS SOS 3 S

O O O 3 3 3 O 3 3 O S C soeo3;;wac

3 3 3

03333300 3 53-:33333Q

3 3 3 3 C 3 2 0

C JO 3 3 3 '53-5T'»-5'33'^3 3333333333

O 9

3 O 9 ooooooooo 0003030309 9 O 3 3 3 3 3 OS 3 O O SO OS O O O OS OS O S O O O 3030^>3 = S00 0000000303 O O O O

9 S

« » « r i » * -e X X (c

«> ^ '«) w

J J -> J W 'J O ^ U l_> u

3

3 33333S33 3S33330S03 3 C O 3 3 O O

3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3 3 3 3 O 3 OCC30C3000 003033033

^ ^

• 1^

> .* ^ ^ y1 ^

3 e 330330000 -533~33r>33 0- 3 3 3 3 3 3 3

3 O 3 3 3 3 O

O 3 O O 3 3 O 3 3 3 3 3 3 O 3333333333 3^33333333 C 3 o o

0 «

O « 3 J o -e c c c s e-B-c-c-ecco

n ^ .* f r

rzrr rrzrrt

rrnzzzTZTr

I 3 3 333333333 3333S3330S O C 3 C O 3 ©

3 3 3 3 3 3 S

3 O 3 O 3 3 O

3 O 3 C C 3 e C0333CQ0O0 333CS30003 3 3 C C

>j ^J "s. ^ ^ -V. -v. "s. ^ ^

% % % -V. -w M N( % ^ -* - %,

•VJ X •»

« a « « www w w w w w w

w w www w w w w w w

w w 09033333 oeoscs3c 9 o o s o o e o o o O 9 O 3 3 O O 3 C 3 09000-03 9 O O 3 3 O —

64
tBATCH

C TRAMSC ARf A SERIES QF PR0CRAH5 CFSIGNCO TO APPLY TH£ METHOD OF

C CHARACTERISTICS SOLUriOH TO THE EQUATIONS DEflNlKC

C OKSTEAOY FLOM IN fAi^TIALLY FILLED CHANNELS.

C

C

01 HENS ION 0P(61>,QIN( 30 I ,T INI 30 1 .VPf bl } ,HP 161 ) .CP ( 61)

OIHENSION V<61)*H(61I

01 HENS I ON Vf^(61),HRf6l),CR<61>«VS(61l«HSt6 1l,CS(6l ).SS<61)

DIMENSION SRl6l>«X<^(6l)«XS(61l,XNtMI ,C(61I

COKMOH/CMl/LTG»UT»OX,D,SO*PL,SEO.XK,iM

C0MK0N/CM2/hi THAX

COMMON /Crt 3 /NPTS»OIN,riN

COMMDN/CM'*/V,H,C,VR,HR ,CR,XR,SP, YS ,HS,CS,XS.SS,XN

COMH0N/CM5/HC »HN

COMMON /C M6/0P » VP, HP tC P C0MM0N/CM7/DL

COMMON/CMfc/ INSOL,NSOL l,NSOL

COMHON/CM'5/SGLVELtHOS ,VDS,CDS.HPLS, VPOStCPDS

COMMON/CMl C/XPCS XSnL,VPS0L,*SrL,O«»SUL.HuStVJS»CuS7HPUS,VPUS,CPUS

COMMON/CMl 1/TL,Th,TT, SH

COMMON/CMli'/Xf AC2, XE AC,TAU,f MC V,f STAT

c

CC

c

CC

C SETINITUL CCNCIT fQr<S

TlME-0.0

VPSOL '0.0

C READ PIPE DESCklPrns DATA

READ! 'itlOC )r , PKtSO

C 0-DlA. PM-HANNIMC C1EFE. SO-PlFt SLOPE. PL-PIPf LfcNGlH

100 FORHAK^FIC.', ) C READ CALCLLAl 1 OH 0*1^

READ(^,103)^»1^AX,TF4C

IF(N.LE.IO) IS2-I

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

C HZ INCREflfhT fiETH'-N OUTPLT hCTES.ALLOhS nORc THA^ lu C CALCULATICh NCCES TT BE USED.

C N-NUMRFR OF PIPE ScCriQNS CChSICEPED, T MA x-Oj k AT 1 0'- OF CalC.

C TFAC-TIhE STEP FA:n-<,l-lC.

103 FORMAT! I3,2F1C.A)

C k£AO INFLCV LATA P-^QFILf. INfLCW PRPrlLE USti) IS dASfL. jN C A LINEAR INTE PPCLATI )N 3ETWEL^ Tr-FSE DATA POINTS. NOIt QftTA C READ IN I^. L/S BUT USED FfOCEAH I« M«*J/S.

RE A0( ^, 102 )h? TS

102 F0RMATU3)

00 50 I-1,NPTS

ftEAD(^«10A)01N(I)«riS(I)

10^ FORMAT! 2F1C.^)

0IN( I )-01h ( I) /lOOO .0

50 CONTINUE

C READ SOLIC CtSCPIPfWE DATA, M St)L- IN 1 T I Al POSITILN,

C TL-LENCTh,TV— »«10TH,TT-THICKNtSS,Sw-SATu»tTEU «T.,

C SEO-SPECifIC ENEPGr -lEOUIl-fL If INlTW.i: ELJ« PAiT

C SOL ID ,XK-ft€'h Tfl SPECIFIC E^^^CY COtrf. AI SP C VALUfS ABOVE SEO

C TAU-SOLID SUfvfACf C^ffF Of SH( A ». , F Hi, y - SL I J 1 NG SOlID

C 10 PlPr N/Ll FRICTins COi F F ^TAT-S TM K i iLiu lu

C Pi PI HALL FFiCTICS C F F .

65
C soLTiM-Tirr vnsiTiaM of sclio ih inflow profile. note

C ZECn VALUfc i: 'JIC/T£S STATIL.^AkY IN PIPE AT OESICNATEO

C SECTION KSr ! , I NSCL-lsniCATCR CF SOLID VEL OC I TT tCT .

0

C INDICATES THAT FLT** ^'ELOCITY AT SECTION NSOL IS ASSUrtED

C AS CALCULATION START PQIMl,

REA0I4,102I HSCL

IF INSOL.EO.OIGCTO 7l<i REAOMtlOO} IL«TM,rT.SH

READM.IO^) SEO«XK

REAOf 4«109)Xf AC«XFAC? .T AU«FnOV «FSTAT

109 F0RrtAT<5F10-^l

READ! <i , 105 ISOLT IM, I KS OL

105 FORKATiFlO.^, 13)

729 CONTINUE

KSOLl-NSOL

IFJINSOL.GT.O I NSOL->> c

c

c , c

c

cC CALCULATIOh OF INITIAL CONCITIThS ALONG THE PIPE

BASED ON GPADU^LLY (VARIED FLOW EUUATIONS. THE C TERMINAL CTNLITIONS IN TH£ PIPE A^F dAStu ON

C CRITICAL DEPTH Al «»If»E EXM.

CALL DEPTH«71ME

)

CALL PROf 1L<C1H»1I .H-itHCXN)

CALL ASSICNIN)

IFITlMf.LE.SCiLTIM.ANJ.lNSCL.CT.C) VSJL>^V(NS0l)

C CALCULATION OP Tllf STfP A^C LFNoTH SECTlON.

CALL TIMINC(N,VP,CPtVN,CN)

OX = PL/FLO>tT(N)

DT-DX/(Tf AC«( CN^VH)

)

DTO=DT

c

C CALCULATION CF SOLID INITIAL PrSiTIO.-t

IFINSOLl.CT.C) XS3L»L)X*FLLAT(hSCLl-l)

IF<NS0L1.CT.C ) XPJS = lIX«FLCAT(NSCL1-1 ) c

c

c

c

C OUTPUT TEST DESCRIPTION PLUS INITIAL CONIjITIjNS.

WRITE ( 3, 202 ID. RM» SO, PL

202 F0RMAT( lHl,/10X,nEST PIPE CONF ICUCAI I Qk: , 1 /lOX, 'OI ANf TE R « •»FlO.'i,« f .

* , I iOX, 'MANNING COEFFICIENT:/-*,? 1C.1t

1 /10X,«PIFE SLCFE - '.FIO.^,' PIPE LENGTH • »,

I FIO.^ ,» «.•,//)

IFJHC.GT.HN) WRITE n.200)Hf,hC

IFfHC.LE.bN) HR ITE ( 3, 201)»-N,HC

200 fOR«AT(lOX,*FLOH SUPERCRITICALf •.•NO^inAL DEPTH - *«flO.<i»

I • H.«,« AND CRITICAL QEPIh « •,F10.'.,' «.'//)

201 FORMAT! lOX, 'FLCH S Uf<C R I T I C AL , '.•NORinAL OEPTri -

1 F10.^,« n. AND CRITICAL DEPTH « •.FlO.'i,* «.*,//!

QiNd »-QIN<l)*10C0.0

WR I Tf « 3, ^03 )tj IH < 1 > tMH ,CN

01Ntli-OINU)/1000.0

203 FORMATdOX, 'INITIAL FLOW RATI - '.FIO.S,' L/>.',

1 • INITIAL DEPTH - '.FlO.'i,' N.',

2 • INITIAL hAVt SPEED - ' , f 1 0 .

* M/S.',/l

Nl-N* I

66
WRITE <3»20«i»L>T,rx,Nl« TFAC

20* fORriAKlOy.'CALCULAriCN TIME STEP - •.FIO.*,* S.*.

1 •LEMCTM iNCREhLNT - •tElO.4,* «.*»• HUMBtR W HOOES - 2 • TFAC -•,E5.?,/»

If (NSOLI.EO.OI GOTO 730

NRITE(3«220)

220 f ORMATClOXt'SCI.fD OFSCRIfTIO* 0*TA.«./l

MftITEI3*221)TL«TM« T T , S M, NS CL 1 • X SOL

221 FORMAK lOX, •LENGTH -•,F7.2«* WIDTH -'.FT.i,

7 • HM.«,« THICKMESS -•,F7.2,» SAlURATtS HT.

2 F7.2,« CM.',//,

j. lOX, "SOLID POSITION, SECTICN HUhBER »»,I3,

4 • DISTANCE FROH ENTRf -•,F7.3,* «.•,/!

MRITEJ3,222)FSTAT,FMQV,TAIJ

222 FORHAT{/ICX,« STATI C FRICIICH FACTOR « •,F6.3,

1 • SLIDING FRICTION FACTOR - •,F6.3,

2 /,10X, 'SOLID SURFACF SHEA^ COEFF. ' •,F7.A,/i

«R1TEC3,2 71)XFAC,XFAC2

271 F0RHAT(10X,«eCLTAHCY FACTCR XFAC •,F5.2,/

I ,10X, 'DOWNThkLST FACTOR XFAC2 ' ' , F . 2 , / ) IF(SOLTIM.CT.C.O) WRI TE ( 3 , 22 3 ) SCLT I

223 FORMAT (/lOX, • SOL 10 PJSITION IN INFLOn PROFILE ^ *,

I F6.2,* SECONDS FROM FLOW I NCC PT I CN .• , / /

)

TL=TL/1C0C.C

TH-Tw/1000.0

TT«TT/1000.C

SH-SH/1 OOC.C

730 CONTINUE

C OUTPUT TEST SirULATIJN RtStLTS.

IF (NSOL .GT.O) GOTO 7i3

WRITE ( 3,205 ) < XM I» ,1 « 1 ,N1 ,H2)

205 FO(<NAT(/iey, ' F0SITT3N FRf^ •,/,13X,» ENTRY N .

' , fc x , 1 If 7 . j , / t WRITE I3,2C*. )T INE ,IHPI 1 » , I-l,Nl,Si»

206 F0RMAT(2x, M1^E - '.Fb.Z,* 5.',* D f p r H • , 7 * ,

• n= * , i i F 7 .

)

HRITE{3,2C7)«VP(H,I='l,Nl,N2)

WRITE ( 3, 3C7)( CP ( 1) , 1= 1 ,N1

)

307 FORNAT( ItJX, • FLOW RATE L/ S= ' , 11 F 7. ) HRITt(3,20E)<CFin,I=l,Ni,N2)

207 FORMATdtiX,' VELOCITY f / S -= • , IIF 7. A ) 203 F0RKATC18X,' WAVE SPD. M/ S

•= ' , 1 IF 7. A , / ) C uOTO 73A

733 CONTINUE

N3-NS0L-2*N2

KA»=NS0L-1.2

H5«NS0L*N2

H6-NS0L-»2«N2

IF(N3.LE.O) N3-NS0L-2

IF(N4.LE.O) N4,-NS0L-1

WRITE I 3,3C^>) VSOL

305 FORMATI/^>«.X,• SOLia PG S I T I CN* , /5^X,« SOL 1 0 VELOCITY',/,

I 5*X,» - • ,F6. 3,» N/S. •

WRITE(3,2 05)XN(l),KN<N3»,XN<N^I,XShL,XS0L,XN(N5»,XNJNbI,*-ifNl)

WRITE(3,2 06)TIHE,HP(L),HP(h3),hP(N<.),HPUS,HPuS,hPtNS),HP<Nb),

1 HP(N1

WRlTE(3,207)VP(lJ,VPlN3»,VP(NAJ,VPUS,VPLJL»VPjN5),<'P(Mo>,Vi'(Ni)

4RlTE(3,30 7lCP(l),aP(S3),GP<N',J,CP5QL,OPSuL,OH(N5),t,P(Nf)).JP(.Nll

WRITE<3,2C6}CPll),CPtNJ),Cr(N^),C.>'US.CPCi,CP(N'-),CP{'io),cP(Nx>

73^ CONTINUE

Z

cC UPDATE TINE ;HC CHfCK CALCtLAllLh LtriiiTn.

600 CONTINUE

67
501 COHTIHUE

CALL TlrlHC(N,VP,Cf ,VN,CNl

OT-DX/f TFAC*«CH-»VN1 » OTO-DT

TIME-T IHE*DT

IFCTIME.CT.TflAX) COT'] 500

c

c

c

c

cC SET Of CALCttATIOHS FOR THIS Tl^£ STEP.

OTRAT-DT/DTO

¥SOL-VPSUL

IF(DTRAT.LT.0.C6) GOTO 500

CALL iHFLOHfT IHE,QI

Jl-l

CALL INTER^^)

Jl = 2 CALL LOSS <VR,VS,HR,HS,M,SR,SS,f ^)

Jl»3

CALL ENTkYITihE)

CALL NODAL I N)

IFdlKE.GE.SLLTlr.lNj.NSOl .CT.C) CALL SOLlOlTIt'tJ

IF(TI MF .GT .SLLT IM. »NJ . INSLL.CT .( ) CALL I h S £ R T ( X N , T I «L

)

IF (TI .Of .SCLT IH. . NSGL .CT . f I XPGS=aSul

IF (HN.CT.HC )CALL E < I T ( T I rtC ) IF{DT .LT.CTC) GOTO 600

UT=DTO

CALL ASSIChtN » IF(TI .Gt .SGLTIM.4'<) .NSOL.CT .0) CfiL- F OK CE < F I HE , XN ,F S Ol I IF (NSOL.GT .C) GCTJ 7il

Jl»7

787 CONTINUE

WRITE(3,206)TlhE,(HP(I),I=l,NI,K^)

Hkl TE < 3,2 07) ( VP ( 1) , I = I ) HRITE{3,3G7)(CF{I),I^1»N1,>>.2)

WRITf(3,2fFHCF(I),I"l,Nl,N^)

GOTO 501

731 COHTINLIE

N3»NS0L-2*N?

N^«NS0L-H?

,M5-NSGL*n2

N6-NS0L*2*h2

IF(N6.GE.hI» H5-NS0L*!

IF(K6.CE.NU ht'=NSaL»2

IFCNSCL.GE .H-U NSOL»0

IFCN3.LE.C> h^-NSO'.-l

IF(N3.LE.0I h3-NSaL-2

IF (NSOL .E 0 .0) KRI T'^( 3t22^)

22^ FORHAT I ///30X , 'SLL 10 HlTHlN ^ SFCTIOMS (If PiPt D I SCHAi' G f . * , / . 1 30X,'SnLlG *SStM£3 TJ CLEAf P I F I , C a L C UL A I i C N T f « h 1 N A H , S JL J I

1 'JN ' 2 »/30X ,

• C.JH7 IhUES Win SPLIT FRFF PIPc Fluh. ',///)

IFIHS0L,tO.C)WFlI£(3,205)XhJl),)M<Nj),xNiM'i),XPGS»tPjS»Xh(N5)»

IF (NSnL.fc CO GOIJ 737

MRITf J 3,3tc ) SOLVEL

303 FOS HAT ( > ,* SLL 10 ve L OC IT Y ' ,/ 5^>, » = •,fb.3»» H/S,'l

IF ( VSGL.^T .C. 0) GOTO 5 77

68
GOTO ^78

577 CONTINUE

WRITE! 3«2C5)XNI 11,XH( (131 «K»4(f<^ },XPOS»XPOS« XNlH51,XH(Nbl «XN(H1 ) $78 CONTINUE

MRITE f 3«206)T1>1E»HPI 1» «HP(»43)..f4PIN« I.HfUS«HPOS«HP f NiltHPtNb ). 1 HP INI I WRITEf 3,207)VPt 1I«VP(N3} «VP(N4 >«VPUS« VPDS . VP! H5 I » V P (H6 ) , VP C N 1

MRlTEi 3t30 7)CP <ll,ClP(p<3)»QPIHAI ,CPSOL.OPSOL«QP(hb* »0PiH6l t QP<N1 I ttRITE(3«2C8)CPCl)*CP(N3)«CP(N^l«CPUStCPDS«CP(N&)«CP(K&) tCPCNII

GOTO 501

300 CONTINUE

£N0

SUBROUTINE TI H1NC< N,VP ,CP« Vh,CM

COMHJN/Cr.e/ IhSCl NSQLI.NSOL

COHMQ N /CHS/ SOLVE L »HDS .VOS .CDStHPOS , VP OS, CP OS

COMMON /C«1C/X PCS, X SOL, VP SCL,VSCL ,0P5 JL , HU

S

, VU

S

, CU

S

, HPUS ,

V

PU

S

, C P uS C THIS SU3kCUTlNE IDENTIFIES THE HIGHEST HAVE SPEED

C AND THE HIGHEST AVERAGE FLCW VFLlCITY IN

C THE SlMUL'TtL FLOW IN ORDER TO EnSUK; THAT THt T I ?i E STEP

C CHOOSFN IS THE SMALLEST, HfhCE ENSURING STABILITY.

DIMENSION VF(61), CPtf)l»

CN«=0. 0 DO 1 I»1,N*1

IFd.EQ.NSCDCCTO 1 IFICPd ).CE.CN) CN»CP( I)

1 CONTINUE

VN-0.

0

DO 2 I -I.N !

IF( I.EQ.NSCL) GOTQ 2 IF ( VP ( I ) .OF .VN I VN«VP I I I 2 CONTINUE

IFICPUS.GT.CN ) CN^-CPJS

IF JCPDS.GT.CM CN-CP3S

IF (VPUS.GT.VN » Vh VJS

IF(VPDS.GT.VhJ VN-VPJS

RETURN

END

C

C

c

cC

C

c

C SUBROUTINE LOSS(VR,VS,HR,HS,N,SR,SS,<n)

01 MENS ION VF(61),VS<(>i),HR(tll,HS(6It,SR(bl),SS<bll

COMMON/CMi'/hCHN

C THIS SUBkCUTlNE CALCULATES THE fOUTViLEM STEADY STATE

C LOSS BASEL CN THE MA'SNInG CCIFFICIEnI.

C DO 1 1-2.

CALL SHAPHT1ME,HR<II ,A,T,P£f<»

SR( I ) - ( ( VM I I «AeSl VR( IM«rr*»2»)/{A/P£P. I«^»1.333

I CONTINUE

HI «N NY-l

69
IF(KH.Le.HC)

IFfHN.LE.HCI NZ-N*I

00 2 l-Nt«HZ

CALL SKAPE<TIK£,HSf IttAtT.PERI

SSI 1 )-UVSI IMABSirS f I > l*RM**2l>/(A/PERI*»l«333

2 CONTIHUE

RETURN

END

c

c

c

G

C

c SUBROUTINE DEPTHITMEI

C THIS SUBROUTINE USES A SECTION Cf ENTkY TO CALCULATE NORMAL

C AND CRITIC/L FLOW DEPTHS.

OIMEMSION CIh( 30), TIN (30J

C0MM0N/CM3/NPTS,OIH,r IN COMMON/Cm!;/HC »HN

CALL ENTKY(TlflE»

RETURN

END

cc

c

cc

c SUBROUTlNf IhFLUH( rue ,QAV ) C THIS SUBRCUTIhE CALC'JLAltS iNFLCt- 'ATfS AT PIPF tNTkT BaScO

C ON THE ENTFY FLCH °^jFlLt LAT*. ^.^Tt IHAI T He 0 LALCoLATtJ

C IS AN AVEfrAGt VALUE rOR THIS Tift STtP.

riMENSION 0( 30 ) ,T ( 1^^ I ,QX t jr ) COMMJN/C^l/r T0,L1 ,0<,D,SO,PL,SFO,)"r,-<<^

C0MM0N/CM3/NPTS»O,

T

Tl-TIMF

TO«=TI ME-DT

J=I

TX = T1

DO 3 I-1,NFTS-1

IF CTX.Ct .T ( I ) . ASC. K.LT .T t Ml I J LQT:^

3 CONTINUE

4 0X( Jl «C( I ) »<&( 1-HJ -0( I» )»( TX-T <I)l/(T(I«l!-ni))

QAV«ax(lJ

P.ETURN

END

C

c

c

c

cc

c SUBROUTINE SHAPE (T I1£ ,H,A,T,PEP

>

C

C DIMENSION OIN no ) , T lr< ( 30 > CuMHON/Cfil/C'?0,LT.l)X ,U,S0,PL»5I (. .XK.K.i

COMM0N/Cfl3/NFTS,O[N,TIN

C0M»<JM/Crt7/CL

C THIS SUp.P.CUTIht CALCJLATIS fLfW AC" ^ t , S U^; f A C £ wILiTH ANLi C HETFED PL^If-LTtR BAScU FLlW tcPTH

70
C THIS SUBROUTINE ALSO C»LCU1.A1E5 MATER SURFACE fPOflLE

C AMD SETS UP PASE CJ><aiTIOMS ALONG THE PIPE AT I I Hfc ZERO.

R-0/2,0

|F«M.LT.R)T»^f TA-2. 0»A T ANtSCRI IH«<0-H* I /IR-HII

IFCH.EQ.RITHETA-Pl

iF<H.CT.r. IT^E TA-PI «2. 0 « ATAM ( (H-Ri/ ( S3 R T C H* (D-H))! ) A-( (04*2 ) /B.O)*(lHETA-SINCTh£TA) J fER-D*THETA/2.0

1-2. 0*< IH«<{>-HI »<).i I 1F«TIME.GT,0.0» COTO 1 0-OIN( 1 > C0H-RH»«2/S0

HCRIT-1.0-<0»»2l»T/f i»A**3 I HNnRh-1.0-|0»«2J»C'lN/ ((A««3.333»/IPEil*»1.33 3n

DL-HC R 1 1/ t UNO RK.*SO } 1 CONTINUE

RETURN

END

C

CC

C

C

C SUBROVTINt »<AVSPD{H,C)

C

C THIS SUBKCUTIhE CALCULAT£5 WAVE SPFEO BASED UN DEPTH AN'j CROSS

C SECTION ShAPE.

CALL SHAPE «TlHE,h, AREA, T,PERJ

C=SQRT(9. fcl*AREA/T)

RETURN

£ND

C

C

cc SUBROUTINE PROFILCO,HN»HC,XN)

C THIS SUBKCUTIHE CALCJLATES ThF INITIAL WATER SURFACE

C PROFILE BASED ON CRITICAL EEPTH AT PIPE cXiT.

DIMENSION HPteD.VPCiD^QPCtlNCPCfal)

DIMENSION X (30 » tOF" ( JO » ,XN (61

)

DIHENSTOn X 1( 3C I tOE* I ( 30J

COMMON/Cnl/Cl 0,DT ,3X,0,S0,PL,SEC,XK,k:<

COMMCN/Cr.i /'N , TPIAX C0WM0N/CMt/OP,VP,HP,CP

C0MMQN/Ch7/tL

COHnON/CMJi/ INSOL ,H SOi. 1 ,NSCL

C0MMUN/Cri9/SCLVEL,HDS»VDS,CrS,^'PLS,VPi)S,l-P0S

COMM0N/C«10/XP0S,XSQL » VPSt L»VSOLtOPSjL,hUS,VUS,CUS,HPL.S,Vr'US.CPUS

c IFCHN.LE.HCJ COTO 900

DH-(HN-HCI/30.0

IS-1

Hl-HC

CALL SHAPE (TIME, HI, A, T, PER

)

X( ll-PL

OEPd »-HC

SL-0.

0

C ' .<ATER SURFACE PROFILE C AlCULAT I CnS . 00 80 I-l,;'CC,2

1S-IS»1

71
M2-HC»DH«FL0A7 U*ll

H3-KC*0H»FL0AT(XI

CALL SHAFE(Tint»Hl«A,T«PEtl

OLl-OL

CALL SHAPE (TlflE»H2«A,T*r€«t>

DL2-0L

CALL SHAPEfTirE*H3«A» TtPERt

0L3-0L

OXP-OH*tOLl«DL2«^.0«JL3 1/3.0

SL-SL-DXP

«l-H2

IFISL.GE.PLI CCTO 91

XUSI-PL-Sl

IF(Hl.CE.HH) GOTO 83

DEPflSI-Hl

60 CONTINUE

81 XIISi'0.0

NIS-I

S

IF CHI . CE.»iN) GOTO 93

OEP( IS l-Hl

GOTO 8^

83 OEPtIS)-HN

GOTO

8A IF (X( I S).CT.C.C1 C3TJ 65

GOTO 6fc 85 X(1S)=0.0

DEP(iS)»riN

NIS«I

S

66 CONTINUE

e

c

C INTERPOLAT ICh PEOUIREO TO CFTtPMNf 0£PTH AT EACn NQjt . 900 CONTINUE

DX»^PL/FLOA T<h ) XN(l)-0.0

00 87 I=2,N-»1

XN{ I )-XN(I-ll*DX

87 CONTINUE

IF(HH.LE.HC) GOTC 901

S2=NIS»1

DO 9*1 J-1,N1S

N2-N2-1

Xl( J»=X(N?J

OEPK Jl-0EP»N2 ) MRI T£ I 3t93 >X1 ( J ),0E PI ( J » 93 FORMA T{ lOX ,2f 10.51

9^ CONTINUE

03 *»5 J"1,NIS

XOI-XKJI

OEPC J)=0£P1 ( J» 95 CONTINUE

HPCD-OEPtn

HP(N*1 l-HC

00 98 1-2,

N

DO 89 K-1,HIS-1

IF ( XN ( n .C T.X tK ) .Ar«n. Xr<( 1 ) .Lt .y <K* U ) CGI (J 9o

89 CONTINUE

90 HP(n-OEP<»<)«IDEP(K»l)-DtP(K))#(X><fI)-X(^l »/lX<K«l)-x(K))

83 CONTINUE

C SET UP PASf CLhCITION* At Tll-£ 2 1 iiD

.

901 CONTINUE

72
CALL :>MAP t I ; 1 r t ,Mr 1 1 5 » A » I » FtR 1 ¥Pt I I -C/A

QP t n -Ci«ic 00.

0

CALL MAVSF0(1!P(I)(CPI !t)

CONTINUE

IF INSOLl.EQ.O » COTO 99

HPUS- HPINSCLI

MPDS='HP<NSCL1J

VPOS" vr<N5CLl

VPDS-VPIH5C1L1)

CP'JS-CrtNSLLl)

CPOS-CPCNSOLII

i2P50L-OP<HSOLl)

99 CONTINUE

RETURN

END

C

Cc

c

cc

c

c SUBROUTINE INTER(N»

C 01 MENS ION V<tl),H<61),C<f>l),VK(tl),H<(6l),CR(61),VS(on

DIMENSION HS(6l),CS(tjl),SF <til,5S(fcH,XN{bl)

DIMENSION >Sl.6l) »XPf bl)

COMMON/Cni/CTO,DT,Dt,0,SO,PL,SFC,XK,<n

COHM0N/Cfl^/V,H,C,VP,HR,CK,)(l-,SP,VStHi,C5,XS,SS,XN

C0MM0N/Cn5/HC -thti CONMON/CMb/ INSOL ,N SOL 1 »NSCL

COMMON/CM^/ SO L VEL » MOS , VOS , CCS ,HPLS, VP OS ,C PDS

COMMON/CNlO/XFCS,XSnL»VPSCL,VS(L,OPSaL,HL,S,VuS,CuS,HPuS,V->US,CP'Ji

C THIS SUBRCUTINE SETS UP, PY i NT t k P OLA T I L N , Trtt BASE ^:ONOiriJNi

C FOR THE NEXT TlrE STEP.

C THETA=DT/DX

N1»=N*1

IF( VSOL.GT.O.OI CCJ .1 6 COTO 7 6 XA*=XSOL-OX

XC-XSOL

DXA«XA-XH( NSOL-1

)

VA-V( HSOL-1 » ( VUS-V (S SOL-1 1 l*D^ */OV

HA»H(NSOL-U-» (HUS-H(NS0L-1 ) )• CXA/Or

CA-:C( NSOL-1 ! (CL'S-C(NS0L-1 ) >*L>*/OX

VCU« V (NSOL-l)*(VLlS-VJ NSOL- 1 ) ) « ( > Si'lL-XH « NS OL-i J )*/ U K MCU-H I NSDL-1 ) » (HUS-H( NSOL-in*(XSnL-AN<NSuL-l))/LX

CCU-C ( NSOL-D* (CUS-CINS0L-1))« ( > SOL-XN ( NS JL-i i i / u K VCO-VDS*( V (HS0L*1 » -V3S > *( XSLL-XK(NSLV ) ) /tX

HCO-HDS*(HtNSOL-»l»-H0S)»(X50L->N(NS0L » )/Ua

CCD-CDS* (C t NSCL^ ll-COS » • ( X SCL -X ^ (N S OL M/LX

xe-xsoL-»ax

0XB-XB-XN<NS0L-»1I

VB-V(NSOL«l>*iV(NSOL*2)-VC^SUL<])»»D<:i3/D*

KB'=H<NSnL*l)«(h(NS1L»2)-H<NSL;L*lJ>^OX6/DX

CB-C<NS0L-»H-»<C<NSlL*2l-C<hS0L-»i)>*DXo/0X

7 CONTINUE

00 1 i-?«Nl

IFt I.E O.NSCL./.M0.VSO;_ .CT.C .C) Ctl" 6 tOTO 9

73
8 ¥<II-VCU

H( n-HCU

Ct I l-CCU

»C I-ll-VA

MCI-l >-HA

CU-1 l-CA

9 COKTIHUE

IF(VSOL.LE.O,0.AH0.I.EO.NSOLI V( IMVUS

IF«VSOL.LE.O.O.AND.I.EQ.NSLL> C« n=CUS

IFIVSOL.LE.C.0.AN0.I.EC.N5CL) h( I)=HJS

If «V50L.L£.0.C.#N0.I.EQ.r*S0L*l) VCI-1»-VDS

IFJVSOL.LE.O.O.ANO.l .EQ.MSOL*! » C(T-l»*CDS

IFI VSOL.LE .C.O.ANO- I . EQ.MSOL* 1 J fit I -1 1- HO S vR(n-=cvii)*ThE7A<'jc{n«v(i-i)-vui*cti-i)n

I /C l.O^TME TA*1 VU I-i ) € < n-C< l-l) ) ) CR( I ) - (Ci 1 )*< 1 .C-VR( I )«THET AMC ( 1-1 I* VRt I 1 »IH£ TA)

1 /( 1,0-»C( 1 )*T^ ETA-C ( I-l ) »rHE TAJ HRC I l-HU )-<h(l l-H( I-l H»1^ETA«< VS (1 »»CR< 1 )1 1 CONTINUE

IF(HN.LE.HC> CGTO ^ 00 2 I^ltN

IFd.EO.NSCL.AhD.VSOL.GT.O.OJ GOTO 10

GOTO 11 10 VdlxVCD

H( I l-HCD

C<I)-CCD

vn*ii-v6

M(I*lJ-Hs

c f 1*1 )'=ce

11 CONTINUE

IFCVSOL.LE .C. C.ANO. I . E3,NSCL) Vtl }=V3S

IF I VSOL.LE .CO. AND. I.EQ.NSCLJ Cn)=Cjj

1 FC VSOL .LE .C. 0. AND. I . EQ.NSCL) ^'( H =HjS

IF(VSOL.Lt.C.O.AN;m.£Q.NSCL-l) V( I<-..»=VuS IF( VS OL .LE . 0. O.ANO. I . EQ.N5,LL-1 ) C ( I +1 =Cb 5 IFtVSOL.LE.CC.AhT.I. E S.-sSCL-l ) h< I i =Ht S VS m* ( V( I )-Th£TA* ( V( i )«C ( 1*1 )-C ( H»» « 1*1 ) » I I / r 1.0-THET A«( V I I )- V( I ! t-c ( n (1 1) J » cs<n»{Ccn-»vs(n>THETa«({.<i)-tu*in>

I /(1.0*7HETA«(C(n-C(l*l) » ) HSUJ=H(U-»ThETA«(*<S(i)-CS(I)J*th([)-H(I*l))

2 CONTINUE

GOTO 3 * DO 5 l'2^h*l

IF ( VSOL .CT . C.O.ANO. I . E Q.NSCL* 1 » CvlT., b IF(NSnL.GT.C,0.AN0.!.EQ.N5CL-»l) CTU 20

GOTO 90

20 V(I-1}«VDS

M« I-l » -HQS

CC I-l )-COS

80 CONTINUE

VS(I)'lvnj»l 1 .0»TH^TA«C( I-IJ J-V( I-1)»THlTA»C(1)I/

I < i.o*thet* J V ( n-v ( i-ij*c u-1 »-c ( 1 n

)

CS(I) = (Ctl)»VS<I)»(C(IJ-C<I-l)))/<l.J»TH, TA«(CU-il-t<I))»

HS(H-H(I)-<hCn-H(I-l)J»THfT^«»yS(I )-CS ( I ) ) 5 CONTINUE

3 CONTINUE

If ( NSCL.EC.OI CCTQ 13

IF (HN.CE . HC 8 GOTO 18 DO 17 I-l,hSLL-l

IF ( VU I .01 .CU ) ) CHTl 17

vs<i)-(vin-in£TA»(v(i»»cn-»i)-c<M»vii*i)))

1 /(1.0-UvT**(V(I)-V(i ll-C<l)*f <iM >>l

74
CSf II-CC<n-»VS<l I*TM^TA»ICI1|-C«l»ll>l

/(1.0«THC TA«(C( I l-r. I] ! ) i I HS<I l-^Hf I l*ThETA*( VSt I >-CS(I> M(H( II-HCI«il>

CONTINUE

CONTI NUE

RETURN

ENO

SUBROUTINE ENTRYtTIHEl

THIS SUBROUTINE CALCJLATES THE UPSTREAM BOUNDARY CONDITIONS

AT EACH TI^E STEP BASED ON A KrChN MfLOW PROFILE.

01 HENS I ON 0P( 61 ) tOINi 30) «TIN4 3C) tVP(b I) ,HP (61 ) tCP I 61i

OINENSION V(bH,H(6n

DIHENSION VR(Cl)«HR(olltCP(&ll«VS(61itHS(6l)tCS(6l)tSS(bl)

OINENSION 5PC61),X«(*>n,XSCtl),XS<'>ll,C(t.l)

C0MM0N/CM1/C7 0,DT,0<, D,SO,PL, SEC .XK,<n

COMMON/Cfli /Nt T^AX

CONHON/C«3/NPT5»CIN,r IN COMHON/CM^/V,h,C »VS»^'< ,CR,yf< »SP » VS ,HS ,CS,XS,Sb»XN

C0NMQN/CM5./hC ,HN

C0MM0N/C«t/Ot ,VP,^P,CP

C-9.81

C0N*RN»«2/SC

|F(T1MF.CT.C.0)CALL INFLOWdl^F ,C)

IFCTIME.CT.C.C) CQTl 600

IKTIME.EC.O.0)O-3fN( I)

CALCULATION Lf CRITICAL DEPTH.

UP«D

0N=0. 0 HC*(UP*DN) /2.0

CONTI NUE

CALL SHAPE »TINE,HC,ARE A,T,PER)

HCRIT-l.C-(0**2)«r/rG«AR£A««3»

IFIHCR ITia^'^.S

DN-HC

GOTO 6

UP"HC

MCN-(UP*1)N )/2.0

IF (ABS ( (HCh-HC )/HC I .L t .0. CCD CCTD 8 HC-HCN

GOTO 7 MC'HCN

CONTINUE

CALCULATION LF NORMAL DEPTH.

UP-D

ON-0.

0

HN-»UP*DN 1/2.

0

CONTINUE

CALL SHAPE«TlME,HN,A8EA,T»PEk)

HNOPM - l.O-l C««2>«C0S/ ((AKtA»*3.353>/(Ptk»*1.333>)

!F(HNORH) 10«lltl2

Dh-hn

GOTO 13

UP "MM HNN«( UP •UN ) /2 .0

I f ( ^ ( < H-^* -t » / » s » .1 f . ^ . r 1 ) m i J <, 75
HN-HNN

GOTO 9

HH-HNN

11 COHTIMUE

IFf TIME.EO. 0.01 COTQ 900

Cc

c

C CALCULATIOW OF BOUNDARY OEfTM.

600 CONTINUE

IFtHN.LE.HC) GOTO 700

UP-0

DM-CO

HB-(UP*DN) /2.0

15 CONTINUE

CALL SHAPE ITIHE.H3, AREA, T»PER

>

X3-G/CS< 1)

X*-VS{1I-G»(S5(1)-S0) »DT-X3*hSfl)

HFL0H-O-IX*i«X3«HB>»A»^E A IFIHFLOKJ It, 17,16

16 UP=H3

GOTO 19

18 0N«H8

l<5 HBB«( UP^Df* )/2 .0

IF { ABS( <HEe-hB )/H^ » .LE .O.CCl ) CCTD ZJ rtS'HQ 3 GOTO 15

20 HB»HbB

17 COHTiNUf

HP(1)»=HB

VPCl) = X',*>3*hP(ll

CALL SHAP t (TI hE ^H"? .ARE A «T, P£K ) OP'1}=AR£A«VP(1»»100j.O

CP( 1) = 50kT (C«AREA/T > 500 CONTINUE

GOTO 800

700 CONTINUE

C CALCULATICN CF NCP1A_ DEPTH.

UP = P DN=0.0

HB«=( UP-»DN ) /2. 0

7<5 COiUINUfc

CALL SHAPE (TI ^E,h9,ft^EA,T, PEP)

HN0RM'=1.0-(C««21«C3N/((AkfcA*«3.3 33 >/(PEk«»1.333))

IFIHNOPrOfcO, 1,82

30 ON»HB

GOTO 83

82 UP-HB

83 HBB»(UP*ON .0

lF(AaS( (riEf-he)/H3 I.LE.0.001) OCT : bt

HB«H6B

GOTO 79

8<i MB-H6B

81 CONTINUE

HP( l)«H8

CALL SHAPE <T IME ,l-!3 , A.^E A,I, P£k ) VP ( 1» -Q/AP f A Cf (i }-Q«iCCi'.0

CP(1»-S0RT«C»ARE*/T»

800 CONTINUE

RE TURN

END

C

76
c

cc SUBROUTINE NCDALIHI

OIKENSIC< OP(6l}tQIHI 30ltTIHI30l«VP((>ll»hPf6ll,CPf 611

OIHENSION Vf6l}tH(6M

01 HENS I ON VK(611«HR{61>«CR(61 )tVSf61)tHS(bl)tCS(611«SS(61)

DiHENSifi !;r(6U,x?(si ),xS(6i>«yNCM>tCft>ii

CCMKGN/CK1/D10«DT,0X«D« S3»PL tSFC*XK«Rn

CaNnON/CN3/hPTS«OI Ntf IN C0MM0N/CR4/V,H«C«VR,^1R«CR*XR«SP«VS*HS ,CS*XS «SS«XM

C0«M0N/CH5/hC »HN

C0HH0N/CN6/OP , VP,HP ,C P C0HM0N/CH8/lNSCL,NSOH.NSOL

C0M«0N/C«10/XP0S,XSai., VPSOL»VS0L,OPSaL fHUS»VUS,CL»i,HPUS»V PUS, CPUS

C THIS SUBROUTINE CALCJLATES THE FLOW VELOCITY AND OEPIH

C AND HAVE SPEED AT EACH OF THE CCES iETMEEN IHE uPSTkEAI

C AND OOHNSTREAfl BOUNOAklES BY SCLUriOi OF THE IWO HAVl

C EQUATIONS.

C-9.81

N2-N

IF(HH.LE.HC» N2-N*l

00 1 I>2,NZ

IF<I.EO.NS0L) GOTO I IF(VSOL.GT.C.0.AND.I.Ea.NSOL-m tOTO 1 X1-G/CR< I ) X3-C/CS{i » X2-X1*HR( I MVRCI >-G*J SR( I )-S0 )*CT

X^ = VS { 1 l-X3«HS{ 1 )-:*( SS I I l-SC)«DT

HP(n = (x2-XA)/(Xl»XlJ

VP(I )=«X1*X3*hP ( I ) CALL HAVSPDihP 11 )i C»( I)

CALL SHAPf <TIrE,H5> (I » ,X1,X3,X^)

OP(I I-X1«VP< I >«1000..}

I CONTINUE

RETURN

END

C

C

C SUBROUTINE FXITITIHE)

01 MENS ION OP Ifcl) ,yo {(s I ) ,HF (fcl) ,CP( Ml

DIMENSION VJfcll ,H'((i>ir . •-

• OIHENSION VRl61),H<^<bl>,CR(61 ),VS(Sl) ,HStoH,CS(cl),SS(6l)

DIMENSION SPI 61) «X^(6I ) ,XS<fcl 1 ,>N( 61 ) tCIbi

)

COMMON /C M 1 /DT C , DT, D X, 0, SO, PL, SfO.XK,R«

COMMON/CM?/N,THAX

C0MM0N/CM5/HC,hN

COlMON/Ch«t/V,h,C,VR,HR,CR,>»i,SP,VS,HS,CS,XS,SS,XN

COMMON/CMfc/OP ,VP,Hf» .CP

C

CC THIS SUBROUTINE CALCULATES Tht FLO« DEPTH AT PIPF OISCHARCf

C BASED ON THE ASSUrtfll^N OF CRITICAL Dt'PTu AT SuCh A BUUNOakY.

CC G-9.81

I

77
J<1-G/CR<H-»1I

X2-Xl*HR« K*IMVR(H»1I-C«CSR««*1I-S0I»0T

UP«0

ON-0.0

HB-CUP«0NI/2.0

CALL SKAPE(TinE«K3«*,T«PERl

HEXII-1.0-(AeStX2-Xl«HB)9CX2-Xl*H8l )*T/(C*A)

JFfHEXITll«2t3

OH«KB

COTO 4

OP -HB

GOTO 4

HB9-I UP*0H)/2 .0

IF(ABS((HCe-MB)/H3I.LT. 0.00001) COTO 2 H6-HBB

COTO 5 CONTI HUE

VP(H* 1 J-X2-X1 HE

MPfM*ll»He

CALL SHAPt<7IHE,H9,A,T,PEF! ) CALL WAySPr<HB»CP(N 1»J

OP<N»l»»100C.O«A«VP(t^-»l)

RETURN

END

SUBROUTINE SCLID(TIHc»

01 KEN 5 I ON CP(fcl)t3lN{30»,TlM3ri,VP(ol»,nP(6l)tCP(t)li

DIMENSION V(tl),^{6n

DI MENS ION VHtll,H?J-,l),CF(tll,VS{ftl»,.HS(&l),CSIfcl),iS(blJ

OlflENSION FS(2),S'<(*>l),X,'»fcl),XS('SU,XN(ol),C(tl)

COri^'ON/Ch 1/C7 0»Dl,D«',D»SO,FL»SfC,>''»'(i

COHMOh/Cni /h ,T^AX

comhon/cm3/npts,cm,t in

COHMON/Cf"./V,h,C,VR,HR,C(<tXk.St,VS.HStCS,Xi,SS,XN

COI1HON/Cn5 /HC thN

COMMON/CMt/CH ,VP,SP,-P

COMMCN/CMt/ INSCL,HSDL l,NStL

COMMON/CMS/ SC L VELtHOS tVOS * CDS* KPLS , VP OS , CP 05

COHMON/CrilO/> POS t<> Ol »VPSOL»VSri tOPSOL.MUS »VUS,CU S»HPUS t Vi>US»CPUS

CO>^haN/Cni l/TL,Tt<t TT, 5 H

£ THIS SUBkCUTIhE CALC'JLATES THE fLOw DEPTrt, LEAKAGE RATE ft ND

; KAVE SPEED AT THE lOi^NSTKtAt BDUKPiPy FOkMED BY AN INITULLT

C ' STATIONARY StiLlC.

C-<>.8 I C0N«Rh««2 /5C

QPSOL-OPSCL/1000.0

IZ-0

SEV-SEO

4-1

HO-HUS

6C CONTINUE

J- I aoOO CALL SHAPE <TIhE,HO,Al,T,P£»;»

X1-C/CR(NSCL)

X2-XI *Hk < hSOL )«VP( HSJL l-C«<SR (hSOL »-S0 )»ul

SE«HJS*(VL'S'>«?)/t.'.0>G)

IF«SE.LT.SEO> GOTO fc
o5 tun I I MUt

U-(X2-VP5CL)»A1/XIC

B-X1»A1 /XK

r-«X2»*2»/l2.0'»CI-SE>-VPS0L«X2/C«IVrS0L*»ZI/C2.0»CI

Z-Xl»VPS0L/C«i.O-X2»Xl/6

Z3-2«0< W*7

Z2-Z»»2*2-©*l»»y

zi«2.o*ir*z«e

ZC-Y«ABS< T>-U

f-Z^#H0*#4*Z3*H0*»3*Z2*HO««2*Zl«HO*Z0

FS<J»-F

IFIJ.EQ.21 CCTO 7000

J-2

H0-H0*0.0001*M0

GOTO BOOO

7000 t>f-CFSC2 »-FS« lU/t^O/ 1 0000. 01

OH-FSf 1 l/DF

Hl-HO-OH

IFIABS MH1-HC)/H0) .LE .0.005) CCTO 70

IFCH1.GT.D)M1-1.1*H0

HO-Hl

iz-rz*i

GOTO 60

70 CALL SHAPE (TI NE,H1,42,T,PEF

)

IF(ABS(<A?-A1)/AU .Lr. 0.005) CCTG 80

Al«(Al<^A2)/2.C

H0*Hi

GOTO 65

80 CONTINUE

MPUS=H1

VPUS»X2-Xl«m

QPS0L»VPUS*/2

IFtVPUS.LT.VFSOL) GOTO 37

GOTO 5 37 CONTINUE

VPUS» VPSOL

HPUS-(X2-VPUS )/Xl

CALL SHAPEfTIhEtHPU^, A2TT,P£k)

OPSOL«VPUS*A2

GOTO 5 5 VPUS=»VPSOL

HPU5- IX2-VPUS )/XI

CALL SHAPE<TlH£,HPUS,A2»r,PEK

)

QPSOL- VPUS*A^

GOTO 5 5 CALL HAVSPD<hPUS,CPUS

)

SEO-SEV

OT-DTO

GOTO 006

•«06 COHTINUE

IFIHN.LE.HC) GOTO dOal

GOTO 8062

8061 CONTINUE

Q-OPSOL

IF«O.LF.O.C) 0-0.05/1 10 . 0* 10. 0*»-M

C CALCULATiCH OF hOKflAL DEPTH.

UP-O

ON-0. 0 HB-(UP»DN)/2.0

7«J CONJIHUL

CALL SHAPE lTlr<E,HB»»K£A,r,Ptk»

79
MN0Rrt-1.0-IC*«2»*C0M/(CARE**«3.333>/tPER»»1.333ll

IF (HNnRH}<)0«91«92

90 08f8B93

92 UP-Hfl

93 HB8«>(UP«0H)/2.0

IFCABSt (HeB-HB}/H3» U.E. 0.001) GOTO <M H8"HAB

GOTO 7«> 94 M8*M8B

91 CONTINUE

HPOS-HB

CAL*. SHAPE(TlhE.Ma,A«EA,T,P£RI

VPOS-O/AREA

OPDS»0*1000.0

OPSOL-OPOS

CPDS» SORT {C*AREA/r

I

GOTO 934

8062 X3=G/CSCNS0L)

X4»VS« NSOL l-&«ISS< HS JL l-SO l*DT-)( 3*HS< NSOL I IFIOPSOL.EC.C.C) C:T3 932

UP-D

ON-0.0

HB-CUP*ON ) /2.

0

15 CONTINUE

CALL SHAPt (TIME ,M3,AREA,T-.PEk » HFLOH*QPSOL-( X4»X3»M8 )«AKFA

IF(HFLOMI lt»17,ie

16 UP^HB

GOTO 19

18 ON=HB

19 KBD-^t UP-»DK )/2 .C

IF CAe5( (HBE-HE)/h3l._E.0.C01» CCIU ZO

MB«HBB

GOTO 15

20 H8=H53

17 CONTINUE

HPOS^HB

VPDS»=XA*X3*HP LS

CALL SHAPE !TIf«E,h3, AREA, T^PEkJ

QPSOL -ARtA«VPDS«10': .0

CPDS* SQPT< G»Af<EA/T » GOTO 933

932 CONTINUE

riP0S»-X4/ X3

VPOS-0.0

CALL HAVSPDOiPCStCPOi ) 933 CONTINUE

IF(VPSOL.CT .C.OI GOTJ 931 • GOTO 935

93* CONTINUE

OXX-DX/< XSLL-XhCHSOL-1

n

VPINSOL J»VPINSCL-l » J X X« ( V PUS-VP < NSOL - 1

U

KPtNSOL }-hP<hSCL-l ) J XX ( HP US-HP

<

NSOl -1 »

I

CP<NSOL » -CP <NSCL-l »*OXX»iCPUS-CP<N>OL-ll)

0XX«»XN(NSCL«l)-XS0LI/(XN<r<SUL»2)-XSJL»

*PJNS0L*l»»VPCS*DX<»(VP(NSCL«2)-VPnS)

MP (NS0L*1 J-HPL)S*DXX»( HP(HSCL«2 l-HPHSi

CP(N>0L«])»CPDS*&X<»iCP<HSLL'2>-CP0SJ

935 CONTINUE

Rf TURN

END

Cc

80
SUBROUTINE INStRTI tM, T iMf

DIMENSION XHfbl ) ,aP(M) ,Vf (61|,Hf C611«CPCbll

C0MMQN/Ct16/0P ,VP,Hf ,CP

CONMON/CHfc/ INS0L,NSDL 1 «NSOL

C0MM0N/C«<^/SDLV£L»ri3S .VDS.CDS,HPDS,VP OS»CP0S

C0NK0N/CnlO/XPOS«XSaL*VPSOL*VSCL«OPS0L«HUS«VUS«CUS«HPUS«VPUStCPUS

c INSOL-0

HSOL-NSOLl

VPUS- VPIhSCL

)

HPUS-HP( NSOL)

CPUS- CPCNSCL

*PDS« VP(NSOL)

HPDS«HP(NSCL

CPOS«CP<NSOL i OPS0L-0P(lsSCL > VSOL-VPUS

VPSOL-VSOL

OX-XNINSGL »-XN<HSOL-l

DXX=DX/4 XSCL-XN(HS3L-1 ) ) VP (NSOL I -VP (NSCL-l I axx*( VPUS-VP4NS0L-II

)

HP(NSOL)'HP <hSCL-l > l)XX* (hPUS-HP (NSOl-I) ) CP (NSOL l-CP <NS0L-1 I JXX»(CPtjS-CP<NSOL-i I » DXX=(XN(NS0L-H)-XSaL»/(XN(NSLL*2J-)(SuL>

VP(HS0L*i)«VPLS*Dx:X*lVP(MSrL*2 )-VPOS»

HP (NS0L*1) "HPUS^CX <«( HP( NStL*2 )-HPDS)

CP(NS0L*1 )«CPDS»DX<*( CP(NSCL-»2)-CP0Sl

RE TUR N END

CC SUBROUTINE f 0RCE(TI1r,XN,f SDL)

C

C THIS SUSRGUTINF CALCULATES ThE fCCE ACTING ON THE SOLID

C AND DETtPhlHtS ITS VELOCITY AND POSITION AT THE tNO Uf 1 n

C NEXT TIME STEP.

C OIMFNSIOS XN(fcl)

COhMON/Crtl/DTO,DT,0)(,0,SO,PL»SECtXK,Rn

CrjM-0N/Cn2 /N , ThAX

COMHON/Cie / INSCLtN SOL I, NSOL

COMMuN/CnS /SLLVEL»HOS » VOS .COS,hPLS, VPOS,CPOS

COMMjN/CNlC/XP0S»XSnL,VPSOL»VS0L tCS OL »HU S t VUS » Lo S » MPUS » V PU S . C P Ui

CUrtMON/Cnll/TL»TW,TT,SW

C0HMGN/CK12, XFAC2,KFAC « TAl^,f hOV ,f ST Al C G-9.3 I RHO-1000.0

F8ICT»FSTAT

FM1«RH0*C« ( HcS-TT/2.0 ) » T W« TT • T T /2.

0

F«2-RH0«o«(TW2.O»»T'<»TT»0.5«( HDS-TT»HDS»

f M3-SK«C*Tl«C.5»C0S( ASIN( (HUS-TT)/TL)

)

TX-(H0S-IT)/SIN(ASIN( ( HUS-TT » / TL )

fM^-XFAC2«P»-L»C*0.5*(HDS-TT)«TH«C.5*lX*«<:

IF (HOS.LE .TT ) FN^-0.0

rP«(0.5/TL)*lFN2*F13*FM'»-rNl)

F P?-( FP-f i0.b*JXU COS ( ASIM <HUS-I T »/TLn

IUVSOL.GI.C.OFKICT-FNOV

H5H-T T»0. b**!-. PUS-IT » CALL SHAPt <T 1 hE ,HSrt.ASH,ISH,PEf. ) FLOAk- ASM-T T« TH

81
CALL SHAPC<TIHt»H«»US,ARUS«TUS,r£RI

QSH-ARUS» (VPUS-VSJLI

SUERV-OSH/ASM

SHFRF-TAU«O.5«RH0»2.0»TL»Tl«SHE*lV*«2

IFtHPUS.LT.TT } Fr2-0.0

FP2-XFAC«f Pi

f SUR-FRIC7*<SW*C*C0S(ASINISO))-^P2l

IF(FSUR.LT.O.O) FSU^»0-0

FS0L»RK0*C*T»»«0.5»<HPUS**2-«PDS«*2l»SK*C*SO-FSUR

VPS0U-VSCL*DT*F5CL/SW

hRITE<3«5 3)FS0L.FPZ.F SUR

53 FORHATI 10X,»F50L-» »Fl0.^t« FP2-' «F 10. 4, • FSUR=* vF 10.4,/

}

IF<VPSOL.LE.O.Ot VPSaL-0.0

X!iOL-0.5»01«< VPSOL»VSOLI*XSOL

IFCVPSOL.Cl.O.O.ANO.VSOL.LE.0.0) CCTO 5 00 4 I-1»N

IF<XNt I I.LE.XSOL.AMO.XMf !<• 1 } . CT .XSOLI HSOL-I

4 CONTINUE

5 CONTINUE

RETURN

END

cc

c

c

cc SUBROUTINE ASSIGN(N»

C THIS SUBRCUTIhE SETS UP THf hEW e*SE CONOITIUNS ALONO THt

C PIPE IN FfFf-ARATION f Of THE ^EXT Tint SIcP.

DIMENSION OP (61 ) ,0 rn( 30 ) ,T l^( 3C ) ,VP< 611 ,riP ( 61 ) tCP ( fcl

»

DIMENSION V(bl),H(6lt

DIMENSION VK 611 ,HR Cijl ) ,CS <fci ) ,VS( »HS ( 61 l,CS< 61 ) ,SS (61 ) DIMENSION SP(fcl)»X'.(t>l} ,XS J6i l,X^C61» ,C(oi» t COMM0^</CM3/NP^S»OIH^ IN C0HMGN/CM^/V,H,C»V-?,H^ ,Ck,XR,S^TVS,HS tCS^XStSS^XN

COMMON/tnt/CP ,VF,HP ,CP

COMMON/CMfc/ iNSOLfNSHL l,NSOL

C0MM0N/C«9/SCLVEL» H3S t VDStCLS»HPl/S. VPOStCP OS

C0MMCN/CM1C/XPCS,XSTL « VPSCL,VSCL «OPSl)L»H.US»VUS»Cu StHPUS,V?USt CPUS

00 1 I-^1,N«1

IF (VSOL.EC.C.C.ANO.I. EQ.NSCL) CCTO 2 v(i) = vp(n

H(I)«HP(I)

c< n-cp< I

»

IF

(

In SOL .CT.O .AND. I .£ Q.NSCLl 1 COTO 6

7 CONTINUE

;F (VSCL.E C.C. C.ANO.NS OLl.CT.t ) COTC 2 GOTO 1

*

' . •..

. 2 HUS«MPUS

VUS-VPUS

CUS-CPUS

HOS-HPOS

VOS-VPOS

COS-CPDS

HPCNSOL )-HPUS

VP INSOD-VPUS

CP(NSOL)-CPLS

H(HSOL)»HPtS

V»NSOL)-VPUS

C<NSOLI«CPUS

core I 6 HPUS-HP(JI

82
vpus-vpc I

)

VPDS-VP<I

CPUS-CP( I

)

CP0S-CP<1

J

COTO 7 1 CONTINUE

SOLVEL-VSOL

VSCL-VPSOL

IFfVPSOL.CE.O.OI GOTO 3

GOTO A

3 VSCL-VPSOL

HUS-HPOS

»US-VPUS

COS-CPUS

HOS-HPDS

VDS-VPOS

COS-CPOS

4 CONTINUE

RETURN

END

SBENO

83

Appendix 2

Deflvation of solid boundary equations

used in Subroutine SOLID

85
Derivation of solid boundary equations employed in Subroutine SOLID.

Available equations at moving solid boundary.

1) Leakage flow past solid.

Leakage

flow, Q

Q = XK {SE - SEq}2

where SE is the specific energy

relative to moving solid.

9 Specific energy h + V^^/Zg

XK, SEq determined by measurement.

2) Flow over solid in terms of flow area

Vg = solid velocity

Relative velocity Vj-g = - Vg

Flow area A = f(h)

Total flow = V^A = V^fCh)

Flow past solid = Vj-^A = Vj.gf(h).

3) Characteristic equation, C*" in both sub and supercritical flow conditions.

P

At

86
C"^ links R at t to P at t + At

C"^ equation Vp = X2 - Zlhp

XI = g/cR

X2 = Vr + ghR/cR - g(SR - So)At

Note charcterlstics equations in terms of absolute velocity.

The flow Q over the solid is given by the empirical relation,

^

2 Q = XK (h + ^re - SE^)^

2g

where Vj.g is the flow velocity relative to the moving solid.

Q = V^gA = (Vabs - Vb)A = (X2 - Xlh - Vb)A

Also, Q = XK {SE - SEq}^

= XK {h + £^abs - ^B^^ - SE^} ^ 2g

Hence

(X2 - Xlh - Vg)A = XK{h + (X2 - Xlh - Vg^^ - SE^}^

2g

Collecting terms

U = (X2 - Vb)A/XK; B = XI A/XK;

W = (Xl2)/2g; Y = X22/2g - SE^ - VgX2/g + V^/Zg;

Z = Xl.vg/g + 1.0 0 X2Xl/g

U - Bh = {Zh + Y + Wh2}2

= z2h2 + y2 + w2h^ + 2YWh2 + 2ZYh + 2ZWh3

W2h^ + 2ZWh3 + (z2 + 2YW)h2 + (2ZY + B)h + (Y.lYl - U) = 0

Note: Y.ABS 1y| retains sign Y.

This quartic is solved by the Newton Raphson method as B and U involve f(h)

through the flow area terra.

87

APPENDIX 3

Typical Output Program TRANSCD

89
90
0 3 3 5 0 3 3 0 3 3 3 3 0033 0 3 0 0 3 3 3 3 0 3 3 3 900 3 3 3 3 0 3 3 3 3 3 3 3 3 0 0 3 3:

f- 0 « «• -r — *

* * * i # if < ^ <A « 1^ « >» e * >» * ^ y «

» —

£ *•«- — n 3 — ^ X

3 ^ ? - — » 0 -3 ->

»- » -

— * y —

3 -J -• — — a 3 — r

-> -v — X y 3 \< -« r^ e

3 ^

> -

3 "^J > 3 >» 3 3 3 C 3 3 0 0 3 0 3 3 3 3 0 3 0 - 0 3 0 0 c 3 e 3 C 3 3 3 3 3 C T C 0 - s -r -r « « 3 * *• a 0 * * * —

0 » -< 0 « .a * -^ a * -< # — «- X M flD 9 -« C » — X r — « ? X s- — — r « X C X. 3 C c e r c ^ 3 0 0 3 3 3 3 3 .-> e 3 m 0 c -. 3 3 3 — 0 0 3 0 e 0 3 3 0

— r 0 "V. 3 ^ - c 0 — - 3 ^ ^ 0 ^ « c e % — 0 0 c- 3 ~> 3 3 3 3 :3 3 3 3 3 3 3 3 3 9 3 3 3 3 3 3 3 3 3 3 3 3 0 3 3 3 3 3 3 1 t * « X * « X r» * * X C # K *«.'>• « — E y 0 J' « s ^ — ^ 0 e •9 ^' 3 r X — 0 — 0 — B y —

0 ^ - «

I > — 0 - <.^ 0 •« 3 - « 33% 0 S ,j 0 <r > 0

SX * 0 J* TC

"V 3 0 3 C 3 3 0 3^33 0 3 3 3 3 3 3 0 0 3 0 e 3 0 3 3 3 3 3 3 3 3 3 0 3 N< s- v. C T S

V. 3 ^ *

* 3

e

c < » r * ^ ^ ^

J> »

>" * X y

<" ©

0

J- 0

<- « 0 *

J- ^

9-c

J» —

rf> m J- < m <ir* c

* ? *

•N. X

« ^

X

r

r-r

*

I. r 5 —

•\l 3 * ^ V

— 3 C r » ^ 0

3 ^ c t^ r - £ < 0 > 3

# 0

3 » 3

* e 3 «

X ? 0 « e C 3 r — X

f 3 "* X -s. e e C e c c c c c c 0 e c e e c e e e ^ e- c c c 0 c c 0 c C c c e e c c c C C r «— a- *i > 0 —

i 0 f» S

3 m c M

e « 0' » » 1^ — > 9- « w »

s 0 «- ^

» * -

< & f

« #

«• < * 3

«

»%

»• « «

< —

r <

? 0 ? « — rf^ » « £ c « ^ <^ r» 0 « r> «

«- 0 r> 0 » 4 0 —

3 «

1^ 0 0 « iT ^ 3 e

0--r c = OS. 5 e 0 r> 0 e e 1" 0 e 0 Q e 0 0 0 S e 0 0 0 0 0 0

X -1 r I E r -J r r ^ c z ,j r c ^ E r c r ^ r c ^ c r c r r r c *r V c » w e ^ »- e >- e >

• >• »•

•e to e w

e V c c Z ~

s

arX z 3 ^ s «S " X 3

u. ^ 0

sr> ^«. or 1^ — « k« Z w

» e s w

z u

0

^ «

ars Ul

ar w

z C

0

car orX

z 3

to 0

a'z X 0

X I

X c > « a >

•J « • ^

X 0 >

<X

>« w ^«

a > u. X

iW ^ < 0 » ^ s a » < W «M ^ A » 1. « a * *te X

Ul Ul a >

« 0 > c >

• • If• • t t • • t 0 M

« ir «

• t • ff • • • • M

• 1 • • a a a a •

UlK

Ul Ul* X

Ulc

Ulc \

r r

91
O * J* o ^ c r A o O O — o o — o 6 ? 3 C • o • o o -

J" tT • 3 o 3 3 £

*• 3 # # — 3

"5 * -3 > — "3

O > *«» 0 > 3 — 9 7-

— 3 — « 3 % • O • O % • C • « C

z> o o 3 3 3 3 o O o o o r O 3 r O r X c « O — * % •V % ? r • 3 • o 3

•r • «' • o O o o o c 3 r r n r T r -) %i z r r >k m < -r % •V, f # o V.

T J* •J

T

r

•\.

v • -5

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107 U.S. GOVERMENT PRINTING OFFICE 1982-0- 366-850/545
NBS-lUA (REV. 2-8C) U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report No. 3. Publication Date BIBLIOGRAPHIC DATA

REPORT NO.

SHEET (See instructions) NBS BSS 139 February 1982

4. TITLE AND SUBTITLE

Application of Method of Characteristics to Model the Transport of Discrete Solids

in Partially-Filled Pipe Flow

5. AUTHOR(S)

J. A. Swaf field

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions) 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D C 20234

Final

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

Library of Congress Catalog Card Number: 81-600198

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 si gnificant information. If document includes a si gnificant

bibliography or literature survey, mention it here)

The flow depth and velocity changes across a moving solid in partially-filled pipe

flow are predicted by means of the application of the method of characteristics to

solve the unsteady flow equations.

Simplified force models are presented which, when used in conjunction with empirical

relationships linking leakage flow past the solid to upstream specific energy, are sufficient to provide the required moving solid boundary conditions that allow solid

velocity prediction.

A wide range of simulated transport conditions are presented that confirm the

applicability of this technique as a basis for the future evaluation of more complex

body force models.

The predicted solid velocity during drain transport is shown to be compatible with

laboratory observations of the influence of solid dimensions and position in inflow

profile on transport characteristics.

12. KEY WORDS (S/x to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) computer based model; drainage; solid transport; unsteady flow.

13. AVAILABILITY

[X] Unlimited

14. NO. OF

PRINTED PAGES

1

1 For Official Distribution. Do Not Release to NTIS 116

I 1 Order From Superintendent of Documents, U.S. Government Printing Office, 20402. Washington, D C

15. Price

["yj Order From National Technical Information Service (NTIS), Springfield, VA 22161 $4.25

USCOMM-DC 6043-P80
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